An optimal error estimate in stochastic homogenization of discrete elliptic equations

TL;DR

This paper provides an optimal error estimate for the approximation of solutions to discrete elliptic equations with random coefficients by their homogenized counterparts, enhancing understanding of stochastic homogenization accuracy.

Contribution

It establishes a sharp error estimate in stochastic homogenization for discrete elliptic equations with i.i.d. random coefficients, extending previous results with optimal bounds.

Findings

Derived an optimal error bound for homogenization approximation

Quantified the rate at which solutions converge to homogenized solutions

Enhanced theoretical understanding of stochastic homogenization accuracy

Abstract

This paper is the companion article to [Ann. Probab. 39 (2011) 779--856]. We consider a discrete elliptic equation on the $d$-dimensional lattice $\mathbb{Z}^d$ with random coefficients $A$ of the simplest type: They are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric "homogenized" matrix $A_{\mathrm{hom}}=a_{\mathrm{hom}}\mathrm{Id}$ is characterized by $\xi\cdot A_{\mathrm{hom}}\xi=<(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)>$ for any direction $\xi\in\mathbb{R}^d$, where the random field $\phi$ (the "corrector") is the unique solution of $-\nabla^*\cdot A(\xi+\nabla\phi)=0$ in $\mathbb{Z}^d$ such that $\phi(0)=0$, $\nabla\phi$ is stationary and $<\nabla\phi>=0$, $<\cdot>$ denoting the ensemble average (or expectation).