Comptes Rendus
no. S1
A GenEO Domain Decomposition method for Saddle Point problems
[Une méthode de décomposition de domaine GenEO pour les problèmes de points selle]
Frédéric Nataf1   ; Pierre-Henri Tournier1
1 Laboratoire J.L. Lions, Sorbonne Université, 4 place Jussieu France
Comptes Rendus. Mécanique, Online first (2023), pp. 1-18.

Nous présentons une méthode de décomposition de domaine (DD) adaptative basée pour résoudre les problèmes de points selle définis comme une matrice bloc 2x2. L’algorithme ne nécessite aucune connaissance de l’espace contraint. Nous supposons que toutes les sous-matrices sont creuses et que les blocs diagonaux sont spectralement équivalents à une somme de matrices semi-définies positives. Cette dernière hypothèse permet de concevoir un espace grossier adaptatif pour les méthodes DD qui étend la théorie de GenEO (Spillane et al., 2014) aux problèmes de points de selle. Des résultats numériques sur des problèmes d’élasticité tridimensionnels pour des structures acier-caoutchouc discrétisées avec une pression continue sont montrés jusqu’à un milliard de degrés de liberté.

We introduce an adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. The algorithm does not require any knowledge of the constrained space. We assume that all sub matrices are sparse and that the diagonal blocks are spectrally equivalent to a sum of positive semi definite matrices. The latter assumption enables the design of adaptive coarse space for DD methods that extends the GenEO theory (Spillane et al., 2014) to saddle point problems. Numerical results on three dimensional elasticity problems for steel-rubber structures discretized by a finite element with continuous pressure are shown for up to one billion degrees of freedom.

Reçu le :
Accepté le :
Première publication :
DOI : 10.5802/crmeca.175
Keywords: domain decomposition method, nearly incompressible elasticity, high performance computing, saddle point problem, coarse space, multiscale finite element, Schur complement
Mot clés : Méthode de décomposition de domaine, élasticité quasi incompressible, calcul haute performance, problème de point selle, espace grossier, éléments finis multi échelle, complément de Schur
Frédéric Nataf 1 ; Pierre-Henri Tournier 1

1 Laboratoire J.L. Lions, Sorbonne Université, 4 place Jussieu France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Fr\'ed\'eric Nataf and Pierre-Henri Tournier},
     title = {A {GenEO} {Domain} {Decomposition} method for {Saddle} {Point} problems},
     journal = {Comptes Rendus. M\'ecanique},
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     year = {2023},
     doi = {10.5802/crmeca.175},
     language = {en},
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Frédéric Nataf; Pierre-Henri Tournier. A GenEO Domain Decomposition method for Saddle Point problems. Comptes Rendus. Mécanique, Online first (2023), pp. 1-18. doi : 10.5802/crmeca.175.

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