A Note on Iterations-based Derivations of High-order Homogenization Correctors for Multiscale Semi-linear Elliptic Equations

TL;DR

This paper introduces a straightforward method combining monotone iterations and two-scale homogenization to derive high-order corrector estimates for semi-linear elliptic equations in perforated domains, enhancing multiscale analysis.

Contribution

It presents a novel, simple procedure for high-order homogenization correctors applicable to complex semi-linear elliptic problems in perforated domains.

Findings

Efficient derivation of high-order correctors for homogenization.

Applicable to nonlinear boundary conditions and coupled vectorial models.

Method simplifies multiscale homogenization analysis.

Abstract

This Note aims at presenting a simple and efficient procedure to derive the structure of high-order corrector estimates for the homogenization limit applied to a semi-linear elliptic equation posed in perforated domains. Our working technique relies on monotone iterations combined with formal two-scale homogenization asymptotics. It can be adapted to handle more complex scenarios including for instance nonlinearities posed at the boundary of perforations and the vectorial case, when the model equations are coupled only through the nonlinear production terms.