Curve shortening flow coupled to lateral diffusion

TL;DR

This paper develops and analyzes a finite element scheme for a coupled system of curve shortening flow with a diffusion process on the evolving curve, providing convergence proofs and numerical validation.

Contribution

It introduces a semi-discrete finite element method for a coupled geometric and parabolic PDE system on an evolving curve, extending previous methods with new error estimates.

Findings

The scheme converges under the proposed estimates.

Numerical results confirm the theoretical convergence.

The analysis includes error estimates for the length element and its time derivative.

Abstract

We present and analyze a semi-discrete finite element scheme for a system consisting of a geometric evolution equation for a curve and a parabolic equation on the evolving curve. More precisely, curve shortening flow with a forcing term that depends on a field defined on the curve is coupled with a diffusion equation for that field. The scheme is based on ideas of \cite{D99} for the curve shortening flow and \cite{DE07} for the parabolic equation on the moving curve. Additional estimates are required in order to show convergence, most notably with respect to the length element: While in \cite{D99} an estimate of its error was sufficient we here also need to estimate the time derivative of the error which arises from the diffusion equation. Numerical simulation results support the theoretical findings.