Error Estimate of Multiscale Finite Element Method for Periodic Media Revisited

TL;DR

This paper provides an optimal energy error estimate for the multiscale finite element method with oversampling in periodic media, including convergence rates in specific norms, accommodating rough microstructures.

Contribution

It derives the first optimal energy error estimate for multiscale finite element methods with oversampling in periodic media with rough microstructures.

Findings

Optimal energy error estimate established

Convergence rate in L^{d/(d-1)}-norm derived

Applicable to media with rough microstructures

Abstract

We derive the optimal energy error estimate for multiscale finite element method with oversampling technique applying to elliptic system with rapidly oscillating periodic coefficients under the assumption that the coefficients are bounded and measurable, which may admit rough microstructures. As a by-product of the energy estimate, we derive the rate of convergence in L$^{d/(d-1)}-$norm.