Optimal error bounds in stochastic homogenization
Seminar Room 1, Newton Institute
We consider one of the simplest set-ups in stochastic homogenization:
A discrete elliptic differential equation on a d-dimensional lattice with identically independently distributed bond conductivities. It is well-known that on scales large w. r. t. the grid size, the resolvent operator behaves like that of a homogeneous, deterministic (and
continuous) elliptic equation. The homogenized coefficients can be characterized by an ensemble average with help of the corrector problem. For a numerical treatment, this formula has to be approximated in two ways: The corrector problem has to be solved on a finite sublattice (with, say, periodic boundary conditions) and the ensemble average has to be replaced by a spatial average. We give estimates on both errors that are optimal in terms of the scaling in the size of the sublattice. This is joint work with Antoine Gloria (INRIA Lille).