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

Published online by Cambridge University Press:  21 January 2014

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

Keywords

Stochastic homogenizationhomogenization errorquantitative estimate
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 2: Multiscale problems and techniques , March 2014 , pp. 325 - 346
Copyright
© EDP Sciences, SMAI, 2014

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