Comptes Rendus
Partial Differential Equations/Numerical Analysis
A robust two-level domain decomposition preconditioner for systems of PDEs
Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1255-1259.

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods. However, it is known that most popular choices of coarse spaces perform rather weakly in presence of heterogeneities in the coefficients in the partial differential equations, especially for systems. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems which isolate the terms responsible for slow convergence. We give a general theoretical result and then some numerical examples on a heterogeneous elasticity problem.

Un moyen efficace pour obtenir des méthodes de décomposition de domaine extensibles ( « scalable » en anglais) est lʼutilisation dʼune grille grossière. Cependant, lorsque les coefficients des équations présentent de grandes hétérogénéités, les méthodes usuelles tombent en défaut, surtout dans le cas des systèmes. Nous introduisons ici, au niveau variationnel, une grille grossière robuste même en présence de telles discontinuités. Pour cela, nous résolvons des problèmes aux valeurs propres généralisés locaux qui isolent les composantes de la solution nuisant à la convergence. Nous présentons un résultat théorique général puis quelques résultats numériques pour un problème dʼélasticité à coefficients discontinus.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.10.021
Nicole Spillane 1, 2; Victorita Dolean 3; Patrice Hauret 2; Frédéric Nataf 1; Clemens Pechstein 4; Robert Scheichl 5

1 Laboratoire J.L. Lions, UMR 7598, UPMC université Paris 6, 75252 Paris cedex 05, France
2 Michelin Technology Center, place des Carmes-Déchaux, 63000 Clermont-Ferrand, France
3 Laboratoire J.-A. Dieudonné, UMR 6621, université de Nice-Sophia Antipolis, 06108 Nice cedex 02, France
4 Institute of Computational Mathematics, Johannes Kepler Universität, Altenberger Str. 69, A-4040 Linz, Austria
5 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
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     author = {Nicole Spillane and Victorita Dolean and Patrice Hauret and Fr\'ed\'eric Nataf and Clemens Pechstein and Robert Scheichl},
     title = {A robust two-level domain decomposition preconditioner for systems of {PDEs}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1255--1259},
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Nicole Spillane; Victorita Dolean; Patrice Hauret; Frédéric Nataf; Clemens Pechstein; Robert Scheichl. A robust two-level domain decomposition preconditioner for systems of PDEs. Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1255-1259. doi : 10.1016/j.crma.2011.10.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.021/

[1] V. Dolean, F. Nataf, R. Scheichl, N. Spillane, Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps, 2011, submitted for publication, . | HAL

[2] Y. Efendiev, J. Galvis, R. Lazarov, J. Willems, Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms, RICAM report, 2011.

[3] F. Hecht FreeFem++ http://www.freefem.org/ff++/ (Laboratoire J.L. Lions, CNRS UMR 7598)

[4] P. Le Tallec Domain decomposition methods in computational mechanics, Comput. Mech. Adv., Volume 1 (1994) no. 2, pp. 121-220

[5] F. Nataf; H. Xiang; V. Dolean; N. Spillane A coarse space construction based on local Dirichlet to Neumann maps, 2011 (SISC) | HAL

[6] C. Pechstein, R. Scheichl, Weighted Poincaré inequalities, Tech. Report NuMa-Report 2010-10, Institute of Computational Mathematics, Johannes Kepler University, Linz, December 2010, submitted for publication.

[7] A. Toselli; O. Widlund Domain Decomposition Methods: Algorithms and Theory, Springer, 2005

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