Computer Methods in Applied Mechanics and Engineering 101 (1992) 73-96North-Holland

CMA 305

A procedure for a posteriori error estimation forh-p finite element methods

Mark Ainsworth I and J. Tinsley OdenThe University of Texas 1lI Allstin. Austin. TX 78712. USA

Re<.:eived 14 October 199 J

A new approach to a posteriori error estimation is outlined which is applicable to general h-p finiteelement approximations of general dasses of boundary value problems. The approach makes usc ofduality arguments and is based on the element residual method (ERM). Important aspects of themethod arc that it provides a systematic approach toward deriving clement boundary conditions for theERM: it leads to an upper bound for the global error in an appropriate energy norm; and it is valid fornon-uniform amI irregular h-p meshes. In the present exposition. a brief outline of the theoreticalfoundations of the method is given together with the results of its application to several representativeproblems. These results show that the appnMch is applicable to gencral linearly elliptic systems.induding unsymmctri<.:al operators. and that the method is valid for broad classes of linear andnon-linear problems.

I. Introduction

IIl a recent paper [1L we developed a general theory for a posteriori error estimation whichhas the following attributes:(1) it employs a special variant of the clement residual method [1-3]:(2) under mild assumptions. it produces estimates ill convcnient energy type norms which

may not be directly associated with the actual bilinear form of the problem underconsideration:

(3) it employs a local duality argument that leads to a guaranteed global upper bound to theerror and which generalizes the duality method of Kelly [4];

(4) it is valid for symmetric and unsymmetric operators:(5) under additional assumptions. the approach can lead to asymptotically exact error

estimators:(6) it is well suited to irregular meshes with non-uniform h-p finite element approximations

and functions indepcndently of the order p of the local clement shape functions:(7) it employs a systematic scheme for nux balancing on element boundaries that substantially

increases the quality of the local and global effectivity indices.

Correspondence 10: Mark Ainsworth. Texas Institute for Computational Mechani<.:s. WRW 305. The Universityof Texas. Austin. TX 7H712. USA.

Ian leave from Mathematics Department. Leicester University. Leicester. UK.

0045-7825/92/$05.00 © 1992 Elsevier Science Publishers BV All rights n;served

74 M. AinslI'orth. J. T. (Jelen. A procedure for (/ posteriori error estimlltion

The thcory gencralizes prcvious work on the ERM. In particular, for thc special casc of lineartriangles (p = I). the conjecture made by Bank and Weiser [2] is confirmed: that a certainvariant of the ERM always providcs an upper bound on the error.

Our purposc in the prescnt papcr is to bridly outline the principal features of thc theory inconnection with a simplc model elliptic boundary value problem and to focus on issues ofimplementation. The robustness and generality of thc method is demonstrated by someapplications including elliptic systcms and unsymmetrical problems. The applicability of themethod to arbit ntry h-p mcshes is also illustrated. In particular. it is dcmonstratcd that themethod yields vcry good local estimates both for meshes with odd and with evcn ordcr shapefunctions on neighboring elemcnts. in contrast to other techniques proposed in recentliterature.

2. Theoretical foundations

2.1. Model prohlem

For clarity. we begin by considcring a simple model elliptic boundary value problem in twodimensions: Find 11= II(X. y) such that

-v· «({VII) + b -VII + Cli = f in 11.

All .a - = g on I ..iJ 11 .\Ii = 0 on I~) .

( I)

where Jl is a connected Lipschitzian domain in ~2 with boundary an = J~, n r~). In (1). thccoeffkicnts a, b. C and dataf, g arc assumed to he such that II exists. is unique. is continuouson the interior of nand dcpends continuously on thc data in appropriate norms. The weakform of (1) is as follows: Find II E?l' such that

B(Il. v) = L(v) 'Iv E ge .wherc

8('= {v E fJl(1l): yv = 0 on I~}

and B : iff x ff-+IR. L : :}'-+IR arc the forms

B(/I. v) = r (aY/I'vv + vb 'Yll + ('/Iv) dxJ!Iand

L(v) = r fv dx + r. gv ds .J!I Jt".

(2)

(3)

(4)

(5)

2.2. Partitioning

We next introducc a partitioning [fJ> of fl into N = N('?P) subdomains (I1nite elements) flKwhcre {] = U {l/\ and construct on '?P a space si' c 2t of piecewise polynomial functions. Thespace :i' could include arbitrary h-p finite element approximations of the type discussed in [5].

M. AinslVurth. J. T. Odell. A procedun' for a posteriori error esrill/mion 75

Following standard finite e1cmcnt procedures. we suppose that a function Ii E ~'is obtainedwhich represents. in some sense. an approximation of the solution 1/ of ( I ). Our goal is to usethe availahle information to calculatc an estimate of the error

e=I/-1/

in an appropriate norm.With regard to the partition r?P. we introduce thc following notation: B K' L K are localiza-

tions of the forms in (4) and (5).

BK(u. v) = r (avu'vv + vb ·vu + cuv) dx.JUKlV

B(u, v) = L BK(I/, u) .K-I

N

L(v) = L L,,(v).K=I

for 1/. V E 2f. Further 2fK is a local subspace of lr. with

.V

2r = EB 2fK•K= I

(6)

(7)

(8)

(9)

( 10)

Therc nm ....ariscs the issue of the norm in which we shall measure thc error e. For thispurpose. we introducc a symmetric. positive definite bilinear form (/: 2f x £-lR.

1l(1/, v) = in (Civl/ ·vv + EI/v) dx. ( 11)

whcre ii and c are constants \vhich are arbitrary except for the requirement that the originalbilincar form B( ... ) of (4) is cocrcive with rcspect to the norm induced by a( '. '). That is.

IB(I/. v)1 ~ f3l1ulL,sup 11 .. 11vEl

where f3 > 0 is a constant. and

IlulL, =Va(u.v).

In addition. we write

N

1l(1/. v) = L llK(Il. v) .K=\

VuE2r . ( 12)

(13)

(14)

76 AI. AinslI'orth. J. r. Odell. A procedllre for II posteriori error estimation

sIlvll, = L IIvIl2,.K·

K=I

(15)

( 16)

(17)

We also introducc the avcraged local flux

(18)

where s is a point on the interelcment boundary r"J = iJn" n anJ sharcd by neighboringelements. "K is the unit vector exterior and normal to an". and a"J is a linear functionassociatcd with cdgc rto.

With this notation now establishcd. we consider the following local problem: Find cPK E ftKsuch that

( 19)

Equation (19) charactcrizcs the local problem providing the basis for the error residualmethod corrcsponding to the norm 11·111' The signiflcancc of (19) is embodied in thefollowing theorem.

THEOREM 1. Let cPK be the solution of the local problem (19) corresponding to the elementfl". Then the functions aKJ of (18) can be chosen so that

(20)

where f3 is the constalll appearing in (12).

PROOF. Sec [1. 61.

REMARK 1. In the case in which the bilinear from B(· .. ) is symmetric and POSItIVCdefinite. we can take a( ' .. ) = B( ' .. ). Thcn the constant f3 = 1 and the norm 11,/1,. reducesto the standard encrgy norm.

REM AR K 2. Thc introduction of the symmetric form a( ... ) is equivalent to symmctrizingthe problcm [7].

REMARK 3. The conditions on the approximate solution ii (that Ii E 2£) can bc weakenedconsidcrably. Let'" bc a standard degree I basis function (a pyramid function) associatcd witha rcgular node in PJ. Thcn we necd only require that Ii E Cllun n 1I2(Pfl) n ff and

B(li. "') = L(",) . (21 )

M. Aillsworth. J. T. Odell. A procedure for a posteriori error estimatioll 77

That is. Ii nced only satisfy an orthogonality condition with respect to the lowest degree basisfunction.

REMARK 4. Let hI; = diam(f1I;) and suppose that

IB(lt. v)1 ~ Mil ltl/,,. I/vll,,. '<Ilt. v E ff£ .

As an additional assumption, suppose that the following condition holds:

(22)

(23)

where p = p(h. p) is a constant possibly dcpending on the mesh parameters hand p. Then itcan be shown [1,2) that

."'.L I/cPl;lI~1.1; ~ M{l + O(l~) + Cp}I/el/~

1;=1

where h = max K h K' Then we havc

N

13l1ell; ~ 2: I/cPKII~,.K ~ Mllell~1;=1

(24)

(25)

whcre lW depends on M, I~ and p. This result establishcs the equivalcnce of the global aposteriori error estimate to the true error. Morcover. if p - 0 as I~- 0 then we have it - M.This shows that the constants appearing in (25) arc asymptotically optimal. In the case ofB( " .) symmetric and positive dcfinite, we have asymptotic exactncss of the error estimator.

3. Implementation

The actual computation of the error estimator may be thought of as consisting of twodistinct stages:(a) the calculation of the linear splitting function fXKJ used in (18) to obtain the boundary

conditions for the local problcm (19).(b) the (approximation of) the solution of the local problem (19).The fundamental critcrion dctermining the choice of flux splittings used in the average (18), isthe following:

(26)

A simple physical interpretation of (26) is secn in the special case a(x) == I. b(x) == 0 andc(x) == O. For this situation (26) bccomes

o=f f(x)dx+f g(S)ds+f (n,,·aV/I)(s)ds.I; ilKnr." ;.K\ IN "

(27)

7H AI. Ainsworth. J. T. Odell. A procedllre for a postaiori ('rror t'.vtinuuion

The condition (27) proclaims that the data for the local problem (19) is in equilibrium. or thatthe fluxes have bcen equilibrated.

Kelly [4] used this same criterion to determine nux splittings in the case of piccewisebilinear finite element approximation of Poisson's cquation in two dimensions. Our approach.while related to Kelly's. differs signifil.:antly in scveral ways. Thc splittings used in ouralgorithm are linear functions. as opposed to constants; our splittings can be obtained usingonly local computations rather than applying a glohal optimization prol.:cdure; and. under mildassumptions. achicve the equilibration exactly (subjcct to rounding error) and are applicableto general linear elliptic systems of second ordcr partial diffcrential equations. In addition. ourapproach applies to general h-p finite element approximations on irregular meshes. withnon-uniform p and is valid for triangular elemcnts. quadrilateral elemcnts or indccd combina-tions of thc two. Importantly, our approach applies equally well to one. two or thrcedimensions.

3. J. The flux spliflillg algorithm

Thc complete details involved in deriving the algorithm to be presented can be found in I~].Here. we restrict ourselves to the bare essentials nccessary to implement the algorithm.

A key role in the algorithm is played by the degrec one basis functions (that is. the pyramidfunctions associated with the regular nodes in the partition). Denote the regular nodes byA. B .... and let l/J" dcnote the pyramid function associated with node A (scaled so that1/1.-\ = 1 at node A).

Thc computations arc localized using the patches of clements over which the functions 1/1"have non-zcro values. For ease of notation. we suppose elements ill' il2 •...• fl;\' constitutethe patch 5 ..\ of elcments on which l/JA docs not vanish. Some examples of possible patches areshown in Figs. 1-3.

Associated with each patch SA is a matrix TA. Thc matrix dcpends only on thc topology ofthe patch SA and not on the geomctry. For this reason we refcr to TA as thc topology matrixfor the patch. T" is a squarc matrix of size N x N. whcrc N is the number of elemcnts in thepatch. Therefore. TA are typically small matril.:cs whosc sizes do not incrcase as the partition isrefined. The entries of TA are given by

TA = [ ~-I-1

o2

-I-I

-1-1

2o

- I.]-I()

2

(Fig. 1). (28)

Fig. I. Topology matrix for interior node on regularmesh.

~aO

Fig. 2. Topology matrix for boundary node.

At. Ainsworth. J. T. Oelell. A procedure for a posteriori error estimation

0, I 0.

°2A

OJ

0.

Fig. 3. Topology matrix for interior node on I-irregular mesh.

I' = [ I - iJ (Fig. 2) . (29)A -I

2 -1 0 0 0 -1-I 3 -I -1 0 0

I'll = I () -1 2 -I () _~ I (Fig. 3). (30)() -I -} 4 -I() 0 0 -I 2 -I

-I 0 0 -I -I 3r if j = i .efA );j = -I. if n; and nj are neighbours in the patch. (31 )

O. otherwise.

wherc C; is the number of elements in the patch which share an edge with element n;. Someexamples of topology matrices for various types of patch in two dimensions are shown in Figs.1-3.

Thc s}ngular matrix TA is then modifkd by adding ] to every entry. thercby giving a newmatrix I'A with entries

if j = i ,if n; and f}j arc neighbours in the patch,otherwise,

(32)

wherc C; is as beforc. It may be shown [8] that fA is non-singular. In fact it is symmetric.positive definite and conscquently simple to invert numerically (using, for example, an LVfactorization) .

Dcfine thc N-vector b A with cntries

(33)whcre

Furthcrmore. for every intcrelemcnt edge rKJ within the patch definc

PKl.A = - (t. 1/111 (s)[1l . (Iv/in ds .JI KJ

where

(34)

(35)

(36)

80 M. A il/slI'onh . 1.T. Ode,1. A procedure for II posteriori error estimllIiofl

Having calculated these various quantities. a set of constants ex;J is computcd using theprocedure outlined in Fig. 4. The case when PKJ.,4 vanishes requires a little extra care, dctailsof which can be found in [8].

The procedurc in Fig. 4 is applied to every regular nodc in thc mesh. The actual nuxsplitting exKJ used in (18) is thcn givcn by

exK/. (.\') = L exKI. .A t/lA (s), s E Tn .A

(37)

Most of the terms in this summation vanish due to ~JA having non-zero values on a smallnumber of edges. For example. in the case of rcgular meshes. only two tcrms in thesummation arc non-zero. namely those corrcsponding to the two nodes forming the endpointsof the edge J~t .. i.e. exKJ is then thc linear function which interpolates to ex"J.A and ex"J.B at theendpoints of thc edge. In the case of irregular meshes the situation is more complicated withat most three non-zero terms appearing in the sum.

It can be shown [81 that. with this choice of splitting. the condition (26) is satisfied. Theprocess described may appear elaborate. However. the computational work entailed is rathersmall by comparison with the cost of performing other standard tasks in the I1nite elementmethod. In [8]. an operation count shows that the process is of optimal order, increasing onlylinearly with the number of elemcnts in the partition.

3.2. Approximation of local problems

Thc approximation of the local problem (19) is performed by means of a Galerkin methodusing a particular set of trial functions. Here. we shall describe the procedure we use forquadrilateral elements.

Lct {P,,(x)} dcnote the usual Legendre polynomials on [-I. I]. It will be neccssary to beable to computc the values of the polynomials thcmselves and their derivatives efficicntly.Unfortunately. in many textbooks it is suggested that these quantities be calculated directlyfrom the expansion in terms of powers of x. This approach is not only unnecessarily expensive

for each regular node A dobegin A

calculate TA; A

calculate an LU factorization LA~\=TA;for every element K E SA do

begincalculate bK,A;calculate PKL,A

end;solve LAUAAA=bA;for every interelement edge TKL in the patch do

begin(tKL,A= 1+ (AK,A-AL,A) / PKL,A

endend;

Fig. 4. Pseudo-code of flux splitting algorithm.

M. AinslI'orth. 1.T. Odell. A procedure for (/ posteriori error e.wimmion 81

but leads to catastrophic build up of round-off errors. Therefore. wc usc thc threc tcrmrecurrence rclations. Thus, for somc fixcd value of x. wc calculatc P,,(x) and P:,(x) as follows:

and

Po(.\') = 1 . P1(x) = X .

2k + 1 kPk+I(X) = I. I 1 xPk(X) - k + 1 Pk_I(X). k ~ 1.

P(I(X) = O. P;(x) = I ,

I 2k + I / k + I .Pkfl(X)= I. xPk(X)--;:-Pk_lr). k~l.

(38)

(38)

To calculate the values of P,,(x) and P:,(x) in this way requires only ordcr II operations.Moreover. as a byproduct. one also obtains the values of all the lowcr order polynomials andthcir dcrivativcs at no extra expcnse.

The values of Po(x) .... 'Pll(x) and P1lr) ..... P:,(x) arc then llsed to computc thcfunctions Xo(x) ..... XlI(x) and xi{r) ..... X:,(x) given by

and

XII(X) = JI PII(x) . XI (x) = ~1PI (x) .

1 ~2k - 1, .xJ\') = /./t._l\ 2 (x'-I)Pk_,(x). k=3 ..... II.

XI')(X) = O. X;(x) =.J1 Po(.\')· X~(.\') = v"1 PI(x).

k = 3..... II .

(40)

(41 )

Lct {] = [-I. I]x 1- I. I] dcnotc the usual refcrencc demcnt. The trial functions will bedefincd on iJ. For simplicity. assume that the finite elcment approximation it is a completcpolynomial of degrcc p on the element under consideration. Let q > 0 be an integer; then wcdefine thc space Vt'·'1(f2) by

vp·qUi)= {X;(Oxk('7J)IO~j.k~p+q and at least one ofj.k>p}. (42)

It is seen that djm( vp,q) = (p + q + 1)2 - (p + I )2. The indcx q controls the number ofincrements in the polynomial degrce of the space and may be used to increase the dimensionof the space.

Thc local approximation space is then t~ken 10 be ii'K = yrK n V1"'!(D}\). The discretizcdversion of the local problem (19) is: Find 4>/\ E;{K sllch that

(43)

Owing to thc above assumptions and the construction of the local space. this problem alwayshas a unique solution.

82 M. Ainsworth. J. T. Odell. A procedure for a posreriori error estilllmi01/

4. NumcricaI cxamplcs

In this scction we present thrce examples to illustrate the performance of the algorithm.In order to compare the estimated error with the true error it is necessary to accurately

calculate the true error ovcr cach elcment. In all our examples. the true error is computcdusing an algorithm which approximates the integral using first a single subdomain and thensubdivides the rcgion of integration into four subdomains and approximates the integral overthe four subregions. Two estimates are thus obtained for the true value of the integral. If thedifference in these two estimates is less than 1% relative error. then the approximation isaccepted. Otherwise, the element is further subdivided into 16 regions and so on untilagreement between consecutive approximations is obtained to less than I% relative error.

4. I. Symmetric elliptic operator with smooth solution

The problem we consider is: Find Ii such that

-illl+ll=(J inn.

subject to the boundary conditions

u(O.y)=(e-\l~-I)sinay. O<y<I/2.

(44)

u(x.O)=ll(x.l/2)=O. 0<x<I/2. all /- = o. x = 1/2, 0 < l' < 1 2.an .(45)

where 0- = 21T'. The geometry of the domain n and the boundary conditions applied are shownin Fig. 5.

The true solution is given by

Il(X. y) = (exp[(x - I rVl + (T21- exp(-xV'i-+ (J'l)) sin ay.

In this case, we have

B( u. v) = ( (V'u' Vv + llU) dxIn

(46)

(47)

and we choose a(l/. v) = B(I/. v). Theorem I predicts that we will obtain guaranteed upperbounds on the true error measured in the energy norm defined by B(' .. ). provided thatequilibration of the fluxes is achieved.

The problem is solved using uniform meshcs of quadrilatcral elemcnts with uniformpolynomial degree. The local problems are approximatcd using an increment q = 2 in the localspace. That is. a degree p finitc clement approximation is analyzed using the space V"·I'+:. Ineach casc. the equilibrution proccdurc is able to reducc the lack of equilibration to the level ofround off error on virtually every element in the partition.

Table I contains the results obtained for finite elemcnt approximations of degree 1-4 on

.\-t. Aillsl\'onh. J. T. Odell. A procedure for a posteriori error estimatioll 83

Fig. 5. Geometry and boundary conditions for smoothmodel problem.

I

U -0 ... x

-.{,;;2 {u=(e -1)sinuy

y

~

dU=Oan

Table 1Global l:ffectivity inuices for model problem withsmooth solution

Degree Number of clements

( p) 16 64 128

I 1.012909 1.003555 l.llO09072 1007872 \.002019 1.0003603 1.(J00265 I.OOll(J9S l.llOOOJI..\ I.OOO2:n \.OO{)\20 l.lJOO()50

meshcs containing 16. 64 and 128 clements. Thc quantity shown in the table is the effectivityindex (the ratio of the estimated crror to the true error). Theorcm I prcdicts that theeffectivity index be greater than unity. This prediction is borne out by the results shown inTable 1.

4.2. Cracked pallel problem

Consider the problem: Find II such that

subject to the boundary conditions

li(r.1T)=O. O<r< 1. iJII / all = O. 0 < r < 1 . H = 0 .

(48)

(49)

with lI(r. e) = rl12 cos ~e on the remaining portion of the boundary. Thc gcometry of thedomain n is shown in Fig. 6.

The true solution is given by

lI(r. 8) = ,11

2 cos !e . (50)

The problem is the analogue of a cracked panel problem in lincar elasticity. with a singularity

_u=o __ au_o_;r;-

Fig. 6. Geometry and boundary conditions for crac·ked panel problem.

~I

Singularity

Fig. 7. Position of clements adjacent to singularity incracked panel.

84 11,4. Aitl.l'lI'or/h. J. T. Odell, A procedure for a posteriori error estimation

at the origin. In this casc. \",e have

B(Ii. v) = 1 Vii' Vv dxJJ

(51 )

and we take a(li. v) = B(u. v). Theorcm 1 again predicts that we \vill obtain uppcr bounds onthe true error. provided that we equilibratc the fluxes.

In analyzing this problem. some care must be takcn with the approximation of the localproblems. Theorem 1 assumes that thc local problems are solved cxactly. which is not the cascin practice. Therefore. for the purposes of illustrating the theory. a sequence of approxi-mations to the true solution of the local problcm is obtained by incremcnting q. That is to say.we compute a sequence of approximations using the spaces

until the difference in the norm of the approximation is sufficiently small.One other feature of this particular example is the difficulty in estimating the error in

elements adjacent to the singular point. Our theory makcs no promiscs concerning theeffcctivity of the estimator in a single clement. However. we present rcsults showing theestimated and thc true error in the clements adjacent to the singularity fl(. and DR (see Fig. 7).

Tables 2-4 contain results of estimating the error in the approximation obtained using auniform mesh of 32 quadrilateral elements with uniform polynomial degree 1. Five incrementsin the local approximation space are needed before agreement is obtained. For purposes ofcomparison. we also give the results obtaincd when no equilibration or balancing of fluxes isperformed (instead. a simple averaging is applied).

Tables 2 and 3 contain the error estimates in the elcments adjacent to the singularity flL andDI{' The estimate of thc error obtained when cquilibration of fluxes is performed is superior tothe estimate obtaincd using a simple averaging. Table 4 contains the global estimates of theerror. It is secn that it is necessary to approximate the local problcms accurately if one is toobtain the upper bound proclaimed hy Theorem I.

Table 2Effect of solving lo<.:alprohlem with increasing a<.:curacy on the estimates of local error inclement III for cracked panel (p = I. 32 elements)

Estimaled lo<.:alerror Local effectivity indexNumber

of Wilh Without With Withoutincrements balancing balancing balancing balancing

0.132644 0.635051 (-I) 0.973984 O.46630H2 0.132644 0.635051 (-I) 0.973984 O.46630H3 0.t50311 0.679112(-( ) t.103710 0.4986614 0.153503 O.686562( -() 1.127149 n.5041325 0.154516 0.689176(- 1) 1.134587 0.506051

True value O.1361R7 O.136187( +0) 1.000000 1.000000

.\1. A il/sI\'orth . J. T. Odell. A procedure for a posteriori error esrima/iol/

Table 3Effcct of solving local problem with increasing accuracv on the estimates of local crror inelement [ll( for cra<.:ked panel (p = 1.32 elements)

Estimated )o<.:alerror Lo<.:al effc<.:tivity indcxNumber

of With Without With \\'ithoutincrcmcnts balancing balancing balancing balancing

I ll.I09140 O.1831NI O.8472lil 1.-t275922 1l.10914ll 0.IR3891 lJ.8472XI 1.-t275923 lJ.11284H 1I.lli41156 11.8701167 1.-t2S8734 0.113189 0.1841162 O.87R715 \.4289205 0.113216 0.184062 0.878924 1.4289211

True value 0.128812 O.12XS12 1.1I01l0lJ() I .OOOOOll

Tablc 4Effe<.:t of solving loc"l problem with in<.:reasing accuracy 011 thc estimates of global error forcracked pancl (p = 1.32 clements)

Estimated global error Global effe<.:tivity indexNumber

of With Without With Withoutincrements halancing balancing balan<.:ing balancing

lI.lS3726 0.206899 1I.921%3 1.03S2482 0.IS3726 0.2116899 0.921963 1.0382483 0.199198 O.20H800 lJ.99l)604 I.0477Sli4 0.201876 0.209060 I.013lJ42 I.0490Y25 0.202664 0.209149 1.0/6997 I.04953Y

Truc value 0.199277 0.19lJ277 I.onoooo l.lJOOO(]lI

Table 5Effect of solving local problem with increasing accura<.:y on the estimates of local error inelement [It for <.:racked panel (p = 1. 128 clements)

Estimated local error Local effectivity indexNumber

of With Without With Withoutincrcmcnts halancing balancing balancing balancing

I 1l.lJ27UI9( -I) O.443S47( -I) O.lJ594U5 O.-t5935.12 O.lJ27019(-I) 0.443847( - I ) O.951J405 O.-t593533 (]. 1f15048(lJ) 0.-t74636( -I ) Ul87 179 1I.-t912184 (].J07279(1I) 0.479842( -I) 1.l10268 11.496605:\ 1I.J07987(O) O.4S I669( - 1) 1.117596 1I.498496

Truc value 1l.lJ66244( - I) O.966244( -() 1.OOlK)()() I .OOOOO()

X5

Tables 5-7 show the corrcsponding rcsults obtaincd when the mesh is relined uniformly to12Hclcments of dcgrce one. The rcsults obtaincd are similar to the case of:n elements. Tables8-10 contain the results obtained when the degrec of thc elements is increased uniformly todegrce 2 on 32 elcments. Once again. thc results show the superiority of the estimate ohtained

86 AI. Aillsll'orth. J. T. ()elell. A procedure for II posTeriori error l'stimilTioll

Tablc 6Effect of solving lo<.:alproblcm with increasing accuracy on thc estimates of local crror inclement nil for cracked panel (p = I. 128 clements)

Estimated local error Local effectivity indexNumber

of With Without With Withoutincrcments balancing balan<.:ing balancing balancing

1 O.762753( -I) 0.128522(0) 0.855679 1.44179X2 0.762753( -I) 0.128522(0) 0.855679 1.4417983 O.788666( -I) 0.128637(0) 0.884749 1.4430884 0.791049(-1 ) 0.128641(0) 0.887422 1.4431335 0.791235(-1 ) 0.128642(0) (1.887631 1.443144

True value 0.!NI401(-I) 0.891401(-1) 1.000000 I.onoooo

Table 7Effect of solving local problem with incrcasing accuracy on the cstimatcs of global crror forcracked panel (p = I. 128 elcments)

Estimated global error Global effectivity indexNumbcr

of With Without With Withoutincrements balancing balan<.:ing balancing balancing

0.129397 0.145495 0.918497 1.0327662 lJ.129397 0.145495 0.918497 1.03276(,3 0.140126 0.146813 11.994655 1.0421214 O.1419X3 0.146992 1.007836 1.0433925 0.142531 0.147053 1.011726 1.043825

True value 0.140879 0.140879 1.000000 1.000UOO

Table 8Effect of solving local problem with increasing accuracy on the estimatcs of local error inclcmcnt ll,. for cra<.:kcd p:lllcl (p = 2. 32 elcmcnts)

Estimated local error Local effectivity indexNumber

ofincrements

I2345

True value

Withbalan<.:ing

O.779887( - I)0.877218(-1)O.916439( - I)0.924515(-1)O.932539( - 1)0.802764( -I)

Withoutbalancing

0,405007( -I )0.4430116(-1)0.457423( -I)0,46606lJ( - t)0,463X73( - I )0.802764( -I)

Withbalancing

0.9715021.0927471.1416051.1516651.16166111.000000

Withoutbalancing

0.5045160.5518510.5698100.5805S00.5778451.000000

using equilibrated fluxes. In cach case. the result of Theorcm I is verified. although it IS

necessary to solve the local problems very accurately.

-J.3. VI/symmetric elliptic system

As a final exam pic . we consider the unsymmetric elliptic system with non-constantconvcction given by: Find III' 112 such that

M. Aillsworth. J. T. Odell. A proced/lre for II posteriori error estil//CIIiOIl

Tahle 9Effect of solving local prohlem with in<.:rcasing accuracy Oil the estimates of local crror inelement [2" for cracked pancl (p = 2. 32 dements)

Estimated local error Local effectivity indexNumber

of With Without With Withoutincrements balan<.:ing balancing balancing balancing

I O.532473( - I ) O.on5535( - I ) 0.686000 ll.R574272 O.6136lN( -I) n.717056( -I) 0.7lJ0633 O.lJ238033 0.628189(-) 0.848990( - I ) n.80lJ313 l.lllJ37774 O.M4079( -[) O.876593( -I) 0.li29785 1.12933lJ5 O.Mli2l)9( -I) O.912130( -)) 0.K15222 1.175122

True value O.7762()[)( -I) n. 776200( -I) I .000000 1.IlUnOOO

R7

Table 10Effect of solving. local problem with int'fcasing accura<.:y on the estimates of global error for cracked panel(p = 2. 32 clements)

Estimated global error Glohal effectivity inJcxNumber

of With Without With Withoutincrements halancing balan<.:ing balancing balancing

O.948095( - I ) O.790lllJ6( - I ) 0.838555 ll.698H 102 0.107495(0) 0.904738(-1) ll.950753 0.8002073 0.111550(0) 0.980423(-1) (l.lJ8661S O.H6714H4 0.113117(0) n. In0715(O) l.tJOO478 0.8907H75 0.114017(0) 0.104075(0) !.OO8438 0.920505

True value O. 113063( 0) O.1l3063( () 1.nOO()OO 1.000000

lUll (Jill- e tlll + X - - V --:-- + XVIl - Il, = 0 .

I aX' ely • I -

all, all,-Etlll,+X~-V~-XYlll+ll,=O inn.

- dx' ay -

whcre E = 1 /100. subject to the boundary conditions

(52)

112 = xy exp[ (x2- yo' - 1) / E I on rD (53)

andalii 2 '

E - = ell . V exp[(x - v- - 1) Ie] .an .iJll, , ,

E --= = Ell' VXV expl(x' - v" - I) h]an . .whcre f2. J~) and J:V are shown in Fig. R.

The true solution to this problem is givcn by

on J.~ .

(54)

"2 = xy exp[ (x2 - / - I) / E] . (55)

AI. Ainsworth. J. T. Odell. A procedllre for {/ postaiori error estimation

rN

(1,0 )

Fig. 8. Geometry and boundary <.:onditions for ul1symmetri<.: ellipti<.: system.

Thc main feature of the solution is the prcsence of a strong boundary layer cffect along theright-hand wall of the domain causcd by the non-constant convection dominated flow fieldh=(x.-y).

In this casco the bilinear form B( ... ) is unsYl11metric. We choose the bilinear form a(· .. )to be

(56)

The theory prcsented in [6] shows that the error estimator bounds the true error measured inthe symmetrized norm. For lhe purposes of illustration. in this example we compute the trueerror in thc symmetrized norm explicitly. It is this quantity which is labclled as the true errorin Tablc 11.

The presence of the boundary layer indicates that an adaptive finite clement analysis basedon refining the mesh and enriching the degree of the approximation is suitable. The sequenceof meshes gencrated during thc analysis is shown in Figs. 9. 12. 15. 18 and 20. The mcshes arenot only irregular but contain clements of differing polynomial degree. Thc final meshcontains elemcnts of degrcc six ncar the houndary laycr. Nevertheless. the behaviour of theerror cstimator remains highly satisfactory as shown by the rcsults ill Table II.

One source of concern when estimating crrors for this type of problem is that thcdistribution of the estimated error will not agree \\lith thc distribution of the truc crror owingto the convective effect. Therefore. in Figs. 9-23. we present plots showing thc distribution ofthe true and cstimatcd errors. It is obscrved that the distribution of the estimated error closelyrcflects the actual error distribution.

Table 11Behaviour of error estimators for unsymmctri<.: elliptic systCI11

Degrees TrueEstimated global error Global cffe<.:tivity index

Mesh of glob;]l With Without With Withoutnumber freedom error balancing balancing balancing balancing

25 U.-toOI29(0) O..t20103(0) 0,420218(0) 1.049919 l.O502062 51 0.145063(0) 0.1 ......794(0) 0.147063(0) 0.998146 1.0137873 III O.5639lJ2( - 1) O.564022( -I) 1I.5rl4159( -I) l.nOOOS3 1.1I00296-I 165 0.646845(-2) O.M7644(-2) lJ.649044( - 2) l.llO1235 1.003400:; 340 0.2(13345(-2) O.272907( - 2) 1I.274071 (- 2) 1.036310 1.040730

M. Ainsworth. J. T. Oden. A procedllre for II posteriori ('I'mI' estimation

MESH 1-----

I

II

I

--.---. I:'" • .,- ~. .- I ~~_~~~_7_~_J D.O.F=25

Fig. 9. Adaptive analysis of unsymmetric elliptic system. Mcsh I.

MESH 1-r-n~ r--'.---,~I

89

I

i 0 0.075-------- -

I_J

I

~--~-~~

MIN= O.570E-03MAX=O.3003809ERROR=0.400 1289D.O.F=25

Fig. 10. Adaptivc analysis of ullsymmetric clliptic systcm. Distribution of true error on Mesh I.

90 M. Ainsworth. 1.T. Odel/. A procedure for (/ posteriori aror estimation

MESH 1

_.-~_._--, ..- -_.-------.;--, .....A:

...

l-ro - 0.075L. ~ _ 0.15 0.25I0.325 I

M1N= o 235E-03MAX=O.3045585ERROR=O.4201032D.O.F=25

Fig. II. Adaptive analysis of unsYllllllctric elliptic system. Distribution of estimatcd error on Mesh I.

MESH 2II

III

.. _~

1---

pc

L_

I

L__-4 6 7 D.O.F=51

Fig. 12. Adaptivc analysis of unsymmetric clliptic system. Mesh 2.

M. Ainsworth. 1.T. Odell. A procedure for a po.\'tl'/'iorierror estimation 91

MESH 2.

-t- -l

- -: -, -lI I I

I

~ ~T--~

I

II ~

o 0.03 0.06 0.09 0.12

MIN= 0.209E-03MAX=0.lI2570 1ERROR=O.!45063!D.O.F=51

Fig. 13. Adaptive analysis of unsymmetric elliptic system. Distribution of true error all Mesh 2.

MESH 2.

MlN= 0.108E-03MAX=O.112969(ERROR=O.1447~D.O.F=5!

I

I1 1

-~-~

I

I , -I, ~+-~--~.-- ~- f~-:;:--I

~-.~:"i40.03 0.06 0.09

---~o

------~ ---1--- -.-- lI -r- ----,

I

I

rig. 14. Adaptive analysis of unsymmetric elliptic system. Distribution of cSlimaled crror on Mesh 2.

92 M. AinslI'orth. J. T. Dc/ell. A procedure for a posteriori error I'stimotio"

MESH 3

~I

- ----;

I II .... I_, .. 1-_ ••• 1

1_

.....--- T I:5f1'J;..'ii 6 7I p= 3 4 5I I 2l.~_~ __

D.O.F= III

Fig. 15. Adaptive analysis of unsymmctric elliptic system. !'vtesh 3.

MESH 3

MIN= O.203E·04MAX=O.0443252ERROR=O.056399.D.O.F= III0.045 l0.033750.02250.01125

1

- -j --

i 1

-1- LJJI

~

I'-

r-i.o

Fig. 16. Adaptive analysis of unsymmetric elliptic system. Distribution of true error on Mesh 3.

A.J. Ainsworth. J. T. Odell. A procedure lol' a posteriori error estill/ation 93

MESH 3

r-----···--

MlN= O.355E-04MAX=O.0444163ERROR=O.056402:D.O.F= III

,---I0.0450.03375

'--r--j

I iL---t--r-lI'llI IIrI

00225~-~-L~~~

o 0.01125

Fig. 17. Adaptive analysis of unsymmetric ellipli<.: system. Distribution of estimatcd error on Mesh 3.

MESH 4-

--'--'r--- l

I

!_-

IIiL- _,I:

I

I- - 1·1[' p= L- - 1 3

J "_ 6 D.O.F= 165

Fig. IS. Adaptive analysis of unsyml11clric ellipti<.: system. Mesh -I.

94 M. AinslI'orth. J. T. ()den. A procedlll'e for {/ postl'liori 1'11'01' estimation

MIN= O.647E-05MAX=O.0042109ERROR=O.OO64684D.O.F= 165

MESH t- -'T-=--I---lI I I

1---1-I I'- --r- --I-I

I r--I1- -,---I II .r-

II--~ -r-EIo O.900E· 0.0021 0.0036 0.0045___ '_ _ __~~ --.J

1'---

Fig. 19. Adaptive analysis of unsymmctric elliptic system. Distribution of true error on Mesh 4.

MESH 4

--L_ MIN= O.532E·05MAX=O.0042149ERROR=O.OO64764D.O.F= 1650.00450.0036

..... --

0.0021

-I-I! I

t-·---·-

I

It

II

1

_-1' .--~.·'\:fS·"'·', h""

.:_ .~ __ • I

o O.900E·

Fig. 20. Adaptive analysis of ullsymmctri<.: ellipti<.: system. Distribution of estimated error on Mesh 4.

iH. Ainsworth. 1. r. Odell. A procedllre for a posreriori ermr estimation

MESH 5------ l-~- 1--1

95

p=' F400 I12345678

Fig. 21. Adaptive analysis of unsymmctric ellipti<.: system. Mesh 5.

MESH 5

D.O.F=340

'----1 i~·~I I I~.~·". :·1

I I I" . ~,.'",-k:.(-"'". ,--,-n

I II

~

~-j--! I I

i ~!+I~- ._1__ 1_1=d I

_ I I ··IOOlIIM-:;~ I : I I--.!~ ~ __ OJ75E._ 0.875E· __ 0.0015 0.001875 ...J

MIN= O.633E-05MAX=O.OO 18429ERROR=0.OO2633·D.O.F= 340

Fig. 22. Adaptive analysis of unsymmetric elliptic system. Distrihution of true error on I'v!csh 5.

M. Ainsworth. J.1'. aden. A proccdure for tl posteriori error estimation

MESH 5

M1N= 0.525E-05MAX=O.OO 19571ERROR=O.OO27291D.O.F= 3400.00210.001650.001050.450E·

II

: I

l l.'- ;o

--f-=~~--r----l: :~·I-~-I

I

Fig. 23. Adaplive ilnalysis of unsymmetric elliptic svstem. Distribution of estimated error on Mesh 5.

Acknowledgment

The support of this work by ONR under grant NOO0l4-89-J-145I. NSF under grantASC91l1540 and ARO under contract DAAL03-89-K-0120 is gratefully acknowledged.

References

III M. Ainsworth and J .T. Oden. A unified approach 10 a posleriori error estimatilln based on element residualmethods. unpublished.

12J R.E. Bank and A. Weiser. Some a posteriori error estimators for ellipti<.: partial differential equations. tvlath.Compo ~~ (1985) 283-301.

13] J.T. Oden. L. Demkowicz. T.F. Strouboulis and Ph. Devloo. Adaptive mcthods lor problems in solid and Iluidmechanics. in: 1. llahuska. O.c. Zienkiewicz. J. Gago ami E.R. de A. Oliveira. cds .. Accuracy Estimatcs andAdaptive Refinemcnts in Finite Element Compulations (\Viley. New York. 1986) 2~9-280.

14) D.W. Kelly, The self equilibration of residuals and complementary error estimates in the tinite element method.Internal. J. Numer. Methods Engrg. 20 (1984) 1491-1506.

'15} .I.T. Oden. 1.. Demkowicz. W. Rachowicz and T.A. Westermann. Towards a universal It-p finite elementstrategy. Part 2. A posteriori error estimation. Comput. tvlethods Appl. Me<.:h. Engrg. 77 (19!{l)) 113-180.

16} M. Ainsworth and J.T. Oden. A posteriori error estimators for second order elliptic systems. Part 1.Theoretical foundation and a posteriori error analysis. Computers in Mathematics and Applications (in press).

171 J.T. Oden. L. Demkowicz. W. Rachowicz and T.A. Westermann. A posteriori error analysis in finite elt:ments:The elemcnt residual method for symmetrizable problcms with applications to compressible Eull'r andNavicr-Stokes equations. Comput. Methods Appl. Mech. Engrg. 82 (19lJO) 183-203.

181 M. Ainsworth and J.T. Oden. A posteriori error estimators for second ordcr elliptic systems. Part 2. Anoptimal ordcr process for <.:alculating self equilibrating fluxes. Computer in Mathematics with Applications (inpress).

19] 1. Bahuska and W.e. Rheinboldl. A posteriori error eSlimales for adaptive finite element computations. SIAMJ. Numer. Anal. 15 (1978) 736-754.