Numerical approximation of level set power mean curvature flow

TL;DR

This paper develops and tests a finite element method for numerically approximating a generalized mean curvature flow of curves, analyzing convergence and demonstrating flow improvements in isoperimetric properties.

Contribution

It introduces a finite element discretization for the level set formulation of power mean curvature flow and verifies convergence rates through numerical experiments.

Findings

Numerical scheme accurately approximates the flow for various k values.

Convergence rates align with theoretical error bounds.

Flow improves the isoperimetric deficit for specific initial curves.

Abstract

In this paper we investigate the numerical approximation of a variant of the mean curvature flow. We consider the evolution of hypersurfaces with normal speed given by $H^k$, $k \ge 1$, where $H$ denotes the mean curvature. We use a level set formulation of this flow and discretize the regularized level set equation with finite elements. In a previous paper we proved an a priori estimate for the approximation error between the finite element solution and the solution of the original level set equation. We obtained an upper bound for this error which is polynomial in the discretization parameter and the reciprocal regularization parameter. The aim of the present paper is the numerical study of the behavior of the evolution and the numerical verification of certain convergence rates. We restrict the consideration to the case that the level set function depends on two variables, i.e. the moving hypersurfaces are curves. Furthermore, we confirm for specific initial curves and different values of $k$ that the flow improves the isoperimetrical deficit.