MsFEM for advection-dominated problems in heterogeneous media: Stabilization via nonconforming variants

TL;DR

This paper develops and compares stabilized multiscale finite element methods for advection-diffusion problems with oscillatory coefficients, demonstrating their effectiveness across different regimes without needing stabilization parameters.

Contribution

It introduces nonconforming multiscale finite element variants with bubble functions for advection-dominated problems, enhancing stability and accuracy without auxiliary parameters.

Findings

Best approach is effective in both diffusion- and advection-dominated regimes.

Method does not require auxiliary stabilization parameters.

Numerical experiments confirm improved stability and accuracy.

Abstract

We study the numerical approximation of advection-diffusion equations with highly oscillatory coefficients and possibly dominant advection terms by means of the Multiscale Finite Element Method. The latter method is a now classical, finite element type method that performs a Galerkin approximation on a problem-dependent basis set, itself pre-computed in an offline stage. The approach is implemented here using basis functions that locally resolve both the diffusion and the advection terms. Variants with additional bubble functions and possibly weak inter-element continuity are proposed. Some theoretical arguments and a comprehensive set of numerical experiments allow to investigate and compare the stability and the accuracy of the approaches. The best approach constructed is shown to be adequate for both the diffusion- and advection-dominated regimes, and does not rely on an auxiliary stabilization parameter that would have to be properly adjusted.