Commun. Comput. Phys., 11 (2012), pp. 456-471.


Matched Asymptotic Expansions of the Eigenvalues of a 3-D Boundary-Value Problem Relative to Two Cavities Linked by a Hole of Small Size

Abderrahmane Bendali 1, M'Barek Fares 2, Abdelkader Tizaoui 3, Sebastien Tordeux 4*

1 Electromagnetism and Acoustics, CERFACS, 42 Avenue Gaspard Coriolis, F-31100 Toulouse, France; and Toulouse University, INSA-Toulouse, Mathematical Institute of Toulouse, UMR-CNRS 5219, 135 avenue de Rangueil, F-31077 Toulouse, France.
2 Electromagnetism and Acoustics, CERFACS, 42 Avenue Gaspard Coriolis, F-31100 Toulouse, France.
3 Toulouse University, INSA-Toulouse, Mathematical Institute of Toulouse, UMR-CNRS 5219, 135 avenue de Rangueil, F-31077 Toulouse, France.
4 Laboratoire de Mathematiques et de leurs Applications, UMR-CNRS 5142, Universite de Pau et des Pays de l'Adour, F-64013 Pau, France; and Project Team MAGIQUE-3D, INRIA Bordeaux-Sud-Ouest, F-64013 Pau, France.

Received 14 December 2009; Accepted (in revised version) 28 June 2010
Available online 24 October 2011
doi:10.4208/cicp.141209.280610s

Abstract

In this article, we consider a domain consisting of two cavities linked by a hole of small size. We derive a numerical method to compute an approximation of the eigenvalues of an elliptic operator without refining in the neighborhood of the hole. Several convergence rates are obtained and illustrated by numerical simulations.

AMS subject classifications: 34E05, 35J05, 65M60, 78M30, 78M35

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Key words: Elliptic operator, matched asymptotic expansions, eigenvalue problem, finite elements.

*Corresponding author.
Email: abendali@insa-toulouse.fr (A. Bendali), fares@cerfacs.fr (M. Fares), abdelkader.tizaoui@gmail.com (A. Tizaoui), sebastien.tordeux@univ-pau.fr (S. Tordeux)
 

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