Multiscale Finite Element methods for advection-dominated problems in perforated domains

TL;DR

This paper investigates multiscale finite element methods for advection-dominated problems in perforated domains, focusing on their behavior, variants, and stabilization techniques to improve numerical solutions.

Contribution

It introduces and compares various multiscale finite element variants, including basis choices, bubble functions, and stabilization for advection-diffusion in perforated domains.

Findings

Different multiscale basis functions are analyzed.

Stabilized formulations improve solution accuracy.

Comparison of methods guides future applications.

Abstract

We consider an advection-diffusion equation that is advection-dominated and posed on a perforated domain. On the boundary of the perforations, we set either homogeneous Dirichlet or homogeneous Neumann conditions. The purpose of this work is to investigate the behavior of several variants of Multiscale Finite Element type methods, all of them based upon local functions satisfying weak continuity conditions in the Crouzeix-Raviart sense on the boundary of mesh elements. In the spirit of our previous works [Le Bris, Legoll and Lozinski, CAM 2013 and MMS 2014] introducing such multiscale basis functions, and of [Le Bris, Legoll and Madiot, M2AN 2017] assessing their interest for advection-diffusion problems, we present, study and compare various options in terms of choice of basis elements, adjunction of bubble functions and stabilized formulations.