Discrete anisotropic curve shortening flow in higher codimension

TL;DR

This paper introduces a new formulation for anisotropic curve shortening flow in higher dimensions, leading to a divergence form PDE and a variational numerical method with proven optimal error estimates.

Contribution

The paper presents a novel divergence form PDE formulation for anisotropic curve shortening flow in higher codimension and develops a finite element method with optimal error bounds.

Findings

Numerical simulations confirm theoretical error estimates.

The method is practical and effective for higher-dimensional curves.

The PDE formulation is strictly parabolic and divergence in form.

Abstract

We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${\mathbb R}^d$, $d\geq2$. The reformulation hinges on a suitable manipulation of the parameterization's tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method. For a fully discrete finite element approximation based on piecewise linear elements we prove optimal error estimates. Numerical simulations confirm the theoretical results and demonstrate the practicality of the method.