Generalized multiscale approximation of mixed finite elements with velocity elimination for subsurface flow

A frame work of the mixed generalized multiscale finite element method (GMsFEM) for solving Darcy's law in heterogeneous media is studied in this paper. Our approach approximates pressure in multiscale function space that is between fine-grid space and coarse-grid space and solves velocity directly in the fine-grid space. To construct multiscale basis functions for each coarse-grid element, three types of snapshot space are raised. The first one is taken as the fine-grid space for pressure and the other two cases need to solve a local problem on each coarse-grid element. We describe a spectral decomposition in the snapshot space motivated by the analysis to further reduce the dimension of the space that is used to approximate the pressure. Since the velocity is directly solved in the fine-grid space, in the linear system for the mixed finite elements, the velocity matrix can be approximated by a diagonal matrix without losing any accuracy. Thus it can be inverted easily. This reduces computational cost greatly and makes our scheme simple and easy for application. Comparing to our previous work of mixed generalized multiscale finite element method (Chung et al. (2015) [14]), both the pressure and velocity space in this approach are bigger. As a consequence, this method enjoys better accuracy. While the computational cost does not increase because of the good property of velocity matrix. Moreover, the proposed method preserves the local mass conservation property that is important for subsurface problems. Numerical examples are presented to illustrate the good properties of the proposed approach. If offline spaces are appropriately selected, one can achieve good accuracy with only a few basis functions per coarse element according to the numerical results.