Numerical Analysis for a System Coupling Curve Evolution to Reaction Diffusion on the Curve

We consider a finite element approximation for a system consisting of the evolution of a closed planar curve by forced curve shortening flow coupled to a reaction-diffusion equation on the evolving curve. The scheme for the curve evolution is based on a parametric description allowing for tangential motion, whereas the discretization for the PDE on the curve uses an idea from [G. Dziuk and C. M. Elliott, IMA J. Numer. Anal., 27 (2007), pp. 262–292]. We prove optimal error bounds for the resulting fully discrete approximation and present numerical experiments. These confirm our estimates and also illustrate the advantage of the tangential motion of the mesh points in practice.