Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations

Gloria, Antoine

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 1, pp. 1-38.

We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice ${\mathbb{Z}}^d$ with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with parameter T) on some box of finite size L. In this article, we replace the regularized corrector (which is the solution of a problem posed on ${\mathbb{Z}}^d$) by some practically computable proxy on some box of size R ≥ L, and quantify the associated additional error. In order to improve the convergence, one may also consider N independent realizations of the computable proxy, and take the empirical average of the associated approximate homogenized coefficients. A natural optimization problem consists in properly choosing T, R, L and N in order to reduce the error at given computational complexity. Our analysis is sharp and sheds some light on this question. In particular, we propose and analyze a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errors we derive. The efficiency of the approach is illustrated by a numerical study in dimension 2.

Classification : 35B27, 39A70, 60H25, 65N99
Mots clés : stochastic homogenization, effective coefficients, difference operator, numerical method

[1] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators, Mathematical Notes 29. Princeton University Press, Princeton, NJ (1982). | MR | Zbl

[2] R. Alicandro, M. Cicalese and A. Gloria, Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity. Arch. Ration. Mech. Anal. 200 (2011) 881-943. | MR | Zbl

[3] A. Bourgeat and A. Piatnitski, Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré 40 (2004) 153-165. | Numdam | MR | Zbl

[4] P. Caputo and D. Ioffe, Finite volume approximation of the effective diffusion matrix: the case of independent bond disorder. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 505-525. | Numdam | MR | Zbl

[5] T. Delmotte, Inégalité de Harnack elliptique sur les graphes. Colloq. Math. 72 (1997) 19-37. | MR | Zbl

[6] A. Dykhne, Conductivity of a two-dimensional two-phase system. Sov. Phys. JETP 32 (1971) 63-65. Russian version: Zh. Eksp. Teor. Fiz. 59 (1970) 110-5.

[7] W. E, P.B. Ming and P.W. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc. 18 (2005) 121-156. | MR | Zbl

[8] A. Gloria, Reduction of the resonance error - Part 1: Approximation of homogenized coefficients. Math. Models Methods Appl. Sci., to appear. | MR | Zbl

[9] A. Gloria and F. Otto, An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39 (2011) 779-856. | MR | Zbl

[10] A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab., to appear. | MR

[11] A. Gloria and F. Otto, Quantitative estimates in stochastic homogenization of linear elliptic equations. In preparation.

[12] T.Y. Hou and X.H. Wu, A Multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | MR | Zbl

[13] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin (1994). | MR | Zbl

[14] T. Kanit, S. Forest, I. Galliet, V. Mounoury and D. Jeulin, Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Sol. Struct. 40 (2003) 3647-3679. | Zbl

[15] S.M. Kozlov, The averaging of random operators. Mat. Sb. (N.S.) 109 (1979) 188-202, 327. | MR | Zbl

[16] S.M. Kozlov, Averaging of difference schemes. Mat. Sb. 57 (1987) 351-369. | Zbl

[17] R. Künnemann, The diffusion limit for reversible jump processes on ${\mathbb{Z}}^d$ with ergodic random bond conductivities. Commun. Math. Phys. 90 (1983) 27-68. | MR | Zbl

[18] J.A. Meijerink and H.A. Van Der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp. 31 (1977) 148-162. | MR | Zbl

[19] A. Naddaf and T. Spencer, Estimates on the variance of some homogenization problems. Preprint (1998).

[20] H. Owhadi, Approximation of the effective conductivity of ergodic media by periodization. Probab. Theory Relat. Fields 125 (2003) 225-258. | MR | Zbl

[21] G.C. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in Random fields I, II (Esztergom, 1979), Colloq. Math. Soc. János Bolyai 27. North-Holland, Amsterdam (1981) 835-873. | MR | Zbl

[22] X. Yue and W. E, The local microscale problem in the multiscale modeling of strongly heterogeneous media: effects of boundary conditions and cell size. J. Comput. Phys. 222 (2007) 556-572. | MR | Zbl

[23] V.V. Yurinskii, Averaging of symmetric diffusion in random medium. Sibirskii Matematicheskii Zhurnal 27 (1986) 167-180. | MR | Zbl