Stability and conditioning of immersed finite element methods
analysis and remedies
Frits de Prenter (TU Delft - Wind Energy)
Clemens V. Verhoosel (Eindhoven University of Technology)
Harald van Brummelen (Eindhoven University of Technology)
Mats Larson (Umeå University)
Santiago Badia (Monash University)
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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.
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