Numerical stochastic homogenization by quasilocal effective diffusion tensors

Abstract

This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The resulting effective deterministic model is given through a quasilocal discrete integral operator, which can be further compressed to an effective partial differential operator. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.