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

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.

Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion

“…Finally, all four of the mentioned contributions address numerical analysis exclusively whereas this work is purely analytic and yields a short-time existence result. We also refer to the recent contribution of Elliott, Garcke and Kovács in [5] who analyze a finite element approximation of (1.1) relying on the existence result presented in this work. In a forthcoming paper, we will discuss several properties of solutions to (1.1), placing emphasis on to what extent the hypersurface in our setting qualitatively evolves as for the usual mean curvature flow.…”

Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion

“…In particular, we prove that a solution of (1.1) fulfills mass conservation. Several of these properties are illustrated in the work of Elliott, Garcke and Kovács [10] that is concerned with numerical analysis for a generalization of our system of equations.…”