[Une méthode de décomposition de domaine à deux niveaux robuste pour les systèmes dʼEDPs]
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.
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.
Accepté le :
Publié le :
Nicole Spillane 1, 2 ; Victorita Dolean 3 ; Patrice Hauret 2 ; Frédéric Nataf 1 ; Clemens Pechstein 4 ; Robert Scheichl 5
@article{CRMATH_2011__349_23-24_1255_0,
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},
publisher = {Elsevier},
volume = {349},
number = {23-24},
year = {2011},
doi = {10.1016/j.crma.2011.10.021},
language = {en},
}
TY - JOUR AU - Nicole Spillane AU - Victorita Dolean AU - Patrice Hauret AU - Frédéric Nataf AU - Clemens Pechstein AU - Robert Scheichl TI - A robust two-level domain decomposition preconditioner for systems of PDEs JO - Comptes Rendus. Mathématique PY - 2011 SP - 1255 EP - 1259 VL - 349 IS - 23-24 PB - Elsevier DO - 10.1016/j.crma.2011.10.021 LA - en ID - CRMATH_2011__349_23-24_1255_0 ER -
%0 Journal Article %A Nicole Spillane %A Victorita Dolean %A Patrice Hauret %A Frédéric Nataf %A Clemens Pechstein %A Robert Scheichl %T A robust two-level domain decomposition preconditioner for systems of PDEs %J Comptes Rendus. Mathématique %D 2011 %P 1255-1259 %V 349 %N 23-24 %I Elsevier %R 10.1016/j.crma.2011.10.021 %G en %F CRMATH_2011__349_23-24_1255_0
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/
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