WS, Mats Larson: Weak and Strong Stabilization of Cut Finite Elements

Speaker
Mats Larson,Umeå University

Abstract
We review stabilization strategies for cut finite element methods and clarify how they relate. The classical approach augments the variational formulation with ghost-penalty terms that control the variation of finite element functions on cut elements. An alternative introduces a discrete extension operator and solves in a subspace of the finite element space, eliminating unstable degrees of freedom while retaining optimal approximation properties. We show that these viewpoints fit into a unified stabilization framework that satisfies standard abstract conditions and yields stable, optimally convergent methods for second-order elliptic problems. Moreover, with a robustly designed ghost penalty, the stabilization parameter can be taken to infinity without locking; in this limit, the method enforces algebraic constraints identical to those used in specific extension-operator formulations, revealing a close connection between stabilization and extension approaches. Several application examples illustrate the concepts.