Space-time unfitted finite element methods for time-dependent problems on moving domains

TL;DR

This paper introduces a novel space-time unfitted finite element method combining discontinuous Galerkin discretization for accurately solving parabolic problems on moving domains, with proven stability and error bounds.

Contribution

It presents a new space-time unfitted finite element scheme with aggregation for robustness, along with comprehensive stability analysis and error estimates for problems on moving domains.

Findings

Method achieves stability and optimal error bounds.

Numerical experiments confirm theoretical predictions.

Applicable to problems with changing topology.

Abstract

We propose a space-time scheme that combines an unfitted finite element method in space with a discontinuous Galerkin time discretisation for the accurate numerical approximation of parabolic problems with moving domains or interfaces. We make use of an aggregated finite element space to attain robustness with respect to the cut locations. The aggregation is performed slab-wise to have a tensor product structure of the space-time discrete space, which is required in the numerical analysis. We analyse the proposed algorithm, providing stability, condition number bounds and anisotropic \emph{a priori} error estimates. A set of numerical experiments confirm the theoretical results for a parabolic problem on a moving domain. The method is applied for a mass transfer problem with changing topology.