Convergence rates for the homogenization of the Poisson problem in randomly perforated domains

TL;DR

This paper establishes convergence rates for the homogenization of the Poisson problem in randomly perforated domains, considering stochastic hole distributions and their impact on solution approximation.

Contribution

It provides the first quantitative convergence rates for homogenization in domains perforated by randomly sized spherical holes generated by point processes.

Findings

Convergence rates depend on the moment condition of the hole radii.

Homogenization is achieved even with overlapping holes for certain parameters.

Rates are explicitly characterized in terms of the parameter eta.

Abstract

In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of $\mathbb{R}^d$, $d \geq 3$. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process $(\Phi, \mathcal{R})$. The point process $\Phi$ generating the centres of the holes is either a Poisson point process or the lattice $\mathbb{Z}^d$; the marks $\mathcal{R}$ generating the radii are unbounded i.i.d random variables having finite $(d-2+\beta)$-moment, for $\beta > 0$. We study the rate of convergence to the homogenized solution in terms of the parameter $\beta$. We stress that, for certain values of $\beta$, the balls generating the holes may overlap with overwhelming probability.