A discrete scheme for regularized anisotropic surface diffusion: a 6th order geometric evolution equation

Abstract

We study anisotropic surface diffusion of curves with a small corner energy regularization. The regularization allows the use of non-convex free energy densities and turns the evolution law into a 6th order geometric equation. Using a semi-implicit time discretization, we present a variational formulation of this equation for parametric curves, leading to a discretization based on linear finite elements. The resulting linear system is shown to be uniquely solvable. Numerical examples include the convergence of closed curves to the Wulff shape and the evolution of a thermodynamically unstable surface into a hill-valley structure and its subsequent coarsening.