In this paper an  a posteriori  error estimate for   hypersingular integral equations is derived by using   hierarchical basis techniques. Based on the  properties of a two-level additive Schwarz method   easily computable local error indicators are obtained.  An algorithm for adaptive error control which allows  anisotropic refinements of the boundary elements  is formulated and numerical results are   included

Mund, Patrick

Stephan, Ernst P.

English

Australian Mathematical Society (AustMS): E-Journals

ANZIAM J. 42 (E) ppC1019{C1033, 2000 C1019An adaptive two-level method forhypersingular integral equations in R3Patrick Mund Ernst P. Stephan(Received 7 August 2000)AbstractIn this paper an a posteriori error estimate for hypersingular inte-gral equations is derived by using hierarchical basis techniques. Basedon the properties of a two-level additive Schwarz method easily com-putable local error indicators are obtained. An algorithm for adaptiveerror control which allows anisotropic renements of the boundaryelements is formulated and numerical results are included.∗Institut fu¨r Angewandte Mathematik, Universita¨t Hannover, Welfengarten 1, 30167Hannover, Germany. mailto:stephan@ifam.uni-hannover.de0See http://anziamj.austms.org.au/V42/CTAC99/Mund for this article and ancillaryservices, c© Austral. Mathematical Soc. 2000. Published 27 Nov 2000.Contents C1020Contents1 A stable two-level subspace decomposition C10202 An a posteriori error estimate C10263 Numerical results C1028References C10321 A stable two-level subspace decompositionIn recent years adaptive hierarchical basis methods in the nite elementmethod (fem) [1, 2, 4] have become increasingly popular. This approachhas meanwhile been also applied to the boundary element method (bem)for weakly singular integral equations [8] and to the fem/bem coupling [7].Here we extend it to hypersingular integral equations on surfaces.We consider the hypersingular integral equationWv(x) := − 14@@nxZΓv(y)@@ny1jx− yj dy = f(x) ; x 2 Γ (1)where Γ is an open plane surface and integration has to be understood in theHadamard sense. W is a bijective mapping from ~H1=2(Γ) onto H−1=2(Γ) and1 A stable two-level subspace decomposition C1021the bilinear form hWu; vi for u; v 2 ~H1=2(Γ) is symmetric and positive denite(cf. Costabel [3], Stephan [9]). Hence the unique solution vN 2 SN  ~H1=2(Γ)of the Galerkin schemehWvN ; i = hf; i 8 2 SN (2)converges quasi-optimally towards the exact solution v 2 ~H1=2(Γ) of (1). Forstandard Sobolev spaces L2(Γ) and H10 (Γ) (which is the completion of C10within H1(Γ)) the space ~H1=2(Γ) is an interpolation space between L2(Γ)and H10 (Γ), and H−1=2(Γ) is the dual space of ~H1=2(Γ). The solution u of (1)for given f 2 H−1=2(Γ) is the jump across Γ of the solution of a Neumannproblem for the Laplacian in R3nΓ, cf. [9].In the following we present a two-level method for the h-version of theGalerkin scheme with bilinear continuous elements. For ease of presentationwe consider Γ = [−1; 1]2.Let γl (0  l  L) be a uniform partition of Γ into squares of side lengthhl = 2−l and letXl = fu 2 C0(Γ) : u piecewise bilinear w.r.t. γl and uj@Γ = 0g (0  l  L)X0l = fu 2 Xl : u = 0 on the nodes of γl−1g (1  l  L):Let bl;i (0  i  nl) be a piecewise bilinear function with value one atan interior node of γl (not belonging to γl−1) and with zero value in allother nodes of γl. Let X0L;i = spanfbL;ig. We have the following two-level1 A stable two-level subspace decomposition C1022decomposition of the space XL:XL = XL−1 X0L;1     X0L;nL (3)LetPL(2) = PL−1 +nLXi=1PL;i (4)be the two-level additive Schwarz operator belonging to the subspace decom-position (3) and the bilinear form hW ; i, i.e.hWPL−1;  i = hW; i 8 2 XL−1;  2 XL;hWPL;i;  i = hW; i 8 2 X0L;i;  2 XL:Then there holdsTheorem 1 There exist constants c1; c2 > 0, independent of L, such thatc1hWu; ui  hWPL(2)u; ui  c2hWu; ui 8u 2 XL : (5)The proof of Theorem 1 is based on the following lemmas where alwaysu 2 XL arbitrary withu = uL−1 +nLXi=1uL;iwhere uL−1 2 XL−1 and uL;i 2 X0L;i. Let IL−1u 2 XL−1 be the bilinearinterpolant of u at the nodes of γL−1.1 A stable two-level subspace decomposition C1023Lemma 2 [6, Lemma 2.5, Lemma 3.10] There exists a constant c, inde-pendent of L and u, such thatku− IL−1ukL2(Γ)  chLkukH1(Γ): (6)Furthermore for any s 2 [0; 1] there exists a constant c = c(s) > 0 such thatkIL−1uk ~Hs(Γ)  ckuk ~Hs(Γ) for any u 2 XL. (7)Lemma 3 [6, Lemma 3.12] Let ΓLi 2 γL be a square with vertices xk(1  k  4). Let v; w be bilinear functions on ΓLi with v(x1) = w(x1) andw(x2) = w(x3) = w(x4) = 0 . Then there holdskwkL2(ΓLi ) 43kvkL2(ΓLi ) :Lemma 4 [5] Let fΓi; i = 1; : : : ; Ng be a nite covering of Γ with rectan-gles Γi and covering constant  2 N, i.e. we can colour fΓi; i = 1; : : : ; Ngby at most  dierent colours such that subdomains with same colour aredisjoint. Let  =PNi=1 i 2 ~Hs(Γ) for s 2 R with i 2 ~Hs(Γi). Then thereholdskk2~Hs(Γ)  NXi=1kik2~Hs(Γi) :1 A stable two-level subspace decomposition C1024Proof: (of Theorem 1) We show that there exist constants c1; c2 > 0 suchthat1c2kuk2~H1=2(Γ)  kuL−1k2~H1=2(Γ) +nLXi=1kuL;ik2~H1=2(Γ) 1c1kuk2~H1=2(Γ): (8)The left inequality follows directly from Lemma 4 with c2 = 9 since to anyx 2 Γ there belong at most 8 dierent bL;i’s with x 2 suppbL;i (1  i  nL)and sincekuL;ik ~H1=2(Γ) = kuL;ik ~H1=2(suppbL;i) :It remains to show the right inequality in (8). Since uL−1 = IL−1u Lemma 2implieskuL−1k ~H1=2(Γ)  ckuk ~H1=2(Γ) : (9)Let fxignLi=1 denote the set of nodes in γL which do not belong to γL−1. Wedecompose the index set f1; 2; : : : ; nLg = M1 [M2 [M3 into 3 disjoint setsMk such that two indices i; j 2 f1; 2; : : : ; nLg belong to the same set Mk ifjxi − xj j is an integer multiple of 2hL. The sets Mk are uniquely determined(up to permutation) (cf. Fig. 1). For k 2 f1; 2; 3g the nodes fxigi2Mk are justthe nodes of the coarse grid γL−1 shifted by hL in the x1- and/or x2-direction.There holds for k 2 f1; 2; 3gmeas(supp bL;i \ supp bL;j) = 0 8i; j 2Mk ; i 6= j1 A stable two-level subspace decomposition C10251 11111333 3332 222Figure 1: Decomposition of the nodes γL into disjoint subsets: Numberk 2 f1; 2; 3g of the node x means x 2Mk. The nodes of the coarse mesh γL−1are marked with .2 An a posteriori error estimate C1026with the Lebesgue-measure meas(). HenceXi2MkkuL;ik20 =∥∥ Xi2MkuL;i∥∥20 169k~uk20 (10)where ~u = u− uL−1 and k  ks = k  k ~Hs(Γ); s 2 IR.Here the last inequality follows from Lemma 3 since w =Pi2Mk uL;i and ~ucoincide at one node of each element in γL and w vanishes in all the othernodes. With the standard inverse inequality for nite elements and (10) and(6) we havenLXi=1kuL;ik21=2  ch−1LPnLi=1 kuL;ik20 = ch−1L3Xk=1Xi2MkkuL;ik20 c0h−1L k~uk20  c0h−1L ku− IL−1uk20 chLkuk21  c kuk21=2 :Together with (9) this implies the right inequality in (8). Thus the proof iscomplete due to the equivalence of the norms kuk1=2 and hWu; ui1=2. ♠2 An a posteriori error estimateNext we need the saturation assumption: (Ah) There exist constants k0 2 Nand 0 <  < 1 such thatkv − vk+1k1=2  kv − vkk1=2 8k  k02 An a posteriori error estimate C1027where vk+1 denotes the Galerkin solution on the level k + 1 and v the exactsolution of (1).Theorem 5 Suppose (Ah) holds. Then there exist constants c1; c2 > 0 suchthat there holds for k  k0c1nLXj=12L;j  kv − vL−1k21=2 = c2nLXj=12L;j (11)withL;j =jhf −WvL−1; bL;jijhWbL−1; bL;ji1=2 (j = 1; 2; : : : ; nL) : (12)Proof: The saturation assumption (Ah) yields the equivalence of normskvL − vL−1k1=2  kv − vL−1k1=2 :Due to Theorem 1 we havec1kvL − vL−1k21=2  kPL−1(vL − vL−1)k21=2 +nLXi=1kPL;i(vL − vL−1)k21=2 c2kvL − vL−1k21=2 :Firstly, we observe that since vL−1 and vL satisfy the Galerkin equation thereholds for any w 2 XL−1hWPL−1vL; wi = hWvL; wi = hf; wi = hWvL−1; wi = hWPL−1vL−1; wi :3 Numerical results C1028HencekPL−1(vL − vL−1)k21=2 = 0:The error indicator L;j in (12) is obtained by solving a linear problem inthe space X0L;j . The function vL;j = PL;j(vL − vL−1) 2 X0L;j solves for anyv 2 X0L;jhWvL;j; vi = hf −WvL−1; vi (13)Hence rstly one solves (13) for 1  j  nL and then one computes the termsL;j = hWvL;j; vL;ji1=2 . Since X0L;j = spanfbL;jg is a one-dimensional space,we have vL;j = cbL;j with coecientc =hf −WvL−1; bL;jihWbL;j; bL;ji :HenceL;j = jcjhWbL;j; bL;ji1=2 :♠3 Numerical resultsAlgorithm 3.1 (Adaptive multilevel algorithm) Let γ0 denote an initialmesh on Γ and X0 the corresponding space of continuous bilinear functions.Furthermore let 0    1 and  > 0 be given.3 Numerical results C10291. Compute the Galerkin solution uk 2 Xk of (2).2. Compute the error indicators k;j, j = 1; : : : ; nk with (12).3. Compute the row error indicators R;m and the column error indica-tors C;n as weighted sums of the single error indicators of one row andone column of the mesh, respectively. The weighting is done by dividingthe quadratic mean of the local error indicators of a row or a columnby the respective number of terms.4. Compute max := maxfR;m; C;ng. Rene all elements of the mth rowin x1-direction ifR;m  max ;rene all elements of the nth column in x2-direction ifC;n  max :Here rening of the element with index i in xk-direction means halvingthe element in xk-direction.5. Thus obtain the space Xk+1. Check whether max <  is satised.Otherwise go to 1.Remark 6 Due to the row and column error indicators the above renementstrategy secures the continuity of the trial functions. No hanging nodes areobtained.3 Numerical results C1030Table 1: Adaptive h -renement with  = 0:6 .L NL EL L L=EL (BhWh) (Wh)1 5 0:559700 0:335091 0:598697 1:24  100 1:24  1002 16 0:466277 0:313657 0:672684 4:86  100 2:36  1003 48 0:311299 0:201438 0:647088 1:39  101 3:74  1004 96 0:214100 0:130477 0:609423 1:65  101 6:99  1005 160 0:149149 0:085045 0:570201 1:77  101 1:37  1016 240 0:105005 0:057504 0:547632 1:80  101 2:86  1017 336 0:074666 0:041280 0:552871 1:83  101 6:13  1018 448 0:053782 0:032152 0:597816 1:84  101 1:34  102Next we choose the model problem Γ to be the L-shaped surface piecein the (x1; x2)-plane with vertices (0; 0; 0), (1; 0; 0), (1;−1; 0), (−1;−1; 0),(−1; 1; 0), (0; 1; 0), and the right hand side f = 1 in (1). In Table 1 the resultsare presented for the adaptive renement strategy. NL denotes the number ofunknowns on level L; EL the relative error in the energy norm; (BhWh) and(Wh) the condition numbers of the preconditioned and unpreconditionedGalerkin matrix of (2), respectively; Bh the preconditioner according to theadditive Schwarz operator in (4); L the sum of the error indicators andL=EL the eciency index of the algorithm. The sequence of correspondingmesh renements is given in Figure 2.3 Numerical results C1031Figure 2: The sequence of rened meshes for  = 0:6 .References C1032Acknowledgments: This work was supported by the German ResearchFoundation (DFG) grants Ste 258/25-9 and Ste573/3-1. The authors wouldlike to thank K.-W. Oertel for performing the numerical experiments andM. Maischak for providing his software package maiprogs.References[1] R.E. Bank and R.K. Smith. A posteriori error estimates based onhierarchical bases. SIAM J. Numer. Anal., 30:921{935, 1993. C1020[2] F.A. Bornemann, B. Erdmann, and R. Kornhuber. A posteriori errorestimates for elliptic problems in two and three space dimensions. SIAMJ. Numer. Anal., 33:1188{1204, 1996. C1020[3] Martin Costabel. Boundary integral operators on Lipschitz domains:Elementary results. SIAM J. Math. Anal., 19(3):613{626, 1988. C1021[4] P. Deuflhard, P. Leinen, and H. Yserentant. Concepts of an adaptivehierarchical nite element code. Impact of Computing in Science andEngineering, 1:3{35, 1989. C1020[5] Norbert Heuer. Additive schwarz methods for weakly singular integralequations in R3|the p-version. In G. Wittum W. Hackbusch, editor,Boundary Elements: Implementation and Analysis of AdvancedAlgorithms, Proceedings of the 12th GAMM{Seminar, Kiel 1996,References C1033volume 54 of Notes on Numerical Fluid Mechanics, pages 126{135,Braunschweig, 1996. Vieweg. C1023[6] Patrick Mund. Zwei-Level-Verfahren fu¨r Randintegralgleichungen mitAnwendungen auf die nichtlineare FEM-BEM-Kopplung. PhD thesis,Hannover, 1997. C1023, C1023[7] Patrick Mund and Ernst Peter Stephan. An adaptive two-level methodfor the coupling of nonlinear fem-bem equations. SIAMJ. Numer.Analysis, 36:1001{1021, 1999. C1020[8] Patrick Mund, Ernst Peter Stephan, and Josha Weie. Two-levelmethods for the single layer potential in R3. Computing, 60:243{266,1998. C1020[9] Ernst Peter Stephan. Boundary integral equations for screen problemsin R3. Integral Equations and Operator Theory, 10:257{263, 1987.C1021, C1021

An adaptive two-level method for hypersingular integral equations in \(R^3\)

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An adaptive two-level method for hypersingular integral equations in $R^3$

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An adaptive two-level method for hypersingular integral equations in $R^3$