Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T19:27:17.400Z Has data issue: false hasContentIssue false
  • English
  • Français

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.. [email protected]
Stefan Neukamm
Affiliation:
Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig, Germany; [email protected]; [email protected] Present address: Weierstraß-Institut Berlin, Germany.
Felix Otto
Affiliation:
Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig, Germany; [email protected]; [email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allaire, G. and Amar, M., Boundary layer tails in periodic homogenization. ESAIM: COCV 4 (1999) 209–243. Google Scholar
Avellaneda, M. and Lin, F.-H., Compactness methods in the theory of homogenization. Commun. Pure Appl. Math. 40 (1987) 803–847. Google Scholar
Bourgeat, A. and Piatnitski, A., 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. Google Scholar
J.G. Conlon and T. Spencer, Strong convergence to the homogenized limit of elliptic equations with random coefficients. Trans. AMS, in press.
Gloria, A., Fluctuation of solutions to linear elliptic equations with noisy diffusion coefficients. Commun. Partial Differ. Eq. 38 (2013) 304–338. Google Scholar
A. Gloria, S. Neukamm, and F. Otto, Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. MPI Preprint 91 (2013).
A. Gloria, S. Neukamm and F. Otto, Approximation of effective coefficients by periodization in stochastic homogenization. In preparation.
A. Gloria and F. Otto, Quantitative results on the corrector equation in stochastic homogenization of linear elliptic equations. In preparation.
Gloria, A. and Otto, F., An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39 (2011) 779–856. Google Scholar
Gloria, A. and Otto, F., An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22 (2012) 1–28. Google Scholar
R.J. Leveque, Finite difference methods for ordinary and partial differential equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2007).
Kozlov, S.M., The averaging of random operators. Mat. Sb. (N.S.) 109 (1979) 188–202, 327. Google Scholar
Künnemann, R., The diffusion limit for reversible jump processes on Zd with ergodic random bond conductivities. Commun. Math. Phys. 90 (1983) 27–68. Google Scholar
D. Marahrens and F. Otto, Annealed estimates on the Green’s function. MPI Preprint 69 (2012).
S.J.N. Mosconi, Discrete regularity for elliptic equations on graphs. CVGMT. Available at http://cvgmt.sns.it/papers/53 (2001).
A. Naddaf and T. Spencer, Estimates on the variance of some homogenization problems. Preprint (1998).
Owhadi, H., Approximation of the effective conductivity of ergodic media by periodization. Probab. Theory Relat. Fields 125 (2003) 225–258. Google Scholar
G.C. Papanicolaou and S.R.S. Varadhan, 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.
Yurinskiĭ, V.V., Averaging of symmetric diffusion in random medium. Sibirskii Matematicheskii Zhurnal 27 (1986) 167–180. Google Scholar