A multiscale virtual element method for elliptic problems in heterogeneous porous media

Abstract

In this paper, we propose a Multiscale Virtual Element Method (MsVEM) for elliptic problems in heterogeneous porous media. The use of very general coarse grids has advantages in subsurface simulations since they provide flexibility and can render more accurate coarse model. It is often necessary to use an irregular coarse grid when highly heterogeneous reservoirs are discretized via irregular anisotropic fine grids. The proposed MsVEM aims to build a coarse model on a flexible grid by integrating Generalized Multiscale Finite Element Method (GMsFEM) with Virtual Element Method (VEM). Because of the irregularity merit of VEM, MsVEM is desirable to simulate complex real-world problems. In the framework of MsVEM, both coarse grid and fine grid may consist of more general polygons. To build multiscale basis functions on coarse grid, a few local problems are solved on each coarse block to enrich multiscale information and get a snapshot space. To significantly reduce degree of freedoms of the coarse multiscale model, we construct a low dimensional multiscale basis space by using greedy selection procedure and penalized matrix decomposition. The greedy selection procedure is utilized to select optimal multiscale basis functions. The penalized matrix decomposition is employed to improve approximation accuracy and get the sparse basis functions. To show the efficiency and accuracy, we present a few numerical examples for flows in heterogeneous porous media. In particular, MsVEM is applied to the subsurface flow in high-contrast porous media and fractured media.