An Accurate and Robust Eulerian Finite Element Method for Partial Differential Equations on Evolving Surfaces

TL;DR

This paper introduces a new Eulerian finite element method for accurately solving PDEs on evolving surfaces, leveraging a fixed bulk mesh and space-time discretization, suitable for smooth and singular topologies.

Contribution

The paper presents a higher order, robust Eulerian finite element method that naturally integrates with level set representations for evolving surfaces, including topological changes.

Findings

Achieves optimal higher order accuracy on smooth evolving surfaces.

Demonstrates robustness in handling topological singularities.

Validates effectiveness through numerical experiments.

Abstract

In this paper we present a new Eulerian finite element method for the discretization of scalar partial differential equations on evolving surfaces. In this method we use the restriction of standard space-time finite element spaces on a fixed bulk mesh to the space-time surface. The structure of the method is such that it naturally fits to a level set representation of the evolving surface. The higher order version of the method is based on a space-time variant of a mesh deformation that has been developed in the literature for stationary surfaces. The discretization method that we present is of (optimal) higher order accuracy for smoothly varying surfaces with sufficiently smooth solutions. Without any modifications the method can be used for the discretization of problems with topological singularities. A numerical study demonstrates both the higher order accuracy for smooth cases and the robustness with respect to toplogical singularities.