A gradient smoothing method and its multiscale variant for flows in heterogeneous porous media

Abstract

In this paper, we study gradient smoothing methods (GSMs) with improved convergence behaviors for high-contrast problems such as the flow in heterogeneous porous media. We propose a GSM that is adaptive to the heterogeneity of the problem in the sense that gradient smoothing is applied inside each of the homogeneous regions separately. The smoothed gradient field of each element is constructed by utilizing the gradient fields of the adjacent elements within the corresponding homogeneous region. In addition, we propose a multiscale variant of the proposed GSM, which results in an accurate numerical solution even if a computational grid coarser than the scale of the heterogeneity is utilized. While existing GSMs have stiffness matrices with larger bandwidth than the standard finite element method (FEM) in general, the proposed multiscale GSM has the same bandwidth as the plain multiscale FEM. Hence, its online computational cost is essentially the same as that of the plain one. Improved performance of the proposed methods is demonstrated through various numerical examples.