Stochastic mixed multiscale finite element methods and their applications in random porous media

We present a framework for stochastic mixed multiscale finite element methods (mixed MsFEMs) for elliptic equations with heterogeneous random coefficients. The use of some global information is necessary in multiscale simulations when there is no scale separation for the heterogeneity. The methods in the proposed framework for the stochastic mixed MsFEMs use some global information. The media properties in a stochastic environment drastically vary among realizations and, thus, many global fields are needed for multiscale simulation. The computations of these global fields on a fine grid can be very expensive. One can utilize upscaling methods to compute the global information on an intermediate coarse grid that reduces the computational cost. We investigate two approaches of stochastic mixed MsFEMs in the framework. First approach entails no stochastic interpolation and the second approach uses stochastic interpolation. If the random media have deterministic features that play significant roles in the flow, we can use the deterministic features of the random media as the global information. This reduces the computational cost of the simulations. We make convergence analysis of the stochastic mixed MsFEMs and investigate their applications to incompressible two-phase flows in random porous media. The numerical results demonstrate the effectiveness of the proposed methods and confirm the convergence.