Analysis of stochastic mimetic finite difference methods and their applications in single-phase stochastic flows

摘要

Stochastic modeling has become a widely accepted approach to quantify uncertainty in applications where diffusion plays a central role. In many of these applications the geometry is complex, and important properties of the underlying deterministic continuum model need to be captured accurately. To address these problems we present a stochastic mimetic finite difference (MFD) method for diffusion equations with random input data. Specifically, we use the MFD methodology for the spatial approximation to ensure the necessary accuracy and robustness is achieved. To treat the high-dimensionality of the stochastic approximation efficiently, we use a stochastic collocation method. We consider the stochastic MFD approximation in hybrid form, and perform a rigorous analysis of its semi-discretization and full discretization for the pressure, flux and Lagrange multipliers. Convergence rates are developed for statistical moments of the quantities of interest. Numerical results are presented for single-phase flow in random porous media, and support the efficiency of the stochastic MFD.