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

Published online by Cambridge University Press:  20 August 2015

Abderrahmane Bendali ,
M’Barek Fares ,
Abdelkader Tizaoui  and
Sébastien Tordeux
Abderrahmane Bendali*
Affiliation:
Electromagnetism and Acoustics, CERFACS, 42 Avenue Gaspard Coriolis, F-31100 Toulouse, France Toulouse University, INSA-Toulouse, Mathematical Institute of Toulouse, UMR-CNRS 5219, 135 avenue de Rangueil, F-31077 Toulouse, France
M’Barek Fares*
Affiliation:
Electromagnetism and Acoustics, CERFACS, 42 Avenue Gaspard Coriolis, F-31100 Toulouse, France
Abdelkader Tizaoui*
Affiliation:
Toulouse University, INSA-Toulouse, Mathematical Institute of Toulouse, UMR-CNRS 5219, 135 avenue de Rangueil, F-31077 Toulouse, France
Sébastien Tordeux*
Affiliation:
Laboratoire de Mathématiques et de leurs Applications, UMR-CNRS 5142, Université de Pau et des Pays de l’Adour, F-64013 Pau, France Project Team MAGIQUE-3D, INRIA Bordeaux-Sud-Ouest, F-64013 Pau, France
*
Email address:abendali@insa-toulouse.fr
Email address:fares@cerfacs.fr
Email address:abdelkader.tizaoui@gmail.com
Corresponding author.Email:sebastien.tordeux@univ-pau.fr
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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.

Keywords

34E0535J0565M6078M3078M35Elliptic operatormatched asymptotic expansionseigenvalue problemfinite elements
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 456 - 471
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Amrouche, C., Girault, V. and Giroire, J., Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator: an approach in weighted Sobolev spaces, J. Math. Pures Appl., 76(9) (1997), 55–81.CrossRefGoogle Scholar
[2]Bendali, A., huard, A., Tizaoui, A., Tordeux, S. and Vila, J. P., Asymptotic expansions of the eigenvalues of a 2-D boundary-value problem relative to two cavities linked by a hole of small size, C. R. Acad. Sci. Paris, Mathématiques, 347 (2009), 1147–1152.CrossRefGoogle Scholar
[3]Bendali, A., Tizaoui, A., Tordeux, S. and Vila, J. P., Matching of asymptotic expansions for an eigenvalue problem with two cavities linked by a narrow hole, SAM Research Report, ETH Zurich, (2009), 74 pages.Google Scholar
[4]Bonnaillie-Noël, V. and Dauge, M., Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners, Ann. Henri Poincaré, 7 (2006), 899–931.CrossRefGoogle Scholar
[5]Brown, R. M., Hislop, P. D. and Martinez, A., Lower bounds on the interaction between cavities connected by a thin tube, Duke Math. J., 73 (1994), 163–176.CrossRefGoogle Scholar
[6]Chechkin, G. A. and Gadyl’shin, R. R., On boundary-value problems for the Laplacian in bounded and in unbounded domains with perforated boundaries, J. Differential Equations, 216 (2005), 502–522.Google Scholar
[7]Costabel, M., Dauge, M. and Nicaise, S., Corner singularities and analytic regularity for linear elliptic systems, in preparation.Google Scholar
[8]Dauge, M., Djurdjevic, I., Faou, E. and Rössle, A., Eigenmode asymptotics in thin elastic plates, J. Math. Pures Appl., 78 (1999), 925–964.Google Scholar
[9]Dautray, R. and Lions, J-L., Mathematical Analysis and Numerical Methods for Science and Technology, Integral Equations and Numerical Methods, Springer-Verlag, Berlin, 1990.Google Scholar
[10]Gadyl’shin, R. R., Surface potentials and the method of matching asymptotic expansions in the Helmholtz resonator problem, (Russian) Algebra i Analiz, 4 (1992), 88–115, translation in St. Petersburg Math. J., 4 (1993), 273–296.Google Scholar
[11]Ilín, A. M., Matching of asymptotic expansions of solutions of boundary value problems, Vol. 102 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992, Translated from the Russian by Minachin, V..Google Scholar
[12]Kondratev, V. A., Boundary-value problems for elliptic equations in domains with conical or angular points, Trans. Moscow. Math. Soc., 16 (1967), 227–313.Google Scholar
[13]Rauch, J. and Taylor, M., Electrostatic screening, J. Math. Phys., 16 (1975), 284–288.Google Scholar
[14]Raviart, P.-A. and Thomas, J.-M., Introduction A l’analyse Numérique des Équations aux Dérivées Partielles, Collection Mathématiques Appliquées pour la Maˆıtrise, [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983.Google Scholar
[15]Rayleigh, J. W. and Strutt, B., The Theory of Sound, 2d ed, Dover Publications, New York, 1945.Google Scholar
[16]Rayleigh, J. W., The Theory of the Helmholtz Resonator, Proceedings of the Royal Society of London, Series A, Containing Papers of a Mathematical and Physical Character, 92(638) (1916), 265–275.Google Scholar
[17]Sanchez-Hubert, J. and Sánchez-Palencia, E., Acoustic fluid flow through holes and permeability of perforated walls, J. Math. Anal. Appl., 87 (1982), 427–453.CrossRefGoogle Scholar
[18]Taflov, A., Umashanker, K., Becker, B., Harfoush, F. and Yee, K. S., Detailed fdtd analysis of electromagnetic fields penetrating narrow slots and lapped joints in thin conducting screens, IEEE T. Antenn. Propag., 36 (1988), 247–257.CrossRefGoogle Scholar
[19]Tuck, E. O., Matching problems involving flow through small holes, Adv. Appl. Mech., 15 (1975), 89–158.Google Scholar
[20]Van Dyke, M., Perturbation Methods in Fluid Mechanics, The Parabolic Press, Stanford, 1975.Google Scholar