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

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