A trace finite element method for PDEs on evolving surfaces

TL;DR

This paper introduces a trace finite element method combined with a fast marching approach for solving PDEs on evolving surfaces, capable of handling topological changes without surface extension.

Contribution

It presents a novel Eulerian framework that uses a fixed background mesh and implicitly defined surfaces, enabling accurate PDE solutions on complex evolving geometries.

Findings

Second order accuracy in space and time for the proposed method

Effective handling of topological changes in evolving surfaces

Numerical experiments confirm convergence and robustness

Abstract

In this paper, we propose an approach for solving PDEs on evolving surfaces using a combination of the trace finite element method and a fast marching method. The numerical approach is based on the Eulerian description of the surface problem and employs a time-independent background mesh that is not fitted to the surface. The surface and its evolution may be given implicitly, for example, by the level set method. Extension of the PDE off the surface is not required. The method introduced in this paper naturally allows a surface to undergo topological changes and experience local geometric singularities. In the simplest setting, the numerical method is second order accurate in space and time. Higher order variants are feasible, but not studied in this paper. We show results of several numerical experiments, which demonstrate the convergence properties of the method and its ability to handle the case of the surface with topological changes.