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

2 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, Inria Lille - Nord Europe, ULB - Université libre de Bruxelles
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.
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Journal articles

https://hal.inria.fr/hal-00863488
Contributor : Antoine Gloria <>
Submitted on : Wednesday, September 3, 2014 - 11:11:49 AM
Last modification on : Friday, November 27, 2020 - 2:18:02 PM
Long-term archiving on: : Friday, April 14, 2017 - 1:26:09 PM

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### Citation

Antoine Gloria, Stefan Neukamm, Felix Otto. An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2014, 48 (2), pp.325-346. ⟨10.1051/m2an/2013110⟩. ⟨hal-00863488v2⟩

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