519

IGPM519.pdf February 2022 
TITLE 
Analysis of optimal preconditioners for CutFEM 
AUTHORS 
Sven Groß, Arnold Reusken 
ABSTRACT 
In this paper we consider a class of unfitted finite element methods for scalar elliptic
problems. These socalled CutFEM methods use standard finite element spaces on a fixed unfitted
triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model
problem we consider the application of such a method to the Poisson interface problem. We introduce
and analyze a new class of preconditioners that is based on a subspace decomposition approach. The
unfitted finite element space is split into two subspaces, where one subspace is the standard finite
element space associated to the background mesh and the second subspace is spanned by all cut basis
functions corresponding to nodes on the cut elements. We will show that this splitting is stable,
uniformly in the discretization parameter and in the location of the interface in the triangulation.
Based on this we introduce an efficient preconditioner that is uniformly spectrally equivalent to the
stiffness matrix. Using a similar splitting, it is shown that the same preconditioning approach can
also be applied to a fictitious domain CutFEM discretization of the Poisson equation. Results of
numerical experiments are included that illustrate optimality of such preconditioners for the Poisson
interface problem and the Poisson fictitious domain problem.

KEYWORDS 
unfitted finite elements, CutFEM, Nitsche method, interface problem, fictitious domain method, preconditioner 