Numerical Analysis/Partial Differential Equations
A multi-domain method for solving numerically multi-scale elliptic problems
Comptes Rendus. Mathématique, Volume 338 (2004) no. 9, pp. 741-746.

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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Published online:
DOI: 10.1016/j.crma.2004.02.014
Glowinski, Roland 1; He, Jiwen 1; Rappaz, Jacques 2; Wagner, Joël 2

1 Dept. 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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Glowinski, Roland; He, Jiwen; Rappaz, Jacques; Wagner, Joël. 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. http://www.numdam.org/articles/10.1016/j.crma.2004.02.014/

[1] Belgacem, F.B. The Mortar finite element method with Lagrange multipliers, Numer. Math., Volume 84 (1999), pp. 173-197

[2] Bernardi, C.; Maday, Y.; Patera, A.T. A new nonconforming approach to domain decomposition: The mortar element method (Brézis, H.; Lions, J.-L., 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] Braess, D.; Dahmen, W. The Mortar element method revisited – what are the right norms? (Debit, N. et al., eds.), Domain Decomposition Methods in Science and Engineering: Thirteenth International Conference on Domain Decomposition Methods, CIMNE, Barcelona, 2002, pp. 27-40

[4] Glowinski, R.; He, J.; Rappaz, J.; Wagner, J. 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] Quarteroni, A.; Valli, A. Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, Oxford, 1999

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