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We consider a numerical scheme for the approximation of a system that couplesthe evolution of a two--dimensional hypersurface to a reaction--diffusionequation on the surface. The surfaces are assumed to be graphs and evolveaccording to forced mean curvature flow. The method uses continuous, piecewiselinear finite elements in space and a backward Euler scheme in time. Assumingthe existence of a smooth solution we prove optimal error bounds both in$L^\infty(L^2)$ and in $L^2(H^1)$. We present several numerical experimentsthat confirm our theoretical findings and apply the method in order to simulatediffusion induced grain boundary motion.

Finite element error analysis for a system coupling surface evolution to diffusion on the surface