Research ArticlePreconditioning and Uniform Convergence forConvection-Diffusion Problems Discretized onShishkin-Type Meshes

Thaacutei Anh Nhan1 and Relja VulanoviT2

1School of Mathematics Statistics and Applied Mathematics National University of Ireland Galway Ireland2Department of Mathematical Sciences Kent State University at Stark 6000 Frank Avenue NW North CantonOH 44720 USA

Correspondence should be addressed toThai Anh Nhan nhananhthaigmailcom

Received 5 September 2015 Revised 9 December 2015 Accepted 26 January 2016

Academic Editor Yinnian He

Copyright copy 2016 T A Nhan and R VulanovicThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

A one-dimensional linear convection-diffusion problem with a perturbation parameter 120576 multiplying the highest derivative isconsidered The problem is solved numerically by using the standard upwind scheme on special layer-adapted meshes It is provedthat the numerical solution is 120576-uniform accurate in themaximumnormThis is done by a newproof technique inwhich the discretesystem is preconditioned in order to enable the use of the principle where ldquo120576-uniform stability plus 120576-uniform consistency implies120576-uniform convergencerdquo Without preconditioning this principle cannot be applied to convection-diffusion problems because theconsistency error is not uniform in 120576 At the same time the condition number of the discrete system becomes independent of 120576 dueto the same preconditioner otherwise the condition number of the discrete system before preconditioning increases when 120576 tendsto 0 We obtained such results in an earlier paper but only for the standard Shishkin mesh In a nontrivial generalization we showhere that the same proof techniques can be applied to the whole class of Shishkin-type meshes

1 Introduction

In this paper we consider a convection-diffusion boundary-value problem in which the coefficient of the diffusion termis a small positive parameter 120576 This makes the problem asingularly perturbed one Singularly perturbed boundary-value problems are characterized by the presence of the per-turbation parameter 120576whichmultiplies the highest derivativein the differential equation considered Therefore when 120576

tends to 0 the degree of the differential equation is reducedand the solution of the reduced equation typically does notsatisfy all the boundary conditions imposed This is why thesolution of the original problem may have boundary andorinterior layers that is the solution may change abruptlyover narrow regions whose size depends directly on 120576 Itis typical that the derivatives of the solution behave in thelayers like O(120576

minus119896

) where 119896 is an appropriate positive number

Because of this the classical numerical methods for solvingboundary-value problems do not work well for singularlyperturbed problems and special numerical methods need tobe constructed [1ndash4] Ideally the goal of these methods is toachieve 120576-uniform pointwise convergenceThe layer-adaptedmeshes of Shishkin or Bakhvalov types are frequently used tomeet this goal

The interest in singular perturbation problems is moti-vated by their numerous applications For instance theintroduction in [4] mentions Navier-Stokes equations withlarge Reynolds number and other convection-diffusion equa-tions modeling water-pollution problems oil-extraction pro-cesses flows in chemical reactors convective heat-transportproblems with large Peclet numbers and semiconductordevices Let us also mention the application to a steadyfully developed laminar flow of alumina-water nanofluid[5]

Hindawi Publishing CorporationAdvances in Numerical AnalysisVolume 2016 Article ID 2161279 11 pageshttpdxdoiorg10115520162161279

2 Advances in Numerical Analysis

In this paper we consider the simplest singularly per-turbed convection-diffusion problem in one dimension

L119906 fl minus12057611990610158401015840

minus 119887 (119909) 1199061015840

+ 119888 (119909) 119906 = 119891 (119909)

119909 isin (0 1) 119906 (0) = 119906 (1) = 0

119887 (119909) ge 120573 gt 0

119888 (119909) ge 0

for 119909 isin 119868 fl [0 1]

(1)

where 0 lt 120576 ≪ 1 and 119887 119888 and 119891 are 1198621

(119868)-functions Theproblem has a unique solution 119906 in 119862

3

(119868) [6 7] In generalthis solution exhibits an exponential boundary layer near 119909 =

0Problem (1) serves as a model problem for discussing

some theoretical aspects related to the proof of 120576-uniformpointwise convergence of numerical methods for singularlyperturbed convection-diffusion problems This proof is notas simple as for the singularly perturbed reaction-diffusionproblems where the proof can be based on the follow-ing classical principle which originates from nonperturbedproblems (cf [8ndash10])

Principle 1 120576-uniform stability and 120576-uniform consistencyboth in the maximum norm imply 120576-uniform pointwiseconvergence

Principle 1 does not work in the proof of 120576-uniformpointwise convergence for convection-diffusion problem (1)When (1) is discretized by using the upwind finite-differencescheme on the Shishkin mesh a careful analysis shows thatthe consistency error in the layer part is not 120576-uniformlyaccurate in the maximum norm (cf [11 Sec 613]) Hencespecial techniques like barrier functions [1ndash4] or the use ofhybrid stability inequalities [2 9 12 13] are required to provethe 120576-uniform pointwise convergence

We show in [14] that it is possible to apply Principle1 to problems of type (1) after the linear system whichresults when (1) is discretized is preconditionedThe upwinddiscretization scheme on the standard Shishkin mesh isconsidered in that paper Although the scheme is 120576-uniformlystable the discrete linear system is ill-posed since its condi-tion number grows unboundedly when 120576 rarr 0 as pointedout by Roos in [15] By using a relatively simple diagonalpreconditioner Roos eliminates the 120576-dependence of thecondition number

It is shown in [14] that a suitable modification of Roosrsquopreconditioner enables the proof based on Principle 1that the pointwise errors of the numerical solution are 120576-uniformly of almost first order Since the standard Shishkinmesh is the only one considered in [14] the question is nowhow to use the same approach and achieve the same resultfor the general class of Shishkin-type meshes in the sense of[2 16] The present paper addresses this question

Therefore for convection-diffusion problem (1) we con-sider the new proof technique of 120576-uniform pointwise con-vergence introduced in [14] and we generalize it to thewhole class of Shishkin-type meshes The generalization is

not entirely straightforward since the general properties ofthe meshes require many new technical details Moreover weslightly modify the preconditioner used in [14]

In this final paragraph of the introduction we give theoutline of the paper In Section 2 we describe the Shishkin-type meshes which are used when discretizing problem (1)We analyze the conditioning of the upwind finite-differencediscretization in Section 3 Then in Section 4 we define thepreconditioner and give a proof of 120576-uniform stability of thepreconditioned discrete system We arrive at the main resultthe proof of 120576-uniformpointwise convergence using Principle1 in Section 5The last section Section 6 contains some finalconclusions

2 Shishkin-Type Meshes

Let 119868119873 be a general discretization mesh with points 119909119894

where

0 = 1199090

lt 1199091

lt sdot sdot sdot lt 119909119873minus1

lt 119909119873

= 1 (2)

We consider the Shishkin-type (or S-type) meshes as intro-duced in [16] see [2] as well Any such mesh is dense in theboundary layer near119909 = 0 and transitions to a coarse uniformmesh at a point constructed like in the standard Shishkinmesh

120590 fl 119886120576119871 (3)

Here 119886 is a user-chosen positive parameter and we think of 119871as of 119871 = ln119873 although a modification of this value can alsobe used like in [17] We assume that 120590 lt 12 otherwise thevalues of 119873 are unreasonably large

The coarse part of the mesh is obtained by uniformlydividing the interval [120590 1] into 119873 minus 119869 subintervals of length119867 = (1 minus 120590)(119873 minus 119869) where 119869 is a positive integer such that119876 fl 119869119873 satisfies 119876 lt 1 and 119876

minus1

le 119862 and where from thispoint on 119862 denotes a generic positive constant independentof both the perturbation parameter 120576 and themesh parameter119873 Inside the layer the mesh points are formed by a mesh-generating function 120601 which is a monotonically increasingfunction satisfying 120601(0) = 0 and 120601(119876) = 119871 In other wordsS-type meshes are defined by

119909119894

=

119886120576120601 (119905119894

) 119894 = 0 1 119869

120590 + (119905119894

minus 119876)

1 minus 120590

1 minus 119876

119894 = 119869 + 1 119873

(4)

where 119905119894

fl 119894119873The mesh steps are defined as ℎ

119894

= 119909119894

minus 119909119894minus1

119894 =

1 2 119873 the steps for 119894 = 1 2 119869 belong to the fine partof the mesh and ℎ

119894

= 119867 for 119894 = 119869+1 119869+2 119873 Let also ℏ119894

=

(ℎ119894

+ℎ119894+1

)2 119894 = 1 2 119873minus1 and let mesh functions on 119868119873

be denoted by119882119873119880119873 and so forth For a function119892 defined

on 119868 119892119894

indicates 119892(119909119894

) and 119892119873 stands for the corresponding

mesh functionWe identifymesh functions119882119873 with (119873+1)-dimensional column vectors 119882119873 = [119882

119873

0

119882119873

1

119882119873

119873

]119879

We use the maximum norm of 11988211987310038171003817100381710038171003817119882119873

10038171003817100381710038171003817= max0le119894le119873

10038161003816100381610038161003816119882119873

119894

10038161003816100381610038161003816 (5)

Advances in Numerical Analysis 3

Table 1 Examples of mesh-generating and mesh-characterising functions of S-type meshes

Mesh 120601 (119905) max1206011015840

120595 max 10038161003816100381610038161205951015840

1003816100381610038161003816

S 119905119871

119876

119871

119876

119873minus119905119876

119871

119876

BS minus ln(1 minus

119905

119876

(1 minus

1

119873

))

119873

119876

1 minus

119905

119876

(1 minus

1

119873

)

1

119876

VS 119905

119876 + 119876119871 minus 119905

21198712

119876

exp(minus

119905

119876 + 119876119871 minus 119905

)

4

119876

The matrix norm induced by the above maximum vectornorm is also denoted by sdot

Following [2 16] let us define the mesh-characterizingfunction 120595 which is related to the mesh-generating function120601 as follows

120595 (119905) = 119890minus120601(119905)

(6)

This function is monotonically decreasing on [0 119876] with120595(0) = 1 and 120595(119876) = 119873

minus1We need to assume some further properties of the mesh

Assumption 1 Let the mesh-generating function 120601(119905) bepiecewise differentiable such that

max119905isin[0119876]

1206011015840

(119905) le 119862119873 (7)

Moreover we also assume that 120601(119905) fulfills

Δ120601

fl min119894=12119869

Δ119894

ge 119862119873minus1

(8)

where Δ119894

fl 120601(119905119894

) minus 120601(119905119894minus1

) 119894 = 1 2 119873

Remark 2 Examples of mesh-generating functions 120601 satisfy-ing the above assumptions are the standard Shishkinmesh (S-mesh) Bakhvalov-Shishkinmesh (BS-mesh) andVulanovic-Shishkin mesh (VS-mesh) They are summarized in Table 1

We proceed to provide some estimates for the step size ofthe S-type meshes generated by 120601 in the layer region Becausefor 119894 = 1 119869

119909119894

= 119886120576120601 (119905119894

) = minus119886120576 ln120595 (119905119894

) lArrrArr 120595(119905119894

) = 119890minus119909119894(119886120576)

(9)

we can bound the mesh step size inside the layer from above

ℎ119894

= 119886120576 (120601 (119905119894

) minus 120601 (119905119894minus1

)) le 119886120576119873minus1 max119905isin[119905119894minus1119905119894]

1206011015840

(119905)

le 119886120576119873minus1

(max119905isin[119905119894minus1 119905119894]

100381610038161003816100381610038161205951015840

(119905)

10038161003816100381610038161003816)

120595 (119905119894

)

= 119886120576119873minus1

( max119905isin[119905119894minus1119905119894]

100381610038161003816100381610038161205951015840

(119905)

10038161003816100381610038161003816) 119890119909119894(119886120576)

le 119886120576119873minus1max 1003816

10038161003816100381610038161205951015840

10038161003816100381610038161003816119890119909119894(119886120576)

(10)

where we use max|1205951015840| fl max119905isin[0119876]

|1205951015840

(119905)| Thus for 1 le 119894 le

119869 minus 1 we have that

ℏ119894

le

119886

2

120576119873minus1max 1003816

10038161003816100381610038161205951015840

10038161003816100381610038161003816(119890119909119894(119886120576)

+ 119890119909119894+1(119886120576)

)

le 119886120576119873minus1max 1003816

10038161003816100381610038161205951015840

10038161003816100381610038161003816119890119909119894+1(119886120576)

(11)

3 The Upwind Scheme andIts Condition Number

We consider the following upwind finite-difference dis-cretization on 119868

119873

119880119873

0

= 0

L119873

119880119873

119894

fl minus12057611986310158401015840

119880119873

119894

minus 119887119894

1198631015840

119880119873

119894

+ 119888119894

119880119873

119894

= 119891119894

119894 = 1 119873 minus 1

119880119873

119873

= 0

(12)

where

11986310158401015840

119882119894

=

1

ℏ119894

(

119882119894+1

minus 119882119894

ℎ119894+1

minus

119882119894

minus 119882119894minus1

ℎ119894

)

1198631015840

119882119894

=

119882119894+1

minus 119882119894

ℎ119894+1

(13)

The matrix form of the linear system (12) is

119860119873

119880119873

=119891

119873 (14)

with a tridiagonal matrix 119860119873

= [119886119894119895

] and 119891

119873

= [0 1198911

1198912

119891119873minus1

0]119879 The nonzero elements of 119860

119873

are 11988600

= 1 119886119873119873

= 1

119897119894

fl 119886119894minus1119894

=

minus

120576

ℏ119894

ℎ119894

1 le 119894 le 119869

minus

120576

1198672

119869 + 1 le 119894 le 119873 minus 1

119903119894

fl 119886119894119894+1

=

minus

120576

ℏ119894

ℎ119894+1

minus

119887119894

ℎ119894+1

1 le 119894 le 119869

minus

120576

1198672

minus

119887119894

119867

119869 + 1 le 119894 le 119873 minus 1

119889119894

fl 119886119894119894

=

1 119894 = 0

minus119897119894

minus 119903119894

+ 119888119894

1 le 119894 le 119873 minus 1

1 119894 = 119873

(15)

4 Advances in Numerical Analysis

Table 2 The condition number of discrete system (12) on the BS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 206119890 + 04 896119890 + 04 372119890 + 05 152119890 + 06 613119890 + 06 246119890 + 07

1119890 minus 3 219119890 + 05 948119890 + 05 393119890 + 06 160119890 + 07 645119890 + 07 259119890 + 08

1119890 minus 4 220119890 + 06 954119890 + 06 396119890 + 07 161119890 + 08 649119890 + 08 261119890 + 09

1119890 minus 5 220119890 + 07 955119890 + 07 396119890 + 08 161119890 + 09 650119890 + 09 261119890 + 10

1119890 minus 6 220119890 + 08 955119890 + 08 396119890 + 09 161119890 + 10 650119890 + 10 261119890 + 11

1119890 minus 7 220119890 + 09 955119890 + 09 396119890 + 10 161119890 + 11 650119890 + 11 261119890 + 12

1119890 minus 8 220119890 + 10 955119890 + 10 396119890 + 11 161119890 + 12 650119890 + 12 261119890 + 13

Table 3 The condition number of discrete system (12) on the VS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 275119890 + 04 123119890 + 05 511119890 + 05 205119890 + 06 812119890 + 06 320119890 + 07

1119890 minus 3 291119890 + 05 130119890 + 06 539119890 + 06 216119890 + 07 855119890 + 07 336119890 + 08

1119890 minus 4 293119890 + 06 131119890 + 07 543119890 + 07 218119890 + 08 861119890 + 08 339119890 + 09

1119890 minus 5 293119890 + 07 131119890 + 08 543119890 + 08 218119890 + 09 862119890 + 09 339119890 + 10

1119890 minus 6 293119890 + 08 131119890 + 09 543119890 + 09 218119890 + 10 862119890 + 10 339119890 + 11

1119890 minus 7 293119890 + 09 131119890 + 10 543119890 + 10 218119890 + 11 862119890 + 11 339119890 + 12

1119890 minus 8 293119890 + 10 131119890 + 11 543119890 + 11 218119890 + 12 862119890 + 12 339119890 + 13

Matrix 119860119873

is an 119871-matrix that is it has positive diagonalentries and nonnegative off-diagonal entries It is also anonsingular matrix satisfying119860

minus1

119873

ge 0 (inequalities involvingmatrices and vectors should be understood componentwise)and therefore inverse monotone In other words 119860

119873

is an119872-matrix (an inverse monotone 119871-matrix) This fact is easyto prove using the following119872-criterion see [8] for instance

Theorem 3 (119872-criterion) Let 119860 be an 119871-matrix and thereexists a vector 119908 such that 119908 gt 0 and 119860119908 ge 120574 for somepositive constant 120574 119860 is then an 119872-matrix and it holds that119860minus1

le 120574minus1

119908

In the above theorem set 119908119894

= 2 minus 119909119894

119894 = 0 1 119873to get that 119860

119873

119908 ge min1 120573 This implies that 119860119873

is an 119872-matrix and that

10038171003817100381710038171003817119860minus1

119873

10038171003817100381710038171003817le

2

min 1 120573

le 119862 (16)

which means that discrete problem (14) is stable uniformly in120576 By examining directly the entries of matrix 119860

119873

and notingthat the lower bounds for ℎ

119894

and ℏ119894

are achieved in the layerregion where ℎ

119894

sim ℏ119894

ge 119862120576Δ120601

we get that

1003817100381710038171003817119860119873

1003817100381710038171003817le 119862

1

120576 (Δ120601

)

2

(17)

Combining this and the 120576-uniform stability (16) we arrive atthe following result related to the condition number 120581(119860

119873

)

of matrix 119860119873

120581(119860119873

) fl 119860119873

119860minus1

119873

Theorem 4 The condition number of matrix 119860119873

satisfies theestimate

120581 (119860119873

) le 119862

1

120576 (Δ120601

)

2

(18)

Let us illustrate Theorem 4 using a simple example takenfrom [2 page 1]

minus12057611990610158401015840

(119909) minus 1199061015840

(119909) = 1 0 lt 119909 lt 1

119906 (0) = 119906 (1) = 0

(19)

It is easy to observe the 120576-dependence of the conditionnumbers in Tables 2 and 3 They clearly show that the boundin Theorem 4 is sharp For the standard Shishkin mesh werefer the readers to the numerical experiments in [15]

4 Preconditioning and Stability

In this section we propose an appropriately designeddiagonal preconditioner which not only eliminates the 120576-dependence of the condition number of the discrete systemsbut also retains its 120576-uniform stability in the sense of (16)Thepreconditioner is not of interest per se but as ameans to prove120576-uniform pointwise convergence via Principle 1

Let 119872 = diag(1198980

1198981

119898119873

) be a diagonal matrix withthe entries

1198980

= 1

119898119894

=

ℏ119894

119867

119894 = 1 2 119873 minus 1

119898119873

= 1

(20)

On noting that ℏ119894

= 119867 for 119894 = 119869 + 1 119873 minus 1 we seethat when the discrete system (12) is multiplied by 119872 this isequivalent to multiplying the equations 1 2 119869 of system(14) by ℏ

119894

119867 119894 = 1 2 119869 The modified system is

119873

119880119873

= 119872119891

119873

(21)

Advances in Numerical Analysis 5

where 119873

= 119872119860119873

Let the entries of 119873

be denoted by 119894119895

the nonzero ones being

119897119894

fl 119894minus1119894

=

minus

120576

ℎ119894

119867

1 le 119894 le 119869

minus

120576

1198672

119869 + 1 le 119894 le 119873 minus 1

119894

fl 119894119894+1

=

minus

120576

ℎ119894+1

119867

minus

119887119894

ℏ119894

ℎ119894+1

119867

1 le 119894 le 119869 minus 1

minus

120576

1198672

minus

119887119894

ℏ119894

1198672

119894 = 119869

minus

120576

1198672

minus

119887119894

119867

119869 + 1 le 119894 le 119873 minus 1

119889119894

fl 119894119894

=

1 119894 = 0

minus119897119894

minus 119894

+ 119888119894

ℏ119894

119867

1 le 119894 le 119869

minus119897119894

minus 119894

+ 119888119894

119869 + 1 le 119894 le 119873 minus 1

1 119894 = 119873

(22)

Matrix 119873

remains an 119871-matrix since the precondi-tioning does not change this property of 119860

119873

We showin Lemma 7 below that

119873

is an 119872-matrix and that themodified discretization (21) is stable uniformly in 120576 But firstwe give a technical result related to the mesh

Lemma 5 Let 119886120573 gt 2 and let in addition to Assumption 1 themesh-generating function 120601 satisfy the following conditions on[0 119876]

(i) 120601(119895)

ge 0 119895 = 1 2 3(ii) 1206011015840

(0) ge 119898 gt 0 where 119898 is a constant independent of 120576and 119873

(iii) 12060110158401015840

le 2(1206011015840

)2

(iv) 1206011015840

le 119862119871119896 for some 119896 gt 0

Then for a sufficiently large119873 independent of 120576 there exists apositive constant 120575 independent of both 120576 and 119873 such that

119878119894

fl120573

2

(1 +

Δ119894

Δ119894+1

) minus

1

119886

(

1

Δ119894

minus

1

Δ119894+1

) ge 120575 gt 0

119894 = 1 119869 minus 1

(23)

Proof Since 119886120573 gt 2 we have that 1205732 ge 1119886 + 120578 for some120578 gt 0 Then

119878119894

ge

1

119886

119875119894

+ 120578 (24)

where

119875119894

=

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

Δ119894

Δ119894+1

(25)

Then because of (i) and (iii)

119875119894

ge 2119875lowast119894

where 119875lowast119894

fl1206011015840

(119905119894minus1

)2

minus 1206011015840

(119905119894+1

)2

1198732

Δ119894

Δ119894+1

(26)

If we show that

119875lowast119894

997888rarr 0 uniformly in 120576 when 119873 997888rarr infin (27)

then we will have (23) provided 119873 is sufficiently largeindependently of 120576 We now show that (27) holds true First

1003816100381610038161003816119875lowast119894

1003816100381610038161003816le 119862

100381610038161003816100381610038161206011015840

(119905119894minus1

)2

minus 1206011015840

(119905119894+1

)2

10038161003816100381610038161003816

(28)

because of (i) and (ii) and then

1003816100381610038161003816119875lowast119894

1003816100381610038161003816le 119862119873

minus1

119871119896

12060110158401015840

(119905119894+1

) le 119862119873minus1

1198713119896 (29)

because of (i) (iii) and (iv)

Themeshes defined inTable 1 satisfy conditions (i)ndash(iv) ofLemma 5 with the exception of the BS-mesh which does notfulfill condition (iv) However estimate (23) still holds truefor the BS-mesh if we assume that radic2 minus 1 le 119876 lt 1 This isnot a serious restriction since119876 = 12 is a typical choiceTheproof of this result stated below in Lemma 6 is given in theappendix

Lemma 6 Let 119886120573 gt 2 and let radic2 minus 1 le 119876 lt 1 Then theBS-mesh satisfies estimate (23)

Lemma 7 Under the assumptions of Lemma 5 (Lemma 6 ifthe BS-mesh is used) matrix

119873

of system (21) satisfies

100381710038171003817100381710038171003817

minus1

119873

100381710038171003817100381710038171003817

le 119862 (30)

Proof We construct a vector V = [V0

V1

V119873

]119879 such that

(a) V119894

gt 0 119894 = 0 1 119873(b) V119894

le 119862 119894 = 0 1 119873

(c) 119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

ge 120579 119894 = 1 2 119873 minus 1 where 120579

is a positive constant independent of both 120576 and 119873

Then according to the 119872-criterion

100381710038171003817100381710038171003817

minus1

119873

100381710038171003817100381710038171003817

le 120579minus1

V le 119862 (31)

We define the vector V as follows (cf [14 15])

V119894

= 120572 minus 119867119894 + 120582min (1 + 120588)119869minus119894

1 (32)

where 120572 and 120582 are appropriately chosen positive constantsand 120588 = 120573119867120576

Since 119867119873 le 119862 there exists a constant 120572 such that V119894

ge

120572minus119867119894 gt 0 and condition (a) is satisfied It also holds true thatV119894

le 120572+120582Therefore condition (b) is satisfied if we prove that120582 le 119862 We do this in next steps while we verify condition (c)

6 Advances in Numerical Analysis

When 1 le 119894 le 119869 minus 1 using Lemma 5 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= (119897119894

+119889119894

+ 119894

) V119894

+119897119894

119867 minus 119894

119867

=

ℏ119894

119867

119888119894

V119894

minus

120576

ℎ119894

+

120576

ℎ119894+1

+

119887119894

ℏ119894

ℎ119894+1

ge minus(

120576

ℎ119894

minus

120576

ℎ119894+1

) +

119887119894

2

+

119887119894

ℎ119894

2ℎ119894+1

ge 119878119894

ge 120575 gt 0

(33)

For 119894 = 119869 condition (c) is verified as follows

119897119869

V119869minus1

+119889119869

V119869

+ 119869

V119869+1

=

ℏ119869

119867

119888119869

V119869

+119897119869

119867 minus 119869

(119867 +

120582120588

1 + 120588

)

ge minus119869

120582120588

1 + 120588

+ (119897119869

minus 119869

)119867

= (

120576

1198672

+

119887119869

ℏ119869

1198672

)

120582120588

1 + 120588

minus

120576

ℎ119869

+

120576

119867

+

119887119869

ℏ119869

119867

= (

2120576 + 119887119869

(ℎ119869

+ 119867)

21198672

)

120582120573119867

120576 + 120573119867

minus

120576

ℎ119869

+

119887119869

2

ge

120582120573

2119867

minus

120576

ℎ119869

+

120573

2

ge

120573

2

(34)

where in the last step invoking (8) we choose a constant 120582 sothat

120582120573

2119867

minus

120576

ℎ119869

ge

120582120573

2119867

minus

120576

119886120576 (120601 (119905119869

) minus 120601 (119905119869minus1

))

ge

120582120573

2119867

minus

1

119886Δ119869

ge

120582120573

4119873minus1

minus

1

119886119862119873minus1

= 119873(

120582120573

4

minus

1

119886119862

) ge 0

(35)

To guarantee this 120582 is chosen so that

120582120573

4

ge

1

119886119862

(36)

Finally when 119869 + 1 le 119894 le 119873 minus 1 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= 119888119894

V119894

+119897119894

119867 minus 119894

119867

+

119897119894

1 + 120588119869

[

120582

(1 + 120588)119894minus1minus119869

minus

120582

(1 + 120588)119894minus119869

]

+

119894

1 + 120588119869

[

120582

(1 + 120588)119894+1minus119869

minus

120582

(1 + 120588)119894minus119869

]

ge 119887119894

+

120588 (1 + 120588)119897119894

minus 120588119894

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

ge 120573 +

(119897119894

minus 119894

+119897119894

120588) 120588

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

= 120573 + (

119887119894

119867

minus

120573

119867

)

120582120588 (1 + 120588)119869minus119894minus1

1 + 120588119869

ge 120573

(37)

By examining the entries of 119873

we easily see that themaximum absolute row sum of

119873

is achieved in the layerregion Therefore

10038171003817100381710038171003817119873

10038171003817100381710038171003817le 119862

119873

Δ120601

(38)

A combination of Lemma 7 and (38) results in thefollowing

Theorem 8 The condition number of matrix 119873

satisfies thefollowing 120576-uniform estimate

120581 (119873

) le 119862

119873

Δ120601

(39)

Corollary 9 For the meshes given in Table 1 one has thefollowing estimates

120581 (119873

) le

119862

1198732

119871

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

1198621198732

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(40)

In Tables 4 and 5 we present the condition numbers ofthe preconditioned linear systems (21) for various values of120576 and 119873 for the BS- and VS-meshes We can observe thatthe condition numbers of the preconditioned systems areindependent of 120576 and that the bound in Corollary 9 is sharp

5 120576-Uniform Convergence

For the proof of 120576-uniform convergence we need to decom-pose the solution 119906 into the smooth and boundary-layer

Advances in Numerical Analysis 7

Table 4 The condition number of matrix 119873

on the BS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 658119890 + 02 268119890 + 03 108119890 + 04 434119890 + 04 174119890 + 05 695119890 + 05

1119890 minus 3 689119890 + 02 278119890 + 03 111119890 + 04 440119890 + 04 175119890 + 05 700119890 + 05

1119890 minus 4 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 702119890 + 05

1119890 minus 5 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 6 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 7 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 8 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

Table 5 The condition number of matrix 119873

on the VS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 748119890 + 02 313119890 + 03 127119890 + 04 510119890 + 04 204119890 + 05 811119890 + 05

1119890 minus 3 780119890 + 02 323119890 + 03 130119890 + 04 520119890 + 04 207119890 + 05 820119890 + 05

1119890 minus 4 785119890 + 02 325119890 + 03 131119890 + 04 522119890 + 04 208119890 + 05 824119890 + 05

1119890 minus 5 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 6 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 7 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 8 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

parts There exist several decompositions of this kind but weuse the version presented in [2 Theorem 348]

119906 (119909) = 119904 (119909) + 119910 (119909) 119909 isin 119868 (41)10038161003816100381610038161003816119904(119896)

(119909)

10038161003816100381610038161003816le 119862 (1 + 120576

2minus119896

)

10038161003816100381610038161003816119910(119896)

(119909)

10038161003816100381610038161003816le 119862120576minus119896

119890minus120573119909120576

119909 isin 119868 119896 = 0 1 2 3

(42)

The details related to the construction of the function 119904 arenot important here they can be found in [2] As for 119910 wenote that it solves the problem

L119910 (119909) = 0 119909 isin (0 1)

119910 (0) = minus119904 (0)

119910 (1) = 0

(43)

We define the consistency error of the finite-differenceoperatorL119873 as

120591119894

= 120591119894

[119906] fl L119873

119906119894

minus 119891119894

119894 = 1 2 119873 minus 1 (44)

It holds true that

120591119894

= L119873

119906119894

minus (L119906)119894

= [119860119873

(119906119873

minus 119880119873

)]119894

(45)

Taylorrsquos expansion gives

1003816100381610038161003816120591119894

[119906]1003816100381610038161003816le 119862ℎ119894+1

(120576

10038171003817100381710038171003817119906101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711990610158401015840

10038171003817100381710038171003817119894) (46)

where 119892119894

fl max119909119894minus1le119909le119909119894+1

|119892(119909)| for any 119862(119868)-function 119892When the diagonal matrix 119872 is multiplied to linear system(14) the consistency error of the multiplied system is

119894

[119906] =

ℏ119894

119867

120591119894

[119906] 1 le 119894 le 119869

120591119894

[119906] 119869 + 1 le 119894 le 119873 minus 1

(47)

Lemma 10 Let 119886120573 ge 2Then the following estimate holds true

1003816100381610038161003816119894

[119906]1003816100381610038161003816le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(48)

Proof We use decomposition (41) to get

119894

[119906] = 119894

[119904] + 119894

[119910] (49)

Then (46) and the derivative-estimates of 119904 given in (42)yield immediately that

1003816100381610038161003816119894

[119904]1003816100381610038161003816le 119862119873

minus1 (50)

and therefore the following remains to be proved

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

119894 = 119869 119873 minus 1

(51)

Note that from condition (7) we easily get

ℎ119894

le 119862120576 119894 = 1 119869

ℏ119894

le 119862120576 119894 = 1 119869 minus 1

(52)

This is the key property of S-type meshes as well as the mainingredient of the following proof

8 Advances in Numerical Analysis

Table 6 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 550119890 + 00 293119890 + 00 151119890 + 00 769119890 minus 01 388119890 minus 01 195119890 minus 01

1119890 minus 3 550119890 + 01 293119890 + 01 151119890 + 01 769119890 + 00 388119890 + 00 195119890 + 00

1119890 minus 4 550119890 + 02 293119890 + 02 151119890 + 02 769119890 + 01 388119890 + 01 195119890 + 01

1119890 minus 5 550119890 + 03 293119890 + 03 151119890 + 03 769119890 + 02 388119890 + 02 195119890 + 02

1119890 minus 6 550119890 + 04 293119890 + 04 151119890 + 04 769119890 + 03 388119890 + 03 195119890 + 03

1119890 minus 7 550119890 + 05 293119890 + 05 151119890 + 05 769119890 + 04 388119890 + 04 195119890 + 04

1119890 minus 8 550119890 + 06 293119890 + 06 151119890 + 06 769119890 + 05 388119890 + 05 195119890 + 05

For 1 le 119894 le 119869 minus 1 (46) estimates (10) and (11) as well asthe derivative-estimates of 119910 in (42) imply

1003816100381610038161003816119894

(119910)1003816100381610038161003816le 119862

ℏ119894

119867

ℎ119894+1

(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894)

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902119909119894+1(119886120576)

119890minus120573119909119894minus1120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890(2119909119894+1minus119886120573119909119894minus1)119886120576

= 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902(119909119894+1minus119909119894minus1)119886120576

119890minus(119886120573minus2)119909119894minus1119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890ℏ119894119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

(53)

where we have used (52) and 119886120573 ge 2 to get

119890ℏ119894(119886120576)

le 119862

119890minus(119886120573minus2)119909119894minus1(119886120576)

le 119862

(54)

When 119869 + 2 le 119894 le 119873 minus 1 (42) and (46) give

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119867(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894) le 119862

119867

1205762

119890minus120573119909119894minus1120576

le 119862

119867

1205762

119890minus120573(120590+119867)120576

le 119862119873

1198672

1205762

119890minus120573119867120576

119890minus120573120590120576

(55)

Estimate (51) follows from the fact that (119867120576minus1

)2

119890minus120573119867120576

le 119862

and because the definition of 120590 implies that

119890minus120573120590120576

= 119890minus120573119886119871

= 119873minus120573119886

le 119873minus2

(56)

We finally prove (51) when 119894 = 119869 119869 + 1 in which case

1003816100381610038161003816119869

[119910]1003816100381610038161003816=

ℏ119869

119867

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816

1003816100381610038161003816119869+1

[119910]1003816100381610038161003816=

1003816100381610038161003816120591119869+1

[119910]1003816100381610038161003816

(57)

We now use the fact that L119910 = 0 and work with a differentform of the consistency error (this is a well-known techniquecf [18 Lemma 5])

1003816100381610038161003816119894

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816le 119877119894

+ 119876119894

+ 119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816 (58)

where

119877119894

= 120576

1003816100381610038161003816100381611986310158401015840

119910119894

10038161003816100381610038161003816

119876119894

= 119887119894

100381610038161003816100381610038161198631015840

119910119894

10038161003816100381610038161003816

(59)

We immediately have that

119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816le 119862119890minus120573119909119894120576

le 119862119890minus120573120590120576

= 119862119890minus120573119886119871

= 119862119873minus119886120573

le 119862119873minus2

(60)

As for 119877119894

it holds true that

119877119894

le ℏminus1

119894

120576 sdot 2

100381710038171003817100381710038171199101015840

10038171003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873119890minus120573120590120576

119890120573ℎ119869120576

le 119862119873minus1

(61)

by (52) Analogously

119876119894

le 119862119867minus1

10038171003817100381710038171199101003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873minus1

(62)

This completes the proof

In general the consistency error 120591119894

of the layer componentof the solution is not bounded pointwise uniformly in 120576When the above proof technique is applied to 120591

119894

[119910] one canonly get

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816

le

119862120576minus1

119873minus1

119871 for the S-mesh

119862120576minus1

119873minus1

for the BS- and VS-meshes

(63)

The proposed preconditioner simplifies the analysis of theconsistency error by introducing an extra 120576 factor needed forthe layer component Hence the approach used in the proofof Lemma 10 is natural and straightforward because it onlyuses the classical Taylor expansion estimate for the truncationerror

To illustrate (63) in Tables 6 and 7 we present the resultsof some numerical experiments on the BS- and VS-meshesrespectivelyWe refer the reader to [14] for the correspondingnumerical results on the S-mesh

By contrast with preconditioner (20) the preconditionedconsistency error converges uniformly in 120576 as shown in Tables8 and 9

In conclusion the preconditioning allows us to usePrinciple 1 and combine Lemmas 7 and 10 to obtain thefollowing main convergence result

Advances in Numerical Analysis 9

Table 7 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 467119890 + 00 247119890 + 00 128119890 + 00 659119890 minus 01 336119890 minus 01 170119890 minus 01

1119890 minus 3 467119890 + 01 247119890 + 01 128119890 + 01 659119890 + 00 336119890 + 00 170119890 + 00

1119890 minus 4 467119890 + 02 247119890 + 02 128119890 + 02 659119890 + 01 336119890 + 01 170119890 + 01

1119890 minus 5 467119890 + 03 247119890 + 03 128119890 + 03 659119890 + 02 336119890 + 02 170119890 + 02

1119890 minus 6 467119890 + 04 247119890 + 04 128119890 + 04 659119890 + 03 336119890 + 03 170119890 + 03

1119890 minus 7 467119890 + 05 247119890 + 05 128119890 + 05 659119890 + 04 336119890 + 04 170119890 + 04

1119890 minus 8 467119890 + 06 247119890 + 06 128119890 + 06 659119890 + 05 336119890 + 05 170119890 + 05

Table 8 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 122119890 minus 01 650119890 minus 02 338119890 minus 02 174119890 minus 02 887119890 minus 03 452119890 minus 03

1119890 minus 3 114119890 minus 01 601119890 minus 02 308119890 minus 02 156119890 minus 02 786119890 minus 03 395119890 minus 03

1119890 minus 4 114119890 minus 01 597119890 minus 02 306119890 minus 02 155119890 minus 02 778119890 minus 03 390119890 minus 03

1119890 minus 5 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 390119890 minus 03

1119890 minus 6 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 7 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 8 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

Table 9 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 160119890 minus 01 912119890 minus 02 502119890 minus 02 270119890 minus 02 143119890 minus 02 747119890 minus 03

1119890 minus 3 150119890 minus 01 844119890 minus 02 458119890 minus 02 242119890 minus 02 126119890 minus 02 653119890 minus 03

1119890 minus 4 149119890 minus 01 837119890 minus 02 454119890 minus 02 240119890 minus 02 125119890 minus 02 645119890 minus 03

1119890 minus 5 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 6 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 7 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 8 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

Theorem 11 The error of the upwind finite-difference dis-cretization on S-type meshes satisfies

10038161003816100381610038161003816119906119873

119894

minus 119880119873

119894

10038161003816100381610038161003816

le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(64)

Corollary 12 For the meshes given in Table 1 one has thefollowing estimates

10038171003817100381710038171003817119906119873

minus 119880119873

10038171003817100381710038171003817

le

119862119873minus1

1198712

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

119862119873minus1

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(65)

Remark 13 The barrier-function technique can be used toobtain similar results for S-type meshes (see [16]) but itrequires another assumption on the mesh-generating func-tion 120601 (cf [16 Theorem 1]) However whether the barrier-function approach can be extended to any Bakhvalov-type

mesh is still an open question [2 Remark 421] By contrastas proved in [19] the preconditioner defined in (20) worksfine for Vulanovicrsquos modification of the Bakhvalov mesh [10]The paper [10] is where the original Bakhvalov mesh [20]is generalized (the main difference between Bakhvalov-typemeshes and S-type meshes is that the former have a smoothtransition from the fine part inside the layer to the coarseuniform part outside the layer the mesh-generating function120601 belongs to 119862

1

(119868))

6 Conclusion

In this paper we present a proof of pointwise uniformconvergence for singularly perturbed convection-diffusionproblem (1) discretized on a general class of Shishkin-typemeshesThe proof is based on preconditioning of the discretesystem which allows us to use the standard ldquoconsistency+ stabilityrdquo principle of convergence This is a conceptuallysimple approach which does not require the use of hybridstability inequalities and discrete Greenrsquos functions In gen-eral new proof techniques are of interest because althougha lot has been achieved in the field of singularly perturbed

10 Advances in Numerical Analysis

problems over the last few decades there are still importantquestions remaining open [21] One of them Question6 is related to the work presented here The question isabout two-dimensional convection-diffusion problems and itpoints out that whereas 120576-uniform pointwise convergence forthe upwind finite-difference discretization on S-type meshesis well-established no similar result for B-type meshes isknownAsmentioned inRemark 13 we already know that thepreconditioning approach works well for B-mesh and one-dimensional convection-diffusion problems The next step isto consider two-dimensional problems

Appendix

Proof of Lemma 6 The proof is different from the proof ofLemma 5because theBS-meshdoes not satisfy condition (iv)Because of (ii) it is sufficient to show that the numerator in(25) is nonnegative for radic2 minus 1 le 119876 lt 1 independently of 120576and 119873 Let 119901 fl 1 minus 119873

minus1 By direct computations we get that

Δ119894

= minus ln(1 minus

119905119894

119901

119876

) + ln(1 minus

119905119894minus1

119901

119876

)

= ln(1 +

1

119869119901 minus 119894

) = ln (1 + 119902)

(A1)

where we denote 119902 fl (119869119901minus119894)minus1 Note that 0 le 119902 lt 1(119876+1)

Analogously

Δ119894+1

= ln(

1

1 minus 119902

) (A2)

Then

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

= Δ119894

(Δ119894+1

+ Δ119894

+ 1) minus Δ119894+1

= ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902)

(A3)

We set

119892 (119902) fl ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902) (A4)

Sinceradic2minus1 le 119876 lt 1 we have 0 le 119902 lt 1radic2 Rearrange 119892(119902)

as

119892 (119902) = [1 + ln(1 +

2119902

1 minus 119902

)] ln (1 + 119902)

+ ln (1 minus 119902)

(A5)

For 119902 isin [0 1) it follows that119902

1 + 1199022

le ln (1 + 119902) le 119902 (A6)

Hence

ln(1 +

2119902

1 minus 119902

) ge 2119902 (A7)

So we are done if we can show that

119892 (119902) ge [1 + 2119902] ln (1 + 119902) + ln (1 minus 119902) ge 0

0 le 119902 lt

1

radic2

(A8)

or

(1 minus 119902) (1 + 119902)2119902+1

= (1 minus 1199022

) (1 + 119902)2119902

ge 1 (A9)

First if 119902 isin [12 1radic2] we have by (1 + 119902)120574

ge 1 + 120574119902 for120574 ge 1

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

) (1 + 21199022

) ge 1 (A10)

Second we have that ln(1+119902) is a concave-down function on[0 12] hence on [0 12]

ln (1 + 119902) ge 2 ln(

3

2

) 119902 ge

4119902

5

(A11)

Then multiplying both sides by 119902 and exponentiating we getthat

(1 + 119902)119902

ge exp(

41199022

5

) ge 1 +

41199022

5

(A12)

Therefore

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

)(1 +

41199022

5

)

2

ge 1 (A13)

which holds true for 119902 isin [0 12]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P A Farrell A F Hegarty J J H Miller E OrsquoRiordan andG I Shishkin Robust Computational Techniques for BoundaryLayers Chapman amp Hall CRC Press Boca Raton Fla USA2000

[2] T Linszlig Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems vol 1985 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[3] J J HMiller E OrsquoRiordan and G I Shishkin Fitted NumericalMethods for Singular Perturbation Problems World ScientificSingapore 1996

[4] H-G Roos M Stynes and L Tobiska Numerical Methodsfor Singularly Perturbed Differential Equations Springer BerlinGermany 2nd edition 2008

[5] A Malvandi and D D Ganji ldquoMixed convection of aluminandashwater nanofluid inside a concentric annulus consideringnanoparticle migrationrdquo Particuology vol 24 pp 113ndash122 2016

[6] R B Kellogg and A Tsan ldquoAnalysis of some difference approx-imations for a singular perturbation problem without turningpointsrdquoMathematics of Computation vol 32 no 144 pp 1025ndash1039 1978

Advances in Numerical Analysis 11

[7] J Lorenz ldquoStability and monotonicity properties of stiff quasi-linear boundary value problemsrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta Serija zaMatematiku vol 12 pp 151ndash175 1982

[8] E Bohl Finite Modelle Gewohnlicher Randwertaufgaben Teub-ner Stuttgart Germany 1981

[9] E C Gartland Jr ldquoGraded-mesh difference schemes for sin-gularly perturbed two-point boundary value problemsrdquo Math-ematics of Computation vol 51 no 184 pp 631ndash657 1988

[10] R Vulanovic ldquoOn a numerical solution of a type of singularlyperturbed boundary value problem by using a special dis-cretization meshrdquo Univerzitet u Novom Sadu Zbornik RadovaPrirodno- Matematichkog Fakulteta Serija za Matematiku vol13 pp 187ndash201 1983

[11] C Grossmann H-G Roos and M Stynes Numerical Treat-ment of Partial Differential Equations Springer Berlin Ger-many 2007

[12] V B Andreev and I A Savin ldquoThe uniform convergence withrespect to a small parameter of A A Samarskiirsquos monotonescheme and its modificationrdquo Computational Mathematics andMathematical Physics vol 35 pp 581ndash591 1995

[13] T LinszligH-G Roos andRVulanovic ldquoUniformpointwise con-vergence on Shishkin-type meshes for quasi-linear convection-diffusion problemsrdquo SIAM Journal on Numerical Analysis vol38 no 3 pp 897ndash912 2000

[14] R Vulanovic and T A Nhan ldquoUniform convergence viapreconditioningrdquo International Journal of Numerical Analysisand Modeling B vol 5 no 4 pp 347ndash356 2014

[15] H-G Roos ldquoA note on the conditioning of upwind schemes onShishkin meshesrdquo IMA Journal of Numerical Analysis vol 16no 4 pp 529ndash538 1996

[16] H-G Roos and T Linszlig ldquoSufficient condition for uniformconvergence on layeradapted gridsrdquo Computing vol 63 no 1pp 27ndash45 1999

[17] R Vulanovic ldquoA higher-order scheme for quasilinear boundaryvalue problems with two small parametersrdquo Computing vol 67no 4 pp 287ndash303 2001

[18] R Vulanovic ldquoA priori meshes for singularly perturbed quasi-linear twominuspoint boundary value problemsrdquo IMA Journal ofNumerical Analysis vol 21 no 1 pp 349ndash366 2001

[19] T A Nhan and R Vulanovic ldquoUniform convergence on aBakhvalov-type mesh using the preconditioning approachTechnical reportrdquo httparxivorgabs150404283

[20] N S Bakhvalov ldquoThe optimization of methods of solvingboundary value problems with a boundary layerrdquo USSR Com-putational Mathematics and Mathematical Physics vol 9 no 4pp 139ndash166 1969

[21] H-G Roos andM Stynes ldquoSome open questions in the numer-ical analysis of singularly perturbed differential equationsrdquoComputational Methods in Applied Mathematics vol 15 no 4pp 531ndash550 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Numerical Analysis

In this paper we consider the simplest singularly per-turbed convection-diffusion problem in one dimension

L119906 fl minus12057611990610158401015840

minus 119887 (119909) 1199061015840

+ 119888 (119909) 119906 = 119891 (119909)

119909 isin (0 1) 119906 (0) = 119906 (1) = 0

119887 (119909) ge 120573 gt 0

119888 (119909) ge 0

for 119909 isin 119868 fl [0 1]

(1)

where 0 lt 120576 ≪ 1 and 119887 119888 and 119891 are 1198621

(119868)-functions Theproblem has a unique solution 119906 in 119862

3

(119868) [6 7] In generalthis solution exhibits an exponential boundary layer near 119909 =

0Problem (1) serves as a model problem for discussing

some theoretical aspects related to the proof of 120576-uniformpointwise convergence of numerical methods for singularlyperturbed convection-diffusion problems This proof is notas simple as for the singularly perturbed reaction-diffusionproblems where the proof can be based on the follow-ing classical principle which originates from nonperturbedproblems (cf [8ndash10])

Principle 1 120576-uniform stability and 120576-uniform consistencyboth in the maximum norm imply 120576-uniform pointwiseconvergence

Principle 1 does not work in the proof of 120576-uniformpointwise convergence for convection-diffusion problem (1)When (1) is discretized by using the upwind finite-differencescheme on the Shishkin mesh a careful analysis shows thatthe consistency error in the layer part is not 120576-uniformlyaccurate in the maximum norm (cf [11 Sec 613]) Hencespecial techniques like barrier functions [1ndash4] or the use ofhybrid stability inequalities [2 9 12 13] are required to provethe 120576-uniform pointwise convergence

We show in [14] that it is possible to apply Principle1 to problems of type (1) after the linear system whichresults when (1) is discretized is preconditionedThe upwinddiscretization scheme on the standard Shishkin mesh isconsidered in that paper Although the scheme is 120576-uniformlystable the discrete linear system is ill-posed since its condi-tion number grows unboundedly when 120576 rarr 0 as pointedout by Roos in [15] By using a relatively simple diagonalpreconditioner Roos eliminates the 120576-dependence of thecondition number

It is shown in [14] that a suitable modification of Roosrsquopreconditioner enables the proof based on Principle 1that the pointwise errors of the numerical solution are 120576-uniformly of almost first order Since the standard Shishkinmesh is the only one considered in [14] the question is nowhow to use the same approach and achieve the same resultfor the general class of Shishkin-type meshes in the sense of[2 16] The present paper addresses this question

Therefore for convection-diffusion problem (1) we con-sider the new proof technique of 120576-uniform pointwise con-vergence introduced in [14] and we generalize it to thewhole class of Shishkin-type meshes The generalization is

not entirely straightforward since the general properties ofthe meshes require many new technical details Moreover weslightly modify the preconditioner used in [14]

In this final paragraph of the introduction we give theoutline of the paper In Section 2 we describe the Shishkin-type meshes which are used when discretizing problem (1)We analyze the conditioning of the upwind finite-differencediscretization in Section 3 Then in Section 4 we define thepreconditioner and give a proof of 120576-uniform stability of thepreconditioned discrete system We arrive at the main resultthe proof of 120576-uniformpointwise convergence using Principle1 in Section 5The last section Section 6 contains some finalconclusions

2 Shishkin-Type Meshes

Let 119868119873 be a general discretization mesh with points 119909119894

where

0 = 1199090

lt 1199091

lt sdot sdot sdot lt 119909119873minus1

lt 119909119873

= 1 (2)

We consider the Shishkin-type (or S-type) meshes as intro-duced in [16] see [2] as well Any such mesh is dense in theboundary layer near119909 = 0 and transitions to a coarse uniformmesh at a point constructed like in the standard Shishkinmesh

120590 fl 119886120576119871 (3)

Here 119886 is a user-chosen positive parameter and we think of 119871as of 119871 = ln119873 although a modification of this value can alsobe used like in [17] We assume that 120590 lt 12 otherwise thevalues of 119873 are unreasonably large

The coarse part of the mesh is obtained by uniformlydividing the interval [120590 1] into 119873 minus 119869 subintervals of length119867 = (1 minus 120590)(119873 minus 119869) where 119869 is a positive integer such that119876 fl 119869119873 satisfies 119876 lt 1 and 119876

minus1

le 119862 and where from thispoint on 119862 denotes a generic positive constant independentof both the perturbation parameter 120576 and themesh parameter119873 Inside the layer the mesh points are formed by a mesh-generating function 120601 which is a monotonically increasingfunction satisfying 120601(0) = 0 and 120601(119876) = 119871 In other wordsS-type meshes are defined by

119909119894

=

119886120576120601 (119905119894

) 119894 = 0 1 119869

120590 + (119905119894

minus 119876)

1 minus 120590

1 minus 119876

119894 = 119869 + 1 119873

(4)

where 119905119894

fl 119894119873The mesh steps are defined as ℎ

119894

= 119909119894

minus 119909119894minus1

119894 =

1 2 119873 the steps for 119894 = 1 2 119869 belong to the fine partof the mesh and ℎ

119894

= 119867 for 119894 = 119869+1 119869+2 119873 Let also ℏ119894

=

(ℎ119894

+ℎ119894+1

)2 119894 = 1 2 119873minus1 and let mesh functions on 119868119873

be denoted by119882119873119880119873 and so forth For a function119892 defined

on 119868 119892119894

indicates 119892(119909119894

) and 119892119873 stands for the corresponding

mesh functionWe identifymesh functions119882119873 with (119873+1)-dimensional column vectors 119882119873 = [119882

119873

0

119882119873

1

119882119873

119873

]119879

We use the maximum norm of 11988211987310038171003817100381710038171003817119882119873

10038171003817100381710038171003817= max0le119894le119873

10038161003816100381610038161003816119882119873

119894

10038161003816100381610038161003816 (5)

Advances in Numerical Analysis 3

Table 1 Examples of mesh-generating and mesh-characterising functions of S-type meshes

Mesh 120601 (119905) max1206011015840

120595 max 10038161003816100381610038161205951015840

1003816100381610038161003816

S 119905119871

119876

119871

119876

119873minus119905119876

119871

119876

BS minus ln(1 minus

119905

119876

(1 minus

1

119873

))

119873

119876

1 minus

119905

119876

(1 minus

1

119873

)

1

119876

VS 119905

119876 + 119876119871 minus 119905

21198712

119876

exp(minus

119905

119876 + 119876119871 minus 119905

)

4

119876

The matrix norm induced by the above maximum vectornorm is also denoted by sdot

Following [2 16] let us define the mesh-characterizingfunction 120595 which is related to the mesh-generating function120601 as follows

120595 (119905) = 119890minus120601(119905)

(6)

This function is monotonically decreasing on [0 119876] with120595(0) = 1 and 120595(119876) = 119873

minus1We need to assume some further properties of the mesh

Assumption 1 Let the mesh-generating function 120601(119905) bepiecewise differentiable such that

max119905isin[0119876]

1206011015840

(119905) le 119862119873 (7)

Moreover we also assume that 120601(119905) fulfills

Δ120601

fl min119894=12119869

Δ119894

ge 119862119873minus1

(8)

where Δ119894

fl 120601(119905119894

) minus 120601(119905119894minus1

) 119894 = 1 2 119873

Remark 2 Examples of mesh-generating functions 120601 satisfy-ing the above assumptions are the standard Shishkinmesh (S-mesh) Bakhvalov-Shishkinmesh (BS-mesh) andVulanovic-Shishkin mesh (VS-mesh) They are summarized in Table 1

We proceed to provide some estimates for the step size ofthe S-type meshes generated by 120601 in the layer region Becausefor 119894 = 1 119869

119909119894

= 119886120576120601 (119905119894

) = minus119886120576 ln120595 (119905119894

) lArrrArr 120595(119905119894

) = 119890minus119909119894(119886120576)

(9)

we can bound the mesh step size inside the layer from above

ℎ119894

= 119886120576 (120601 (119905119894

) minus 120601 (119905119894minus1

)) le 119886120576119873minus1 max119905isin[119905119894minus1119905119894]

1206011015840

(119905)

le 119886120576119873minus1

(max119905isin[119905119894minus1 119905119894]

100381610038161003816100381610038161205951015840

(119905)

10038161003816100381610038161003816)

120595 (119905119894

)

= 119886120576119873minus1

( max119905isin[119905119894minus1119905119894]

100381610038161003816100381610038161205951015840

(119905)

10038161003816100381610038161003816) 119890119909119894(119886120576)

le 119886120576119873minus1max 1003816

10038161003816100381610038161205951015840

10038161003816100381610038161003816119890119909119894(119886120576)

(10)

where we use max|1205951015840| fl max119905isin[0119876]

|1205951015840

(119905)| Thus for 1 le 119894 le

119869 minus 1 we have that

ℏ119894

le

119886

2

120576119873minus1max 1003816

10038161003816100381610038161205951015840

10038161003816100381610038161003816(119890119909119894(119886120576)

+ 119890119909119894+1(119886120576)

)

le 119886120576119873minus1max 1003816

10038161003816100381610038161205951015840

10038161003816100381610038161003816119890119909119894+1(119886120576)

(11)

3 The Upwind Scheme andIts Condition Number

We consider the following upwind finite-difference dis-cretization on 119868

119873

119880119873

0

= 0

L119873

119880119873

119894

fl minus12057611986310158401015840

119880119873

119894

minus 119887119894

1198631015840

119880119873

119894

+ 119888119894

119880119873

119894

= 119891119894

119894 = 1 119873 minus 1

119880119873

119873

= 0

(12)

where

11986310158401015840

119882119894

=

1

ℏ119894

(

119882119894+1

minus 119882119894

ℎ119894+1

minus

119882119894

minus 119882119894minus1

ℎ119894

)

1198631015840

119882119894

=

119882119894+1

minus 119882119894

ℎ119894+1

(13)

The matrix form of the linear system (12) is

119860119873

119880119873

=119891

119873 (14)

with a tridiagonal matrix 119860119873

= [119886119894119895

] and 119891

119873

= [0 1198911

1198912

119891119873minus1

0]119879 The nonzero elements of 119860

119873

are 11988600

= 1 119886119873119873

= 1

119897119894

fl 119886119894minus1119894

=

minus

120576

ℏ119894

ℎ119894

1 le 119894 le 119869

minus

120576

1198672

119869 + 1 le 119894 le 119873 minus 1

119903119894

fl 119886119894119894+1

=

minus

120576

ℏ119894

ℎ119894+1

minus

119887119894

ℎ119894+1

1 le 119894 le 119869

minus

120576

1198672

minus

119887119894

119867

119869 + 1 le 119894 le 119873 minus 1

119889119894

fl 119886119894119894

=

1 119894 = 0

minus119897119894

minus 119903119894

+ 119888119894

1 le 119894 le 119873 minus 1

1 119894 = 119873

(15)

4 Advances in Numerical Analysis

Table 2 The condition number of discrete system (12) on the BS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 206119890 + 04 896119890 + 04 372119890 + 05 152119890 + 06 613119890 + 06 246119890 + 07

1119890 minus 3 219119890 + 05 948119890 + 05 393119890 + 06 160119890 + 07 645119890 + 07 259119890 + 08

1119890 minus 4 220119890 + 06 954119890 + 06 396119890 + 07 161119890 + 08 649119890 + 08 261119890 + 09

1119890 minus 5 220119890 + 07 955119890 + 07 396119890 + 08 161119890 + 09 650119890 + 09 261119890 + 10

1119890 minus 6 220119890 + 08 955119890 + 08 396119890 + 09 161119890 + 10 650119890 + 10 261119890 + 11

1119890 minus 7 220119890 + 09 955119890 + 09 396119890 + 10 161119890 + 11 650119890 + 11 261119890 + 12

1119890 minus 8 220119890 + 10 955119890 + 10 396119890 + 11 161119890 + 12 650119890 + 12 261119890 + 13

Table 3 The condition number of discrete system (12) on the VS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 275119890 + 04 123119890 + 05 511119890 + 05 205119890 + 06 812119890 + 06 320119890 + 07

1119890 minus 3 291119890 + 05 130119890 + 06 539119890 + 06 216119890 + 07 855119890 + 07 336119890 + 08

1119890 minus 4 293119890 + 06 131119890 + 07 543119890 + 07 218119890 + 08 861119890 + 08 339119890 + 09

1119890 minus 5 293119890 + 07 131119890 + 08 543119890 + 08 218119890 + 09 862119890 + 09 339119890 + 10

1119890 minus 6 293119890 + 08 131119890 + 09 543119890 + 09 218119890 + 10 862119890 + 10 339119890 + 11

1119890 minus 7 293119890 + 09 131119890 + 10 543119890 + 10 218119890 + 11 862119890 + 11 339119890 + 12

1119890 minus 8 293119890 + 10 131119890 + 11 543119890 + 11 218119890 + 12 862119890 + 12 339119890 + 13

Matrix 119860119873

is an 119871-matrix that is it has positive diagonalentries and nonnegative off-diagonal entries It is also anonsingular matrix satisfying119860

minus1

119873

ge 0 (inequalities involvingmatrices and vectors should be understood componentwise)and therefore inverse monotone In other words 119860

119873

is an119872-matrix (an inverse monotone 119871-matrix) This fact is easyto prove using the following119872-criterion see [8] for instance

Theorem 3 (119872-criterion) Let 119860 be an 119871-matrix and thereexists a vector 119908 such that 119908 gt 0 and 119860119908 ge 120574 for somepositive constant 120574 119860 is then an 119872-matrix and it holds that119860minus1

le 120574minus1

119908

In the above theorem set 119908119894

= 2 minus 119909119894

119894 = 0 1 119873to get that 119860

119873

119908 ge min1 120573 This implies that 119860119873

is an 119872-matrix and that

10038171003817100381710038171003817119860minus1

119873

10038171003817100381710038171003817le

2

min 1 120573

le 119862 (16)

which means that discrete problem (14) is stable uniformly in120576 By examining directly the entries of matrix 119860

119873

and notingthat the lower bounds for ℎ

119894

and ℏ119894

are achieved in the layerregion where ℎ

119894

sim ℏ119894

ge 119862120576Δ120601

we get that

1003817100381710038171003817119860119873

1003817100381710038171003817le 119862

1

120576 (Δ120601

)

2

(17)

Combining this and the 120576-uniform stability (16) we arrive atthe following result related to the condition number 120581(119860

119873

)

of matrix 119860119873

120581(119860119873

) fl 119860119873

119860minus1

119873

Theorem 4 The condition number of matrix 119860119873

satisfies theestimate

120581 (119860119873

) le 119862

1

120576 (Δ120601

)

2

(18)

Let us illustrate Theorem 4 using a simple example takenfrom [2 page 1]

minus12057611990610158401015840

(119909) minus 1199061015840

(119909) = 1 0 lt 119909 lt 1

119906 (0) = 119906 (1) = 0

(19)

It is easy to observe the 120576-dependence of the conditionnumbers in Tables 2 and 3 They clearly show that the boundin Theorem 4 is sharp For the standard Shishkin mesh werefer the readers to the numerical experiments in [15]

4 Preconditioning and Stability

In this section we propose an appropriately designeddiagonal preconditioner which not only eliminates the 120576-dependence of the condition number of the discrete systemsbut also retains its 120576-uniform stability in the sense of (16)Thepreconditioner is not of interest per se but as ameans to prove120576-uniform pointwise convergence via Principle 1

Let 119872 = diag(1198980

1198981

119898119873

) be a diagonal matrix withthe entries

1198980

= 1

119898119894

=

ℏ119894

119867

119894 = 1 2 119873 minus 1

119898119873

= 1

(20)

On noting that ℏ119894

= 119867 for 119894 = 119869 + 1 119873 minus 1 we seethat when the discrete system (12) is multiplied by 119872 this isequivalent to multiplying the equations 1 2 119869 of system(14) by ℏ

119894

119867 119894 = 1 2 119869 The modified system is

119873

119880119873

= 119872119891

119873

(21)

Advances in Numerical Analysis 5

where 119873

= 119872119860119873

Let the entries of 119873

be denoted by 119894119895

the nonzero ones being

119897119894

fl 119894minus1119894

=

minus

120576

ℎ119894

119867

1 le 119894 le 119869

minus

120576

1198672

119869 + 1 le 119894 le 119873 minus 1

119894

fl 119894119894+1

=

minus

120576

ℎ119894+1

119867

minus

119887119894

ℏ119894

ℎ119894+1

119867

1 le 119894 le 119869 minus 1

minus

120576

1198672

minus

119887119894

ℏ119894

1198672

119894 = 119869

minus

120576

1198672

minus

119887119894

119867

119869 + 1 le 119894 le 119873 minus 1

119889119894

fl 119894119894

=

1 119894 = 0

minus119897119894

minus 119894

+ 119888119894

ℏ119894

119867

1 le 119894 le 119869

minus119897119894

minus 119894

+ 119888119894

119869 + 1 le 119894 le 119873 minus 1

1 119894 = 119873

(22)

Matrix 119873

remains an 119871-matrix since the precondi-tioning does not change this property of 119860

119873

We showin Lemma 7 below that

119873

is an 119872-matrix and that themodified discretization (21) is stable uniformly in 120576 But firstwe give a technical result related to the mesh

Lemma 5 Let 119886120573 gt 2 and let in addition to Assumption 1 themesh-generating function 120601 satisfy the following conditions on[0 119876]

(i) 120601(119895)

ge 0 119895 = 1 2 3(ii) 1206011015840

(0) ge 119898 gt 0 where 119898 is a constant independent of 120576and 119873

(iii) 12060110158401015840

le 2(1206011015840

)2

(iv) 1206011015840

le 119862119871119896 for some 119896 gt 0

Then for a sufficiently large119873 independent of 120576 there exists apositive constant 120575 independent of both 120576 and 119873 such that

119878119894

fl120573

2

(1 +

Δ119894

Δ119894+1

) minus

1

119886

(

1

Δ119894

minus

1

Δ119894+1

) ge 120575 gt 0

119894 = 1 119869 minus 1

(23)

Proof Since 119886120573 gt 2 we have that 1205732 ge 1119886 + 120578 for some120578 gt 0 Then

119878119894

ge

1

119886

119875119894

+ 120578 (24)

where

119875119894

=

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

Δ119894

Δ119894+1

(25)

Then because of (i) and (iii)

119875119894

ge 2119875lowast119894

where 119875lowast119894

fl1206011015840

(119905119894minus1

)2

minus 1206011015840

(119905119894+1

)2

1198732

Δ119894

Δ119894+1

(26)

If we show that

119875lowast119894

997888rarr 0 uniformly in 120576 when 119873 997888rarr infin (27)

then we will have (23) provided 119873 is sufficiently largeindependently of 120576 We now show that (27) holds true First

1003816100381610038161003816119875lowast119894

1003816100381610038161003816le 119862

100381610038161003816100381610038161206011015840

(119905119894minus1

)2

minus 1206011015840

(119905119894+1

)2

10038161003816100381610038161003816

(28)

because of (i) and (ii) and then

1003816100381610038161003816119875lowast119894

1003816100381610038161003816le 119862119873

minus1

119871119896

12060110158401015840

(119905119894+1

) le 119862119873minus1

1198713119896 (29)

because of (i) (iii) and (iv)

Themeshes defined inTable 1 satisfy conditions (i)ndash(iv) ofLemma 5 with the exception of the BS-mesh which does notfulfill condition (iv) However estimate (23) still holds truefor the BS-mesh if we assume that radic2 minus 1 le 119876 lt 1 This isnot a serious restriction since119876 = 12 is a typical choiceTheproof of this result stated below in Lemma 6 is given in theappendix

Lemma 6 Let 119886120573 gt 2 and let radic2 minus 1 le 119876 lt 1 Then theBS-mesh satisfies estimate (23)

Lemma 7 Under the assumptions of Lemma 5 (Lemma 6 ifthe BS-mesh is used) matrix

119873

of system (21) satisfies

100381710038171003817100381710038171003817

minus1

119873

100381710038171003817100381710038171003817

le 119862 (30)

Proof We construct a vector V = [V0

V1

V119873

]119879 such that

(a) V119894

gt 0 119894 = 0 1 119873(b) V119894

le 119862 119894 = 0 1 119873

(c) 119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

ge 120579 119894 = 1 2 119873 minus 1 where 120579

is a positive constant independent of both 120576 and 119873

Then according to the 119872-criterion

100381710038171003817100381710038171003817

minus1

119873

100381710038171003817100381710038171003817

le 120579minus1

V le 119862 (31)

We define the vector V as follows (cf [14 15])

V119894

= 120572 minus 119867119894 + 120582min (1 + 120588)119869minus119894

1 (32)

where 120572 and 120582 are appropriately chosen positive constantsand 120588 = 120573119867120576

Since 119867119873 le 119862 there exists a constant 120572 such that V119894

ge

120572minus119867119894 gt 0 and condition (a) is satisfied It also holds true thatV119894

le 120572+120582Therefore condition (b) is satisfied if we prove that120582 le 119862 We do this in next steps while we verify condition (c)

6 Advances in Numerical Analysis

When 1 le 119894 le 119869 minus 1 using Lemma 5 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= (119897119894

+119889119894

+ 119894

) V119894

+119897119894

119867 minus 119894

119867

=

ℏ119894

119867

119888119894

V119894

minus

120576

ℎ119894

+

120576

ℎ119894+1

+

119887119894

ℏ119894

ℎ119894+1

ge minus(

120576

ℎ119894

minus

120576

ℎ119894+1

) +

119887119894

2

+

119887119894

ℎ119894

2ℎ119894+1

ge 119878119894

ge 120575 gt 0

(33)

For 119894 = 119869 condition (c) is verified as follows

119897119869

V119869minus1

+119889119869

V119869

+ 119869

V119869+1

=

ℏ119869

119867

119888119869

V119869

+119897119869

119867 minus 119869

(119867 +

120582120588

1 + 120588

)

ge minus119869

120582120588

1 + 120588

+ (119897119869

minus 119869

)119867

= (

120576

1198672

+

119887119869

ℏ119869

1198672

)

120582120588

1 + 120588

minus

120576

ℎ119869

+

120576

119867

+

119887119869

ℏ119869

119867

= (

2120576 + 119887119869

(ℎ119869

+ 119867)

21198672

)

120582120573119867

120576 + 120573119867

minus

120576

ℎ119869

+

119887119869

2

ge

120582120573

2119867

minus

120576

ℎ119869

+

120573

2

ge

120573

2

(34)

where in the last step invoking (8) we choose a constant 120582 sothat

120582120573

2119867

minus

120576

ℎ119869

ge

120582120573

2119867

minus

120576

119886120576 (120601 (119905119869

) minus 120601 (119905119869minus1

))

ge

120582120573

2119867

minus

1

119886Δ119869

ge

120582120573

4119873minus1

minus

1

119886119862119873minus1

= 119873(

120582120573

4

minus

1

119886119862

) ge 0

(35)

To guarantee this 120582 is chosen so that

120582120573

4

ge

1

119886119862

(36)

Finally when 119869 + 1 le 119894 le 119873 minus 1 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= 119888119894

V119894

+119897119894

119867 minus 119894

119867

+

119897119894

1 + 120588119869

[

120582

(1 + 120588)119894minus1minus119869

minus

120582

(1 + 120588)119894minus119869

]

+

119894

1 + 120588119869

[

120582

(1 + 120588)119894+1minus119869

minus

120582

(1 + 120588)119894minus119869

]

ge 119887119894

+

120588 (1 + 120588)119897119894

minus 120588119894

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

ge 120573 +

(119897119894

minus 119894

+119897119894

120588) 120588

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

= 120573 + (

119887119894

119867

minus

120573

119867

)

120582120588 (1 + 120588)119869minus119894minus1

1 + 120588119869

ge 120573

(37)

By examining the entries of 119873

we easily see that themaximum absolute row sum of

119873

is achieved in the layerregion Therefore

10038171003817100381710038171003817119873

10038171003817100381710038171003817le 119862

119873

Δ120601

(38)

A combination of Lemma 7 and (38) results in thefollowing

Theorem 8 The condition number of matrix 119873

satisfies thefollowing 120576-uniform estimate

120581 (119873

) le 119862

119873

Δ120601

(39)

Corollary 9 For the meshes given in Table 1 one has thefollowing estimates

120581 (119873

) le

119862

1198732

119871

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

1198621198732

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(40)

In Tables 4 and 5 we present the condition numbers ofthe preconditioned linear systems (21) for various values of120576 and 119873 for the BS- and VS-meshes We can observe thatthe condition numbers of the preconditioned systems areindependent of 120576 and that the bound in Corollary 9 is sharp

5 120576-Uniform Convergence

For the proof of 120576-uniform convergence we need to decom-pose the solution 119906 into the smooth and boundary-layer

Advances in Numerical Analysis 7

Table 4 The condition number of matrix 119873

on the BS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 658119890 + 02 268119890 + 03 108119890 + 04 434119890 + 04 174119890 + 05 695119890 + 05

1119890 minus 3 689119890 + 02 278119890 + 03 111119890 + 04 440119890 + 04 175119890 + 05 700119890 + 05

1119890 minus 4 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 702119890 + 05

1119890 minus 5 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 6 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 7 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 8 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

Table 5 The condition number of matrix 119873

on the VS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 748119890 + 02 313119890 + 03 127119890 + 04 510119890 + 04 204119890 + 05 811119890 + 05

1119890 minus 3 780119890 + 02 323119890 + 03 130119890 + 04 520119890 + 04 207119890 + 05 820119890 + 05

1119890 minus 4 785119890 + 02 325119890 + 03 131119890 + 04 522119890 + 04 208119890 + 05 824119890 + 05

1119890 minus 5 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 6 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 7 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 8 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

parts There exist several decompositions of this kind but weuse the version presented in [2 Theorem 348]

119906 (119909) = 119904 (119909) + 119910 (119909) 119909 isin 119868 (41)10038161003816100381610038161003816119904(119896)

(119909)

10038161003816100381610038161003816le 119862 (1 + 120576

2minus119896

)

10038161003816100381610038161003816119910(119896)

(119909)

10038161003816100381610038161003816le 119862120576minus119896

119890minus120573119909120576

119909 isin 119868 119896 = 0 1 2 3

(42)

The details related to the construction of the function 119904 arenot important here they can be found in [2] As for 119910 wenote that it solves the problem

L119910 (119909) = 0 119909 isin (0 1)

119910 (0) = minus119904 (0)

119910 (1) = 0

(43)

We define the consistency error of the finite-differenceoperatorL119873 as

120591119894

= 120591119894

[119906] fl L119873

119906119894

minus 119891119894

119894 = 1 2 119873 minus 1 (44)

It holds true that

120591119894

= L119873

119906119894

minus (L119906)119894

= [119860119873

(119906119873

minus 119880119873

)]119894

(45)

Taylorrsquos expansion gives

1003816100381610038161003816120591119894

[119906]1003816100381610038161003816le 119862ℎ119894+1

(120576

10038171003817100381710038171003817119906101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711990610158401015840

10038171003817100381710038171003817119894) (46)

where 119892119894

fl max119909119894minus1le119909le119909119894+1

|119892(119909)| for any 119862(119868)-function 119892When the diagonal matrix 119872 is multiplied to linear system(14) the consistency error of the multiplied system is

119894

[119906] =

ℏ119894

119867

120591119894

[119906] 1 le 119894 le 119869

120591119894

[119906] 119869 + 1 le 119894 le 119873 minus 1

(47)

Lemma 10 Let 119886120573 ge 2Then the following estimate holds true

1003816100381610038161003816119894

[119906]1003816100381610038161003816le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(48)

Proof We use decomposition (41) to get

119894

[119906] = 119894

[119904] + 119894

[119910] (49)

Then (46) and the derivative-estimates of 119904 given in (42)yield immediately that

1003816100381610038161003816119894

[119904]1003816100381610038161003816le 119862119873

minus1 (50)

and therefore the following remains to be proved

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

119894 = 119869 119873 minus 1

(51)

Note that from condition (7) we easily get

ℎ119894

le 119862120576 119894 = 1 119869

ℏ119894

le 119862120576 119894 = 1 119869 minus 1

(52)

This is the key property of S-type meshes as well as the mainingredient of the following proof

8 Advances in Numerical Analysis

Table 6 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 550119890 + 00 293119890 + 00 151119890 + 00 769119890 minus 01 388119890 minus 01 195119890 minus 01

1119890 minus 3 550119890 + 01 293119890 + 01 151119890 + 01 769119890 + 00 388119890 + 00 195119890 + 00

1119890 minus 4 550119890 + 02 293119890 + 02 151119890 + 02 769119890 + 01 388119890 + 01 195119890 + 01

1119890 minus 5 550119890 + 03 293119890 + 03 151119890 + 03 769119890 + 02 388119890 + 02 195119890 + 02

1119890 minus 6 550119890 + 04 293119890 + 04 151119890 + 04 769119890 + 03 388119890 + 03 195119890 + 03

1119890 minus 7 550119890 + 05 293119890 + 05 151119890 + 05 769119890 + 04 388119890 + 04 195119890 + 04

1119890 minus 8 550119890 + 06 293119890 + 06 151119890 + 06 769119890 + 05 388119890 + 05 195119890 + 05

For 1 le 119894 le 119869 minus 1 (46) estimates (10) and (11) as well asthe derivative-estimates of 119910 in (42) imply

1003816100381610038161003816119894

(119910)1003816100381610038161003816le 119862

ℏ119894

119867

ℎ119894+1

(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894)

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902119909119894+1(119886120576)

119890minus120573119909119894minus1120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890(2119909119894+1minus119886120573119909119894minus1)119886120576

= 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902(119909119894+1minus119909119894minus1)119886120576

119890minus(119886120573minus2)119909119894minus1119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890ℏ119894119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

(53)

where we have used (52) and 119886120573 ge 2 to get

119890ℏ119894(119886120576)

le 119862

119890minus(119886120573minus2)119909119894minus1(119886120576)

le 119862

(54)

When 119869 + 2 le 119894 le 119873 minus 1 (42) and (46) give

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119867(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894) le 119862

119867

1205762

119890minus120573119909119894minus1120576

le 119862

119867

1205762

119890minus120573(120590+119867)120576

le 119862119873

1198672

1205762

119890minus120573119867120576

119890minus120573120590120576

(55)

Estimate (51) follows from the fact that (119867120576minus1

)2

119890minus120573119867120576

le 119862

and because the definition of 120590 implies that

119890minus120573120590120576

= 119890minus120573119886119871

= 119873minus120573119886

le 119873minus2

(56)

We finally prove (51) when 119894 = 119869 119869 + 1 in which case

1003816100381610038161003816119869

[119910]1003816100381610038161003816=

ℏ119869

119867

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816

1003816100381610038161003816119869+1

[119910]1003816100381610038161003816=

1003816100381610038161003816120591119869+1

[119910]1003816100381610038161003816

(57)

We now use the fact that L119910 = 0 and work with a differentform of the consistency error (this is a well-known techniquecf [18 Lemma 5])

1003816100381610038161003816119894

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816le 119877119894

+ 119876119894

+ 119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816 (58)

where

119877119894

= 120576

1003816100381610038161003816100381611986310158401015840

119910119894

10038161003816100381610038161003816

119876119894

= 119887119894

100381610038161003816100381610038161198631015840

119910119894

10038161003816100381610038161003816

(59)

We immediately have that

119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816le 119862119890minus120573119909119894120576

le 119862119890minus120573120590120576

= 119862119890minus120573119886119871

= 119862119873minus119886120573

le 119862119873minus2

(60)

As for 119877119894

it holds true that

119877119894

le ℏminus1

119894

120576 sdot 2

100381710038171003817100381710038171199101015840

10038171003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873119890minus120573120590120576

119890120573ℎ119869120576

le 119862119873minus1

(61)

by (52) Analogously

119876119894

le 119862119867minus1

10038171003817100381710038171199101003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873minus1

(62)

This completes the proof

In general the consistency error 120591119894

of the layer componentof the solution is not bounded pointwise uniformly in 120576When the above proof technique is applied to 120591

119894

[119910] one canonly get

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816

le

119862120576minus1

119873minus1

119871 for the S-mesh

119862120576minus1

119873minus1

for the BS- and VS-meshes

(63)

The proposed preconditioner simplifies the analysis of theconsistency error by introducing an extra 120576 factor needed forthe layer component Hence the approach used in the proofof Lemma 10 is natural and straightforward because it onlyuses the classical Taylor expansion estimate for the truncationerror

To illustrate (63) in Tables 6 and 7 we present the resultsof some numerical experiments on the BS- and VS-meshesrespectivelyWe refer the reader to [14] for the correspondingnumerical results on the S-mesh

By contrast with preconditioner (20) the preconditionedconsistency error converges uniformly in 120576 as shown in Tables8 and 9

In conclusion the preconditioning allows us to usePrinciple 1 and combine Lemmas 7 and 10 to obtain thefollowing main convergence result

Advances in Numerical Analysis 9

Table 7 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 467119890 + 00 247119890 + 00 128119890 + 00 659119890 minus 01 336119890 minus 01 170119890 minus 01

1119890 minus 3 467119890 + 01 247119890 + 01 128119890 + 01 659119890 + 00 336119890 + 00 170119890 + 00

1119890 minus 4 467119890 + 02 247119890 + 02 128119890 + 02 659119890 + 01 336119890 + 01 170119890 + 01

1119890 minus 5 467119890 + 03 247119890 + 03 128119890 + 03 659119890 + 02 336119890 + 02 170119890 + 02

1119890 minus 6 467119890 + 04 247119890 + 04 128119890 + 04 659119890 + 03 336119890 + 03 170119890 + 03

1119890 minus 7 467119890 + 05 247119890 + 05 128119890 + 05 659119890 + 04 336119890 + 04 170119890 + 04

1119890 minus 8 467119890 + 06 247119890 + 06 128119890 + 06 659119890 + 05 336119890 + 05 170119890 + 05

Table 8 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 122119890 minus 01 650119890 minus 02 338119890 minus 02 174119890 minus 02 887119890 minus 03 452119890 minus 03

1119890 minus 3 114119890 minus 01 601119890 minus 02 308119890 minus 02 156119890 minus 02 786119890 minus 03 395119890 minus 03

1119890 minus 4 114119890 minus 01 597119890 minus 02 306119890 minus 02 155119890 minus 02 778119890 minus 03 390119890 minus 03

1119890 minus 5 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 390119890 minus 03

1119890 minus 6 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 7 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 8 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

Table 9 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 160119890 minus 01 912119890 minus 02 502119890 minus 02 270119890 minus 02 143119890 minus 02 747119890 minus 03

1119890 minus 3 150119890 minus 01 844119890 minus 02 458119890 minus 02 242119890 minus 02 126119890 minus 02 653119890 minus 03

1119890 minus 4 149119890 minus 01 837119890 minus 02 454119890 minus 02 240119890 minus 02 125119890 minus 02 645119890 minus 03

1119890 minus 5 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 6 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 7 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 8 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

Theorem 11 The error of the upwind finite-difference dis-cretization on S-type meshes satisfies

10038161003816100381610038161003816119906119873

119894

minus 119880119873

119894

10038161003816100381610038161003816

le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(64)

Corollary 12 For the meshes given in Table 1 one has thefollowing estimates

10038171003817100381710038171003817119906119873

minus 119880119873

10038171003817100381710038171003817

le

119862119873minus1

1198712

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

119862119873minus1

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(65)

Remark 13 The barrier-function technique can be used toobtain similar results for S-type meshes (see [16]) but itrequires another assumption on the mesh-generating func-tion 120601 (cf [16 Theorem 1]) However whether the barrier-function approach can be extended to any Bakhvalov-type

mesh is still an open question [2 Remark 421] By contrastas proved in [19] the preconditioner defined in (20) worksfine for Vulanovicrsquos modification of the Bakhvalov mesh [10]The paper [10] is where the original Bakhvalov mesh [20]is generalized (the main difference between Bakhvalov-typemeshes and S-type meshes is that the former have a smoothtransition from the fine part inside the layer to the coarseuniform part outside the layer the mesh-generating function120601 belongs to 119862

1

(119868))

6 Conclusion

In this paper we present a proof of pointwise uniformconvergence for singularly perturbed convection-diffusionproblem (1) discretized on a general class of Shishkin-typemeshesThe proof is based on preconditioning of the discretesystem which allows us to use the standard ldquoconsistency+ stabilityrdquo principle of convergence This is a conceptuallysimple approach which does not require the use of hybridstability inequalities and discrete Greenrsquos functions In gen-eral new proof techniques are of interest because althougha lot has been achieved in the field of singularly perturbed

10 Advances in Numerical Analysis

problems over the last few decades there are still importantquestions remaining open [21] One of them Question6 is related to the work presented here The question isabout two-dimensional convection-diffusion problems and itpoints out that whereas 120576-uniform pointwise convergence forthe upwind finite-difference discretization on S-type meshesis well-established no similar result for B-type meshes isknownAsmentioned inRemark 13 we already know that thepreconditioning approach works well for B-mesh and one-dimensional convection-diffusion problems The next step isto consider two-dimensional problems

Appendix

Proof of Lemma 6 The proof is different from the proof ofLemma 5because theBS-meshdoes not satisfy condition (iv)Because of (ii) it is sufficient to show that the numerator in(25) is nonnegative for radic2 minus 1 le 119876 lt 1 independently of 120576and 119873 Let 119901 fl 1 minus 119873

minus1 By direct computations we get that

Δ119894

= minus ln(1 minus

119905119894

119901

119876

) + ln(1 minus

119905119894minus1

119901

119876

)

= ln(1 +

1

119869119901 minus 119894

) = ln (1 + 119902)

(A1)

where we denote 119902 fl (119869119901minus119894)minus1 Note that 0 le 119902 lt 1(119876+1)

Analogously

Δ119894+1

= ln(

1

1 minus 119902

) (A2)

Then

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

= Δ119894

(Δ119894+1

+ Δ119894

+ 1) minus Δ119894+1

= ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902)

(A3)

We set

119892 (119902) fl ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902) (A4)

Sinceradic2minus1 le 119876 lt 1 we have 0 le 119902 lt 1radic2 Rearrange 119892(119902)

as

119892 (119902) = [1 + ln(1 +

2119902

1 minus 119902

)] ln (1 + 119902)

+ ln (1 minus 119902)

(A5)

For 119902 isin [0 1) it follows that119902

1 + 1199022

le ln (1 + 119902) le 119902 (A6)

Hence

ln(1 +

2119902

1 minus 119902

) ge 2119902 (A7)

So we are done if we can show that

119892 (119902) ge [1 + 2119902] ln (1 + 119902) + ln (1 minus 119902) ge 0

0 le 119902 lt

1

radic2

(A8)

or

(1 minus 119902) (1 + 119902)2119902+1

= (1 minus 1199022

) (1 + 119902)2119902

ge 1 (A9)

First if 119902 isin [12 1radic2] we have by (1 + 119902)120574

ge 1 + 120574119902 for120574 ge 1

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

) (1 + 21199022

) ge 1 (A10)

Second we have that ln(1+119902) is a concave-down function on[0 12] hence on [0 12]

ln (1 + 119902) ge 2 ln(

3

2

) 119902 ge

4119902

5

(A11)

Then multiplying both sides by 119902 and exponentiating we getthat

(1 + 119902)119902

ge exp(

41199022

5

) ge 1 +

41199022

5

(A12)

Therefore

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

)(1 +

41199022

5

)

2

ge 1 (A13)

which holds true for 119902 isin [0 12]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P A Farrell A F Hegarty J J H Miller E OrsquoRiordan andG I Shishkin Robust Computational Techniques for BoundaryLayers Chapman amp Hall CRC Press Boca Raton Fla USA2000

[2] T Linszlig Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems vol 1985 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[3] J J HMiller E OrsquoRiordan and G I Shishkin Fitted NumericalMethods for Singular Perturbation Problems World ScientificSingapore 1996

[4] H-G Roos M Stynes and L Tobiska Numerical Methodsfor Singularly Perturbed Differential Equations Springer BerlinGermany 2nd edition 2008

[5] A Malvandi and D D Ganji ldquoMixed convection of aluminandashwater nanofluid inside a concentric annulus consideringnanoparticle migrationrdquo Particuology vol 24 pp 113ndash122 2016

[6] R B Kellogg and A Tsan ldquoAnalysis of some difference approx-imations for a singular perturbation problem without turningpointsrdquoMathematics of Computation vol 32 no 144 pp 1025ndash1039 1978

Advances in Numerical Analysis 11

[7] J Lorenz ldquoStability and monotonicity properties of stiff quasi-linear boundary value problemsrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta Serija zaMatematiku vol 12 pp 151ndash175 1982

[8] E Bohl Finite Modelle Gewohnlicher Randwertaufgaben Teub-ner Stuttgart Germany 1981

[9] E C Gartland Jr ldquoGraded-mesh difference schemes for sin-gularly perturbed two-point boundary value problemsrdquo Math-ematics of Computation vol 51 no 184 pp 631ndash657 1988

[10] R Vulanovic ldquoOn a numerical solution of a type of singularlyperturbed boundary value problem by using a special dis-cretization meshrdquo Univerzitet u Novom Sadu Zbornik RadovaPrirodno- Matematichkog Fakulteta Serija za Matematiku vol13 pp 187ndash201 1983

[11] C Grossmann H-G Roos and M Stynes Numerical Treat-ment of Partial Differential Equations Springer Berlin Ger-many 2007

[12] V B Andreev and I A Savin ldquoThe uniform convergence withrespect to a small parameter of A A Samarskiirsquos monotonescheme and its modificationrdquo Computational Mathematics andMathematical Physics vol 35 pp 581ndash591 1995

[13] T LinszligH-G Roos andRVulanovic ldquoUniformpointwise con-vergence on Shishkin-type meshes for quasi-linear convection-diffusion problemsrdquo SIAM Journal on Numerical Analysis vol38 no 3 pp 897ndash912 2000

[14] R Vulanovic and T A Nhan ldquoUniform convergence viapreconditioningrdquo International Journal of Numerical Analysisand Modeling B vol 5 no 4 pp 347ndash356 2014

[15] H-G Roos ldquoA note on the conditioning of upwind schemes onShishkin meshesrdquo IMA Journal of Numerical Analysis vol 16no 4 pp 529ndash538 1996

[16] H-G Roos and T Linszlig ldquoSufficient condition for uniformconvergence on layeradapted gridsrdquo Computing vol 63 no 1pp 27ndash45 1999

[17] R Vulanovic ldquoA higher-order scheme for quasilinear boundaryvalue problems with two small parametersrdquo Computing vol 67no 4 pp 287ndash303 2001

[18] R Vulanovic ldquoA priori meshes for singularly perturbed quasi-linear twominuspoint boundary value problemsrdquo IMA Journal ofNumerical Analysis vol 21 no 1 pp 349ndash366 2001

[19] T A Nhan and R Vulanovic ldquoUniform convergence on aBakhvalov-type mesh using the preconditioning approachTechnical reportrdquo httparxivorgabs150404283

[20] N S Bakhvalov ldquoThe optimization of methods of solvingboundary value problems with a boundary layerrdquo USSR Com-putational Mathematics and Mathematical Physics vol 9 no 4pp 139ndash166 1969

[21] H-G Roos andM Stynes ldquoSome open questions in the numer-ical analysis of singularly perturbed differential equationsrdquoComputational Methods in Applied Mathematics vol 15 no 4pp 531ndash550 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Numerical Analysis 3

Table 1 Examples of mesh-generating and mesh-characterising functions of S-type meshes

Mesh 120601 (119905) max1206011015840

120595 max 10038161003816100381610038161205951015840

1003816100381610038161003816

S 119905119871

119876

119871

119876

119873minus119905119876

119871

119876

BS minus ln(1 minus

119905

119876

(1 minus

1

119873

))

119873

119876

1 minus

119905

119876

(1 minus

1

119873

)

1

119876

VS 119905

119876 + 119876119871 minus 119905

21198712

119876

exp(minus

119905

119876 + 119876119871 minus 119905

)

4

119876

The matrix norm induced by the above maximum vectornorm is also denoted by sdot

Following [2 16] let us define the mesh-characterizingfunction 120595 which is related to the mesh-generating function120601 as follows

120595 (119905) = 119890minus120601(119905)

(6)

This function is monotonically decreasing on [0 119876] with120595(0) = 1 and 120595(119876) = 119873

minus1We need to assume some further properties of the mesh

Assumption 1 Let the mesh-generating function 120601(119905) bepiecewise differentiable such that

max119905isin[0119876]

1206011015840

(119905) le 119862119873 (7)

Moreover we also assume that 120601(119905) fulfills

Δ120601

fl min119894=12119869

Δ119894

ge 119862119873minus1

(8)

where Δ119894

fl 120601(119905119894

) minus 120601(119905119894minus1

) 119894 = 1 2 119873

Remark 2 Examples of mesh-generating functions 120601 satisfy-ing the above assumptions are the standard Shishkinmesh (S-mesh) Bakhvalov-Shishkinmesh (BS-mesh) andVulanovic-Shishkin mesh (VS-mesh) They are summarized in Table 1

We proceed to provide some estimates for the step size ofthe S-type meshes generated by 120601 in the layer region Becausefor 119894 = 1 119869

119909119894

= 119886120576120601 (119905119894

) = minus119886120576 ln120595 (119905119894

) lArrrArr 120595(119905119894

) = 119890minus119909119894(119886120576)

(9)

we can bound the mesh step size inside the layer from above

ℎ119894

= 119886120576 (120601 (119905119894

) minus 120601 (119905119894minus1

)) le 119886120576119873minus1 max119905isin[119905119894minus1119905119894]

1206011015840

(119905)

le 119886120576119873minus1

(max119905isin[119905119894minus1 119905119894]

100381610038161003816100381610038161205951015840

(119905)

10038161003816100381610038161003816)

120595 (119905119894

)

= 119886120576119873minus1

( max119905isin[119905119894minus1119905119894]

100381610038161003816100381610038161205951015840

(119905)

10038161003816100381610038161003816) 119890119909119894(119886120576)

le 119886120576119873minus1max 1003816

10038161003816100381610038161205951015840

10038161003816100381610038161003816119890119909119894(119886120576)

(10)

where we use max|1205951015840| fl max119905isin[0119876]

|1205951015840

(119905)| Thus for 1 le 119894 le

119869 minus 1 we have that

ℏ119894

le

119886

2

120576119873minus1max 1003816

10038161003816100381610038161205951015840

10038161003816100381610038161003816(119890119909119894(119886120576)

+ 119890119909119894+1(119886120576)

)

le 119886120576119873minus1max 1003816

10038161003816100381610038161205951015840

10038161003816100381610038161003816119890119909119894+1(119886120576)

(11)

3 The Upwind Scheme andIts Condition Number

We consider the following upwind finite-difference dis-cretization on 119868

119873

119880119873

0

= 0

L119873

119880119873

119894

fl minus12057611986310158401015840

119880119873

119894

minus 119887119894

1198631015840

119880119873

119894

+ 119888119894

119880119873

119894

= 119891119894

119894 = 1 119873 minus 1

119880119873

119873

= 0

(12)

where

11986310158401015840

119882119894

=

1

ℏ119894

(

119882119894+1

minus 119882119894

ℎ119894+1

minus

119882119894

minus 119882119894minus1

ℎ119894

)

1198631015840

119882119894

=

119882119894+1

minus 119882119894

ℎ119894+1

(13)

The matrix form of the linear system (12) is

119860119873

119880119873

=119891

119873 (14)

with a tridiagonal matrix 119860119873

= [119886119894119895

] and 119891

119873

= [0 1198911

1198912

119891119873minus1

0]119879 The nonzero elements of 119860

119873

are 11988600

= 1 119886119873119873

= 1

119897119894

fl 119886119894minus1119894

=

minus

120576

ℏ119894

ℎ119894

1 le 119894 le 119869

minus

120576

1198672

119869 + 1 le 119894 le 119873 minus 1

119903119894

fl 119886119894119894+1

=

minus

120576

ℏ119894

ℎ119894+1

minus

119887119894

ℎ119894+1

1 le 119894 le 119869

minus

120576

1198672

minus

119887119894

119867

119869 + 1 le 119894 le 119873 minus 1

119889119894

fl 119886119894119894

=

1 119894 = 0

minus119897119894

minus 119903119894

+ 119888119894

1 le 119894 le 119873 minus 1

1 119894 = 119873

(15)

4 Advances in Numerical Analysis

Table 2 The condition number of discrete system (12) on the BS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 206119890 + 04 896119890 + 04 372119890 + 05 152119890 + 06 613119890 + 06 246119890 + 07

1119890 minus 3 219119890 + 05 948119890 + 05 393119890 + 06 160119890 + 07 645119890 + 07 259119890 + 08

1119890 minus 4 220119890 + 06 954119890 + 06 396119890 + 07 161119890 + 08 649119890 + 08 261119890 + 09

1119890 minus 5 220119890 + 07 955119890 + 07 396119890 + 08 161119890 + 09 650119890 + 09 261119890 + 10

1119890 minus 6 220119890 + 08 955119890 + 08 396119890 + 09 161119890 + 10 650119890 + 10 261119890 + 11

1119890 minus 7 220119890 + 09 955119890 + 09 396119890 + 10 161119890 + 11 650119890 + 11 261119890 + 12

1119890 minus 8 220119890 + 10 955119890 + 10 396119890 + 11 161119890 + 12 650119890 + 12 261119890 + 13

Table 3 The condition number of discrete system (12) on the VS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 275119890 + 04 123119890 + 05 511119890 + 05 205119890 + 06 812119890 + 06 320119890 + 07

1119890 minus 3 291119890 + 05 130119890 + 06 539119890 + 06 216119890 + 07 855119890 + 07 336119890 + 08

1119890 minus 4 293119890 + 06 131119890 + 07 543119890 + 07 218119890 + 08 861119890 + 08 339119890 + 09

1119890 minus 5 293119890 + 07 131119890 + 08 543119890 + 08 218119890 + 09 862119890 + 09 339119890 + 10

1119890 minus 6 293119890 + 08 131119890 + 09 543119890 + 09 218119890 + 10 862119890 + 10 339119890 + 11

1119890 minus 7 293119890 + 09 131119890 + 10 543119890 + 10 218119890 + 11 862119890 + 11 339119890 + 12

1119890 minus 8 293119890 + 10 131119890 + 11 543119890 + 11 218119890 + 12 862119890 + 12 339119890 + 13

Matrix 119860119873

is an 119871-matrix that is it has positive diagonalentries and nonnegative off-diagonal entries It is also anonsingular matrix satisfying119860

minus1

119873

ge 0 (inequalities involvingmatrices and vectors should be understood componentwise)and therefore inverse monotone In other words 119860

119873

is an119872-matrix (an inverse monotone 119871-matrix) This fact is easyto prove using the following119872-criterion see [8] for instance

Theorem 3 (119872-criterion) Let 119860 be an 119871-matrix and thereexists a vector 119908 such that 119908 gt 0 and 119860119908 ge 120574 for somepositive constant 120574 119860 is then an 119872-matrix and it holds that119860minus1

le 120574minus1

119908

In the above theorem set 119908119894

= 2 minus 119909119894

119894 = 0 1 119873to get that 119860

119873

119908 ge min1 120573 This implies that 119860119873

is an 119872-matrix and that

10038171003817100381710038171003817119860minus1

119873

10038171003817100381710038171003817le

2

min 1 120573

le 119862 (16)

which means that discrete problem (14) is stable uniformly in120576 By examining directly the entries of matrix 119860

119873

and notingthat the lower bounds for ℎ

119894

and ℏ119894

are achieved in the layerregion where ℎ

119894

sim ℏ119894

ge 119862120576Δ120601

we get that

1003817100381710038171003817119860119873

1003817100381710038171003817le 119862

1

120576 (Δ120601

)

2

(17)

Combining this and the 120576-uniform stability (16) we arrive atthe following result related to the condition number 120581(119860

119873

)

of matrix 119860119873

120581(119860119873

) fl 119860119873

119860minus1

119873

Theorem 4 The condition number of matrix 119860119873

satisfies theestimate

120581 (119860119873

) le 119862

1

120576 (Δ120601

)

2

(18)

Let us illustrate Theorem 4 using a simple example takenfrom [2 page 1]

minus12057611990610158401015840

(119909) minus 1199061015840

(119909) = 1 0 lt 119909 lt 1

119906 (0) = 119906 (1) = 0

(19)

It is easy to observe the 120576-dependence of the conditionnumbers in Tables 2 and 3 They clearly show that the boundin Theorem 4 is sharp For the standard Shishkin mesh werefer the readers to the numerical experiments in [15]

4 Preconditioning and Stability

In this section we propose an appropriately designeddiagonal preconditioner which not only eliminates the 120576-dependence of the condition number of the discrete systemsbut also retains its 120576-uniform stability in the sense of (16)Thepreconditioner is not of interest per se but as ameans to prove120576-uniform pointwise convergence via Principle 1

Let 119872 = diag(1198980

1198981

119898119873

) be a diagonal matrix withthe entries

1198980

= 1

119898119894

=

ℏ119894

119867

119894 = 1 2 119873 minus 1

119898119873

= 1

(20)

On noting that ℏ119894

= 119867 for 119894 = 119869 + 1 119873 minus 1 we seethat when the discrete system (12) is multiplied by 119872 this isequivalent to multiplying the equations 1 2 119869 of system(14) by ℏ

119894

119867 119894 = 1 2 119869 The modified system is

119873

119880119873

= 119872119891

119873

(21)

Advances in Numerical Analysis 5

where 119873

= 119872119860119873

Let the entries of 119873

be denoted by 119894119895

the nonzero ones being

119897119894

fl 119894minus1119894

=

minus

120576

ℎ119894

119867

1 le 119894 le 119869

minus

120576

1198672

119869 + 1 le 119894 le 119873 minus 1

119894

fl 119894119894+1

=

minus

120576

ℎ119894+1

119867

minus

119887119894

ℏ119894

ℎ119894+1

119867

1 le 119894 le 119869 minus 1

minus

120576

1198672

minus

119887119894

ℏ119894

1198672

119894 = 119869

minus

120576

1198672

minus

119887119894

119867

119869 + 1 le 119894 le 119873 minus 1

119889119894

fl 119894119894

=

1 119894 = 0

minus119897119894

minus 119894

+ 119888119894

ℏ119894

119867

1 le 119894 le 119869

minus119897119894

minus 119894

+ 119888119894

119869 + 1 le 119894 le 119873 minus 1

1 119894 = 119873

(22)

Matrix 119873

remains an 119871-matrix since the precondi-tioning does not change this property of 119860

119873

We showin Lemma 7 below that

119873

is an 119872-matrix and that themodified discretization (21) is stable uniformly in 120576 But firstwe give a technical result related to the mesh

Lemma 5 Let 119886120573 gt 2 and let in addition to Assumption 1 themesh-generating function 120601 satisfy the following conditions on[0 119876]

(i) 120601(119895)

ge 0 119895 = 1 2 3(ii) 1206011015840

(0) ge 119898 gt 0 where 119898 is a constant independent of 120576and 119873

(iii) 12060110158401015840

le 2(1206011015840

)2

(iv) 1206011015840

le 119862119871119896 for some 119896 gt 0

Then for a sufficiently large119873 independent of 120576 there exists apositive constant 120575 independent of both 120576 and 119873 such that

119878119894

fl120573

2

(1 +

Δ119894

Δ119894+1

) minus

1

119886

(

1

Δ119894

minus

1

Δ119894+1

) ge 120575 gt 0

119894 = 1 119869 minus 1

(23)

Proof Since 119886120573 gt 2 we have that 1205732 ge 1119886 + 120578 for some120578 gt 0 Then

119878119894

ge

1

119886

119875119894

+ 120578 (24)

where

119875119894

=

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

Δ119894

Δ119894+1

(25)

Then because of (i) and (iii)

119875119894

ge 2119875lowast119894

where 119875lowast119894

fl1206011015840

(119905119894minus1

)2

minus 1206011015840

(119905119894+1

)2

1198732

Δ119894

Δ119894+1

(26)

If we show that

119875lowast119894

997888rarr 0 uniformly in 120576 when 119873 997888rarr infin (27)

then we will have (23) provided 119873 is sufficiently largeindependently of 120576 We now show that (27) holds true First

1003816100381610038161003816119875lowast119894

1003816100381610038161003816le 119862

100381610038161003816100381610038161206011015840

(119905119894minus1

)2

minus 1206011015840

(119905119894+1

)2

10038161003816100381610038161003816

(28)

because of (i) and (ii) and then

1003816100381610038161003816119875lowast119894

1003816100381610038161003816le 119862119873

minus1

119871119896

12060110158401015840

(119905119894+1

) le 119862119873minus1

1198713119896 (29)

because of (i) (iii) and (iv)

Themeshes defined inTable 1 satisfy conditions (i)ndash(iv) ofLemma 5 with the exception of the BS-mesh which does notfulfill condition (iv) However estimate (23) still holds truefor the BS-mesh if we assume that radic2 minus 1 le 119876 lt 1 This isnot a serious restriction since119876 = 12 is a typical choiceTheproof of this result stated below in Lemma 6 is given in theappendix

Lemma 6 Let 119886120573 gt 2 and let radic2 minus 1 le 119876 lt 1 Then theBS-mesh satisfies estimate (23)

Lemma 7 Under the assumptions of Lemma 5 (Lemma 6 ifthe BS-mesh is used) matrix

119873

of system (21) satisfies

100381710038171003817100381710038171003817

minus1

119873

100381710038171003817100381710038171003817

le 119862 (30)

Proof We construct a vector V = [V0

V1

V119873

]119879 such that

(a) V119894

gt 0 119894 = 0 1 119873(b) V119894

le 119862 119894 = 0 1 119873

(c) 119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

ge 120579 119894 = 1 2 119873 minus 1 where 120579

is a positive constant independent of both 120576 and 119873

Then according to the 119872-criterion

100381710038171003817100381710038171003817

minus1

119873

100381710038171003817100381710038171003817

le 120579minus1

V le 119862 (31)

We define the vector V as follows (cf [14 15])

V119894

= 120572 minus 119867119894 + 120582min (1 + 120588)119869minus119894

1 (32)

where 120572 and 120582 are appropriately chosen positive constantsand 120588 = 120573119867120576

Since 119867119873 le 119862 there exists a constant 120572 such that V119894

ge

120572minus119867119894 gt 0 and condition (a) is satisfied It also holds true thatV119894

le 120572+120582Therefore condition (b) is satisfied if we prove that120582 le 119862 We do this in next steps while we verify condition (c)

6 Advances in Numerical Analysis

When 1 le 119894 le 119869 minus 1 using Lemma 5 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= (119897119894

+119889119894

+ 119894

) V119894

+119897119894

119867 minus 119894

119867

=

ℏ119894

119867

119888119894

V119894

minus

120576

ℎ119894

+

120576

ℎ119894+1

+

119887119894

ℏ119894

ℎ119894+1

ge minus(

120576

ℎ119894

minus

120576

ℎ119894+1

) +

119887119894

2

+

119887119894

ℎ119894

2ℎ119894+1

ge 119878119894

ge 120575 gt 0

(33)

For 119894 = 119869 condition (c) is verified as follows

119897119869

V119869minus1

+119889119869

V119869

+ 119869

V119869+1

=

ℏ119869

119867

119888119869

V119869

+119897119869

119867 minus 119869

(119867 +

120582120588

1 + 120588

)

ge minus119869

120582120588

1 + 120588

+ (119897119869

minus 119869

)119867

= (

120576

1198672

+

119887119869

ℏ119869

1198672

)

120582120588

1 + 120588

minus

120576

ℎ119869

+

120576

119867

+

119887119869

ℏ119869

119867

= (

2120576 + 119887119869

(ℎ119869

+ 119867)

21198672

)

120582120573119867

120576 + 120573119867

minus

120576

ℎ119869

+

119887119869

2

ge

120582120573

2119867

minus

120576

ℎ119869

+

120573

2

ge

120573

2

(34)

where in the last step invoking (8) we choose a constant 120582 sothat

120582120573

2119867

minus

120576

ℎ119869

ge

120582120573

2119867

minus

120576

119886120576 (120601 (119905119869

) minus 120601 (119905119869minus1

))

ge

120582120573

2119867

minus

1

119886Δ119869

ge

120582120573

4119873minus1

minus

1

119886119862119873minus1

= 119873(

120582120573

4

minus

1

119886119862

) ge 0

(35)

To guarantee this 120582 is chosen so that

120582120573

4

ge

1

119886119862

(36)

Finally when 119869 + 1 le 119894 le 119873 minus 1 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= 119888119894

V119894

+119897119894

119867 minus 119894

119867

+

119897119894

1 + 120588119869

[

120582

(1 + 120588)119894minus1minus119869

minus

120582

(1 + 120588)119894minus119869

]

+

119894

1 + 120588119869

[

120582

(1 + 120588)119894+1minus119869

minus

120582

(1 + 120588)119894minus119869

]

ge 119887119894

+

120588 (1 + 120588)119897119894

minus 120588119894

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

ge 120573 +

(119897119894

minus 119894

+119897119894

120588) 120588

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

= 120573 + (

119887119894

119867

minus

120573

119867

)

120582120588 (1 + 120588)119869minus119894minus1

1 + 120588119869

ge 120573

(37)

By examining the entries of 119873

we easily see that themaximum absolute row sum of

119873

is achieved in the layerregion Therefore

10038171003817100381710038171003817119873

10038171003817100381710038171003817le 119862

119873

Δ120601

(38)

A combination of Lemma 7 and (38) results in thefollowing

Theorem 8 The condition number of matrix 119873

satisfies thefollowing 120576-uniform estimate

120581 (119873

) le 119862

119873

Δ120601

(39)

Corollary 9 For the meshes given in Table 1 one has thefollowing estimates

120581 (119873

) le

119862

1198732

119871

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

1198621198732

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(40)

In Tables 4 and 5 we present the condition numbers ofthe preconditioned linear systems (21) for various values of120576 and 119873 for the BS- and VS-meshes We can observe thatthe condition numbers of the preconditioned systems areindependent of 120576 and that the bound in Corollary 9 is sharp

5 120576-Uniform Convergence

For the proof of 120576-uniform convergence we need to decom-pose the solution 119906 into the smooth and boundary-layer

Advances in Numerical Analysis 7

Table 4 The condition number of matrix 119873

on the BS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 658119890 + 02 268119890 + 03 108119890 + 04 434119890 + 04 174119890 + 05 695119890 + 05

1119890 minus 3 689119890 + 02 278119890 + 03 111119890 + 04 440119890 + 04 175119890 + 05 700119890 + 05

1119890 minus 4 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 702119890 + 05

1119890 minus 5 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 6 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 7 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 8 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

Table 5 The condition number of matrix 119873

on the VS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 748119890 + 02 313119890 + 03 127119890 + 04 510119890 + 04 204119890 + 05 811119890 + 05

1119890 minus 3 780119890 + 02 323119890 + 03 130119890 + 04 520119890 + 04 207119890 + 05 820119890 + 05

1119890 minus 4 785119890 + 02 325119890 + 03 131119890 + 04 522119890 + 04 208119890 + 05 824119890 + 05

1119890 minus 5 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 6 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 7 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 8 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

parts There exist several decompositions of this kind but weuse the version presented in [2 Theorem 348]

119906 (119909) = 119904 (119909) + 119910 (119909) 119909 isin 119868 (41)10038161003816100381610038161003816119904(119896)

(119909)

10038161003816100381610038161003816le 119862 (1 + 120576

2minus119896

)

10038161003816100381610038161003816119910(119896)

(119909)

10038161003816100381610038161003816le 119862120576minus119896

119890minus120573119909120576

119909 isin 119868 119896 = 0 1 2 3

(42)

The details related to the construction of the function 119904 arenot important here they can be found in [2] As for 119910 wenote that it solves the problem

L119910 (119909) = 0 119909 isin (0 1)

119910 (0) = minus119904 (0)

119910 (1) = 0

(43)

We define the consistency error of the finite-differenceoperatorL119873 as

120591119894

= 120591119894

[119906] fl L119873

119906119894

minus 119891119894

119894 = 1 2 119873 minus 1 (44)

It holds true that

120591119894

= L119873

119906119894

minus (L119906)119894

= [119860119873

(119906119873

minus 119880119873

)]119894

(45)

Taylorrsquos expansion gives

1003816100381610038161003816120591119894

[119906]1003816100381610038161003816le 119862ℎ119894+1

(120576

10038171003817100381710038171003817119906101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711990610158401015840

10038171003817100381710038171003817119894) (46)

where 119892119894

fl max119909119894minus1le119909le119909119894+1

|119892(119909)| for any 119862(119868)-function 119892When the diagonal matrix 119872 is multiplied to linear system(14) the consistency error of the multiplied system is

119894

[119906] =

ℏ119894

119867

120591119894

[119906] 1 le 119894 le 119869

120591119894

[119906] 119869 + 1 le 119894 le 119873 minus 1

(47)

Lemma 10 Let 119886120573 ge 2Then the following estimate holds true

1003816100381610038161003816119894

[119906]1003816100381610038161003816le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(48)

Proof We use decomposition (41) to get

119894

[119906] = 119894

[119904] + 119894

[119910] (49)

Then (46) and the derivative-estimates of 119904 given in (42)yield immediately that

1003816100381610038161003816119894

[119904]1003816100381610038161003816le 119862119873

minus1 (50)

and therefore the following remains to be proved

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

119894 = 119869 119873 minus 1

(51)

Note that from condition (7) we easily get

ℎ119894

le 119862120576 119894 = 1 119869

ℏ119894

le 119862120576 119894 = 1 119869 minus 1

(52)

This is the key property of S-type meshes as well as the mainingredient of the following proof

8 Advances in Numerical Analysis

Table 6 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 550119890 + 00 293119890 + 00 151119890 + 00 769119890 minus 01 388119890 minus 01 195119890 minus 01

1119890 minus 3 550119890 + 01 293119890 + 01 151119890 + 01 769119890 + 00 388119890 + 00 195119890 + 00

1119890 minus 4 550119890 + 02 293119890 + 02 151119890 + 02 769119890 + 01 388119890 + 01 195119890 + 01

1119890 minus 5 550119890 + 03 293119890 + 03 151119890 + 03 769119890 + 02 388119890 + 02 195119890 + 02

1119890 minus 6 550119890 + 04 293119890 + 04 151119890 + 04 769119890 + 03 388119890 + 03 195119890 + 03

1119890 minus 7 550119890 + 05 293119890 + 05 151119890 + 05 769119890 + 04 388119890 + 04 195119890 + 04

1119890 minus 8 550119890 + 06 293119890 + 06 151119890 + 06 769119890 + 05 388119890 + 05 195119890 + 05

For 1 le 119894 le 119869 minus 1 (46) estimates (10) and (11) as well asthe derivative-estimates of 119910 in (42) imply

1003816100381610038161003816119894

(119910)1003816100381610038161003816le 119862

ℏ119894

119867

ℎ119894+1

(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894)

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902119909119894+1(119886120576)

119890minus120573119909119894minus1120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890(2119909119894+1minus119886120573119909119894minus1)119886120576

= 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902(119909119894+1minus119909119894minus1)119886120576

119890minus(119886120573minus2)119909119894minus1119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890ℏ119894119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

(53)

where we have used (52) and 119886120573 ge 2 to get

119890ℏ119894(119886120576)

le 119862

119890minus(119886120573minus2)119909119894minus1(119886120576)

le 119862

(54)

When 119869 + 2 le 119894 le 119873 minus 1 (42) and (46) give

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119867(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894) le 119862

119867

1205762

119890minus120573119909119894minus1120576

le 119862

119867

1205762

119890minus120573(120590+119867)120576

le 119862119873

1198672

1205762

119890minus120573119867120576

119890minus120573120590120576

(55)

Estimate (51) follows from the fact that (119867120576minus1

)2

119890minus120573119867120576

le 119862

and because the definition of 120590 implies that

119890minus120573120590120576

= 119890minus120573119886119871

= 119873minus120573119886

le 119873minus2

(56)

We finally prove (51) when 119894 = 119869 119869 + 1 in which case

1003816100381610038161003816119869

[119910]1003816100381610038161003816=

ℏ119869

119867

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816

1003816100381610038161003816119869+1

[119910]1003816100381610038161003816=

1003816100381610038161003816120591119869+1

[119910]1003816100381610038161003816

(57)

We now use the fact that L119910 = 0 and work with a differentform of the consistency error (this is a well-known techniquecf [18 Lemma 5])

1003816100381610038161003816119894

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816le 119877119894

+ 119876119894

+ 119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816 (58)

where

119877119894

= 120576

1003816100381610038161003816100381611986310158401015840

119910119894

10038161003816100381610038161003816

119876119894

= 119887119894

100381610038161003816100381610038161198631015840

119910119894

10038161003816100381610038161003816

(59)

We immediately have that

119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816le 119862119890minus120573119909119894120576

le 119862119890minus120573120590120576

= 119862119890minus120573119886119871

= 119862119873minus119886120573

le 119862119873minus2

(60)

As for 119877119894

it holds true that

119877119894

le ℏminus1

119894

120576 sdot 2

100381710038171003817100381710038171199101015840

10038171003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873119890minus120573120590120576

119890120573ℎ119869120576

le 119862119873minus1

(61)

by (52) Analogously

119876119894

le 119862119867minus1

10038171003817100381710038171199101003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873minus1

(62)

This completes the proof

In general the consistency error 120591119894

of the layer componentof the solution is not bounded pointwise uniformly in 120576When the above proof technique is applied to 120591

119894

[119910] one canonly get

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816

le

119862120576minus1

119873minus1

119871 for the S-mesh

119862120576minus1

119873minus1

for the BS- and VS-meshes

(63)

The proposed preconditioner simplifies the analysis of theconsistency error by introducing an extra 120576 factor needed forthe layer component Hence the approach used in the proofof Lemma 10 is natural and straightforward because it onlyuses the classical Taylor expansion estimate for the truncationerror

To illustrate (63) in Tables 6 and 7 we present the resultsof some numerical experiments on the BS- and VS-meshesrespectivelyWe refer the reader to [14] for the correspondingnumerical results on the S-mesh

By contrast with preconditioner (20) the preconditionedconsistency error converges uniformly in 120576 as shown in Tables8 and 9

In conclusion the preconditioning allows us to usePrinciple 1 and combine Lemmas 7 and 10 to obtain thefollowing main convergence result

Advances in Numerical Analysis 9

Table 7 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 467119890 + 00 247119890 + 00 128119890 + 00 659119890 minus 01 336119890 minus 01 170119890 minus 01

1119890 minus 3 467119890 + 01 247119890 + 01 128119890 + 01 659119890 + 00 336119890 + 00 170119890 + 00

1119890 minus 4 467119890 + 02 247119890 + 02 128119890 + 02 659119890 + 01 336119890 + 01 170119890 + 01

1119890 minus 5 467119890 + 03 247119890 + 03 128119890 + 03 659119890 + 02 336119890 + 02 170119890 + 02

1119890 minus 6 467119890 + 04 247119890 + 04 128119890 + 04 659119890 + 03 336119890 + 03 170119890 + 03

1119890 minus 7 467119890 + 05 247119890 + 05 128119890 + 05 659119890 + 04 336119890 + 04 170119890 + 04

1119890 minus 8 467119890 + 06 247119890 + 06 128119890 + 06 659119890 + 05 336119890 + 05 170119890 + 05

Table 8 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 122119890 minus 01 650119890 minus 02 338119890 minus 02 174119890 minus 02 887119890 minus 03 452119890 minus 03

1119890 minus 3 114119890 minus 01 601119890 minus 02 308119890 minus 02 156119890 minus 02 786119890 minus 03 395119890 minus 03

1119890 minus 4 114119890 minus 01 597119890 minus 02 306119890 minus 02 155119890 minus 02 778119890 minus 03 390119890 minus 03

1119890 minus 5 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 390119890 minus 03

1119890 minus 6 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 7 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 8 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

Table 9 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 160119890 minus 01 912119890 minus 02 502119890 minus 02 270119890 minus 02 143119890 minus 02 747119890 minus 03

1119890 minus 3 150119890 minus 01 844119890 minus 02 458119890 minus 02 242119890 minus 02 126119890 minus 02 653119890 minus 03

1119890 minus 4 149119890 minus 01 837119890 minus 02 454119890 minus 02 240119890 minus 02 125119890 minus 02 645119890 minus 03

1119890 minus 5 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 6 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 7 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 8 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

Theorem 11 The error of the upwind finite-difference dis-cretization on S-type meshes satisfies

10038161003816100381610038161003816119906119873

119894

minus 119880119873

119894

10038161003816100381610038161003816

le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(64)

Corollary 12 For the meshes given in Table 1 one has thefollowing estimates

10038171003817100381710038171003817119906119873

minus 119880119873

10038171003817100381710038171003817

le

119862119873minus1

1198712

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

119862119873minus1

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(65)

Remark 13 The barrier-function technique can be used toobtain similar results for S-type meshes (see [16]) but itrequires another assumption on the mesh-generating func-tion 120601 (cf [16 Theorem 1]) However whether the barrier-function approach can be extended to any Bakhvalov-type

mesh is still an open question [2 Remark 421] By contrastas proved in [19] the preconditioner defined in (20) worksfine for Vulanovicrsquos modification of the Bakhvalov mesh [10]The paper [10] is where the original Bakhvalov mesh [20]is generalized (the main difference between Bakhvalov-typemeshes and S-type meshes is that the former have a smoothtransition from the fine part inside the layer to the coarseuniform part outside the layer the mesh-generating function120601 belongs to 119862

1

(119868))

6 Conclusion

In this paper we present a proof of pointwise uniformconvergence for singularly perturbed convection-diffusionproblem (1) discretized on a general class of Shishkin-typemeshesThe proof is based on preconditioning of the discretesystem which allows us to use the standard ldquoconsistency+ stabilityrdquo principle of convergence This is a conceptuallysimple approach which does not require the use of hybridstability inequalities and discrete Greenrsquos functions In gen-eral new proof techniques are of interest because althougha lot has been achieved in the field of singularly perturbed

10 Advances in Numerical Analysis

problems over the last few decades there are still importantquestions remaining open [21] One of them Question6 is related to the work presented here The question isabout two-dimensional convection-diffusion problems and itpoints out that whereas 120576-uniform pointwise convergence forthe upwind finite-difference discretization on S-type meshesis well-established no similar result for B-type meshes isknownAsmentioned inRemark 13 we already know that thepreconditioning approach works well for B-mesh and one-dimensional convection-diffusion problems The next step isto consider two-dimensional problems

Appendix

Proof of Lemma 6 The proof is different from the proof ofLemma 5because theBS-meshdoes not satisfy condition (iv)Because of (ii) it is sufficient to show that the numerator in(25) is nonnegative for radic2 minus 1 le 119876 lt 1 independently of 120576and 119873 Let 119901 fl 1 minus 119873

minus1 By direct computations we get that

Δ119894

= minus ln(1 minus

119905119894

119901

119876

) + ln(1 minus

119905119894minus1

119901

119876

)

= ln(1 +

1

119869119901 minus 119894

) = ln (1 + 119902)

(A1)

where we denote 119902 fl (119869119901minus119894)minus1 Note that 0 le 119902 lt 1(119876+1)

Analogously

Δ119894+1

= ln(

1

1 minus 119902

) (A2)

Then

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

= Δ119894

(Δ119894+1

+ Δ119894

+ 1) minus Δ119894+1

= ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902)

(A3)

We set

119892 (119902) fl ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902) (A4)

Sinceradic2minus1 le 119876 lt 1 we have 0 le 119902 lt 1radic2 Rearrange 119892(119902)

as

119892 (119902) = [1 + ln(1 +

2119902

1 minus 119902

)] ln (1 + 119902)

+ ln (1 minus 119902)

(A5)

For 119902 isin [0 1) it follows that119902

1 + 1199022

le ln (1 + 119902) le 119902 (A6)

Hence

ln(1 +

2119902

1 minus 119902

) ge 2119902 (A7)

So we are done if we can show that

119892 (119902) ge [1 + 2119902] ln (1 + 119902) + ln (1 minus 119902) ge 0

0 le 119902 lt

1

radic2

(A8)

or

(1 minus 119902) (1 + 119902)2119902+1

= (1 minus 1199022

) (1 + 119902)2119902

ge 1 (A9)

First if 119902 isin [12 1radic2] we have by (1 + 119902)120574

ge 1 + 120574119902 for120574 ge 1

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

) (1 + 21199022

) ge 1 (A10)

Second we have that ln(1+119902) is a concave-down function on[0 12] hence on [0 12]

ln (1 + 119902) ge 2 ln(

3

2

) 119902 ge

4119902

5

(A11)

Then multiplying both sides by 119902 and exponentiating we getthat

(1 + 119902)119902

ge exp(

41199022

5

) ge 1 +

41199022

5

(A12)

Therefore

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

)(1 +

41199022

5

)

2

ge 1 (A13)

which holds true for 119902 isin [0 12]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P A Farrell A F Hegarty J J H Miller E OrsquoRiordan andG I Shishkin Robust Computational Techniques for BoundaryLayers Chapman amp Hall CRC Press Boca Raton Fla USA2000

[2] T Linszlig Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems vol 1985 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[3] J J HMiller E OrsquoRiordan and G I Shishkin Fitted NumericalMethods for Singular Perturbation Problems World ScientificSingapore 1996

[4] H-G Roos M Stynes and L Tobiska Numerical Methodsfor Singularly Perturbed Differential Equations Springer BerlinGermany 2nd edition 2008

[5] A Malvandi and D D Ganji ldquoMixed convection of aluminandashwater nanofluid inside a concentric annulus consideringnanoparticle migrationrdquo Particuology vol 24 pp 113ndash122 2016

[6] R B Kellogg and A Tsan ldquoAnalysis of some difference approx-imations for a singular perturbation problem without turningpointsrdquoMathematics of Computation vol 32 no 144 pp 1025ndash1039 1978

Advances in Numerical Analysis 11

[7] J Lorenz ldquoStability and monotonicity properties of stiff quasi-linear boundary value problemsrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta Serija zaMatematiku vol 12 pp 151ndash175 1982

[8] E Bohl Finite Modelle Gewohnlicher Randwertaufgaben Teub-ner Stuttgart Germany 1981

[9] E C Gartland Jr ldquoGraded-mesh difference schemes for sin-gularly perturbed two-point boundary value problemsrdquo Math-ematics of Computation vol 51 no 184 pp 631ndash657 1988

[10] R Vulanovic ldquoOn a numerical solution of a type of singularlyperturbed boundary value problem by using a special dis-cretization meshrdquo Univerzitet u Novom Sadu Zbornik RadovaPrirodno- Matematichkog Fakulteta Serija za Matematiku vol13 pp 187ndash201 1983

[11] C Grossmann H-G Roos and M Stynes Numerical Treat-ment of Partial Differential Equations Springer Berlin Ger-many 2007

[12] V B Andreev and I A Savin ldquoThe uniform convergence withrespect to a small parameter of A A Samarskiirsquos monotonescheme and its modificationrdquo Computational Mathematics andMathematical Physics vol 35 pp 581ndash591 1995

[13] T LinszligH-G Roos andRVulanovic ldquoUniformpointwise con-vergence on Shishkin-type meshes for quasi-linear convection-diffusion problemsrdquo SIAM Journal on Numerical Analysis vol38 no 3 pp 897ndash912 2000

[14] R Vulanovic and T A Nhan ldquoUniform convergence viapreconditioningrdquo International Journal of Numerical Analysisand Modeling B vol 5 no 4 pp 347ndash356 2014

[15] H-G Roos ldquoA note on the conditioning of upwind schemes onShishkin meshesrdquo IMA Journal of Numerical Analysis vol 16no 4 pp 529ndash538 1996

[16] H-G Roos and T Linszlig ldquoSufficient condition for uniformconvergence on layeradapted gridsrdquo Computing vol 63 no 1pp 27ndash45 1999

[17] R Vulanovic ldquoA higher-order scheme for quasilinear boundaryvalue problems with two small parametersrdquo Computing vol 67no 4 pp 287ndash303 2001

[18] R Vulanovic ldquoA priori meshes for singularly perturbed quasi-linear twominuspoint boundary value problemsrdquo IMA Journal ofNumerical Analysis vol 21 no 1 pp 349ndash366 2001

[19] T A Nhan and R Vulanovic ldquoUniform convergence on aBakhvalov-type mesh using the preconditioning approachTechnical reportrdquo httparxivorgabs150404283

[20] N S Bakhvalov ldquoThe optimization of methods of solvingboundary value problems with a boundary layerrdquo USSR Com-putational Mathematics and Mathematical Physics vol 9 no 4pp 139ndash166 1969

[21] H-G Roos andM Stynes ldquoSome open questions in the numer-ical analysis of singularly perturbed differential equationsrdquoComputational Methods in Applied Mathematics vol 15 no 4pp 531ndash550 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Numerical Analysis

Table 2 The condition number of discrete system (12) on the BS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 206119890 + 04 896119890 + 04 372119890 + 05 152119890 + 06 613119890 + 06 246119890 + 07

1119890 minus 3 219119890 + 05 948119890 + 05 393119890 + 06 160119890 + 07 645119890 + 07 259119890 + 08

1119890 minus 4 220119890 + 06 954119890 + 06 396119890 + 07 161119890 + 08 649119890 + 08 261119890 + 09

1119890 minus 5 220119890 + 07 955119890 + 07 396119890 + 08 161119890 + 09 650119890 + 09 261119890 + 10

1119890 minus 6 220119890 + 08 955119890 + 08 396119890 + 09 161119890 + 10 650119890 + 10 261119890 + 11

1119890 minus 7 220119890 + 09 955119890 + 09 396119890 + 10 161119890 + 11 650119890 + 11 261119890 + 12

1119890 minus 8 220119890 + 10 955119890 + 10 396119890 + 11 161119890 + 12 650119890 + 12 261119890 + 13

Table 3 The condition number of discrete system (12) on the VS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 275119890 + 04 123119890 + 05 511119890 + 05 205119890 + 06 812119890 + 06 320119890 + 07

1119890 minus 3 291119890 + 05 130119890 + 06 539119890 + 06 216119890 + 07 855119890 + 07 336119890 + 08

1119890 minus 4 293119890 + 06 131119890 + 07 543119890 + 07 218119890 + 08 861119890 + 08 339119890 + 09

1119890 minus 5 293119890 + 07 131119890 + 08 543119890 + 08 218119890 + 09 862119890 + 09 339119890 + 10

1119890 minus 6 293119890 + 08 131119890 + 09 543119890 + 09 218119890 + 10 862119890 + 10 339119890 + 11

1119890 minus 7 293119890 + 09 131119890 + 10 543119890 + 10 218119890 + 11 862119890 + 11 339119890 + 12

1119890 minus 8 293119890 + 10 131119890 + 11 543119890 + 11 218119890 + 12 862119890 + 12 339119890 + 13

Matrix 119860119873

is an 119871-matrix that is it has positive diagonalentries and nonnegative off-diagonal entries It is also anonsingular matrix satisfying119860

minus1

119873

ge 0 (inequalities involvingmatrices and vectors should be understood componentwise)and therefore inverse monotone In other words 119860

119873

is an119872-matrix (an inverse monotone 119871-matrix) This fact is easyto prove using the following119872-criterion see [8] for instance

Theorem 3 (119872-criterion) Let 119860 be an 119871-matrix and thereexists a vector 119908 such that 119908 gt 0 and 119860119908 ge 120574 for somepositive constant 120574 119860 is then an 119872-matrix and it holds that119860minus1

le 120574minus1

119908

In the above theorem set 119908119894

= 2 minus 119909119894

119894 = 0 1 119873to get that 119860

119873

119908 ge min1 120573 This implies that 119860119873

is an 119872-matrix and that

10038171003817100381710038171003817119860minus1

119873

10038171003817100381710038171003817le

2

min 1 120573

le 119862 (16)

which means that discrete problem (14) is stable uniformly in120576 By examining directly the entries of matrix 119860

119873

and notingthat the lower bounds for ℎ

119894

and ℏ119894

are achieved in the layerregion where ℎ

119894

sim ℏ119894

ge 119862120576Δ120601

we get that

1003817100381710038171003817119860119873

1003817100381710038171003817le 119862

1

120576 (Δ120601

)

2

(17)

Combining this and the 120576-uniform stability (16) we arrive atthe following result related to the condition number 120581(119860

119873

)

of matrix 119860119873

120581(119860119873

) fl 119860119873

119860minus1

119873

Theorem 4 The condition number of matrix 119860119873

satisfies theestimate

120581 (119860119873

) le 119862

1

120576 (Δ120601

)

2

(18)

Let us illustrate Theorem 4 using a simple example takenfrom [2 page 1]

minus12057611990610158401015840

(119909) minus 1199061015840

(119909) = 1 0 lt 119909 lt 1

119906 (0) = 119906 (1) = 0

(19)

It is easy to observe the 120576-dependence of the conditionnumbers in Tables 2 and 3 They clearly show that the boundin Theorem 4 is sharp For the standard Shishkin mesh werefer the readers to the numerical experiments in [15]

4 Preconditioning and Stability

In this section we propose an appropriately designeddiagonal preconditioner which not only eliminates the 120576-dependence of the condition number of the discrete systemsbut also retains its 120576-uniform stability in the sense of (16)Thepreconditioner is not of interest per se but as ameans to prove120576-uniform pointwise convergence via Principle 1

Let 119872 = diag(1198980

1198981

119898119873

) be a diagonal matrix withthe entries

1198980

= 1

119898119894

=

ℏ119894

119867

119894 = 1 2 119873 minus 1

119898119873

= 1

(20)

On noting that ℏ119894

= 119867 for 119894 = 119869 + 1 119873 minus 1 we seethat when the discrete system (12) is multiplied by 119872 this isequivalent to multiplying the equations 1 2 119869 of system(14) by ℏ

119894

119867 119894 = 1 2 119869 The modified system is

119873

119880119873

= 119872119891

119873

(21)

Advances in Numerical Analysis 5

where 119873

= 119872119860119873

Let the entries of 119873

be denoted by 119894119895

the nonzero ones being

119897119894

fl 119894minus1119894

=

minus

120576

ℎ119894

119867

1 le 119894 le 119869

minus

120576

1198672

119869 + 1 le 119894 le 119873 minus 1

119894

fl 119894119894+1

=

minus

120576

ℎ119894+1

119867

minus

119887119894

ℏ119894

ℎ119894+1

119867

1 le 119894 le 119869 minus 1

minus

120576

1198672

minus

119887119894

ℏ119894

1198672

119894 = 119869

minus

120576

1198672

minus

119887119894

119867

119869 + 1 le 119894 le 119873 minus 1

119889119894

fl 119894119894

=

1 119894 = 0

minus119897119894

minus 119894

+ 119888119894

ℏ119894

119867

1 le 119894 le 119869

minus119897119894

minus 119894

+ 119888119894

119869 + 1 le 119894 le 119873 minus 1

1 119894 = 119873

(22)

Matrix 119873

remains an 119871-matrix since the precondi-tioning does not change this property of 119860

119873

We showin Lemma 7 below that

119873

is an 119872-matrix and that themodified discretization (21) is stable uniformly in 120576 But firstwe give a technical result related to the mesh

Lemma 5 Let 119886120573 gt 2 and let in addition to Assumption 1 themesh-generating function 120601 satisfy the following conditions on[0 119876]

(i) 120601(119895)

ge 0 119895 = 1 2 3(ii) 1206011015840

(0) ge 119898 gt 0 where 119898 is a constant independent of 120576and 119873

(iii) 12060110158401015840

le 2(1206011015840

)2

(iv) 1206011015840

le 119862119871119896 for some 119896 gt 0

Then for a sufficiently large119873 independent of 120576 there exists apositive constant 120575 independent of both 120576 and 119873 such that

119878119894

fl120573

2

(1 +

Δ119894

Δ119894+1

) minus

1

119886

(

1

Δ119894

minus

1

Δ119894+1

) ge 120575 gt 0

119894 = 1 119869 minus 1

(23)

Proof Since 119886120573 gt 2 we have that 1205732 ge 1119886 + 120578 for some120578 gt 0 Then

119878119894

ge

1

119886

119875119894

+ 120578 (24)

where

119875119894

=

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

Δ119894

Δ119894+1

(25)

Then because of (i) and (iii)

119875119894

ge 2119875lowast119894

where 119875lowast119894

fl1206011015840

(119905119894minus1

)2

minus 1206011015840

(119905119894+1

)2

1198732

Δ119894

Δ119894+1

(26)

If we show that

119875lowast119894

997888rarr 0 uniformly in 120576 when 119873 997888rarr infin (27)

then we will have (23) provided 119873 is sufficiently largeindependently of 120576 We now show that (27) holds true First

1003816100381610038161003816119875lowast119894

1003816100381610038161003816le 119862

100381610038161003816100381610038161206011015840

(119905119894minus1

)2

minus 1206011015840

(119905119894+1

)2

10038161003816100381610038161003816

(28)

because of (i) and (ii) and then

1003816100381610038161003816119875lowast119894

1003816100381610038161003816le 119862119873

minus1

119871119896

12060110158401015840

(119905119894+1

) le 119862119873minus1

1198713119896 (29)

because of (i) (iii) and (iv)

Themeshes defined inTable 1 satisfy conditions (i)ndash(iv) ofLemma 5 with the exception of the BS-mesh which does notfulfill condition (iv) However estimate (23) still holds truefor the BS-mesh if we assume that radic2 minus 1 le 119876 lt 1 This isnot a serious restriction since119876 = 12 is a typical choiceTheproof of this result stated below in Lemma 6 is given in theappendix

Lemma 6 Let 119886120573 gt 2 and let radic2 minus 1 le 119876 lt 1 Then theBS-mesh satisfies estimate (23)

Lemma 7 Under the assumptions of Lemma 5 (Lemma 6 ifthe BS-mesh is used) matrix

119873

of system (21) satisfies

100381710038171003817100381710038171003817

minus1

119873

100381710038171003817100381710038171003817

le 119862 (30)

Proof We construct a vector V = [V0

V1

V119873

]119879 such that

(a) V119894

gt 0 119894 = 0 1 119873(b) V119894

le 119862 119894 = 0 1 119873

(c) 119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

ge 120579 119894 = 1 2 119873 minus 1 where 120579

is a positive constant independent of both 120576 and 119873

Then according to the 119872-criterion

100381710038171003817100381710038171003817

minus1

119873

100381710038171003817100381710038171003817

le 120579minus1

V le 119862 (31)

We define the vector V as follows (cf [14 15])

V119894

= 120572 minus 119867119894 + 120582min (1 + 120588)119869minus119894

1 (32)

where 120572 and 120582 are appropriately chosen positive constantsand 120588 = 120573119867120576

Since 119867119873 le 119862 there exists a constant 120572 such that V119894

ge

120572minus119867119894 gt 0 and condition (a) is satisfied It also holds true thatV119894

le 120572+120582Therefore condition (b) is satisfied if we prove that120582 le 119862 We do this in next steps while we verify condition (c)

6 Advances in Numerical Analysis

When 1 le 119894 le 119869 minus 1 using Lemma 5 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= (119897119894

+119889119894

+ 119894

) V119894

+119897119894

119867 minus 119894

119867

=

ℏ119894

119867

119888119894

V119894

minus

120576

ℎ119894

+

120576

ℎ119894+1

+

119887119894

ℏ119894

ℎ119894+1

ge minus(

120576

ℎ119894

minus

120576

ℎ119894+1

) +

119887119894

2

+

119887119894

ℎ119894

2ℎ119894+1

ge 119878119894

ge 120575 gt 0

(33)

For 119894 = 119869 condition (c) is verified as follows

119897119869

V119869minus1

+119889119869

V119869

+ 119869

V119869+1

=

ℏ119869

119867

119888119869

V119869

+119897119869

119867 minus 119869

(119867 +

120582120588

1 + 120588

)

ge minus119869

120582120588

1 + 120588

+ (119897119869

minus 119869

)119867

= (

120576

1198672

+

119887119869

ℏ119869

1198672

)

120582120588

1 + 120588

minus

120576

ℎ119869

+

120576

119867

+

119887119869

ℏ119869

119867

= (

2120576 + 119887119869

(ℎ119869

+ 119867)

21198672

)

120582120573119867

120576 + 120573119867

minus

120576

ℎ119869

+

119887119869

2

ge

120582120573

2119867

minus

120576

ℎ119869

+

120573

2

ge

120573

2

(34)

where in the last step invoking (8) we choose a constant 120582 sothat

120582120573

2119867

minus

120576

ℎ119869

ge

120582120573

2119867

minus

120576

119886120576 (120601 (119905119869

) minus 120601 (119905119869minus1

))

ge

120582120573

2119867

minus

1

119886Δ119869

ge

120582120573

4119873minus1

minus

1

119886119862119873minus1

= 119873(

120582120573

4

minus

1

119886119862

) ge 0

(35)

To guarantee this 120582 is chosen so that

120582120573

4

ge

1

119886119862

(36)

Finally when 119869 + 1 le 119894 le 119873 minus 1 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= 119888119894

V119894

+119897119894

119867 minus 119894

119867

+

119897119894

1 + 120588119869

[

120582

(1 + 120588)119894minus1minus119869

minus

120582

(1 + 120588)119894minus119869

]

+

119894

1 + 120588119869

[

120582

(1 + 120588)119894+1minus119869

minus

120582

(1 + 120588)119894minus119869

]

ge 119887119894

+

120588 (1 + 120588)119897119894

minus 120588119894

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

ge 120573 +

(119897119894

minus 119894

+119897119894

120588) 120588

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

= 120573 + (

119887119894

119867

minus

120573

119867

)

120582120588 (1 + 120588)119869minus119894minus1

1 + 120588119869

ge 120573

(37)

By examining the entries of 119873

we easily see that themaximum absolute row sum of

119873

is achieved in the layerregion Therefore

10038171003817100381710038171003817119873

10038171003817100381710038171003817le 119862

119873

Δ120601

(38)

A combination of Lemma 7 and (38) results in thefollowing

Theorem 8 The condition number of matrix 119873

satisfies thefollowing 120576-uniform estimate

120581 (119873

) le 119862

119873

Δ120601

(39)

Corollary 9 For the meshes given in Table 1 one has thefollowing estimates

120581 (119873

) le

119862

1198732

119871

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

1198621198732

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(40)

In Tables 4 and 5 we present the condition numbers ofthe preconditioned linear systems (21) for various values of120576 and 119873 for the BS- and VS-meshes We can observe thatthe condition numbers of the preconditioned systems areindependent of 120576 and that the bound in Corollary 9 is sharp

5 120576-Uniform Convergence

For the proof of 120576-uniform convergence we need to decom-pose the solution 119906 into the smooth and boundary-layer

Advances in Numerical Analysis 7

Table 4 The condition number of matrix 119873

on the BS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 658119890 + 02 268119890 + 03 108119890 + 04 434119890 + 04 174119890 + 05 695119890 + 05

1119890 minus 3 689119890 + 02 278119890 + 03 111119890 + 04 440119890 + 04 175119890 + 05 700119890 + 05

1119890 minus 4 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 702119890 + 05

1119890 minus 5 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 6 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 7 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 8 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

Table 5 The condition number of matrix 119873

on the VS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 748119890 + 02 313119890 + 03 127119890 + 04 510119890 + 04 204119890 + 05 811119890 + 05

1119890 minus 3 780119890 + 02 323119890 + 03 130119890 + 04 520119890 + 04 207119890 + 05 820119890 + 05

1119890 minus 4 785119890 + 02 325119890 + 03 131119890 + 04 522119890 + 04 208119890 + 05 824119890 + 05

1119890 minus 5 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 6 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 7 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 8 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

parts There exist several decompositions of this kind but weuse the version presented in [2 Theorem 348]

119906 (119909) = 119904 (119909) + 119910 (119909) 119909 isin 119868 (41)10038161003816100381610038161003816119904(119896)

(119909)

10038161003816100381610038161003816le 119862 (1 + 120576

2minus119896

)

10038161003816100381610038161003816119910(119896)

(119909)

10038161003816100381610038161003816le 119862120576minus119896

119890minus120573119909120576

119909 isin 119868 119896 = 0 1 2 3

(42)

The details related to the construction of the function 119904 arenot important here they can be found in [2] As for 119910 wenote that it solves the problem

L119910 (119909) = 0 119909 isin (0 1)

119910 (0) = minus119904 (0)

119910 (1) = 0

(43)

We define the consistency error of the finite-differenceoperatorL119873 as

120591119894

= 120591119894

[119906] fl L119873

119906119894

minus 119891119894

119894 = 1 2 119873 minus 1 (44)

It holds true that

120591119894

= L119873

119906119894

minus (L119906)119894

= [119860119873

(119906119873

minus 119880119873

)]119894

(45)

Taylorrsquos expansion gives

1003816100381610038161003816120591119894

[119906]1003816100381610038161003816le 119862ℎ119894+1

(120576

10038171003817100381710038171003817119906101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711990610158401015840

10038171003817100381710038171003817119894) (46)

where 119892119894

fl max119909119894minus1le119909le119909119894+1

|119892(119909)| for any 119862(119868)-function 119892When the diagonal matrix 119872 is multiplied to linear system(14) the consistency error of the multiplied system is

119894

[119906] =

ℏ119894

119867

120591119894

[119906] 1 le 119894 le 119869

120591119894

[119906] 119869 + 1 le 119894 le 119873 minus 1

(47)

Lemma 10 Let 119886120573 ge 2Then the following estimate holds true

1003816100381610038161003816119894

[119906]1003816100381610038161003816le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(48)

Proof We use decomposition (41) to get

119894

[119906] = 119894

[119904] + 119894

[119910] (49)

Then (46) and the derivative-estimates of 119904 given in (42)yield immediately that

1003816100381610038161003816119894

[119904]1003816100381610038161003816le 119862119873

minus1 (50)

and therefore the following remains to be proved

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

119894 = 119869 119873 minus 1

(51)

Note that from condition (7) we easily get

ℎ119894

le 119862120576 119894 = 1 119869

ℏ119894

le 119862120576 119894 = 1 119869 minus 1

(52)

This is the key property of S-type meshes as well as the mainingredient of the following proof

8 Advances in Numerical Analysis

Table 6 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 550119890 + 00 293119890 + 00 151119890 + 00 769119890 minus 01 388119890 minus 01 195119890 minus 01

1119890 minus 3 550119890 + 01 293119890 + 01 151119890 + 01 769119890 + 00 388119890 + 00 195119890 + 00

1119890 minus 4 550119890 + 02 293119890 + 02 151119890 + 02 769119890 + 01 388119890 + 01 195119890 + 01

1119890 minus 5 550119890 + 03 293119890 + 03 151119890 + 03 769119890 + 02 388119890 + 02 195119890 + 02

1119890 minus 6 550119890 + 04 293119890 + 04 151119890 + 04 769119890 + 03 388119890 + 03 195119890 + 03

1119890 minus 7 550119890 + 05 293119890 + 05 151119890 + 05 769119890 + 04 388119890 + 04 195119890 + 04

1119890 minus 8 550119890 + 06 293119890 + 06 151119890 + 06 769119890 + 05 388119890 + 05 195119890 + 05

For 1 le 119894 le 119869 minus 1 (46) estimates (10) and (11) as well asthe derivative-estimates of 119910 in (42) imply

1003816100381610038161003816119894

(119910)1003816100381610038161003816le 119862

ℏ119894

119867

ℎ119894+1

(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894)

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902119909119894+1(119886120576)

119890minus120573119909119894minus1120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890(2119909119894+1minus119886120573119909119894minus1)119886120576

= 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902(119909119894+1minus119909119894minus1)119886120576

119890minus(119886120573minus2)119909119894minus1119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890ℏ119894119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

(53)

where we have used (52) and 119886120573 ge 2 to get

119890ℏ119894(119886120576)

le 119862

119890minus(119886120573minus2)119909119894minus1(119886120576)

le 119862

(54)

When 119869 + 2 le 119894 le 119873 minus 1 (42) and (46) give

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119867(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894) le 119862

119867

1205762

119890minus120573119909119894minus1120576

le 119862

119867

1205762

119890minus120573(120590+119867)120576

le 119862119873

1198672

1205762

119890minus120573119867120576

119890minus120573120590120576

(55)

Estimate (51) follows from the fact that (119867120576minus1

)2

119890minus120573119867120576

le 119862

and because the definition of 120590 implies that

119890minus120573120590120576

= 119890minus120573119886119871

= 119873minus120573119886

le 119873minus2

(56)

We finally prove (51) when 119894 = 119869 119869 + 1 in which case

1003816100381610038161003816119869

[119910]1003816100381610038161003816=

ℏ119869

119867

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816

1003816100381610038161003816119869+1

[119910]1003816100381610038161003816=

1003816100381610038161003816120591119869+1

[119910]1003816100381610038161003816

(57)

We now use the fact that L119910 = 0 and work with a differentform of the consistency error (this is a well-known techniquecf [18 Lemma 5])

1003816100381610038161003816119894

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816le 119877119894

+ 119876119894

+ 119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816 (58)

where

119877119894

= 120576

1003816100381610038161003816100381611986310158401015840

119910119894

10038161003816100381610038161003816

119876119894

= 119887119894

100381610038161003816100381610038161198631015840

119910119894

10038161003816100381610038161003816

(59)

We immediately have that

119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816le 119862119890minus120573119909119894120576

le 119862119890minus120573120590120576

= 119862119890minus120573119886119871

= 119862119873minus119886120573

le 119862119873minus2

(60)

As for 119877119894

it holds true that

119877119894

le ℏminus1

119894

120576 sdot 2

100381710038171003817100381710038171199101015840

10038171003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873119890minus120573120590120576

119890120573ℎ119869120576

le 119862119873minus1

(61)

by (52) Analogously

119876119894

le 119862119867minus1

10038171003817100381710038171199101003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873minus1

(62)

This completes the proof

In general the consistency error 120591119894

of the layer componentof the solution is not bounded pointwise uniformly in 120576When the above proof technique is applied to 120591

119894

[119910] one canonly get

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816

le

119862120576minus1

119873minus1

119871 for the S-mesh

119862120576minus1

119873minus1

for the BS- and VS-meshes

(63)

The proposed preconditioner simplifies the analysis of theconsistency error by introducing an extra 120576 factor needed forthe layer component Hence the approach used in the proofof Lemma 10 is natural and straightforward because it onlyuses the classical Taylor expansion estimate for the truncationerror

To illustrate (63) in Tables 6 and 7 we present the resultsof some numerical experiments on the BS- and VS-meshesrespectivelyWe refer the reader to [14] for the correspondingnumerical results on the S-mesh

By contrast with preconditioner (20) the preconditionedconsistency error converges uniformly in 120576 as shown in Tables8 and 9

In conclusion the preconditioning allows us to usePrinciple 1 and combine Lemmas 7 and 10 to obtain thefollowing main convergence result

Advances in Numerical Analysis 9

Table 7 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 467119890 + 00 247119890 + 00 128119890 + 00 659119890 minus 01 336119890 minus 01 170119890 minus 01

1119890 minus 3 467119890 + 01 247119890 + 01 128119890 + 01 659119890 + 00 336119890 + 00 170119890 + 00

1119890 minus 4 467119890 + 02 247119890 + 02 128119890 + 02 659119890 + 01 336119890 + 01 170119890 + 01

1119890 minus 5 467119890 + 03 247119890 + 03 128119890 + 03 659119890 + 02 336119890 + 02 170119890 + 02

1119890 minus 6 467119890 + 04 247119890 + 04 128119890 + 04 659119890 + 03 336119890 + 03 170119890 + 03

1119890 minus 7 467119890 + 05 247119890 + 05 128119890 + 05 659119890 + 04 336119890 + 04 170119890 + 04

1119890 minus 8 467119890 + 06 247119890 + 06 128119890 + 06 659119890 + 05 336119890 + 05 170119890 + 05

Table 8 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 122119890 minus 01 650119890 minus 02 338119890 minus 02 174119890 minus 02 887119890 minus 03 452119890 minus 03

1119890 minus 3 114119890 minus 01 601119890 minus 02 308119890 minus 02 156119890 minus 02 786119890 minus 03 395119890 minus 03

1119890 minus 4 114119890 minus 01 597119890 minus 02 306119890 minus 02 155119890 minus 02 778119890 minus 03 390119890 minus 03

1119890 minus 5 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 390119890 minus 03

1119890 minus 6 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 7 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 8 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

Table 9 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 160119890 minus 01 912119890 minus 02 502119890 minus 02 270119890 minus 02 143119890 minus 02 747119890 minus 03

1119890 minus 3 150119890 minus 01 844119890 minus 02 458119890 minus 02 242119890 minus 02 126119890 minus 02 653119890 minus 03

1119890 minus 4 149119890 minus 01 837119890 minus 02 454119890 minus 02 240119890 minus 02 125119890 minus 02 645119890 minus 03

1119890 minus 5 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 6 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 7 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 8 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

Theorem 11 The error of the upwind finite-difference dis-cretization on S-type meshes satisfies

10038161003816100381610038161003816119906119873

119894

minus 119880119873

119894

10038161003816100381610038161003816

le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(64)

Corollary 12 For the meshes given in Table 1 one has thefollowing estimates

10038171003817100381710038171003817119906119873

minus 119880119873

10038171003817100381710038171003817

le

119862119873minus1

1198712

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

119862119873minus1

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(65)

Remark 13 The barrier-function technique can be used toobtain similar results for S-type meshes (see [16]) but itrequires another assumption on the mesh-generating func-tion 120601 (cf [16 Theorem 1]) However whether the barrier-function approach can be extended to any Bakhvalov-type

mesh is still an open question [2 Remark 421] By contrastas proved in [19] the preconditioner defined in (20) worksfine for Vulanovicrsquos modification of the Bakhvalov mesh [10]The paper [10] is where the original Bakhvalov mesh [20]is generalized (the main difference between Bakhvalov-typemeshes and S-type meshes is that the former have a smoothtransition from the fine part inside the layer to the coarseuniform part outside the layer the mesh-generating function120601 belongs to 119862

1

(119868))

6 Conclusion

In this paper we present a proof of pointwise uniformconvergence for singularly perturbed convection-diffusionproblem (1) discretized on a general class of Shishkin-typemeshesThe proof is based on preconditioning of the discretesystem which allows us to use the standard ldquoconsistency+ stabilityrdquo principle of convergence This is a conceptuallysimple approach which does not require the use of hybridstability inequalities and discrete Greenrsquos functions In gen-eral new proof techniques are of interest because althougha lot has been achieved in the field of singularly perturbed

10 Advances in Numerical Analysis

problems over the last few decades there are still importantquestions remaining open [21] One of them Question6 is related to the work presented here The question isabout two-dimensional convection-diffusion problems and itpoints out that whereas 120576-uniform pointwise convergence forthe upwind finite-difference discretization on S-type meshesis well-established no similar result for B-type meshes isknownAsmentioned inRemark 13 we already know that thepreconditioning approach works well for B-mesh and one-dimensional convection-diffusion problems The next step isto consider two-dimensional problems

Appendix

Proof of Lemma 6 The proof is different from the proof ofLemma 5because theBS-meshdoes not satisfy condition (iv)Because of (ii) it is sufficient to show that the numerator in(25) is nonnegative for radic2 minus 1 le 119876 lt 1 independently of 120576and 119873 Let 119901 fl 1 minus 119873

minus1 By direct computations we get that

Δ119894

= minus ln(1 minus

119905119894

119901

119876

) + ln(1 minus

119905119894minus1

119901

119876

)

= ln(1 +

1

119869119901 minus 119894

) = ln (1 + 119902)

(A1)

where we denote 119902 fl (119869119901minus119894)minus1 Note that 0 le 119902 lt 1(119876+1)

Analogously

Δ119894+1

= ln(

1

1 minus 119902

) (A2)

Then

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

= Δ119894

(Δ119894+1

+ Δ119894

+ 1) minus Δ119894+1

= ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902)

(A3)

We set

119892 (119902) fl ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902) (A4)

Sinceradic2minus1 le 119876 lt 1 we have 0 le 119902 lt 1radic2 Rearrange 119892(119902)

as

119892 (119902) = [1 + ln(1 +

2119902

1 minus 119902

)] ln (1 + 119902)

+ ln (1 minus 119902)

(A5)

For 119902 isin [0 1) it follows that119902

1 + 1199022

le ln (1 + 119902) le 119902 (A6)

Hence

ln(1 +

2119902

1 minus 119902

) ge 2119902 (A7)

So we are done if we can show that

119892 (119902) ge [1 + 2119902] ln (1 + 119902) + ln (1 minus 119902) ge 0

0 le 119902 lt

1

radic2

(A8)

or

(1 minus 119902) (1 + 119902)2119902+1

= (1 minus 1199022

) (1 + 119902)2119902

ge 1 (A9)

First if 119902 isin [12 1radic2] we have by (1 + 119902)120574

ge 1 + 120574119902 for120574 ge 1

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

) (1 + 21199022

) ge 1 (A10)

Second we have that ln(1+119902) is a concave-down function on[0 12] hence on [0 12]

ln (1 + 119902) ge 2 ln(

3

2

) 119902 ge

4119902

5

(A11)

Then multiplying both sides by 119902 and exponentiating we getthat

(1 + 119902)119902

ge exp(

41199022

5

) ge 1 +

41199022

5

(A12)

Therefore

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

)(1 +

41199022

5

)

2

ge 1 (A13)

which holds true for 119902 isin [0 12]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P A Farrell A F Hegarty J J H Miller E OrsquoRiordan andG I Shishkin Robust Computational Techniques for BoundaryLayers Chapman amp Hall CRC Press Boca Raton Fla USA2000

[2] T Linszlig Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems vol 1985 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[3] J J HMiller E OrsquoRiordan and G I Shishkin Fitted NumericalMethods for Singular Perturbation Problems World ScientificSingapore 1996

[4] H-G Roos M Stynes and L Tobiska Numerical Methodsfor Singularly Perturbed Differential Equations Springer BerlinGermany 2nd edition 2008

[5] A Malvandi and D D Ganji ldquoMixed convection of aluminandashwater nanofluid inside a concentric annulus consideringnanoparticle migrationrdquo Particuology vol 24 pp 113ndash122 2016

[6] R B Kellogg and A Tsan ldquoAnalysis of some difference approx-imations for a singular perturbation problem without turningpointsrdquoMathematics of Computation vol 32 no 144 pp 1025ndash1039 1978

Advances in Numerical Analysis 11

[7] J Lorenz ldquoStability and monotonicity properties of stiff quasi-linear boundary value problemsrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta Serija zaMatematiku vol 12 pp 151ndash175 1982

[8] E Bohl Finite Modelle Gewohnlicher Randwertaufgaben Teub-ner Stuttgart Germany 1981

[9] E C Gartland Jr ldquoGraded-mesh difference schemes for sin-gularly perturbed two-point boundary value problemsrdquo Math-ematics of Computation vol 51 no 184 pp 631ndash657 1988

[10] R Vulanovic ldquoOn a numerical solution of a type of singularlyperturbed boundary value problem by using a special dis-cretization meshrdquo Univerzitet u Novom Sadu Zbornik RadovaPrirodno- Matematichkog Fakulteta Serija za Matematiku vol13 pp 187ndash201 1983

[11] C Grossmann H-G Roos and M Stynes Numerical Treat-ment of Partial Differential Equations Springer Berlin Ger-many 2007

[12] V B Andreev and I A Savin ldquoThe uniform convergence withrespect to a small parameter of A A Samarskiirsquos monotonescheme and its modificationrdquo Computational Mathematics andMathematical Physics vol 35 pp 581ndash591 1995

[13] T LinszligH-G Roos andRVulanovic ldquoUniformpointwise con-vergence on Shishkin-type meshes for quasi-linear convection-diffusion problemsrdquo SIAM Journal on Numerical Analysis vol38 no 3 pp 897ndash912 2000

[14] R Vulanovic and T A Nhan ldquoUniform convergence viapreconditioningrdquo International Journal of Numerical Analysisand Modeling B vol 5 no 4 pp 347ndash356 2014

[15] H-G Roos ldquoA note on the conditioning of upwind schemes onShishkin meshesrdquo IMA Journal of Numerical Analysis vol 16no 4 pp 529ndash538 1996

[16] H-G Roos and T Linszlig ldquoSufficient condition for uniformconvergence on layeradapted gridsrdquo Computing vol 63 no 1pp 27ndash45 1999

[17] R Vulanovic ldquoA higher-order scheme for quasilinear boundaryvalue problems with two small parametersrdquo Computing vol 67no 4 pp 287ndash303 2001

[18] R Vulanovic ldquoA priori meshes for singularly perturbed quasi-linear twominuspoint boundary value problemsrdquo IMA Journal ofNumerical Analysis vol 21 no 1 pp 349ndash366 2001

[19] T A Nhan and R Vulanovic ldquoUniform convergence on aBakhvalov-type mesh using the preconditioning approachTechnical reportrdquo httparxivorgabs150404283

[20] N S Bakhvalov ldquoThe optimization of methods of solvingboundary value problems with a boundary layerrdquo USSR Com-putational Mathematics and Mathematical Physics vol 9 no 4pp 139ndash166 1969

[21] H-G Roos andM Stynes ldquoSome open questions in the numer-ical analysis of singularly perturbed differential equationsrdquoComputational Methods in Applied Mathematics vol 15 no 4pp 531ndash550 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Numerical Analysis 5

where 119873

= 119872119860119873

Let the entries of 119873

be denoted by 119894119895

the nonzero ones being

119897119894

fl 119894minus1119894

=

minus

120576

ℎ119894

119867

1 le 119894 le 119869

minus

120576

1198672

119869 + 1 le 119894 le 119873 minus 1

119894

fl 119894119894+1

=

minus

120576

ℎ119894+1

119867

minus

119887119894

ℏ119894

ℎ119894+1

119867

1 le 119894 le 119869 minus 1

minus

120576

1198672

minus

119887119894

ℏ119894

1198672

119894 = 119869

minus

120576

1198672

minus

119887119894

119867

119869 + 1 le 119894 le 119873 minus 1

119889119894

fl 119894119894

=

1 119894 = 0

minus119897119894

minus 119894

+ 119888119894

ℏ119894

119867

1 le 119894 le 119869

minus119897119894

minus 119894

+ 119888119894

119869 + 1 le 119894 le 119873 minus 1

1 119894 = 119873

(22)

Matrix 119873

remains an 119871-matrix since the precondi-tioning does not change this property of 119860

119873

We showin Lemma 7 below that

119873

is an 119872-matrix and that themodified discretization (21) is stable uniformly in 120576 But firstwe give a technical result related to the mesh

Lemma 5 Let 119886120573 gt 2 and let in addition to Assumption 1 themesh-generating function 120601 satisfy the following conditions on[0 119876]

(i) 120601(119895)

ge 0 119895 = 1 2 3(ii) 1206011015840

(0) ge 119898 gt 0 where 119898 is a constant independent of 120576and 119873

(iii) 12060110158401015840

le 2(1206011015840

)2

(iv) 1206011015840

le 119862119871119896 for some 119896 gt 0

Then for a sufficiently large119873 independent of 120576 there exists apositive constant 120575 independent of both 120576 and 119873 such that

119878119894

fl120573

2

(1 +

Δ119894

Δ119894+1

) minus

1

119886

(

1

Δ119894

minus

1

Δ119894+1

) ge 120575 gt 0

119894 = 1 119869 minus 1

(23)

Proof Since 119886120573 gt 2 we have that 1205732 ge 1119886 + 120578 for some120578 gt 0 Then

119878119894

ge

1

119886

119875119894

+ 120578 (24)

where

119875119894

=

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

Δ119894

Δ119894+1

(25)

Then because of (i) and (iii)

119875119894

ge 2119875lowast119894

where 119875lowast119894

fl1206011015840

(119905119894minus1

)2

minus 1206011015840

(119905119894+1

)2

1198732

Δ119894

Δ119894+1

(26)

If we show that

119875lowast119894

997888rarr 0 uniformly in 120576 when 119873 997888rarr infin (27)

then we will have (23) provided 119873 is sufficiently largeindependently of 120576 We now show that (27) holds true First

1003816100381610038161003816119875lowast119894

1003816100381610038161003816le 119862

100381610038161003816100381610038161206011015840

(119905119894minus1

)2

minus 1206011015840

(119905119894+1

)2

10038161003816100381610038161003816

(28)

because of (i) and (ii) and then

1003816100381610038161003816119875lowast119894

1003816100381610038161003816le 119862119873

minus1

119871119896

12060110158401015840

(119905119894+1

) le 119862119873minus1

1198713119896 (29)

because of (i) (iii) and (iv)

Themeshes defined inTable 1 satisfy conditions (i)ndash(iv) ofLemma 5 with the exception of the BS-mesh which does notfulfill condition (iv) However estimate (23) still holds truefor the BS-mesh if we assume that radic2 minus 1 le 119876 lt 1 This isnot a serious restriction since119876 = 12 is a typical choiceTheproof of this result stated below in Lemma 6 is given in theappendix

Lemma 6 Let 119886120573 gt 2 and let radic2 minus 1 le 119876 lt 1 Then theBS-mesh satisfies estimate (23)

Lemma 7 Under the assumptions of Lemma 5 (Lemma 6 ifthe BS-mesh is used) matrix

119873

of system (21) satisfies

100381710038171003817100381710038171003817

minus1

119873

100381710038171003817100381710038171003817

le 119862 (30)

Proof We construct a vector V = [V0

V1

V119873

]119879 such that

(a) V119894

gt 0 119894 = 0 1 119873(b) V119894

le 119862 119894 = 0 1 119873

(c) 119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

ge 120579 119894 = 1 2 119873 minus 1 where 120579

is a positive constant independent of both 120576 and 119873

Then according to the 119872-criterion

100381710038171003817100381710038171003817

minus1

119873

100381710038171003817100381710038171003817

le 120579minus1

V le 119862 (31)

We define the vector V as follows (cf [14 15])

V119894

= 120572 minus 119867119894 + 120582min (1 + 120588)119869minus119894

1 (32)

where 120572 and 120582 are appropriately chosen positive constantsand 120588 = 120573119867120576

Since 119867119873 le 119862 there exists a constant 120572 such that V119894

ge

120572minus119867119894 gt 0 and condition (a) is satisfied It also holds true thatV119894

le 120572+120582Therefore condition (b) is satisfied if we prove that120582 le 119862 We do this in next steps while we verify condition (c)

6 Advances in Numerical Analysis

When 1 le 119894 le 119869 minus 1 using Lemma 5 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= (119897119894

+119889119894

+ 119894

) V119894

+119897119894

119867 minus 119894

119867

=

ℏ119894

119867

119888119894

V119894

minus

120576

ℎ119894

+

120576

ℎ119894+1

+

119887119894

ℏ119894

ℎ119894+1

ge minus(

120576

ℎ119894

minus

120576

ℎ119894+1

) +

119887119894

2

+

119887119894

ℎ119894

2ℎ119894+1

ge 119878119894

ge 120575 gt 0

(33)

For 119894 = 119869 condition (c) is verified as follows

119897119869

V119869minus1

+119889119869

V119869

+ 119869

V119869+1

=

ℏ119869

119867

119888119869

V119869

+119897119869

119867 minus 119869

(119867 +

120582120588

1 + 120588

)

ge minus119869

120582120588

1 + 120588

+ (119897119869

minus 119869

)119867

= (

120576

1198672

+

119887119869

ℏ119869

1198672

)

120582120588

1 + 120588

minus

120576

ℎ119869

+

120576

119867

+

119887119869

ℏ119869

119867

= (

2120576 + 119887119869

(ℎ119869

+ 119867)

21198672

)

120582120573119867

120576 + 120573119867

minus

120576

ℎ119869

+

119887119869

2

ge

120582120573

2119867

minus

120576

ℎ119869

+

120573

2

ge

120573

2

(34)

where in the last step invoking (8) we choose a constant 120582 sothat

120582120573

2119867

minus

120576

ℎ119869

ge

120582120573

2119867

minus

120576

119886120576 (120601 (119905119869

) minus 120601 (119905119869minus1

))

ge

120582120573

2119867

minus

1

119886Δ119869

ge

120582120573

4119873minus1

minus

1

119886119862119873minus1

= 119873(

120582120573

4

minus

1

119886119862

) ge 0

(35)

To guarantee this 120582 is chosen so that

120582120573

4

ge

1

119886119862

(36)

Finally when 119869 + 1 le 119894 le 119873 minus 1 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= 119888119894

V119894

+119897119894

119867 minus 119894

119867

+

119897119894

1 + 120588119869

[

120582

(1 + 120588)119894minus1minus119869

minus

120582

(1 + 120588)119894minus119869

]

+

119894

1 + 120588119869

[

120582

(1 + 120588)119894+1minus119869

minus

120582

(1 + 120588)119894minus119869

]

ge 119887119894

+

120588 (1 + 120588)119897119894

minus 120588119894

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

ge 120573 +

(119897119894

minus 119894

+119897119894

120588) 120588

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

= 120573 + (

119887119894

119867

minus

120573

119867

)

120582120588 (1 + 120588)119869minus119894minus1

1 + 120588119869

ge 120573

(37)

By examining the entries of 119873

we easily see that themaximum absolute row sum of

119873

is achieved in the layerregion Therefore

10038171003817100381710038171003817119873

10038171003817100381710038171003817le 119862

119873

Δ120601

(38)

A combination of Lemma 7 and (38) results in thefollowing

Theorem 8 The condition number of matrix 119873

satisfies thefollowing 120576-uniform estimate

120581 (119873

) le 119862

119873

Δ120601

(39)

Corollary 9 For the meshes given in Table 1 one has thefollowing estimates

120581 (119873

) le

119862

1198732

119871

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

1198621198732

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(40)

In Tables 4 and 5 we present the condition numbers ofthe preconditioned linear systems (21) for various values of120576 and 119873 for the BS- and VS-meshes We can observe thatthe condition numbers of the preconditioned systems areindependent of 120576 and that the bound in Corollary 9 is sharp

5 120576-Uniform Convergence

For the proof of 120576-uniform convergence we need to decom-pose the solution 119906 into the smooth and boundary-layer

Advances in Numerical Analysis 7

Table 4 The condition number of matrix 119873

on the BS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 658119890 + 02 268119890 + 03 108119890 + 04 434119890 + 04 174119890 + 05 695119890 + 05

1119890 minus 3 689119890 + 02 278119890 + 03 111119890 + 04 440119890 + 04 175119890 + 05 700119890 + 05

1119890 minus 4 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 702119890 + 05

1119890 minus 5 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 6 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 7 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 8 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

Table 5 The condition number of matrix 119873

on the VS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 748119890 + 02 313119890 + 03 127119890 + 04 510119890 + 04 204119890 + 05 811119890 + 05

1119890 minus 3 780119890 + 02 323119890 + 03 130119890 + 04 520119890 + 04 207119890 + 05 820119890 + 05

1119890 minus 4 785119890 + 02 325119890 + 03 131119890 + 04 522119890 + 04 208119890 + 05 824119890 + 05

1119890 minus 5 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 6 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 7 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 8 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

parts There exist several decompositions of this kind but weuse the version presented in [2 Theorem 348]

119906 (119909) = 119904 (119909) + 119910 (119909) 119909 isin 119868 (41)10038161003816100381610038161003816119904(119896)

(119909)

10038161003816100381610038161003816le 119862 (1 + 120576

2minus119896

)

10038161003816100381610038161003816119910(119896)

(119909)

10038161003816100381610038161003816le 119862120576minus119896

119890minus120573119909120576

119909 isin 119868 119896 = 0 1 2 3

(42)

The details related to the construction of the function 119904 arenot important here they can be found in [2] As for 119910 wenote that it solves the problem

L119910 (119909) = 0 119909 isin (0 1)

119910 (0) = minus119904 (0)

119910 (1) = 0

(43)

We define the consistency error of the finite-differenceoperatorL119873 as

120591119894

= 120591119894

[119906] fl L119873

119906119894

minus 119891119894

119894 = 1 2 119873 minus 1 (44)

It holds true that

120591119894

= L119873

119906119894

minus (L119906)119894

= [119860119873

(119906119873

minus 119880119873

)]119894

(45)

Taylorrsquos expansion gives

1003816100381610038161003816120591119894

[119906]1003816100381610038161003816le 119862ℎ119894+1

(120576

10038171003817100381710038171003817119906101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711990610158401015840

10038171003817100381710038171003817119894) (46)

where 119892119894

fl max119909119894minus1le119909le119909119894+1

|119892(119909)| for any 119862(119868)-function 119892When the diagonal matrix 119872 is multiplied to linear system(14) the consistency error of the multiplied system is

119894

[119906] =

ℏ119894

119867

120591119894

[119906] 1 le 119894 le 119869

120591119894

[119906] 119869 + 1 le 119894 le 119873 minus 1

(47)

Lemma 10 Let 119886120573 ge 2Then the following estimate holds true

1003816100381610038161003816119894

[119906]1003816100381610038161003816le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(48)

Proof We use decomposition (41) to get

119894

[119906] = 119894

[119904] + 119894

[119910] (49)

Then (46) and the derivative-estimates of 119904 given in (42)yield immediately that

1003816100381610038161003816119894

[119904]1003816100381610038161003816le 119862119873

minus1 (50)

and therefore the following remains to be proved

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

119894 = 119869 119873 minus 1

(51)

Note that from condition (7) we easily get

ℎ119894

le 119862120576 119894 = 1 119869

ℏ119894

le 119862120576 119894 = 1 119869 minus 1

(52)

This is the key property of S-type meshes as well as the mainingredient of the following proof

8 Advances in Numerical Analysis

Table 6 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 550119890 + 00 293119890 + 00 151119890 + 00 769119890 minus 01 388119890 minus 01 195119890 minus 01

1119890 minus 3 550119890 + 01 293119890 + 01 151119890 + 01 769119890 + 00 388119890 + 00 195119890 + 00

1119890 minus 4 550119890 + 02 293119890 + 02 151119890 + 02 769119890 + 01 388119890 + 01 195119890 + 01

1119890 minus 5 550119890 + 03 293119890 + 03 151119890 + 03 769119890 + 02 388119890 + 02 195119890 + 02

1119890 minus 6 550119890 + 04 293119890 + 04 151119890 + 04 769119890 + 03 388119890 + 03 195119890 + 03

1119890 minus 7 550119890 + 05 293119890 + 05 151119890 + 05 769119890 + 04 388119890 + 04 195119890 + 04

1119890 minus 8 550119890 + 06 293119890 + 06 151119890 + 06 769119890 + 05 388119890 + 05 195119890 + 05

For 1 le 119894 le 119869 minus 1 (46) estimates (10) and (11) as well asthe derivative-estimates of 119910 in (42) imply

1003816100381610038161003816119894

(119910)1003816100381610038161003816le 119862

ℏ119894

119867

ℎ119894+1

(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894)

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902119909119894+1(119886120576)

119890minus120573119909119894minus1120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890(2119909119894+1minus119886120573119909119894minus1)119886120576

= 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902(119909119894+1minus119909119894minus1)119886120576

119890minus(119886120573minus2)119909119894minus1119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890ℏ119894119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

(53)

where we have used (52) and 119886120573 ge 2 to get

119890ℏ119894(119886120576)

le 119862

119890minus(119886120573minus2)119909119894minus1(119886120576)

le 119862

(54)

When 119869 + 2 le 119894 le 119873 minus 1 (42) and (46) give

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119867(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894) le 119862

119867

1205762

119890minus120573119909119894minus1120576

le 119862

119867

1205762

119890minus120573(120590+119867)120576

le 119862119873

1198672

1205762

119890minus120573119867120576

119890minus120573120590120576

(55)

Estimate (51) follows from the fact that (119867120576minus1

)2

119890minus120573119867120576

le 119862

and because the definition of 120590 implies that

119890minus120573120590120576

= 119890minus120573119886119871

= 119873minus120573119886

le 119873minus2

(56)

We finally prove (51) when 119894 = 119869 119869 + 1 in which case

1003816100381610038161003816119869

[119910]1003816100381610038161003816=

ℏ119869

119867

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816

1003816100381610038161003816119869+1

[119910]1003816100381610038161003816=

1003816100381610038161003816120591119869+1

[119910]1003816100381610038161003816

(57)

We now use the fact that L119910 = 0 and work with a differentform of the consistency error (this is a well-known techniquecf [18 Lemma 5])

1003816100381610038161003816119894

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816le 119877119894

+ 119876119894

+ 119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816 (58)

where

119877119894

= 120576

1003816100381610038161003816100381611986310158401015840

119910119894

10038161003816100381610038161003816

119876119894

= 119887119894

100381610038161003816100381610038161198631015840

119910119894

10038161003816100381610038161003816

(59)

We immediately have that

119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816le 119862119890minus120573119909119894120576

le 119862119890minus120573120590120576

= 119862119890minus120573119886119871

= 119862119873minus119886120573

le 119862119873minus2

(60)

As for 119877119894

it holds true that

119877119894

le ℏminus1

119894

120576 sdot 2

100381710038171003817100381710038171199101015840

10038171003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873119890minus120573120590120576

119890120573ℎ119869120576

le 119862119873minus1

(61)

by (52) Analogously

119876119894

le 119862119867minus1

10038171003817100381710038171199101003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873minus1

(62)

This completes the proof

In general the consistency error 120591119894

of the layer componentof the solution is not bounded pointwise uniformly in 120576When the above proof technique is applied to 120591

119894

[119910] one canonly get

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816

le

119862120576minus1

119873minus1

119871 for the S-mesh

119862120576minus1

119873minus1

for the BS- and VS-meshes

(63)

The proposed preconditioner simplifies the analysis of theconsistency error by introducing an extra 120576 factor needed forthe layer component Hence the approach used in the proofof Lemma 10 is natural and straightforward because it onlyuses the classical Taylor expansion estimate for the truncationerror

To illustrate (63) in Tables 6 and 7 we present the resultsof some numerical experiments on the BS- and VS-meshesrespectivelyWe refer the reader to [14] for the correspondingnumerical results on the S-mesh

By contrast with preconditioner (20) the preconditionedconsistency error converges uniformly in 120576 as shown in Tables8 and 9

In conclusion the preconditioning allows us to usePrinciple 1 and combine Lemmas 7 and 10 to obtain thefollowing main convergence result

Advances in Numerical Analysis 9

Table 7 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 467119890 + 00 247119890 + 00 128119890 + 00 659119890 minus 01 336119890 minus 01 170119890 minus 01

1119890 minus 3 467119890 + 01 247119890 + 01 128119890 + 01 659119890 + 00 336119890 + 00 170119890 + 00

1119890 minus 4 467119890 + 02 247119890 + 02 128119890 + 02 659119890 + 01 336119890 + 01 170119890 + 01

1119890 minus 5 467119890 + 03 247119890 + 03 128119890 + 03 659119890 + 02 336119890 + 02 170119890 + 02

1119890 minus 6 467119890 + 04 247119890 + 04 128119890 + 04 659119890 + 03 336119890 + 03 170119890 + 03

1119890 minus 7 467119890 + 05 247119890 + 05 128119890 + 05 659119890 + 04 336119890 + 04 170119890 + 04

1119890 minus 8 467119890 + 06 247119890 + 06 128119890 + 06 659119890 + 05 336119890 + 05 170119890 + 05

Table 8 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 122119890 minus 01 650119890 minus 02 338119890 minus 02 174119890 minus 02 887119890 minus 03 452119890 minus 03

1119890 minus 3 114119890 minus 01 601119890 minus 02 308119890 minus 02 156119890 minus 02 786119890 minus 03 395119890 minus 03

1119890 minus 4 114119890 minus 01 597119890 minus 02 306119890 minus 02 155119890 minus 02 778119890 minus 03 390119890 minus 03

1119890 minus 5 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 390119890 minus 03

1119890 minus 6 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 7 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 8 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

Table 9 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 160119890 minus 01 912119890 minus 02 502119890 minus 02 270119890 minus 02 143119890 minus 02 747119890 minus 03

1119890 minus 3 150119890 minus 01 844119890 minus 02 458119890 minus 02 242119890 minus 02 126119890 minus 02 653119890 minus 03

1119890 minus 4 149119890 minus 01 837119890 minus 02 454119890 minus 02 240119890 minus 02 125119890 minus 02 645119890 minus 03

1119890 minus 5 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 6 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 7 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 8 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

Theorem 11 The error of the upwind finite-difference dis-cretization on S-type meshes satisfies

10038161003816100381610038161003816119906119873

119894

minus 119880119873

119894

10038161003816100381610038161003816

le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(64)

Corollary 12 For the meshes given in Table 1 one has thefollowing estimates

10038171003817100381710038171003817119906119873

minus 119880119873

10038171003817100381710038171003817

le

119862119873minus1

1198712

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

119862119873minus1

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(65)

Remark 13 The barrier-function technique can be used toobtain similar results for S-type meshes (see [16]) but itrequires another assumption on the mesh-generating func-tion 120601 (cf [16 Theorem 1]) However whether the barrier-function approach can be extended to any Bakhvalov-type

mesh is still an open question [2 Remark 421] By contrastas proved in [19] the preconditioner defined in (20) worksfine for Vulanovicrsquos modification of the Bakhvalov mesh [10]The paper [10] is where the original Bakhvalov mesh [20]is generalized (the main difference between Bakhvalov-typemeshes and S-type meshes is that the former have a smoothtransition from the fine part inside the layer to the coarseuniform part outside the layer the mesh-generating function120601 belongs to 119862

1

(119868))

6 Conclusion

In this paper we present a proof of pointwise uniformconvergence for singularly perturbed convection-diffusionproblem (1) discretized on a general class of Shishkin-typemeshesThe proof is based on preconditioning of the discretesystem which allows us to use the standard ldquoconsistency+ stabilityrdquo principle of convergence This is a conceptuallysimple approach which does not require the use of hybridstability inequalities and discrete Greenrsquos functions In gen-eral new proof techniques are of interest because althougha lot has been achieved in the field of singularly perturbed

10 Advances in Numerical Analysis

problems over the last few decades there are still importantquestions remaining open [21] One of them Question6 is related to the work presented here The question isabout two-dimensional convection-diffusion problems and itpoints out that whereas 120576-uniform pointwise convergence forthe upwind finite-difference discretization on S-type meshesis well-established no similar result for B-type meshes isknownAsmentioned inRemark 13 we already know that thepreconditioning approach works well for B-mesh and one-dimensional convection-diffusion problems The next step isto consider two-dimensional problems

Appendix

Proof of Lemma 6 The proof is different from the proof ofLemma 5because theBS-meshdoes not satisfy condition (iv)Because of (ii) it is sufficient to show that the numerator in(25) is nonnegative for radic2 minus 1 le 119876 lt 1 independently of 120576and 119873 Let 119901 fl 1 minus 119873

minus1 By direct computations we get that

Δ119894

= minus ln(1 minus

119905119894

119901

119876

) + ln(1 minus

119905119894minus1

119901

119876

)

= ln(1 +

1

119869119901 minus 119894

) = ln (1 + 119902)

(A1)

where we denote 119902 fl (119869119901minus119894)minus1 Note that 0 le 119902 lt 1(119876+1)

Analogously

Δ119894+1

= ln(

1

1 minus 119902

) (A2)

Then

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

= Δ119894

(Δ119894+1

+ Δ119894

+ 1) minus Δ119894+1

= ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902)

(A3)

We set

119892 (119902) fl ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902) (A4)

Sinceradic2minus1 le 119876 lt 1 we have 0 le 119902 lt 1radic2 Rearrange 119892(119902)

as

119892 (119902) = [1 + ln(1 +

2119902

1 minus 119902

)] ln (1 + 119902)

+ ln (1 minus 119902)

(A5)

For 119902 isin [0 1) it follows that119902

1 + 1199022

le ln (1 + 119902) le 119902 (A6)

Hence

ln(1 +

2119902

1 minus 119902

) ge 2119902 (A7)

So we are done if we can show that

119892 (119902) ge [1 + 2119902] ln (1 + 119902) + ln (1 minus 119902) ge 0

0 le 119902 lt

1

radic2

(A8)

or

(1 minus 119902) (1 + 119902)2119902+1

= (1 minus 1199022

) (1 + 119902)2119902

ge 1 (A9)

First if 119902 isin [12 1radic2] we have by (1 + 119902)120574

ge 1 + 120574119902 for120574 ge 1

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

) (1 + 21199022

) ge 1 (A10)

Second we have that ln(1+119902) is a concave-down function on[0 12] hence on [0 12]

ln (1 + 119902) ge 2 ln(

3

2

) 119902 ge

4119902

5

(A11)

Then multiplying both sides by 119902 and exponentiating we getthat

(1 + 119902)119902

ge exp(

41199022

5

) ge 1 +

41199022

5

(A12)

Therefore

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

)(1 +

41199022

5

)

2

ge 1 (A13)

which holds true for 119902 isin [0 12]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P A Farrell A F Hegarty J J H Miller E OrsquoRiordan andG I Shishkin Robust Computational Techniques for BoundaryLayers Chapman amp Hall CRC Press Boca Raton Fla USA2000

[2] T Linszlig Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems vol 1985 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[3] J J HMiller E OrsquoRiordan and G I Shishkin Fitted NumericalMethods for Singular Perturbation Problems World ScientificSingapore 1996

[4] H-G Roos M Stynes and L Tobiska Numerical Methodsfor Singularly Perturbed Differential Equations Springer BerlinGermany 2nd edition 2008

[5] A Malvandi and D D Ganji ldquoMixed convection of aluminandashwater nanofluid inside a concentric annulus consideringnanoparticle migrationrdquo Particuology vol 24 pp 113ndash122 2016

[6] R B Kellogg and A Tsan ldquoAnalysis of some difference approx-imations for a singular perturbation problem without turningpointsrdquoMathematics of Computation vol 32 no 144 pp 1025ndash1039 1978

Advances in Numerical Analysis 11

[7] J Lorenz ldquoStability and monotonicity properties of stiff quasi-linear boundary value problemsrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta Serija zaMatematiku vol 12 pp 151ndash175 1982

[8] E Bohl Finite Modelle Gewohnlicher Randwertaufgaben Teub-ner Stuttgart Germany 1981

[9] E C Gartland Jr ldquoGraded-mesh difference schemes for sin-gularly perturbed two-point boundary value problemsrdquo Math-ematics of Computation vol 51 no 184 pp 631ndash657 1988

[10] R Vulanovic ldquoOn a numerical solution of a type of singularlyperturbed boundary value problem by using a special dis-cretization meshrdquo Univerzitet u Novom Sadu Zbornik RadovaPrirodno- Matematichkog Fakulteta Serija za Matematiku vol13 pp 187ndash201 1983

[11] C Grossmann H-G Roos and M Stynes Numerical Treat-ment of Partial Differential Equations Springer Berlin Ger-many 2007

[12] V B Andreev and I A Savin ldquoThe uniform convergence withrespect to a small parameter of A A Samarskiirsquos monotonescheme and its modificationrdquo Computational Mathematics andMathematical Physics vol 35 pp 581ndash591 1995

[13] T LinszligH-G Roos andRVulanovic ldquoUniformpointwise con-vergence on Shishkin-type meshes for quasi-linear convection-diffusion problemsrdquo SIAM Journal on Numerical Analysis vol38 no 3 pp 897ndash912 2000

[14] R Vulanovic and T A Nhan ldquoUniform convergence viapreconditioningrdquo International Journal of Numerical Analysisand Modeling B vol 5 no 4 pp 347ndash356 2014

[15] H-G Roos ldquoA note on the conditioning of upwind schemes onShishkin meshesrdquo IMA Journal of Numerical Analysis vol 16no 4 pp 529ndash538 1996

[16] H-G Roos and T Linszlig ldquoSufficient condition for uniformconvergence on layeradapted gridsrdquo Computing vol 63 no 1pp 27ndash45 1999

[17] R Vulanovic ldquoA higher-order scheme for quasilinear boundaryvalue problems with two small parametersrdquo Computing vol 67no 4 pp 287ndash303 2001

[18] R Vulanovic ldquoA priori meshes for singularly perturbed quasi-linear twominuspoint boundary value problemsrdquo IMA Journal ofNumerical Analysis vol 21 no 1 pp 349ndash366 2001

[19] T A Nhan and R Vulanovic ldquoUniform convergence on aBakhvalov-type mesh using the preconditioning approachTechnical reportrdquo httparxivorgabs150404283

[20] N S Bakhvalov ldquoThe optimization of methods of solvingboundary value problems with a boundary layerrdquo USSR Com-putational Mathematics and Mathematical Physics vol 9 no 4pp 139ndash166 1969

[21] H-G Roos andM Stynes ldquoSome open questions in the numer-ical analysis of singularly perturbed differential equationsrdquoComputational Methods in Applied Mathematics vol 15 no 4pp 531ndash550 2015

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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6 Advances in Numerical Analysis

When 1 le 119894 le 119869 minus 1 using Lemma 5 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= (119897119894

+119889119894

+ 119894

) V119894

+119897119894

119867 minus 119894

119867

=

ℏ119894

119867

119888119894

V119894

minus

120576

ℎ119894

+

120576

ℎ119894+1

+

119887119894

ℏ119894

ℎ119894+1

ge minus(

120576

ℎ119894

minus

120576

ℎ119894+1

) +

119887119894

2

+

119887119894

ℎ119894

2ℎ119894+1

ge 119878119894

ge 120575 gt 0

(33)

For 119894 = 119869 condition (c) is verified as follows

119897119869

V119869minus1

+119889119869

V119869

+ 119869

V119869+1

=

ℏ119869

119867

119888119869

V119869

+119897119869

119867 minus 119869

(119867 +

120582120588

1 + 120588

)

ge minus119869

120582120588

1 + 120588

+ (119897119869

minus 119869

)119867

= (

120576

1198672

+

119887119869

ℏ119869

1198672

)

120582120588

1 + 120588

minus

120576

ℎ119869

+

120576

119867

+

119887119869

ℏ119869

119867

= (

2120576 + 119887119869

(ℎ119869

+ 119867)

21198672

)

120582120573119867

120576 + 120573119867

minus

120576

ℎ119869

+

119887119869

2

ge

120582120573

2119867

minus

120576

ℎ119869

+

120573

2

ge

120573

2

(34)

where in the last step invoking (8) we choose a constant 120582 sothat

120582120573

2119867

minus

120576

ℎ119869

ge

120582120573

2119867

minus

120576

119886120576 (120601 (119905119869

) minus 120601 (119905119869minus1

))

ge

120582120573

2119867

minus

1

119886Δ119869

ge

120582120573

4119873minus1

minus

1

119886119862119873minus1

= 119873(

120582120573

4

minus

1

119886119862

) ge 0

(35)

To guarantee this 120582 is chosen so that

120582120573

4

ge

1

119886119862

(36)

Finally when 119869 + 1 le 119894 le 119873 minus 1 we have

119897119894

V119894minus1

+119889119894

V119894

+ 119894

V119894+1

= 119888119894

V119894

+119897119894

119867 minus 119894

119867

+

119897119894

1 + 120588119869

[

120582

(1 + 120588)119894minus1minus119869

minus

120582

(1 + 120588)119894minus119869

]

+

119894

1 + 120588119869

[

120582

(1 + 120588)119894+1minus119869

minus

120582

(1 + 120588)119894minus119869

]

ge 119887119894

+

120588 (1 + 120588)119897119894

minus 120588119894

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

ge 120573 +

(119897119894

minus 119894

+119897119894

120588) 120588

(1 + 120588119869

) (1 + 120588)119894+1minus119869

120582

= 120573 + (

119887119894

119867

minus

120573

119867

)

120582120588 (1 + 120588)119869minus119894minus1

1 + 120588119869

ge 120573

(37)

By examining the entries of 119873

we easily see that themaximum absolute row sum of

119873

is achieved in the layerregion Therefore

10038171003817100381710038171003817119873

10038171003817100381710038171003817le 119862

119873

Δ120601

(38)

A combination of Lemma 7 and (38) results in thefollowing

Theorem 8 The condition number of matrix 119873

satisfies thefollowing 120576-uniform estimate

120581 (119873

) le 119862

119873

Δ120601

(39)

Corollary 9 For the meshes given in Table 1 one has thefollowing estimates

120581 (119873

) le

119862

1198732

119871

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

1198621198732

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(40)

In Tables 4 and 5 we present the condition numbers ofthe preconditioned linear systems (21) for various values of120576 and 119873 for the BS- and VS-meshes We can observe thatthe condition numbers of the preconditioned systems areindependent of 120576 and that the bound in Corollary 9 is sharp

5 120576-Uniform Convergence

For the proof of 120576-uniform convergence we need to decom-pose the solution 119906 into the smooth and boundary-layer

Advances in Numerical Analysis 7

Table 4 The condition number of matrix 119873

on the BS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 658119890 + 02 268119890 + 03 108119890 + 04 434119890 + 04 174119890 + 05 695119890 + 05

1119890 minus 3 689119890 + 02 278119890 + 03 111119890 + 04 440119890 + 04 175119890 + 05 700119890 + 05

1119890 minus 4 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 702119890 + 05

1119890 minus 5 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 6 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 7 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 8 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

Table 5 The condition number of matrix 119873

on the VS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 748119890 + 02 313119890 + 03 127119890 + 04 510119890 + 04 204119890 + 05 811119890 + 05

1119890 minus 3 780119890 + 02 323119890 + 03 130119890 + 04 520119890 + 04 207119890 + 05 820119890 + 05

1119890 minus 4 785119890 + 02 325119890 + 03 131119890 + 04 522119890 + 04 208119890 + 05 824119890 + 05

1119890 minus 5 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 6 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 7 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 8 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

parts There exist several decompositions of this kind but weuse the version presented in [2 Theorem 348]

119906 (119909) = 119904 (119909) + 119910 (119909) 119909 isin 119868 (41)10038161003816100381610038161003816119904(119896)

(119909)

10038161003816100381610038161003816le 119862 (1 + 120576

2minus119896

)

10038161003816100381610038161003816119910(119896)

(119909)

10038161003816100381610038161003816le 119862120576minus119896

119890minus120573119909120576

119909 isin 119868 119896 = 0 1 2 3

(42)

The details related to the construction of the function 119904 arenot important here they can be found in [2] As for 119910 wenote that it solves the problem

L119910 (119909) = 0 119909 isin (0 1)

119910 (0) = minus119904 (0)

119910 (1) = 0

(43)

We define the consistency error of the finite-differenceoperatorL119873 as

120591119894

= 120591119894

[119906] fl L119873

119906119894

minus 119891119894

119894 = 1 2 119873 minus 1 (44)

It holds true that

120591119894

= L119873

119906119894

minus (L119906)119894

= [119860119873

(119906119873

minus 119880119873

)]119894

(45)

Taylorrsquos expansion gives

1003816100381610038161003816120591119894

[119906]1003816100381610038161003816le 119862ℎ119894+1

(120576

10038171003817100381710038171003817119906101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711990610158401015840

10038171003817100381710038171003817119894) (46)

where 119892119894

fl max119909119894minus1le119909le119909119894+1

|119892(119909)| for any 119862(119868)-function 119892When the diagonal matrix 119872 is multiplied to linear system(14) the consistency error of the multiplied system is

119894

[119906] =

ℏ119894

119867

120591119894

[119906] 1 le 119894 le 119869

120591119894

[119906] 119869 + 1 le 119894 le 119873 minus 1

(47)

Lemma 10 Let 119886120573 ge 2Then the following estimate holds true

1003816100381610038161003816119894

[119906]1003816100381610038161003816le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(48)

Proof We use decomposition (41) to get

119894

[119906] = 119894

[119904] + 119894

[119910] (49)

Then (46) and the derivative-estimates of 119904 given in (42)yield immediately that

1003816100381610038161003816119894

[119904]1003816100381610038161003816le 119862119873

minus1 (50)

and therefore the following remains to be proved

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

119894 = 119869 119873 minus 1

(51)

Note that from condition (7) we easily get

ℎ119894

le 119862120576 119894 = 1 119869

ℏ119894

le 119862120576 119894 = 1 119869 minus 1

(52)

This is the key property of S-type meshes as well as the mainingredient of the following proof

8 Advances in Numerical Analysis

Table 6 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 550119890 + 00 293119890 + 00 151119890 + 00 769119890 minus 01 388119890 minus 01 195119890 minus 01

1119890 minus 3 550119890 + 01 293119890 + 01 151119890 + 01 769119890 + 00 388119890 + 00 195119890 + 00

1119890 minus 4 550119890 + 02 293119890 + 02 151119890 + 02 769119890 + 01 388119890 + 01 195119890 + 01

1119890 minus 5 550119890 + 03 293119890 + 03 151119890 + 03 769119890 + 02 388119890 + 02 195119890 + 02

1119890 minus 6 550119890 + 04 293119890 + 04 151119890 + 04 769119890 + 03 388119890 + 03 195119890 + 03

1119890 minus 7 550119890 + 05 293119890 + 05 151119890 + 05 769119890 + 04 388119890 + 04 195119890 + 04

1119890 minus 8 550119890 + 06 293119890 + 06 151119890 + 06 769119890 + 05 388119890 + 05 195119890 + 05

For 1 le 119894 le 119869 minus 1 (46) estimates (10) and (11) as well asthe derivative-estimates of 119910 in (42) imply

1003816100381610038161003816119894

(119910)1003816100381610038161003816le 119862

ℏ119894

119867

ℎ119894+1

(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894)

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902119909119894+1(119886120576)

119890minus120573119909119894minus1120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890(2119909119894+1minus119886120573119909119894minus1)119886120576

= 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902(119909119894+1minus119909119894minus1)119886120576

119890minus(119886120573minus2)119909119894minus1119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890ℏ119894119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

(53)

where we have used (52) and 119886120573 ge 2 to get

119890ℏ119894(119886120576)

le 119862

119890minus(119886120573minus2)119909119894minus1(119886120576)

le 119862

(54)

When 119869 + 2 le 119894 le 119873 minus 1 (42) and (46) give

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119867(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894) le 119862

119867

1205762

119890minus120573119909119894minus1120576

le 119862

119867

1205762

119890minus120573(120590+119867)120576

le 119862119873

1198672

1205762

119890minus120573119867120576

119890minus120573120590120576

(55)

Estimate (51) follows from the fact that (119867120576minus1

)2

119890minus120573119867120576

le 119862

and because the definition of 120590 implies that

119890minus120573120590120576

= 119890minus120573119886119871

= 119873minus120573119886

le 119873minus2

(56)

We finally prove (51) when 119894 = 119869 119869 + 1 in which case

1003816100381610038161003816119869

[119910]1003816100381610038161003816=

ℏ119869

119867

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816

1003816100381610038161003816119869+1

[119910]1003816100381610038161003816=

1003816100381610038161003816120591119869+1

[119910]1003816100381610038161003816

(57)

We now use the fact that L119910 = 0 and work with a differentform of the consistency error (this is a well-known techniquecf [18 Lemma 5])

1003816100381610038161003816119894

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816le 119877119894

+ 119876119894

+ 119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816 (58)

where

119877119894

= 120576

1003816100381610038161003816100381611986310158401015840

119910119894

10038161003816100381610038161003816

119876119894

= 119887119894

100381610038161003816100381610038161198631015840

119910119894

10038161003816100381610038161003816

(59)

We immediately have that

119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816le 119862119890minus120573119909119894120576

le 119862119890minus120573120590120576

= 119862119890minus120573119886119871

= 119862119873minus119886120573

le 119862119873minus2

(60)

As for 119877119894

it holds true that

119877119894

le ℏminus1

119894

120576 sdot 2

100381710038171003817100381710038171199101015840

10038171003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873119890minus120573120590120576

119890120573ℎ119869120576

le 119862119873minus1

(61)

by (52) Analogously

119876119894

le 119862119867minus1

10038171003817100381710038171199101003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873minus1

(62)

This completes the proof

In general the consistency error 120591119894

of the layer componentof the solution is not bounded pointwise uniformly in 120576When the above proof technique is applied to 120591

119894

[119910] one canonly get

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816

le

119862120576minus1

119873minus1

119871 for the S-mesh

119862120576minus1

119873minus1

for the BS- and VS-meshes

(63)

The proposed preconditioner simplifies the analysis of theconsistency error by introducing an extra 120576 factor needed forthe layer component Hence the approach used in the proofof Lemma 10 is natural and straightforward because it onlyuses the classical Taylor expansion estimate for the truncationerror

To illustrate (63) in Tables 6 and 7 we present the resultsof some numerical experiments on the BS- and VS-meshesrespectivelyWe refer the reader to [14] for the correspondingnumerical results on the S-mesh

By contrast with preconditioner (20) the preconditionedconsistency error converges uniformly in 120576 as shown in Tables8 and 9

In conclusion the preconditioning allows us to usePrinciple 1 and combine Lemmas 7 and 10 to obtain thefollowing main convergence result

Advances in Numerical Analysis 9

Table 7 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 467119890 + 00 247119890 + 00 128119890 + 00 659119890 minus 01 336119890 minus 01 170119890 minus 01

1119890 minus 3 467119890 + 01 247119890 + 01 128119890 + 01 659119890 + 00 336119890 + 00 170119890 + 00

1119890 minus 4 467119890 + 02 247119890 + 02 128119890 + 02 659119890 + 01 336119890 + 01 170119890 + 01

1119890 minus 5 467119890 + 03 247119890 + 03 128119890 + 03 659119890 + 02 336119890 + 02 170119890 + 02

1119890 minus 6 467119890 + 04 247119890 + 04 128119890 + 04 659119890 + 03 336119890 + 03 170119890 + 03

1119890 minus 7 467119890 + 05 247119890 + 05 128119890 + 05 659119890 + 04 336119890 + 04 170119890 + 04

1119890 minus 8 467119890 + 06 247119890 + 06 128119890 + 06 659119890 + 05 336119890 + 05 170119890 + 05

Table 8 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 122119890 minus 01 650119890 minus 02 338119890 minus 02 174119890 minus 02 887119890 minus 03 452119890 minus 03

1119890 minus 3 114119890 minus 01 601119890 minus 02 308119890 minus 02 156119890 minus 02 786119890 minus 03 395119890 minus 03

1119890 minus 4 114119890 minus 01 597119890 minus 02 306119890 minus 02 155119890 minus 02 778119890 minus 03 390119890 minus 03

1119890 minus 5 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 390119890 minus 03

1119890 minus 6 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 7 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 8 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

Table 9 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 160119890 minus 01 912119890 minus 02 502119890 minus 02 270119890 minus 02 143119890 minus 02 747119890 minus 03

1119890 minus 3 150119890 minus 01 844119890 minus 02 458119890 minus 02 242119890 minus 02 126119890 minus 02 653119890 minus 03

1119890 minus 4 149119890 minus 01 837119890 minus 02 454119890 minus 02 240119890 minus 02 125119890 minus 02 645119890 minus 03

1119890 minus 5 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 6 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 7 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 8 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

Theorem 11 The error of the upwind finite-difference dis-cretization on S-type meshes satisfies

10038161003816100381610038161003816119906119873

119894

minus 119880119873

119894

10038161003816100381610038161003816

le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(64)

Corollary 12 For the meshes given in Table 1 one has thefollowing estimates

10038171003817100381710038171003817119906119873

minus 119880119873

10038171003817100381710038171003817

le

119862119873minus1

1198712

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

119862119873minus1

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(65)

Remark 13 The barrier-function technique can be used toobtain similar results for S-type meshes (see [16]) but itrequires another assumption on the mesh-generating func-tion 120601 (cf [16 Theorem 1]) However whether the barrier-function approach can be extended to any Bakhvalov-type

mesh is still an open question [2 Remark 421] By contrastas proved in [19] the preconditioner defined in (20) worksfine for Vulanovicrsquos modification of the Bakhvalov mesh [10]The paper [10] is where the original Bakhvalov mesh [20]is generalized (the main difference between Bakhvalov-typemeshes and S-type meshes is that the former have a smoothtransition from the fine part inside the layer to the coarseuniform part outside the layer the mesh-generating function120601 belongs to 119862

1

(119868))

6 Conclusion

In this paper we present a proof of pointwise uniformconvergence for singularly perturbed convection-diffusionproblem (1) discretized on a general class of Shishkin-typemeshesThe proof is based on preconditioning of the discretesystem which allows us to use the standard ldquoconsistency+ stabilityrdquo principle of convergence This is a conceptuallysimple approach which does not require the use of hybridstability inequalities and discrete Greenrsquos functions In gen-eral new proof techniques are of interest because althougha lot has been achieved in the field of singularly perturbed

10 Advances in Numerical Analysis

problems over the last few decades there are still importantquestions remaining open [21] One of them Question6 is related to the work presented here The question isabout two-dimensional convection-diffusion problems and itpoints out that whereas 120576-uniform pointwise convergence forthe upwind finite-difference discretization on S-type meshesis well-established no similar result for B-type meshes isknownAsmentioned inRemark 13 we already know that thepreconditioning approach works well for B-mesh and one-dimensional convection-diffusion problems The next step isto consider two-dimensional problems

Appendix

Proof of Lemma 6 The proof is different from the proof ofLemma 5because theBS-meshdoes not satisfy condition (iv)Because of (ii) it is sufficient to show that the numerator in(25) is nonnegative for radic2 minus 1 le 119876 lt 1 independently of 120576and 119873 Let 119901 fl 1 minus 119873

minus1 By direct computations we get that

Δ119894

= minus ln(1 minus

119905119894

119901

119876

) + ln(1 minus

119905119894minus1

119901

119876

)

= ln(1 +

1

119869119901 minus 119894

) = ln (1 + 119902)

(A1)

where we denote 119902 fl (119869119901minus119894)minus1 Note that 0 le 119902 lt 1(119876+1)

Analogously

Δ119894+1

= ln(

1

1 minus 119902

) (A2)

Then

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

= Δ119894

(Δ119894+1

+ Δ119894

+ 1) minus Δ119894+1

= ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902)

(A3)

We set

119892 (119902) fl ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902) (A4)

Sinceradic2minus1 le 119876 lt 1 we have 0 le 119902 lt 1radic2 Rearrange 119892(119902)

as

119892 (119902) = [1 + ln(1 +

2119902

1 minus 119902

)] ln (1 + 119902)

+ ln (1 minus 119902)

(A5)

For 119902 isin [0 1) it follows that119902

1 + 1199022

le ln (1 + 119902) le 119902 (A6)

Hence

ln(1 +

2119902

1 minus 119902

) ge 2119902 (A7)

So we are done if we can show that

119892 (119902) ge [1 + 2119902] ln (1 + 119902) + ln (1 minus 119902) ge 0

0 le 119902 lt

1

radic2

(A8)

or

(1 minus 119902) (1 + 119902)2119902+1

= (1 minus 1199022

) (1 + 119902)2119902

ge 1 (A9)

First if 119902 isin [12 1radic2] we have by (1 + 119902)120574

ge 1 + 120574119902 for120574 ge 1

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

) (1 + 21199022

) ge 1 (A10)

Second we have that ln(1+119902) is a concave-down function on[0 12] hence on [0 12]

ln (1 + 119902) ge 2 ln(

3

2

) 119902 ge

4119902

5

(A11)

Then multiplying both sides by 119902 and exponentiating we getthat

(1 + 119902)119902

ge exp(

41199022

5

) ge 1 +

41199022

5

(A12)

Therefore

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

)(1 +

41199022

5

)

2

ge 1 (A13)

which holds true for 119902 isin [0 12]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P A Farrell A F Hegarty J J H Miller E OrsquoRiordan andG I Shishkin Robust Computational Techniques for BoundaryLayers Chapman amp Hall CRC Press Boca Raton Fla USA2000

[2] T Linszlig Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems vol 1985 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[3] J J HMiller E OrsquoRiordan and G I Shishkin Fitted NumericalMethods for Singular Perturbation Problems World ScientificSingapore 1996

[4] H-G Roos M Stynes and L Tobiska Numerical Methodsfor Singularly Perturbed Differential Equations Springer BerlinGermany 2nd edition 2008

[5] A Malvandi and D D Ganji ldquoMixed convection of aluminandashwater nanofluid inside a concentric annulus consideringnanoparticle migrationrdquo Particuology vol 24 pp 113ndash122 2016

[6] R B Kellogg and A Tsan ldquoAnalysis of some difference approx-imations for a singular perturbation problem without turningpointsrdquoMathematics of Computation vol 32 no 144 pp 1025ndash1039 1978

Advances in Numerical Analysis 11

[7] J Lorenz ldquoStability and monotonicity properties of stiff quasi-linear boundary value problemsrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta Serija zaMatematiku vol 12 pp 151ndash175 1982

[8] E Bohl Finite Modelle Gewohnlicher Randwertaufgaben Teub-ner Stuttgart Germany 1981

[9] E C Gartland Jr ldquoGraded-mesh difference schemes for sin-gularly perturbed two-point boundary value problemsrdquo Math-ematics of Computation vol 51 no 184 pp 631ndash657 1988

[10] R Vulanovic ldquoOn a numerical solution of a type of singularlyperturbed boundary value problem by using a special dis-cretization meshrdquo Univerzitet u Novom Sadu Zbornik RadovaPrirodno- Matematichkog Fakulteta Serija za Matematiku vol13 pp 187ndash201 1983

[11] C Grossmann H-G Roos and M Stynes Numerical Treat-ment of Partial Differential Equations Springer Berlin Ger-many 2007

[12] V B Andreev and I A Savin ldquoThe uniform convergence withrespect to a small parameter of A A Samarskiirsquos monotonescheme and its modificationrdquo Computational Mathematics andMathematical Physics vol 35 pp 581ndash591 1995

[13] T LinszligH-G Roos andRVulanovic ldquoUniformpointwise con-vergence on Shishkin-type meshes for quasi-linear convection-diffusion problemsrdquo SIAM Journal on Numerical Analysis vol38 no 3 pp 897ndash912 2000

[14] R Vulanovic and T A Nhan ldquoUniform convergence viapreconditioningrdquo International Journal of Numerical Analysisand Modeling B vol 5 no 4 pp 347ndash356 2014

[15] H-G Roos ldquoA note on the conditioning of upwind schemes onShishkin meshesrdquo IMA Journal of Numerical Analysis vol 16no 4 pp 529ndash538 1996

[16] H-G Roos and T Linszlig ldquoSufficient condition for uniformconvergence on layeradapted gridsrdquo Computing vol 63 no 1pp 27ndash45 1999

[17] R Vulanovic ldquoA higher-order scheme for quasilinear boundaryvalue problems with two small parametersrdquo Computing vol 67no 4 pp 287ndash303 2001

[18] R Vulanovic ldquoA priori meshes for singularly perturbed quasi-linear twominuspoint boundary value problemsrdquo IMA Journal ofNumerical Analysis vol 21 no 1 pp 349ndash366 2001

[19] T A Nhan and R Vulanovic ldquoUniform convergence on aBakhvalov-type mesh using the preconditioning approachTechnical reportrdquo httparxivorgabs150404283

[20] N S Bakhvalov ldquoThe optimization of methods of solvingboundary value problems with a boundary layerrdquo USSR Com-putational Mathematics and Mathematical Physics vol 9 no 4pp 139ndash166 1969

[21] H-G Roos andM Stynes ldquoSome open questions in the numer-ical analysis of singularly perturbed differential equationsrdquoComputational Methods in Applied Mathematics vol 15 no 4pp 531ndash550 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Numerical Analysis 7

Table 4 The condition number of matrix 119873

on the BS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 658119890 + 02 268119890 + 03 108119890 + 04 434119890 + 04 174119890 + 05 695119890 + 05

1119890 minus 3 689119890 + 02 278119890 + 03 111119890 + 04 440119890 + 04 175119890 + 05 700119890 + 05

1119890 minus 4 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 702119890 + 05

1119890 minus 5 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 6 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 7 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

1119890 minus 8 694119890 + 02 280119890 + 03 112119890 + 04 443119890 + 04 176119890 + 05 703119890 + 05

Table 5 The condition number of matrix 119873

on the VS-mesh for problem (19)

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 748119890 + 02 313119890 + 03 127119890 + 04 510119890 + 04 204119890 + 05 811119890 + 05

1119890 minus 3 780119890 + 02 323119890 + 03 130119890 + 04 520119890 + 04 207119890 + 05 820119890 + 05

1119890 minus 4 785119890 + 02 325119890 + 03 131119890 + 04 522119890 + 04 208119890 + 05 824119890 + 05

1119890 minus 5 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 6 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 7 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

1119890 minus 8 785119890 + 02 325119890 + 03 131119890 + 04 523119890 + 04 208119890 + 05 825119890 + 05

parts There exist several decompositions of this kind but weuse the version presented in [2 Theorem 348]

119906 (119909) = 119904 (119909) + 119910 (119909) 119909 isin 119868 (41)10038161003816100381610038161003816119904(119896)

(119909)

10038161003816100381610038161003816le 119862 (1 + 120576

2minus119896

)

10038161003816100381610038161003816119910(119896)

(119909)

10038161003816100381610038161003816le 119862120576minus119896

119890minus120573119909120576

119909 isin 119868 119896 = 0 1 2 3

(42)

The details related to the construction of the function 119904 arenot important here they can be found in [2] As for 119910 wenote that it solves the problem

L119910 (119909) = 0 119909 isin (0 1)

119910 (0) = minus119904 (0)

119910 (1) = 0

(43)

We define the consistency error of the finite-differenceoperatorL119873 as

120591119894

= 120591119894

[119906] fl L119873

119906119894

minus 119891119894

119894 = 1 2 119873 minus 1 (44)

It holds true that

120591119894

= L119873

119906119894

minus (L119906)119894

= [119860119873

(119906119873

minus 119880119873

)]119894

(45)

Taylorrsquos expansion gives

1003816100381610038161003816120591119894

[119906]1003816100381610038161003816le 119862ℎ119894+1

(120576

10038171003817100381710038171003817119906101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711990610158401015840

10038171003817100381710038171003817119894) (46)

where 119892119894

fl max119909119894minus1le119909le119909119894+1

|119892(119909)| for any 119862(119868)-function 119892When the diagonal matrix 119872 is multiplied to linear system(14) the consistency error of the multiplied system is

119894

[119906] =

ℏ119894

119867

120591119894

[119906] 1 le 119894 le 119869

120591119894

[119906] 119869 + 1 le 119894 le 119873 minus 1

(47)

Lemma 10 Let 119886120573 ge 2Then the following estimate holds true

1003816100381610038161003816119894

[119906]1003816100381610038161003816le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(48)

Proof We use decomposition (41) to get

119894

[119906] = 119894

[119904] + 119894

[119910] (49)

Then (46) and the derivative-estimates of 119904 given in (42)yield immediately that

1003816100381610038161003816119894

[119904]1003816100381610038161003816le 119862119873

minus1 (50)

and therefore the following remains to be proved

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119873

minus1

119894 = 119869 119873 minus 1

(51)

Note that from condition (7) we easily get

ℎ119894

le 119862120576 119894 = 1 119869

ℏ119894

le 119862120576 119894 = 1 119869 minus 1

(52)

This is the key property of S-type meshes as well as the mainingredient of the following proof

8 Advances in Numerical Analysis

Table 6 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 550119890 + 00 293119890 + 00 151119890 + 00 769119890 minus 01 388119890 minus 01 195119890 minus 01

1119890 minus 3 550119890 + 01 293119890 + 01 151119890 + 01 769119890 + 00 388119890 + 00 195119890 + 00

1119890 minus 4 550119890 + 02 293119890 + 02 151119890 + 02 769119890 + 01 388119890 + 01 195119890 + 01

1119890 minus 5 550119890 + 03 293119890 + 03 151119890 + 03 769119890 + 02 388119890 + 02 195119890 + 02

1119890 minus 6 550119890 + 04 293119890 + 04 151119890 + 04 769119890 + 03 388119890 + 03 195119890 + 03

1119890 minus 7 550119890 + 05 293119890 + 05 151119890 + 05 769119890 + 04 388119890 + 04 195119890 + 04

1119890 minus 8 550119890 + 06 293119890 + 06 151119890 + 06 769119890 + 05 388119890 + 05 195119890 + 05

For 1 le 119894 le 119869 minus 1 (46) estimates (10) and (11) as well asthe derivative-estimates of 119910 in (42) imply

1003816100381610038161003816119894

(119910)1003816100381610038161003816le 119862

ℏ119894

119867

ℎ119894+1

(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894)

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902119909119894+1(119886120576)

119890minus120573119909119894minus1120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890(2119909119894+1minus119886120573119909119894minus1)119886120576

= 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902(119909119894+1minus119909119894minus1)119886120576

119890minus(119886120573minus2)119909119894minus1119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890ℏ119894119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

(53)

where we have used (52) and 119886120573 ge 2 to get

119890ℏ119894(119886120576)

le 119862

119890minus(119886120573minus2)119909119894minus1(119886120576)

le 119862

(54)

When 119869 + 2 le 119894 le 119873 minus 1 (42) and (46) give

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119867(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894) le 119862

119867

1205762

119890minus120573119909119894minus1120576

le 119862

119867

1205762

119890minus120573(120590+119867)120576

le 119862119873

1198672

1205762

119890minus120573119867120576

119890minus120573120590120576

(55)

Estimate (51) follows from the fact that (119867120576minus1

)2

119890minus120573119867120576

le 119862

and because the definition of 120590 implies that

119890minus120573120590120576

= 119890minus120573119886119871

= 119873minus120573119886

le 119873minus2

(56)

We finally prove (51) when 119894 = 119869 119869 + 1 in which case

1003816100381610038161003816119869

[119910]1003816100381610038161003816=

ℏ119869

119867

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816

1003816100381610038161003816119869+1

[119910]1003816100381610038161003816=

1003816100381610038161003816120591119869+1

[119910]1003816100381610038161003816

(57)

We now use the fact that L119910 = 0 and work with a differentform of the consistency error (this is a well-known techniquecf [18 Lemma 5])

1003816100381610038161003816119894

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816le 119877119894

+ 119876119894

+ 119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816 (58)

where

119877119894

= 120576

1003816100381610038161003816100381611986310158401015840

119910119894

10038161003816100381610038161003816

119876119894

= 119887119894

100381610038161003816100381610038161198631015840

119910119894

10038161003816100381610038161003816

(59)

We immediately have that

119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816le 119862119890minus120573119909119894120576

le 119862119890minus120573120590120576

= 119862119890minus120573119886119871

= 119862119873minus119886120573

le 119862119873minus2

(60)

As for 119877119894

it holds true that

119877119894

le ℏminus1

119894

120576 sdot 2

100381710038171003817100381710038171199101015840

10038171003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873119890minus120573120590120576

119890120573ℎ119869120576

le 119862119873minus1

(61)

by (52) Analogously

119876119894

le 119862119867minus1

10038171003817100381710038171199101003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873minus1

(62)

This completes the proof

In general the consistency error 120591119894

of the layer componentof the solution is not bounded pointwise uniformly in 120576When the above proof technique is applied to 120591

119894

[119910] one canonly get

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816

le

119862120576minus1

119873minus1

119871 for the S-mesh

119862120576minus1

119873minus1

for the BS- and VS-meshes

(63)

The proposed preconditioner simplifies the analysis of theconsistency error by introducing an extra 120576 factor needed forthe layer component Hence the approach used in the proofof Lemma 10 is natural and straightforward because it onlyuses the classical Taylor expansion estimate for the truncationerror

To illustrate (63) in Tables 6 and 7 we present the resultsof some numerical experiments on the BS- and VS-meshesrespectivelyWe refer the reader to [14] for the correspondingnumerical results on the S-mesh

By contrast with preconditioner (20) the preconditionedconsistency error converges uniformly in 120576 as shown in Tables8 and 9

In conclusion the preconditioning allows us to usePrinciple 1 and combine Lemmas 7 and 10 to obtain thefollowing main convergence result

Advances in Numerical Analysis 9

Table 7 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 467119890 + 00 247119890 + 00 128119890 + 00 659119890 minus 01 336119890 minus 01 170119890 minus 01

1119890 minus 3 467119890 + 01 247119890 + 01 128119890 + 01 659119890 + 00 336119890 + 00 170119890 + 00

1119890 minus 4 467119890 + 02 247119890 + 02 128119890 + 02 659119890 + 01 336119890 + 01 170119890 + 01

1119890 minus 5 467119890 + 03 247119890 + 03 128119890 + 03 659119890 + 02 336119890 + 02 170119890 + 02

1119890 minus 6 467119890 + 04 247119890 + 04 128119890 + 04 659119890 + 03 336119890 + 03 170119890 + 03

1119890 minus 7 467119890 + 05 247119890 + 05 128119890 + 05 659119890 + 04 336119890 + 04 170119890 + 04

1119890 minus 8 467119890 + 06 247119890 + 06 128119890 + 06 659119890 + 05 336119890 + 05 170119890 + 05

Table 8 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 122119890 minus 01 650119890 minus 02 338119890 minus 02 174119890 minus 02 887119890 minus 03 452119890 minus 03

1119890 minus 3 114119890 minus 01 601119890 minus 02 308119890 minus 02 156119890 minus 02 786119890 minus 03 395119890 minus 03

1119890 minus 4 114119890 minus 01 597119890 minus 02 306119890 minus 02 155119890 minus 02 778119890 minus 03 390119890 minus 03

1119890 minus 5 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 390119890 minus 03

1119890 minus 6 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 7 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 8 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

Table 9 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 160119890 minus 01 912119890 minus 02 502119890 minus 02 270119890 minus 02 143119890 minus 02 747119890 minus 03

1119890 minus 3 150119890 minus 01 844119890 minus 02 458119890 minus 02 242119890 minus 02 126119890 minus 02 653119890 minus 03

1119890 minus 4 149119890 minus 01 837119890 minus 02 454119890 minus 02 240119890 minus 02 125119890 minus 02 645119890 minus 03

1119890 minus 5 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 6 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 7 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 8 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

Theorem 11 The error of the upwind finite-difference dis-cretization on S-type meshes satisfies

10038161003816100381610038161003816119906119873

119894

minus 119880119873

119894

10038161003816100381610038161003816

le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(64)

Corollary 12 For the meshes given in Table 1 one has thefollowing estimates

10038171003817100381710038171003817119906119873

minus 119880119873

10038171003817100381710038171003817

le

119862119873minus1

1198712

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

119862119873minus1

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(65)

Remark 13 The barrier-function technique can be used toobtain similar results for S-type meshes (see [16]) but itrequires another assumption on the mesh-generating func-tion 120601 (cf [16 Theorem 1]) However whether the barrier-function approach can be extended to any Bakhvalov-type

mesh is still an open question [2 Remark 421] By contrastas proved in [19] the preconditioner defined in (20) worksfine for Vulanovicrsquos modification of the Bakhvalov mesh [10]The paper [10] is where the original Bakhvalov mesh [20]is generalized (the main difference between Bakhvalov-typemeshes and S-type meshes is that the former have a smoothtransition from the fine part inside the layer to the coarseuniform part outside the layer the mesh-generating function120601 belongs to 119862

1

(119868))

6 Conclusion

In this paper we present a proof of pointwise uniformconvergence for singularly perturbed convection-diffusionproblem (1) discretized on a general class of Shishkin-typemeshesThe proof is based on preconditioning of the discretesystem which allows us to use the standard ldquoconsistency+ stabilityrdquo principle of convergence This is a conceptuallysimple approach which does not require the use of hybridstability inequalities and discrete Greenrsquos functions In gen-eral new proof techniques are of interest because althougha lot has been achieved in the field of singularly perturbed

10 Advances in Numerical Analysis

problems over the last few decades there are still importantquestions remaining open [21] One of them Question6 is related to the work presented here The question isabout two-dimensional convection-diffusion problems and itpoints out that whereas 120576-uniform pointwise convergence forthe upwind finite-difference discretization on S-type meshesis well-established no similar result for B-type meshes isknownAsmentioned inRemark 13 we already know that thepreconditioning approach works well for B-mesh and one-dimensional convection-diffusion problems The next step isto consider two-dimensional problems

Appendix

Proof of Lemma 6 The proof is different from the proof ofLemma 5because theBS-meshdoes not satisfy condition (iv)Because of (ii) it is sufficient to show that the numerator in(25) is nonnegative for radic2 minus 1 le 119876 lt 1 independently of 120576and 119873 Let 119901 fl 1 minus 119873

minus1 By direct computations we get that

Δ119894

= minus ln(1 minus

119905119894

119901

119876

) + ln(1 minus

119905119894minus1

119901

119876

)

= ln(1 +

1

119869119901 minus 119894

) = ln (1 + 119902)

(A1)

where we denote 119902 fl (119869119901minus119894)minus1 Note that 0 le 119902 lt 1(119876+1)

Analogously

Δ119894+1

= ln(

1

1 minus 119902

) (A2)

Then

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

= Δ119894

(Δ119894+1

+ Δ119894

+ 1) minus Δ119894+1

= ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902)

(A3)

We set

119892 (119902) fl ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902) (A4)

Sinceradic2minus1 le 119876 lt 1 we have 0 le 119902 lt 1radic2 Rearrange 119892(119902)

as

119892 (119902) = [1 + ln(1 +

2119902

1 minus 119902

)] ln (1 + 119902)

+ ln (1 minus 119902)

(A5)

For 119902 isin [0 1) it follows that119902

1 + 1199022

le ln (1 + 119902) le 119902 (A6)

Hence

ln(1 +

2119902

1 minus 119902

) ge 2119902 (A7)

So we are done if we can show that

119892 (119902) ge [1 + 2119902] ln (1 + 119902) + ln (1 minus 119902) ge 0

0 le 119902 lt

1

radic2

(A8)

or

(1 minus 119902) (1 + 119902)2119902+1

= (1 minus 1199022

) (1 + 119902)2119902

ge 1 (A9)

First if 119902 isin [12 1radic2] we have by (1 + 119902)120574

ge 1 + 120574119902 for120574 ge 1

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

) (1 + 21199022

) ge 1 (A10)

Second we have that ln(1+119902) is a concave-down function on[0 12] hence on [0 12]

ln (1 + 119902) ge 2 ln(

3

2

) 119902 ge

4119902

5

(A11)

Then multiplying both sides by 119902 and exponentiating we getthat

(1 + 119902)119902

ge exp(

41199022

5

) ge 1 +

41199022

5

(A12)

Therefore

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

)(1 +

41199022

5

)

2

ge 1 (A13)

which holds true for 119902 isin [0 12]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P A Farrell A F Hegarty J J H Miller E OrsquoRiordan andG I Shishkin Robust Computational Techniques for BoundaryLayers Chapman amp Hall CRC Press Boca Raton Fla USA2000

[2] T Linszlig Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems vol 1985 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[3] J J HMiller E OrsquoRiordan and G I Shishkin Fitted NumericalMethods for Singular Perturbation Problems World ScientificSingapore 1996

[4] H-G Roos M Stynes and L Tobiska Numerical Methodsfor Singularly Perturbed Differential Equations Springer BerlinGermany 2nd edition 2008

[5] A Malvandi and D D Ganji ldquoMixed convection of aluminandashwater nanofluid inside a concentric annulus consideringnanoparticle migrationrdquo Particuology vol 24 pp 113ndash122 2016

[6] R B Kellogg and A Tsan ldquoAnalysis of some difference approx-imations for a singular perturbation problem without turningpointsrdquoMathematics of Computation vol 32 no 144 pp 1025ndash1039 1978

Advances in Numerical Analysis 11

[7] J Lorenz ldquoStability and monotonicity properties of stiff quasi-linear boundary value problemsrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta Serija zaMatematiku vol 12 pp 151ndash175 1982

[8] E Bohl Finite Modelle Gewohnlicher Randwertaufgaben Teub-ner Stuttgart Germany 1981

[9] E C Gartland Jr ldquoGraded-mesh difference schemes for sin-gularly perturbed two-point boundary value problemsrdquo Math-ematics of Computation vol 51 no 184 pp 631ndash657 1988

[10] R Vulanovic ldquoOn a numerical solution of a type of singularlyperturbed boundary value problem by using a special dis-cretization meshrdquo Univerzitet u Novom Sadu Zbornik RadovaPrirodno- Matematichkog Fakulteta Serija za Matematiku vol13 pp 187ndash201 1983

[11] C Grossmann H-G Roos and M Stynes Numerical Treat-ment of Partial Differential Equations Springer Berlin Ger-many 2007

[12] V B Andreev and I A Savin ldquoThe uniform convergence withrespect to a small parameter of A A Samarskiirsquos monotonescheme and its modificationrdquo Computational Mathematics andMathematical Physics vol 35 pp 581ndash591 1995

[13] T LinszligH-G Roos andRVulanovic ldquoUniformpointwise con-vergence on Shishkin-type meshes for quasi-linear convection-diffusion problemsrdquo SIAM Journal on Numerical Analysis vol38 no 3 pp 897ndash912 2000

[14] R Vulanovic and T A Nhan ldquoUniform convergence viapreconditioningrdquo International Journal of Numerical Analysisand Modeling B vol 5 no 4 pp 347ndash356 2014

[15] H-G Roos ldquoA note on the conditioning of upwind schemes onShishkin meshesrdquo IMA Journal of Numerical Analysis vol 16no 4 pp 529ndash538 1996

[16] H-G Roos and T Linszlig ldquoSufficient condition for uniformconvergence on layeradapted gridsrdquo Computing vol 63 no 1pp 27ndash45 1999

[17] R Vulanovic ldquoA higher-order scheme for quasilinear boundaryvalue problems with two small parametersrdquo Computing vol 67no 4 pp 287ndash303 2001

[18] R Vulanovic ldquoA priori meshes for singularly perturbed quasi-linear twominuspoint boundary value problemsrdquo IMA Journal ofNumerical Analysis vol 21 no 1 pp 349ndash366 2001

[19] T A Nhan and R Vulanovic ldquoUniform convergence on aBakhvalov-type mesh using the preconditioning approachTechnical reportrdquo httparxivorgabs150404283

[20] N S Bakhvalov ldquoThe optimization of methods of solvingboundary value problems with a boundary layerrdquo USSR Com-putational Mathematics and Mathematical Physics vol 9 no 4pp 139ndash166 1969

[21] H-G Roos andM Stynes ldquoSome open questions in the numer-ical analysis of singularly perturbed differential equationsrdquoComputational Methods in Applied Mathematics vol 15 no 4pp 531ndash550 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Numerical Analysis

Table 6 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 550119890 + 00 293119890 + 00 151119890 + 00 769119890 minus 01 388119890 minus 01 195119890 minus 01

1119890 minus 3 550119890 + 01 293119890 + 01 151119890 + 01 769119890 + 00 388119890 + 00 195119890 + 00

1119890 minus 4 550119890 + 02 293119890 + 02 151119890 + 02 769119890 + 01 388119890 + 01 195119890 + 01

1119890 minus 5 550119890 + 03 293119890 + 03 151119890 + 03 769119890 + 02 388119890 + 02 195119890 + 02

1119890 minus 6 550119890 + 04 293119890 + 04 151119890 + 04 769119890 + 03 388119890 + 03 195119890 + 03

1119890 minus 7 550119890 + 05 293119890 + 05 151119890 + 05 769119890 + 04 388119890 + 04 195119890 + 04

1119890 minus 8 550119890 + 06 293119890 + 06 151119890 + 06 769119890 + 05 388119890 + 05 195119890 + 05

For 1 le 119894 le 119869 minus 1 (46) estimates (10) and (11) as well asthe derivative-estimates of 119910 in (42) imply

1003816100381610038161003816119894

(119910)1003816100381610038161003816le 119862

ℏ119894

119867

ℎ119894+1

(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894)

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902119909119894+1(119886120576)

119890minus120573119909119894minus1120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890(2119909119894+1minus119886120573119909119894minus1)119886120576

= 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

1198902(119909119894+1minus119909119894minus1)119886120576

119890minus(119886120573minus2)119909119894minus1119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119890ℏ119894119886120576

le 119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

(53)

where we have used (52) and 119886120573 ge 2 to get

119890ℏ119894(119886120576)

le 119862

119890minus(119886120573minus2)119909119894minus1(119886120576)

le 119862

(54)

When 119869 + 2 le 119894 le 119873 minus 1 (42) and (46) give

1003816100381610038161003816119894

[119910]1003816100381610038161003816le 119862119867(120576

10038171003817100381710038171003817119910101584010158401015840

10038171003817100381710038171003817119894

+

1003817100381710038171003817100381711991010158401015840

10038171003817100381710038171003817119894) le 119862

119867

1205762

119890minus120573119909119894minus1120576

le 119862

119867

1205762

119890minus120573(120590+119867)120576

le 119862119873

1198672

1205762

119890minus120573119867120576

119890minus120573120590120576

(55)

Estimate (51) follows from the fact that (119867120576minus1

)2

119890minus120573119867120576

le 119862

and because the definition of 120590 implies that

119890minus120573120590120576

= 119890minus120573119886119871

= 119873minus120573119886

le 119873minus2

(56)

We finally prove (51) when 119894 = 119869 119869 + 1 in which case

1003816100381610038161003816119869

[119910]1003816100381610038161003816=

ℏ119869

119867

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119869

[119910]1003816100381610038161003816

1003816100381610038161003816119869+1

[119910]1003816100381610038161003816=

1003816100381610038161003816120591119869+1

[119910]1003816100381610038161003816

(57)

We now use the fact that L119910 = 0 and work with a differentform of the consistency error (this is a well-known techniquecf [18 Lemma 5])

1003816100381610038161003816119894

[119910]1003816100381610038161003816le

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816le 119877119894

+ 119876119894

+ 119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816 (58)

where

119877119894

= 120576

1003816100381610038161003816100381611986310158401015840

119910119894

10038161003816100381610038161003816

119876119894

= 119887119894

100381610038161003816100381610038161198631015840

119910119894

10038161003816100381610038161003816

(59)

We immediately have that

119888119894

1003816100381610038161003816119910119894

1003816100381610038161003816le 119862119890minus120573119909119894120576

le 119862119890minus120573120590120576

= 119862119890minus120573119886119871

= 119862119873minus119886120573

le 119862119873minus2

(60)

As for 119877119894

it holds true that

119877119894

le ℏminus1

119894

120576 sdot 2

100381710038171003817100381710038171199101015840

10038171003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873119890minus120573120590120576

119890120573ℎ119869120576

le 119862119873minus1

(61)

by (52) Analogously

119876119894

le 119862119867minus1

10038171003817100381710038171199101003817100381710038171003817119894

le 119862119873119890minus120573(120590minusℎ119869)120576

le 119862119873minus1

(62)

This completes the proof

In general the consistency error 120591119894

of the layer componentof the solution is not bounded pointwise uniformly in 120576When the above proof technique is applied to 120591

119894

[119910] one canonly get

1003816100381610038161003816120591119894

[119910]1003816100381610038161003816

le

119862120576minus1

119873minus1

119871 for the S-mesh

119862120576minus1

119873minus1

for the BS- and VS-meshes

(63)

The proposed preconditioner simplifies the analysis of theconsistency error by introducing an extra 120576 factor needed forthe layer component Hence the approach used in the proofof Lemma 10 is natural and straightforward because it onlyuses the classical Taylor expansion estimate for the truncationerror

To illustrate (63) in Tables 6 and 7 we present the resultsof some numerical experiments on the BS- and VS-meshesrespectivelyWe refer the reader to [14] for the correspondingnumerical results on the S-mesh

By contrast with preconditioner (20) the preconditionedconsistency error converges uniformly in 120576 as shown in Tables8 and 9

In conclusion the preconditioning allows us to usePrinciple 1 and combine Lemmas 7 and 10 to obtain thefollowing main convergence result

Advances in Numerical Analysis 9

Table 7 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 467119890 + 00 247119890 + 00 128119890 + 00 659119890 minus 01 336119890 minus 01 170119890 minus 01

1119890 minus 3 467119890 + 01 247119890 + 01 128119890 + 01 659119890 + 00 336119890 + 00 170119890 + 00

1119890 minus 4 467119890 + 02 247119890 + 02 128119890 + 02 659119890 + 01 336119890 + 01 170119890 + 01

1119890 minus 5 467119890 + 03 247119890 + 03 128119890 + 03 659119890 + 02 336119890 + 02 170119890 + 02

1119890 minus 6 467119890 + 04 247119890 + 04 128119890 + 04 659119890 + 03 336119890 + 03 170119890 + 03

1119890 minus 7 467119890 + 05 247119890 + 05 128119890 + 05 659119890 + 04 336119890 + 04 170119890 + 04

1119890 minus 8 467119890 + 06 247119890 + 06 128119890 + 06 659119890 + 05 336119890 + 05 170119890 + 05

Table 8 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 122119890 minus 01 650119890 minus 02 338119890 minus 02 174119890 minus 02 887119890 minus 03 452119890 minus 03

1119890 minus 3 114119890 minus 01 601119890 minus 02 308119890 minus 02 156119890 minus 02 786119890 minus 03 395119890 minus 03

1119890 minus 4 114119890 minus 01 597119890 minus 02 306119890 minus 02 155119890 minus 02 778119890 minus 03 390119890 minus 03

1119890 minus 5 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 390119890 minus 03

1119890 minus 6 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 7 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 8 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

Table 9 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 160119890 minus 01 912119890 minus 02 502119890 minus 02 270119890 minus 02 143119890 minus 02 747119890 minus 03

1119890 minus 3 150119890 minus 01 844119890 minus 02 458119890 minus 02 242119890 minus 02 126119890 minus 02 653119890 minus 03

1119890 minus 4 149119890 minus 01 837119890 minus 02 454119890 minus 02 240119890 minus 02 125119890 minus 02 645119890 minus 03

1119890 minus 5 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 6 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 7 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 8 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

Theorem 11 The error of the upwind finite-difference dis-cretization on S-type meshes satisfies

10038161003816100381610038161003816119906119873

119894

minus 119880119873

119894

10038161003816100381610038161003816

le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(64)

Corollary 12 For the meshes given in Table 1 one has thefollowing estimates

10038171003817100381710038171003817119906119873

minus 119880119873

10038171003817100381710038171003817

le

119862119873minus1

1198712

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

119862119873minus1

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(65)

Remark 13 The barrier-function technique can be used toobtain similar results for S-type meshes (see [16]) but itrequires another assumption on the mesh-generating func-tion 120601 (cf [16 Theorem 1]) However whether the barrier-function approach can be extended to any Bakhvalov-type

mesh is still an open question [2 Remark 421] By contrastas proved in [19] the preconditioner defined in (20) worksfine for Vulanovicrsquos modification of the Bakhvalov mesh [10]The paper [10] is where the original Bakhvalov mesh [20]is generalized (the main difference between Bakhvalov-typemeshes and S-type meshes is that the former have a smoothtransition from the fine part inside the layer to the coarseuniform part outside the layer the mesh-generating function120601 belongs to 119862

1

(119868))

6 Conclusion

In this paper we present a proof of pointwise uniformconvergence for singularly perturbed convection-diffusionproblem (1) discretized on a general class of Shishkin-typemeshesThe proof is based on preconditioning of the discretesystem which allows us to use the standard ldquoconsistency+ stabilityrdquo principle of convergence This is a conceptuallysimple approach which does not require the use of hybridstability inequalities and discrete Greenrsquos functions In gen-eral new proof techniques are of interest because althougha lot has been achieved in the field of singularly perturbed

10 Advances in Numerical Analysis

problems over the last few decades there are still importantquestions remaining open [21] One of them Question6 is related to the work presented here The question isabout two-dimensional convection-diffusion problems and itpoints out that whereas 120576-uniform pointwise convergence forthe upwind finite-difference discretization on S-type meshesis well-established no similar result for B-type meshes isknownAsmentioned inRemark 13 we already know that thepreconditioning approach works well for B-mesh and one-dimensional convection-diffusion problems The next step isto consider two-dimensional problems

Appendix

Proof of Lemma 6 The proof is different from the proof ofLemma 5because theBS-meshdoes not satisfy condition (iv)Because of (ii) it is sufficient to show that the numerator in(25) is nonnegative for radic2 minus 1 le 119876 lt 1 independently of 120576and 119873 Let 119901 fl 1 minus 119873

minus1 By direct computations we get that

Δ119894

= minus ln(1 minus

119905119894

119901

119876

) + ln(1 minus

119905119894minus1

119901

119876

)

= ln(1 +

1

119869119901 minus 119894

) = ln (1 + 119902)

(A1)

where we denote 119902 fl (119869119901minus119894)minus1 Note that 0 le 119902 lt 1(119876+1)

Analogously

Δ119894+1

= ln(

1

1 minus 119902

) (A2)

Then

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

= Δ119894

(Δ119894+1

+ Δ119894

+ 1) minus Δ119894+1

= ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902)

(A3)

We set

119892 (119902) fl ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902) (A4)

Sinceradic2minus1 le 119876 lt 1 we have 0 le 119902 lt 1radic2 Rearrange 119892(119902)

as

119892 (119902) = [1 + ln(1 +

2119902

1 minus 119902

)] ln (1 + 119902)

+ ln (1 minus 119902)

(A5)

For 119902 isin [0 1) it follows that119902

1 + 1199022

le ln (1 + 119902) le 119902 (A6)

Hence

ln(1 +

2119902

1 minus 119902

) ge 2119902 (A7)

So we are done if we can show that

119892 (119902) ge [1 + 2119902] ln (1 + 119902) + ln (1 minus 119902) ge 0

0 le 119902 lt

1

radic2

(A8)

or

(1 minus 119902) (1 + 119902)2119902+1

= (1 minus 1199022

) (1 + 119902)2119902

ge 1 (A9)

First if 119902 isin [12 1radic2] we have by (1 + 119902)120574

ge 1 + 120574119902 for120574 ge 1

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

) (1 + 21199022

) ge 1 (A10)

Second we have that ln(1+119902) is a concave-down function on[0 12] hence on [0 12]

ln (1 + 119902) ge 2 ln(

3

2

) 119902 ge

4119902

5

(A11)

Then multiplying both sides by 119902 and exponentiating we getthat

(1 + 119902)119902

ge exp(

41199022

5

) ge 1 +

41199022

5

(A12)

Therefore

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

)(1 +

41199022

5

)

2

ge 1 (A13)

which holds true for 119902 isin [0 12]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P A Farrell A F Hegarty J J H Miller E OrsquoRiordan andG I Shishkin Robust Computational Techniques for BoundaryLayers Chapman amp Hall CRC Press Boca Raton Fla USA2000

[2] T Linszlig Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems vol 1985 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[3] J J HMiller E OrsquoRiordan and G I Shishkin Fitted NumericalMethods for Singular Perturbation Problems World ScientificSingapore 1996

[4] H-G Roos M Stynes and L Tobiska Numerical Methodsfor Singularly Perturbed Differential Equations Springer BerlinGermany 2nd edition 2008

[5] A Malvandi and D D Ganji ldquoMixed convection of aluminandashwater nanofluid inside a concentric annulus consideringnanoparticle migrationrdquo Particuology vol 24 pp 113ndash122 2016

[6] R B Kellogg and A Tsan ldquoAnalysis of some difference approx-imations for a singular perturbation problem without turningpointsrdquoMathematics of Computation vol 32 no 144 pp 1025ndash1039 1978

Advances in Numerical Analysis 11

[7] J Lorenz ldquoStability and monotonicity properties of stiff quasi-linear boundary value problemsrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta Serija zaMatematiku vol 12 pp 151ndash175 1982

[8] E Bohl Finite Modelle Gewohnlicher Randwertaufgaben Teub-ner Stuttgart Germany 1981

[9] E C Gartland Jr ldquoGraded-mesh difference schemes for sin-gularly perturbed two-point boundary value problemsrdquo Math-ematics of Computation vol 51 no 184 pp 631ndash657 1988

[10] R Vulanovic ldquoOn a numerical solution of a type of singularlyperturbed boundary value problem by using a special dis-cretization meshrdquo Univerzitet u Novom Sadu Zbornik RadovaPrirodno- Matematichkog Fakulteta Serija za Matematiku vol13 pp 187ndash201 1983

[11] C Grossmann H-G Roos and M Stynes Numerical Treat-ment of Partial Differential Equations Springer Berlin Ger-many 2007

[12] V B Andreev and I A Savin ldquoThe uniform convergence withrespect to a small parameter of A A Samarskiirsquos monotonescheme and its modificationrdquo Computational Mathematics andMathematical Physics vol 35 pp 581ndash591 1995

[13] T LinszligH-G Roos andRVulanovic ldquoUniformpointwise con-vergence on Shishkin-type meshes for quasi-linear convection-diffusion problemsrdquo SIAM Journal on Numerical Analysis vol38 no 3 pp 897ndash912 2000

[14] R Vulanovic and T A Nhan ldquoUniform convergence viapreconditioningrdquo International Journal of Numerical Analysisand Modeling B vol 5 no 4 pp 347ndash356 2014

[15] H-G Roos ldquoA note on the conditioning of upwind schemes onShishkin meshesrdquo IMA Journal of Numerical Analysis vol 16no 4 pp 529ndash538 1996

[16] H-G Roos and T Linszlig ldquoSufficient condition for uniformconvergence on layeradapted gridsrdquo Computing vol 63 no 1pp 27ndash45 1999

[17] R Vulanovic ldquoA higher-order scheme for quasilinear boundaryvalue problems with two small parametersrdquo Computing vol 67no 4 pp 287ndash303 2001

[18] R Vulanovic ldquoA priori meshes for singularly perturbed quasi-linear twominuspoint boundary value problemsrdquo IMA Journal ofNumerical Analysis vol 21 no 1 pp 349ndash366 2001

[19] T A Nhan and R Vulanovic ldquoUniform convergence on aBakhvalov-type mesh using the preconditioning approachTechnical reportrdquo httparxivorgabs150404283

[20] N S Bakhvalov ldquoThe optimization of methods of solvingboundary value problems with a boundary layerrdquo USSR Com-putational Mathematics and Mathematical Physics vol 9 no 4pp 139ndash166 1969

[21] H-G Roos andM Stynes ldquoSome open questions in the numer-ical analysis of singularly perturbed differential equationsrdquoComputational Methods in Applied Mathematics vol 15 no 4pp 531ndash550 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Numerical Analysis 9

Table 7 The maximum norm of the consistency error [119860119873

119906119873

minus119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 467119890 + 00 247119890 + 00 128119890 + 00 659119890 minus 01 336119890 minus 01 170119890 minus 01

1119890 minus 3 467119890 + 01 247119890 + 01 128119890 + 01 659119890 + 00 336119890 + 00 170119890 + 00

1119890 minus 4 467119890 + 02 247119890 + 02 128119890 + 02 659119890 + 01 336119890 + 01 170119890 + 01

1119890 minus 5 467119890 + 03 247119890 + 03 128119890 + 03 659119890 + 02 336119890 + 02 170119890 + 02

1119890 minus 6 467119890 + 04 247119890 + 04 128119890 + 04 659119890 + 03 336119890 + 03 170119890 + 03

1119890 minus 7 467119890 + 05 247119890 + 05 128119890 + 05 659119890 + 04 336119890 + 04 170119890 + 04

1119890 minus 8 467119890 + 06 247119890 + 06 128119890 + 06 659119890 + 05 336119890 + 05 170119890 + 05

Table 8 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the BS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 122119890 minus 01 650119890 minus 02 338119890 minus 02 174119890 minus 02 887119890 minus 03 452119890 minus 03

1119890 minus 3 114119890 minus 01 601119890 minus 02 308119890 minus 02 156119890 minus 02 786119890 minus 03 395119890 minus 03

1119890 minus 4 114119890 minus 01 597119890 minus 02 306119890 minus 02 155119890 minus 02 778119890 minus 03 390119890 minus 03

1119890 minus 5 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 390119890 minus 03

1119890 minus 6 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 7 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

1119890 minus 8 114119890 minus 01 596119890 minus 02 305119890 minus 02 154119890 minus 02 777119890 minus 03 389119890 minus 03

Table 9 The maximum norm of the preconditioned consistency error [119873

119906119873

minus 119872119891

119873

] on the VS-mesh

120576 119873 = 32 119873 = 64 119873 = 128 119873 = 256 119873 = 512 119873 = 1024

1119890 minus 2 160119890 minus 01 912119890 minus 02 502119890 minus 02 270119890 minus 02 143119890 minus 02 747119890 minus 03

1119890 minus 3 150119890 minus 01 844119890 minus 02 458119890 minus 02 242119890 minus 02 126119890 minus 02 653119890 minus 03

1119890 minus 4 149119890 minus 01 837119890 minus 02 454119890 minus 02 240119890 minus 02 125119890 minus 02 645119890 minus 03

1119890 minus 5 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 6 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 7 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

1119890 minus 8 149119890 minus 01 837119890 minus 02 453119890 minus 02 240119890 minus 02 125119890 minus 02 644119890 minus 03

Theorem 11 The error of the upwind finite-difference dis-cretization on S-type meshes satisfies

10038161003816100381610038161003816119906119873

119894

minus 119880119873

119894

10038161003816100381610038161003816

le

119862119873minus1

(max 100381610038161003816100381610038161205951015840

10038161003816100381610038161003816)

2

119894 = 1 119869 minus 1

119862119873minus1

119894 = 119869 119873 minus 1

(64)

Corollary 12 For the meshes given in Table 1 one has thefollowing estimates

10038171003817100381710038171003817119906119873

minus 119880119873

10038171003817100381710038171003817

le

119862119873minus1

1198712

119891119900119903 119905ℎ119890 119878-119898119890119904ℎ

119862119873minus1

119891119900119903 119905ℎ119890 119861119878- 119886119899119889 119881119878-119898119890119904ℎ119890119904

(65)

Remark 13 The barrier-function technique can be used toobtain similar results for S-type meshes (see [16]) but itrequires another assumption on the mesh-generating func-tion 120601 (cf [16 Theorem 1]) However whether the barrier-function approach can be extended to any Bakhvalov-type

mesh is still an open question [2 Remark 421] By contrastas proved in [19] the preconditioner defined in (20) worksfine for Vulanovicrsquos modification of the Bakhvalov mesh [10]The paper [10] is where the original Bakhvalov mesh [20]is generalized (the main difference between Bakhvalov-typemeshes and S-type meshes is that the former have a smoothtransition from the fine part inside the layer to the coarseuniform part outside the layer the mesh-generating function120601 belongs to 119862

1

(119868))

6 Conclusion

In this paper we present a proof of pointwise uniformconvergence for singularly perturbed convection-diffusionproblem (1) discretized on a general class of Shishkin-typemeshesThe proof is based on preconditioning of the discretesystem which allows us to use the standard ldquoconsistency+ stabilityrdquo principle of convergence This is a conceptuallysimple approach which does not require the use of hybridstability inequalities and discrete Greenrsquos functions In gen-eral new proof techniques are of interest because althougha lot has been achieved in the field of singularly perturbed

10 Advances in Numerical Analysis

problems over the last few decades there are still importantquestions remaining open [21] One of them Question6 is related to the work presented here The question isabout two-dimensional convection-diffusion problems and itpoints out that whereas 120576-uniform pointwise convergence forthe upwind finite-difference discretization on S-type meshesis well-established no similar result for B-type meshes isknownAsmentioned inRemark 13 we already know that thepreconditioning approach works well for B-mesh and one-dimensional convection-diffusion problems The next step isto consider two-dimensional problems

Appendix

Proof of Lemma 6 The proof is different from the proof ofLemma 5because theBS-meshdoes not satisfy condition (iv)Because of (ii) it is sufficient to show that the numerator in(25) is nonnegative for radic2 minus 1 le 119876 lt 1 independently of 120576and 119873 Let 119901 fl 1 minus 119873

minus1 By direct computations we get that

Δ119894

= minus ln(1 minus

119905119894

119901

119876

) + ln(1 minus

119905119894minus1

119901

119876

)

= ln(1 +

1

119869119901 minus 119894

) = ln (1 + 119902)

(A1)

where we denote 119902 fl (119869119901minus119894)minus1 Note that 0 le 119902 lt 1(119876+1)

Analogously

Δ119894+1

= ln(

1

1 minus 119902

) (A2)

Then

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

= Δ119894

(Δ119894+1

+ Δ119894

+ 1) minus Δ119894+1

= ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902)

(A3)

We set

119892 (119902) fl ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902) (A4)

Sinceradic2minus1 le 119876 lt 1 we have 0 le 119902 lt 1radic2 Rearrange 119892(119902)

as

119892 (119902) = [1 + ln(1 +

2119902

1 minus 119902

)] ln (1 + 119902)

+ ln (1 minus 119902)

(A5)

For 119902 isin [0 1) it follows that119902

1 + 1199022

le ln (1 + 119902) le 119902 (A6)

Hence

ln(1 +

2119902

1 minus 119902

) ge 2119902 (A7)

So we are done if we can show that

119892 (119902) ge [1 + 2119902] ln (1 + 119902) + ln (1 minus 119902) ge 0

0 le 119902 lt

1

radic2

(A8)

or

(1 minus 119902) (1 + 119902)2119902+1

= (1 minus 1199022

) (1 + 119902)2119902

ge 1 (A9)

First if 119902 isin [12 1radic2] we have by (1 + 119902)120574

ge 1 + 120574119902 for120574 ge 1

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

) (1 + 21199022

) ge 1 (A10)

Second we have that ln(1+119902) is a concave-down function on[0 12] hence on [0 12]

ln (1 + 119902) ge 2 ln(

3

2

) 119902 ge

4119902

5

(A11)

Then multiplying both sides by 119902 and exponentiating we getthat

(1 + 119902)119902

ge exp(

41199022

5

) ge 1 +

41199022

5

(A12)

Therefore

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

)(1 +

41199022

5

)

2

ge 1 (A13)

which holds true for 119902 isin [0 12]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P A Farrell A F Hegarty J J H Miller E OrsquoRiordan andG I Shishkin Robust Computational Techniques for BoundaryLayers Chapman amp Hall CRC Press Boca Raton Fla USA2000

[2] T Linszlig Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems vol 1985 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[3] J J HMiller E OrsquoRiordan and G I Shishkin Fitted NumericalMethods for Singular Perturbation Problems World ScientificSingapore 1996

[4] H-G Roos M Stynes and L Tobiska Numerical Methodsfor Singularly Perturbed Differential Equations Springer BerlinGermany 2nd edition 2008

[5] A Malvandi and D D Ganji ldquoMixed convection of aluminandashwater nanofluid inside a concentric annulus consideringnanoparticle migrationrdquo Particuology vol 24 pp 113ndash122 2016

[6] R B Kellogg and A Tsan ldquoAnalysis of some difference approx-imations for a singular perturbation problem without turningpointsrdquoMathematics of Computation vol 32 no 144 pp 1025ndash1039 1978

Advances in Numerical Analysis 11

[7] J Lorenz ldquoStability and monotonicity properties of stiff quasi-linear boundary value problemsrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta Serija zaMatematiku vol 12 pp 151ndash175 1982

[8] E Bohl Finite Modelle Gewohnlicher Randwertaufgaben Teub-ner Stuttgart Germany 1981

[9] E C Gartland Jr ldquoGraded-mesh difference schemes for sin-gularly perturbed two-point boundary value problemsrdquo Math-ematics of Computation vol 51 no 184 pp 631ndash657 1988

[10] R Vulanovic ldquoOn a numerical solution of a type of singularlyperturbed boundary value problem by using a special dis-cretization meshrdquo Univerzitet u Novom Sadu Zbornik RadovaPrirodno- Matematichkog Fakulteta Serija za Matematiku vol13 pp 187ndash201 1983

[11] C Grossmann H-G Roos and M Stynes Numerical Treat-ment of Partial Differential Equations Springer Berlin Ger-many 2007

[12] V B Andreev and I A Savin ldquoThe uniform convergence withrespect to a small parameter of A A Samarskiirsquos monotonescheme and its modificationrdquo Computational Mathematics andMathematical Physics vol 35 pp 581ndash591 1995

[13] T LinszligH-G Roos andRVulanovic ldquoUniformpointwise con-vergence on Shishkin-type meshes for quasi-linear convection-diffusion problemsrdquo SIAM Journal on Numerical Analysis vol38 no 3 pp 897ndash912 2000

[14] R Vulanovic and T A Nhan ldquoUniform convergence viapreconditioningrdquo International Journal of Numerical Analysisand Modeling B vol 5 no 4 pp 347ndash356 2014

[15] H-G Roos ldquoA note on the conditioning of upwind schemes onShishkin meshesrdquo IMA Journal of Numerical Analysis vol 16no 4 pp 529ndash538 1996

[16] H-G Roos and T Linszlig ldquoSufficient condition for uniformconvergence on layeradapted gridsrdquo Computing vol 63 no 1pp 27ndash45 1999

[17] R Vulanovic ldquoA higher-order scheme for quasilinear boundaryvalue problems with two small parametersrdquo Computing vol 67no 4 pp 287ndash303 2001

[18] R Vulanovic ldquoA priori meshes for singularly perturbed quasi-linear twominuspoint boundary value problemsrdquo IMA Journal ofNumerical Analysis vol 21 no 1 pp 349ndash366 2001

[19] T A Nhan and R Vulanovic ldquoUniform convergence on aBakhvalov-type mesh using the preconditioning approachTechnical reportrdquo httparxivorgabs150404283

[20] N S Bakhvalov ldquoThe optimization of methods of solvingboundary value problems with a boundary layerrdquo USSR Com-putational Mathematics and Mathematical Physics vol 9 no 4pp 139ndash166 1969

[21] H-G Roos andM Stynes ldquoSome open questions in the numer-ical analysis of singularly perturbed differential equationsrdquoComputational Methods in Applied Mathematics vol 15 no 4pp 531ndash550 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Numerical Analysis

problems over the last few decades there are still importantquestions remaining open [21] One of them Question6 is related to the work presented here The question isabout two-dimensional convection-diffusion problems and itpoints out that whereas 120576-uniform pointwise convergence forthe upwind finite-difference discretization on S-type meshesis well-established no similar result for B-type meshes isknownAsmentioned inRemark 13 we already know that thepreconditioning approach works well for B-mesh and one-dimensional convection-diffusion problems The next step isto consider two-dimensional problems

Appendix

Proof of Lemma 6 The proof is different from the proof ofLemma 5because theBS-meshdoes not satisfy condition (iv)Because of (ii) it is sufficient to show that the numerator in(25) is nonnegative for radic2 minus 1 le 119876 lt 1 independently of 120576and 119873 Let 119901 fl 1 minus 119873

minus1 By direct computations we get that

Δ119894

= minus ln(1 minus

119905119894

119901

119876

) + ln(1 minus

119905119894minus1

119901

119876

)

= ln(1 +

1

119869119901 minus 119894

) = ln (1 + 119902)

(A1)

where we denote 119902 fl (119869119901minus119894)minus1 Note that 0 le 119902 lt 1(119876+1)

Analogously

Δ119894+1

= ln(

1

1 minus 119902

) (A2)

Then

Δ119894

Δ119894+1

+ Δ2

119894

+ Δ119894

minus Δ119894+1

= Δ119894

(Δ119894+1

+ Δ119894

+ 1) minus Δ119894+1

= ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902)

(A3)

We set

119892 (119902) fl ln (1 + 119902) (ln1 + 119902

1 minus 119902

+ 1) + ln (1 minus 119902) (A4)

Sinceradic2minus1 le 119876 lt 1 we have 0 le 119902 lt 1radic2 Rearrange 119892(119902)

as

119892 (119902) = [1 + ln(1 +

2119902

1 minus 119902

)] ln (1 + 119902)

+ ln (1 minus 119902)

(A5)

For 119902 isin [0 1) it follows that119902

1 + 1199022

le ln (1 + 119902) le 119902 (A6)

Hence

ln(1 +

2119902

1 minus 119902

) ge 2119902 (A7)

So we are done if we can show that

119892 (119902) ge [1 + 2119902] ln (1 + 119902) + ln (1 minus 119902) ge 0

0 le 119902 lt

1

radic2

(A8)

or

(1 minus 119902) (1 + 119902)2119902+1

= (1 minus 1199022

) (1 + 119902)2119902

ge 1 (A9)

First if 119902 isin [12 1radic2] we have by (1 + 119902)120574

ge 1 + 120574119902 for120574 ge 1

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

) (1 + 21199022

) ge 1 (A10)

Second we have that ln(1+119902) is a concave-down function on[0 12] hence on [0 12]

ln (1 + 119902) ge 2 ln(

3

2

) 119902 ge

4119902

5

(A11)

Then multiplying both sides by 119902 and exponentiating we getthat

(1 + 119902)119902

ge exp(

41199022

5

) ge 1 +

41199022

5

(A12)

Therefore

(1 minus 1199022

) (1 + 119902)2119902

ge (1 minus 1199022

)(1 +

41199022

5

)

2

ge 1 (A13)

which holds true for 119902 isin [0 12]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] P A Farrell A F Hegarty J J H Miller E OrsquoRiordan andG I Shishkin Robust Computational Techniques for BoundaryLayers Chapman amp Hall CRC Press Boca Raton Fla USA2000

[2] T Linszlig Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems vol 1985 of Lecture Notes in MathematicsSpringer Berlin Germany 2010

[3] J J HMiller E OrsquoRiordan and G I Shishkin Fitted NumericalMethods for Singular Perturbation Problems World ScientificSingapore 1996

[4] H-G Roos M Stynes and L Tobiska Numerical Methodsfor Singularly Perturbed Differential Equations Springer BerlinGermany 2nd edition 2008

[5] A Malvandi and D D Ganji ldquoMixed convection of aluminandashwater nanofluid inside a concentric annulus consideringnanoparticle migrationrdquo Particuology vol 24 pp 113ndash122 2016

[6] R B Kellogg and A Tsan ldquoAnalysis of some difference approx-imations for a singular perturbation problem without turningpointsrdquoMathematics of Computation vol 32 no 144 pp 1025ndash1039 1978

Advances in Numerical Analysis 11

[7] J Lorenz ldquoStability and monotonicity properties of stiff quasi-linear boundary value problemsrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta Serija zaMatematiku vol 12 pp 151ndash175 1982

[8] E Bohl Finite Modelle Gewohnlicher Randwertaufgaben Teub-ner Stuttgart Germany 1981

[9] E C Gartland Jr ldquoGraded-mesh difference schemes for sin-gularly perturbed two-point boundary value problemsrdquo Math-ematics of Computation vol 51 no 184 pp 631ndash657 1988

[10] R Vulanovic ldquoOn a numerical solution of a type of singularlyperturbed boundary value problem by using a special dis-cretization meshrdquo Univerzitet u Novom Sadu Zbornik RadovaPrirodno- Matematichkog Fakulteta Serija za Matematiku vol13 pp 187ndash201 1983

[11] C Grossmann H-G Roos and M Stynes Numerical Treat-ment of Partial Differential Equations Springer Berlin Ger-many 2007

[12] V B Andreev and I A Savin ldquoThe uniform convergence withrespect to a small parameter of A A Samarskiirsquos monotonescheme and its modificationrdquo Computational Mathematics andMathematical Physics vol 35 pp 581ndash591 1995

[13] T LinszligH-G Roos andRVulanovic ldquoUniformpointwise con-vergence on Shishkin-type meshes for quasi-linear convection-diffusion problemsrdquo SIAM Journal on Numerical Analysis vol38 no 3 pp 897ndash912 2000

[14] R Vulanovic and T A Nhan ldquoUniform convergence viapreconditioningrdquo International Journal of Numerical Analysisand Modeling B vol 5 no 4 pp 347ndash356 2014

[15] H-G Roos ldquoA note on the conditioning of upwind schemes onShishkin meshesrdquo IMA Journal of Numerical Analysis vol 16no 4 pp 529ndash538 1996

[16] H-G Roos and T Linszlig ldquoSufficient condition for uniformconvergence on layeradapted gridsrdquo Computing vol 63 no 1pp 27ndash45 1999

[17] R Vulanovic ldquoA higher-order scheme for quasilinear boundaryvalue problems with two small parametersrdquo Computing vol 67no 4 pp 287ndash303 2001

[18] R Vulanovic ldquoA priori meshes for singularly perturbed quasi-linear twominuspoint boundary value problemsrdquo IMA Journal ofNumerical Analysis vol 21 no 1 pp 349ndash366 2001

[19] T A Nhan and R Vulanovic ldquoUniform convergence on aBakhvalov-type mesh using the preconditioning approachTechnical reportrdquo httparxivorgabs150404283

[20] N S Bakhvalov ldquoThe optimization of methods of solvingboundary value problems with a boundary layerrdquo USSR Com-putational Mathematics and Mathematical Physics vol 9 no 4pp 139ndash166 1969

[21] H-G Roos andM Stynes ldquoSome open questions in the numer-ical analysis of singularly perturbed differential equationsrdquoComputational Methods in Applied Mathematics vol 15 no 4pp 531ndash550 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Numerical Analysis 11

[7] J Lorenz ldquoStability and monotonicity properties of stiff quasi-linear boundary value problemsrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta Serija zaMatematiku vol 12 pp 151ndash175 1982

[8] E Bohl Finite Modelle Gewohnlicher Randwertaufgaben Teub-ner Stuttgart Germany 1981

[9] E C Gartland Jr ldquoGraded-mesh difference schemes for sin-gularly perturbed two-point boundary value problemsrdquo Math-ematics of Computation vol 51 no 184 pp 631ndash657 1988

[10] R Vulanovic ldquoOn a numerical solution of a type of singularlyperturbed boundary value problem by using a special dis-cretization meshrdquo Univerzitet u Novom Sadu Zbornik RadovaPrirodno- Matematichkog Fakulteta Serija za Matematiku vol13 pp 187ndash201 1983

[11] C Grossmann H-G Roos and M Stynes Numerical Treat-ment of Partial Differential Equations Springer Berlin Ger-many 2007

[12] V B Andreev and I A Savin ldquoThe uniform convergence withrespect to a small parameter of A A Samarskiirsquos monotonescheme and its modificationrdquo Computational Mathematics andMathematical Physics vol 35 pp 581ndash591 1995

[13] T LinszligH-G Roos andRVulanovic ldquoUniformpointwise con-vergence on Shishkin-type meshes for quasi-linear convection-diffusion problemsrdquo SIAM Journal on Numerical Analysis vol38 no 3 pp 897ndash912 2000

[14] R Vulanovic and T A Nhan ldquoUniform convergence viapreconditioningrdquo International Journal of Numerical Analysisand Modeling B vol 5 no 4 pp 347ndash356 2014

[15] H-G Roos ldquoA note on the conditioning of upwind schemes onShishkin meshesrdquo IMA Journal of Numerical Analysis vol 16no 4 pp 529ndash538 1996

[16] H-G Roos and T Linszlig ldquoSufficient condition for uniformconvergence on layeradapted gridsrdquo Computing vol 63 no 1pp 27ndash45 1999

[17] R Vulanovic ldquoA higher-order scheme for quasilinear boundaryvalue problems with two small parametersrdquo Computing vol 67no 4 pp 287ndash303 2001

[18] R Vulanovic ldquoA priori meshes for singularly perturbed quasi-linear twominuspoint boundary value problemsrdquo IMA Journal ofNumerical Analysis vol 21 no 1 pp 349ndash366 2001

[19] T A Nhan and R Vulanovic ldquoUniform convergence on aBakhvalov-type mesh using the preconditioning approachTechnical reportrdquo httparxivorgabs150404283

[20] N S Bakhvalov ldquoThe optimization of methods of solvingboundary value problems with a boundary layerrdquo USSR Com-putational Mathematics and Mathematical Physics vol 9 no 4pp 139ndash166 1969

[21] H-G Roos andM Stynes ldquoSome open questions in the numer-ical analysis of singularly perturbed differential equationsrdquoComputational Methods in Applied Mathematics vol 15 no 4pp 531ndash550 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of