A priori error analysis of a numerical stochastic homogenization method

TL;DR

This paper analyzes the expected error of a localized orthogonal decomposition method for stochastic homogenization, showing it depends on mesh size and correlation length under certain statistical assumptions.

Contribution

It provides the first a priori error estimate for the LOD method in stochastic homogenization with quantitative decorrelation assumptions.

Findings

Expected $L^2$ error estimated as $H+(rac{ ext{small correlation length}}{H})^{d/2}$

Error bound includes logarithmic factors and applies under spectral gap inequality

Bridges recent numerical homogenization and stochastic homogenization results

Abstract

This paper provides an a~priori error analysis of a localized orthogonal decomposition method (LOD) for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in form of the spectral gap inequality, then the expected $L^2$ error of the method can be estimated, up to logarithmic factors, by $H+(\varepsilon/H)^{d/2}$; $\varepsilon$ being the small correlation length of the random coefficient and $H$ the width of the coarse finite element mesh that determines the spatial resolution. The proof bridges recent results of numerical homogenization and quantitative stochastic homogenization.