An Optimal Variance Estimate in Stochastic Homogenization of Discrete Elliptic Equations
Antoine Gloria, Félix Otto, Antoine Gloria, Félix Otto · 2011
Abstract. We consider a discrete elliptic equation on the d-dimensional lattice Zd 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 Ahom = ahomId is characterized by ξ · Ahomξ = 〈(ξ + ∇φ) · A(ξ + ∇φ) 〉 for any direction ξ ∈ Rd, where the random field φ (the “corrector”) is the unique stationary solution of −∇ ∗ · A(ξ +∇φ) = 0 normalized by 〈φ 〉 = 0, and 〈· 〉 denotes the ensemble average. It is known (“by ergodicity”) that the above ensemble average of the energy density E = (ξ +∇φ) · A(ξ +∇φ), which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of E on length scales L satisfies the optimal estimate, i. e. var [ ∑ EηL]. L−d, where the averaging function (i. e. ηL = 1, supp (ηL) ⊂ {|x | ≤ L}) has to be smooth in the sense that |∇ηL |. L−1−d. In two space dimensions (i. e. d = 2), there is a logarithmic correction. This estimate is optimal since it shows that smooth averages of the energy density E decay in L as if E would be independent from edge to edge (which it is not for d> 1). This result is of practical significance, since it allows to estimate the dominant error when numerically computing ahom.