Comptes Rendus
no. 10
Numerical Analysis
Lower bounds for additive Schwarz methods with mortars
[Bornes inférieures pour méthode de Schwarz additives et éléments finis avec joints.]

Présenté par : Olivier Pironneau

Dan Stefanica1
1 Baruch College, City University of New York, One Bernard Baruch Way, Box B 6-230, New York, NY 10010, USA
Comptes Rendus. Mathématique, Volume 339 (2004) no. 10, pp. 739-743.

On détermine des bornes inférieures de conditionnement d'algorithmes de type Schwarz additif pour des problèmes elliptiques discrétisés par éléments finis avec joints. Ces limites sont identiques à des constantes multiplicatives près, aux bornes supérieures classiques établies par ailleurs. L'optimalité du conditionnement est ainsi démontré.

We establish lower bounds for the condition number of overlapping additive Schwarz algorithms for elliptic problems discretized by mortar finite elements. These bounds coincide, up to constants, with the classical upper bounds from the literature. The optimality of the condition number estimates is thus established.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.09.016
Dan Stefanica 1

1 Baruch College, City University of New York, One Bernard Baruch Way, Box B 6-230, New York, NY 10010, USA 
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Dan Stefanica. Lower bounds for additive Schwarz methods with mortars. Comptes Rendus. Mathématique, Volume 339 (2004) no. 10, pp. 739-743. doi : 10.1016/j.crma.2004.09.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.016/

[1] C. Bernardi; Y. Maday; A. Patera A new nonconforming approach to domain decomposition: the mortar element method, Collège de France Seminar, Pitman, 1994

[2] S. Brenner Lower bounds for two-level additive Schwarz preconditioners with small overlap, SIAM J. Sci. Comput., Volume 21 (2000), pp. 1657-1669

[3] B. Wohlmuth A mortar finite element method using dual spaces for the Lagrange multiplier, SIAM J. Numer. Anal., Volume 3 (2000), pp. 989-1012

[4] M. Dryja; O. Widlund Domain decomposition algorithms with small overlap, SIAM J. Sci. Comput., Volume 15 (1994), pp. 604-620

[5] O. Widlund, Two-level Schwarz algorithms, using overlapping subregions, for mortar finite element methods, Technical report, Computer Science Department, Courant Institute of Mathematical Sciences, in preparation

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