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

¹ UniversitéLibre de Bruxelles (ULB) Brussels, Belgium and Project-team SIMPAF Inria Lille - Nord Europe Villeneuve d’Ascq, France.
agloria@ulb.ac.be
² Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig, Germany
neukamm@mis.mpg.de; otto@mis.mpg.de
³ Present address: Weierstraß-Institut Berlin, Germany.

Received: 3 August 2013

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 L²-norm in probability of the H¹-norm in space of this error scales like ε, where ε 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.