An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations. (21st January 2014)

• Record Type: Journal Article
• Authors: Gloria, Antoine; Neukamm, Stefan; Otto, Felix
• Is Part Of: Mathematical modelling and numerical analysis. Volume 48:Part 2(2014)
• Page Start: 325
• Page End: 346
• Publication Date: 2014-01-21
• Subjects: Numerical analysis -- Periodicals; Mathematical models -- Periodicals; 510
• DOI: 10.1051/m2an/2013110
• Languages: English
• ISSNs: 0764-583X

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>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 <italic>L</italic>²-norm in probability of the <italic>H</italic>¹-norm in space of this error scales like <italic>ε</italic>, where <italic>ε</italic> 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.</p> </abstract>