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Comptes Rendus Mathématique
Volume 347, n° 19-20
pages 1147-1152 (octobre 2009)
Doi : 10.1016/j.crma.2009.09.005
Received : 5 May 2009 ;  accepted : 2 September 2009
Asymptotic expansions of the eigenvalues of a 2-D boundary-value problem relative to two cavities linked by a hole of small size
Développement asymptotique des valeurs et fonctions propres d’un problème aux limites 2-D relatif à deux cavités reliées par un trou de petite taille.1
 

Abderrahmane Bendali , Alain Huard , Abdelkader Tizaoui , Sébastien Tordeux , Jean-Paul Vila
Toulouse University, INSA, Mathematical Institute of Toulouse (UMR-CNRS 5219), 135, avenue de Rangueil, 31077 Toulouse, France 

Abstract

This Note presents the derivation of the 2nd-order asymptotic expansion of the eigenvalues and the eigenfunctions of the operator associated to an interior elliptic equation supplemented by a Dirichlet boundary condition on a domain consisting of two cavities linked by a hole of small size. The asymptotic expansion is carried out with respect to the size of the hole. The main feature of the method is to yield a robust numerical procedure making it possible to compute the eigenvalues without resorting to a refined mesh around the hole. To cite this article: A. Bendali et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.
Résumé

Cette Note présente la dérivation du développement asymptotique au 2nd ordre des valeurs et des fonctions propres de l’opérateur associé à une équation elliptique complétée par une condition aux limites de Dirichlet sur un domaine formé de deux cavités reliées par un trou de petite taille. Le développement asymptotique est effectué relativement à la taille du trou. La principale caractéristique de la méthode est de donner lieu à une procédure numérique permettant de calculer les valeurs propres sans recourir à un maillage raffiné autour du trou. Pour citer cet article : A. Bendali et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.

 This work was supported by the French National Research Agency under grant no. ANR-08-SYSC-001.

1  Ce travail a bénéficié d’un soutien de l’Agence Nationale de la Recherche portant la référence ANR-08-SYSC-001.


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