An optimal error estimate in stochastic homogenization of discrete elliptic equations

Abstract

This paper is the companion article to [Ann. Probab. 39 (2011) 779–856]. We consider a discrete elliptic equation on the d-dimensional lattice ℤ^d with random coefficients A of the simplest type: They are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric “homogenized” matrix A_hom = a_homId is characterized by ξ ⋅ A_homξ = 〈(ξ + ∇ϕ) ⋅ A(ξ + ∇ϕ)〉 for any direction ξ ∈ ℝ^d, where the random field ϕ (the “corrector”) is the unique solution of −∇* ⋅ A(ξ + ∇ϕ) = 0 in ℤ^d such that ϕ(0) = 0, ∇ϕ is stationary and 〈∇ϕ〉 = 0, 〈⋅〉 denoting the ensemble average (or expectation).

In order to approximate the homogenized coefficients A_hom, the corrector problem is usually solved in a box Q_L = [−L, L)^d of size 2L with periodic boundary conditions, and the space averaged energy on Q_L defines an approximation A_L of A_hom. Although the statistics is modified (independence is replaced by periodic correlations) and the ensemble average is replaced by a space average, the approximation A_L converges almost surely to A_hom as L ↑ ∞. In this paper, we give estimates on both errors. To be more precise, we do not consider periodic boundary conditions on a box of size 2L, but replace the elliptic operator by T⁻¹ − ∇^∗ ⋅ A∇ with (typically) T ∼ L², as standard in the homogenization literature. We then replace the ensemble average by a space average on Q_L, and estimate the overall error on the homogenized coefficients in terms of L and T.