The Method of Reflections, Homogenization and Screening for Poisson and Stokes Equations in Perforated Domains

TL;DR

This paper analyzes the convergence of the Method of Reflections for solving Poisson and Stokes equations in perforated domains, providing conditions for convergence, modifications for divergence cases, and new proofs of classical homogenization results.

Contribution

It introduces a modified Method of Reflections that converges in broader scenarios and offers new proofs for classical homogenization results in perforated domains.

Findings

Method converges if capacity density is small and domain is bounded.

Method diverges with large capacity density or unbounded domains.

Modified method achieves convergence in previously divergent cases.

Abstract

We study the convergence of the Method of Reflections for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains which consist in the exterior of balls. We prove that the method converges if the balls are contained in a bounded region and the density of the electrostatic capacity of the balls is sufficiently small. If the capacity density is too large or the balls extend to the whole space, the method diverges, but we provide a suitable modification of the method that converges to the solution of the Dirichlet problem also in this case. We give new proofs of classical homogenization results for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains using the (modified) Method of Reflections.