Approximation of homogenized coefficients in deterministic homogenization and convergence rates in the asymptotic almost periodic setting

TL;DR

This paper develops an approximation scheme for homogenized coefficients in deterministic homogenization, proves the existence of a distributional corrector, and analyzes near-optimal convergence rates in the asymptotic almost periodic setting, supported by numerical simulations.

Contribution

It introduces a new approximation scheme for homogenized coefficients and establishes convergence rates in a non-periodic setting, advancing numerical methods in homogenization theory.

Findings

Existence of a distributional corrector for the homogenization problem.

Near-optimal convergence rates in the asymptotic almost periodic setting.

Numerical simulations validating theoretical results.

Abstract

For a homogenization problem associated to a linear elliptic operator, we prove the existence of a distributional corrector and we find an approximation scheme for the homogenized coefficients. We also study the convergence rates in the asymptotic almost periodic setting, and we show that the rates of convergence for the zero order approximation, are near optimal. The results obtained constitute a step towards the numerical implementation of results from the deterministic homogenization theory beyond the periodic setting. To illustrate this, numerical simulations based on finite volume method are provided to sustain our theoretical results.