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MiS Preprint
41/2013

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

Antoine Gloria, Stefan Neukamm and Felix Otto

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 $H^1$-norm in space of this error scales like $\varepsilon$, where $\varepsilon$ 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.

Received:
Apr 16, 2013
Published:
Apr 19, 2013
MSC Codes:
35B27, 39A70, 60H25, 60F99
Keywords:
stochastic homogenization, homogenization error, quantitative estimate

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inJournal
2014 Repository Open Access
Antoine Gloria, Stefan Neukamm and Felix Otto

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

In: ESAIM / Mathematical modelling and numerical analysis, 48 (2014) 2, pp. 325-346