Non shape regular domain decompositions: an analysis using a stable decomposition in H 1 0

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1 Non shape regular doman decompostons: an analyss usng a stable decomposton n H 1 0 Martn Gander 1, Laurence Halpern, and Kévn Santugn Repquet 3 Abstract In ths paper, we establsh the exstence of a stable decomposton n the Sobolev space H 1 0 for doman decompostons whch are not shape regular n the usual sense. In partcular, we consder doman decompostons where the largest subdoman s sgnfcantly larger than the smallest subdoman. We provde an explct upper bound for the stable decomposton that s ndependant of the rato between the dameter of the largest and the smallest subdoman. 1 Introducton One of the great success stores n doman decomposton methods s the nventon and analyss of the addtve Schwarz method by Dryja and Wdlund [1987]. Even before the seres of nternatonal conferences on doman decomposton methods started, Dryja and Wdlund [1987] presented a varant of the classcal alternatng Schwarz method see Schwarz [1870], whch has the advantage of beng symmetrc for symmetrc problems, and t also contans a coarse space component. In a fully dscrete analyss, Dryja and Wdlund [1987] proved, based on a stable decomposton result for shape regular decompostons, that the condton number of the precondtoned operator wth a decomposton nto many subdomans only grows as a functon of H δ, where H s the subdoman dameter, and δ s the overlap between subdomans. Martn Gander Unversté de Genève e-mal: Martn.Gander@unge.ch Laurence Halpern Unversté Pars 13 e-mal: halpern@math.unv-pars13.fr Kévn Santugn Repquet Insttut Mathmatques de Bordeaux, CNRS UMR551, MC, INRIA Bordeaux - Sud-Ouest e-mal: Kevn.Santugn@math.u-bordeaux1.fr 1

2 Martn Gander, Laurence Halpern, and Kévn Santugn Repquet Ths analyss nspred a generaton of numercal analysts, who used these technques n order to analyze many other doman decomposton methods, see the reference books Smth et al. [1996], Quarteron and Vall [1999], Tosell and Wdlund [004], or the monographs Xu [199], Chan and Mathew [1994], and references theren. The key assumpton that the decomposton s shape regular s however often not satsfed n practce: because of load balancng, hghly refned subdomans are often physcally much smaller than subdomans contanng less refned elements, and t s therefore of nterest to consder doman decompostons that are only locally shape regular,.e. doman decompostons where the largest subdoman can be consderably larger than the smallest subdoman. In such a doman decomposton, the rato H δ can be gven at least two dfferent meanngs: let H refer to the dameter of subdoman number and δ refer to the wdth of the overlap around subdoman number. Is the explct upper bound of the stable decomposton lnear n max H δ or s t only lnear n maxh mn δ? The latter estmate s much more pessmstc than the former when the subdomans are of wldly dfferent sze, and the general analyss based on a shape regular decomposton of the addtve Schwarz method does not permt to answer ths queston. In Gander et al. [011], we establshed the exstence of a stable decomposton n the contnuous settng wth an explct upper bound and a quanttatve defnton of shape regularty. The explct upper bound s also lnear n H δ, and the result s lmted to shape regular doman decompostons where all subdomans have smlar sze and where the overlap wdth s unform over all subdomans. Havng explct upper bounds however allows us now, usng smlar technques, to establsh the exstence of a stable decomposton n the contnuous settng wth explct upper bounds when max H mn H. We provde an explct upper bound whch s lnear n max H /δ. To get ths result, only a few of the nequaltes establshed n Gander et al. [011] need to be reworked, and t would be very dffcult to obtan such a result wthout the explct upper bounds from the contnuous analyss n Gander et al. [011]. Geometrc parameters and man theorem In the remander of ths paper, we always consder a doman decomposton that has the followng propertes: Ω s a bounded doman of R. The U 1 N are a non overlappng doman decomposton of Ω,.e. N U = Ω. The U are bounded connected open sets of R and for all subdomans U the measure of U \ U s zero. We set H := damu.

3 Non shape regular doman decomposton 3 Two dstnct subdomans U and U j are sad to be neghbors f U U j. For each subdoman U, let δ > 0 be such that δ mn dstu j,u U j=, U j. We set Ω := {x Ω, dstx, U < δ }. The Ω form an overlappng doman decomposton of Ω. When subdomans U and U j are neghbors, then the overlap between Ω and Ω j s δ + δ j wde. The ntersecton Ω Ω j s empty f and only f the dstance between U and U j s postve. We set δ s = mn j,u δ U j j and δ l = max j,u δ U j j. The doman decomposton has N c colors: there exsts a partton of N [1, N] nto N c sets I k such that Ω Ω j s empty whenever j and and j belong to the same color I k. T s a coarse trangular mesh of Ω: one node x per subdoman Ω not countng the nodes located on Ω. Let θ mn be the mnmum of all angles of mesh T. No node ncludng the nodes located on Ω of the coarse mesh has more than K neghbors. Let d be the length of the largest edge orgnatng from node x n the mesh T. Let H h, be the length of the shortest heght through x of any trangle n the coarse mesh T that connects to x. We also set H h, as the mnmum of H h,j over and ts drect neghbors n mesh T. We suppose that for each subdoman U, there exsts r > 0 such that U s star-shaped wth respect to any pont n the ball Bx, r. We also suppose r H h, 4K+1 and r H h, /. For all, we set l = 1 πr Bx,r uxdx = 1 π B0,1 ux + r ydy. We suppose that for each U there exsts an open layer L contanng U, a vector feld X contnuous on L U, C on L U such that DX xx x = 0, X x = 1, and ε 0 > 0 such that for all postve ε < ε 0 and for all ˆx n U, ˆx + εx ˆx U and ˆx εx ˆx / U. We set, for all postve δ, U δ = {x U, dstx, U < δ }, and V δ = {ˆx + sx ˆx, ˆx U, 0 < s < δ }. We assume there exst ˆR > 0, θ X, 0 < θ X π/, and δ 0, 0 < δ 0 ˆR sn θ X such that V ˆR L U and U δ V δ / sn θ X for all postve δ δ 0. Set R := 1/ dvx. We suppose δ 0 > δ l. We now state our man result, the exstence of a stable decomposton of H 1 0 Ω whose upper bound s ndependant of maxh mn H. Theorem 1. For u n H 1 0 Ω, there exsts a stable decomposton u 0 N of u,.e. u = N =0 u, u 0 n P 1 T and u H 1 0 Ω such that

4 4 Martn Gander, Laurence Halpern, and Kévn Santugn Repquet u L Ω C u L Ω, where C = C C 1 C and max r H C 1 = h, K 5 6π max d r + π tan θ mn 1 K K + 1 max r H h, max r H h,, C = + 8λ N c max λ N c max max H r wth λ beng a unversal constant. ˆR δ l max R + 1 δ s max ˆR δ l max R δ s max H 4 r ˆR δ s sn θ X r δ s ˆR sn θ X 4 1 H r H4 r 4 Note that the condton r H h, 4K+1 ensures that 1 K K + 1 max r /H h, max r /H h, remans postve., 3 Proof of Theorem Constructng the fne component We begn by establshng a stable decomposton when there s no coarse mesh. Lemma 1. Let u be n H0 1 Ω. Then, there exst u 1 N, u n H0 1 Ω such that u = N u, and u L Ω u L Ω + N 8λ N c ˆR δl ˆR R δ s δ s sn θ u L U X N + 8λ N c ˆR δl 1 R δ s ˆR u L sn θ U, X where λ s a unversal constant that depends only on the dmenson. We further have, for all η > 0, δ s 1

5 Non shape regular doman decomposton 5 u L Ω u L Ω + 8λ N c η + λ 3 N c 1 H r + 1 N H 4 r N 1 + ˆR R δl δ s 1 + ˆR δl r R δ s ˆR sn θ X δ s 4 1 H r δ s ˆR δ s sn θ u L U X H4 r 4 u L U N η πλ N c ˆR δl H R δ s ˆR l u. sn θ X Proof. We follow the proof of [Gander et al., 011, Th. 4.6]. Let ρ be a C non negatve functon whose support s ncluded n the closed unt ball of R and whose L 1 norm s 1. Let ρ ε x = ρx/ε/ε for all ε > 0. Let h be the characterstc functon of the set {x R, dstx, U < δ /}. Let φ = ρ δ/ h. The functon φ s equal to 1 nsde U, vanshes outsde of {x R, dstx, U < δ }, and φ ρ L1 R /δ. For n N [1, N], 1 let ψ = φ k=1 1 φ k. We have 0 ψ 1, ψ zero n Ω\Ω and ψ = 1 n Ω. Set u = ψ u. The functon u s n H0 1 Ω and u = u. Followng the proof of [Gander et al., 011, Lemma 4.3], we get N ψ x N C 1 N φ x. Therefore, for all x n Ω, ψ x 8N c 1 ρ L 1 R 1 Ω\U x δ. Snce u L Ω u L Ω dx + Ω ux ψ x dx, we get u L Ω u L Ω dx+4λ N c 1 N U 1{dstx, U < δ wth λ := ρ L1 R. To get 1, we apply Lemma 4.5 n Gander et al. [011] to each U, and to obtan, we apply Lemma 5.10 from the same reference. To obtan a stable decomposton wth a coarse component, we want to construct u 0 n P 1 T such that for all, l u 0 = l u. l} ux δ s dx,

6 6 Martn Gander, Laurence Halpern, and Kévn Santugn Repquet 3. Constructng the coarse component To construct u 0, we follow the deas of [Gander et al., 011, 5.]. Frst, we defne a specal norm. Defnton 1. Let T be the coarse mesh of doman Ω. Let V be the set of pars of neghborng nodes n T, and B be the set of boundary nodes 1 of T. We defne V,B : R N R +, y y j,j V y + y. B When u s n P 1 T H 1 0 Ω, set u V,B := ux 1 N V,B, where the x are the nteror nodes of the mesh T. Lemma. For u n H 1 0 Ω, there exsts u 0 n P 1 T H 1 0 Ω such that, for all n {1,..., N}, l u 0 = l u and u 0 L Ω 1 tan θ mn 1 + max r H h, K 5 6π max d r + π 1 K K + 1 max r H h, max r H h,. Proof. The results of [Gander et al., 011, Lemmas 5.6,and 5.8] stand wthout modfcatons. Therefore u 0 exsts, and we have u 0 L Ω max H h, tan θ mn 1 K K + 1 max r H h, max r H h, u V,B. Note that the condton r H h, 4K+1 ensures that 1 K K + 1 max r /H h, max r /H h, remans postve. It remans to compare u V,B and u L Ω. We need to adapt the proof of [Gander et al., 011, Lemma 5.7]. We can suppose wthout any loss of generalty that u s n C Ω. Let, j n {1,..., N} be ndces of neghborng nodes of T. Let d j = x x j, and d j = d j. We have for all, j V r 1 The nodes that are located on Ω are not numbered among {1,..., N}, and B contans only the nodes whch are neghbor to a node located on Ω.

7 Non shape regular doman decomposton 7 l u l j u = 1 π B0,1ux + r y ux j + r j ydy 1 π B0,1 d j + r r j π d j + r r j π 0 u tx + r y + 1 tx j + r j y x x j + r r j y dtdy B0,1 0 T,j uy u tx + r y + 1 tx j + r j y dtdy 0 1{ y tx 1 tx j tr + 1 tr j } tr + 1 tr j dtdy, where the tube T,j s the convex hull of Bx, r Bx j, r j. We get max y R 0 = max s,σ R = max = 1{ y tx 1 tx j tr + 1 tr j } tr + 1 tr j dt 0 s [ r j,d j+r ] max s [ r j,d j+r ] 1{ s td j + σ tr + 1 tr j } tr + 1 tr j dt 1{ s td j tr + 1 tr j } 0 tr + 1 tr j dt d j r j + sr r j mnr, r j d j r r j. Snce d j H h, 4 maxr, r j, l u l j u 5d j /6π mnr, r j u L T. j If s n the boundary set of the coarse mesh, then the node x s neghbor to a node x located on Ω. Note that les outsde of the range {1,..., N}. Usng [Gander et al., 011, Eqs 5.7 and 5.9], we get l u B B 4 x x πr T ux dx + Kπ u L Ω, 3 where T s the convex hull of Bx, r Bx, r. We sum l u l j u 5d j /6π mnr, r j u L T j over all, j n the neghbor set and combne t wth equaton 3. Snce maxr, r j H h, / mnh h,, H h,j /, no pont can belong to more than K tubes T,j or T. Therefore u V,B K 5 max d /r /6π + π u L Ω. Ths concludes the proof. To prove Theorem 1, we use Lemma to construct the coarse component u 0. We then apply Lemma 1 to u u 0 to get the fne components u. The terms n l u vansh.

8 8 Martn Gander, Laurence Halpern, and Kévn Santugn Repquet Concluson We have proven the exstence of a stable decomposton of the Sobolev space H 1 0 Ω n the presence of a coarse mesh when the doman decomposton s only guaranteed to be locally shape regular. We provded an explct upper bound for the stable decomposton that depends nether on max H / mn H, nor on the number of subdomans. Establshng the exstence of a stable decomposton wth a unform upper bound that does not explode when max H / mn H does would not have been possble wthout the explct upper bounds provded n Gander et al. [011]. Ths shows that dervng such explct upper bounds can be mportant for problems arsng naturally n applcatons,.e. load balanced doman decompostons wth local refnement. References Tony F. Chan and Tarek P. Mathew. Doman decomposton algorthms. In Acta Numerca 1994, pages Cambrdge Unversty Press, Maksymlan Dryja and Olof B. Wdlund. An addtve varant of the Schwarz alternatng method for the case of many subregons. Techncal Report 339, also Ultracomputer Note 131, Department of Computer Scence, Courant Insttute, Martn Gander, Laurence Halpern, and Kévn Santugn-Repquet. Contnuous Analyss of the Addtve Schwarz Method: a Stable Decomposton n H 1. Submtted, 011. URL hal /fr/. Alfo Quarteron and Alberto Vall. Doman Decomposton Methods for Partal Dfferental Equatons. Oxford Scence Publcatons, H. A. Schwarz. Über enen Grenzübergang durch alternerendes Verfahren. Verteljahrsschrft der Naturforschenden Gesellschaft n Zürch, 15:7 86, May Barry F. Smth, Petter E. Bjørstad, and Wllam Gropp. Doman Decomposton: Parallel Multlevel Methods for Ellptc Partal Dfferental Equatons. Cambrdge Unversty Press, Andrea Tosell and Olof Wdlund. Doman Decomposton Methods - Algorthms and Theory, volume 34 of Sprnger Seres n Computatonal Mathematcs. Sprnger, 004. Jnchao Xu. Iteratve methods by space decomposton and subspace correcton. SIAM Revew, 344: , December 199.