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

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.

Edge multiscale methods for elliptic problems with heterogeneous coefficients

“…Motivated by [13], the local multiscale basis functions restricted on ∂ω i , which can approximate u| ∂ω i plays a vital role in approximating the solution u ∈ V in (2.1) efficiently. In view that u| ∂ω i ∈ H s (∂ω i ) for some positive constant s ≥ 1/2 and the approximation properties of the Haar wavelets, cf.…”

“…Here, we denote ψ k,j for j = 1, · · · 2 k+2 as the Haar wavelets defined on the four edges of ω i of level k and the local operator L i is defined as in Algorithm 2. The convergence of the edge spectral basis functions is a direct consequence of the results in [13]. One main observation in [13] is that the edge spectral basis functions play the critical role in the convergence analysis should the convergence rate of O(H) be after.…”

“…One of the critical challenges in [13] is to make every estimate independent of the heterogeneity in the coefficient, e.g., the mulple scales and large deviation of values. As proved in [13], among the three types of multiscale basis functions to approximate each component of the solution over each local region, the local multiscale basis functions over the coarse edges play a critical role. Its energy error estimate poses a certain difficulty in the proof, which relies mainly on the regularity properties [3,12] of the high-contrast problems and the transposition method [15].…”

A Multiscale Method for the Heterogeneous Signorini Problem

“…The following weighted Poincaré inequality is presented in [23] under Assumption 1. One may also refer to [24,25]. In this work, we assume that κ is a piecewise constant function.…”

Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions

“…Moreover, approximation in a novel space H 1/2 00 (e) is employed for each edge e, which leads to the desired guarantee of accuracy. We remark that some of the recent works [17,25] also consider enriching edge basis functions and provide a rigorous theoretical guarantee. They use L 2 (e) space for the edge, which corresponds to a weaker norm compared to H 1/2 00 (e).…”