INTERACTIONS BETWEEN 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 studied extensively for a single inclusion or well-separated inclusions. In two-dimensional situations, we investigate 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. We also present numerical experiments implementing a multiscale superposition method based on our first order expansion.