Convergence Rate of Multiscale Finite Element Method for Various Boundary Problems

TL;DR

This paper analyzes the convergence rate of the multiscale finite element method (MsFEM) for various boundary problems, emphasizing the first-order expansion structure over boundary correctors, and provides improved theoretical estimates.

Contribution

It introduces a novel approach focusing on first-order expansion structures, leading to milder assumptions and clearer convergence rate estimates for MsFEM.

Findings

Convergence rates are established under less restrictive assumptions.

The first-order expansion structure is effective for analyzing MsFEM.

Results apply to mixed Dirichlet-Neumann, Robin, and hemivariational inequality problems.

Abstract

In this paper, we examine the effectiveness of classic multiscale finite element method (MsFEM) (Hou and Wu, 1997; Hou et al., 1999) for mixed Dirichlet-Neumann, Robin and hemivariational inequality boundary problems. Constructing so-called boundary correctors is a common technique in existing methods to prove the convergence rate of MsFEM, while we think not reflects the essence of those problems. Instead, we focus on the first-order expansion structure. Through recently developed estimations in homogenization theory, our convergence rate is provided with milder assumptions and in neat forms.