A convergent algorithm for forced mean curvature flow driven by diffusion on the surface

TL;DR

This paper introduces a convergent numerical algorithm for simulating the evolution of surfaces driven by mean curvature flow coupled with a surface reaction-diffusion process, validated through theoretical proofs and numerical experiments.

Contribution

It develops a new convergent finite element-based algorithm for coupled surface evolution and reaction-diffusion systems, with proven convergence for polynomial degree at least two.

Findings

The algorithm converges for polynomial degree ≥ 2 and BDF orders 2-5.

Numerical experiments confirm convergence and effectiveness.

Application demonstrated in 3D tumour growth modeling.

Abstract

The evolution of a closed two-dimensional surface driven by both mean curvature flow and a reaction--diffusion process on the surface is formulated into a system, which couples the velocity law not only to the surface partial differential equation but also to the evolution equations for the geometric quantities, namely the normal vector and the mean curvature on the surface. Two algorithms are considered for the obtained system. Both methods combine surface finite elements as a space discretisation and linearly implicit backward difference formulae for time integration. Based on our recent results for mean curvature flow, one of the algorithms directly admits a convergence proof for its full discretisation in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. Numerical examples are provided to support and complement the theoretical convergence results (demonstrating the convergence properties of the method without error estimate), and demonstrate the effectiveness of the methods in simulating a three-dimensional tumour growth model.