A Higher Order Unfitted Space-Time Finite Element Method for Coupled Surface-Bulk problems

TL;DR

This paper introduces a higher order unfitted finite element method for coupled surface-bulk convection-diffusion problems with evolving geometries, combining existing methods and extending them to higher orders with confirmed convergence.

Contribution

It develops a higher order space-time unfitted finite element approach for coupled surface-bulk problems with dynamic geometries, extending previous methods to arbitrary high orders.

Findings

Numerical convergence confirmed for higher order discretizations.

Method effectively handles evolving geometries with isoparametric mapping.

Generalizes previous lower order results to higher orders.

Abstract

We present a higher order space-time unfitted finite element method for convection-diffusion problems on coupled (surface and bulk) domains. In that way, we combine a method suggested by Heimann, Lehrenfeld, Preu{\ss} (SIAM J. Sci. Comput. 45(2), 2023, B139 - B165) for the bulk case with a method suggested by Sass, Reusken (Comput. Math. Appl. 146(15), 2023, 253-270) for the surface case. The geometry is allowed to change with time, and the higher order discrete approximation of this geometry is ensured by a time-dependent isoparametric mapping. The space-time discretisation approach allows for straightforward handling of arbitrary high orders. In that way, we also generalise results of Hansbo, Larson, Zahedi (Comput. Methods Appl. Mech. Engrg. 307, 2016, 96-116) to higher orders. The convergence of the proposed higher order discretisations is confirmed numerically.