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

TL;DR

This paper introduces a numerical method for solving the Poisson equation in domains with small inclusions, avoiding mesh refinement around tiny holes, and provides stability and error analysis supported by numerical experiments.

Contribution

The paper presents a novel asymptotic-based numerical scheme that efficiently approximates solutions in domains with small inclusions without meshing the small hole.

Findings

The method is stable and provides accurate approximations as the inclusion size tends to zero.

Error estimates are established for the proposed numerical scheme.

Numerical experiments demonstrate the method's efficiency and robustness.

Abstract

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.