A Feed-Back Approach to Error Control in Finite Element. Methods: Basic Analysis and Examples. Roland Becker and Rolf Rannacher 1

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1 A Feed-Back Approac to Error Control in Finite Element Metods: Basic Analysis and Examples Roland Becker and Rolf Rannacer 1 Summary. A rened approac to residual-based error control in nite element (FE) discretizations is presented. Te conventional strategies for adaptive mes renement in FE metods are mostly based on a posteriori error estimates in te global energy or L 2 -norm involving local residuals of te computed solution. Te mes renement process ten aims at equilibrating tese local error indicators. Suc estimates reect te approximation properties of te nite element space by local interpolation constants wile te stability properties of te continuous model enter troug a global stability constant, wic may be known explicitly in simple cases. Meses generated on te basis of suc global error estimates may not be appropriate in cases of strongly varying coecients and for te computation of local quantities as, for example, point values or contour integrals. More detailed information about te mecanism of error propagation can be obtained by employing duality arguments specially adapted to te quantity of interest. Tis results in a posteriori error estimates in wic te local information derived from te dual solution is used in te form of weigts multiplied by local residuals. On te basis of suc estimates, a feed-back process in wic te weigts are numerically computed wit increasing accuracy leads to almost optimal meses for various kinds of error functionals. Tis approac is developed ere for a simple model problem, namely te Poisson equation in two dimensions, in order to empasize its basic features. However, te underlying concept is rater universal and as, on a euristic basis, already been successfully applied to muc more complex problems in structural and uid mecanics as well as in astropysics. 1 Introduction Te strategies for mes renement conventionally used in FE metods are mostly based on a posteriori error estimates in global norms, e.g., te energy or te L 2 -norm, involving local residuals of te computed solution. Te mes renement process ten aims at equilibrating tese local error indicators. However, meses generated on te basis of suc global error estimates may not be appropriate in cases of strongly varying coecients and for controlling te accuracy in approximating local quantities as, e.g., point values or contour integrals. For tis one needs more detailed information on te mecanism of error propagation wic can be obtained by employing suitable duality arguments known from te a priori error analysis as te so-called Aubin-Nitsce trick (see, e.g., [11]). Te corresponding dual solutions ten yield te appropriate weigt factors to be used in te a posteriori error estimates. Consider, for example, te Poisson equation? u = f in ; u = 0 ; (1.2) on a bounded domain R 2 wic, for te moment, is assumed to be convex polygonal. Te restriction to two dimensions is only for simplicity as te extension to arbitrary dimensions is 1 Tis work as been supported by te German Researc Foundation (DFG), SFB 359 Reactive Flow, Diusion and Transport, Institute of Applied Matematics, University of Heidelberg, INF 293, D Heidelberg, Germany 1

2 straigtforward. By (; ) we denote te L 2 inner product and by k k te corresponding norm on. Let (1.2) be discretized by a standard nite element Galerkin metod using piecewise polynomial, say, linear or bilinear sape functions on meses T = fg satisfying te usual regularity conditions (see, e.g., [11]). To facilitate local mes renement, anging nodes may be allowed. For eac T, let max = max 2T, were = diam(). Te corresponding nite element subspaces are V V := H 1 0 (). Teir approximation properties are caracterized in terms of local approximation estimates (see ([11]), o max nkv? I vk ; 1=2 kv? I C i; 1+r kr1+r vk ~ ; r 2 f0; 1g ; (1.3) for v 2 V \ H 1+r (), were I v 2 V is some locally dened approximation to v, and k k B denotes te L 2 -norm over a set B. For r = 0, ~ is te union of all neigboring elements of, wile for r = 1, one simply as ~ =. Starting from te variational formulation of (1.2), te nite element metod seeks to determine approximations u 2 V, suc tat (ru ; r' ) = (f; ' ) 8 ' 2 V : (1.4) Subtracting (1.4) from te variational formulation of (1.2) results in te following ortogonality relation for te error e = u? u, (re; r' ) = 0 8 ' 2 V : (1.5) Tis discretization allows for optimal-order a priori estimates in te energy and L 2 -norm, kek + max krek C 2 maxkr 2 uk; (1.6) provided tat te solution u is suciently regular. Corresponding estimates also old wit respect to te maximum norm (for references see, e.g., [11]). For error control in te energy or L 2 -norm, one may proceed as follows. Using Galerkin ortogonality (1.5) and integration by parts on eac element yields (re; rz) = 2T n (f + u ; z? z )? 1 2 (n [ru ]; z? z o ; (1.7) for any z 2 V, were [ru ] denotes te jump of ru across te element boundary, and z 2 V is a suitable approximation of z. On edges along te boundary, we set [ru ] = ru. Ten, using Holder's inequality on eac element, we obtain j(re; rz)j 2T! ; (1.8) wit local residuals and weigts! dened by := kf + u k =2 kn [ru ; (1.9) n o! := max?1 kz? z k ;?1=2 kz? z : (1.10) By virtue of (1.3), te approximation z may be cosen suc tat for r 2 f0; 1g, tere olds! C i; r kr1+r zk ~ : (1.11) 2

3 Consequently, it follows tat j(re; rz)j C i 2T 2r 2 1=2kr 1+r zk ; (1.12) for z 2 H 1+r (). Te interpolation constant C i is usually of size 0:1 C i 1, depending on te sape of te elements. Taking now te supremum over z 2 V \ H 1+r (), one obtains te following a posteriori error bound in te energy norm, for r = 0, or in te L 2 -norm, for r = 1, kr 1?r ek C s C i 2T 2r 2 1=2 =: (u ) : (1.13) Te stability constant C s measures te stability properties of te dual problem in terms of te global a priori estimate (r'; rz) = (r 1?r e; r 1?r ') 8 ' 2 V; (1.14) kr 1+r zk C s kr 1?r ek ; (1.15) wic, of course, is trivial for r = 0. Clearly, by analogous arguments a posteriori error estimates may also be derived wit respect to L q -norms including te limit case q = 1. For earlier work on a posteriori error estimates for FE metods, we refer to te pioneering papers of Babuska and Reinboldt [1], Bank and Weiser [3], and Babuska and Gui [2]. Te underlying approaces ave been surveyed in Verfurt [28] and [30]. Te concept of a posteriori error estimation by duality arguments as been developed by Eriksson, Jonson and teir coworkers (see [13], [21], [15], and te literature cited terein). Heuristically based error indicators for elliptic problems ave been devised, e.g., by ienkiewicz and u [34]. Based on an a posteriori error estimate of te type (1.13) te mes renement process may be organized according to te so-called error per cell strategy. For some prescribed error tolerance T OL, te goal is to reac a most economical mes T on wic (u ) T OL. Accordingly, te mes renement process aims at equilibrating te local error indicators := r by rening (or coarsening) te elements 2 T according to te criterion T OL p NCi C s ; N = #f 2 T g : (1.16) Te renement strategies based on te conventional a posteriori error estimate (1.13) rely on te assumption tat te local error indicators := r properly describe te dependence of te global error on te local mes size. However, tis may not be true in certain situations since te a posteriori error estimate (1.13) contains information about te mecanism of error propagation only troug te global stability constant C s. To overcome tis deciency, it as been proposed in [7] to use te quantities! in te estimate (1.8) as weigt-factors multiplied by te local residuals and to compute tem numerically. Tese weigts contain all information about te local approximation properties of te spaces V, as well as te local stability properties of te underlying continuous problem. Tis correspondence can be used for te mes renement algoritm troug a feed-back process. In te course of te renement te dual solution is calculated on te current mes yielding approximate weigts. On te basis of te resulting 3

4 a posteriori error estimate te mes is rened according to one of te criteria described below. Tis process is repeated yielding more and more accurate weigts, i.e., a posteriori error bounds, until te prescribed stopping criterion is fullled. Tis approac allows one to construct almost optimal meses for various kinds of error quantities, were \optimal" can mean \most economical for acieving a prescribed accuracy T OL" or \most accurate for a given maximal number N max of mes points". 2 Residual based weigted a posteriori error estimates First, we illustrate our concept at a simple model situation. Let = (?1; 1) 2 and te pointvalue u(0) be te quantity to be computed wit best accuracy. It is not likely tat te meses obtained troug energy or global L 1 -error control are optimal for tis purpose. To do better, we ave rst to derive an a posteriori error estimate for te point-error. Tis can be done by a duality argument. Let z be te Green function corresponding to te origin, i.e., te solution of te \dual" problem (r'; rz) = '(0) 8 ' 2 V \ C( ) ; (2.2) were, in tis special situation, z(x) log(jxj). Ten, we ave e(0) = (re; r(z? z )); (2.3) wit te nodal interpolant z 2 V, appropriately regularized at x = 0. Consequently, in view of te error identity (1.7), tere olds je(0)j 2T! ; (2.4) wit residuals and weigts! as dened above. Using te particular form of te dual solution, and te usual local interpolation properties of nite elements, tis error estimator takes te form je(0)j C i 2 2T r 2 were r := maxfdist(; 0); g and C i 1. =: point (u ) ; (2.5) Suppose tat for some prescribed tolerance T OL te renement process as led to a nal mes T opt on wic te local error indicators =! are equilibrated, Tis implies te relation T OL N opt ; N opt = #f T opt g : (2.6) T OL r 2 N opt 1=4 ; (2.7) if we assume tat 2. Te latter assumption means tat on te current mes, tere olds 1=2 kn [ru 2 max jn?1 [ru ]j c(u) 2 ; (2.8)

5 a ypotesis wic is supported by computational experiments (see te te examples, below). As consequence, te number N opt of elements of te nal mes T opt is given by N opt 2?2 = N 1=2 2 opt 2T opt 2T opt T OL 1=2 r : (2.9) Since r?1 is integrable over, it follows tat N opt TOL?1 and te \optimal" mes-size distribution is (TOLr ) 1=2. In contrast, te global energy-error estimator (1.13), for r = 0, would generate meses of complexity N opt T OL?2, in order to guarantee an asymptotic beavior krek TOL. Te corresponding global L 2 -error estimator 1=2 =: 2 (u ) (2.10) kek C s C i 2T 2 2 seems to be more appropriate as, at least, it as te rigt asymptotic beavior as! 0. Alternatively, one may use te L 1 -error estimator wic is obtained by an L 1 =L 1 -duality argument (see te proof of Lemma 4.2 in Section 4). It takes te form max jej C i C s j log 0 j 1=2 max 2T 1; =: 1 (u ) ; (2.11) were 0 is te diameter of te element containing te point x = 0, and 1; := 2 max jf + u j + max jn [ru ]j Tis estimator also refers to a global error norm and does not particularly empasize te point x = 0 at wic te solution u is to be computed. Terefore, it will not be able to produce a most economical mes for tis particular purpose. However, asymptotically, it yields meses of te same complexity in terms of TOL, N opt TOL?1, as te point-value estimator point (). Tis situation canges wen point-values of te gradient ru(0) (\stress values") are to be computed. Ten, te global error bounds require N opt TOL?2 cells to acieve a corresponding error of size TOL, wile te weigted error estimator acieves tis again wit only N opt TOL?1 cells (see Example 1, below). Of course, in more general situations, explicit bounds for te dual solution z are not known and ave to be computed numerically. Tis essential question will be discussed in more detail in Section 4. Te simplest approac is based on te local interpolation estimate (1.11) for r = 1. Estimates for te weigts! may be obtained from te approximate dual solution ~z 2 V, computed on te current mes, by taking appropriate 2nd-order dierence quotients, were x denotes te mid-point of element.! ~! := 2 jr2 ~z (x )j ; (2.12) Heuristically, te meaning of our a posteriori error estimate is as follows. We expect tat te weigts! converge to certain limits, as! 0, in te sense?1=2!! jd 2 z(x )j, were x denotes a point contained in an innite sequence of nested elements. Furter, te quantity?1 n [ru ] can be viewed as a certain second-order dierence quotient of u. Hence, it may be expected tat, for a sequence of properly rened meses, te residual terms in (2.4) also converge to certain limits as! 0, jj?1=2 kf + u k! j(f + u)(x )j = 0; j@j?1=2 k?1 n [ru jd 2 u(x )j: 5

6 In tis sense, te a posteriori error estimate (2.4) asymptotically, for! 0, takes te form je(0)j 2T 4 jd 2 z(x )jjd 2 u(x )j: (2.13) Te local residuals consist of two parts, te domain residual kf + u k and te normal-jump terms 1=2 kn [ru For linear elements, it can be argued tat, in te case of smoot f, te contribution of te normal jumps asymptotically dominates tat of te domain residual, and te latter may terefore be neglected. Tis aspect will be discussed in more detail in Section 4. Hence, wit te above notation, we are led to te practical error estimator were je(0)j C i ~! ~ =: approx 2T point (u ) ; (2.14) ~ := 1=2 kn [ru ; ~! := 3=2 kn [r~z ; (2.15) and te interpolation constant may be set to C i 1. Numerical test. Te weigted error estimator (2.5) and its approximate version (2.14) ave been tested, wit C i = 0:1, for te solution u = 10 sin(2x 1 +x 2 +2) using bilinear elements. For successively reduced tolerances TOL, te resulting balanced meses T opt and tose obtained by te global L 2 -error estimator (2.10) and te L 1 -estimator (2.11) are sown in Figure 1. Te corresponding errors are listed in Table 1. Here, L is te number of renement levels of te nal mes T opt and I ef f := je(0)j=(u ). For te problem considered, te predicted dependence N opt TOL?1 for te weigted a posteriori estimate (2.4) is conrmed. Te sligt tendency of our error estimator to underestimate te true error could be compensated by using a larger interpolation constant C i. 6

7 2 1 point Figure 1: Rened meses (left to rigt) for computing u(0) for u = 10 sin(2x 1 + x 2 + 2), using te L 2 - and L 1 -error estimators 2 and 1, respectively, compared to te weigted error estimator point Te above argument directly generalizes to te case of an arbitrary linear error functional J() dened on te space V, or on a suitable subspace containing te nite element space V and te exact solution u. Oter relevant cases are, for instance, te moments of u over (torsion moment), te value of its derivative ru(x 0 ) at some point x 0 2 (stress values), or te weigted integral of te normal n u over te (total surface tension), J(') = ' dx; J(') = r'(x 0 ); n ' ds : (2.16) 7

8 2 TOL N L je(0)j I ef f e e ? e e ? e e ? e e ? e e ? e e ? e e TOL N L je(0)j I ef f e e ? e e ? e e ? e e ? e e ? e e ? e e point TOL N L je(0)j I e e e ? e e ? e e ? e e ? e e ? e e ? e e approx point TOL N L je(0)j I e e e ? e e ? e e ? e e ? e e ? e e ? e e Table 1: Results obtained for computing u(0) for u = 10 sin(2x 1 + x 2 + 2), using te L 2 - and 1 L - error estimators 2 and 1, respectively, compared to te weigted estimator point and its approximate analogue approx point, all wit C i = 0:1 Ten, following te above approac, we ave te a posteriori error estimate jj(e)j 2T! ; (2.17) wit weigts! and residuals as dened above. We like to empasize tat, in tis particular problem, it is crucial to utilize te approximation property of te nite element space to its full extent also in te evaluation of te weigts!, in order to obtain economical meses. Tis applies even toug tere may be no a priori bound for te resulting norms of te continuous dual solution z (see [24], [10] and [19]). We will come back to tis point in Example 4 at te end of tis paper. In many interesting cases, te functional J() is \singular", i.e., not properly dened on H 1 0 () but only on some subspace containing te solution u and on te nite element space V. Typical examples are te evaluation of point-values or of contour integrals of derivatives. In suc a case, it is advisable to regularize te functional according to te prescribed tolerance, in order to avoid over-renement near te singularities. Te type of regularization depends on te particular situation and sould be suc tat jj(u)? J T OL (u)j TOL : (2.18) For example, te point-value estimator (2.14) may be used in te regularized form je(0)j 2T 2 (r + TOL) 2 : (2.19) 8

9 Furter examples of tis type will be discussed in Section 5. Since some of te examples presented below deal wit domains wit a curved boundary and also iger order nite elements, we ave to extend our a posteriori error analysis to tis situation. Now, te domain is assumed eiter to ave a smoot boundary or to be convex polygonal, in order to ensure tat te usual L 2 =H 2 -sift teorem olds true. Accordingly, along te te elements may be curved (isoparametric elements). For te mes size, we use te same notation, etc., as above. In addition, we now introduce te mes domain := [f 2 T g and assume tat all nodal points lie Tis means tat, in general, 6 for a non-convex domain. In suc a case, one as to cope wit a boundary strip f n g [ f n g of local widt ; m+1. Te nite element spaces are dened by V := fv 2 H 1 0 ( ); v j 2 P ()g ; were P () is a suitable vector space of polynomials over containing P m (). Tey approximate te solution space V = H0(), 1 suc tat local approximation properties (1.3) old for v 2 V \ H r+1 ( [ ) wit 1 r m. We introduce te subset of elements adjacent to te boundary, := f 2 T ; meas(@ ) 6= 0g and accordingly set T 0 := T n To eac element 2 tere naturally belongs a section of te strip f n g [ f n g denoted by S. By a variant of te Poincare inequality, tere olds In tis case te nite element sceme reads kvk S C i; m+1 krvk : (2.20) (ru ; r' ) = ( f; ' ) 8 ' 2 V ; (2.21) were f (if necessary) is a suitable extension of f to [. For te following, we extend also te nite element solution u as well as te dual solution z by zero to [ (in accordance wit omogeneous boundary conditions). Using tis convention, tere olds (re; rz) = (f; z)? (ru ; rz) \? (ru ; rz) n = (f; z) n + ( f; z)? (ru ; rz) = (f; z) n + ( f; z? z )? (ru ; r(z? z )) ; from wic by integration by parts, we conclude te following analogue of te identity (1.7), o (re; rz) = n(f + u ; z? z )? 12 (n [ru ]; z? z (2.22) 2T 0 + o n( f + u ; z? z )? 12 (n [ru ]; z? z + (f; z) S : Ten, te weigted a posteriori estimate (2.17) olds wit te local residuals and weigts as dened above, wit te only modication tat for 2 := kf + u k =2 kn [ru + kfk S ; (2.23) n o! := max?1 kz? z k ;?1=2 kz? z ; kzk S : (2.24) In view of (2.20), we ave! C i; m+1 kzk H m+1 () for 2 For a detailed analysis of te eect of boundary approximation using similar arguments, we refer to [12]. 9

10 3 A feed-back algoritm for adaptive mes renement We briey discuss ow a mes renement process may be organized on te basis of an a posteriori error estimate of te type (2.17) or (2.14). Suppose tat some error tolerance T OL and maximum number N max of mes points are given. Te goal is to nd a most economical mes T on wic jj(e)j (u ) = 2T T OL ; (3.2) wit te local error indicators :=!. Usually, one starts from an initial coarse mes wic is ten successively rened. Tere are essentially tree alternative strategies: 1. Error per cell strategy. Te mes generation aims at equilibrating te local error indicators, by rening (or coarsening) te elements 2 T according to te criterion T OL N ; N = #f 2 T g: Since N depends on te result of te renement decision, tis strategy is implicit and usually needs iteration. It is common practice to work wit a varying value N on eac renement level wic is permanently updated according to te renement process. Te result is a mes on wic (u ) T OL, provided tat N max is not exceeded. 2. Fixed fraction strategy. In eac renement cycle, te elements are ordered according to te size of and eiter a xed portion (say 30%) of te elements wit largest or te portion of elements wic make up for a certain part of te estimator, (u ), is rened. Te appropriate coice of te parameter is crucial and depends very muc on te particular situation. For \regular" functionals, one may coose = 0:6?0:8, wile for \singular" functions a smaller coice = 0:1? 0:2 is advisable, in order to enance local renement. Tis process is repeated until te stopping criterion (u ) T OL is fullled, or N max is exceeded. 3. Tolerance reduction strategy. One works wit a varying tolerance T OL var. If on a mes T a discrete solution u old as been obtained wit corresponding error estimator (u old ), te tolerance is set to T OL var := (u old ) ; wit some reduction factor 2 (0; 1) (usually = 0:5). In te next step, one (or more) cycles of te error per cell strategy are applied wit tolerance T OL var yielding a rened mes T new and te new solution u new wit corresponding error bound (u new ). Ten, te tolerance is reduced again and a new renement cycle begins. Tis process is repeated until T OL var T OL, or N max is exceeded. Te error per cell strategy is very delicate as its performance strongly depends on te parameters in te renement decision. In certain cases it may appen tat an inappropriate coice leads to very slow mes renement and in turn to an inecient overall solution. On te oter and, te xed fraction strategy guarantees tat in eac renement cycle a suciently large number of elements is rened. However, in its pure form, it does not allow for mes coarsening and in certain cases may tend to over-rene te mes. Terefore, for practical computations, we recommend te tolerance reduction strategy wic seems to be te most robust and accurate strategy among te tree. Its use is particularly advisable if te weigt-function!(x) as 10

11 nonintegrable singularities wic would oterwise dominate te renement process (see te examples at te end). However, for simplicity, in all te test calculations presented below, te \optimal" meses ave been generated by te xed fraction strategy based on te simplied estimator (2.14) T opt ~(u ) := C i kn [ru kn [rz (1) 2T ; were z (1) is te bilinear approximation to z obtained on te current mes. Te interpolation constant is usually set C i = 0:1, in order to compensate for te over-estimation in tis bound. Accordingly, te local error indicators are dened as := C i kn [ru kn [rz (1) : (3.3) 4 Model analysis of te weigted a posteriori error estimation We will give some teoretical support for te proposed approac to adaptive error control by looking in more detail at te model problem introduced in te previous section. Te mes renement process is organized as described above using (isoparametric) bilinear sape functions on quadrilateral elements. Eac element is allowed to ave just one anging node on eac of its edges. Te crucial question is weter te a posteriori error bounds obtained by te proposed approac are not only asymptotically optimal (order-optimal) but also asymptotically sarp in te sense jj(e)j I e := lim T OL!0 (u ) = 1 ; possibly depending on te work te user is willing to invest in te evaluation of te weigts!, i.e., in te numerical solution of te dual problem. First, we run some computational experiments in order to see wat can be expected. Unfortunately, te result is rater negative, i.e., wit acceptable eort asymptotic sarpness does not seem to be acievable. Wit all te metods considered te eciency index I e never really tends to one, but in most relevant cases stays well below 2, wat may actually be considered as good enoug. Tere are two separate aspects to be considered: Te sarpness of te global error bound (u ), and te optimality of te local error indicators =! wic are used in te mes renement process. Starting from te error representation (1.7), J(e) = 2T n (f + u ; z? z )? 1 2 (n [ru ]; z? z o ; (4.2) tere are various steps in evaluating te rigt and side. First, one estimates jj(e)j 2T ; (4.3) wit te (positive) local error indicators := (f + u ; z? z )? 1 2 (n [ru ]; z? z ; (4.4) 11

12 wic are used in te mes renement process. By tis step, in general, te asymptotic sarpness of te global error estimate is lost. Tis is seen, for instance, in te case were te exact as well as te approximate solution are anti-symmetric wit respect to te x-axis meaning tat e(0) = 0, but (u ) 6= 0. Terefore, we will analyze various ways of directly evaluating te error identity (4.2). Here, z 2 V may be taken as te bilinear interpolation of te approximation for z on te current mes, or simply be set zero, as its global contributions (if exactly evaluated) cancel out to zero under summation. Te approximate values obtained for J(e) are denoted by (i) (u ). 1. Approximation by iger order metods: Te dual problem is solved by using biquadratic nite elements on te current mes yielding an approximation z (2) to z. Te error estimator obtained by using tis approximation in (4.3) and (4.4) is denoted by (1) (). Computing approximations to z by using a iger order nite element sceme does not appear very economical in estimating te error in te low-order sceme. Hence, one may try to work wit te given metod, i.e., piecewise bilinear approximation, and increase its accuracy troug defect correction wit biquadratic interpolation on a coarser mes. Notice tat simple Ricardson extrapolation using two consecutive meses yields improved approximations only at te nodes of te coarser mes. However, since tis procedure requires to solve at least twice on te current mes, it appears too costly for practical purposes and is not furter considered. 2. Approximation by iger order interpolation: A furter simplication is acieved by simple local biquadratic interpolation of te bilinear approximation z (1) on te current mes yielding an approximation I (2) z(1) to z. Te resulting global error estimator is denoted by (2) (). Tis requires some special care on elements wit anging nodes, in order to preserve te iger order accuracy of te interpolation process. 3. Approximation by iger order dierence quotients: Witout loss of generality, consider a rectangular element oriented in te cartesian directions wit side lengts 1 = ;1 and 2 = ;2. Its vertices are numbered in te counter-clockwise sense as a i = (x i 1; x i 2) (i = 0; :::; 3). On te error in te bilinear interpolation can be expanded like (z? I (1) z)(x) = E (x) 2 + R ( ; x) ; (4.5) wit an expansion coecient E (x) and a remainder term R ( ; x) = O( 3 ). Tere olds E (x) = n'(' o?2? 1)( z 11 (a 0 ) + z 11 (a 3 )) + (? 1)('z 11 (a 1 ) + ' z 11 (a 2 )) ?2 n (? 1)( z 22 (a 0 ) + 'z 22 (a 1 )) + (? 1)( z 22 (a 1 ) + z 22 (a 2 )) were ' = (x 1? x 0 1)=(x 1 1? x 0 1), = (x 1 1? x 1 )=(x 1 1? x 0 1), = (x 2? x 0 2)=(x 1 2? x 0 2), = (x 1 2? x 2 )=(x 1 2? x0 2 ), and jr ( ; x)j 3 max jr 3 zj. Te elementary but tedious computation is omitted. According to (4.5), te local error indicators are estimated in te form 2 (f + u ; E )? 1(n [ru 2 ]; E : (4.6) Te second derivatives of z in te expansion coecient E are ten approximated by suitable central dierence quotients of its bilinear approximation z (1) obtained on te current mes. Again at anging nodes special care is required. Te resulting global error estimator is denoted by (3) (). o ; 12

13 4. Approximation by local problems: Following te idea already used in te energy-error estimator of Bank and Weiser [3], on eac element te local problems (rv ; r' ) = (f + u ; ' )? 1 2 (n [ru ]; ' 8 ' 2 V ; (4.7) are solved, were V = Q 2 () te space of biquadratic polynomials on. Using te resulting solutions, te local residual terms are replaced by ~ := j(rv ; r(z? z )) j : (4.8) Te resulting global error estimator is denoted by (4) (). Te related error indicator based on solving patc-wise Diriclet problems is not considered ere, since it turns out to be too expensive in practice. From te various results obtained in our tests, we quote tose for te point-value computation discussed above, as well as tose for te error measured in te global L 2 -norm. Te observed eciency indices are listed in Tables 2 and 3. Tese results indicate tat even for te simplest model situations, an eciency index I e = 1 is acievable only at te expense of unacceptably ig cost, e.g., by approximating te dual solution using iger-order elements. L N J(e) = e(0) J(e)= (1) J(e)= (2) J(e)= (3) J(e)= (4) e e e e e Table 2: Eciency of various weigted error indicators obtained for te point-error J(e) = je(0)j L N J(e) = kek J(e)= (1) J(e)= (2) J(e)= (3) J(e)= (4) e e e e e Table 3: Eciency of various weigted error indicators obtained for te L 2 -error J(e) = kek To prove tat te approximation strategies described above actually work in constructing \optimal" meses seems very dicult, as for tis one would need precise information about te local (on te element level) approximation properties of te nite element Ritz projection on general unstructured meses. Unfortunately, results of tis type are not available in literature and may not even be true at all. Below, we will prove a result for te point-error estimator wic is not quite wat one would like to ave, but provides some partial justication. It states tat, at least, on an optimally rened mes on te basis of te \exact" estimator (u ), te perturbed estimator ~(u ) using a iger-order approximation to te dual solution z sows te 13

14 rigt beavior in terms of te tolerance TOL. To tis end, we ave to make some reasonable tecnical assumptions. First, suppose tat on te \optimal" meses T, tere olds c 2 n o max jf + u j + max jn r[u ]?1 j C(u) 2 2 T ; wic indicates te (global) stability of te nite element Ritz projection wit respect to te discrete W1 2 -norm on general locally rened meses. On quasi-uniform meses (satisfying te uniform sape and size condition) tis immediately follows by te known error estimates in te W1-norm 1 and te inverse properties of nite elements. For more general meses, tis as to be left as a conjecture. However, (4.9) may be cecked in te course of te mes renement process. Lemma 4.1. For te model problem (1.2) on te square = (?1; 1) 2, let te dual solution z, i.e., te Green function to te point x = 0, be approximated on te current mes by its Ritz projection z (2) into te subspace of piecewise quadratic functions. Furter, suppose tat for a sequence of tolerances, TOL! 0, optimally balanced meses T opt ave been generated on te basis of te exact estimator point (u ), suc tat point (u ) TOL and TOL N opt ; 2 T opt : (4.10) Ten, te approximate error estimator ~ point (u ) := n 2T is asymptotically exact in te sense (f + u ; z (2) )? 1 2 (n [ru ]; z (2) (4.11) j point (u )? ~ point (u )j = o(t OL) : (4.12) Proof. We only give a sketc of te proof wic is based on results from local nite element error analysis (see [11]). Using te abbreviation w := z? z (2), we ave by Galerkin ortogonality point (u )? ~ point (u ) = 2T n (f + u ; w? I w )? 1 2 (n [ru ]; w? I w o ; and, consequently, by te local approximation properties of te interpolation operator I, Tis is furter estimated by point (u )? ~ point (u ) point (u )? ~ point (u ) C i Te second factor on te rigt is bounded by 2T kr 2 w k : 1=2 C i C(u) 2 2 r?4 rkr 4 2 w k 2 2T 2T 2T r 4 kr2 w k 2 1=2 14 r 4 jr 2 w j 2 dx 1=2 ; 1=2 : (4.13)

15 were r(x) := maxfjxj; 0 g, 0 = min being te diameter of te element containing te point x = 0. By standard arguments from local nite element analysis, one can prove tat r 4 jr 2 w j 2 dx 1=2 cmax c max r 4 jr 3 zj 2 dx 1=2 r?2 dx 1=2 cmax jlog( 0 )j 1=2 : We omit te rater tecnical details and refer to [18] and [11] for similar arguments. Under te assumptions (4.10) and (4.9), for te optimal mes T, tere olds N T OL?1 and (T OLr ) 1=2. Terefore, te rst factor on te rigt of (4.13) can be bounded by 1=2 1=2 C(u)T OL C(u) jlog(0 )j 1=2 T OL : 2 2T 2 r?4 r?2 2T Combining te foregoing estimates in (4.13) yields point (u )? ~ point (u ) C i C(u) jlog( 0 )j max T OL : In view of max T OL 1=2, tis implies te assertion. In view of te computational results sown in Table 3, te statement of Lemma 4.1 does not seem to be true for te approximation of te global L 2 -error. Since asymptotically sarp a posteriori control does not seem to be feasible in general, we ave to live wit error estimates involving at least certain interpolation constants C i. Tese constants may usually be assumed to be of moderate size C i = 0:1?1, but certainly represent some factor of uncertainty. In particular situations, one may try to calibrate tese constants in te course of te renement process wic, owever, is a very delicate matter. Te full teoretical justication of te error estimator ~() would require an analysis of te eect of replacing te exact second derivatives of te dual solution z by suitable dierence quotients of some approximation z obtained numerically on te current mes. Yet, tis is strongly supported by numerical evidence, even in te present simple case te proof must be left as an open problem. For te residual-based energy error estimator (1.13) one as te two-sided bound krek energy (u ) fkrek + kf? f kg ; (4.14) were f is a suitable approximation of f. Tis sows asymptotic optimality. An analogous bound cannot old in general for local error quantities like te point-error. Wat we can ope to acieve is only an optimality estimate of te type presented in te following lemma. Lemma 4.2. For te nite element approximation of te model problem (1.2) on te square = (?1; 1) 2, te error estimator point () dened in (2.4) is asymptotically optimal in te sense je(0)j point (u ) c jlog( 0 )j max 2T max were 0 is te diameter of te element containing te point x = 0. jej + 2 max jr2 uj ; (4.15) 15

16 Proof. Starting from (2.4), we use te Holder inequality to obtain point (u ) 2 2 2T r 2 1=2 2T?2 r2!2 1=2; (4.16) were r := maxfdist(; 0); g. Te second factor can be estimated by (see [18] or [11]) 2T?2 r2! 2 c f 0 <jxj<1g jxj 2 jr 2 zj 2 dx + cjlog( 0 )j c jlog( 0 )j : (4.17) Furter, using te usual inverse inequality for nite elements togeter wit te local interpolation estimate (1.3), we conclude in a standard way tat kek =2 kn c?1 kek ~ + c kr 2 uk ~ ; were ~ denotes te union of all cells in T intersecting. Tis implies 2 2T r 2 Combining (4.17) and (4.18), te assertion follows. 2 c jlog( 0 )j max max 2T jej + 2 ~ 2 max ~ uj jr2 : (4.18) For a special situation (uniform tensor-product mes) it as been sown in [32] and [33] tat in te energy-error estimate (1.13) eiter te domain residual terms kf + u k or te jump terms kn [ru may be asymptotically neglected depending on weter te degree of te elements used is odd or even, respectively. Tis result as recently been extended in [31] to general adaptively rened meses for lowest-order elements (odd degree). We will derive a corresponding result for te present situation wic to some extent justies te use of te simplied weigted error estimator ~() dened in (2.14). Lemma 4.3. Suppose tat f 2 H 1 () in te model problem (1.2). Ten, for piecewise bilinear elements, tere olds jj(e)j C i 2T were ~ is te union of all cells in T intersecting. 3 krfk + ~ 3=2 kn [ru kr 2 zk ; (4.19) ~ Proof. For simplicity, we give te proof only for rectangular meses. Te generalization to more general meses requires some more tecnical work. In tis case u j 0 on eac element. Since te mes T is generated from a coarse mes by local cell-wise bisection (i.e., subdivision into four sub-cells), it can be decomposed into macro-cells consisting of four sub-cells eac. Tese macro-cells form a coarser mes named ~ T. Te case of cells wit anging nodes requires obvious modications and will not be discussed in detail. On ~T we dene an interpolation operator ~I : C( )! V, by requiring ~ I v(a) = v(a), for vertices a of ~T, and R ~ ~ I v dx = R ~ v dx, for eac macro-cell ~ 2 ~T. By standard arguments (Bramble-Hilbert lemma) we conclude te estimate kv? ~ I vk + 1=2 kv? ~ I C i; 2 kr2 vk ~ ; (4.20) 16

17 wit certain interpolation constants C i; independent of. Now, we coose z = ~ I z in te error representation (4.2) and obtain J(e) = 2T n (f? ~ f ; z? ~ I z )? 1 2 (n [ru ]; z? ~ I z o ; (4.21) were ~ f denotes te piecewise constant interpolation f on te macro-mes ~T. Hence, in view of (4.20), we conclude tat jj(u )j C i wic proves te assertion. 2T 3 krfk + ~ 3=2 kn [ru kr 2 zk ; (4.22) ~ 5 Applications Te following examples are intended to support some of te claims made above on te basis of teoretical analysis. In te tables sown below, we list te results obtained for a sequence of successively reduced values of TOL, te number of points, N opt, of te nal mes T opt, on wic te error indicators are balanced, te maximum number of renement levels, L, in tis mes, and te eciency index I e := jj(e)j=(u ). Example 1. As rst example, we consider te computation of isolated stress values, i.e., point-values of te gradient ru, in te model Poisson problem (1.2) on te square = (?1; 1) 2. Let, for example, te error functional be cosen as J(u) 1 u(0) : Since te corresponding dual solution does not exist in te sense of H 1 0(), we ave to regularize te functional according to J (u) := jb (0)j?1 B 1 u dx ; were B (0) := fx 2 ; jxj < g, and = TOL. Ten, jj(u)? J (u)j TOL, and te corresponding dual solution z is well dened in H0() 1. Observing tat z (x) x 1 =(jxj + ) 2, we are led to te weigted a posteriori error estimate j@ 1 e(0)j C i 2T 2 (r + TOL) 3 + ctol ; (5.2) were r = maxfdist(; 0); g. We again assume tat te local residuals satisfy 2. Ten, on a mes T opt on wic te local error indicators =! are equilibrated, tere olds T OL N opt ; N opt = #f T opt g ; (5.3) T OL (r + TOL) 3 1=4 ; (5.4) N opt 17

18 Consequently, te number N opt of elements of T opt is given by N opt 2T opt 2?2 = 2T opt 2 N 1=2 opt : (5.5) TOL 1=2 3=2 (r + TOL) Since r?3=2 is integrable over, it follows tat N opt TOL?1, as for te evaluation of te point value u(0). Furter, te mes size is distributed like TOL 1=2 (r + TOL) 3=4 ; and, consequently, min TOL 5=4 and max TOL 1=2. We note tat if te functional J() is not regularized, te same reasoning yields te mes-size distribution TOL 1=2 r 3=4 and, in tis case, min TOL 2, meaning a strong over-renement at x = 0. Tis eect is conrmed in our test computations. In contrast to te weigted estimator (5.2), te global energy-error estimator (1.13), for r = 0, or te analogue of te L 1 -error estimator (2.11), wit 1=2 krek C s C i 2 =: energy (u ) ; (5.6) 2T max j@ 1 ej C i C s j log 0 j 1=2 max 2T 1; =: 1 (u ) ; (5.7) 1; := max jf + u j + max jn [ru ]j would generate meses of complexity N opt T OL?2, in order to guarantee (u ) TOL. Figure 2 sows te balanced mes for TOL = 4?4 and te approximation to te dual solution z, = TOL, computed on tis mes. Te corresponding errors for a sequence of tolerances are listed in Table 4. For te problem considered, te weigted a posteriori estimate (2.4) for te point error is obviously asymptotically optimal and te predicted dependence N opt TOL?1 is conrmed. Example 2. integral As a very particular example, we consider te computation of te contour n un 1 ds ; (5.8) for te solution of te model problem (1.2) on te unit disc, = B 1 = fx 2 R 2 ; jxj < 1g, were n 1 is te x 1 -component of te outer unit vector Error functionals of tis type occur, for instance, in te computation of drag and lift coecients of blunt bodies in viscous ows modeled by te Navier-Stokes equations (see [7]). In view of te pointwise analogue of te a priori error estimate (1.6), we ave J(e) = O( max ), so tat convergence is no problem. Te question is ow te mes renement process sould be organized in order to obtain J(u) wit best accuracy. We will see tat te answer is somewat surprising and seems, at te rst glance, to conict wit wat one is used to from a nite element metod. 18

19 Figure 2: Rened mes and approximate dual solution for 1 u(0) in Example 1, using te weigted error estimator point, wit TOL = 4?4 energy TOL N L j@ 1 e(0)j I e e e ? e e ? e e ;1 TOL N L j@ 1 e(0)j I e e e ? e e ? e e ;point TOL N L j@ 1 e(0)j I e e e ? e e ? e e ? e e ? e e ? e e ? e e regular 1;point TOL N L j@ 1 e(0)j I e e e-2-4? e e-2-4? e e-2-4? e e ? e e ? e e ? e e Table 4: Results for 1 u(0) in Example 1, using te energy- and W 1 -error estimators energy and 1;1, respectively, compared to te weigted estimator 1;point and its regularized variant regular 1;point First, in oder to avoid aving to deal wit measures, we regularize te functional J(). Wit te cut-o function ( 1 ; 0 jxj < 1? ; (x) =?1 (1? jxj) ; 1? jxj < 1 ; we make te following ansatz for te dual solution z (x) =?x 1 (x) ; 19

20 were = TOL. Let ' 2 V [ C 2 ( ) be arbitrary. We split te integration over B 1 according to B 1 = B 1? [ S wit S = B 1 n B 1? and obtain, using polar coordinates (r; ), n o rz r' dx n z ' ds r r ' + r B 1 ' dx 1? By construction, tere n z =?@ r x 1 =?cos 1 n B 1?, and consequently, observing x 1 = r cos(), n z ' ds =? cos()'(1? ; ) ds = O() : 0 Furter, we ave and r r ' dx =? Finally, observing tat we nd 1? x r ' dx? r ' dx =? S S?1 x r ' dx + O() ; S S ' dx =?1?1 S x r ' dx = S r?2 (1? r)sin()@ ' dx = O() 1 n n ' ds + O() ; J(') = (rz ; r') + O() : (5.9) Tis implies tat te approximate weigts ~! kr 2 z k in te a posteriori error estimate (2.17) corresponding to te functional J (u ) := (rz ; r') are all zero on elements outside a boundary strip of widt, i.e., tere is no contribution to te error J(e) from tose interior elements. Tis remains true if we let! 0. Terefore, starting from some coarse mes, te \optimal" renement strategy would be in eac step only to rene tose elements wic are adjacent to te boundary and to leave te oters uncanged. Te resulting meses are maximally rened along te boundary down to a minimal mes size min, wile in te interior of te elements are kept of size max 1. If only local mes bisection is allowed in eac renement step, te mes obtained after L steps as minimal size min 2?L and consists of N L 2 L elements. To reac an error of size jj(e)j T OL, about k log 2 (1=T OL) renement cycles are needed leading to an \optimal" mes wit N 1=T OL elements. However, to realize te full accuracy on te constructed mes, te rigt and side consisting of integrals over te force f multiplied by te nodal basis functions as to be calculated almost exactly also on te coarse elements. Figure 3 sows te te nest mes obtained after 7 renement steps and te approximation to te dual solution z ; = min, computed on tis mes. In Table 5 te results obtained by te usual energy error estimator energy (u ) are compared to tose of te \optimal" renement strategy. Example 3. By te tird example, we demonstrate tat carrying weigts in te a posteriori error estimates can also be advantageous in estimating te error in global norms. Tis applies 20

21 Figure 3: Rened mes and approximate dual solution for Example 2 obtained by te approximate weigted error estimator approx ; after 7 renement steps weigt energy L N J(e) : : : :49? :77? :73? 1 6 memory exausted 7 memory exausted \optimal" strategy L N J(e) : : : :48? :62? :67? :93? :03? 2 Table 5: Results obtained wit te energy norm estimator energy strategy for Example 2 and by te \optimal" renement particularly wen global a priori bounds for te dual solution do not appropriately reect te local stability properties of te underlying problem, e.g., for problems wit strongly varying coecients. Consider, for example te boundary value problem? divfarug = f in ; u = 0 ; (5.10) on te square = (?1; 1) 2 wit a coecient function, a(x) = 0:1 + e 3(x 1+x 2) : (5.11) We discretize tis problem by bilinear nite elements and recall te weigted L 2 -a posteriori error estimates following from (1.8), kek 2T! ; (5.12) were te residuals and weigts! are dened by (1.9) and (1.10), were z is te solution 21

22 of te dual problem,? divfarzg = e=kek in ; z = 0 ; (5.13) First, te weigts ~! are computed by taking second dierence quotients of te discrete dual solution z (1) 2 V on te current mes T, wile on te rigt and side of (5.13) te exact error e is used. Te reference solution is as before u = 10 sin(2x 1 + x 2 + 2). Tis gives te approximate stability constant 1=2 ~C s := : 2T ~! 2 Tis way, we obtain a rst approximate weigted L 2 -error estimator ~ weigt (u ) := 2T ~! as well as te corresponding global L 2 -error estimator ~ 2 (u ) := C i ~ Cs 2T 2 2 te interpolation constant being cosen as C i = 0:2. Next, te unknown error e in (5.13) is approximated by ~e := I (2) u? u, were I (2) u is te patc-wise biquadratic interpolation of u. Te resulting error estimator is denoted by ~ approx weigt. Te results obtained for te above test problem by using tese tree L 2 -error estimators are listed in Table 6. Figure 4 sows te errors on te corresponding \optimal" meses for te global estimator ~ 2 (u ) and te weigted estimator ~ weigt (u ). Tese tests clearly demonstrate te superiority of ~ weigt over ~ 2. ; Figure 4: Errors on balanced meses wit N for Example 3 by te L 2 -error estimator ~ 2 (u ) (left, scaled by 1:10) and te weigted error estimator ~ approx weigt (u ) (rigt, scaled by 1:30) Example 4. Te nal example is intended to support our claim tat, in using iger order nite elements, one sould indeed use te approximation property of te nite element space to its full extent in generating te weigts! in te a posteriori estimate (2.17). First, we 22

23 ~ 2 TOL N L kek I e Cs ~ e e ? e e ? e e ? e e ? e e ? e e ~ weigt TOL N L kek I e e e ? e e ? e e ? e e ? e e ? e e ? e e ? e e ? e e ? e e ~ approx weigt TOL N L kek I e e e ? e e ? e e ? e e ? e e ? e e ? e e ? e e ? e e ? e e Table 6: Results obtained for Example 3 by te global L 2 -error estimator ~ 2, compared to te weigted estimators ~ weigt and ~ approx weigt illustrate tis point by an analytical argument. Consider again te model Poisson problem (1.2) on te unit circle = fx 2 R 2 ; jxj < 1g. Suppose tat te mean value J(u) := u(x) dx as to be computed. In tis case te corresponding dual solution is just te quadratic polynomial z(x) = 1? 1 2 jxj2, wic satises?z = 1 and z j@ = 0. We tink tis problem to be solved by biquadratic nite elements (case p = 2). Ten, te \optimal" a posteriori error bound jj(e)j C i 2T 2 kr 3 zk would indicate tat, similar as in Example 2, te error is concentrated along te boundary, wile tere is no contribution from interior mes cells. In contrast, te sub-optimal a posteriori bound jj(e)j C i 2T kr 2 zk would yield contributions to te error from all mes cells. For furter illustration, we consider again te computation of te point value u(0) in te above model situation. Accordingly, we compare te following two a posteriori error estimates, je(0)j C i 3 2T r 3 23 =: (3) point (u ) ; (5.14)

24 and je(0)j C i 2 2T r 2 Te results sown in Table 7 clearly support our claim. =: (2) point (u ) : (5.15) (3) point TOL N L 4? e-1 4? e-2 4? e-3 4? e-3 4? e-4 4? e-4 4? e-5 4? e-5 4? e-6 4? e-7 (2) point TOL N L 4? e-1 4? e-2 4? e-2 4? e-3 4? e-4 4? e-4 4? e-5 4? e-5 4? e-6 4? e-7 Table 7: Results for computing e(0) by quadratic elements in Example 4, obtained wit te local estimators (3) point (u ) and (2) point (u ) 6 Generalization to nonlinear problems Te approac to residual-based error estimation described above can be extended to general nonlinear problems; for ideas in te same direction see [14], [22], [23], [16], as well as [29] and [26]. Related energy-error estimators ave been derived in [29]. We outline our general concept in an abstract setting. Let V be a Hilbert space wit inner product (; ) and corresponding norm k k, and a(; ) a semi-linear form continuously dened on V V. We seek a solution to te abstract variational problem u 2 V : a(u; ') = 0 8 ' 2 V: (6.2) Tis problem is approximated by a Galerkin metod using a sequence of nite dimensional subspaces V V parameterized by a discretization parameter. Te discrete problems read u 2 V : a(u ; ' ) = 0 8 ' 2 V : (6.3) Wit te derivative a 0 (; ; ) of a(; ), we ave te following ortogonality relation for te error e = u? u : 1 0 a 0 (tu + (1? t)u ; e; ' ) dt = a(u; ' )? a(u ; ' ) = 0 8' 2 V : (6.4) Tis suggests te use of te bilinear form L(u; u ; '; z) = 1 0 a 0 (tu + (1? t)u ; '; z) dt ; (6.5) 24