CONVERGENCE CRITERIA FOR ITERATIVE METHODS IN SOLVINGCONVECTION-DIFFUSION EQUATIONS ON ADAPTIVE MESHES

CHIN-TIEN WU ∗ AND HOWARD C. ELMAN †

Abstract. In this work, sparse linear systems obtained from the streamline diffusion finite element discretizationof the convection-diffusion equations are solved by a multigrid method and the generalized minimal residule method.Adaptive mesh refinement process is considered as an integral part of the solution process. We propose some stoppingcriteria for iterative solvers to ensure the iterative errors are within the range of the a posteriori error bound. Underthe assumption that the error indicators do not change rapidly during mesh refinement processes, we also showthat the error indicators computed from iterative solutions satisfying the proposed stopping criteria are as reliableand efficient as the error indicators computed from directive solutions. Moreover, our numerical results show thatiterative steps are reduced significantly for the multigridsolver to satisfy the proposed stopping criteria. The refinedmeshes obtained from such iterative solutions are almost indistinguishable with the refined meshes obtained fromdirective solutions.

1. Introduction. In solving a partial differential equation (PDE), spatial discretizationof the PDE on a finite dimensional subspace often results in a large sparse linear systemAu = f . Iterative methods are natural choices for solving sparse linear systems. It is clearthat the iterative solution will not be a good approximationto the exact solution if the iter-ations are stopped too early. On the other hand, since discretization errors exist a priori, ahighly accurate iterative solution may require too many iterations and simply waste compu-tation time without increasing the overall solution qualities. Thus, an important aspect of anyiterative method is to determine when the iterations shouldbe stopped.

It has been shown by M. Arioli et al that the algebraic norm of the residual may give amisleading information about the convergence and theA−1 norms of the residual reveals atrue evaluation of the error for self-adjoint elliptic problems [2] [4]. Letuk denote the iter-ative solution,rk = f − Auk be the residual andek = u − uk be the iterative error after kiterations. By considering the quadratic formrt

kArk as a Stieltjes integral and approximatingthe integral by some Gauss quadrature rules, lower and upperbounds of‖ek‖A = ‖rk‖A−1

can be derived naturally from Lanczos process as shown in papers [9] [10] [11] by G. Golubet al. Base on these important works, practical stopping criteria for the conjugate-gradientmethod (CG) and symmetric LQ method (SYMMLQ) have been proposed by M. Arioli [1]and G. Meurant [16], and D. Calvetti, et al [5], respectively. For nonsymmetric positive def-inite problems, stopping criteria for the generalized minimal residual method (GMRES) isrecently developed by M. Arioli, D. Loghin and J. Wathen [3].The iterative errors are guar-anteed to be confined in the a priori error bounds when the iterative solutions satisfy stoppingcriteria in [1] [3].

In this paper, we consider the convection-diffusion problems,

−ǫu+ b · ∇u = f, onΩu = g on∂Ω,

(1.1)

whereb andf are sufficiently smooth and the domainΩ is convex with Lipschitz boundary∂Ω. The linear system resulting from SDFEM discretization is denoted byAsdu = fsd. It iswell known that the solution typically has boundary layers caused by Dirichlet conditions on

∗National center for theoretical sciences, Mathematics devision, Tsing Hua University Hsing Chu, Taiwan 30043,ctw@math.umd.edu

†Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, Col-lege Park, MD 20742.elman@cs.umd.edu.

1

the outflow boundaries or internal layers caused by discontinuities in the inflow boundaries,for the convection dominant case, i.e.|b| ≫ ǫ. In order to improve the accuracy of thesolution particularly in layer regions at a reasonable cost, adaptive mesh refinement processes[14] [18] [21] [22], are generally employed, in which some a posteriori error indicators areused to pinpoint the regions where errors are large and grid resolution is increased locallyin the identified regions. Theoretically, the error indicators are to be computed from exactfinite element solutions. Clearly, the reliability of any error indicator may be deterioratedwhen it is computed from an inaccurate iterative solution. Thus, instead of directly usingstopping criteria that enforce iterative errors to be less than some a priori error bounds, ourgoal here is to find stopping criteria for iterative solvers such that not only the iterative errorscan be confined in some a posteriori error bounds but also the error indicators, computedfrom iterative soultions, can effectively be used to identify the regions where errors are large.Evidently, as we shall see in Section 3 and Section 4, the new stopping criteria should bedetermined by the a posteriori error estimations and the strategy used to increase the gridresolution. Nevertheless, we would like to note that the stopping criteria (18) or (20) in [3]can be applied directly to GMRES for solvingAsdu = fsd due to the fact that the a prioriestimation

|‖u− uh‖| ≤ c

√

h

ǫ|‖u‖| , for some constant c, (1.2)

holds [22] (Chapter 2), where|‖·‖| denotes the energy norm induced by the symmetric partof the matrixAsd.

This paper is structured as follows. In Section 2, we introduce the SDFEM discretization,the a posteriori error estimations and the so-called maximum marking strategy which is usedto label elements for mesh refinement. For the a posteriori error estimations, the Neumann-type of error indicator proposed by Kay and Silvester [14] are introduced. In Section 3, twotypes of stopping criteria associated with both the markingstrategy and Kay and Silvester’serror indicator are presented. The first type of criteria concerns that the iterative errors shouldbe bounded by the upper bound in the a posteriori error estimations and the second type ofcriteria concerns that severe over-refinement should not occur when the error indicators arecomputed from iterative solutions. In our analysis, for simplicity, only one mesh refinementis performed and the error indicator is computed from the exact finite element solution oncoarse grid level in our analysis. Since the second type of stopping criteria has direct impactto the adaptive mesh refinement process, we present our numerical studies that support theeffectiveness of the second type of stopping crieria in Section 4. In section 5, we draw ourconclusions.

2. Discretization and A Posteriori Error Estimation. Letℑh be a given quasi-uniformmesh of triangles onΩ and letVh be the linear finite element space onℑh. It is well knownthat the standard Galerkin finite element discretization onuniform grids produces inaccurateoscillatory solutions to convection-diffusion problems.Here, the equation (1.1) is discretizedby the streamline diffusion finite element method (SDFEM) [7] [12] [13], a variant of thestandard Galerkin method, where extra diffusion in the streamline direction is introduced.The SDFEM formulation is to finduh ∈ Vh such that

Bsd(uh, vh) = fsd(vh), for all vh ∈ Vh, (2.1)

2

where

Bsd(uh, vh) = ǫ(∇uh,∇vh) + (b · ∇uh, vh) +∑

T∈ℑh

δT (b · ∇uh, b · ∇vh)T ,

fsd(vh) = (fh, vh) +∑

T∈ℑh

(f, δT b · ∇vh),

andδT is a stabilization parameter. With a carefully chosen valueof δT , the streamline diffu-sion finite element discretization is able to eliminate mostoscillations and produce accuratesolutions in the regions where no layers are present [13]. Asshown in [8], a good choice ofδT is

δT =

1

2‖b‖T

(

1 − 1PeT

)

if PeT > 1,

0 if PeT < 1,(2.2)

where

PeT =‖b‖T hT

2ǫ, for T ∈ ℑh with diameterhT ,

is the mesh Peclet number.

This strategy does not produce accurate solutions in regions containing layers that arenot resolved by the grid. Accuracy can be achieved at reasonable cost in such regions byadaptive mesh refinement. In general, the adaptive mesh refinement process consists of loopsof the following form:

Solve︸ ︷︷ ︸

1

→ Compute error indicator︸ ︷︷ ︸

2

→ Refine mesh︸ ︷︷ ︸

3

in which the coarse grid solution is first computed in step 1, an a posteriori error estimator iscomputed from the coarse grid solution and is used to identify regions where errors are largein step 2, and a marking strategy is used in step 3 to select elements with large a posteriorierror values. Once decisions on where to refine are made, a common mesh refinement schemesuch as regular refinement or the longest side bisection refinement [19] [20], is applied to thecoarse grid and a more accurate solutions can be computed on the refined mesh. Clearly,a reliable computable a posteriori error estimator in step 2is the key for the adaptive meshrefinement process to succeed. For the convection-diffusion equation discretized by SDFEM,the first a posteriori error estimation where the error is estimated by computing the residualwas proposed by Verfurth [21], and the first a posteriori error estimation where the error isestimated by solving a local problem was developed by Kay andSilvester [14]. In this study,we use the Kay and Silvester’s a posteriori error estimator which we have found to be a moreeffective choice in [22]. Hereafter, we call this indicatorthe KS-indicator, denoted byηh,T

for any elementT ∈ ℑh. In step 3, we use the heuristicmaximum marking strategy, wherean elementT ∗ is marked for refinement if

ηT∗ > θ maxT∈ℑh

ηT , (2.3)

with a prescribed threshold0 < θ < 1. More complicated marking and refinement strategiesthat lead to convergence of the adaptive solution process 2 can be seen in [6][15].

3

For introducing the a posteriori error estimation, let us introduce the following abbrevi-ations. Let‖·‖0,Ω and‖·‖0,T denote theL2 norm on domainΩ and element T, respectively.Let E(T ) be the set of edges of T andωT = ∪T ′∩T∈E(T )T

′. Let π0T be theL2 projection

onto the space of polynomials of degree 0 on element T. The interior residualRT of elementT and the inter-element flux jumpRE of edge E are defined as follows:

RT = (f − b · ∇uh)|T ,R0

T = π0T (RT ),

RE =

[ ∂uh

∂nE]E if E ∈ Ω

0 if E ∈ ∂Ω,

where[·]E is the jump across edge E.1 Let Φ be the element affine mapping from the physicaldomain to the computational domain andχi be the nodal basis function of node i. Theapproximation space is denoted asQT = QT

⊕BT , where

QT = spanψE Φ−1| ψE = 4χiχj , i,j are the endpoints ofE andE ∈ ∂T⋂

(Ω⋃

ΓN )

is the space spanned by quadratic edge bubble functions and

BT = spanψT Φ−1| ψT = 273∏

i=1

χi

is the space spanned by cubic interior bubble functions. Fordetails, see [14]. On each elementT, the error estimator is then given byηh,T = ‖∇eT ‖0,T , whereeT ∈ QT satisfies

ǫ(∇eT ,∇v)T = (R0T , v)T − 1

2ǫ∑

E∈∂T

(RE , v)E . (2.4)

Let eh = u− uh. The a posteriori error estimation in [14] is specified as follows:

(Global Upper Bound):

‖∇(eh)‖0,Ω ≤ C(∑

T∈ℑh

η2h,T +

∑

T∈ℑh

(h

ǫ)2∥∥RT −R0

T

∥∥

2

0,T)1/2 (2.5)

(Local Lower Bound):

ηh,T ≤ c

(

‖eh‖0,ωT+∑

T⊂ωT

hT

ǫ‖b · ∇eh‖0,T +

∑

T⊂ωT

hT

ǫ

∥∥RT −R0

T

∥∥

0,T

)

, (2.6)

where constantsC andc are independent of the diffusion parameterǫ and mesh size h. In thefollowing, we assume the interpolation errors are high order terms and can be ignored. As aresult, the term

∥∥RT −R0

T

∥∥

0,Tin the a posteriori bounds is ignored in our analysis.

1This definition ofRE for E ∈ ∂Ω is for Dirichlet boundary conditions. We don’t consider Neumann condi-tions in this study; in caseE ∈ ∂Ω and a Neumann condition holds, the flux jumpRE would be set to−2( ∂uh

∂nE)

[14].

4

The following lemma measures how close two functions have tobe in order to ensurethat the error estimators, computed from these two functions, have similar profiles. We thenutilize this result to derive our stopping criteria in next section.

LEMMA 2.1. Suppose the mesh Peclet numberPeT≫ 1 for all T ∈ ℑh. Letη1

h,T andη2

h,T be the error indicators computed fromu1 andu2 ∈ Vh on element T respectively. If

‖∇(u1 − u2)‖0,ωT≤ ch,ωT

η1h,T , wherech,ωT

= O(ǫ

h), (2.7)

then

1

2η1

h,T ≤ η2h,T ≤ 3

2η1

h,T . (2.8)

Proof: From (2.4), we have

ǫ〈∇(e1,T − e2,T ),∇v〉T = 〈R01,T −R0

2,T , v〉T − 1

2ǫ∑

E∈E(T )

〈R1,E −R2,E , v〉E . (2.9)

Let v = e1,T − e2,T . From the Schwartz inequality, (2.9) implies

ǫ ‖∇(e1,T − e2,T )‖20,T ≤

∥∥R0

1,T −R02,T

∥∥

0,T‖e1,T − e2,T‖0,T

︸ ︷︷ ︸

I

+1

2ǫ∑

E∈E(T )

‖R1,E −R2,E‖0,E ‖e1,T − e2,T‖0,E

︸ ︷︷ ︸

II

.(2.10)

First, let’s estimate∥∥R0

1,T −R02,T

∥∥

0,T:

∥∥R0

1,T −R02,T

∥∥

0,T=∥∥π0

T (f − b · ∇u1) − π0T (f − b · ∇u2)

∥∥

0,T

=∥∥π0

T (b · (∇(u2 − u1)))∥∥

0,T

‖b · ∇(u1 − u2)‖0,T

≤ ‖b‖∞,T ‖∇(u1 − u2)‖0,T .

(2.11)

Sincee1,T − e2,T ∈ QT , from a scaling argument, we have

‖e1,T − e2,T ‖0,T ≤ C(θT )hT ‖∇(e1,T − e2,T )‖0,T . (2.12)

From (2.11) and (2.12), it is clear that

(I) ≤ C(θT ) ‖b‖∞,T hT ‖∇(u1 − u2)‖0,T ‖∇(e1,T − e2,T )‖0,T (2.13)

5

Now, let’s estimate‖R1,E −R2,E‖0,E . ForE ∈ Eh,n, using the trace inequality,

‖R1,E −R2,E‖0,E =

∥∥∥∥[| ∂u1

∂nE|]E − [| ∂u2

∂nE|]E∥∥∥∥

0,E

=

∥∥∥∥[| ∂u1

∂nE− ∂u2

∂nE|]E∥∥∥∥

0,E

≤ h−1/2T

∥∥∥∥[| ∂(u1 − u2)

∂nE|]E∥∥∥∥

0,T

≤ h−1/2T (‖∇(u1 − u2)‖0,T + ‖∇(u1 − u2)‖0,Tnb

),

(2.14)

whereTnb is the triangle sharing edge E with T, ie,Tnb ∩ T = E.Again, from a scaling argument, we have

‖e1,T − e2,T ‖0,E ≤ C(θ)h1/2E ‖∇(e1,T − e2,T )‖0,T . (2.15)

By (2.14) and (2.15), we have

(II) ≤ 1

2ǫ∑

E∈E

C(θT )h1/2E h

−1/2T [‖∇(u1 − u2)‖0,T + ‖∇(u1 − u2)‖0,Tnb

] ‖∇(e1,T − e2,T )‖0,T

≤ 3

2C(θT )max

E∈E(hE

hT)1/2ǫ ‖∇(u1 − u2)‖0,ωT

‖∇(e1,T − e2,T )‖0,T .

(2.16)

Let CI = C(θT ) ‖b‖∞,T (hT

ǫ ) = C(θT )PeTandCII = 3

2C(θT )maxE∈E(T ) (hE

hT)1/2. By

combining (2.10), (2.13) and (2.16), we have

‖∇(e1,T − e2,T )‖0,T ≤ [CI + CII ] ‖∇(u1 − u2)‖0,ωT≈ CI ‖∇(u1 − u2)‖0,ωT

,

becauseCI is the dominating term whenǫ ≪ h. Recallη1h,T = ‖∇e1,T ‖0,T andη2

h,T =

‖∇e2,T‖0,T . The above inequality implies

|η1h,T − η2

h,T | CI ‖∇(u1 − u2)‖0,ωT. (2.17)

Clearly, if ‖∇(u1 − u2)‖0,ωT≤ 1

2CIη1

h,T , we have

1

2η1

h,T ≤ η2h,T ≤ 3

2η1

h,T . (2.18)

2

Letηnh,T denote the error estimator computed from the solution obtained after n iterations

of a chosen iterative linear solver. Replacingη1h,T andη2

h,T by ηh,T andηnh,T , the following

corollaries can be derived from Lemma 2.1 naturally.

COROLLARY 2.2. Letuh be the finite element solution andunh be the iterative solution

after n iterations. If n is large enough such that

‖∇(uh − unh)‖0,ωT

≤ ch,ωTηh,T , wherech,ωT

= O(ǫ

h), (2.19)

6

then

1

2ηh,T ≤ ηn

h,T ≤ 3

2ηh,T . (2.20)

COROLLARY 2.3. Letuh be the finite element solution andunh be the iterative solution

after n iterations. If n is large enough such that

(∑

T∈ℑ∗

h

‖∇(uh − unh)‖2

0,ωT)1/2 ≤ min

T∈ℑ∗

h

ch,ωT(∑

T∈ℑ∗

h

η2h,T )1/2 (2.21)

, whereℑ∗h ⊆ ℑh andch,ωT

= O( ǫh ) then

1

2(∑

T∈ℑ∗

h

η2h,T )1/2 ≤ (

∑

T∈ℑ∗

h

ηnh,T

2)1/2 ≤ 3

2(∑

T∈ℑ∗

h

η2h,T )1/2. (2.22)

3. Stopping Criteria. In the adaptive solution processes, when iterative solversare usedto solve the linear system obtained from SDFEM discretization of the convection-diffusionequations, it is natural to require large enough iterative steps such that the iterative error|‖uh − un

h‖| satisfies

|‖uh − unh‖| ≤ c

∑

T∈ℑh

η2h,T for some constantc > 0,

and the error indicatorηnh,T is close enough toηh,T for all T ∈ ℑh. Corollary 2.2 and

Corollary 2.3 try to give a discription on how large the iterative steps should be. However,since the error estimatorηh,T can not be computed without knowing the exact solution oflinear systemAsdu = fsd, as a result, one can not use the inequalities (2.19) and (2.21) toconstruct the stopping criteria of the iterative solvers. In order to obtain computable stoppingcriteria, the following inequalities

1

2√

2<

|‖u− uh‖|Ω∣∣∥∥u− uhp

∥∥∣∣Ω

, (3.1)

and

0 ≪ maxT∈ℑhηh,T

maxTp∈ℑhpηhp,Tp

, (3.2)

are needed, whereℑhpis the coarse grid triangulation,Tp ∈ ℑhp

is the parent triangle ofthe elementT ∈ ℑh, hp is the diameter ofTp andηhp,Tp

is the error indicator computedfrom the coarse grid solution. The inequality (3.1) basically follows from the a priori errorestimation of the SDFEM solution which takes the form:|‖u− uh‖| ≤ chk−1/2|u|k where| · |k is theHk−semi norm fork = 1 or 2. The inequality (3.2) assumes that the maximumof the error indicator will not reduced dramatically as the mesh being refined. For problemswith exponential boundary layers due to the Dirichlet outflow boundary condition, the locallower bound (2.6) is dominated by the termhT

ǫ ‖b · ∇eT ‖ and the pointwise error estimationin [17] implies

‖b · ∇eh‖ ≤ ‖b‖h−1T ‖u− uh‖0,T

≤ chT |u− uh|∞,T , for some constantc > 0,

≤ c ∗ Chk−1+11/8T |log(hT )| ‖u‖k,Br∪Ω , for some constantC > 0,

7

here,Br is a small ball with the center at the point where the maximum|u − uh|∞,T is

reached and radiusr = O(h3/4T |log(hT )|). Clearly, for the casek = 1, we have

ηh,T ≤ C1

ǫh

1+11/8T |log(hT )| ‖u‖1,Ω0

, C is a constant independent withhT andǫ,

≤ C1

ǫ2h

1+11/8T |log(hT )|, C is a constant independent withhT andǫ,

whenever the element T is in the exponential boundary layer regions. As a result, for suchproblems, the inequality (3.2) holds. For other problems such as problems with only paraboliclayers, we will present some numerical evidence to support the legitimacy of the inequality(3.2) in next section. Here, we simply assume the inequalities (3.1) and (3.2) hold and deriveour stopping criteria as follows.

First, letrnh be the residual of the iterative solutionun

h, i.e. rnh = fsd −Asdu

nh. Since

‖rnh‖ωT

= ‖fsd −Asdunh‖ωT

= ‖Asd(uh − unh)‖ωT

≥ min Λ(AsdA∗sd)1/2 ‖uh − un

h‖ωT

√ǫh−1

T ‖uh − unh‖0,ωT

√ǫ ‖∇(uh − uh,n)‖0,ωT

, by inverse inequality,

we have

‖∇(uh − unh)‖0,ωT

ǫ−1/2 ‖rnh‖ωT

. (3.3)

The same analysis also gives

‖∇(uh − unh)‖0,Ω ǫ−1/2 ‖rn

h‖Ω . (3.4)

Now, we present two stopping criteria based on the inequalities (3.1) and (3.2) in the follow-ing theorems.

THEOREM 3.1. Assume (3.1) holds. If n is large enough such that the residual rnh of nth

iterative solution satisfying

‖rnh‖Ω ǫ

hmax(∑

T∈ℑhp

η2hp,Tp

)1/2, (3.5)

wherehmax is the maximum diameter of triangles inℑhp, we have

‖∇(uh − unh)‖0,Ω (

∑

T∈ℑh

η2h,T )1/2.

8

Proof: From the local lower bound (2.6), we have

(∑

Tp∈ℑhp

ηhp,Tp

2) ∑

T∈ℑhp

(∥∥∇(u− uhp

)∥∥

0,ωTp

+hp

ǫ

∑

T∈ωTp

∥∥b · ∇(u− uhp

)∥∥

0,T)2

4(hmax

ǫ)2

∑

Tp∈ℑhp

∥∥b · ∇(u− uhp

)∥∥

2

0,Tp

≤ 4(hmax

ǫ)2∣∣∥∥u− uhp

∥∥∣∣2

Ω

≤ 32(hmax

ǫ)2 ‖u− uh‖2

Ω , by (3.1),

≤ 32(hmax

ǫ)2∑

T∈ℑh

η2h,T .

(3.6)

By plugging the above estimates (3.5) and (3.6) into (3.4), the theorem holds.2

THEOREM 3.2. Letαη,∞ be a constant satisfying

αη,∞ ≤ maxT∈ℑhηh,T

maxTp∈ℑhpηhp,Tp

. (3.7)

Assume the maximum marking strategy is used with threshold valueθ. If

‖rnh‖ωT

(ǫ3/2

4hp)αη,∞θ max

Tp∈ℑhp

ηhp,Tp, for all T ∈ ℑh, (3.8)

then

1

2ηh,T ≤ ηn

h,T ≤ 3

2ηh,T , (3.9)

for any marked element T. On the other hand, for elementT satisfying

ηh,T <3θ

8maxT∈ℑh

ηh,T (3.10)

will not be marked by the same marking strategy withηh,T replaced byηnh,T .

Proof: First, for any elementT ∈ ℑh, (3.3) and (3.8) imply

‖∇(uh − unh)‖0,ωT

ǫ

4hpαη,∞θ max

Tp∈ℑhp

ηhp,Tp

<ǫ

8hθ max

T∈ℑh

ηh,T .(3.11)

Let T be a marked element satisfying

ηh,T ≥ θ maxT∈ℑh

ηh,T . (3.12)

From (3.11), we have

‖∇(uh − unh)‖0,ωT

<ǫ

hηh,T .

9

By Corollary 2.2, the inequality (3.9) holds. Now, letT be an element satisfying (3.10).Recall that

ǫ

h|ηh,T − ηn

h,T | ≤ ‖∇(uh − unh)‖0,ωT

, (3.13)

from (2.17). By combining (3.11) and (3.13), we have

|ηh,T − ηnh,T | ≤

θ

8maxT∈ℑh

ηh,T .

Therefore,

ηnh,T ≤ ηh,T +

θ

8maxT∈ℑh

ηh,T

≤ θ

2maxT∈ℑh

ηh,T by (3.10),

≤ θ maxT∈ℑh

ηnh,T , by (3.9).

The second part of the theorem is proved.2

REMARK 3.3. Theorem 3.1 ensures that the global a posteriori upper boundwill not beviolated when it is computed from the iterative soultion that satisfies (3.5). On the other hand,Theorem 3.2 guarrantees that the mesh generated fromηn

h,T will not produce serious over-refinement compared to the mesh generated fromηh,T when the maximum marking strategyis used. Moreover,ηn

h,T ≈ ηh,T in the regions that can be marked by usingηh,T and themarking strategy (2.3),

REMARK 3.4. For deriving stopping criteria, the inequality (3.2) is needed here becauseof the maximum marking strategy. It has been shown in [22] Theorem 5.2.7 that when themarking strategy in [6] is used, a stopping criterion similar to (3.8) can be derived by utiliz-ing the Corollary 2.3 without assuming the inequality (3.2).

4. Numerical Studies. In this section, we show that the refined meshes are almost thesame no matter the meshes are generated fromηh,T or ηn

h,T for some benchmark problems.Our numerical results seem to support the comments in Remark3.3 about Theorem 3.2. Sincethe criterion (3.8) in Theorem 3.2 needs to test the residuallocally on each element patch, wealso like to investigate how does this criterion affect the iteration counts for various iterativesolvers. The following two benchmark problems are considered.

Problem 1: Constant flow with characteristic and downstreamlayers: The equation(1.1) is given with the coefficientb = (0, 1) and the right hand sidef = 0 on the domainΩ = [−1, 1] × [−1, 1]. The Dirichlet boundary condition is set asg = 1 on the segmentsy = 0 ∩ x > 0 andx = 1, andg = 0 elsewhere.

Problem 2: Flow with closed characteristics:Here, the coefficient vector(b1, b2) is(2(2y− 1)(1− (2x− 1)2),−2(2x− 1)(1− (2y− 1)2)) and the righthand sidef = 0 on thedomainΩ = [0, 1]× [0, 1]. The Dirichlet boundary condition isg = 1 on the segmentsy = 1andg = 0 elsewhere.

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SDFEM solutions of the above problems for the caseǫ = 10−3 are shown in Figure 4. Ineach problem, the linear systems are directly solved on the coarsest 4x4 grid first. Then thefollowing procedures are followed:

−1

−0.5

0

0.5

1

−1−0.8−0.6−0.4−0.200.20.40.60.81

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

(a) Solution of Problem 1 on a32 × 32 grid

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

10

0.2

0.4

0.6

0.8

1

(b) Solution of Problem 2 on a64 × 64 grid

FIG. 4.1.SDFEM solutions of Problem 1 and Problem 2 for the caseǫ = 10−3

1. Compute error estimatorη.2. Select elements according to the maximum marking strategy.3. Refine selected elements and generate a new mesh.4. Obtain the initial guess by interpolating the current solution to the new mesh.5. Solve linear system so that a given stopping criterionSi is satisfied.

For the linear solvers, three methods, Gauss elimination (GE), the multigrid method(MG) and the generalized minimal residual method (GMRES), are used to solve linear sys-tems. In the MG solver, the prolongation operator is the standard linear interpolation, therestriction operator is the adjoint of the prolongation operator and the correction operator isobtained from direct SDFEM discretization on the coarse grid. We use the same Gauss-Seidel(GS) preconditioner for both MG and GMRES. In Problem 1, the preconditioner is one stepof vertical GS sweep. In Problem 2, the preconditioner consists of one bottom-to-top verticalGS sweep, one top-to-bottom vertical GS sweep, one left-to-right horizontal GS sweep andone right-to-left horizontal GS sweep. Three different criteriaSi, i = 0, 1, 2, are chosen.WhenS0 is given, the linear systems are solved directly by GE. When using MG and GM-RES solvers, the stopping criterionS1 represents the heuristic stopping tolerance, i.e., theL2

norm of the residual of iterative solutions less than10−6 and the stopping criteriionS2 is thecriterion (3.8) in Theorem 3.2.

For the mesh refinement, the thresholdθ in the maximum strategy and the number ofrefinement steps are carefully chosen so that more detail layer structures of the solutions canbe seen during each refinement step. For both problems, the threshold is set to 0.1. For thenumber of refinement steps, four steps are performed for the caseǫ = 10−2, seven steps areperformed for the caseǫ = 10−3, and eight steps are performed for the cases10−4.

In the following, we compare the refined meshes generated by the error indicators com-puted from GE, MG and GMRES solutions for each problem with 3 for different values ofǫ = 10−2, 10−3 and10−4. The iteration steps among different linear solvers and stoppingcriteria are also compared. The error indicators computed from solutions satisfying criterion

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S1 ,S1 andS2 are denoted asη0, η1 andη2, respectively. Since, Table 4.1 and Table 4.4 showthat the constantαη,∞ ≫ 0 in (3.2), we setαη,∞ = 0.5 in the stopping criterionS2. Table 4.2and Table 4.5 show the number of iterations for MG and GMRES tosatisfy different stoppingcriteria on various mesh levels. Table 4.3 and Table 4.6 showthe number of nodes generatedby error indicators,η0, η1 andη2 computed from MG solutions, along the refinement process.From, Table 4.2 and Table 4.5, it is clear that fewer iterations are needed for MG to satisfy thecriterionS2, i.e., the criterion (3.8) in Theorem 3.2. In contrast, moreiterations are neededfor GMRES to satisfyS2 comparing to the iterative steps needed for GMRES to satisfyS1.The total amount of work of MG with our stopping criterion is about half of the amount ofwork of MG with the heuristic stopping criterion. Finally, from Table 4.3 and Table 4.6, wecan see that the number of nodes of each refined mesh remains almost unchanged no matterwhich error indicator is used to generate the meshes.

Numerical results for Problem 1:

ǫ maxT∈ℑh

ηT

ηTpon refined meshes

10−2 0.5 0.5 0.5 0.49910−3 0.5 0.5 0.5 0.5 0.5 0.5 0.510−4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

TABLE 4.1Verification of the assumption (3.2) of the new stopping criteria

ǫ Tol Iterations

10−2 S1 12 10 9 12S2 4 4 4 7

10−3 S1 16 13 11 9 8 8 15S2 7 4 4 4 3 4 13

10−4 S1 16 14 12 10 9 8 8 8S2 9 6 5 5 4 3 3 3

(a) MG iteration steps

ǫ Tol Iterations

10−2 S1 11 12 13 17S2 26 26 26 30

10−3 S1 11 12 12 12 13 15 28S2 26 26 26 26 27 27 35

10−4 S1 11 12 12 12 13 15 16 17S2 26 26 26 26 27 27 28 28

(b) GMRES iteration steps

TABLE 4.2Comparison of iteration steps for different stopping criteria

ǫ Error Indicator Node number10−2 η0, η1, η2 50 97 190 394

10−3 η0, η1 50 91 174 343 697 1350 2702η2 50 91 174 343 683 1359 2702

10−4 η0, η1 50 91 174 343 679 1346 2674 5331η2 50 91 174 343 679 1346 2688 5334

TABLE 4.3Comparison of number of nodes of refined meshes from MG solutions

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Numerical results for Problem 2:

ǫ maxT∈ℑh

ηT

ηTpon refined meshes

10−2 0.34 0.30 0.36 0.2410−3 0.34 0.37 0.51 0.68 0.75 0.44 0.1710−4 0.34 0.31 0.33 0.24 0.71 0.54 0.82 0.52

TABLE 4.4Verification of the assumption (3.2)of the new stopping criteria

ǫ Tol Iterations

10−2 S1 20 10 8 6S2 6 5 4 3

10−3 S1 41 21 16 14 18 18 10S2 13 11 10 8 11 10 6

10−4 S1 52 27 22 24 17 15 15 25S2 22 17 16 16 9 9 11 21

(a) MG iteration steps

ǫ Tol Iterations

10−2 S1 15 23 20 36S2 27 30 33 36

10−3 S1 28 34 37 39 45 54 53S2 28 34 37 39 45 54 53

10−4 S1 28 24 39 32 35 42 56 69S2 28 35 39 32 35 42 55 69

(b) GMRES iteration steps

TABLE 4.5Comparison of iteration steps for different stopping criteria

ǫ Error Indicator Node number10−2 η0, η1, η2 70 168 390 911

10−3 η0, η1 70 176 345 592 948 1458 2391η2 70 176 345 592 948 1458 2391

10−4 η0, η1 70 176 354 764 1143 1752 2674 4093η2 70 176 354 764 1143 1750 2688 4082

TABLE 4.6Comparison of number of nodes of refined meshes from MG solutions

5. Conclusion. In this work, we have presented two stopping criteria, (3.5)and (3.8),for the iterative linear solvers based on the a posteriori error estimations and the maximummarking strategy. We have shown that the iterative error canbe bounded by the upper boundof the a posteriori error when the iterative solution satisfies the criterion (3.5). Numericalstudies in [22] indicate that the a posteriori error estimations by Kay and Silvester are opti-mal. Therefore, in the sense of measuring the global true errors between numerical solutionsand the weak solution of the PDE, the exact SDFEM solution andthe iterative solutions sat-isfying criterion (3.5) are indistinguishable. On the other hand, although the global errors ofiterative soultions can be confined by using cirterion (3.5), this criterion does not guarran-tee the effectiveness of the mesh refinement process since the marking strategy is essentiallybased on the local information about the error indicator. Here, we have also shown that whenthe error indicator is computed from iterative solutions, the criterion (3.8) can be used toavoid refinement over wrong locations or over-refinement. Numerical results in Section 4support the effectiveness of the criterion (3.8). Moreover, we have found significant amount

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of work can be saved by using the MG solver with Gauss-Seidel smoother that satisfies cri-terion (]refs3:eq18). According to this study, we suggest that the stopping criterion (3.8) canbe used to achieve the efficiency of the mesh refinement process and stopping criterion (3.5)should be verified at the finest mesh where a reliable solutionis expected.

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