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

TL;DR

This paper introduces self-similar shrinking solutions for the crystalline mean curvature flow in three dimensions, using them to evaluate a new finite element-based numerical algorithm that improves edge handling.

Contribution

It presents novel self-similar solutions for crystalline mean curvature flow in 3D and demonstrates an improved finite element method for simulating the flow.

Findings

Self-similar solutions for crystalline mean curvature flow in 3D are constructed.

The finite element discretization enhances edge handling in simulations.

Numerical experiments illustrate the effectiveness of the new method.

Abstract

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.