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Raccordement de développements asymptotiques pour des problèmes aux valeurs propres comportant deux cavités reliées par un petit trou en dimensions 2 et 3

Abstract : In a lot of physical problems, the boundary of the computational domain is perforated. This configura- tion can lead to numerical difficulties when the diameter of the holes are really smaller than the other characteristic lengths. Indeed, it can be very costly to compute a sharp numerical approximation of the solution of such problems for two main reasons: With a standard method like finite elements or finite differences, a refined mesh cannot be avoided in the neighborhood of the hole; the mesh generation of a perforated structure can be a hard task. Many authors have studied the effect of the perforation of the boundaries both from the theoretical and the numerical point of views, see for example [3-5]. However, less results have been obtained for the eigenvalue problem in the case of a three dimensional domain. In [2], Gadyl'shin considered a two dimensional domain consisting of two domains linked by a small hole. He derived a complete asymptotic expansion of the scattering frequencies of the Laplacian operator equipped with Dirichlet boundary condition. In [1], these results were extended to the eigenvalues and eigenvectors of an elliptic operator with varying coefficients. In this talk, we are interested to a three dimensional configuration with varying coefficients and Neumann boundary condition. References : [1] A. Bendali, A. Huard, A. Tizaoui, S. Tordeux and J. P. Vila, 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, Mathmatiques, 347 (2009), pp. 1147-1152. [2] R. R. Gadyl'shin, Surface potentials and the method of matching asymptotic expansions in the Helmholtz resonator problem, (Russian) Algebra i Analiz 4 (1992), no. 2, 88-115, translation in St. Petersburg Math. J. 4 (1993), no. 2, 273-296. [3] J. Sanchez-Hubert and E. Sanchez-Palencia, Acoustic fluid flow through holes and permeability of perforated walls, J. Math. Anal. Appl., 87 (1982), pp. 427-453. [4] A. Taflov, K. Umashanker, B. Becker, F. Harfoush, and K. S. Yee, Detailed fdtd analysis of electromagnetic fields penetrating narrow slots and lapped joints in think conducting screens, IEEE Trans Antenna and Propagation, 36 (1988), pp. 247257. [5] E. O. Tuck, Matching problems involving flow through small holes, in Advances in applied mechan- ics, Vol. 15, Academic Press, New York, (1975), pp. 89-158.
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Sébastien Tordeux. Raccordement de développements asymptotiques pour des problèmes aux valeurs propres comportant deux cavités reliées par un petit trou en dimensions 2 et 3. Séminaire d'analyse numérique, LJK, 2009, Grenoble, France. ⟨inria-00528075⟩

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