Interactions between moderately close inclusions for the Laplace equation

The presence of small inclusions modi¯es the solution of the Laplace equation posed in a reference domain 0. 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 ¯rst consider two perfectly insulated inclusions. In this con¯guration we give a complete mul-tiscale 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 @0. We also present numerical experiments implementing a multiscale superposition method based on our ¯rst order expansion