Dan Stefanica

Comptes Rendus Mathématique

1/-rature.esecpar ailleurs.uritionC. R. Acad. Sci. Paris, Ser. I 339 (2004) 739–743http://france.elsevier.com/direct/CRASSNumerical AnalysisLower bounds for additive Schwarz methods with mortarsDan StefanicaBaruch College, City University of New York, One Bernard Baruch Way, Box B 6-230, New York, NY 10010, USAReceived 3 March 2004; accepted after revision 3 September 2004Available online 27 October 2004Presented by Olivier PironneauAbstractWe establish lower bounds for thecondition number of overlapping additive Schwarz algorithms for elliptic problems discretized by mortar finite elements. These bounds coincide, up to constants, with the classical upper bounds from the liteThe optimality of the condition number stimates is thus established.To cite this article: D. Stefanica, C. R. Acad. Sci. Paris,Ser. I 339 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméBornes inférieures pour méthode de Schwarz additives et éléments finis avec joints.On détermine des bornes inférieurde conditionnement d’algorithmes de type Schwarz additif pour des problèmes elliptiques discrétisés par éléments finis avjoints. Ces limites sont identiques à des constantes multiplicatives près, aux bornes supérieures classiques établiesL’optimalité du conditionnement est ainsi démontré.Pour citer cet article : D. Stefanica, C. R. Acad. Sci. Paris, Ser. I 339(2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Version française abrégée1Nous montrons que les bornes supérieures [5] du conditionnement d’algorithmes additifs de Schwarz pol’équation de Poisson avec condition aux limites de Neumann–Dirichlet sur un domaineΩ sont optimales. Ceproblème est discrétisé en utilisant un espaceV d’éléments finis du premier ordre avec joints sur une decompospolygonale{Dk}k=1:K deΩ .Pour construire un opérateur de Schwarz pour le problème discret (1), nous utilisons une décomposition derecouvrement{Ωi}i=1:N deΩ , avec chevauchementδ, et un espaceV0 fait d’élémentsP 1 ou Q1, de diamètreH .E-mail address: dstefan@math.mit.edu (D. Stefanica).1 Les références font référence à la version anglaise.1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2004.09.016740 D. Stefanica / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 739–743tionsementsme,ellble meshnded toloss-singh mixedt ellipticislu-ethecetrsalLa restriction deV àΩi est notéeVi et la restriction auxVi × Vi deaΓ , le produitH 1 décomposé sur{Dk}k=1:K ,est notéeaΓi , i = 1 : N . L’opérateur de Schwarz additifTas est reconstitué à partir des opérateurs des solulocalesTi correspondant àaΓi comme suit :Tas = T0 + T1 + · · · + TN ; voir aussi (2).Pour un petit chevauchement (δ/H < 1/4), il est bien connu queκ(Tas)  C(1 + H/δ) ; voir [4] pour leséléments finis continus et [5] pour les éléments avec joints. Nous montrons queκ(Tas)  C(1 + H/δ) pour leséléments avec joints.De (8), il suit immédiatement queλmax(Tas)  1. Pour montrer queλmin(Tas)  Cδ/H , voir (9), nous con-struisons une fonctionv ∈ V telle que pour toute décompositionv = v0 + v1 + · · · + vN , vi ∈ V , l’inégalitéaΓ (v, v)  C δH(aΓ1 (v1, v1) + aΓ2 (v2, v2)) est satisfaite.1. IntroductionMortar finite elements, first introduced by Bernardi, Maday and Patera in [1], are nonconforming finite elthat allow for a geometrically nonconforming decompositi n of the computational domain and, at the same tifor the optimal coupling of different variational approximations in different subregions. Mortar elements are wsuited for parallel computing and have several advantages over the conforming finite elements, e.g., flexigeneration and straightforward local refinement. Many domain decomposition algorithms have been extemortar element discretization in order to take advantage of the inherent flexibility of the mortars, without anyin convergence properties.We establish here that the upper bounds [5] on the conditionnumbers of the overlappingadditive Schwarz algorithms for mortars are sharp. To do so, we prove estimates for the lower bounds on these condition numbers umethods pioneered by Brenner [2]. We restrict our attention to the two dimensional Poisson equation witNeumann–Dirichlet boundary conditions, discretized by first order mortar elements on a polygonal domainΩ .The extensions to three dimensional curvilinear domains, higher order mortar elements, and self-adjoinequations are standard.Let f ∈ L2(Ω). We look for a solutionu ∈ H 1(Ω) to the problem−u = f on Ω , with u = 0 on ∂ΩDand∂u/∂n = 0 on ∂ΩN , where∂Ω = ∂ΩN ∪ ∂ΩD and∂ΩD has positive Lebesgue measure. This problemdiscretized using a mortar finite element spaceV on a nonoverlapping polygonal partition {Dk}k=1:K of Ω ; seeSection 2 for details on the definition ofV . The resulting discrete problem consists of findinguh ∈ V such thataΓ (uh, vh) = f (vh), ∀vh ∈ V, (1)whereaΓ (wh, vh) = ∑k ∫Dk ∇wh · ∇vh dx andf (vh) = ∫Ω f vh dx. The existence and uniqueness of the sotion uh of (1) follows from the Lax–Milgram lemma and a Friedrichs inequality for mortar finite elements.To assemble the additive Schwarz preconditioner, w introduce an overlapping decomposition{Ωi}i=1:N of Ω .Let δ be the maximum overlap between the subdomains{Ωi}i=1:N . Let Vi ⊂ V , i = 1 : N , be local spaces madof the mortar functionsvi ∈ V which vanish at all the genuine degrees of freedom outsideΩi . A coarse spaceV0made ofP 1 or Q1 elements of diameterH is introduced onΩ . It may be unstructured with respect to bothmortar partition and the overlappingpartition. Without a coarse space, thenumber of iterations to convergenwould depend on the number of overlapping subdomains.Let Ii :Vi → V be embedding operators,i = 0 : N , let aΓ0 :V0 × V0 → R be theH 1 inner-product, and leaΓi :Vi × Vi → R be the restriction ofaΓ to Vi × Vi , i = 1 : N . The corresponding projection-like operatoT̃i : V → Vi are defined by:aΓi(T̃iv, vi) = aΓ (v, Iivi), ∀vi ∈ Vi, v ∈ V, i = 0 : N. (2)Let Ti = Ii T̃i andTas = T0 + T1 + · · · + TN . The additive Schwarz method for (1) consists of solvingTasu = gaswithout further preconditioning by a conjugate gradient algorithm; heregas can be computed by solving locproblems without previous knowledge ofu.D. Stefanica / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 739–743 741blem,ortars,canfbyrdes ofrnctiondi-oints-se-For small overlap, i.e.,δ/H < 1/4, the following upper bound for the condition numberκ(Tas) of the additiveSchwarz operator was proved in [4] for conforming finite elements, and for mortar finite elements in [5]:κ(Tas)  C(1+ Hδ). (3)Here, and throughout the rest of the paper,C denotes a constant independent of any parameters of the proe.g.,h, H , δ, K, N .2. Special choice of mortar functionsThe bound (3) was shown in [2] to be sharp for conforming finite elements. To prove a similar result for mwe introduce a special partition of the computational domain with a structured coarse space. Our constructionbe extended to the nonconforming case with unstructured overlapping subdomains and coarse space.Let {Dk}k=1:K be a geometrically conforming morta decomposition of a polygonal domainΩ into rectangles odiameter of orderH . Let {Ωk}k=1:K be the overlapping subdomains, whereΩk is the smallest rectangle obtainedincluding all the nodes at distanceδ/2 from∂Dk . The diameter of the overlapping subdomains is thus of ordeH .The coarse spaceV0 is theQ1 finite element space defined on the coarse grid determined by the corner no{Dk}k=1:K .The restriction of the mortar finite element spaceV to any rectangleDk is a Q1 finite element function on amesh of diameterh. Weak continuity is required acrossΓ , the interface between the subregions{Dk}k=1:K . Wechoose a set of edges of{Dk}k=1:K , called nonmortars, whichform a disjoint partition ofΓ . For each nonmortasideγ there exists exactly one side opposite to it, which is called a mortar side. The jump of a mortar fuw ∈ V across any nonmortarγ must be orthogonal to a space of test functionsΨ (γ ):∫γ[w]ψ ds = 0, ∀ψ ∈ Ψ (γ ). (4)In [1], Ψ (γ ) was chosen to be a subspace of codimension two ofV (γ ), the restriction ofV to γ . It consists ofcontinuous, piecewise linear functions onγ that are constant in the first and last mesh intervals ofγ . A biorthogonalmortar space was introduced in [3], corresponding to discontinuous piecewise linear test functions.For either mortar space, the values ofw at the end points ofγ cannot be determined from the mortar contions (4) and therefore are genuine degrees of freedom. We denote byπα,β : L2(γ ) → V (γ ) the mortar projectionoperator corresponding to (4), which takes valuesα andβ , respectively, at the end pointsA andB of γ .It is easy to see thatπα1,β1(w1) − πα2,β2(w2) = πα1−α2,β1−β2(w1 − w2), (5)for all wi ∈ V , αi,βi ∈ R, i = 1,2. Note thatπ0,0, the mortar projection operator which vanishes at the end pof γ , is stable inL2(γ ), H 10 (γ ), and, by interpolation, inH1/200 (γ ); see, e.g., [3].To establish an upper bound onλmin(Tas), we introduce a special mortar functionv ∈ V . Following a construction from [2], let v̂ ∈ H 1([0,1]) be a piecewise linear function on the uniform mesh of size 1/8 on[0,1] such thatv̂ vanishes outside(1/4,3/4) and is orthogonal in theL2([0,1]) inner product to any linear function. We choov ∈ V to be discrete harmonic on every subregionDk and vanish onΓ except onζ andγ , opposite mortar andnonmortar sides of subregionsD1 andD2, respectively. We also choose the local mesh onD1 such that its restriction to ζ is a dyadic subdivision ofζ . Let A andB be the common end points ofζ andγ , and letQ1 andQ2 bethe mesh nodes onAB obtained by scaling from the nodes 1/4 and 3/4 on [0,1]. The restrictionvζ of v to ζ is742 D. Stefanica / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 739–743tofrznobtained by scalinĝv, while vγ , the restriction ofv to γ , is obtained via a mortar projection with 0 values aAandB, i.e.,vγ = π0,0(vζ ). Note thatvζ is defined by scalinĝv and has compact supportQ1Q2 ⊂ [0,1]. Therefore,‖vζ ‖2H1/200 (ζ )= |vζ |2H1/2(ζ ) =CH‖vζ ‖2L2(ζ ) =CH‖vζ ‖2L2(Q1Q2). (6)Sincev is discrete harmonic and using the stability properties ofπ0,0, we find thataΓ (v, v) = ‖vζ ‖2H1/200 (ζ )+ ‖vγ ‖2H1/200 (γ ) C‖vζ ‖2H1/200 (ζ )= CH‖vζ ‖2L2(Q1Q2). (7)3. Lower bound on the condition numberWe recall thatλmax(Tas) = maxv∈V,v =0aΓ (v, v)minv=I0v0+···+IN vN∑Ni=0 aΓi (vi , vi)(8)λmin(Tas) = minv∈V,v =0aΓ (v, v)minv=I0v0+···+IN vN∑Ni=0 aΓi (vi , vi). (9)The lower boundλmax(Tas)  1 follows immediately by choosingv in the spaceV1 and a decomposition ofvwith v1 = v andvi = 0 for i = 1.To prove an upper bound of the formλmin(Tas)  Cδ/H , it is enough to show that, for any decompositionthe special functionv ∈ V from Section 2 of the formv = v0 + v1 +· · ·+ vN , with vi ∈ Vi , i = 0 : N , the followinginequality holds:aΓ (v, v)  C δH(aΓ1 (v1, v1) + aΓ2 (v2, v2)). Using (7), we only need to show that‖vζ ‖2L2(Q1Q2)  Cδ(aΓ1 (v1, v1) + aΓ2 (v2, v2)). (10)Let vi,ζ andvi,γ be the restrictions ofvi to ζ andγ , respectively, fori = 1,2. Note thatQ1Q2 belongs to onlytwo overlapping subdomains,Ω1 andΩ2, sinceδ/H  1/4. Therefore, onQ1Q2, vζ = v0 + v1 + v2. Recall thatv0 is linear onζ , and that, by construction,vζ is L2-orthogonal onζ to any linear function. Using the Schwainequality, we obtain:‖vζ ‖2L2(Q1Q2)  2(‖v1,ζ ‖2L2(Q1Q2) + ‖v2,ζ ‖2L2(Q1Q2)). (11)By definition, v2 ∈ V2 vanishes on∂Ω2, i.e., at distanceδ/2 from Q1Q2. From the Friedrichs inequality oD1 ∩ Ω2, we find that‖v2,ζ‖2L2(Q1Q2)  CδaΓ2 (v2, v2). (12)Let Iγ :L2(AB) → V (γ ) be the piecewise linear interpolation ontoγ . Thenv1,ζ =(v1,ζ − Iγ (v1,ζ )) + (Iγ (v1,ζ ) − v1,γ ) + v1,γ . (13)From a classical interpolation inequality, we find that∥∥v1,ζ − Iγ (v1,ζ )∥∥2L2(Q1Q2)  Ch|v1|2H1(D1). (14)Sincev1 ∈ V1 vanishes at distanceδ/2 fromQ1Q2, we use Friedrichs inequality to obtain:‖v1,γ ‖2L2(Q1Q2)  CδaΓ1 (v1, v1). (15)To estimate theL2(Q1Q2) norm of Iγ (v1,ζ ) − v1,γ , recall thatv1,γ = πv1,γ (A),v1,γ (B)(v1,ζ ). SinceIγ (v1,ζ )is piecewise linear onγ , its mortar projection onγ with valuesv1,ζ (A) andv1,ζ (B) at A andB is itself, i.e.,D. Stefanica / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 739–743 743,.adant PSC-ernardide669.2.uterIγ (v1,ζ ) = πv1,ζ (A),v1,ζ (B)(Iγ (v1,ζ )). Let [v1] = v1,ζ − v1,γ be the jump ofv1 acrossγ . Applying (5) repeatedlywe find thatIγ (v1,ζ ) − v1,γ = π[v1](A),[v1](B)(Iγ (v1,ζ ) − v1,ζ) = π0,0(Iγ (v1,ζ ) − v1,ζ ) + π[v1](A),[v1](B)(0). (16)From the stability of the mortar projectionπ0,0 and an interpolation estimate, we find that∥∥π0,0(Iγ (v1,ζ ) − v1,ζ )∥∥2L2(Q1Q2)  C∥∥v1,ζ − Iγ (v1,ζ )∥∥2L2(AB)  Ch|v1|2H1(D1). (17)For biorthogonal mortars, the mortar projection is a local operator, and thusπ[v1](A),[v1](B)(0) vanishes onQ1Q2. For classical mortars, the nodal values ofπ[v1](A),0(0) andπ0,[v1](B)(0) on Q1Q2 decline exponentiallyFrom (5), it is easy to see thatπ[v1](A),[v1](B)(0) = π[v1](A),0(0) + π0,[v1](B)(0). We use Friedrichs inequality andSobolev-type inequality for finite element functions to bound[v1](A) and[v1](B) in terms of theH 1 seminorm ofv1 onD1 andD2. We obtain:∥∥π[v1](A),[v1](B)(0)∥∥2L2(Q1Q2)  Ch(|v1|2H1(D1) + |v1|2H1(D2))  ChaΓ1 (v1, v1). (18)We conclude from (16)–(18) that‖Iγ (v1,ζ ) − v1,γ ‖2L2(Q1Q2)  ChaΓ1 (v1, v1), and, using (13)–(15), that‖v1,ζ‖2L2(Q1Q2)  CδaΓ1 (v1, v1). (19)From (11), (12) and (19), we find that‖vζ ‖2L2(Q1Q2)  Cδ(aΓ1 (v1, v1) + aΓ2 (v2, v2)). Thus (10) holds true antherefore, as explained above,λmin(Tas)  Cδ/H . Since 1 λmax(Tas), we conclude thatC(1+ H/δ)  κ(Tas).AcknowledgementsCe travail a été soutenu par le Grant NSF-DMS-0103588 de la National Science Foundation et par le GrCUNY 63461-00-32 de la Research Foundation de la CUNY. L’auteur remercie les professeurs Christine Bet Yvon Maday de leur hospitalité lors d’une visite à l’Université Pierre et Marie Curie.References[1] C. Bernardi, Y. Maday, A. Patera, A newnonconforming approach to domain decomposition:the mortar element method, in: CollègeFrance Seminar, Pitman, 1994.[2] S. Brenner, Lower bounds for two-level additive Schwarz preconditioners with small overlap, SIAM J. Sci. Comput. 21 (2000) 1657–1[3] B. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier, SIAM J. Numer. Anal. 3 (2000) 989–101[4] M. Dryja, O. Widlund, Domain decomposition algorithms with small overlap, SIAM J. Sci. Comput. 15 (1994) 604–620.[5] O. Widlund, Two-level Schwarz algorithms, using overlapping subregions, for mortar finite element methods, Technical report, CompScience Department, Courant Institute of Mathematical Sciences, in preparation.

Lower bounds for additive Schwarz methods with mortars

Stefanica, Dan

Numérisation de Documents Anciens Mathématiques

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2004.09.016.pdf

Lower bounds for additive Schwarz methods with mortars

Abstract

Similar works

Full text

Available Versions

Comptes Rendus Mathématique

Numérisation de Documents Anciens Mathématiques