An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation

TL;DR

This paper introduces an embedded corrector problem approach for approximating homogenized coefficients in elliptic equations with oscillatory coefficients, providing convergent approximations and discussing efficient numerical methods.

Contribution

It proposes a novel embedded corrector problem method as an alternative to standard approaches for homogenization, with proven convergence and practical numerical strategies.

Findings

Three new approximations of homogenized coefficients are introduced.

The approximations converge as the radius R tends to infinity.

Efficient numerical methods for solving the embedded corrector problem are discussed.

Abstract

We consider a diffusion equation with highly oscillatory coefficients that admits a homogenized limit. As an alternative to standard corrector problems, we introduce here an embedded corrector problem, written as a diffusion equation in the whole space in which the diffusion matrix is uniform outside some ball of radius $R$. Using that problem, we next introduce three approximations of the homogenized coefficients. These approximations, which are variants of the standard approximations obtained using truncated (supercell) corrector problems, are shown to converge when $R \to \infty$. We also discuss efficient numerical methods to solve the embedded corrector problem.