Numerical surgery for mean curvature flow of surfaces

TL;DR

This paper introduces a finite element numerical algorithm for simulating mean curvature flow of convex surfaces, incorporating a surgery process inspired by analytical techniques to handle singularities.

Contribution

It presents a novel numerical surgery method for mean curvature flow, closely following analytical surgery procedures, with detailed implementation and extensions to other flows.

Findings

Effective numerical surgery process demonstrated through experiments

Algorithm accurately approximates mean curvature and normals

Extensions to other geometric flows discussed

Abstract

A numerical algorithm for mean curvature flow of closed mean convex surfaces with surgery is proposed. The method uses a finite element based mean curvature flow algorithm based on a coupled partial differential equation system which directly provides an approximation for mean curvature and outward unit normal. The proposed numerical surgery process closely follows the analytical surgery of Huisken \& Sinestrari, and Brendle \& Huisken. The numerical surgery approach is described in detail, along with extensions to other geometric flows and methods. Numerical experiments report on the performance of the numerical surgery process.