Stability and conditioning of immersed finite element methods: analysis and remedies

TL;DR

This review analyzes the stability and conditioning issues in immersed finite element methods, discussing remedies like Schwarz preconditioning, element aggregation, and ghost penalty, and outlines future research directions.

Contribution

It provides a comprehensive overview of stability and conditioning challenges in immersed finite element methods and reviews developed remedies and methodologies.

Findings

Schwarz preconditioning improves stability

Element aggregation enhances conditioning

Ghost penalty formulation mitigates adverse effects

Abstract

This review paper discusses the developments in immersed or unfitted finite element methods over the past decade. The main focus is the analysis and the treatment of the adverse effects of small cut elements. We distinguish between adverse effects regarding the stability and adverse effects regarding the conditioning of the system, and we present an overview of the developed remedies. In particular, we provide a detailed explanation of Schwarz preconditioning, element aggregation, and the ghost penalty formulation. Furthermore, we outline the methodologies developed for quadrature and weak enforcement of Dirichlet conditions, and we discuss open questions and future research directions.