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 hole of size $\delta$. We present a simple numerical method, based on an asymptotic analysis, which allows to approximate robustly the far field of the solution as $\delta$ goes to zero without meshing the small hole. We prove the stability of the scheme and provide error estimates. We end the paper with numerical experiments illustrating the efficiency of the technique.