On the Convergence Rates of GMsFEMs for Heterogeneous Elliptic Problems without Oversampling Techniques

TL;DR

This paper provides a rigorous analysis of the convergence rates of GMsFEMs for high-contrast heterogeneous elliptic problems, establishing optimal convergence without oversampling techniques and offering insights into their efficiency.

Contribution

It introduces a theoretical framework proving optimal convergence of GMsFEMs with local spectral and harmonic basis functions without oversampling.

Findings

Proves optimal convergence in energy norm for GMsFEMs.

Shows convergence of model order reduction for harmonic basis.

Provides theoretical justification for GMsFEM efficiency.

Abstract

This work is concerned with the rigorous analysis on the Generalized Multiscale Finite Element Methods (GMsFEMs) for elliptic problems with high-contrast heterogeneous coefficients. GMsFEMs are popular numerical methods for solving flow problems with heterogeneous high-contrast coefficients, and it has demonstrated extremely promising numerical results for a wide range of applications. However, the mathematical justification of the efficiency of the method is still largely missing. In this work, we analyze two types of multiscale basis functions, i.e., local spectral basis functions and basis functions of local harmonic extension type, within the GMsFEM framework. These constructions have found many applications in the past few years. We establish their optimal convergence in the energy norm under a very mild assumption that the source term belongs to some weighted $L^2$ space, and without the help of any oversampling technique. Furthermore, we analyze the model order reduction of the local harmonic extension basis and prove its convergence in the energy norm. These theoretical findings shed insights into the mechanism behind the efficiency of the GMsFEMs.