A cut finite element method for coupled bulk-surface problems on time-dependent domains

TL;DR

This paper introduces a novel cut finite element method for accurately and stably solving coupled bulk-surface problems on moving domains using a space-time formulation and level set representation.

Contribution

The paper presents a new cut finite element approach with stabilization for coupled bulk-surface problems on time-dependent domains, ensuring accuracy and stability.

Findings

Method achieves high accuracy in numerical tests.

Algebraic systems are well-conditioned regardless of interface position.

Stabilization effectively handles convection and system stability.

Abstract

In this contribution we present a new computational method for coupled bulk-surface problems on time-dependent domains. The method is based on a space-time formulation using discontinuous piecewise linear elements in time and continuous piecewise linear elements in space on a fixed background mesh. The domain is represented using a piecewise linear level set function on the background mesh and a cut finite element method is used to discretize the bulk and surface problems. In the cut finite element method the bilinear forms associated with the weak formulation of the problem are directly evaluated on the bulk domain and the surface defined by the level set, essentially using the restrictions of the piecewise linear functions to the computational domain. In addition a stabilization term is added to stabilize convection as well as the resulting algebraic system that is solved in each time step. We show in numerical examples that the resulting method is accurate and stable and results in well conditioned algebraic systems independent of the position of the interface relative to the background mesh.