Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces

TL;DR

This paper develops and analyzes finite element methods for simulating the coupled evolution of a surface under mean curvature flow and a reaction-diffusion process, demonstrating convergence and qualitative flow properties.

Contribution

It introduces two algorithms for coupled surface and diffusion evolution, proving convergence of one method and illustrating key flow behaviors through numerical experiments.

Findings

Convergence of the proposed finite element method in the $H^1$ norm.

Preservation and loss of convexity observed in simulations.

Flow exhibits maximum principles and self-intersections.

Abstract

An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction--diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the $H^1$ norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.