Numerical Analysis for a System Coupling Curve Evolution to Reaction-Diffusion on the Curve

TL;DR

This paper develops a finite element method for simulating the evolution of a closed curve coupled with a reaction-diffusion process on it, providing error bounds and demonstrating the benefits of tangential mesh motion.

Contribution

It introduces a novel finite element scheme for coupled curve evolution and reaction-diffusion equations, with proven optimal error bounds and practical advantages of tangential mesh movement.

Findings

Numerical experiments confirm the theoretical error estimates.

Tangential motion of mesh points improves numerical stability and accuracy.

The method effectively captures the coupled dynamics of curve evolution and reaction-diffusion.

Abstract

We consider a finite element approximation for a system consisting of the evolution of a closed planar curve by forced curve shortening flow coupled to a reaction-diffusion equation on the evolving curve. The scheme for the curve evolution is based on a parametric description allowing for tangential motion, whereas the discretisation for the PDE on the curve uses an idea from [6]. We prove optimal error bounds for the resulting fully discrete approximation and present numerical experiments. These confirm our estimates and also illustrate the advantage of the tangential motion of the mesh points in practice.