An optimal error estimate in stochastic homogenization of discrete elliptic equations

An optimal error estimate in stochastic homogenization of discrete elliptic equations

Abstract. This paper is the companion article of [6]. We consider a discrete elliptic equation on the d-dimensional lattice Z 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...

An optimal variance estimate in stochastic homogenization of discrete elliptic equations

An Optimal Error Estimate in Stochastic Homogenization of Discrete Elliptic Equations1 by Antoine Gloria and Felix Otto

This paper is the companion article to [Ann. Probab. 39 (2011) 779–856]. 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 (co...

An Optimal Variance Estimate in Stochastic Homogenization of Discrete Elliptic Equations by Antoine Gloria and Felix Otto

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. Thi...

An Optimal Quantitative Two-scale Expansion in Stochastic Homogenization of Discrete Elliptic Equations

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...