Small obstacle asymptotics for a 2D semi-linear convex problem

Abstract

We study a 2D semi-linear equation in a domain with a small Dirichlet obstacle of size . Using the method of matched asymptotic expansions, we compute an asymptotic expansion of the solution as tends to zero. Its relevance is justified by proving a rigorous error estimate. Then we construct an approximate model, based on an equation set in the limit domain without the small obstacle, which provides a good approximation of the far field of the solution of the original problem. The interest of this approximate model lies in the fact that it leads to a variational formulation which is very simple to discretize. We present numerical experiments to illustrate the analysis.

Keywords:

• Small obstacle
• semi-linear convex problem
• asymptotic analysis
• singular perturbation

Notes

No potential conflict of interest was reported by the authors.

1 See also the beginning of Section3.2 for an explanation of this choice.

2 FreeFem++, http://www.freefem.org/ff++/.

Additional information

Funding

The research of L. C. was supported by the FMJH through the [grant number ANR-10-CAMP-0151-02] in the ‘Programme des Investissements d’Avenir’. The research of S.A. N. was supported by the Russian Foundation for Basic Research [grant number 15-01-02175].