  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.
