Alternative to evolving surface finite element method

TL;DR

This paper introduces an alternative numerical approach to the evolving surface finite element method (ESFEM) for solving linear advection-diffusion equations on evolving surfaces, transforming the problem to a fixed surface for easier computation.

Contribution

The paper proposes a transformed equation approach that simplifies solving PDEs on evolving surfaces by mapping them onto a fixed initial surface, offering a comparable accuracy to ESFEM.

Findings

Both approaches achieve similar accuracy in numerical experiments.

The transformed equation method simplifies computations on evolving surfaces.

Numerical examples validate the effectiveness of the proposed method.

Abstract

ESFEM is a method introduced in order to solve a linear advection-diffusion equation on an evolving two-dimensional surface with finite elements by using a moving grid with nodes sitting on and evolving with the surface. The evolution of the surface is assumed to be given as a smooth one-parameter family of embeddings of a fixed initial surface into $\mathbb{R}^3$ satisfying uniform $C^4$ bounds. We calculate an equivalent transformed equation which is defined on the fixed initial surface and can hence be solved numerically on a fixed grid. We present numerical examples which indicate that both approaches are essentially of the same accuracy.