A numerical approach for the Poisson equation in a planar domain with a small inclusion

We consider the Poisson equation in a domain with a small inclusion. We present a simple numerical method, based on asymptotic analysis, which allows to approximate robustly the far field of the solution as the size of the inclusion goes to zero without any mesh adaptation procedure. The discretization is based on a fully standard Galerkin approach such as finite elements. We prove stability and consistency of the numerical method and provide error estimates. We end the paper with numerical experiments illustrating the efficiency of the technique.