On the self-similar solutions of the crystalline mean curvature flow in three dimensions

Understand

We present two types of self-similar shrinking solutions of positive genus for the crystalline mean curvature flow in three dimensions analogous to the solutions known for the standard mean curvature flow.

• We use them to test a numerical implementation of a level set algorithm for the crystalline mean curvature flow in three dimensions based on the minimizing movements scheme of A.~Chambolle, \textit{Interfaces Free Bound.~6}~(2004).
• We implement a finite element method discretization that seems to improve the handling of edges in three dimensions compared to the standard finite difference method and illustrate its behavior on a few examples.