  For a parabolic surface partial differential equation coupled to surface
evolution, convergence of the spatial semidiscretization is studied in this
paper. The velocity of the evolving surface is not given explicitly, but
depends on the solution of the parabolic equation on the surface. Various
velocity laws are considered: elliptic regularization of a direct pointwise
coupling, a regularized mean curvature flow and a dynamic velocity law. A novel
stability and convergence analysis for evolving surface finite elements for the
coupled problem of surface diffusion and surface evolution is developed. The
stability analysis works with the matrix-vector formulation of the method and
does not use geometric arguments. The geometry enters only into the consistency
estimates. Numerical experiments complement the theoretical results.
