[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]
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.
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.
Accepté le :
Publié le :
@article{CRMATH_2009__347_19-20_1147_0,
author = {Bendali, Abderrahmane and Huard, Alain and Tizaoui, Abdelkader and Tordeux, S\'ebastien and Vila, Jean-Paul},
title = {Asymptotic expansions of the eigenvalues of a {2-D} boundary-value problem relative to two cavities linked by a hole of small size},
journal = {Comptes Rendus. Math\'ematique},
pages = {1147--1152},
publisher = {Elsevier},
volume = {347},
number = {19-20},
year = {2009},
doi = {10.1016/j.crma.2009.09.005},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.09.005/}
}
TY - JOUR AU - Bendali, Abderrahmane AU - Huard, Alain AU - Tizaoui, Abdelkader AU - Tordeux, Sébastien AU - Vila, Jean-Paul TI - Asymptotic expansions of the eigenvalues of a 2-D boundary-value problem relative to two cavities linked by a hole of small size JO - Comptes Rendus. Mathématique PY - 2009 SP - 1147 EP - 1152 VL - 347 IS - 19-20 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.09.005/ DO - 10.1016/j.crma.2009.09.005 LA - en ID - CRMATH_2009__347_19-20_1147_0 ER -
%0 Journal Article %A Bendali, Abderrahmane %A Huard, Alain %A Tizaoui, Abdelkader %A Tordeux, Sébastien %A Vila, Jean-Paul %T Asymptotic expansions of the eigenvalues of a 2-D boundary-value problem relative to two cavities linked by a hole of small size %J Comptes Rendus. Mathématique %D 2009 %P 1147-1152 %V 347 %N 19-20 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.09.005/ %R 10.1016/j.crma.2009.09.005 %G en %F CRMATH_2009__347_19-20_1147_0
Bendali, Abderrahmane; Huard, Alain; Tizaoui, Abdelkader; Tordeux, Sébastien; Vila, Jean-Paul. Asymptotic expansions of the eigenvalues of a 2-D boundary-value problem relative to two cavities linked by a hole of small size. Comptes Rendus. Mathématique, Tome 347 (2009) no. 19-20, pp. 1147-1152. doi : 10.1016/j.crma.2009.09.005. http://www.numdam.org/articles/10.1016/j.crma.2009.09.005/
[1] A note on the generalized Dumbbell problem, Proc. Amer. Math. Soc., Volume 123 (1995), pp. 2595-2599
[2] Scattering frequencies of resonators, Comm. Pure Appl. Math., Volume 26 (1973), pp. 549-563
[3] A. Bendali, A. Tizaoui, S. Tordeux, J.P. Vila, Matching of asymptotic expansions for an eigenvalue problem with two cavities linked by a narrow hole, SAM Research Report, ETH Zürich, 2009, no. 17
[4] Lower bounds on the interaction between cavities connected by a thin tube, Duke Math. J., Volume 73 (1994), pp. 163-176
[5] Surface potentials and the method of matching asymptotic expansions in the Helmholtz resonator problem, Algebra i Analiz, Volume 4 (1992) no. 2, pp. 88-115 (in Russian), translation in St. Petersburg Math. J., 4, 2, 1993, pp. 273-296
[6] Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Translations of Mathematical Monographs, vol. 102, American Mathematical Society, Providence, RI, 1992 (translated from the Russian by V. Minachin)
[7] Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion, Multiscale Model. Simul., Volume 5 (2006), pp. 304-336
[8] Matching of asymptotic expansions for waves propagation in media with thin slots. II. The error estimates, M2AN Math. Model. Numer. Anal., Volume 42 (2008), pp. 193-221
[9] Electrostatic screening, J. Math. Phys., Volume 16 (1975), pp. 284-288
[10] Acoustic fluid flow through holes and permeability of perforated walls, J. Math. Anal. Appl., Volume 87 (1982), pp. 427-453
[11] Matching problems involving flow through small holes, Advances in Applied Mechanics, vol. 15, Academic Press, New York, 1975, pp. 89-158
[12] Perturbation Methods in Fluid Mechanics, The Parabolic Press, Stanford, CA, 1975 (annotated)
Cité par Sources :
☆ This work was supported by the French National Research Agency under grant no. ANR-08-SYSC-001.
