An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations
Antoine Gloria, Stefan Neukamm, and Felix Otto
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Submission date: 16. Apr. 2013
published in: ESAIM / Mathematical modelling and numerical analysis, 48 (2014) 2, p. 325-346
DOI number (of the published article): 10.1051/m2an/2013110
MSC-Numbers: 35B27, 39A70, 60H25, 60F99
Keywords and phrases: stochastic homogenization, homogenization error, quantitative estimate
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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 Greens function by Marahrens and the third author.