Stochastic homogenization of linear elliptic equations: Higher-order error estimates in weak norms via second-order correctors

TL;DR

This paper develops higher-order error estimates for the stochastic homogenization of linear elliptic equations with random coefficients, using second-order correctors and weak norms, applicable to symmetric and nonsymmetric cases, including systems.

Contribution

It introduces novel estimates for second-order homogenization correctors and extends error analysis to weak norms for both symmetric and nonsymmetric coefficient fields.

Findings

Higher-order approximation accuracy in weak norms

Optimal error estimates for second-order two-scale expansion

Extension to elliptic systems and nonsymmetric coefficients

Abstract

We are concerned with the homogenization of second-order linear elliptic equations with random coefficient fields. For symmetric coefficient fields with only short-range correlations, quantified through a logarithmic Sobolev inequality for the ensemble, we prove that when measured in weak spatial norms, the solution to the homogenized equation provides a higher-order approximation of the solution to the equation with oscillating coefficients. In the case of nonsymmetric coefficient fields, we provide a higher-order approximation (in weak spatial norms) of the solution to the equation with oscillating coefficients in terms of solutions to constant-coefficient equations. In both settings, we also provide optimal error estimates for the two-scale expansion truncated at second order. Our results rely on novel estimates on the second-order homogenization corrector, which we establish via sensitivity estimates for the second-order corrector and a large-scale $L^p$ theory for elliptic equations with random coefficients. Our results also cover the case of elliptic systems.