Mixed multiscale finite element methods for stochastic porous media flows

In this paper, we propose a stochastic mixed multiscale finite element method. The proposed method solves the stochastic porous media flow equation on the coarse grid using a set of precomputed basis functions. The precomputed basis functions are constructed based on selected realizations of the stochastic permeability field, and furthermore the solution is projected onto the finite-dimensional space spanned by these basis functions. We employ multiscale methods using limited global information since the permeability fields do not have apparent scale separation. The proposed approach does not require any interpolation in stochastic space and can easily be coupled with interpolation-based approaches to predict the solution on the coarse grid. Numerical results are presented for permeability fields with normal and exponential variograms.