Based on earlier work by the authors, in this paper we introduce novel fully
discrete, fully practical parametric finite element approximations for geometric evolution
equations of curves in the plane. The fully implicit approximations are unconditionally
stable and intrinsically equidistribute the vertices at each time level.
We present iterative solution methods for the systems of nonlinear equations arising
at each time level and present several numerical results. The ideas easily generalize
to the evolution of curve networks and to anisotropic surface energies.