Adaptive generalized multiscale approximation of a mixed finite element method with velocity elimination

TL;DR

This paper develops and analyzes offline and online adaptive algorithms for a multiscale mixed finite element method with velocity elimination, effectively solving high-contrast subsurface flow problems with proven convergence and improved accuracy.

Contribution

It introduces novel adaptive enrichment algorithms with theoretical convergence analysis for multiscale mixed finite element methods in heterogeneous media.

Findings

Both adaptive methods significantly reduce error.

Online adaptive method outperforms offline in accuracy.

Convergence rate is independent of permeability contrast with proper initial space.

Abstract

In this paper, we propose offline and online adaptive enrichment algorithms for the generalized multiscale approximation of a mixed finite element method with velocity elimination to solve the subsurface flow problem in high-contrast and heterogeneous porous media. We give the theoretical analysis for the convergence of these two adaptive methods, which shows that sufficient initial basis functions (belong to the offline space) leads to a faster convergence rate. A series of numerical examples are provided to highlight the performance of both these two adaptive methods and also validate the theoretical analysis. Both offline and online adaptive methods are effective that can reduce the relative error substantially. In addition, the online adaptive method generally performs better than the offline adaptive method as online basis functions contain important global information such as distant effects that cannot be captured by offline basis functions. The numerical results also show that with a suitable initial multiscale space that includes all offline basis functions corresponding to relative smaller eigenvalues of local spectral decompositions in the offline stage, the convergence rate of the online enrichment is independent of the permeability contrast.