A combined finite element and multiscale finite element method for the multiscale elliptic problems

TL;DR

This paper introduces a hybrid finite element and multiscale finite element method for efficiently solving multiscale elliptic problems, especially in challenging regions where traditional MsFEM struggles, with proven convergence and demonstrated accuracy.

Contribution

The paper develops a combined FE-MsFEM approach that integrates standard finite elements with oversampling MsFEM, improving applicability and efficiency in complex multiscale domains.

Findings

The method achieves rigorous convergence under periodic diffusion coefficients.

Numerical results show high accuracy for periodic and random coefficients.

The approach effectively handles high contrast channels in multiscale problems.

Abstract

The oversampling multiscale finite element method (MsFEM) is one of the most popular methods for simulating composite materials and flows in porous media which may have many scales. But the method may be inapplicable or inefficient in some portions of the computational domain, e.g., near the domain boundary or near long narrow channels inside the domain due to the lack of permeability information outside of the domain or the fact that the high-conductivity features cannot be localized within a coarse-grid block. In this paper we develop a combined finite element and multiscale finite element method (FE-MsFEM), which deals with such portions by using the standard finite element method on a fine mesh and the other portions by the oversampling MsFEM. The transmission conditions across the FE-MSFE interface is treated by the penalty technique. A rigorous convergence analysis for this special FE-MsFEM is given under the assumption that the diffusion coefficient is periodic. Numerical experiments are carried out for the elliptic equations with periodic and random highly oscillating coefficients, as well as multiscale problems with high contrast channels, to demonstrate the accuracy and efficiency of the proposed method.