On the convergence rates of GMSFEMs for heterogeneous elliptic problems without oversampling techniques

Abstract

\bigcirc c 2019 Society for Industrial and Applied Mathematics This work is concerned with the rigorous analysis of 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 they have 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 or H1 seminorm under a very mild assumption that the source term belongs to some weighted L2 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 offer insight into the mechanism behind the efficiency of the GMsFEMs.