On moderately close inclusions for the Laplace equation

Abstract

The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω0. This question has been deeply studied for a single inclusion or well separated inclusions. We investigate in this note the case where the distance between the holes tends to zero but remains large with respect to their characteristic size. We first consider two perfectly insulated inclusions. In this configuration we give a complete multiscale asymptotic expansion of the solution to the Laplace equation. We also address the situation of a single inclusion close to a singular perturbation of the boundary ∂Ω0. To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005). RésuméInteractions entre inclusions relativement proches pour l'équation de Laplace. La présence de petites inclusions dans un domaine de référence Ω0 modifie la solution de l'équation de Laplace dans ce domaine. Les cas d'une inclusion isolée ou de plusieurs bien séparées ontété largementétudiés. Dans cette note, nous considérons le cas où la distance entre deux inclusions tends vers zéro mais reste grande par rapportà leur taille caractéristique. Nous donnons un dévelopement asymptotique multi-échelle complet de la solution de l'équation de Laplace dans la situation de deux inclusions parfaitement isolantes. Nous présentonségalement le cas d'une seule inclusion proche du bord ∂Ω0 qui est lui même perturbé. Pour citer cet article : A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).Email addresses: Virginie.Bonnaillie@bretagne.ens-cachan.fr (V. Bonnaillie-Noël), Marc.Dambrine@utc.fr (M. Dambrine), Sebastien.Tordeux@insa-toulouse.fr (S. Tordeux), Gregory.Vial@bretagne.ens-cachan.fr (G. Vial). Preprint submitted to the Académie des sciences 4 mai 2007Version française abrégéeSoit Ω 0 un domaine borné de R 2 . Pour ε > 0 suffisamment petit, on considère le domaine perturbé Ω ε obtenu en enlevantà Ω 0 deux inclusions de taille ε et séparées d'une distance 2η ε . Précisément le domaine Ω ε est défini par la relation (1), où les motifs ω ± , contenant l'origine, sont dilatésà l'échelle ε, et translatés en x ± ε = 0 ± η ε d, voir Figure 1. Notre objectif est de donner un développement asymptotique de la solution du problème aux limites (2) posé dans Ω ε .Le cas d'une seule inclusion aété largementétudié dans [8,6,7,9,3,4,1,5]. Ces travaux s'appuient sur la notion essentielle de profil, solution normalisée de l'équation de Laplace dans le domaine extérieur obtenu par blow-up de la perturbation (voir (4)). Exprimé en variable rapide x ε , le profil V 0 décrit le comportement local de la solution au voisinage de l'inclusion. La convergence du développement estétablie grâceà la décroissance de V 0à l'infini. Ainsi, dans le cas de conditions aux limites de type Neumann sur le bord de l'inclusion, le début du développement est donné par u ε (x) = u 0 (x) + εV 0 (x/ε) + r 1 ε (x), où u 0 est la solution de l'équation de Laplace dans Ω 0 et V 0 résout (4). Le reste satisfait r 1 ε H 1 (Ωε) = O(ε 2 ). Cetteétude...

A numerical approach for the Poisson equation in a planar domain with a small inclusion

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“…These techniques provide satisfying results in many cases, but they require careful and thorough implementation efforts, and/or rely on strong assumptions such as homogeneity of the coefficients of the equation under consideration. Other numerical strategies are based on an approximation of u δ of the form "u 0 + corrector", where both terms of this sum are computed separately (see for example [18,7]). These approaches may induce substantial additional computational cost in a real life simulation.…”

Second-order topological expansion for electrical impedance tomography

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The Dirichlet problem in a planar domain with two moderately close holes

“…The asymptotic analysis of elliptic boundary value problems in domains with many holes which collapse one to the other while shrinking their sizes is a topic of growing interest and several authors have recently proposed different techniques and points of view. We mention for example the method based on multiscale asymptotic expansions which have been used by Bonnaillie-Noël, Dambrine, Tordeux, and Vial [5,6], Bonnaillie-Noël and Dambrine [3], and Bonnaillie-Noël, Dambrine, and Lacave [4] to study problems with two moderately close holes, i.e., problems with two holes whose mutual distance tends to zero while their size tends to zero at faster speed. The case when the number of holes is large has been considered by Maz'ya, Movchan, and Nieves in a series of papers where they propose a mesoscale approximation method to analyse problems for the Laplace operator and for the system of linear elasticity.…”