Oversampling for the Multiscale Finite Element Method

TL;DR

This paper introduces a novel constrained oversampling strategy for the Multiscale Finite Element Method, providing the first rigorous convergence proof and connecting it to the Variational Multiscale Method in homogenization.

Contribution

It proposes a new oversampling approach with linear independence constraints, reinterprets MsFEM via the Variational Multiscale Method, and offers the first convergence proof for oversampling in MsFEM.

Findings

Reinterpreted MsFEM within the Variational Multiscale framework.

Developed a fully discrete error analysis for the constrained oversampling method.

Provided the first rigorous convergence proof for MsFEM with oversampling.

Abstract

This paper reviews standard oversampling strategies as performed in the Multiscale Finite Element Method (MsFEM). Common to those approaches is that the oversampling is performed in the full space restricted to a patch but including coarse finite element functions. We suggest, by contrast, to perform local computations with the additional constraint that trial and test functions are linear independent from coarse finite element functions. This approach re-interprets the Variational Multiscale Method in the context of computational homogenization. This connection gives rise to a general fully discrete error analysis for the proposed multiscale method with constrained oversampling without any resonance effects. In particular, we are able to give the first rigorous proof of convergence for a MsFEM with oversampling.