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STUDIES IN APPLIED MECHANICS 47

Advances in AdaptiveComputational Methodsin Mechanics

Edited by

P. LadevezeLMT - CachanENS de Cachan, CNRS, Universite Paris 661, avenue du President Wi/son94235 Cachan Cedex, France

J.T. OdenTICAMThe University of Texas at Austin3500 West Ba/cones Center DriveAustin, Texas 78759, USA

1998

ELSEVIERAmsterdam - Lausanne - New York - Oxford - Shannon - Singapore - Tokyo

Advances in,Adaptive Computational Methods in MechanicsP. Ladeveze and J.T. Oden (Editors)© 1998 Elsevier Science Ltd. All rights reserved. 43

A technique for a posteriori error estimation of h-p approximationsof the Stokes equations

J.T. Oden • and S. Prudhomme t

Texas Institute for Computational and Applied Mathematics,,The University of Texas at Austin, Taylor Hall TAY 2.400, Austin TX 78712, USA

This paper deals with a posteriori error estimation for finite element approximationsof the Stokes problem. The ultimate objective is to analyse how given error quantities ofinterest are influenced by the residuals, which are viewed as the sources of the numericalerror. First, a global error estimator is proposed in terms of the norms of the residuals.Then, techniques to efficiently estimate these norms are advocated. This is followed bythe investigation of a general approach to evaluate the numerical error in pointwise, localor global quantities of interest.

1. INTRODUCTION

Error estimation in computational processes has been a subject of interest for morethan two decades since the pioneering work of Babuska and Rheinboldt [3]. We referto Ainsworth and Oden [2] and Verfiirth [19] for an extended account of the subject.In the particular case of the Stokes problem, several approaches have been investigatedin [18,8,6,16,2).

The method we propose here belongs to the family of Implicit Error Residual Meth-ods. The errors in the velocity and pressure variables are driven by the residuals n'hand n~in the momentum equation and the continuity equation respectively. The resid-uals represent the sources of error in the finite element approximations, and as such, arepost-processed to provide meaningful error estimates in global "energy" norms. The eval-uation of n~is shown to be exact, local and cheap. The calculation of the error measure

~ n'h is however more demanding. A new technique has been developed which providesaccurate approximations of R'h through a global but inexpensive iterative process. First,error estimates are obtained using spaces of low-order bubble functions as perturbations.

• These are then corrected by using enriched spaces constructed through an adaptive proce-dure. This approach avoids the major difficulty of prescribing proper boundary conditionsfor each subproblem in the Element Residual method [11,7,2). The effectiveness of themethodology is demonstrated on various test cases of the Stokes problem with smoothand un smooth solutions.

In recent years, the success of a posteriori error estimation has prompted users todemand error estimates in quantities of interest other than the classical energy norm.

'Director, TICAM, Cockrell Family Regents Chair in Engineering.t Gradua.te Research Assistant.

44

Works in this field have been undertaken by Babuska and Strouboulis [4] and Becketand Rannacher [10,9) in order to estimate and control the error by adapting the meshparameters with respect to these quantities of interest. We extend the ideas to the Stokesproblem and perform a preliminary investigation of the performance of such techniquesto evaluate the error in pointwise, local or global quantities of interest.

2. PRELIMINARIES

Let 0 denote an open bounded Lipschitz domain in IRn, n = 2 or 3, with boundaryao. We consider the Stokes equations:

-~u+ \lp = f\l·u = 0

in 0in 51 (1)

with Dirichlet boundary condition:

u = g, on ao, (2)

where u = u(x) and p = p(x) are respectively a vector-valued and a scalar-valuedfunction defined at point x = (Xl, X2 •.... xn) in O. Since the Stokes equations can bederived as a linearization of the steady-state Navier-Stokes equations, the variable u and pare referred to as the velocity and pressure. The source term f = f( x) is a prescribed bodyforce and g is a function defined on an which must satisfy the compatibility condition,

1 g. nds = 0lan . (3)

In what follows, we restrict ourselves to the case of homogeneous boundary conditions,i.e. g = 0 on 80. This simplifies somewhat the theoretical analysis of the Stokes equa-tions, while retaining all their interesting features. We shall use standard notations forvarious Sobolev spaces of functions defined on O. We begin by introducing the trial spacesof velocities V and pressures Q defined by:

v = HMn) = (Hci(n)t,

with corresponding norms:

Ivl~= in \lv : \lv dx,

Q = {q E L 2 (51) : in q dx = O},

..

We also introduce the bilinear forms a and b,

a: Ht(O) X HI(O) --t utj a(u,v) = in \lu: \lvdx

b: Ht(O) X U(O) --t IR; b(v, q) = -in q\l. v dx.

The bilinear form (j is the inner product associated to the norm 1·ll in V, so that

a(v,v) = lvii, 'r/v E V.

45

Lemma 1 The bilinear f01'm b is continuous on V x Q, and in particular, there exists apositive constant !vh such that

b(v,q) $ Mb Ivill/qllo, Vv E V, Vq E Q,

. where lvIb = Vn, where n is the geometrical dimension of the problem.

Proof Let v and q be arbitrary functions in V and Q. Since \7. v E Q, we have:

~ b(v, q) $1/\7. vl/o IIqllo'It is now sufficient to show there exists iVh such that IIV, vllo $ lvI. Ivlt. Indeed,

IIV·vll~ = f (t~V~)2 dx $ f nt(~v~)2dX $ n f\7v:Vvdx = nlvli.in .=1 uX, in .=1 uX, in (4)

so that 11\7 . vllo $ Vn Ivll and M. = Vn'Moreover, it can be shown that b satisfies the standard LBB condition (see Oirault and

Raviart [12)), i.e. there exists a constant {3 > 0 such that

sup Ib(v,q)1v E V\{O} ~;::: {3 I/ql/o' Vq E Q.

The Stokes problem is now reformulated in the equivalent weak form:

(5)

For f E VI given, find (u,p) E V x Q, such thata(u, v) + b(v,p)

b(u,q) =(f,v), Vv E V0, Vq E Q

(6)

Extensive results concerning the existence, uniqueness and regularity of solutions (u, p)in V x Q of the Stokes problem can be found in Oirault and Raviart [12].

Let Vh C V and Qh C Q denote finite element spaces, possibly h-p finite elementspaces [14], of the spaces V and Q. Approximate solutions to problem (6) are thenobtained by solving the following system of discrete equations:

For f E V' given, find (Uh, Ph) E Vh x Qh, such thata(uh' v) + b(V,ph) =

b(Uh,q)(f, v), Vv E Vh

0, Vq E Qh(7)

Let us consider a pair (Uh,ph) E Vh x Qh, not necessarily a solution of (7). The• numerical error (e, E) E V x Q in (Uh, Ph) is defined as

(e, E) = (u,p) - (Uh,ph). (8)

Then, substituting U and P in (6) by Uh + e and Ph + E respectively, we show that thedistribution of (e, E) is governed by the system of equations:

a(e,v) + b(v, E) = n'h(v), Vv E Vb(e,q) = n~(q), VqEQ (9)

46

where the linear functionals R'h : V ~ IR and R~ ; Q ~ IR

R'h(v) = (Rh',v) == (f,v} - a(uh'v) - b(V,Ph)nHq) = (R~,q) == - b(Uh,q)

are respectively the residual in the momentum equation and the residual in the con-tinuity equation. The residuals R'h and n~are viewed as the sou'rces of error. Indeed,whenever they are zero, the numerical error is zero as well. We show in the next sectionhow the residuals can provide reliable global error estimates in specific norms.

3. GLOBAL ERROR ESTIMATION

The objective here is to relate the residuals to the numerical errors expressed in someglobal norms. We recall (see [12, p.22)) that the space V = HMn) can be decomposed I

into the direct sum V = J EElJl., where the spaces J and Jl. are defined as:

J = {vEV; b(v,q)=O, VqEQ},

Jl. = {v E V; a(v,w) = 0, Vw E J}.

The space Jl. is the orthogonal complement of J with respect to Hb(n) for the innerproduct a(·, .). It follows that the error in the velocity variable e E V can be uniquelydecomposed into a sum of two vectors, ed E J and el. E Jl., such that e = ed + el..

We naturally measure the error (e, E) in the usual global norm

lI(e, E)1I2 = lei; + IIEII~= led/; + /el.l; + IIEII~· (10)and consider the following norms for the residuals n'h and R~

lin}, II. = sup IR'h(v)1v E V\{O} Ivll

IIRhll. = sup \RX(q)!q E Q\ {oJ Ilqllo . (11)

(12)

Lemma 2 With the above definitions and assumptions:

f3lel.ll ~ IIRhll. ~ Mb le.J.I!·Proof From the definition of the residual RX, we have, for all q E Q

nh(q) = b(e,q) = b(el.,q) ~ Mb lel.\lllqllo'The upper bound in (12) follows as

c RHq) Mb lel./1 IIqllo At I IIIRhll. = sup -II II ~ sup II II ~ b e.J.I'qEQ qo qEQ qo

The divergence operator is an isomorphism of J.J. onto Q, so the function '\7. e.J. belongsto Q and therefore

11\7·el.lI~ = -nh(\7· e.J.) ~ IIRxll.II\7· e.lllowhich yields 11\7·e.J.llo~ IInxll •. From Girault and Raviart [12, p.24, p.81], we obtainthe lower bound in (12) as

f3le.J.II ~ 11'\7·e.J.llo ~ IIRxll.·

47

Lemma 3 With the above definitions and assumptions, we have:

(1) ledll ::; IIRh'II.·(2) IIR'h II. ::; lell + Mb IIElla'(9) f3IIElla::; IIRh'II. + je.L11 ,

Proof

1. By simple algebra, we get:

which shows (13).

2. We also have

(13)(14)(15)

IIRh'II. = R'h(v) < a(e, v) + b(v, E)sup - supv E V Ivll - v E V Ivll

a(e,v) b(v,E)sup -I I + sup -, I ::; lell + Mb IIElla .

vEV VI vEV VI

3. The upper bound for the error in the pressure variable follows from the inf-supcondition on the bilinear form b, i.e.

•

f3I1Ella < b(v, E) < b(v, E) 'Rh'(v) - a(e, v)sup - sup -< sup

- v E V Ivll - v E J.L Ivll - v E J.L Ivll

< R'h(v) a(e,v) Rh'(v) a(e.L,v)sup -+ sup -< sup -+ sup

- v E J.L Ivll V E J.L Ivll - v E V Ivll V E J.L Ivll< IIRh'II. + le.LII .

• Theorem 1 Let (e, E) be the numerical error in an approximation (Uh, Ph) of the Stokesproblem. Then there exist positive constants Ct and O2 such that

where CI and O2 depend only on the constants f3 and Mb respectiuely:

(16)

C2 = max(2 + Mf,2Mf). (17)

48

Proof The proof of this theorem directly follows from Lemma 2 and Lemma 3. Inparticular, the lower bound is obtained from inequalities (12), (13) and (15), while theupper bound is derived from the inequalities (12) and (14).

This theorem is similar to the Theorem 6.1 introduced by Ainsworth and aden in [2].It shows that the global quantity IIR~II~+ IIR'hII~is equivalent to the global error measure •II (e, E) 112. We emphasize that it represents a meaningful quantitative global estimate aslong as the constants Ct and C2 do not take values far from one, Here the constant C2

takes the value 4 in two-dimensions and 6 in three-dimensions since Nh = ..;n. Therefore,the accuracy and robustness of this error estimate essentially depends on the value ofCj, a fortiori, /3, which depends itself on the problem in consideration. Moreover, weemphasize that such a global result does not provide any reliable information about thelocal (elementwise or pointwise) error, as we know that the error can propagate far awayfrom their sources R'h and R~. This actually motivated the work undertaken by Babuska, I

Strouboulis and co-workers [4] and aden and Feng [15) on pollution error in order to esti-mate the contributions of the residuals to the error outside the region of interest. However,the quantity IInh'll~+ IInhll~provides information about the location and intensity of thesources of errors, and as such, should be used to determine local refinement indicators.

4. EVALUATING NORMS OF RESIDUALS

This section is devoted to the evaluation of IIn'hll. and IIn~II•. The objectives are, onone hand, to minimize the cost of the computations while retaining a certain accuracy,and, on the other hand, to obtain elementwise contributions of these quantities.

4.1. Residual in the continuity equationThe residual R~ gives us information about whether the discrete velocity Uh does or

does not satisfy the incompressibility constraint. This is stated in the next lemma, wherethe norm of the residual R~ is expressed in terms of the divergence of Uh.

Lemma 4 Let Uh E Vh be the discrete velocity of the Stokes problem. Then,

(18)

Proof From the definition of the residual R'h, we have, for all q in Q:

It follows that

(19)

We also have

that is, IIV', Uhllo $IIR~II., and the equality (18) has just been proven.

49

The result of Lemma 4 is remarkable in the sense that the evaluation of IIR~II.isboth exact and cheap, as it is equivalent to compute the L2(n)-norm of the function\7. Uh- In addition, it is straightforward to decompose IIR~II.into a sum of elementwisecontributions as:

where LK represents the sum over all the elements in a finite element partition.

4.2. Residual in the momentum equationSimple considerations reveal that the evaluation of IIR'h II. is neither cheap nor exact. At

best, one obtains approximations of it. There exist basically two approaches to compute.. such approximations. The first one always delivers lower bounds on IIRh'II. while the

other provides upper bounds. Here we only consider the former, which is referred to asthe conforming method.

4.2.1. Exact evaluationSince V is a Hilbert space, the Riesz Representation Theorem tells us there exists a

unique element <P in V which satisfies:

a(<p, v) = nh'(v), Vv E V,

as well as:

1<pll = Ja(<p,<p) = IIRh'II.·

(20)

(21)

In other words, the residual R'h E V'is identified with an element <P E V of equalnorm. Moreover, the function <P gives information about the local intensity of the residualn'h, such as:

However, the problem (20) cannot be solved numerically due to the infinite dimension• of the space V.

4.2.2. Conforming finite element approximationsThe objective here is to construct a finite element subspace VI> C V, hence the label

• conforming method, in which to approximate problem (20), that is, to find <Ph E VI> suchthat:

(22)

We show that the norm of <Ph is always smaller than the norm of R'h and that bothquantities are indeed equivalent. First, we observe from (20) and (22) that:

(23)

so

In consequence, ifill is simply the orthogonal projection of IfI onto the space VII withrespect to the inner product a(·, .) associated to '.11, It follows that:

(24)

which yields the inequality Ilfl - IfIllb ~ IlfIb· Such a result can be sharpened, as statedin the next lemma.

Lemma 5 Let IfIE V and IfIh E vh satisfy (20) and (22) respectively. If cP :f. 0, let Vh

be such that CPIl :f. O. Then, there exists a constant (1, 0 ~ (1 < 1 such that:

(25)

Proof Since Icphll :f. 0, Ilfl - 1fI1l11is strictly less than IIfIIIby (24).

Theorem 2 Let IfI E V, ifill E VII and Vh satisfy the conditions of Lemma 5. Then,there exists a constant (1) 0 ~ (1 < 1, such that:

(26)

Proof The upper bound follows from equality (24). Indeed, Icpllil~ IIfIII= IIn'hI\.. Now,making use of (24) and (25), we have:

that is

(27)

from which the lower bound follows.Theorem 2 shows that the quantities IlfIhll and IIR'hIl. are equivalent. However,llfIhll

provides a valuable approximation of Ilnh'lI. a.~ long as v'f'=(T2 is close to one. Theconstant (1 clearly depends on how rich the finite element space Vh is, but, at the sametime, we point out that even a crude approximation ifill of cP may be sufficient to obtainan acceptable estimate of IIR'h II•. For instance, by committing a relati\'e error (1 = 40% •in approximating 1fI,we still approximate 1I'R.'hIi.with an effectivity of 91.6%.

The finite element space Vh has to be chose~ 80 that the action of the residual R'h isdifferent from zero for at least one element of Vh. In the case where the pair (Uh, Ph) is •the solution of problem (7), we observe that the residual vanishes on VII, that is:

ni:'(v) = 0, "Iv E Vh. (28)

It follows that the space Vh should be larger than Vh itself. It is then constructed byenriching Vh with elements of a space Wh

, called space of perturbations, which satisfies

(29)

51

so that yh C yh = yh +Wh C Y. Since yh C yh, problem (22) is more expensive tosolve than the original one. At this point, we want to reduce the cost of the computationsby taking advantage of the fact that the residual vanishes on the space yh. In other words,we would like to approximate IIn'h/1. by the norm of the function "ph E Wh satisfying

(30)

•

Following Bank [5], we suppose that a Strengthened Cauchy-Schwartz Inequality holdswith respect to the spaces yh and Wh, in the sense that there exists a positive constant, < 1 such that for all Vh E yh and for all Wh E Wh,

(31)

which implies, using Young's inequality, that

IVh + whl~ = IVhl~ + 2a(vh,wh) + IWhl~ ~ IVhl~- 2, IVh\llwh\] + IWhl~> IVhli -Ivhli -,2lwhli + IWhii = (1 _,2) IWhl~.

Thus,

(32)

The Strengthened Cauchy-Schwartz Inequality allows us to relate the accuracy of theapproximation l"phll ~ l'Phll to" the cosine of the angle between the spaces yh and Wh:

Theorem 3 Let 'P E y, 'Ph E yh and yh satisfy the conditions of Lemma 5. Letthe strengthened Cauchy-Schwartz inequality (31) hold fOl' the spaces yh and Wh, whereyh = yh + Wh and R'h = 0 on yh. Let"ph E Wh satisfy (30). Then

(33)

Proof The upper bound is readily obtained as wh C Y. In order to prove the lowerbound, we write 'Ph = 'Pi.+1/1';., where 'Pi. E yh and 1/1';. E Wh. Then, from equation (22),we have, since 'Pi. E yh:

a('Ph,'PjJ = Rh'('P';.) = 0,

and from (22) and (30), we get:

Hence,

I'Ph I~ = a ('Ph, 'Ph) = a( 'Ph, 'Pi. + 1/1i.) = a ( 'Ph, 1/1';.) = a (1/1h, 1/1i.)< j"phlll1/Ji.ll·

(34)

(35)

(36)

52

Applying the strengthened Cauchy-schwartz inequality to solution r.ph yields:

~ l'lJlhll;:; Ir.pi. + 'lJlhll = Ir.phll'which, combined to (36), gives

Ir.phl~ ;:; l'lJlhI11'lJli.ll;:; ~ l1fIhI11r.phll'

that is,

(37)

(38)

(39)

1Ir.phll ;:; ~ l1fIhl!·

Then, using inequality (27) shown in Theorem 2, allows us to write:

1VI - (J21r.p1! ;:; Ir.phll ~ ~ l1fIhl! .

The lower bound is proved.The above theorem shows that the accuracy of the approximation l1fIhll ~ Ir.pll only

depends on the constants (J and ,.We recall that (J depends on the richness of the space Vh, or Wh

. The main difficulty isto construct Wh so that the approximations 1fIh E Wh deliver reliable estimates of IIR'h II.at the lowest cost. In h-p finite element methods, the perturbations can be convenientlyconstructed from layers of piecewise polynomial basis functions involving monomials ofdegree between p+ 1 and p+ q, q ~ 1, where p is defined as the maximal degree of the basisfunctions in Vh. Obviously, in two-dimensional problems, such basis functions consist ofedge and/or interior bubble functions. In three-dimensional problems, they would consistof edge, face and/or inte-rior bubble functions. One question one has to answer is "whatis the best value for q ?" Actually, this is problem dependent, and one should consider amethod in which the value of q is increased as needed.

We recall that, represents the angle between Vh and Wh. Its value is directly related

to the choice of the shape functions used to construct the basis functions of Vh and Wh.

Here we use hierarchic shape functions based on integrated Legendre polynomials (seeSzabo and Babuska [17, chapt. 6)), which satisfy the orthogonality property with respectto the inner product a(-, .).

Meanwhile, the cost in solving (30) is controlled by the dimension of Who However,the bilinear form a(-,·) is symmetric and positive definite so that the finite system (30)can be cheaply solved using the iterative Conjugate Gradient method (eG). Moreover, weemphasize that the solution does not have to be highly accurate as shown by Theorem 2,so that it is usually sufficient to perform a few iterations.

Finally, conforming methods always deliver lower bounds on R'h. In order to constructupper bounds, one has to approximate problem (20) in a space larger than V. This lead tothe equilib-rated residual method developed by Ainsworth and Oden [1,2) or Ladeveze andLeguillon [13). In this approach, the global problem (20) is decoupled into a collection oflocal problems, usually constructed on each of the elements in the partition. Our presentconforming method avoids the major difficulty of prescribing boundary conditions for eachsubproblem typical of the equilibrium methods at comparable cost.

53

5. ESTIMATION AND CONTROL OF ERROR QUANTITIES OF INTER-EST

Let L denote a linear functional defined on the product space V x Q. We supposethat we are interested in the error quantity L(e, E) E IR, which we want to estimateand control. For example, L(-,·) may represent something more localized than the globalestimate (16), such as the error in u and p or their gradients at a point :CoE !1 or alonga curve r c !1; e.g. L( e, E) = E( :Co)01' L( e, E) = fr. e .n ds, etc. We cite more examples

.. later. The main objective is then to relate L(e, E) to the sources of error n'h and n~; inother words, we would like to find linear functionals wln and we, if they exist, such that

(40)

These functions are viewed as influence junctions as they indicate the influence of the• residuals on the quantity L(e, E). They are defined on the bidual of V and Q, and since

these spaces are reflexive, we have:

and we E Q. (41)

This is understood in the sense that for each wm E V" and each we E Q", there existwm E V and we E Q respectively such that

By identifying wm and WC with wm and WC respectively, the relation (40) becomes

(42)

Substituting for the terms R'h(wm) and n~(wC) in (42) using (9), rearranging andassuming a symmetric, we finally get

L(e,E) = a(e,wm) +b(wm,E) +b(e,wC)

= a(wm,e)+b(e,wC)+b(wm,E).(43)(44)

The influence functions can thus be obtained as solutions of the global dual problem:

IFind (wm, wC) E V x Q, such that

• a(wm,v) + b(v,wC) + b(wm,q) = L(v,q), V(v,q) E V x Q

which is equivalent to the generalized Stokes problem:

Find (wm ,WC) E V x Q, such thata(wm,v)+b(v,wC) = L(v,O), VvEV

b(wm,q) = L(D,q), Vq E Q

(45)

(46)

It readily follows that the functions wm and WC do exist and are unique as L(v,O) EV' and L(O, q) E Q' (see Girault and Raviart [12]). Obviously, (46) cannot be solvedexactly for the functions wm and wC in the general case. At best, one seeks numericalapproximations of wm and we. This raises two questions, which should be simulatenouslyinvestigated:

54

1. How do we effectively calculate finite element approximations of wm and WC? Inparticular, is it necessary to solve a global problem?

2. How do we utilize the relation (42) to derive elementwise refinement indicators inorder to control the error quantity L(e, E) ?

From a computational point of view, we observe that the cost involved in approximatingwm and WC in the spaces yh and Qh is almost negligible, as the resulting finite systemhas already been factorized once to calculate the solution (UI" Ph)' The cost thereforereduces to perform one backward and one forward substitutions. However, we recall thatthe residuals are identically zero on yh x Qh whenever the pair (Uh,ph) is the solution ofthe finite system (7). Therefore, if we approximate wm and WCby wi:' E yh and wh E Qh,we obtain that

which implies

and RJ.{wh) = 0, (47)•

In this case, the quantity L(e, E) would be estimated by a quantity which proves to beO. This reveals that wm and WC should be approximated in spaces of greater dimensionsthan Vh and Qh.

One strategy, proposed by Becker and Rannacher [10], consists in using (47) such that

L(e, E) = Ri:'(wm) + Ri,(wC) - Ri:'(wi:') - Ri,(wh)= Ri:'(wm - w'h) + RJ.{wc - wh)o

Then, an upper bound can be derived as:

IL(e,E)1 = IRi:'(wm-w'h)+R;;(wC-w/')1 = /a(<p,wm-w'h)-b(Uh,WC-Wh)J

= l~aK(<p,wm - wi:') - ~bJ«wC - w/,)I

< E laK(<p,Wm - w'hll + E IbK(wC - wi.) IK K

< E (1<pll,I( Iwm - wh'lu(+ IIV, uhllo,J( IIwc - whllo,K) (48)K

< ~ VI<pli,K + IIV, uhll~.K Vlwm - w'hli,K + IIwc - wXII~.K' (49)

where <P is the function in Y satisfying (20), Becker and Rannacher [10] use (48) toestimate the quantities L( e, E) using local interpolation properties to evaluate the errorquantity /wm - w'hll K and IIwc -wXllo K' They thus introduce an interpolation constantthey arbitrarily choo~e to be 1. Here, w~ could estimate L( e, E) by introducing the global

,..,.om<mNm

...

..

55

error estimates developed in section 3 into (49). However, those estimates are not localand do not reflect the error propagation in w'h and who In general, such a procedureoverestimates IL(e, E)I by several orders of magnitude. If one is interested in accuratevalues of L(e, E), one has to consider another approach.

The strategy we propose is to solve a global problem defined in some finite elementspaces yh and {Jh satisfying

yh C yh C y, Qh c cl c Q .

Obviously it is more expensive to solve this global problem than the solution problemin yh x Qh. In order to optimize the selection of yh and (Jh, we propose to constructthem by adapting yh and Qh according to a combination of global error estimates on(w'h,wK) as well as error estimates on L(e, E) using (49). Thus, we can utilize the globalerror estimates (very cheap) developed in the previous sections. Let (wi:' ,wX) denote thesolution in Vh X (Jh. Then we have

L(e, E) ~ R'h(w'h) + nh(Wh) (50)

::; L: Jlcpli'K + IIV', uhll~,K Jlwi:' - w'hli,K + IIwl; - wl;lI~.K· (51)K

Therefore, the quantity L(e, E) is estimated using (50) and controlled by adapting themesh according to the elementwise indicators produced by (51).

We now provide some examples for which we would be interested in calculating thequantity L(e, E) E JR. The error measure L(e, E) obviously represents the numericalerror we do in computing L(Uh,Ph) instead of L(u,p). Indeed,

L(e, E) = L(u,p) - L(Uh,Ph)'Linear quantities of potential interest in the variable (1.1" p) are pointwise values of the

velocity component u, L(u,p) = u(xo), Xo E n, volume flow rates through a surface r,L(u,p) = Iru, nds, where n is the unit normal to r, or directional derivatives of thevelocity averaged over one element fl./{, L(u,p) = InK V'u·l dx, where l is a given unitvector. In the case one is interested in nonlinear quantities N (1.1" p) E JR, one performsthe expansion:

N(u,p) = N(Uh,Ph) + N'(Uh,Ph)' (1.1, - Uh,p - Ph) + ...so that the error quantity N(u,p) - N(Uh,ph) may be approximated by

N(u,p) - N(Uh,Ph) ~ N'(Uh,ph)' (e,E).

Then, we denote the linear quantity N'(Uh, Ph) . (e, E) by L(e, E) and all the analysisdescribed above applies. For example, let us suppose that one is interested in the kineticenergy K. in some subregion fl.K of fl.. We employ

N(u,p) = K.(u) = -21 r 1.1,2 dx.inK

It results that the error in the kinetic energy may be approximated by

[(.(1.1,) - K.(Uh) ~ L(e, E) = r Uh' e dx.inK

56

6. NUMERICAL EXPERIMENTS

6.1. Global error estimationIn the first example, the Stokes problem is solved on the unit square n = [0,1) x [0,1]

with the data f chosen such that the exact solution is given by the velocity profile u =(u, v) shown in Figure 2 and by the pressure p = O. We consider uniform meshes formedof squared elements with size h x h, h = 1/2,1/4,1/8,1/16,1/32. The polynomial degreep of the basis functions in Vh is uniformly set to 2 or 3.

Figure 1. Exact velocity component u. Figure 2. Exact velocity component v.

We consider the global error estimator 7} defined as:

where rp E V is approximated by a function rph E Vh or a function t/Jh E Wh satisfy-ing (22) or (30). We characterize the extra degree (with respect to p) of the perturbationsin Wh by the pair q = (qedge, qind, qedge being the extra degree for the edge bubblesand qint the extra degree for the interior bubbles. We recall that the cost of this errorestimator is controlled by the cost in solving (22) or (30), that is, by the size of Wh

and the number of iterations performed using the Conjugate Gradient method. What issought is a trade-off between accuracy in the approximations t/Jh and cost (time) spentto solve for them. Values of approximations of rp and corresponding times are given inTable 1 and Table 2 for various values of h, p and q. The approximations are normalizedwith respect to overkills using q = (2,2) while the times are normalized with respectto the times spent to solve for the solutions (Uh,Ph). The number of iterations in theCG algorithm are determined according to a preset tolerance in solution accuracy. \Vefirst observe that the performances greatly increases as the size of the problem increases.For example, for h = 1/16, P = 2, the residual is estimated within a 1% accuracy whilespending less than 7% of the solution time. We also observe that it is approximated with

•

Table 1Normalized values of approximations of IIR'h II•.

Wh

h

1/2

1/41/8

1/16

1/4

1/8

P2

3

q=(O,I)0.7620

0.7300

0.8154

0.8184

0.8406

0.8815

q = (1,0)

0.5235

0.8282

0.9684

0.9941

0.8952

0.9781

q = (1,1)0.8520

0.8983

0.9797

0.9947

0.9679

0.9895

57

vh

q=(I,I)0.8571

0,9040

0.9805

0.9948

0.9693

0.9896

Table 2Normalized times and number of iterations 0 performed using the CG method.

Wh Vh

h1/2

1/4

1/8

1/16

1/4

1/8

P2

3

q = (0,1)0.169 (2)

0.065 (2)

0.015 (2)

0.006 (2)

0.071 (2)

0.004 (2)

q = (1,0)0.196 (2)

0.117(4)

0.042 (4)

0.018 (3)

0.013 (4)

0.004 (4)

q = (1,1)0.695 (6)

0.443 (7)

0.241 (8)

0.069 (5)

0.123 (9)

0.088 (9)

q=(l,l)3.454 (8)

2.925 (9)

1.309 (9)

0.474 (8)

0.921 (9)

0.339 (9)

less than one percent of error using Wh instead of Vh with q = (1,1). Moreover thenumber of iterations is kept very small while requiring reasonable accuracy of the order10-2• We also compute effectivity indices, defined as:

u.P _ --.!l.I - lI(e, E) II

as well as elementwise effectivity indices. Results are collected in Table 3 for the case.,ph E wh, q = (1,1). We also show in Figure 3 and Figure 4 the elementwise distributionof the exact errors and of the effectivity indices for h = 1/8 and p = 2. The globaleffectivity indices are excellent as they are all close to unity. On the other hand, the localeffectivity indices, as it is expected for the case of a global error estimator, may behavepoorly, and in some cases drop to a value as low as 0.012. However, these are always nearunity in elements with large error, and diverge from unity only in elements with rathersmall error.

6.2. Backward facing step Stokes problemThe second example is the classical backward facing step Stokes problem. The problem

is slightly different than in Bank and Welfert [8), as we consider a homogeneous Neumann

58

Table 3Global and elementwise effectivity indices.

h p ,U,l' min/'ll max/~l'

1/2 2 0,760 0.717 0.9211/4 0.927 0.274 1.1031/8 1.038 0.069 1.3211/16 1.066 0.033 1.3881/32 1.073 0.012 1.408

1/4 3 1.028 0.519 1.0511/8 1.084 0.167 1.3391/16 1.096 0.025 1.381

-------_._.~ ,--------

---EXT-ERR 1 EFF-IND

2.274 F11.2252.081 -- 1.129

0.75\ I 1.\ 1~i4.. \ 'I!'!!l I. 1.889 0.751.033

1.697 ~.'.. 0.9361.504 --- 0.840

0.51 I I I I I I I I 1.1.3120.5

0.7441.119 0.6470.927 0.5510.734 0.455

0.251 I I I I I I I I~ 0.542 0.25 0.3580.349 0.262

L~ 0.1660' , , I , ,

0 0.25 0.5 0.75 1 0.25 0.5 0_75

Figure 3. Elementwise distribution of theerrors.

Figure 4. Elementwise distribution of theeffectivity indices.

boundary condition downstream of the domain. The solution develops a discontinuity inthe pressure variable due to the discontinuity in the geometry of the boundary. The normof the residual R'h is then expected to be very large in the near region of the reentrantcorner and small in the farfield, as illustrated in Figure 5, where the solution has beencomputed on a pre-adapted mesh of 120 elements.

We are then motivated to construct Wh in an adaptive manner in order to minimizeits dimension, that is, the number of degrees of freedom associated to problem (30). Asa first guess, the residual is approximated by .,ph E Wh with q = (1,0). Then the spaceis successively enriched with interior and edge bubble functions of higher degrees only inthe elements for which the contribution to !.,phll is large. Results are shown in Table 4where the approximations of Iln'hIl., computed either in the complete space W~mp, withuniform q = (qedge,qint), or in the adapted space W~dp with q varying between (0,0)

59

0.0590.0520.0450.0380.0320.025

.;", 0.0180.011

-, 0.004

Figure 5. Elementwise distribution of the resicluaIIIR'hIl •.

to (qedge' qint) depending on the element, are compared. It is clear that the number ofdegrees of freedom, and a fortiori the cost of the error estimator, is greatly reduced forcomparable accuracy. The global effectivity index for this problem has been computed as"'fu,'p = 0.884 and 0.915 for p = 2 and 3 respectively, noting that the exact solutions areactually approximated by overkills of degree p + 3.

VV~mp0.1361(2980)0.0612(3940)

q=(2,2)VV~dp

0.1344(922)

0.0607(854)

VV~mp

0.1301(1780)0.0539(2260)

q = (2,1)

VV~dp

0.1271(716)

0.0535(686)

3

Table 4Approximations of IIRh'II. in complete or adapted spaces of perturbations with corre-sponding numbers of degrees of freedom O.

q=(I,O) q=(l,l)

VV~mp VV~dp VV~mp

0.1078 0.1247 0.1275(530) (650) (1250)

0.0291 0.0527 0.0531(530) (650) (1730)

P2

•

..

In Figures 6 and 7, we respectively show an adapted mesh and the correspondingsolution (Uh,Ph). Such a mesh has been automatically adapted from a given coarse meshin order to achieve a preset tolerance of 1.2% in the global relative error.

6.3. Error estimation in the quantity of interestThe following results constitute a preliminary study of error estimation in local quan-

tities of interest. We consider the test case described in section 6.1 and suppose we areinterested in evaluating the quantity

1 1 auL(u,p) = InK! OK ay dx E IR

where InKI represents the area of a given subdomain nK C n. Then, the error in thecomputed quantity L(Uh,Ph), where (Uh,Ph) is the finite element solution in Vh x Q\ is

60

2I

I ~II

- -

>- 1

o2 4 x 6 8 10

Figure 6. Example of an adapted mesh (201 elements).

given by L(e, E). We consider a uniform mesh with h = 1/8 and p = 2 and choose thesub domain f1K to be either element 1 located in the lower left corner of the domain, orelement 55 the next to last element in the upper right corner. Then, L(e, E) is estimatedby three quantities in IR denoted 7/l> 1]2 and 1J3· The first one, '711 is obtained as:

where the discrete influence functions w'h and wX belong to the spaces Vii and Qhdefined using basis functions up to the degree p + 1 and are solutions of the generalizedStokes problem (46). The second one, 1]2, is calculated as

where Iwm- w'h It,/( and Ilwe

- wI; lIo,K are approximated using the global error estima-tor. We also note that 1/JhE Wh, q = (1,1), is an approximation of cp E V, that is, ofthe residual n'h. Finally the last approximation, 1]3, is computed as

Tl3 = L: JI1/Jhl~.K + 11'1 . 'Uhll~,K Jlwi: - w'hI~.K + IlwX - wXIl~,KK

where w'h and wI; have been defined earlier. The results are displayed in Table 5. Theelementwise distribution of 172 and 173 are shown in Figures 8-9 and Figures 10-11 forelement 1 and element 55 respectively.

Table 5Estimates 7/1, 7)2 and 173 of the error quantity L( e, E) for elements 1 and 55.

Element L( '1£, p) L( e, E) 171 1]2

1 -0.000364 0.000088 0.000081 0.00330455 1.6699 0.0258 0.0271 0.3733

1'/30.015205

0.4783

•

61

Figure 7. Velocity and pressure on adapted mesh with relative estimated error of lessthan 1.2%.

The conclusions of this preliminary study are twofold. First, we observe that only thequantity 771 represents an accurate approximation of L(e, E). The other two quantities7}2 and 7}3 provide very pessimistic estimates as expected, but provide local refinementindicators as they reflect the intensity of the elementwise contribution to the error quantityL(e, E). Secondly, we remark that the distribution 7}2, relative to the case involvingelement 1 (see Figures 8 and 9) is very different from the one of 7}3. This can be explainedby the fact that the quantities Iwm - w'h 11,K and IIwc - wKllo,K are approximated using aglobal error estimator, which does not take in account the propagation of the errors awayfrom their sources. On the other hand, the discrepancy is less noticeable for element 55,because the region of large numerical errors in the influence functions superpose with theregions of large sources of errors R'h and RX'

ACKNOWLEDGEMENT The support of this work by the Office of Naval Researchunder contract NOOOI4-95-1-0401is gratefully acknowledged.

REFERENCES

1. M. AINSWORTH AND J. T. ODEN, A unified approach to a posteriori error estimationusing element residual methods, Numer. Math., 65 (1993), pp. 23-50.

2. -, A posteriori error estimation in finite element analysis, Compo Meth. in Appl.Mech. and Eng" 142 (1997), pp. 1-88.

62

0.00044 0.001250.00041 0.00115

0.751 I I I I I 1 I I I. g:~g~~ 0.75 g:::~~0.00030 0.00084

I, '''F' c II I I I I~ 0.00026 0 5 0.000730.5 " 0.00022' 0.00063~. 0.00018 0.00052

_,,_. ' .. 0.00015 0.000420.25 t "'-!-+-+-+-+-I I 0.00011 0.25 0.00031

0.00007 0.000210.00004 0.00011

I I I I n0.5 0.75

Figure 8. Elementwise contributions of TJ2for element 1.

Figure 9. Elementwise contributions of TJ3for element 1.

3. I. BABUSKAANDW. RHEINBOLDT, A posteriori error estimates for the finite elementmethod, Int. J. for Num. Meth. in Eng., 12 (1978), pp. 1597-1615.

4. 1. BABUSKA, T. STROUBOULIS, C. S. UPADHYAY,AND S. K. GANGARAJ, A pos-teriori estimation and adaptive control of the pollution error in the h-version of thefinite element method, Int. J. for Num. Meth. in Eng., 38 (1995), pp. 4207-4235.

5. R. E. BANK, Hierarchical bases and the finite element method. To Appear in ActaNumerica.

6. R. E. BANK AND R. K. SMITH, A posteriori error estimates based on hierarchicalbases, SIAM J. Numer. Anal., 30 (1993), pp. 921-935.

7. R. E. BANK AND A, WEISER, Some a posteriori error estimates for elliptic partialdifferential equations, Math. Comp., 44 (1985), pp. 283-301.

8. R. E. BANK AND B. D. WELFERT, A posteriori error estimates for the Stokesproblem, SIAM J. Numer. Anal., 28 (1991), pp. 591-623.

9. R. BECKER AND R. RANNACHER,A feedback approach to error control in finite ele-ments methods: Basic analysis and examples, Preprint 96-52, Institut fiir AngewandteMathematik, Universitiit Heidelberg, (1996).

10. -, Weighted a posteriori error control in FE method, in ENUMATH-95, Paris,Sept. 1995.

11. L. DEMKOWICZ, J. T. ODEN, AND T. STROUBOULIS,Adaptive finite elements forflow problems with moving boundaries. Part 1: Variational principles and a posteriorierror estimates, Compo Meth. in AppL Mech. and Eng., 46 (1984), pp. 217-251.

12. V. GIRAULT AND P .-A. RAVIART,Finite Element Methods for Navier-Stokes Equa-tions, Springer-Verlag, 1986. .

13. P. LADEVEZE AND D. LEGUILLON, Error estimate procedure in the finite elementmethod and applications, SIAM J. Numer. AnaL, 20 (1983), pp. 485-509.

14. J. T. ODEN, L. DEMI<OWICZ,W. RACHOWICZ,ANDO. HARDY, Toward a universalh-p adaptive finite element strategy. Part 1: Constrained approximation and datastructure, Compo Meth. in AppL Mech. and Eng., 77 (1989), pp. 113-180.

"

63

0.06477 0.068080.05937 0.06240

0.751 I I I I I- I. 0.05397 0.75 0.05673"._- 0.04858 0.05106

0.04318 0.04538

0.51-1 I 1 I I I I0.03778 0.03971

I '.0.03238 0.5 0.034040.02699 0.02837

0.251- I i I-H IW 002159 0.02269. 0.01619 0.25 0.01702

0.01079 0.01135_, 0.00540 0.00567

0' I I I I I I I000 0.25 0.5 0.75 1 0.25 0.5 0.75

Figure 10. Elementwise contributions ofT/2 for element 55.

Figure 11. Elementwise contributions ofT/3 for element 55.

15. J. T. ODEN AND Y. FENG, Local and pollution error estimation for finite element ap-proximations of elliptic boundary value problems, J. Comput. Appl. Math., 74 (1996),pp. 245-293.

16. J. T. ODEN, W. Wu, AND M, AINSWORTH, An a posteriori error estimate for finiteelement approximations of the Navier-Stokes equations, Compo Meth. in Appl. Mech.and Eng., 111 (1993), pp. 185-202.

17. B. SZABO AND 1. BABUSI<A,Finite Element Analysis, John Wiley and Sons, 1991.18. R. VERFURTH, A posteriori error estimators for the Stokes equations, Numer. Math.,

55 (1989), pp. 309-325.19. R. VERFURTH, A Review of A Posteriori Error Estimation and Adaptive Mesh-

refinement Techniques, Wiley-Teubner, 1996.