On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients

Legoll, Frédéric; Thomines, Florian

ESAIM: Mathematical Modelling and Numerical Analysis , Multiscale problems and techniques. Special Issue, Tome 48 (2014) no. 2, pp. 347-386.

We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717-724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl. 88 (2007) 34-63.]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717-724.]. We first establish, in the one-dimensional case, a convergence result (with an explicit rate) on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem. We next return to the multidimensional situation. As often in random homogenization, the homogenized matrix is defined from a so-called corrector function, which is the solution to a problem set on the entire space. We describe and prove the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem.

[1] A. Anantharaman, R. Costaouec, C. Le Bris, F. Legoll and F. Thomines, Introduction to numerical stochastic homogenization and the related computational challenges: some recent developments. In vol. 22 of Lect. Not. Ser., edited by W. Bao and Q. Du. Institute for Mathematical Sciences, National University of Singapore, (2011) 197-272. | MR

[2] G. Bal, J. Garnier, Y. Gu and W. Jing, Corrector theory for elliptic equations with long-range correlated random potential. Asymptot. Anal. 77 (2012) 123-145. | MR | Zbl

[3] G. Bal, J. Garnier, S. Motsch and V. Perrier, Random integrals and correctors in homogenization. Asymptot. Anal. 59 (2008) 1-26. | MR | Zbl

[4] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977) 337-403. | MR | Zbl

[5] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures, vol. 5 of Studi. Math. Appl. North-Holland Publishing Co., Amsterdam-New York (1978). | MR | Zbl

[6] P. Billingsley, Convergence of Probability Measures. John Wiley & Sons Inc (1968). | MR | Zbl

[7] X. Blanc, C. Le Bris and P.-L. Lions, Une variante de la théorie de l'homogénéisation stochastique des opérateurs elliptiques [A variant of stochastic homogenization theory for elliptic operators]. C. R. Acad. Sci. Série I 343 (2006) 717-724. | MR | Zbl

[8] X. Blanc, C. Le Bris and P.-L. Lions, Stochastic homogenization and random lattices. J. Math. Pures Appl. 88 (2007) 34-63. | MR | Zbl

[9] A. Bourgeat and A. Piatnitski, Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator. Asymptot. Anal. 21 (1999) 303-315. | MR | Zbl

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

[11] D. Cioranescu and P. Donato, An introduction to homogenization, vol. 17 of Oxford Lect. Ser. Math. Appl. Oxford University Press, New York (1999). | MR | Zbl

[12] R. Costaouec, C. Le Bris and F. Legoll, Approximation numérique d'une classe de problèmes en homogénéisation stochastique [Numerical approximation of a class of problems in stochastic homogenization]. C. R. Acad. Sci. Série I 348 (2010) 99-103. | MR | Zbl

[13] B. Engquist and P.E. Souganidis, Asymptotic and numerical homogenization. Acta Numerica 17 (2008) 147-190. | MR | Zbl

[14] D. Henao and C. Mora-Corral, Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch. Ration. Mech. Anal. 197 (2010) 619-655. | MR | Zbl

[15] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag (1994). | MR | Zbl

[16] U. Krengel, Ergodic theorems, vol. 6 of De Gruyter Studies in Mathematics. De Gruyter (1985). | MR | Zbl

[17] G.C. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in Proc. Colloq. on Random Fields: Rigorous Results in Statistical Mechanics and Quantum Field Theory (1979). In vol. 10 of Colloquia Mathematica Societatis Janos Bolyai, edited by J. Fritz, J.L. Lebaritz and D. Szasz. North-Holland (1981) 835-873. | MR | Zbl

[18] A.N. Shiryaev, Probability, vol. 95 of Graduate Texts in Mathematics. Springer (1984). | Zbl

[19] A.A. Tempel'Man, Ergodic theorems for general dynamical systems. Trudy Moskov. Mat. Obsc. 26 (1972) 94-132. | MR | Zbl

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