On the numerical approximation of hyperbolic mean curvature flows for surfaces

TL;DR

This paper develops finite element and finite difference methods for numerically approximating hyperbolic mean curvature flows of surfaces in three-dimensional space, including convergence analysis and singularity simulations.

Contribution

It introduces novel numerical schemes for hyperbolic mean curvature flows and applies them to both general and axially symmetric surfaces.

Findings

Numerical schemes demonstrate convergence in tests.

Simulations reveal potential singularity formation.

Methods effectively handle different surface symmetries.

Abstract

The paper addresses the numerical approximation of two variants of hyperbolic mean curvature flow of surfaces in $\mathbb R^3$. For each evolution law we propose both a finite element method, as well as a finite difference scheme in the case of axially symmetric surfaces. We present a number of numerical simulations, including convergence tests as well as simulations suggesting the onset of singularities.