Meso-Scale Approximations of Fields Around Clusters of Defects

Abstract

We present a review of recent results on asymptotic approximations of solutions to boundary value problems in domains with large clusters of small defects. There are no assumptions which require periodicity within the cluster. The asymptotic approximations we discuss are uniform and include the boundary layers occurring in neighborhoods of singularly perturbed boundaries of the domains concerned. The term “meso-scale” is used to describe these approximations since they go beyond the conventional constraints of the homogenization theory.