In this paper we present a family of iterative methods to solve numerically second order elliptic problems with multi-scale data using multiple levels of grids. These methods are based upon the introduction of a Lagrange multiplier to enforce the continuity of the solution and its fluxes across interfaces. This family of methods can be interpreted as a mortar element method with complete overlapping domain decomposition for solving numerically multi-scale elliptic problems.
Dans cette Note nous présentons une famille de méthodes itératives pour résoudre numériquement des problèmes elliptiques du deuxième ordre à données multi-échelles utilisant plusieurs niveaux de grilles. Ces méthodes sont basées sur l'introduction d'un multiplicateur de Lagrange pour imposer la continuité de la solution et de ses flux à travers les interfaces. Ces méthodes peuvent être interprétées comme des méthodes de décomposition de domaines avec recouvrement total, de type mortier, pour résoudre numériquement des problèmes elliptiques multi-échelles.
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Roland Glowinski 1; Jiwen He 1; Jacques Rappaz 2; Joël Wagner 2
@article{CRMATH_2004__338_9_741_0,
author = {Roland Glowinski and Jiwen He and Jacques Rappaz and Jo\"el Wagner},
title = {A multi-domain method for solving numerically multi-scale elliptic problems},
journal = {Comptes Rendus. Math\'ematique},
pages = {741--746},
publisher = {Elsevier},
volume = {338},
number = {9},
year = {2004},
doi = {10.1016/j.crma.2004.02.014},
language = {en},
}
TY - JOUR AU - Roland Glowinski AU - Jiwen He AU - Jacques Rappaz AU - Joël Wagner TI - A multi-domain method for solving numerically multi-scale elliptic problems JO - Comptes Rendus. Mathématique PY - 2004 SP - 741 EP - 746 VL - 338 IS - 9 PB - Elsevier DO - 10.1016/j.crma.2004.02.014 LA - en ID - CRMATH_2004__338_9_741_0 ER -
%0 Journal Article %A Roland Glowinski %A Jiwen He %A Jacques Rappaz %A Joël Wagner %T A multi-domain method for solving numerically multi-scale elliptic problems %J Comptes Rendus. Mathématique %D 2004 %P 741-746 %V 338 %N 9 %I Elsevier %R 10.1016/j.crma.2004.02.014 %G en %F CRMATH_2004__338_9_741_0
Roland Glowinski; Jiwen He; Jacques Rappaz; Joël Wagner. A multi-domain method for solving numerically multi-scale elliptic problems. Comptes Rendus. Mathématique, Volume 338 (2004) no. 9, pp. 741-746. doi : 10.1016/j.crma.2004.02.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.02.014/
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[2] A new nonconforming approach to domain decomposition: The mortar element method (H. Brézis; J.-L. Lions, eds.), Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, vol. 11, Longman Scientific and Technical, Harlow, UK, 1994, pp. 13-51
[3] The Mortar element method revisited – what are the right norms? (N. Debit et al., eds.), Domain Decomposition Methods in Science and Engineering: Thirteenth International Conference on Domain Decomposition Methods, CIMNE, Barcelona, 2002, pp. 27-40
[4] Approximation of multi-scale elliptic problems using patches of finite elements, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003), pp. 679-684
[5] R. Glowinski, J. He, J. Rappaz, J. Wagner, A multi-domain method for numerical solution of multi-scale elliptic problems, in preparation
[6] Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, Oxford, 1999
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