Mixed Generalized Multiscale Finite Element Methods and Applications

TL;DR

This paper introduces a mixed GMsFEM framework for simulating flow in heterogeneous media, constructing multiscale basis functions that ensure mass conservation and improve accuracy with few basis functions.

Contribution

The paper develops a novel mixed GMsFEM approach with specialized basis functions and oversampling techniques, enhancing accuracy and efficiency in flow simulations in complex media.

Findings

Achieves good accuracy with few basis functions per coarse edge.

Oversampling improves the accuracy of the mixed GMsFEM.

Numerical results validate the effectiveness for two-phase flow.

Abstract

In this paper, we present a mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving flow in heterogeneous media. Our approach constructs multiscale basis functions following a GMsFEM framework and couples these basis functions using a mixed finite element method, which allows us to obtain a mass conservative velocity field. To construct multiscale basis functions for each coarse edge, we design a snapshot space that consists of fine-scale velocity fields supported in a union of two coarse regions that share the common interface. The snapshot vectors have zero Neumann boundary conditions on the outer boundaries and we prescribe their values on the common interface. We describe several spectral decompositions in the snapshot space motivated by the analysis. In the paper, we also study oversampling approaches that enhance the accuracy of mixed GMsFEM. A main idea of oversampling techniques is to introduce a small dimensional snapshot space. We present numerical results for two-phase flow and transport, without updating basis functions in time. Our numerical results show that one can achieve good accuracy with a few basis functions per coarse edge if one selects appropriate offline spaces.