A multi-scale patch approximation for Poisson problems with a small inhomogeneous inclusion

Abstract

The paper deals with the multi-scale approximation of the influence of a small inhomogeneity of arbitrary shape in an elastic medium. A new multi-scale patch method is introduced, whose caracteristic is to deal with a large scale problem without inclusion, a small-scale problem on a patch surrounding the inclusion defining a corrector and an iterative procedure between these two problems. Theoretical results of convergence of the iterations, a posteriori error estimate and comparison of the corrector with the asymptotic expansion are provided. The finite element approximation is also addressed together with some numerical tests