When a thin periodic layer meets corners: asymptotic analysis of a singular Poisson problem

TL;DR

This paper develops a high-order asymptotic analysis for the Poisson equation in a domain with a thin, periodic layer, accounting for boundary layer effects and corner singularities using matched asymptotic expansions and homogenization.

Contribution

It introduces a comprehensive asymptotic expansion that incorporates boundary layer effects and corner singularities for the Poisson problem with a thin periodic layer.

Findings

Derivation of a high-order asymptotic expansion.

Justification of the asymptotic method including boundary layers.

Analysis of corner singularities near layer extremities.

Abstract

The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization, and a complete justification is included in the paper or its appendix.