A numerical approach for the Poisson equation in a planar domain with a small inclusion
This work provides a practical numerical technique for handling small geometric features in PDE problems, which is relevant for engineers and scientists simulating domains with small holes or inclusions.
The paper presents a numerical method for solving the Poisson equation in a domain with a small inclusion, using asymptotic analysis to approximate the far field without meshing the small hole. The method is proven stable with error estimates, and numerical experiments demonstrate its efficiency.
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.