Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields

TL;DR

This paper extends the analysis of an exponential correction method for computing effective properties in multiscale stochastic PDEs, demonstrating improved accuracy and efficiency through theoretical and numerical validation.

Contribution

It provides the first mathematical and numerical analysis of a modified elliptic corrector problem for stochastic homogenization, including well-posedness and bias analysis.

Findings

Exponential correction reduces resonance error in stochastic homogenization.

The modified corrector problem is well-posed under stationarity and ergodicity.

Numerical results confirm theoretical error decay and efficiency improvements.

Abstract

A central question in numerical homogenization of partial differential equations with multiscale coefficients is the accurate computation of effective quantities, such as the homogenized coefficients. Computing homogenized coefficients requires solving local corrector problems followed by upscaling relevant local data. The most naive way of computing homogenized coefficients is by solving a local elliptic problem, which is known to suffer from the so-called resonance error dominating all other errors inherent in multiscale computations. A far more efficient modelling strategy, based on adding an exponential correction term to the standard local elliptic problem, has recently been proved to result in exponentially decaying error bounds with respect to the size of the local geometry. The questions in relation with the accuracy and computational efficiency of this approach has been previously addressed in the context of periodic homogenization. The present article concerns the extension of mathematical and numerical study of this modified elliptic corrector problem to stochastic homogenization problems. In particular, we assume a stationary, ergodic micro-structure and i) establish the well-posedness of the corrector equation, ii) analyse the bias (or the systematic error) originating from additional exponential correction term in the model. Numerical results corroborating our theoretical findings are presented.