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.
@article{M2AN_2014__48_2_325_0,
author = {Gloria, Antoine and Neukamm, Stefan and Otto, Felix},
title = {An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {48},
year = {2014},
pages = {325-346},
doi = {10.1051/m2an/2013110},
mrnumber = {3177848},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2014__48_2_325_0}
}
Gloria, Antoine; Neukamm, Stefan; Otto, Felix. An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) pp. 325-346. doi : 10.1051/m2an/2013110. http://gdmltest.u-ga.fr/item/M2AN_2014__48_2_325_0/
[1] and , Boundary layer tails in periodic homogenization. ESAIM: COCV 4 (1999) 209-243. | Numdam | MR 1696289 | Zbl 0922.35014
[2] and , Compactness methods in the theory of homogenization. Commun. Pure Appl. Math. 40 (1987) 803-847. | MR 910954 | Zbl 0632.35018
[3] and , Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator. Asymptotic Anal. 21 (1999) 303-315. | MR 1728027 | Zbl 0960.60057
[4] and , Strong convergence to the homogenized limit of elliptic equations with random coefficients. Trans. AMS, in press. | Zbl 1283.81102
[5] , Fluctuation of solutions to linear elliptic equations with noisy diffusion coefficients. Commun. Partial Differ. Eq. 38 (2013) 304-338. | MR 3009082 | Zbl 1270.35063
[6] , , and , Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. MPI Preprint 91 (2013). | MR 3302119
[7] , and , Approximation of effective coefficients by periodization in stochastic homogenization. In preparation.
[8] and , Quantitative results on the corrector equation in stochastic homogenization of linear elliptic equations. In preparation.
[9] and , An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39 (2011) 779-856. | MR 2789576 | Zbl 1215.35025
[10] and , An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22 (2012) 1-28. | MR 2932541 | Zbl pre06026087
[11] , Finite difference methods for ordinary and partial differential equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2007). | MR 2378550 | Zbl 1127.65080
[12] , The averaging of random operators. Mat. Sb. (N.S.) 109 (1979) 188-202, 327. | MR 542557 | Zbl 0415.60059
[13] , The diffusion limit for reversible jump processes on Zd with ergodic random bond conductivities. Commun. Math. Phys. 90 (1983) 27-68. | MR 714611 | Zbl 0523.60097
[14] and , Annealed estimates on the Green's function. MPI Preprint 69 (2012).
[15] , Discrete regularity for elliptic equations on graphs. CVGMT. Available at http://cvgmt.sns.it/papers/53 (2001).
[16] and , Estimates on the variance of some homogenization problems. Preprint (1998).
[17] , Approximation of the effective conductivity of ergodic media by periodization. Probab. Theory Relat. Fields 125 (2003) 225-258. | MR 1961343 | Zbl 1040.60025
[18] and , Boundary value problems with rapidly oscillating random coefficients. In Random fields, Vol. I, II (Esztergom, 1979), vol. 27 of Colloq. Math. Soc. János Bolyai. North-Holland, Amsterdam (1981) 835-873. | MR 712714 | Zbl 0499.60059
[19] , Averaging of symmetric diffusion in random medium. Sibirskii Matematicheskii Zhurnal 27 (1986) 167-180. | MR 867870 | Zbl 0614.60051