Anne Laure Bessoud

Françoise Krasucki

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 343 (2006) 279–282http://france.elsevier.com/direct/CRASS1/Numerical Analysis/Mathematical Problems in MechanicsQ-superlinear convergence of the GMRES algorithmfor multi-materials with strong interfaceAnne Laure Bessoud, Françoise KrasuckiLaboratoire de Mécanique et de Génie Civil, UMR 5508, Université Montpellier II, place Eugène-Bataillon, 34695 Montpellier cedex 5, FranceReceived 30 May 2006; accepted 21 June 2006Available online 21 July 2006Presented by Philippe G. CiarletAbstractWe consider a model non-classical transmission problem corresponding to a multistructure composed of two bodies bondedby a thin strong layer. By using a domain decomposition, the problem is reduced to an equation defined on the interface of theform (I − G)g = F . We prove that G is compact of Carleman class Cs , and hence the q-superlinearly convergence of the GMRESalgorithm (in exact arithmetic). To cite this article: A.L. Bessoud, F. Krasucki, C. R. Acad. Sci. Paris, Ser. I 343 (2006).© 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméConvergence q-superlinéaire de l’algorithme GMRES pour un multimatériau à liaison forte. On étudie un problèmemodèle non-classique de transmission, décrivant une multistructure composée de deux solides reliés par une jonction forte. Enutilisant une méthode de décomposition de domaines, le problème se ramène à une équation définie sur l’interface, de la forme :(I − G)g = F . On montre que G est compact de classe de Carleman Cs , et on en déduit la convergence q-superlinéaire de l’algo-rithme GMRES. Pour citer cet article : A.L. Bessoud, F. Krasucki, C. R. Acad. Sci. Paris, Ser. I 343 (2006).© 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.1. Position of the problemFor the sake of simplicity we shall consider a model problem, such as the heat conduction through two bodies,bonded by a thin strong layer whose thermal properties are large with respect to those of the two bodies. The sameresults hold for other situations, e.g. for linear elasticity or linear plates theories. With an asymptotic analysis, similarto that in [2,9,7], the thin strong layer ‘disappears’ from a geometrical point of view, but remains through a surfaceenergy which gives rise to a non-classical transmission condition. In order to state this limit problem, we denoteby Ω+ and Ω− the two bodies, which we assume to be open connected subsets of R3 with Lipschitz-continuousboundaries ∂Ω+ and ∂Ω− and we assume that S = ∂Ω+ ∩ ∂Ω− is a non-empty regular surface of positive measure.Let Ω be the interior of (Ω+ ∪ Ω−), with boundary ∂Ω . We assume that the following boundary conditions of mixedtype hold: the temperature is zero on Γ −0 ∪ Γ +0 and the heat flux is given on the part Γ +ϕ with Γ −0 = ∂Ω− ∩ ∂Ω ,E-mail addresses: Anne-Laure.bessoud@laposte.net (A.L. Bessoud), krasucki@lmgc.univ-montp2.fr (F. Krasucki).1631-073X/$ – see front matter © 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2006.06.031280 A.L. Bessoud, F. Krasucki / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 279–282Γ +0 ∪ Γ +ϕ = ∂Ω+ ∩ ∂Ω and ∂S ⊂ Γ −0 ∪ Γ +0 . For a function w defined on Ω , let w+ and w− denote the restrictionsof w to Ω+ and Ω−, respectively. Let us define the space V :V = {v = (v+, v−); v+ ∈ H 1(Ω+), v− ∈ H 1(Ω−), v+|S = v−|S ∈ H 10 (S) and v± = 0 on Γ ±0 } (1)whose associate norm is‖v‖2V = ‖∇v+‖2L2(Ω+) + ‖∇v−‖2L2(Ω−) + ‖∇τ v‖2L2(S),where ∇τ denotes the surface gradient operator. The problem can then be stated as follows:{Find u ∈ V so thatJ (u)  J (v), ∀v ∈ V, (2)whereJ (v) = 12a+(v+, v+) + 12a−(v−, v−) + 12as(v, v) − L+(v+) − L−(v−), (3)a±(v±, v±) =∫Ω±K±∇v±∇v± dΩ, (4)as(v, v) =∫SKs∇τ v∇τ v dS, (5)L+(v+) =∫Ω+f +v+ dΩ −∫Γ +ϕϕ+v dΓ ; L−(v−) =∫Ω−f −v− dΩ. (6)Under standard assumptions on the regularity of the loading and the positivity of the coefficients K± and Ks , we caneasily prove the existence and the uniqueness of a solution to this problem.Let us remark that the transmission conditions on S are u+ = u− and the non-classical one:K+ ∂u+∂n++ K− ∂u−∂n−= Ksτu. (7)2. Multi-domains formulationWe cannot apply directly the iterative methods used for multi-domains to take into account the transmission condi-tion (7) on the strong interface S. We propose a suitable domain decomposition method that reduces the problem to anintegral equation on S. As in e.g. [6], we solve at first a classical non-homogeneous Dirichlet problem or a mixed prob-lem on each subdomain with a boundary datum g on the interface S. To this end, let g ∈ H 10 (S) be given and definethe spaces V +g = {v ∈ H 1(Ω+); v = g on S, v = 0 on Γ +0 } and V −g = {v ∈ H 1(Ω−); v = g on S, v = 0 on Γ −0 }.Let u±g ∈ V ±g be such that:J±(u±g) J±(v±g), ∀v±g ∈ V ±g , (8)whereJ±(v±g) = 12a±(v±g , v±g) − L±(v±g ). (9)Then ug = (u+g , u−g ) is the solution to the global minimization problem (2) if and only if g is the (unique) solution tothe following problem:{Find g ∈ H 10 (S), such that for all g∗ ∈ H 10 (S)as(g, g∗) + a+(u+g , v+g∗) + a−(u−g , v−g∗) − L+(v+g∗) − L−(v−g∗) = 0. (10)Using a Green formula, we can see that the previous problem can be written as:⎧⎪⎨⎪⎩Find g ∈ H 10 (S), such that for all g∗ ∈ H 10 (S)as(g, g∗) =〈K+∂u+g+ + K−∂u−g− , g∗〉,(11)∂n ∂nA.L. Bessoud, F. Krasucki / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 279–282 281where 〈·, ·〉 denotes the duality pairing between H−1/2(S) and H 1/2(S) ∩ H 10 (S). Let us denote by G :H−1(S) →H 10 (S) the solution operator corresponding to the Dirichlet problem for the form as . Hence g solves (11) if and onlyif:g = G(K+∂u+g∂n++ K− ∂u−g∂n−). (12)Since the maps g → K±∂u±g /∂n± from H 10 (S) into H−1/2(S) are affine, we can writeG(K+∂u+g∂n++ K− ∂u−g∂n−)= Gg + F,whereGg = G(K+(∂u+g∂n+− ∂u+0∂n+)+ K−(∂u−g∂n−− ∂u−0∂n−)), F = G(K+∂u−0∂n++ K− ∂u−0∂n−),and u±0 denotes the solution of (9) for g = 0.The solution of problem (11) is hence reduced to solving the linear equation:Ag ≡ (I − G)g = F in H 10 (S). (13)The existence and uniqueness of the solution to problem (2) implies that (13) has a unique solution in H 10 (S).In the GMRES method [11], the kth iterate uk minimizes the norm of the residual rk = F − Auk overthe affine space u0 + Kk where Kk is the Krylov space generated by r0, Ar0, . . . ,Ak−1r0. As a consequence‖rk‖  ‖p(A)‖‖r0‖, where p(A) is any polynomial of degree k with p(0) = 1. The rate of the decay of the spec-trum of G can be used to choose a special sequence of polynomials in order to obtain convergence rate estimates.Here, the main aim is to use the regularizing character of the operator G to obtain such convergence rate estimatesfor (13). For that, the following result is essential:Theorem 2.1. There exists t with 1/4 < t  1/2, such that the linear map g → Gg is continuous from H 10 (S) toH 10 (S) ∩ H 1+t (S).Proof. SinceK+∂u+g∂n++ K− ∂u−g∂n−∈ H−1/2(S),it follows from results of P. Grisvard [8] and by interpolation thatG(K+∂u+g∂n++ K− ∂u−g∂n−)∈ H 1+t (S)for some 1/4 < t  1/2, the exact value of which depends on the regularity of the surface S. Corollary 2.2. The linear map g → Gg is compact of Carleman class Cs in H 10 (S) for every s  2/t .Proof. Since S is contained in R2, the imbedding of H 1+t (S) with t > 1/4 in H 1(S) is compact of Carleman classCs for every s  2/t , cf. [1]. The result then follows from Theorem 2.1 and the property that the composition of abounded operator and an operator of class Cs is an operator of class Cs . The classes Cs of compact operators in Hilbert spaces are studied in depth in ([5], Chapter XI), where are estab-lished completeness properties of the eigenfunctions generalizing earlier work of Carleman [4] for Hilbert–Schmidtoperators.These results allow to obtain the following result of convergence:Theorem 2.3. The unique solution of Eq. (13) can be approximated by the GMRES method, which convergesq-superlinearly in exact arithmetic.282 A.L. Bessoud, F. Krasucki / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 279–282Proof. The q-superlinearity convergence, i.e., the property that the sequence {rn} of the residuals satisfies:‖rn+1‖/‖rn‖ → 0, follows from Corollary 2.2 and a straightforward application of Theorem 1 of [10]. The same reasoning based on the regularizing character of the analogous of the operator G can be used to prove theq-superlinear convergence of the GMRES method for others non-classical transmission conditions, for instance thoseappearing in the case of a bonded structure with a weak interface, as in [6].With the same method, one can also prove the convergence of the GMRES algorithm for the finite element approx-imation (see [3] where suitable error estimates and numerical results are given also for linear elasticity problems).References[1] S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math. XV (1962)119–147.[2] A. Ait Moussa, Comportement asymptotique de bandes élastiques minces de faible ou grande rigidité, Thèse de Sciences Mathématiques del’Université Mohamed 1er, Oujda, Maroc, 1995.[3] A.L. Bessoud, L. Daridon, F. Krasucki, in preparation.[4] T. Carleman, Sur les équations intégrales singulières à noyau réel et symétrique. Almquist and Wiksells, Uppsala, 1923.[5] N. Dunford, J.T. Schwartz, Linear Operators, Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space, Wiley Interscience, New York,1963.[6] G. Geymonat, F. Krasucki, D. Marini, M. Vidrascu, A domain decomposition method for a bonded structure, Math. Models Methods Appl.Sci. 8 (1998) 1387–1402.[7] G. Geymonat, F. Krasucki, S. Lenci, Mathematical analysis of a bonded joint with a soft thin adhesive, Math. Mech. Solids 4 (1999) 201–255.[8] P. Grisvard, Singularities in Boundary Value Problems, Masson–Springer-Verlag, 1992.[9] E. Malachanne, Modélisation du comportement de multimatériaux avec liaisons fortes, Mémoire de D.E.A. “Mécanique des Matériaux et desMilieux complexes, des Structures et des Systèmes” de l’Université Montpellier II, 2005.[10] I. Moret, A note on the superlinear convergence of GMRES, SIAM J. Numer. Anal. 34 (1997) 513–516.[11] Y. Saad, M. Schultz, GMRES a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comp. 7(1986) 856–869.

Q-superlinear convergence of the GMRES algorithm for multi-materials with strong interface

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Q-superlinear convergence of the GMRES algorithm for multi-materials with strong interface

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