An MsFEM type approach for perforated domains

TL;DR

This paper develops a multiscale finite element method tailored for perforated domains, enhancing accuracy with bubble functions and outperforming existing MsFEM variants in theoretical and numerical assessments.

Contribution

It introduces a novel MsFEM approach for perforated domains with bubble function enrichment, providing theoretical error estimates and demonstrating superior performance.

Findings

The method achieves improved accuracy over existing MsFEM variants.

Theoretical error estimates validate the approach.

Numerical results confirm the method's efficiency.

Abstract

We follow up on our previous work [C. Le Bris, F. Legoll and A. Lozinski, Chinese Annals of Mathematics 2013] where we have studied a multiscale finite element (MsFEM) type method in the vein of the classical Crouzeix-Raviart finite element method that is specifically adapted for highly oscillatory elliptic problems. We adapt the approach to address here a multiscale problem on a perforated domain. An additional ingredient of our approach is the enrichment of the multiscale finite element space using bubble functions. We first establish a theoretical error estimate. We next show that, on the problem we consider, the approach we propose outperforms all dedicated existing variants of MsFEM we are aware of.