A Space-Time Cut Finite Element Method with quadrature in time

TL;DR

This paper introduces a space-time cut finite element method with quadrature in time for convection-diffusion problems on evolving domains, featuring a new stabilization to ensure well-conditioned systems without remeshing.

Contribution

It extends previous cut finite element methods to higher order elements on evolving surfaces and introduces a novel stabilization term for better numerical stability.

Findings

The method avoids remeshing by embedding the evolving domain in a fixed mesh.

A new stabilization guarantees well-conditioned systems regardless of domain position.

Extension to higher order elements improves accuracy on evolving surfaces.

Abstract

We consider convection-diffusion problems in time-dependent domains and present a space-time finite element method based on quadrature in time which is simple to implement and avoids remeshing procedures as the domain is moving. The evolving domain is embedded in a domain with fixed mesh and a cut finite element method with continuous elements in space and discontinuous elements in time is proposed. The method allows the evolving geometry to cut through the fixed background mesh arbitrarily and thus avoids remeshing procedures. However, the arbitrary cuts may lead to ill-conditioned algebraic systems. A stabilization term is added to the weak form which guarantees well-conditioned linear systems independently of the position of the geometry relative to the fixed mesh and in addition makes it possible to use quadrature rules in time to approximate the space-time integrals. We review here the space-time cut finite element method presented in [13] where linear elements are used in both space and time and extend the method to higher order elements for problems on evolving surfaces (or interfaces). We present a new stabilization term which also when higher order elements are used controls the condition number of the linear systems from cut finite element methods on evolving surfaces. The new stabilization combines the consistent ghost penalty stabilization [1] with a term controlling normal derivatives at the interface.