An MsFEM approach enriched using Legendre polynomials

TL;DR

This paper introduces an enriched MsFEM method using Legendre polynomials, providing convergence analysis, error estimates, and demonstrating reduced errors and resonance issues in numerical experiments.

Contribution

It presents a novel MsFEM variant with Legendre polynomial enrichments, improving accuracy and robustness against resonance errors.

Findings

Significant error reduction shown in numerical experiments

Approach less prone to resonance errors when coarse mesh size is comparable to oscillation scale

Provides convergence analysis and a posteriori error estimates

Abstract

We consider a variant of the conventional MsFEM approach with enrichments based on Legendre polynomials, both in the bulk of mesh elements and on their interfaces. A convergence analysis of the approach is presented. Residue-type a posteriori error estimates are also established. Numerical experiments show a significant reduction in the error at a limited additional off-line cost. In particular, the approach developed here is less prone to resonance errors in the regime where the coarse mesh size $H$ is of the order of the small scale $\varepsilon$ of the oscillations.