An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations

TL;DR

This paper proves an optimal convergence rate for stochastic homogenization of discrete elliptic equations, showing that the error between the random solution and the two-scale expansion scales linearly with the discretization parameter.

Contribution

It establishes the first optimal, linear convergence rate for the stochastic homogenization of discrete linear elliptic equations with i.i.d. coefficients.

Findings

Error in $L^2$-norm scales like $oxed{ ext{discretization parameter } oldsymbol{ ext{ extit{ extepsilon}}}}$

Convergence rate matches the periodic case, indicating optimality

Uses advanced Green's function estimates and previous homogenization results

Abstract

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the $L^2$-norm in probability of the \mbox{$H^1$-norm} in space of this error scales like $\epsilon$, where $\epsilon$ is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green's function by Marahrens and the third author.