CONVERGENCE OF A SEMIDISCRETE SCHEME FOR THE CURVE SHORTENING FLOW

Convergence for a spatial discretization of the curvature flow for curves in possibly higher codimension is proved in L infinity ((0, T), L(2) (R/2 pi)) boolean AND L(2) ((0, T), H-1 (R/2 pi)). Asymptotic convergence in these norms is achieved for the position vector and its time derivative which is proportional to curvature. The underlying algorithm rests on a formulation of mean curvature how which uses the Laplace-Beltrami operator and leads to tridiagonal linear systems which can be easily solved.