A filtered time stepping scheme for curve shortening flow for open and closed curves

Abstract

We propose a filtered time stepping finite element scheme for curve shortening flow of open and closed curves in arbitrary codimension that is second-order accurate in time. Open curves are assumed to evolve inside a given domain $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, and meet the external boundary $\partial \Omega$ orthogonally. We prove optimal error bounds for the $L^2$-- and $H^1$--norms. In practice only a single linear system needs to be solved at each time step. Numerical experiments confirm the accuracy and practicality of the introduced method, including an asymptotic equidistribution property.

arXiv comment26 pages, 6 figures