A variational formulation of anisotropic geometric evolution equations in higher dimensions

Abstract

We present a novel variational formulation of fully anisotropic motion by surface diffusion and mean curvature flow in $$\mathbb {R}^d$$, d ≥ 2. This new formulation leads to an unconditionally stable, fully discrete, parametric finite element approximation in the case d = 2 or 3. The resulting scheme has very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments for d = 3, including regularized crystalline mean curvature flow and regularized crystalline surface diffusion.