Numerical Approximation of Anisotropic Geometric Evolution Equations

Abstract

We present a variational formulation of fully anisotropic motion by surface diffusion and mean curvature flow, as well as related flows. The proposed scheme covers both the closed curve case, and the case of curves that are connected via triple junction points. On introducing a parametric finite element approximation, we prove stability bounds and report on numerical experiments, including crystalline mean curvature flow and crystalline surface diffusion. The presented scheme has very good properties with respect to the equidistribution of mesh points and, if applicable, area conservation.