On the convergence analysis of MsFEM with oversampling: Interpolation error

TL;DR

This paper provides a new interpolation error analysis for MsFEM with oversampling, applicable to highly oscillatory coefficients, showing error bounds depend only on the homogenized solution and mesh sizes.

Contribution

It introduces a scale-separation-independent interpolation error analysis for MsFEM with oversampling, applicable to rough coefficients without regularity assumptions.

Findings

Derived interpolation error of order H + epsilon/H

Analysis is independent of corrector regularity

Applicable to highly oscillatory periodic coefficients

Abstract

In this paper, we investigate the approximation properties of two types of multiscale finite element methods with oversampling as proposed in [Hou \& Wu, {\textit{J. Comput. Phys.}}, 1997] and [Efendiev, Hou \& Wu, \textit{SIAM J. Numer. Anal.}, 2000] without scale separation. We develop a general interpolation error analysis for elliptic problems with highly oscillatory rough coefficients, under the assumption of the existence of a macroscopic problem with suitable $L^2$-accuracy. The distinct features of the analysis, in the setting of highly oscillatory periodic coefficients, include: (i) The analysis is independent of the first-order corrector or the solutions to the cell problems, and thus independent of their regularity properties; (ii) The analysis only involves the homogenized solution and its minimal regularity. We derive an interpolation error $\mathcal{O}\left(H+\frac{\epsilon}{H}\right)$ with $\epsilon$ and $H$ being the period size and the coarse mesh size, respectively, when the oversampling domain includes one layer of elements from the target coarse element.