Numerical approximation of gradient flows for closed curves in Rd

Abstract

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