Numerical approximation of gradient flows for closed curves in Rd

We present parametric finite element approximations of curvature flows for in Rd, d ≥ 2, as well as for curves on two-dimensional manifolds in R3. we consider the curve shortening flow, curve diffusion and the elastic flow. It demonstrated that the curve shortening and the elastic flows on manifolds can be to compute nontrivial geodesics, and that the corresponding geodesic curve diffusion leads to solutions of partitioning problems on two-dimensional manifolds R3. In addition, we extend these schemes to anisotropic surface energy densities. presented schemes have very good properties with respect to stability and the of mesh points, and hence no remeshing is needed in practice.