on shape regular doman decompostons: an analyss usng a stable decomposton n 1 0 Martn J. Gander 1, Laurence alpern, and Kévn Santugn Repquet 3 1 Unversté de Genève, Secton de Mathématques, Martn.Gander@unge.ch Unversté Pars 13, LAGA UMR 7539 CRS halpern@math.unv-pars13.fr 3 Unversté Bordeaux, IMB, UMR 551 CRS, IRIA, F-33400 Talence, France. Kevn.Santugn@math.u-bordeaux1.fr hal-00607043, verson - 10 Feb 01 Summary. In ths paper, we establsh the exstence of a stable decomposton n the Sobolev space 0 1 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 ndependent 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 n []. Even before the seres of nternatonal conferences on doman decomposton methods started, Dryja and Wdlund presented a varant of the hstorcal alternatng Schwarz method nvented by Schwarz n [5] to prove the Drchlet prncple on general domans. Ths varant, called the addtve Schwarz method, has the advantage of beng symmetrc for symmetrc problems, and t also contans a coarse space component. In a fully dscrete analyss n [], Dryja and Wdlund 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 lnearly as a functon of δ, where s the subdoman dameter, and the overlap between subdomans. 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 [6, 4, 7], or the monographs [8, 1], 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
Martn J. Gander, Laurence alpern, and Kévn Santugn Repquet hal-00607043, verson - 10 Feb 01 be consderably larger than the smallest subdoman, and therefore the subdoman dameter and overlap parameters depend strongly on the subdoman ndex. In such a doman decomposton, the generc rato δ from the classcal convergence result of the addtve Schwarz method can be gven at least two dfferent meanngs: let refer to the dameter of subdoman number and δ refer to the wdth of the overlap around subdoman number. Then n the classcal convergence result from [], one could replace the generc rato δ by max mn δ, but ths s lkely to lead to a very pessmstc estmate for the condton number growth. The general analyss of the addtve Schwarz method based on a shape regular decomposton does unfortunately not permt to answer the queston f the condton number growth for a locally shape regular decomposton s n fact only lnear n the quantty max δ, whch s much smaller than max mn δ n the case of subdomans and overlaps of wdely dfferent szes, a case of great nterest n applcatons. In [3], we establshed the exstence of a stable decomposton n the contnuous settng wth an explct upper bound and a quanttatve defnton of shape regularty n two spatal dmensons. The explct upper bound s also lnear n the generc quantty 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. avng 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 mn, and we provde an explct upper bound whch s lnear n max /δ for problems n two spatal dmensons. To get ths result, only a few of the nequaltes establshed n [3] 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 [3]. We state frst n our man theorem along wth the assumptons we make on the doman decomposton. We then prove the man theorem n 3 n two steps: frst, we show n Lemma 1 how to construct the fne component n 3.1, whch s an extenson of the result [3, Theorem 4.6] for the case where subdoman szes and overlaps δ can strongly depend on the subdoman ndex. The major contrbuton s however n the second step, presented n Lemma n 3., where we show how to construct the coarse component n the case of strongly varyng and δ between subdomans. Ths result s a substantal generalzaton of [3, Lemma 5.7]. Usng these two new results, and the remanng estmates from [3] whch are stll vald, we can prove our man theorem. We fnally summarze our results n the conclusons n 4. 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.
on shape regular doman decomposton 3 hal-00607043, verson - 10 Feb 01 The U 1 are a non-overlappng doman decomposton of Ω,.e., satsfy =1 U = Ω and U U j = /0 when j. The U are bounded connected open sets of R and for all subdomans U the measure of U \U s zero. We set := damu. Two dstnct subdomans U and U j are sad to be neghbors f U U j /0. For each subdoman U, let δ > 0 be such that δ mn j,u U j =/0 dstu,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 = mn j,u U j /0 δ j and δ l = max j,u U j /0 δ j. The doman decomposton has c colors: there exsts a partton of [1,] nto 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. By P 1 T, we denote the standard fnte element space of contnuous functons that are pecewse lnear over each trangular cell of T. Let θ mn be the mnmum of all angles of mesh T. o 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, 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, as the mnmum of 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 starshaped wth respect to any pont n the ball Bx,r. We also suppose r h, 4K+1 and r h, /. We also assume the exstence of both a pseudo normal X and of a pseudo curvature radus R for the doman U,.e., 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+ε X ˆx U and ˆx ε X 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 snθ X such that V ˆR L U and U δ V δ /snθ X for all postve δ δ 0. Set R := 1/ dvx. We suppose δ 0 > δ l We fnally defne, for all, the lnear form on 0 1 Ω by l u := 1 πr uxdx = 1 ux + r ydy. Bx,r π B0,1 We can now state our man theorem, namely the exstence of a stable decomposton of 0 1Ω whose upper bound s ndependent of max mn. Ths theorem therefore leads to a substantally sharper condton number estmate n the mportant case.
4 Martn J. Gander, Laurence alpern, and Kévn Santugn Repquet of an only locally shape regular decomposton, and s a major mprovement of [3, Theorem 5.1], whch only consdered shape regular decompostons, albet at the contnuous level, n contrast to []. Theorem 1. For u n 1 0 Ω, there exsts a stable decomposton u 0 of u,.e., u = =0 u, u 0 n P 1 T 1 0 Ω and u 1 0 Ω such that =0 u L Ω C u L Ω, hal-00607043, verson - 10 Feb 01 where C = C 1 + 1 +C 1 C and 1 C 1 = tanθ mn 1 + max r K 5 h, 6π max d r + π 1 K + 1 + 4K + 1max r max h, r, h, C = + 8λ c 1 1 + max + 8 3 λ c 1 1 + max r ˆR max R ˆR max R δ l δ l max max max + 1 1 4 + 4 r ˆR snθ X δ s 4 r ˆR snθ X 1 r 4 r 4, wth λ a unversal constant dependng only on the dmenson, and beng smaller than 6 n the two dmensonal case we consder here. ote that the condton r h, 4K+1 mples that the denomnator of C 1 s postve. The value of C s also always postve. 3 Proof of Theorem 1 The proof s based on the contnuous analyss n [3], but two results must be adapted to the stuaton of only locally shape regular decompostons: we frst show n 3.1 how to construct the fne component, whch s a techncal extenson of the result [3, Theorem 4.6] for the case where subdoman szes and overlaps δ can strongly depend on the subdoman ndex. Second, we explan n 3. the constructon of the coarse component n the case of strongly varyng and δ between subdomans, whch s a non-trval generalzaton of [3, Lemma 5.7]. Wth these two new results, and the remanng estmates from [3], the proof can be completed. 3.1 Constructng the fne component We begn by establshng a stable decomposton when there s no coarse mesh.
on shape regular doman decomposton 5 Lemma 1. Let u be n 0 1Ω. Then, there exst u 1, u n 0 1Ω such that u = =1 u, and =1 u L Ω u L Ω + 8λ c 1 1 + ˆR δ l ˆR =1 R δ s snθ u L X U + 8λ c 1 1 + ˆR δ l 1 =1 R δ s u ˆR L snθ U, X where λ s the unversal constant of Theorem 1. We further have, for all η > 0, 1 hal-00607043, verson - 10 Feb 01 =1 u L Ω u L Ω + 8λ c 1 1 + ˆR δ l R =1 81 + η + λ c 1 3 1 + ˆR δ l =1 R δ s + 1 1 4 + 4 r r 4 δ s 1 r + 81 + 1 η πλ c 1 1 + ˆR δ l =1 R δ s ˆR snθ X r ˆR snθ X u L U 4 r 4 u L U l u ˆR. snθ X Proof. We follow the proof of [3, 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 L R ρ L 1 R ;R, /δ. ere, ρ L 1 R ;R, means R =1 ρ dx. For n [1,], let ψ = φ 1 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 1 0 Ω and u = u. Followng the proof of [3, Lemma 4.3], we get =1 ψ x C 1 =1 φ x. Therefore, for all x n Ω, =1 ψ x 8 c 1 ρ L 1 R ;R, =1 1 Ω \U x, where 1 O s the ndcator functon for the set O. Snce u L Ω u L Ω + Ω ux ψ x dx, we get =1 u L Ω u L Ω + 4λ c 1 =1 δ U 1 {dstx, U <δ l ux } dx,
6 Martn J. Gander, Laurence alpern, and Kévn Santugn Repquet wth λ := ρ L 1 R ;R,. Usng the W 1,1 R functon ρx = 1 x, we obtan the estmate λ = 6. To get 1, we apply Lemma 4.5 n [3] 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. 3. Constructng the coarse component To construct u 0, we follow the deas of [3, 5.]. Frst, we defne a specal norm. Defnton 1. Let T be the coarse mesh of the doman Ω. Let B be the set of ndces of the nodes of T located on the boundary 4 Ω. Let B be the set of the ndces of the nodes that are neghbors to the nodes wth ndex n B. Let V be the set of pars of ndces of neghborng nodes n T whch are not on Ω. We defne hal-00607043, verson - 10 Feb 01 V,B : R R +, y, j V y y j + y. B When u s n P 1 T 1 0 Ω, set u V,B := ux 1 V,B, where the x are the nteror nodes of the mesh T. Lemma. For u n 0 1Ω, there exsts u 0 n P 1 T 0 1 Ω such that, for all n {1,...,}, l u 0 = l u and u 0 L Ω 1 tanθ mn 1 + max r K 5 h, 6π max d r + π 1 K + 1 + 4K + 1max r max h, r. h, Proof. The results of [3, Lemmas 5.6,and 5.8] stand wthout modfcatons. Therefore u 0 exsts, and we have r h, u 0 L Ω 1 1 + max tanθ mn 1 K + 1 + 4K + 1max r max h, r u V,B. h, ote that the condton r h, 4K+1 mples the second denomnator n the above equaton postve. It remans to compare u V,B and u L. We need to adapt the proof of [3, Ω Lemma 5.7]. We can suppose wthout any loss of generalty that u s n C Ω. Let, j n {1,...,} be ndces of neghborng nodes of T. Let d j = x x j, and d j = d j. We have for all, j V 4 Because of the homogenous Drchlet condton on the boundary Ω, the nodes whose ndces are n B are not assocated to a degree of freedom, therefore B and {1,...,} have empty ntersecton.
on shape regular doman decomposton 7 l u l j u = 1 π ux + r y ux j + r j ydy B0,1 1 1 u tx + r y + 1 tx j + r j y π x x j + r r j y dtdy B0,1 0 d j + r r j 1 u tx + r y + 1 tx j + r j y π dtdy B0,1 0 d j + r r j 1 uy 1 { y tx 1 tx j tr +1 tr j } π T, j tr + 1 tr j dtdy, where the tube T, j s the convex hull of Bx,r Bx j,r j. We get 0 hal-00607043, verson - 10 Feb 01 1 max y R 0 1 { y tx 1 tx j tr +1 tr j } tr + 1 tr j dt 1 = max s,s R 0 = max s [ r j,d j +r ] max s [ r j,d j +r ] 1 { s td j +s tr +1 tr j } 1 0 tr + 1 tr j 1 { s td j tr +1 tr j } tr + 1 tr j dt s+r j d j r r j s r j d j +r r j = max 1 s [ r j,d j +r ] r r j = max s [ r j,d j +r ] = dt 1 tr + 1 tr j dt [ 1 tr + 1 tr j d j r j + sr r j mnr,r j d j r r j. Snce d j h, 4maxr,r j, we have l u l j u ] s+r j d j r r j s r j d j +r r j 5d j 6π mnr,r j u L T j. 3 If s n the boundary set of the coarse mesh, then the node x s neghbor to a node x located on Ω. ote that les outsde of the range {1,...,}. Usng [3, Eqs 5.7 and 5.9], we get l u B B 4 x x πr T ux dx + Kπ u L Ω, 4 where T s the convex hull of Bx,r Bx,r. We sum nequalty 3 over all, j n the neghbor set and combne the resultng nequalty wth equaton 4. Snce
8 Martn J. Gander, Laurence alpern, and Kévn Santugn Repquet maxr,r j h, / mn h,, h, j /, no pont can belong to more than K tubes T, j or T. Therefore, u V,B K 5max d /r /6π + π u L. Ths concludes the proof. Ω 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. 4 Concluson hal-00607043, verson - 10 Feb 01 We have proved the exstence of a stable decomposton of the Sobolev space 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 /mn, nor on the number of subdomans. Ths would not have been possble wthout the explct upper bounds provded n [3]. Ths shows that dervng such explct upper bounds can be mportant for problems arsng naturally n applcatons, e.g., load balanced doman decompostons wth local refnement. References [1] Tony F. Chan and Tarek P. Mathew. Doman decomposton algorthms. In Acta umerca 1994, pages 61 143. Cambrdge Unversty Press, 1994. [] 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 ote 131, Department of Computer Scence, Courant Insttute, 1987. [3] Martn J. Gander, Laurence alpern, and Kévn Santugn-Repquet. Contnuous Analyss of the Addtve Schwarz Method: a Stable Decomposton n 1. Submtted, 011. URL http://hal.archves-ouvertes.fr/ hal-0046006/fr/. [4] Alfo Quarteron and Alberto Vall. Doman Decomposton Methods for Partal Dfferental Equatons. Oxford Scence Publcatons, 1999. [5] ermann A. Schwarz. Über enen Grenzübergang durch alternerendes Verfahren. Verteljahrsschrft der aturforschenden Gesellschaft n Zürch, 15: 7 86, May 1870. [6] Barry F. Smth, Petter E. Bjørstad, and Wllam Gropp. Doman Decomposton: Parallel Multlevel Methods for Ellptc Partal Dfferental Equatons. Cambrdge Unversty Press, 1996. [7] Andrea Tosell and Olof Wdlund. Doman Decomposton Methods - Algorthms and Theory, volume 34 of Sprnger Seres n Computatonal Mathematcs. Sprnger, 004. [8] Jnchao Xu. Iteratve methods by space decomposton and subspace correcton. SIAM Revew, 344:581 613, December 199.