Error analysis for a finite difference scheme for axisymmetric mean curvature flow of genus-0 surfaces

TL;DR

This paper develops and analyzes a finite difference scheme for axisymmetric mean curvature flow of genus-0 surfaces, providing error bounds and numerical validation for the method.

Contribution

It introduces a novel finite difference approach that carefully handles degeneracy at the axis, with proven error bounds and numerical experiments confirming accuracy.

Findings

Error bounds established in $L^2$ and $H^1$ norms.

Numerical convergence experiments validate theoretical results.

Simulations include non-embedded self-shrinkers for mean curvature flow.

Abstract

We consider a finite difference approximation of mean curvature flow for axisymmetric surfaces of genus zero. A careful treatment of the degeneracy at the axis of rotation for the one dimensional partial differential equation for a parameterization of the generating curve allows us to prove error bounds with respect to discrete $L^2$- and $H^1$-norms for a fully discrete approximation. The theoretical results are confirmed with the help of numerical convergence experiments. We also present numerical simulations for some genus-0 surfaces, including for a non-embedded self-shrinker for mean curvature flow.