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

Lucas Chesnel (1, 2) , Xavier Claeys (3, 4)

1 DeFI - Shape reconstruction and identification
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 LJLL - Laboratoire Jacques-Louis Lions
4 ALPINES - Algorithms and parallel tools for integrated numerical simulations

We consider the Poisson equation in a domain with a small hole of size δ. We present a simple numerical method, based on an asymptotic analysis, which allows to approximate robustly the far field of the solution as δ 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.

Mots clés: singular perturbation; finite element method; asymptotic analysis; Small hole

Domaines: Analyse numérique [math.NA]

Lucas Chesnel, Xavier Claeys. A numerical approach for the Poisson equation in a planar domain with a small inclusion. BIT Numerical Mathematics, 2016. ⟨hal-01109552v2⟩