Convergence rate for the homogenization of stationary diffusions in dilutely perforated domains with reflecting boundaries

TL;DR

This paper investigates the homogenization of stationary diffusion in perforated domains with reflecting boundaries, focusing on the convergence rates and the transition between dilute and fixed-volume regimes.

Contribution

It introduces new convergence rates for homogenization in dilute perforated domains and analyzes the continuity of effective models as the hole volume fraction varies.

Findings

Established convergence rates for dilute perforated domains.

Demonstrated the continuity of effective models with respect to hole volume fraction.

Applied layer potential theory to asymptotic analysis of cell problems.

Abstract

We revisit the homogenization problem for the Poisson equation in periodically perforated domains with zero Neumann data at the boundary of the holes and prescribed Dirichlet data at the outer boundary. It is known that, if the periodicity of the holes goes to zero but their volume fraction remains fixed and positive, the limit problem is a Dirichlet boundary value problem posed in the domain without the holes, and the effective diffusion coefficients are non-trivially modified; if that volume fraction goes to zero instead, i.e. the holes are dilute, the effective operator remains the Laplacian (that is, unmodified). Our main results contain the study of a "continuity" in those effective models with respect to the volume fraction of the holes and some new convergence rates for homogenization in the dilute setting. Our method explores the classical two-scale expansion ansatz and relies on asymptotic analysis of the rescaled cell problems using layer potential theory.