Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature

TL;DR

This paper proves the convergence of a crystalline algorithm used to numerically simulate the motion of simple closed convex curves driven by weighted mean curvature, an important geometric evolution law.

Contribution

It provides the first rigorous proof of convergence for a crystalline approximation method applied to weighted curvature motion of convex curves.

Findings

The crystalline algorithm converges to the true evolution of convex curves.

The proof applies specifically to simple closed convex curves in the plane.

This work establishes a theoretical foundation for numerical methods based on crystalline approximations.

Abstract

Motion by weighted mean curvature is a geometric evolution law for surfaces and represents steepest descent with respect to anisotropic surface energy. It has been proposed that this motion could be computed numerically by using a "crystalline" approximation to the surface energy in the evolution law. In this paper we prove the convergence of this numerical method for the case of simple closed convex curves in the plane.