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.
2 Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig, Germany
3 Present address: Weierstraß-Institut Berlin, Germany.
Received: 3 August 2013
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.
Mathematics Subject Classification: 35B27 / 39A70 / 60H25 / 60F99
Key words: Stochastic homogenization / homogenization error / quantitative estimate
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