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

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.

Cited by in corpus (8)

• The additive structure of elliptic homogenization
• Lipschitz regularity for elliptic equations with random coefficients
• Mesoscopic higher regularity and subadditivity in elliptic homogenization
• Quantitative estimates in stochastic homogenization for correlated coefficient fields
• Higher-order pathwise theory of fluctuations in stochastic homogenization
• The choice of representative volumes in the approximation of effective properties of random materials
• Quantitative nonlinear homogenization: control of oscillations
• Uniform estimate of an iterative method for elliptic problems with rapidly oscillating coefficients