_N'H~~~

ELSEVIER

Abstract

JOURNAL OFCOMPUTATIONAL ANDAPPUED MATHEMATICS

Journal of Computational and Applied Mathematics 74 (19961245 293

Local and pollution error estimation for finiteelement approxill1ations of elliptic boundary value

problell1sJ. Tinsley Oden*. Yushcng Feng

Texas University. TlC4M Tay 2-4000, A/lstin. TX 787/2, USA

Received 15 July 1995

This paper addresses the issue of local clementwise error estimation of finite element approximations of ellipticboundary value problems. The characterization of element error as local and pollution error component is presented andthe relationship between energy norms of local errors and that predicted by means of a posteriori error estimators isinvestigated. In addition. techniques for calculating element indicators of local error. pollution error. and other errorindicators are presented.

Ke\'lI'O/,{/s' A posteriori error estimation: Finite elements; Pollution error

1. Introduction

A posteriori error estimation is viewed as a fundamental component of modern adaptive finiteelement methods. In virtually all sllch methods. local (element) error indicators are computed andthe mesh is then refined in the regions where the error indicators exceed a preset tolerance. Thus.the success of adaptive schemes may depend upon the accuracy and eflkiency with which sllch localindications of error can be made.

It is well known. however. that available a posteriori error estimates yield only global bounds onthe error. For example. the element residual method (ERM) for a posteriori error estimations forfinite element approximations of elliptic boundary-value problems (e.g. [2J) yields global bounds

*Corresponding author.

0377-0427/96($15.00 {:' 1996 Elsevier Science B.V. All rights reservedPII S0377-0427(96)00027-1

246 J.7: Odc/1. r. Ft'ng/Joll/'/lal of' Compurational alld Applied Alathematics 74 (199M 245-293

of the type

1ieilE ~ L,tl 11~f/2where lIellf: is the global error in an energy norm and 111\ is a local error indicator for element Q1\ ina mesh containing NI; elements. The numbers 111\ may be a poor approximation of the actual errorin cnergy of element !h.

In reccnt work. Babuska et a!. [5. 6J have pointed out that local error estimation techniques suchas the ERM. may be incapable of detecting "pollution crror" produced by singularitics in thesolution at points in thc domain remote from the element in question. In order to obtain accuratclocal estimates in energy norms, it was argued in [5] that this pollution error in each region ofinterest must also be estimated.

The related investigations earlier by Nitsche et a\. on a priori local estimatcs indicate that thecrror on a region Q{J c D in the Lz - or Lor -norm can be bounded by the sum of two terms. onedetermined by local interpolation properties and the regularity of the solution ncar Do. and theother (the pollution error) determined by the regularity of the solution outsidc of the region. theregularity of the hOllndary. and the particular boundary operator involvcd in thc approximation(cf. [9. 11-13]).

In this paper. a gcncral approach toward a posteriori estimations of local and pollution errorsfor a class of linear clliptic boundary value problems is presented. A new characterization of localand element wise pollution error is given and new local error indicators arc introduced whichprovide measures of the local and pollution errors. Also. the relationship betwecn the local error ina finite element. measured in the energy norm. and the error indicators delivercd by the ERM isestablished.

The analysis described here is based on the assumption that if till is a finite elcment approxima-tion of the solution It to an elliptic boundary value problem on a coarse mesh. and It I, is anapproximation on a finer mcsh. thcn Llil is close enough to LI in an energy norm that (till - L1lf) isa reasonable approximation to the crror. The fine mesh approximation It" is actually nevercomputed. but is inferred from computations typical of standard error residual methods fora posteriori error cstimation. When this assumption is valid. the entire process of local and globalerror estimation reduces to a finite dimensional problem on the fine mesh. In fact. it is shown thatthe combination of local ERM and pollution error estimates yields lower and upper bounds fordiscrete local error (tlh - L1H lin,. i.e..

IIfP~IILJ. + IIfP~·I'O'lll.Q,~ lltll - 1t1l)Ii.fJ, ~ (1 + jl1\llfP~IIi.Q. + IIfP~·polllru,where fPZ and q>~.I'''1 arc ERM and pollution estimators respectively. and ..111\ is a constant generallydcpending on mesh parameters II and p. Numerical experimcnts on specilic model problemsindicate that .111\ is essentially constant and. does not vary dramatically with thc problem size. Thenumerical results also illustratc the role of various components of the local error.

2. Notation and preliminaries

To set the stage for our stlldy. we first record some notations and preliminary results.

J. T Odell. r. FClIg/Joumal ~rComputational alld Applied Mathematics 74 (/996) 245-293 247

2. /. Model pl'Ohiem

We begin by considering a model class of the problems of the following form:

I Find II E V such that a(u, u) = L(v) \I v E v·1Here.

V = {v E HI (Q): j'o I' = ° on r D}'

a: V x V -+ IR: L: V -+ IR.

a(lI. v) = L [W I;)TAF II + mw] dx,

L(v) = j' fll dx + J' Yl' ds.Q f',

(2.1)

(2.2)\Ix E !2

Q = an open bOllnded. Lipschitz domain in 1R"./l= 1.:2 (Fig. I). with boundary ()Q = [DUrN./"DnL, = 0. meas [D > 0, dx = dX1 ... dxll.fE 1.2(Q): () E L2([ ....).

We employ the usual notation Hm(Q) for Sobolev spaces of functions with norms 1I!JIlIn.a andwith generalized derivations of order ~ III in L2jQ): the trace of v E HI(Q) on the boundary isdenoted }'o I'.

The coctlkients A and 0" are assumed to satisfy conditions suOlcient to guarantee that (2.1)corresponds to a well-posed. symmetric. linear elliptic boundary-vallie problem. For example, thefollowing are sufficient for our purposes:

3AJ J .D, 0"\ E IR+. 0"0 E IRsuch that

AI 11'1' I' ~ I,T A (x) I' ~ DIIT I': A(xl = A J'lx)

A E (Co(Q))" ,", I' E IR"

x = (XI,X2 ..... xll)

Fig. I. A l\\'l)·dimensional Lipschitz domain.

248 J.T Odell. r. Fellgl.lournal ofComp/ltatiollal and Applied A!mhematics 74 (/996) 145-293

The model problem 12.1) is clearly cq uivalcnl to the boundary-valuc problem

All =I in Q.

II = 0 on I lJ

A J7 II • 11 = q on IN

whcre 11(x) is the unit exterior normal at x E cQ and A is thc linear symmctric operator:

A (x) = - J7 • A (x)J7 + ('j(x)I.

Then. we have.

(2.3)

(2.4)

ProJlosition 2.1. Under cOl1ditiol1s (2.2) the /Jilil1('m'/orlll II ('. ,) is umtilll/Ous. sYlI/lI/etric al1d coercit:e.till' lincor ./illlctiol1(// L is cOl1til1uous, IIlId prob/ell/ (2, I) possesscs (/ IInil/ue so/utioll liE V.

The energy norm of functions v E V is defined by

Ilr"E.n = ,/a(u. r).

.,., Discretizatio/l

(2.5)

Next, we use finite clement methods to construct discrete (finite-dimensional) approximations ofV and of problem (2. I). A family !}JH of partitions of Q into N f; subdomains fh is considered suchthat

Sf

!2 = U {l1\. Q1\nQt. = O.1\ = I

K#:L

H 1\ = diam(Qd. H= min 111\.1 ~ K ~ j\'1

We refer to Q1\ (or its closure !21\) as a finite element. Now. we construct a polynomial subspace~/ H of V such that the global basis functions are denoted /..,V and dim V 11 = Nil:

II f. l'vIIV = span \/...\'f,V= I'

Functions "11 E VII are represented as linear combinations of the functions /..",:\'

rJ/(x) = ('ill.\'(X).

(2.6)

(2.71

Here (/nd throlly/wllt this paper lI'e lise the slllllmatioll cOIl/'cntioll: repeated iI/dices are Slllllll/(///{/throuqllOlit their r(/Il!lc (1 ~ N ~ N 11 ).

The discrete approximation of (2.1) is characterized by the following problem:

Find U/1 E V II such that a(lIu, pu) = L(1111) V I'u E V 11. (2.8)

J.T OdclI. r. Fellg!JvlI/'lIul a/Computational and Applied Mathemalics 74 (ll)l}tij 245-293 249

2.3. A Posteriori errol' estimation

Given the solution l;/t to (2.8), we now consider thc question of estimating the approximationerror.

e = II - LIlt (2.9)

in an appropriate norm. One approach toward resolving this question is provided by the errorresidual method (ERM) introduced in [10J and generalized in [1.2] which yields global a poste-riori bounds in energy norm. The major fcatures of the ERM of interest here for problem (2.1) aresummarized in thc following five remarks:

(I) E/ellle/ll hOl/lu/ories: Consider element QK in the partition YJI of Q (Fig. 2). The boundary('QK of this clement is made up of sets In = c'QK(\?QL, 1 ~ K.L ~ N£ consisting of p(K,L)smooth arcs ri~. and isolated points Sn such that r "I. = Uf/~{'r~~.uSn: we set1'0" = DQ(\(;Q". Thcn the element boundary in the interior domain on the partition ·:9'1t is definedas

s,J~. J' ,il U r"t= (.YIt) = n·

".1.= I." > t.I ~.\I",'IK.I.1

The total clcmcnt boundary. including r and the segment on cQ. is dcnotcd 1'0. and we set.\",

cr~ = aQ" \ U S"I.·1.=0

,...'

Fig. 2. A typical clement Q,. in domain Q.

250 J. T. Odell. l'. Feng /Jo/{I'I/al {)( Computatiol/al {[lid Applied Alathematics 74 (199M 245 --293

(2) Bro/.;ell SP{/(,l'S (sP{/(,l's 01/ (I p(lrtitiol/ .-1'1/): We introduced the spacc [-J1II(:1'/() on thc partition:3'H of Q defincd by

H III(.JPII) = {,;E L2(Q): Il/o:= vlQl E HIII(Q,,): I ~ K ~ Ndand

V (;Y'/() = {r E H 1 (.1)1/1): '/0 I' = ° on In}

The functions in II (YII) can sulfer jllmp discontinuities across inter-elemcnt bOllndary segmentsri~..Indeed, if rK E HI (QKl and VI. E HI (QLl . and if (~Q/o:rH""QL =1= 0 . then the inter-element jumpacross the common segment I~~_ is denoted by

where i.I.:!. = - i.l•K = I if K > L or - I if K < L. We denote by V /0: the subspace of V (.1P1l )

containing functions of the form

ria, == 0, L =I- J\..

whcre /'0 is the tracc operator mapping HI (Q/o:l onto H tl2 (aQf.:). Then.

"',V (·3'H) = L V /0: •

/0:= I

(2.10)

For simplicity. we follow standard practice and ass lime that each clement Q" is thc image ofa master element (} under an affine invertible map F /0:. We denote by QP(D) the space of tensorproducts of integrated Legendre polynomials of dcgrce ~ p on Q; the function in QP(Q) isdescribed in dctail for rJ c 1R1in the Appendix. Also, .3'p(Q) is the space of polynomials of degree~ p defined on (1 For clement !l/o:. we denote

.3'p(!h) =: 1,'111 = v"t'(xl: x E !l/o:. vllp = t: F;: 1. t E.3' p(Q)}

and

QP(Qd = {l/IP = vhp(xl: xE Q/o:. vhp = {;:F;:I, [E QP(Q)}.

For definiteness. we shall focus on cases in which Q = ( - 1. I) x ( - I. I). although the theory andapproach are applicable to morc general element families.

(3) Local hi IiI/ear (lm/lillear forllls: Denoting

aK(lt.l') = j' [(VI;f' AI'" + 0'11/:] dx.11,

\ve have

(2.11 )

S,

(1(11, v) = L (/,,(11. V),"=1

.\',.

L(v) = L L,,(p) ."=1

(2.12)

1. T. Odell. Y. FClIg/Journal ~rComputational alld Applied Mathematics 74 (/996) 245- 293 251

(4) Elemel/t residuals: If Uu is the solution of (2.8). we detlne the following rcsiduals correspond-ing to clement 52,,:

Interior e1emelll residual for dell/em 521.::

o III U [JJ\fll.:' 1 ~ K ~ NE·J

Boundary elemellT residual for element 521.::

(2.13)

I·M1Il T,\n 01.:. I ~ K.L ~ NJc' 1 ~ AI ~ /I(K.O).

o

R~ = ~1[a~]r~L + On in Tt~.. I ~ K. L ~ t·.jE. 1 ~ 1\1 ~ p(K. L).

111 ( 0 (l~h)\ar~.L=1

Here a~ is the approximate normal flux associated with Uff,

a~ = "K ·AfuIl

(2.14)

(2.15)

"K being the unit exterior normal to a~h.and On are functions designed to have the property

(2.16)

and

(2.17)

where p is the order of the polynomial basis functions X'" over element Ql\' The residuaifullctionalfor 521.:is defined by

or. equivalently.

(2.18)

Sf.

" II'L... I~I.: ('j/'-'.'1L=O

1".\1 .. plI'.J.1

(2.19)

where ~nl.: represents a distribution corresponding to the residual functional ~H".and l5n".is theDirac delta (the single laycr distribution) concentrated on segment /,~ft. (i.c.. ({)r:',. cp) = <p(s).S E riA. cp E .Q(Q), wherc [t(Q) is the space of test functions defined in Q).

252 J.T. OdCII, r. Fcng/Joumal o/Computatiol/al rind Applied Mathematics 74 (1990) 245-]93

(5) The 10('£11 errol' illdicators: In the ERM. we considcr local boundary vaillc problems overcach clement QI\ or the form.

(2.20)

Notice that because of the way wc have defined the spaces V K' (2,20) is equivalent toa(C{JK,17)= ~HI\(r)VI' E II1\. Then. in (2.20), it is understood that ~nK is the restriction of the functionalin (2.19) to the subspace VK C VU.1'Il)' The following thcorem summarizes thc key a posteriorierror estimation property of the ER l'vl.

Theorem 2.2. (Ainsworth and Oden [L 2]). Let

'1K = ...../aK( C{JI\.C{J,,). 1 ~ K ~ N E (2.21)

(2.22)

wh/.'r/.' C{JI\is tile solutiOIl o(rhe local prohlem (2.20). Theil the app/'oximatioll errore = II - 1I11.'i(lli,~fi/.'s

1IeIIE.Q ~ {"tl 'd }'!2Thus, the ERM. with local error indicators given by (2.21), provides a rigorous. global. upper

bound to the error in the energy norm.

3. Fine mesh approximations

3. J. Approximation (~rthe error indicators

In practical calculations of the error indicators C{J" • the local problem (2.20) cannot be solvedexactly. and approximate indicators C{J~ must be lIsed. Thus. instead or (2.20). we address thediscrete problems.

(3.1)

H ere V~ is afille- or /.'Ilriched-space approximat iOIl oj V K :.fC)/· example. ifpolYllo11lilli hasis JlInctionsl.:\' (?ldegree pal'/.' IIsed ill definillff the approximate soilitioll space Vlt to which lilt be/OllflS. the 10('£11£'1'1'01'illdicators are compl/ted ill a space V~ obtained by addillfilO this basis Ilell'jilllctiolls of degreep + k, k > O. AI or£' specUica/ly. we effecl iuel,v allgmellt V 11with a space of' pert I/rhatiolls

V 1111 = span {()' 1 ~'Y. ~ N'rlt}

where C E V are the .filiI.' mesh pertl//'bal iOIl jimct;olls. We then have

V" = Vlf + VllltC V.

Vh=span{{I.:\'}.{C}: I ~N~NIt.1 ~c:L~Nhlt}.

II" Sp'111{' 01 vOl ~" =., Is ti.· \,) Ii. J •

(3.2)

(3.3)

1. T Odell. Y. Fellg/Jurlrlwl of Compwational and Applied Mathematics 74 (/996) 245-293 253

where .%,~I(l,.Glla, are the functions obtained by extending by zero to the restrictions ;(vI6, and ~"Itl,to Q/flK• and dim V" = N" = Nil + N',II' Similarly. we dcnotc

S1

J;d'('-¥H)= L VZ·K= I

The approximate error indicators are thcn

Ir J (', Ir)'IK = (/K CPK' CPK

and the global error is estimated by the quantity

(3.4)

(3.5)

(3.6)

Remark 3.1. The functions cP~ may, of course. be a poor approximation of CPK' If /..,"are piecewiselinear or bilincar functions. experience indicates that it is often sullicient to take /.:= 1 or 2 toproduce acceptable indicators [2], provided the solution is regular. In [I], a procedure is givcn inwhich /.: is increased locally until/l~ remains essentially unchanged. In examples involving strongsingularities. such as crack problems. this process may require using k = 3 or 4.

Implicit here is the assumption that the approximation error in the energy norm is adequatelyapproximated by the encrgy norm of the fine mesh error

(3.7)

where II" is the fine mesh approximation and is the solution of the problem.

(3.8)

Every function v" in the enriched space Vir is of the form

v" (x) = V h" l.,dx) + iif. (" (x). X E awhere Vi~' and Vh are real coellicients and repeated indices are summed throughollt their ranges:1 ::::;;N ::::;;Nil. 1 ::::;;~ ::::;;N/,II' Let

A.\'.\I = a(;(". 1..\1 ). A.\'M = a- t (b.!..\[).

Then. without loss in generality. we may introduce the new basis functions

11,,(x) = (,(x) - A'LA L.\t bdx).

We then have the orthogonality condition

a(b·./I,,) = 0, 1::::;; N ::::;;Nil. I ::::;;,'Yo ::::;; NIrH (3.9)

254 1. T Odell. Y. Fellg/Joumal of Computational aild Applied Mathematics 74 (1996) 245-293

and

Vh = span{ {x,d. {A,}: I ~ N ~ NH• I ~ 'Y. ~ Nhll}.

V'I(X) = Vi} Xx (x) + V,~11.(x), 'V Vh E J;·h. X E Q. (3.10)

Remark 3.2. The orthogonality condition (3.9) is introduced only for simplicity in notation. it isnot essential in any subsequcnt development. With it. we have V', = Vll EEl V'IH. VII n V"II = 0.

Since eh E Vh. it follows that constants ES and e" exist such that

e'l(x) = EN X,v(x) + e' /1, (x); x E 51 (3.11)

where. again. repeated indiccs are summed. 0111'goal is (0 characterize (he coe.lficiellts E'" and eJ andto characterize the local demelltwise error alld its relatiollship (0 cp~ (and 11~). 111 particlliar. we willexplore how (he residuals ouTside all elell/ellf poillue (he esfill/ate of the local error.

3.2. Calclliatioll of the errol'

In order to analyze local and pollution components of e,,(x). we next establish a useful theorem.

Theorem 3.3. For every K. I ~ K ~ Nil. defille W~ E 1"1 as (he sollttioll (~rthe bOlllldary wlueproblems

Then(i)

s,.a(e,!,v,,) = L a(J;V~,v,,) 'Vr"E V".

1\=1

(ii) If ~v~(x) = Wi x..,,(x) + \\'~,,1.(x). x E Q, we have

(3.12)

(3.13)

N,

e" = L ~Vi'1\ = I

Proof. (i) Clearly.

N.

and e' = L w~ .1\=1

(3.14)

a(e'l' r,,) = a(It". lih) - a(ltll' rh)

= L(Vt,) - a(III1' V'I)

Xc= L (Ldl;h) - adltH, rh))

1\=1

~ (J" f)= .L (J- Altll)r"d.\' + . a~ V'lds1\ = 1 a. . O'll.

J.T. Odell. r. Fel/g/JOIlrlwl ojCompl{(Cltiol/a/ and Applied A/athematics 74 (/Y96) 245H293 255

N,

= L ~HdL'h)1:= I

1\"

= L lI(W~.l''')1:=1

as asserted.(ii) If 1.'" is given by (3.11). insertion into (3.13) gives

EN t Nt; IPSAf N.\( = L I: = I 'I' I: N ,It·

'h ......., '/e '(I = L. 1:"= 1 II'K J'{I'

where

h1P=a(A1.Ap). I ~'J..fJ~N"ll'

Solving (3.15) for EN and e' yields (3.14). 0

Clearly. e,,(x) is the solution of the following problem:

.... t;

Find 1.'1, E Vi' such that a(e". 1'1') = L ~HKll'll)' "if VI, E V" .1:=1

As a consequence of Theorem 3.1, we have

Corollury 3.4. Let

(3.15)

(3.16)

(3.17)

[A.I/N J - I = A ;\'.11.

~HI:(zX) = ~HK.\"J.b _ ,xc 'I'

.71.\' - L.K= I . 'KS·

[h,/I] - J = h,/I.

~Hd;11)= Pl:l'

,Xlp, = L.K= I PK1'

I~ALN~Nll'

(3.18)

Theil. error (3.11) at allY poillt x E fl is Y;VCIl hy

(3.19)

256 J. T. Odel/, Y. Fel/gjJo/l/'l1al tJ( COlllputatiol/al and Applied Mathematics 74 (1996) 245 -293

Proof. If J'F~ is characterized by (3.12). it follows that

H/ S _ ~ S.\t ,1} andn l' - /1 :/11'.\1 '

Thus. (3.19) follows from (3.14). D

Remark 3.5 (Discrere Greell's,/llllctioll). The Green's function G~ of the problem (2.3) satisfies, forv E 9(Q).

(l(G~, v) = r(~).

Let ct be the discrete approximation of G~in V lL

II . Gil .Go: (x) = 0: X,v/x)

(/(GrX"rl = Zlrl~), I:::; AI:::; NIl'

Thus,

I G:(x) = AN.\fbt(~)XN(X).

If ~ = xR is a (vertex) nodc and f.,\t(xR) = c)J~' then

G.~""= A !\lR

and

GII() ASR - (v)Ix' X = I..N ., .

(3.20)

This relation provides an interpretation of the first tcrm in (3.19) as the action of the discreteGreen's function on the rcsiduals ~HN;likewise, b%p/1p(x) rcprescnts the fine-mesh correction of thediscrete Green's function on p,. D

Eg. (3.19) provides a charactcrization of the pointwise finc-mesh error. It remains to examine thebehavior of the local (element wise) error and its relationship to the error indicators.

4. Local and pollution errors

4.1. Characterization of local and pol/lIlion errors

Once again we concentrate on the partition '~JJ of Q. the enriched fine mesh space of functionsVh defined by (3.10), the error e/. given by (3.19), and we focus on a typical finite element [21" Weintroduce the sets of integers.

SK = {N E N: X",(x) =1= O. x E QK' 1 :::; N :::;Nil}'

TK = {::t. EN: /l.(x) =t= O. XE {iK. I:::; 1:1.:::; Nw}. (4.1)

1.T. Oden. r Fel/g /Journal of Computational and Applied Mathematics 74 (/996) 145-]9_~ 257

Then, the fine mesh approximation of the 1.'1'1'0,. due /() local residuals in elemem Q" is defined by

e~~(.\") = L ~ll#.',\'.4'\··\[Lu(X) + L p",b,/lAf/(x)S ..\[ ES. ,. /lE T/.

(4.2)

The remaining fine mesh error at a point x is due to residuals outside of Q" and is defined as thepolll/tioll er,.or.

ek~,J(x) = L L: ~H}NA'\',\( X.\t(x) + I L flJ.b,/l A/l(xlN . .\IE.h}"#.' ,./lET.,},,"

(4.3)

Comparing (4.2) and (4.3) with (3.19). we observe that the error at a point in Q" is given by

(4.4)

A further characterization of the local and pollution errors can be defined by introducing thefollowing coetlicients:

{O.

M _ \'.\tEK.loe - ~HK.\ A' .M¢SK'N.AfESK· {

O./l - ,/le".loe - PK,b .

fJ¢T".1.. II E T,,_

L Phb,/I. 1../l¢T1\.}"K

0.

E·\t =:'K.pul

"'I' AS,\( ." ·\/~SL. .\}.\' . ''','1''1\'}""

0.(I _

('1\.pnl -

fi¢T1\.(4.5)

Then. for x E Q. we have the following compact form (repeated indices are slimmed as notedearlier):

e~,~(x) = Etlocl..dx) + e~.loe A,(x)

ek~,I(x) = EtPOll..v(X) + e~.poIA,(x)

xEQ. 1 ~ K ~ Nt:. 1 ~ M.N ~ Nu. 1 ~1. ~ N"1I'

4.2. Algebraic sTructure (?t' error componenTS

(4.6)

An alternative interpretation of the characterization of the local and pollution errors is to uscmatrix notation.

258 ),1. Odell. r. FellgjJolI/'/lal ofComplItatiol1al Cll1dApplied Mathematics 74 (/996j 245-293

Fig. 3. An interior subdomain in Q.

SlIppose a is decomposed into two subdomains. an intcrior sllbdomain fh and a sllbdomainaK outside of aK. such that

Q = aKulluQi:.UcQ

where QK is an elcment (or a patch of elements). and lt is the interface between QK and Qi:. (secFig. 3). Let

VII = span {t/JII(X): 11 = 1, .... Ndwhere t/J/I are generic basis functions representing those in either (3.3) or (3.10): then the errorequation (3.] 7) can be expressed in terms of matrix notation as

A&=R

N"ell(x) = L 8/1,p/I(x),

/1= ]

(4.7)

Eq. (4.7) can be rewritten in terms of submatrices associated with degrees of freedom insubdomains QK, Qi:. and interface lt (the subscripts K. K and I indicating the correspondingsubmatrices).lA

K AKI 0rK R

KA~K Al Ari:. ~( = RI \ (4.8)

Aj;:l Ai:. 8i:. Rj;:

with

t,K] rK

8 = I ~J • R = Rr

8i; Ri;

(4.9)

J.T. Odell. r. Fellg/Jouf'I1al q(Compuwtional and Applied Alathell/atics 74 (/9961 245-293 259

In light of Thcorem 3.1. thl: residual vc<:t()J'R can he considercd as the sUln-of t\VOterms: oneassociated with degrees of frccdom in QK. the other associated with degrees of freedom in Qi. Theresidual componcnts (cf. Definition 2.14) on interface rl are split into two parts using flux splittingschemcs (e.g. [4J), i.e..

fRK '( r RK 1 I' o. 1

R = RI' = (R~K) J + ' R~KI \ .lRi J l 0 l Ri J

Then the local and pollution crror can be characterized in thc same way as in (4.2) and (4.3):

AK AKI'J' '

o jl&t<1 IR,AIK At A - 61K) - R(K)

IK 1'11-, I0 Ail Ai 6~o' 0

and

A" AKI 0 8 pOll 0'K

AIK AI AtK t(i) = R(i)? I 1 I

0 Ail Ai ttl Ri)

(4.10)

(4.11 )

The solutions of (4.10) and (4.11) provide the complcte relative representation of crrors on Q. andreveal how thc residuals on Qi elTect the crrors on Q", and vicc vcrsa. The local error componentere is due to the rcsiduals on the interior dcgrees of freedom on Q/\: the error component O'~OI is thepollution error over Qi from the residuals on Q,,: while the error component f,~") is the local errorwhen it is associatcd with QK and is the pollution error when it is associated with Qi. Analogousinterpretations apply to pollution and local errors on Qi. Thus, the local and pollution errorcomponents on Q/\ can be written in the following form:

{8

10<}'7loe _ K..{;" - (j~KI .

(4.12)

and the total error on Q/\ (incillding its boundary ft I is

(4.13)

(4.15)

It is important to note that the matrix At-1 need not to be computed cxpli<:itly. A procedureavoiding inversion will be described in the next section.

We recall that our goal is to characterize 6/\ and to establish the relationship between the localand pollution errors and the ERM estimator cpi. Towards that end. we further investigate Eqs.(4.10) and (4.11). From (4.10), we have

[4 A J { E loe } {R },'IK Kt hK K

AIK .4, t~/\I = R~KI (4.14)

where/., =AI-AJJ~AfIAil' and.l'~ol_ A:-1 •• $(/\1(J" - - K .'1 /\ t () t .

260 1. T. Odell. r. Fellg/Jo/{mal of' Computational alld Applied Mathematics 74 (/996) 245-]93

Similarly. we have from (4.11).

[ - 'J {(tl} {ll.:l}AI An Ifl R,Atl At If"t = RK

where AI = AI - A,/\AK IAK/. and,pol _ -I (Kl8/\ - -AI.: A/\,lfl .

Therefore, the pollution component on QI.: is obtained from (4.12) and (4.17) .

."-1'01 = {If.r:'} = { - AK 1 AK/} -"ftlOK L'1"1 °1 .

o t .ff

To summarize. the local and polllltion components on QK are characterized as follows:

and

{1f"l'tl'} { 4 -1 A }~pol = ~" = -/tK KI ,(11:18" 'IKl IfltS , .ff

wherc of is the idcntity matrix. and tS~K) is given by (4.16) .To represent error functions on Q". we introduce a set of integrals similar to (4.1),

./IK = {/1 EN: 1/1,,(X) =1= O. x E Q". 1 :::; /1 :::; Nhl·

Then.

(4.16)

(4.17)

(4.18)

(4.19)

(4.20)

(4.21)

liE .11"

and

(4.22)

(4.23)

Clearly. from (4.22) und (4.23) (also (4.16). (4.19) and 4.20)). both local and polllltion errorsinvolve a global solution process on V". thus the local error is, in this sense. a global quantity. Thepollution error consists of errors on the interelement boundary r 1 (di/'ect polll/t ion e/'/'or) and errorsin thc interior of QI.: polluted by errors in QK (j,e, olltside of Q,,) transmitted through interelementboundary (i/1di/'e('( {wlll/tio/1 er/'or).

J. r Odel/. Y. Fel/g/Jo/{rJ/al <?f COlllpwatiol/al al/d Applied Mathematics 74 (199M 145-293 261

It is importa nt to observe tha t (4. IlJ)and (4,20) provide not on Iy Ihe characterization of the localand pollution errors. but also a relationship for assessing local and pollution crror estimation. Infact. the effectiveness of the local error estimation depends on the approximation of the Schurcomplcment AI, viz.. it depcnds on how one can choose another matrix BI (generated by a bilinearform) spectrally close to Al sllch that a linear system of equations Ihr = b is locally solyable. Theestimation of the pollution error. however. depends totally on the approximation of 6'~K) after theset of basis functions are chosen.

Now. it becomes transparent why the ERM estimator cannot always detect the pollution errorsince the element residual method simply lakes Hr = Al ::::=. I'it and ignores e~~1completely. Thus. wehave the following remark relating local error component and thc ERM estimator.

Remark 4.1. Let g~c be defined as in (4.19). and denote (P~ (x) = I"EIf. (PK/r 1/1" (x) 16•. where lfJ ~ is thesolution to (3.1), then

(4.24)

Thc proof of (4.24) is straightforward. Eq. (3.1) can be written in tcrms of matrix notation as

. = [AK AK/J- I { RK }tPK" A..4 n(I.:I·nIl.: nl fit

The desired result (4.24) is obtained by comparing (4.25) with (4.19).

5. Algorithms for Iwllution error estimation

(4.25)

One of the esscntial criteria in adaptive methods is that an error estimator can be computedlocally. We propose two algorithms to estimate the pollution error which involve minimumelement-to-element interaction.

5.1. The equivalent pollution residual method IEPRM)

Recall. from (117) and (4.4). that

Nta(e". Fir) = L (a(e~~. Fir) + a(e~~l . Fir))

1.:=1

i.V,

= L ~IlK (1',,)K=I

'\'f

= ~HK(Vh) + L ~Il.tll;/r)' I,ll E V',J=IJ#K

(5.1)

262 J. T Odell. Y. Fellg/Jo/lmal of ComfJlItatiollal alld Applied lv/athematiC's 74 (/996j 245-293

and we have the following local characterizations.

Xl

L a(e~~.l'h) = ~Hdr/,), 'VVhE V".K=t

Nc "',.

L (I(e~~,I.V,.)= I ~HJ(r,.). 'VVIIEV". 1 ~K~NIt·K= I J= I

J .. "

In analogy to the ERM estimator. we introduce the local polllttioll indicator cp ~POI.

(5.2)

{I ,((II h.po) I' ) _ 'I'rol(I' )K '/'" ". -. \" 'll· (5.3)

Our strategies for estimating local pollution error is based on the following observations:(1) The coarse mesh solution IIIl and the associated stiffness matrix All are assumed to be known

and available for use in error estimation. and All is readily filctored into upper and lower triangularmatrices: All = LnUH·

(2) Globally, if Ah is the fine-mesh stiffness matrix. then. for any element Q" (symbolically. withvector notations displayed momentarily),

4 eloc + 4 ePol - '1' . + ,ph. pol• h K • h K -. II.: .IK .

and

t'pol = .4 - I '1"" polK .• " .1" .

(3) A readily calculable coarse mesh approximation of efol is then

A-I '1' II. polH • \"

where ~H~'polis the submatrix of~H~'po'corresponding to the coarse mesh degrees of freedom due tothe orthogonality condition (3.9): otherwise. a Schur complement type of operation is involved inthe process to calculate ~H~·polfrom ~H~·"OI.

(4) The local problem

A .11I"."01 - A .(4 -I ,pl/.POI) ,},,,olI.: '/' " - K' II . I" Q. • I" (5.4)

then yields a local pollution error indicator.The calculation of Ail I ~H~'PO) involves only a back substitution using UII. Moreover the

calculation of~ll~·polcan be accomplished using D/SAXPY operation on the ERM error indicators.as discussed below. The cost of the entire process, beginning with a kno\vledge of AI/ and the coarsemesh residuals is (!) (N Ih a nd this can be reduced to (11 (N II log N H) by fast matrix-vector multiplica-tion algorithms. The full algorithm is as follows:

Algorithm IStep I. Solve the coarse mesh problem (2.8) for !til E V H.

Step 2. Construct perturbation space V',H and compute residuals r~. R~ using (2.13) and(2.14).

Step 3. Complltc local error indicator. cp~ (1 ~ K ~ N E) using (3.1).

(5.6)

1. T Ddt·n. ): FengjJournal {?( Computational and Applied iHarhematics 74 (/996) 245-293 263

Step 4. Calculate global pollution residllals from element QJ. J =j:. K.

N"~l\~POI(l'h) = L lIJ(q/'. L'h) 'V I'll E Jill. (5.5)

J= IJ,.K

Step 5. Calculate equivalent pollution residual 9\kol on QK based on ~ll~·rol. ~H~'POI -. ~Hfol .(a) Compute ~H~'po' from ~H~·pol via a Schur complement type of operations.(b) Compute equivalent pollution rcsidual ~Hfol := AK(Aiil ~H~'POI )a. on f21\.Step 6. Solve for pollution estimatc q>~.Pol using (5.3).

Remark 5.1. Up to step 3. no more calculations havc becn done than are ordinarily required tocompute error indicators in convcntional schemes based on the ERr.". techniques.

Remark 5.2. The key steps in this algorithm are steps 4 and 5. We observe that the pollution erroris carried. in part. by the global residual functional which is restricted on the intcrclemcnt boundaryrt (direct polllltion as indicatcd prcviously). and the crror transmitted through rl (indircctpollution). In this algorithm. both the direct and indirect pollution crror components arc taken intoaccount by constructing a global residual functional (vector) 'J\~'POI. and then calculating theequivalent pollution residual ~l\fol on QK (thus the name).

Remark 5.3. As noted earlier. the matrix operation equivalent to step 4 is a D/SAXPY operation.which can be executed efficiently using BLAS library routines. Various parallcl computingtechniques can be adopted in this step.

5.2. The inter-element residual method (fERM) lor calculating pol/lition error

Another algorithm to estimate the pollution error on f2K can be derived dircctly from (4.16). Thekey is to e~timatc 6;K) on the interfacc rl. The pollution crror is readily availablc by (4.20) and(4.23) if 8;KI is known, To be specific. wc have, from (4.16). that

.!ilK) (A A A-LA A ./4~1 .4.)-I(R .• -.4~IR-)() I = 1 - t K K KI - 1K I K '~K t 1 - ,~t K' I K 1\ .

Solving (5.6) exactly is equivalent to solving for the fine mesh solution IIh(X). which defeatsthe purpose of error estimation. Therefore, an approximation needs to be introduced toestimate o<Y;:'. Toward that end. we consider an element QK surrounded by elements Q~ whichinclude all elements neighboring f2K. Here we have a l-irrcgular hp data structllre in mind (see thcAppendix). but thc algorithm below is very general and can be applied to other types of datastructures.

Let- -,.

WI\. = Ql\.uQi;

Define

"V'K = {J EN: flJn(l)K =j:. (~. 1 ~ J ~ N f:}'

A typical patch WK (including Q/\ and its neighbors) is shown in Fig. 4.

264 J.T. Odell, r. Fellg/Jo/l,.,,{// {~(C()mp/ltatiollal alld Applied A/athematic,\' 74 (/996j 245-293

(5.7)

Fig. 4. A typical patch WI( in hp finite element discretization.

Notice that AIK and A~I have non-zero entries only for the degrees of freedom on w" for allJ E .1"/\. i.c .. Q/\ interacts only with its neighbors. The pollution crror. however. includes thecontribution from the residuals on thc entire domain except those on QI.:. sincc thc discrete Green'sfunction Ai I is. in fact. a global function.

The idea to approximate ~:K) is to replace the global discrete Green's function on Vh by the oneon l!/I. \vhich is already available in solving for 11/1. Thus. the approximation to 8i~l. denotcd0":*" I. is defined as

[

AI/ All - ] {8(*KI} {R/*Kl}·'"IT /(*/\1 ., T _ 'I>(f.pol _ IIt /I 11 -. \/\ - 11

_ A(*K)J A/*Kl tS'1*KI RI*K)

where ~llf·pol is the global residual functional (vector) on Vlt as discussed in Algorithm 1: R~*K) isthe residual associated with the degrees of freedom on r t, and R:~K I represents the residualassociated with the rest of the degrces of frcedom. Then. the local pollution indicator equation canbe rewritten as

(P~poJ(xl = L cr~~""',,(x) x E ti/\, I ~ K ~ Nf;. I ~ /I ~ Nillie .II"

where

(5.8)

(5.9)

and .J is the identity matrix.Eqs. (5.8) and (5_9) provide an cxplicit expression for the pollution error indicator. The major

steps to calculate the pollution error indicator discussed above are summarized as follows.

Algorithm 2Step 1 Compute local crror estimate (P~ on QK (I ~ K ~ ~rJ using (3.1).Step 2. Calculatc the direct polllltion error componcnt 8)*/\1 on the element interface fl lIsing

(5.7) and construct the discretc intcraction function AKI.

(5.10)

(5.11 )

1.1. Oelell, r. Fen~!Joll/'J/al ~rComputatiolla! and Applied Idathematin 74 (/996) 245-293 265

Step 3. CalclIlate the indirect pollution error component by solving a local problem

A Fpol -4 F(*KI1\ (J * K = -, 1\/ (J / '

Step 4. Construct 8}*K I using (5.9) and local pollution indicator (P~' pol using (5.8).

Remark 5.4. Algorithms I and 2 are essentially equivalcnt since both algorithms utilize the globalresidual function 9i~' pol on II II. The second algorithm differs from the first by recognizing thedirect pollution error component, which is associated with degrees of freedom on rt and theindirect pollution error component. which is transmitted to the interior degrees of freedom onQI\ through the degrees of freedom on the interelement boundary 1'/.

5.3. Local a posteriori er/'OI'estimation

For simplicity. let Eq. (4.7) be written in terms of submatrices associated with degrees of freedomin QK. and those in the subdol11ain Qi; outside of QK by appropriate choice of boundary elementfluxcs (e.g .. e<-averaging tlllX for some'J. [4]):

[A~ A~i;J{~~} = {R~}.An AK 81\ RK

The residual vector R can be considered as the sum of two terms: one associated with degrees offreedom in QK. the other associated with degrees of freedom Qk.

R={R~}={RK}+{O_}def R~+R~.RK 0 RK

Using formula for inverse of a partitioned matrix. the local and pollution error components onQK can be written in the following form:

ploc u' - I (A ) R D,pol - .. - I A - OJ - I ( .• -) R - (5 12)f) K =.7 K 1\. f) K - - /I KKK oJ ,'1 K 1\ .

andp.' • _ §·Joc + FpolVA-VA oK (5.13)

whereS"(AK) and 9' (Ai; ) denotes Schur complement of AI\ and Ai; respectively.Now. let us examine the behavior of local error (1111 - lIulln,. Before establishing a posteriori

error bounds. we introduce the following technical lemmas.

Lemma 5.5. SlIppose Ill. 11.1E N alld II = m + I. LeTA he II symmetric alld lltJ/lsillfllllar maTrix with thefO/'1l1

[

AllA-A21

where A. A II. alld A 22 lire II x II. III X m, lI/I(/ 1 x IlIllltrices. respect irely. Let .Y" (A 22) dCllote Schurcomplemellt t~rA 22. i.e..

S" (A 22) = A 22 - A 2 I A III A I 2

266 1. T. Odell. r. Feng! }oumal of Computational and Applied A1arhematics 74 (/99(') 245-293

alld f/ - 1 (. ) denote its inverse. ~rA 11 and A22 are nOllsillf}lIlar, then

f/-l(A22) -A221 =A221A21AI11AI2,9"-I(A22)'

Proof. It is equivalent to show that

1-A 2-21 .'I'(A22) = A 221 A21 A 111A t2.

Obviously,

1- A 221 .9"(A22) - 1 - A 221 (A22 - A21Allt A 12) = A 221 A21Alii A12.

This completes the proof. 0

(5.14)

Lemma 5.6. Let (' .. ) dell ate J2-illner product of tlVO vectors ill jR". Let the condit iOlls in Lemma 5.1hold: then/or lIny x = (Xl,X2)T E jR". XI E JRIII. and X2 E 1R'. II = m + I.

Proof. (cf. COllie [7]). It is well known that

A-I = [ .~- 1(A I .J - A III A 129"- I (A 22 ) ]

-A22IA2I.'1'-I(Ald ,~-I(A22) .

Consider

(x.A - Ix) = (x2.A221 X2) = (XI' .9"- I (A I dxd - 2(X1.A 1,1A 129"-1 (A22)X2)

+ (X2.9"-I(A22)X2).

(5.15)

On the other hand.

((XI - A 12A221 X2). y- 1 (A 11 )(Xt - A 12A2i X2))

=(XI' Y' - 1(A II )xll -2(X1' 5/'-' (A 11)A 12A221 X2) + (A12A221 X2'.9" -I (A I.JA I 2A 2-21X2)

=(XI. y- 1(A II )XI ) -2(xI.A III A21 9" - I (A22)X2) + (A 12A221 x2.A 1-\1A 219" - I (A22)X2)

since A = AT implies A - I = A -T. The result follows immediately from Lemma 5.1. 0

Remark 5.7. According to [7], the result in Lemma 5.1 was known earlier to Woodbury [14].

Now. recall from (3.7) and (3.17) that

e" = U,' - U/(.

and

then.

1.T. Odell, Y. FellgfJoumal o.fComputatiollaf alld Applied Mathematics 74 (/996) 245-293 267

Similarly. we have

lIe~/~1I1.!h = (R~.A -[ R~) and lIek~,1IILh= (R~.A -I R~). (5.16)

Also. recall from (3.1) and (5.3) that

a(cp~. 1'/.) = ~l\dvl') V VII E V~

and

a (fflil. poi., ) = '1' pol (L' )'t''' • L.', .1" 'II

then

IICP~II~.n.= (R",Ai 1RK) and

Notice that

ePol - fflil. pol/\11 - 't'K

if

Rft'= -AKRg-J(AK)RR.

IIfflll.poll12 - (RpoJ A-IRpol)'1'" E.n. - ". K K'

(5.17)

In view of assumptions (2.2), the bilinear form a(',') is continuous and coercive. Thus, there existconstants iV! 0 and~o. such that for any rEV.

~ollvIIII'(m ~ a(r.r) ~ iHollvlllI'LP.l· (5.]8)

Let BK = Ai IAKRy-1 (AK)AKKAi I and A" = Ai I. Let }'~ax denote the largest eigenvalue ofthe following generalized eigen-problem:

BKc1>K = AAKc1>K

where c1>K E [Rm is a vector. Then, we have the following local estimates.

(5.19)

Theorem 5.8. Let eKI. = (UII - ltll 1I,J.' cp~ (l1lt1cp~' pol he defilled in (3. I) and (S.3). respectively. LetRkol be defined (IS ill (5.17); assume thar

(i) the discrete error equation is of the form (5.10):(ii) AK allti AR are symmetric positive defillite:

(iii) there exists a constam ./IK E lR such that )'~ax ~ JIK• where ).I~ax is the largest eigenvalue o(theeigell-problem (5,19).

Then

(5.20)

Proof. Based on the inverse formula of partitioned matrix and Lemma 5.2.

IleKlllli,Q. = (R~,A -I R~) + (RfOJ,Ai I RrOJ)

= (RK.Ai I R/\) + (RR, ,Cf' -I (AR)Rd + (Rrol.Ai IRrOJ)

- II fflh 112 + (R-' (0 - I (A -)R-') + II fflll.poll12- 't'/\ E.n. ". ·.7 K K 't'K E.!h

268 J. T. Odell. r. FengjJoumal 0.( Computational and Applied Mathematics 74 (/996) 245-293

where

Rpol A - (',L' -I (A·)R·K = - KK'-" K K

and

R-· - A- A-tRK - - KK K K·

Since cih, /;/'-I (Ai{) ih.l ~ 0,

IIcKlllliLl. ~ IIcp~IILh + IIcp~'pol IIi. u•.

On the other hand. by assumption (iii),

(ilK' tl' -1 (AKlih) = (RKAi: 1AKK ..<,/'-1 (AK)Ai{KAi: 1RK)

= (RK• BKRK)

~ ..//dRK• AKRK)

= .ilK IIcp~lli.o•.Therefore.

IIeKhlllQ, ~ (1 + ./IK)lIcp~·IIi.o, + IIcp~·pollll.!J,'

This completes the proof. 0

5.4. Local and pol/ltfioll error indicators

So far. we have analyzed the local and pollution errors and their attributes. To summarize. welist the following local error indicators corresponding to each of the various error components.Local error i"dicator:

••11 = [ll .(,)Ioc l,loe)J 1!2rK K <-Kt,· Ktr .

Local ERl\,-1 i"dicator:

II [ (II II )J 1/2'1K = aK CPK' C{JK .

Pol/ution error indiL'Cltor:

.".h = [(I .("pOI ePol)Jl/2"K K <-Kh' Kh .

Local pollutio" error i"dicator:

II~ ~ [ad cP~'pol • cp~'POl) J 1/2.

Here we understand that cp~.pol is to he approximated as suggested in (5.4). We note that

IIell IIi. 0, ~ (i,~)2 + (1t~)2:= (e~)2

thus. we denote

Local total error inclinltor:

(\,~)2 = (ll~)2 + 111~)2.

(5.21)

(5.22)

(5.23)

(5.24)

(5.25)

J.T. Oden. Y. Feng;Joumal of ComputaTional and Applied Mathematics 74 (/990) 245-293 269

We shall explore thc relative valucs of thcse error cstimators through numerical expcriments onmodel problems.

6. Numerical eXI)criments

6.1. Numerical results in one space dimension

Wc first considcr a family of two-point boundary value problcms of the form:

- II" (x) + GII(X) = fIx) x E 1O. 1).

11(0) = 11(1) = O.

with/chosen thalll(x) = (x - 0.441)2/3 - 0.09919x - 0.5794 (0' = 10). and lhe solution has a sin-gularity at x = 0.441.

In this examplc. the flux jump on the intcrface of elcments is split by using a version of the nuxbalancing scheme lip to order p. which ensures that condition (2.17) holds (cf. [3. 4. 6. 8J).

The coarse mesh (II 11) approximation consists of functions Xx (N = 1..... N H) which areorthogonal to the perturbation functions 11, in the sense of(3.9) with a(lI.r) = J ~(II' r' + mw) dx. Weconsider a mesh consisting of only eight element of equal length with seven interior nodes. NH = 7.The pollution error is estimated using Algorithm 1.

To compare the various errors and error indicators. data collected in Tablcs 1-4 are dcsigned todemonstrate the relationship between error indicators and error e and eh The following cases areconsidered:

Case I: l.s linear. 11, cubic.Case II: XN linear. 11, quartic.Case Ill: XN quadratic. 11, cubic.Case IV: I.:v quadratic, A. quartic.

Table IComparison of the errors and the error indicators for a two-point boundary-value problem (case II

K IleIlE.ll, lleh liE.11. i'~/llellf:,II. rrUlielif:.l1, 11~/llchllf.lJ, II~/Ilchllr..", 11~h'~ II~! rr~

0.207E - I 0.209E - I 0.976 0.026 10]5 0.023 t.042 1.0212 0.236E - I 0228E - 1 0.943 0.058 0.954 0.047 ].064 1.3523 0.412E - ] 0.401E - I 0.919 0.080 0996 0.093 ].102 1.2144 0.129E + a 0.117E + () 0.887 0.114 0.956 0.128 0.943 0.9955 O.574E - I O.594E - I 0.944 0.056 0.912 0163 0.954 0.9996 0.368E - ] 0.359E - I 0.965 0.036 0.871 0.1 ]6 1.083 1.3287 0.3t5E - 1 0.309E - I 0.979 0.022 0.885 0.099 l.\ 89 1.3718 0.244E - I O.235E - 1 0.983 0.016 0.904 0.0]8 1152 1.315

Global 0.160E + () 0.150E + 0 0.952 0.(>4') 0,961 0.093 1113 1.247

270 J.T. Odell, Y. Fellg/Jou/'Ilal ojColllputational alld Applied Mathematics 74 (1996) 2-15-293

Table 2Comparison of the errors and the error indicators for a two-point houndary-value prohlcm (case II)

K Ilellul, l!chIll.II, i'~/llellr..u, 7r~/lIelll.u. '/~/lhllr..u, J1Ulkhllr..u, '/~/i'~ JI~/ 7r~

1 0.207E - I 0.210E - I 0.992 0.00 1.036 0.012 1.024 1.1292 0.236E - I 0.229E - I 0.967 0.034 093R 0.029 1.043 1.4553 0.4l2E - I UAU5E - I 0.935 0.065 0.995 0.057 1.089 1.3234 0.129E + 0 0.120E+O 0.899 0.102 0.906 0.134 0.968 0.9895 0.574E - I O.581E - I 0.953 0.048 0.934 0.129 0.989 0.9796 0.368E - 1 0.362E - I 0.971 0.030 0.891 0.14t 1.016 1.2407 0.315E - I 0.312E - I 0.988 0.013 0.986 0.1 t2 1.094 1.2178 0.244E - I 0.23RE - I 0.997 0.004 0.977 0.093 1.082 1.328

Global 0.160E +0 a.152E + 0 0.982 0.019 0.972 0.104 1.085 1.409

Tablc 3Comparison of the errors and the error indicators for a two-point boundary-valuc problem (casc [[II

K lie Ill. II, /fehllr..u, i'~/IIe11LU, ltUlieIlE.ll, 'IUlkhllr..u, J{Ulleh liE.II, '/~/i'~ JI~/lt~

I 0.126E - 2 0.135E - 2 0.965 0.036 1.052 0.032 1.023 1.2112 0.465E - 2 OA31E - 2 0.947 0.054 0.949 0.044 1.054 1.1523 0.65RE - 2 0.624E - 2 0.925 0.078 0.985 0.059 1.071 1.1034 0.R84E - I 0.859E - I 0.874 0.125 0.974 0.088 0.929 0,9935 0.735E - 2 0.708E - 2 0.931 0.070 0.958 0.101 0.952 0.9916 0.397E - 2 0.382E - 2 0.942 0.057 0.906 0.t26 1.064 1.1327 0.268E - 2 0.275E - 2 0.962 0.039 0.992 0.127 1.103 l.t708 0.294E - 2 0.283E - 2 0.979 0.022 0.976 0.089 1.082 1.085

Global 0.893E - I O.S67E - I 0.946 0.055 0.981 0.043 1.110 1.219

Table 4Comparison of thc errors and the error indicators for a two-point boundary-value problcm (casc IV)

K lIellr..u. llehllr..u, i'~/liellr.,u, 7r~/lieIlE.u. ',~/llehllr.l1. JIUllehllr..II, '/~/i'~ µUlt~

1 0.126E - 2 0.129E - 2 0.982 0.019 1.041 0.025 1.035 1.4372 0.465E - 2 0.459E - 2 0.968 0.033 0.991 0.021 1.013 1.2013 0.658E - 2 0.627E - 2 0.956 0.046 0.979 0.028 1.098 1.1204 0.884E - 1 0.870E - I 0.902 0.099 0.941 0.058 0.956 1.0215 0.735E - 2 0.712E - 2 0.949 0.051 0.935 0.083 0.976 0.9636 0.397E - 2 0.389E - 2 0.963 0.037 0.974 0.055 1.093 1.1057 0.268E - 2 0.273E - 2 0.972 0.029 0.945 0.014 1.041 1.1868 0.294E - 2 O.285E - 2 0.985 0.015 0.972 0.017 1.032 1.254

Global 0.893E - I 0.8781: - I 0.946 0.055 0.982 0.044 1.091 1.317

J.r Odell. Y. FellgjJollrnal o.fComp/ltaliollal alld Applied Mathematics 74 (/996) 245-293 271

Table 5Local and pollution error cstimation on Elcment 7 for a lwo-point boundary-valuc problcm underIr-refinement

1/21/41/81/161/321/64

Error on f1, Loc. Err. Loc. Est.-

III'IIE.D, IhIlE.D, .,h1l~ 1/4 JI~17

OJOIE - I O.237E - I 0.212E - ] 0.969E - 2 0.294E - I 0.103E - I0.214E - I 0.198E - I 0.167E - I 0.935E - 2 0.205E - 1 O.t54E - I0.105E - 1 0.1 toE - I 0.J05E - 1 0.873E -1 0.119E -I 0.975E - 20.649E - 2 OA95E - 2 0.381E - 2 0.691 E - 2 0.5821' - 2 0.763E - 20.298E - 2 O.363E - 2 0.IS2E - 2 0.309E - 2 0.397F - 2 0.595E - 20.291E - 2 O.255E - 2 0.104E - 2 0.234E - 2 0.3741' - 2 OAI2E - 2

Tablc 6Local and pollution error estimation on Element 7 for a two-point boundary-value problem underp-rcfinement

p

I23456

Error on f17 Loc. Err. Loc, Esl.-

111'11£.11. IleI> 111',/l. .." 114 "~ JI~17

0.315E - 1 0.299E - 1 0.254E - I 0.130E - I 0.343F - I 0.156E - I0.203E - I 0.176E - I 0.149E - I 0.981 E - 2 0.147E - I 0.112E - I0.968E - 2 0, 121E - t 0,894E - 2 0.756F. - 2 0.808E - 2 0.941E - 20.392E - 2 O.263E - 2 0.222E - 2 0.258E - 2 0.213\:-2 0.454E - 20.204E - 2 O.2toE - 2 0.109E - 2 O.203E - 2 0.140E - 2 0.302E - 2O.254E - 2 0.239E - 2 0.101E-2 0.216E - 2 0.938E - 3 O.299E - 2

Thc results show that the pollution componcnts are relativcly small compared to thc local errorcomponent (<20% in this example). Although the ER M estimator oilers an upper error boundglobally. local error estimates can vary widely. Both the local ERM error estimator and the localERM pollution error estimator deliver very good estimates for local and pollution errors for theeight-element mesh, as illustrated by the ratios II~/"l~ and 11~/1T.~.

Another experiment is designed to illustrate the effcct of the singularity on pollution error insome elements. Considering Element 7. Tables 5-6 demonstrate the reduction of the errors and theerror indicators as Element 7 is refined while keeping othcr elements unchanged. The local error forElement 7 is not reduced under 11- and p-reflnement. Moreover. the error estimators indicate thatthe pollution component is morc dominant as mesh is relincd. This coincides with the observationin [5].

272 1.T. Odell. Y. Feng,IJo/ll'llal (i( Computational and Applied Mathematics 74 (19%) 245-293

To determinc the behavior of the paramctcr ,/IK in the estimatc 15.20) for thcse model problems,we observe that

Table 7The quotientll('hlli:.l/J(\'~lfor various cases

K lIehIlE,llJ(\'~ )

Case I Case II Case III Case IV

0.9849 0.9652 0.9501 0.96032 1.0469 1.0656 t.0526 t.00893 0.9997 1.0034 1.0134 1.02104 1.0367 1.0919 1.0225 1.06075 1.0794 1.0606 1.0381 1.06536 1.1381 1.1085 1.0932 1.02517 1.1230 1.0078 0.9999 1.058tl! 1.1059 1.0189 1.0204 1.0286

1.500

t.375

lip

Qrnax 1.250

1.125

1.000

25 50 75 100

No. of degrees of frccdom

Fig. 5. The plot of Q~, vs. dcgrces of freedom.

J. T. Odell. r. FellgfJolI/'llal of Complltatiollal alld Applied Mathematics 74 (1996) 245-193 273

so that

1~ lIehIlE,I1, ~ )1 + /1....., (,I,)"'" . f,;.II\'

Let Q~f:lXdenotethe maximum quotient of the total error estimate (\'~) and the discrete error inenergy norm on element QI\. for every K, i.e..

Qllp. _ . IIeh liE.!},max - n1.lX II

K (\'K)

Therefore. the quotient Ile/IIIE.I1J(\'~)and the quantity Q~f.1Xcan deliver reasonable indications forthe behavior of the parameter. /IK. Table 7 summarizes the quotients Ilct,IIE.!1j(\'~) for cases I-IV.and Fig. 5 shows how the quantity Q::faxdepends upon the number of degree of frecdom for variouscases of the discretization. The numerical results demonstrate that JIK does not vary dramaticallywith mcsh parameter. Notice that the quotients Iletllllul.l(V~) are close to I. and thosc entries lessthan 1 in Table 7 contradict thc estimate (5.20) and. we belicve. result from inaccuracies inherent incomputing approximations of R~ol on the coarse mesh.

6.2. Numerical results ill two space dimellsion

Consider a Poisson problem on an L-shaped domain in [R2. The source term and boundaryconditions are specified such that the solution is II(X) = 1r2j3sinliG). with (1',0) polar coordinatescentered at the re-entrant corner. To model this problem. we use I-irregular lip finite clement

8

6

5

4

3

2.,. ,. ~...... ' ' ;' .. ~.

I~~~~J~;' Id-,·.-··l

;.~ "':'._' I

.' ~ L ~c jI • - ~ ..;-.

~_I .. ~~' .,::,'::'

Fig. 6. The hI' I\lcsh I.

D.O.f--43

274 1.T. Odell. Y. Fellg/Jou/"I/al of Complltatimwl alld Applied Mathematics 74 (/996) 245-293

8

7

6

5

2

D.O.F=86

Fig. 7. The Irp Mesh II.

c Ii

8

1

6

5

4

3

2 I

D.O.F=IOO

Fig, 8. The Irp Mesh III.

J. T Odell. Y. Feng/.loumal of Computatiollal and Applied Mathematics 74 (1996) 145-293 275

8

7

6

4 ji

~ ..

Fig. 9. The hp t\lesh IV.

1l.0.f=:>93

meshes (see the Appendix). Four hierarchical levels of lip meshes (Figs. 6-9) are constructed in thecalclllation. There are thrce quadratic elements in the corner of the domain in Mesh I (Fig. 6).Those three elements are enriched to cubic elements and the rest of the elcments on the outer layerof the L-shaped domain are enriched to quadratic elements in Meshes II and III (Figs. 7 and 8).Thedifference between Meshes II and III is that there are finer grids (II-refincment) around thesingularity in Mcsh III as shown in Fig. 8, and Mesh IV has more II-refinement around thesingularity and p-rcfinement on the boundary of the domain. as ilIustratcd in Fig. 9.

The error and various error components are shown for the following cases:(a) The exact errors: Figs. 10. 16.22, and 28 for Mesh I. II. III. and IV. respectively.(b) The ERM error estimates 11~: Figs. 11. 17.23. and 29 for Mesh I. II. III. and IV. respectively.(c) The pollution error estimates Jl~: Figs. 12. 18. 24. and 30 for Mesh I. II. 111. and IV.

respectively.(d) The total error estimates v~: Figs. 13. 19.25, and 31 for Mesh I. II. III. and IV. respectively.(c) The etTectivity of the ERM error estimates: Figs. 14.20.26. and 32 for Mesh L II. III. and IV.

respectively.(f) The effectivity of total error estimates: Figs. 15. 21. 27. and 33 for Mesh 1. 11. III. and IV.

respectively.Globally. the exact errors (normalized with respect to the solution) in the HI (Q)-norm are 9%.

7%.3% and 1°;;,for Mesh I. II, III and IV. respectively. The result indicates, by comparing MeshesI and II that local p-refinement on the outer layer of the domain does little on improving the globalcrror since the error around thc singularity is not resolvcd. It may be. howcver. evidence thatthe pollution error. from the singularity to thc olltcr layer rcgion of the domain. cannot be

276 J. T. Odell. Y. Fellg/.!ou/'llal ~,. Compllfational and Applied Mathematics 74 (/996) 245-293

.9

.675

.45

225

r-~~:~.. '.:. i

~~jL

Fig. 10. The exact error in the H IIQJ.; I-norm for I\fesh I.

(;)(ACT ERROR

MIN=.09384HMAX=-886320

GWDAI. =2.243764

.9

.7

.3

00

Fig. II. The E R1'\'1error indicator I/~ for Mesh I.

ERRUK ESllMATEMIN=.\145Z7MAX=.87604\

GI OIlA!. =2.2790674

1.T. Odell. r. Feng '.Iof{/'llal of Compu({/tiollal alld Applied Mathematics 74 (/996) 245- 293 277

.10

.04 r~-~i'1.':;:.1, .1'·1

II _

Fig. 12. The local pollution error indicator II~ for rvtesh I.

l'OIJ.ITrlO!\ EST,MIN=.05176

MAX=-192S8

GLOBAL =.39134743

.9

.n

.45

.IS

TOTAL ERR. EST.MIN=~108738

MAX=.89S37Q

GLOBAL =2.3124232

Fig. 13. The local total error il1lJicalor "~ for Mesh I (compare with Fig. 10).

278 J. r. Odell, Y. FengiJoumal of Computational and Applied Mathematics 74 (1996) 245-293

1.225

1.15

1.015

1.

.925F.FfT:CTIVITY INDEX

MlN=.943276MAX=1.2 \36.~9

GLOBAL =1.015134

Fig. 14. The effectivity index 11~lIellll'lfI,1 for the ERM error indicator for Mesh I.

1225

1.15

1.015

.925

II

11·:.'''··•.l.,-,'.~.i

i TOT AI. EFFECfIVITY~IIN=.964732MAX=122 J59I

(jI.OUAL =1.030612

Fig. 15. The effectivity index \.~iliellll'U1d for the total error indicator for Mesh I.

J. T. Odel/. r. Feng/Joll/'llal (J{ Computational al/d Applied Matlu!matics 74 (/996) 245-293 279

.9

.675

,45

225

.9

.675

.45

.225

[

'f ..._.':r',

~IFig. 16. The exact error in the H1(H"I-norm for Ivlesh II.

Fig. 17. The ER M error indicator II~ for Mesh II.

EXACJ' ERROR

MlN=.078421

MAX=.867323GLOBAL :2.247325

[cRIlOR F$T1MA TEMIN=. 103465MAX=.8411623

GUlllAI. =2.2783Q44

280 J.T. Odell. r. FClIg;Jo/{I'I/a/ ofComp/{tatiollal ami Applied Mathematics 74 (1996) 245-293

.20

.16

.12

08

POU.IlTION F.5T.MIN=.OSI18MAX=. 18924

GW8AI. =.39881'311

Fig. 18. The local pollution error indicator Il~ for Mesh II.

.1

.'ns

.798

.443

.171

'"

TOTAL ERR EST.ML'I:=. I 01255MAX=.972S49

fiI.o8AI. =2.3129469

Fig. 19. The locallotal error indil:ator \'~ ror l'vksh II (compare with Fig. 1(,).

J.T. Odell. Y. "'englJol//'IIal ofComp/ltariollal and Applied Marhemlltics 74 (f996) 145-2'J3 281

1.2S5

1.195

1.075

915

925

t",''~::.,~-,-

I

EFFECJ'rvlTY INDEXML'I:=.9421192MAX=I.237641

GLOBAL=\.0137¥5

Fig. 20. The ell'cctivity index Il~ 11e111l'1U. for the FRM error indicator for !\\esh II.

1.2S5

1.195

~IIi .L.

1.075

~'....~-,~

.915I

dFig. 21.

TOTAL EFfECTI VITYMIN=O.972114~MAX=1.24623 1

GUlUAI. =1.02915M

The clh;ctivity index l'~lIklln'IUd for the lotal error indicalOr ror Mesh [I.

282 1. T. Odell. Y. Fellg,Jouf'l/al or Complllatiollal alld Applied Marl1emalics 74 (/996) 245-293

.975

.78 I~~'.

;

.4i75

~~,1·'~:~I~#r.. .r~"-:

.195

EXACfERRORMII'=.037 1695MAX",9584523

GWDAL =2.35978 16

Fig. 22. The exact error in the II I (Q,d-norm for Mesh III.

.975

.758

.433

.217 ~

!.. ,:.:.'.1'-',.,.

I"·;.'~;l

Fig. 23. The ERM error indicator '1~ for Mesh Ill.

ERHOR liSTlMA TEMIN=042S731MAX=.9594274

GLODAL =2.3617638

1. T. OdclI. r. FellgjJvIIl'I/al of Complltatiollal alld Applied Mathematics 74 (/996) ].15 -293 283

,12

096

.06

.024

~ .•.'.. ,~........,

r;-.•. ·1

"..:

roU.lmON EST.MIN=.023171

MAX=.112443

GLODAL =.3504776

Fig. 24. The local pollution error indicator Il~ for Mesh III.

.'115

.78

.4l75

195

Ip'.}.:..

~i'.,''7:

TOTAL ERR. ESTMIN=.0404522

MAX=.96147S6

OLOIlA!. =2.311621

Fig. 25. The local total error indicator \'~ for Mesh 11I (compare with Fig. 22).

284 JJ Odt'l/. r. Feng/Jou/'I/a/ v(('u/II/lutatiol/a/ and Applied ,Harhel/larics 74 (/99ti) 2./5-293

I~

If~11.17 ~j, .!

.~I..1M

.98

.9EIHiel'IVrrv INDEX

MIN=.9218341MAX=\.201031

GLOIlAL =1.0084

Fig. 26. The effectivity index '/~/lk"ll'(Qd for the ERM error indicator for ~Iesh III.

1.2

\.125

\.05

0.915

.9

TOT Ai. EFFECTIVITY~lN=.9454652MAX=I.2024761

GLOIlAL =1.011822

Fig. 27. The etTectivity index \'~/IIe"II'IlI. I for the lotal error indicator for Mesh III.

J. T Odell. Y. Fellg! Jou,."al of Computatiollal alld Applied il-/uthematics 74 (/996) 245-293 2H5

.405

:n

.135

EXACfERRORMIN=.OOO315~IAX=-5234621

GWDAL =1.9805433

Fig. 28. The exact error in the H1(QKl-norm for l\lesh IV.

"

52

.m

.26

.127

ERROR FSTIMATE

MIN=.I034225MAX=5198374

GI.DIlAL =1.9819297

Fig. 29. The ER1',t error indicator II~ Ii.)r Mesh IV.

J.T. Odell. Y. FClIg/Joumal olComp/ltational alld Applied Mathematics 74 (/996) 245-293

..

.25

.125

POIl.lrrJON EST.MIN ... OOOO46MAX=.0974:l65

GLOOAL =.30395533

Fig. 30. The local pollution error indicator JI~ for rvlesh IV.

.58

.446

.268

.134

TOTAL ERR EST.MlN=.OOO417MAX=.5732 145

GLOOAL =2.005 102

Fig. 31. The local total error indicator \'~ for Mesh IV (compare with Fig. 2SI.

1. T. Odell. Y. FengjJolll'/lal of Computational alld Applied Mathematics 74 (/996) 245-293 287

1.11

1.1225

1.065

1.0075

.95El-l'lieflVITY INDEX

MIN ...95J4556

MAX=I.168951GLODAL =1.0007

Fig. 32. The effectivity index ,,~lllellll'tlld for the ERM error indicator for t-.lesh IV.

1.2

1.15

1.015

\.0

.95TOTAl. F.FFECTIVTTY

MIN=.954316

MAX=1.1951192GlJ18AL =1.0124

Fig, 33. The errectivity index I'Uliellu'ClId for the total error indicator for Mesh IV.

288 1. 7: Odell. ~~FeI/8/Jouma{ of COlllputaliollal alld Applied AfatlzelllCllics 74 (l99f1) 245--293

improving by IOt:al refinement. Therefore, both local and pollution error components are impor-tant in the a posteriori error estimation.

In all the results below, the \lux balancing scheme of [4J is used in applying the ERM. The localerror components are estimated by the ERM error indicators which are obtained by solving thelocal problem with spectral order PK + 2 for each element. where !'K is the spectral order of thelocal shapc functions defined on QK in V1/. The pollution error components are estimated byAlgorithm 2.

For ease of the notation. we introduce three global quantities as follows:

(L~'= I (ll~)2)112EFF ERM = (6.1)lilt - 1/1/111/'(01

("I", (.h)2 )li2L.K=\ ,)\ ..,

EFF!o! = (6._)lilt - It It III/I WI

( \"' .'>_ (Jlh.)2) 112

/"/0/ _ L.K-I K 63)

/0 - \. h' l" . ( .(L~';"I(\'d-) ,-

The global ellcctivities and the pollution error percentage arc summarized in Table 8.We observe that the amendment of the pollution error estimates to the local ERM esti-mates increases the total error effectivity indices and makes them grcatcr than one for allfour meshes.

Locally. from Figs. 14,.20,26. and 32, the ERM error indicators alone may not characterize thelocal error components well enough to guide adaptive strategies; they can lInderestimate trueerrors (about 10% on somc elements in this example.) The numerical results. however. indicate thatthe ERM cstimates combincd with the pollution estimates deliver very good local error estimatesin the energy norm (see Figs. 15. 21. 27, and 33). The local total error indicators predict the localtrue errors within 2°A. on all elements.

To illustrate the global nature of the pollution errors, surface plots of the pointwise pollutionerror estimate lpll.llOI (with lp'I.POI!f1

K= lp~'POI) for Meshes III and IV are shown in Figs. 34 and 35. We

observe that the pollution error functions are rather smooth across the domain. and the maximumpollution error decreases rapidly as the mesh is refined. The pollution error estimate in Lx-normII lph. pol ilL , vs. degrees of freedom is shown in Fig. 36.

Table 8SUlIlmary of the global elfectivities and the pollution percentage

~Iesh I II III IV

r:n;l'IlM \.0157 \.0138 \.0084 1.0007EFF 1.0306 1.0292 1.0118 1.0124,.-,ll'y" 1.47% 1.52% 0,31 'X, 1.17%

1. T. Oden. Y. Fel/glJournal of Computatiul/al and Applied Mathematics 74 (/996) 245-293 289

MIN= 0.00000ooMAX=O.0147603

Fig. 34. The surface plot of the pointwise pollution error for Mesh III..

MIN20.0000000MAX=O.0098625

Fig. 35. The surface plot of the pointwise pollution error for Mesh IV.

290 1.T. Odell. Y. Fellg/.!olll'l/{/l of Computational <lndApplied Mathematics 74 (1996) 245-293

0.025

Eg 0.02E"E'xCDE.Een!CD.§ 0.015OJCD

~~c:.2~'0Q

0.01

50 100degrees of freedom

200 300

Fig. 36. COilvergence graph of pollution error estimates in maximum norm.

6.3. Computational costs

It is interesting to measure the relative costs of error estimates for the example problems studiedhere, For this purpose. we introduce the following parameters:

C = CPU time (in seconds) required to solve the boundary-value problem on the "coarsemesh" (in jlH).

T'l = CPU time (in seconds) required to compute 11~using the ERM (with local spectral orderPK + 2).

T pol = CPU time (in seconds) required to compute JI~ using Algorithm 1 or 2.

The results for Meshes I-IV are summarized in Table 9 (the measurement is taken on an IBMRISC/6000 Model 590 workstation). It is obvious that the computational costs for evaluating thepollution error are equivalent to that for the ERM.

Tahle 9CPU time ratio of error estimation VS. solution process

Mesh

0.250.27

II

0.260.27

III

0.28{),29

IV

0.280.30

J.T Odell. Y. FellgfJoumal ~rComp/{(atiollal alld Applied Mathematics 74 (/996) 245-293 291

7. Conclusions

In this paper. an approach to\vard the calculation of a local a posteriori error estimates fora class of linear elliptic boundary value problems is proposed. The local elementwise error ischaracterized as the slim of local and pollution error components. and can be fllrther characterizedexplicitly in terms of residual functionals. It is understood that the pollution error componentconsists of direct and indirect pollution parts. Two algorithms to estimate the pollution errorlocally are proposed. in which local ERM-type error estimates are employed. Various errors anderror indicators in the energy norm for model problems are computed and compared. Numericalresults show that inclusion of a pollution error component in the local error estimation may benecessary in order to obtain reliable local error estimates. The numerical results also demonstratethat the proposed local polllltion error estimators can give reasonable estimates for the pollutionerror component.

Acknowledgemen.s

The authors wish to thank Professors l. Babuska and T. Strouboulis for many helpful dis-cussions on local and pollution errors and for suggestions which improved the present paper. Thesupport of this work by the ONR LInder contract no. N00014-89-J-3t09 and of ARPA undercontract no. DABT63-92-C-0042 is gratefully acknowledged.

Appcndix: hp fini'e elcmcnt setting

We outline here a general hp-finite element structure that permits the construction of generallip-bases for ~lIt and Vh. Let Q be an L-shaped domain in 1R2. We consider a I-irreglllar mesh QII ofquadrilateral (reclangular) elements Q" that are affine images ofa master square.Q = [ - I. + If.On .Q we introduce the shape functions:Vertices (Nodes):

i/i;(¢.lll = 1(1 ± ~)(l ± III i = I.2. 3. 4.

Edyes:

?k (~ )1,,; s·11 =

(-I);(I+ ) (J;) . - t ~. ') k i~ _ I} Pk'" 1 - . -. ~ ~ ~ Pk.

( - 1); ( I ") () . 2 4 2 k i~. ± S Pk 11 1 = .: ~. ~ Pk'

1nterior (Bubble):

bijK 1/) = Pi(~)Pj(lll 2 ~ i ~Pkfll' 2 ~ j ~ Pkfl2'

~J'~pd~) =,.;~ _I Lk- ds)ds.

292 .I. T. Odel/. r. Fel/g/Jou/'I/al of CO/llpntational and Applied lv/arhemarics 74 (/9IJfi) 245-11J3

8

76

5P = 4

3H I l-LJJ \~(1.1)

2W I I L'~o, i ~,,~~I

Fig. 37, L-shapl:d domain with mesh QI/ consisting of nonuniform Irp-distributions of conforming elements.

and Lk- 1 is the Legendre polynomial of degrce k - 1. The shape function on elcment QK are then

,1,(1.:)(" v) = ,L.oF;-l(\. \ l VI")I = ~k)F;-l(\ \')'1'; ,'1,,'2 '1'. K . I .. 2 . I.,k (,.1'" I .. 2 .....

where FI.: is an invertible affine map from .Q onto Q", The global basis functions are thenconstructed by matching shape functions on adjacent elcments in QH so that full continuity of theX,./s is achieved across element interfaces.

Constructions of this type can be used to produce nonuniform lip-meshes of the type illustratedin Fig. 37. Special cases would be 1..\' (1 ~ N ~ Nul piecewise bilinear. biquadratic. etc.

Refcrcnccs

r 1J I'v\'Ainsworth and IT. Oden. A procedure for a posteriori error estimation for Ir-p finite element methods. Compul.A/etllOds Appl. Mel'li. fnljl/g. 101 (1992173-96.

[2] 1\1.Ainsworth and J.T. aden. A unified approach to a posteriori error estimation using element residual methods.Numer. Marh. 65 (1993123-50.

[JJ I'v\. Ainsworth and .I.T. aden. A posterior; error estimators for second order elliptic systcms: Part I. Theoreticalfoundations and {I/wsteriori error analysis. Comput. Marh. Appl. 25 (19931101-113.

r4J rv\. Ainsworth and J.T. Oden. A posteriori error estimators for second order elliptic systems: Part 2. An optimalorder process for calculating self-cquilibrating fluxes. COlllpllt. lvlatlr. Appl. 26 (1993) 75-X7.

[5] I. Babuska. T. Sirouboulis. A, Tvlathur and C.S. Upadhyay, Pollutioll error in the Ir-version of the finitc elcmentmethod and the local quality of a posierilll'i errol' estimators. Fil/itl' 1~/emellts AI/al. Dl'si!ll/ 17 (1994) 273-321.

[6] I. Babuska. T. Strouboulis. C.S. LJpadhyay and S.K. Gangaraj. A posteriori estimation and adaptive control of Ihepollution-error in the h-version of the finite element method. Tech. Note BN-1175 and C\IC Report No. 94-06.August 1994.

[7] R.W. COllIe. 1\lanifestations of the Schur complement. Linea/' AII/eh/'a Appl. 8 (19741189-211.[8] D.W. Kelly. The self-equilibration of residuals and complementary a postcriori crror estimales in the tlnitc elcment

met hod. Illte/'lwt ./. N/I/I/('r. M ethat/s. I~I/.clrl/, 20 (1984) 1491-1506,[9] l.A. Nitsche and A.I-I. Schatz. Intcrior estimates for Ritz-Galcrkin Methods. Math. COlli/I. 211119741 937-958.

J. T. Oden. Y. Fel/glJo/{mal of Compmational and Applied Mathematics 74 (/996j 2-15-293 293

[10] J.T. Oden. L. Demkowicz .. T. Strouboulis and 1'. Devloo. Adaptive methods for problcms in solid and fiuidmechanics. in: 1. Babuska. O.c. Zienkicwicz,.I. Gago and de A.E.R. Olivcira. Eds .. AcclIrlIc,r Estimares al/d Adapc;!'£'R~fil/£'mel/ts ill Fillite E1emel/t Comp/{ca/iolls IWilcy. New York. 19X6). 249-2RO.

[IIJ A.II. Schatz and L.B. Wahlbin. Interior maximum norm estimates for finitc element methods. Math. Compo 3t(977)414-442.

[I2J A.H. Schatz and L.B. Wahlbin. Maximum norm estimates in the finite clement method on plane polygonaldomains. Part I. Ma/Ir. Compo 32 (197X1 73-109.

[13] L.B. Wahlbin. On the sharpness of certain local estimates for H? projections into finite clement spaces: intluence ofa rccntrant corner. ;\fa/II. Com[l. 42 (1984)1-8.

[14] M.A. Woodhury. Invcrting modified matrices. ~vlemo. Rep. No. 42. Stat. Res. Group. Princcton University.1950.