[Approximation de problèmes elliptiques multi-échelles utilisant des patches d'éléments finis]
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.
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@article{CRMATH_2003__337_10_679_0,
author = {Glowinski, Roland and He, Jiwen and Rappaz, Jacques and Wagner, Jo\"el},
title = {Approximation of multi-scale elliptic problems using patches of finite elements},
journal = {Comptes Rendus. Math\'ematique},
pages = {679--684},
publisher = {Elsevier},
volume = {337},
number = {10},
year = {2003},
doi = {10.1016/j.crma.2003.09.029},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2003.09.029/}
}
TY - JOUR AU - Glowinski, Roland AU - He, Jiwen AU - Rappaz, Jacques AU - Wagner, Joël TI - Approximation of multi-scale elliptic problems using patches of finite elements JO - Comptes Rendus. Mathématique PY - 2003 SP - 679 EP - 684 VL - 337 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2003.09.029/ DO - 10.1016/j.crma.2003.09.029 LA - en ID - CRMATH_2003__337_10_679_0 ER -
%0 Journal Article %A Glowinski, Roland %A He, Jiwen %A Rappaz, Jacques %A Wagner, Joël %T Approximation of multi-scale elliptic problems using patches of finite elements %J Comptes Rendus. Mathématique %D 2003 %P 679-684 %V 337 %N 10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2003.09.029/ %R 10.1016/j.crma.2003.09.029 %G en %F CRMATH_2003__337_10_679_0
Glowinski, Roland; He, Jiwen; Rappaz, Jacques; Wagner, Joël. Approximation of multi-scale elliptic problems using patches of finite elements. Comptes Rendus. Mathématique, Tome 337 (2003) no. 10, pp. 679-684. doi : 10.1016/j.crma.2003.09.029. http://www.numdam.org/articles/10.1016/j.crma.2003.09.029/
[1] The mortar element method with overlapping subdomains, SIAM J. Numer. Anal., Volume 40 (2002) no. 2, pp. 601-628
[2] A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal., Volume 30 (1993) no. 4, pp. 921-935
[3] 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] 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
[7] The fast adaptive composite-grid (FAC) method for elliptic equations, Math. Comp., Volume 46 (1986) no. 174, pp. 439-456
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