Stability and Conditioning of Immersed Finite Element Methods: Analysis and Remedies

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.

Geometrical discretisations for unfitted finite elements on explicit boundary representations

“…In ghost penalty formulations, one can readily use the non-aggregated spaces on the active mesh and add stabilization terms (see, e.g., [33]) to make the problem robust with respect to cut locations. Thus, the space-time FE spaces to be used in this case is simply V 𝑛 ℎ .…”

Algebra preconditionings for 2D Riesz distributed‐order space‐fractional diffusion equations on convex domains

“…u(x, y, 0) = 1 Ω (x, y)u 0 (x, y), (x, y) ∈ Ω, u(x, y, t) = 0, (x, y) ∈ R 2 ⧵ Ω and t ∈ (0, T], (17) then the computation of the symbol can still be performed by resorting to the GLT theory. Indeed, the discretization of Equation ( 17) in matrix form reads as (10) and (11), and it can be easily shown that…”

Unfittted finite element methods for explicit boundary representations

“…In ghost penalty formulations, one can readily use the non-aggregated spaces on the active mesh and add stabilization terms (see, e.g., [42]) to make the problem robust with respect to cut locations. Thus, the space-time FE spaces to be used in this case is simply V n h .…”