A multiscale correction method for local singular perturbations of the boundary

Abstract

Abstract.In this work, we consider singular perturbations of the boundary of a smooth domain.We describe the asymptotic behavior of the solution uε of a second order elliptic equation posed in the perturbed domain with respect to the size parameter ε of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of uε based on a multiscale superposition of the unperturbed solution u0 and a profile defined in a model domain. We conclude with numerical results.Mathematics Subject Classification. 35B25, 35B40, 35J25, 49Q10, 65N30.

On moderately close inclusions for the Laplace equation

“…To get round this difficulty, it is possible to consider perturbation which are selfsimilar after straightening, see [10]. For a perturbation self-similar in the physical coordinates in a locally convex domain, an expansion has been obtained up to order 2 in [5]; -the case of Dirichlet conditions on the inclusions in dimension 2 requires a more evolved analysis because of the logarithmic potential. Non-decaying profiles appear, and the expansion also contains powers of log 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)).…”

“…The case of a single inclusion ω, centered at the origin 0 being either in Ω 0 or in Γ, has been deeply studied, see [8,6,7,9,3,4,1,5]. The techniques rely on the notion of profile, a normalized solution of the Laplace equation in the exterior domain obtained by blow-up of the perturbation, see (4).…”

“…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)).…”

“…The boundary of Ω 0 is perturbed in such a way that Ω ε ⊂ Ω 0 , we hence need cut-off functions (see [5,11]). Precisely, the asymptotic expansion takes the form…”