Comptes Rendus
no. 10
Numerical Analysis
Approximation of multi-scale elliptic problems using patches of finite elements
[Approximation de problèmes elliptiques multi-échelles utilisant des patches d'éléments finis]

Présenté par : Roland Glowinski

Roland Glowinski1  ; Jiwen He1 ; Jacques Rappaz2 ; Joël Wagner2
1 Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX 77204-3008, USA
2 Section of Mathematics, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland
Comptes Rendus. Mathématique, Volume 337 (2003) no. 10, pp. 679-684.

Dans cette Note nous présentons une méthode faisant apparaı̂tre plusieurs niveaux de grilles non nécessairement emboı̂tées pour résoudre numériquement des problèmes elliptiques à données multi-échelles. La méthode consiste à calculer des corrections successives de la solution par sous-domaines discrétisés de façon non nécessairement conforme. Elle s'apparente à la méthode FAC (voir Math. Comp. 46 (174) (1986) 439–456) et sa convergence s'obtient par une technique de décomposition de domaines (voir Math. Comp. 57 (195) (1991) 1–21). Toutefois elle permet une plus grande souplesse d'utilisation que ces dernières citées.

In this paper we present a method to solve numerically elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. It resembles the FAC method (see Math. Comp. 46 (174) (1986) 439–456) and its convergence is obtained by a domain decomposition technique (see Math. Comp. 57 (195) (1991) 1–21). However it is of much more flexible use in comparison to the latter.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2003.09.029
Roland Glowinski 1 ; Jiwen He 1 ; Jacques Rappaz 2 ; Joël Wagner 2

1 Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX 77204-3008, USA
2 Section of Mathematics, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland 
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Roland Glowinski; Jiwen He; Jacques Rappaz; Joël Wagner. Approximation of multi-scale elliptic problems using patches of finite elements. Comptes Rendus. Mathématique, Volume 337 (2003) no. 10, pp. 679-684. doi : 10.1016/j.crma.2003.09.029. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.029/

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[3] J.H. Bramble; J.E. Pasciak; J. Wang; J. Xu Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp., Volume 57 (1991) no. 195, pp. 1-21

[4] R. Glowinski, J. He, J. Rappaz, J. Wagner, Finite element approximation of multi-scale elliptic problems using patches of elements, in preparation

[5] F. Hecht, O. Pironneau, K. Ohtsuka, Freefem++, ver. 1.28, http://www.freefem.org

[6] M. Jung; J.-F. Maitre Some remarks on the constant in the strengthened CBS inequality: estimate for hierarchical finite element discretizations of elasticity problems, Numer. Methods Partial Differential Equations, Volume 15 (1999) no. 4, pp. 469-487

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