An Eulerian space-time finite element method for diffusion problems on evolving surfaces

TL;DR

This paper develops and analyzes a novel Eulerian space-time finite element method for solving diffusion equations on evolving surfaces, providing stability analysis and numerical experiments demonstrating its effectiveness.

Contribution

It introduces a new variational formulation and a DG space-time finite element discretization for diffusion problems on evolving surfaces, with stability proofs.

Findings

The method is stable under the derived conditions.

Numerical experiments confirm the method's accuracy and robustness.

The approach effectively handles complex surface evolutions.

Abstract

In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in $\Bbb{R}^d$ defines a $d$-dimensional space-time manifold in the space-time continuum $\Bbb{R}^{d+1}$. We derive and analyze a variational formulation for a class of diffusion problems on the space-time manifold. For this variational formulation new well-posedness and stability results are derived. The analysis is based on an inf-sup condition and involves some natural, but non-standard, (anisotropic) function spaces. Based on this formulation a discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous Galerkin (DG) space-time finite element discretization. This DG space-time method is explained and results of numerical experiments are presented that illustrate its properties.