Discrete anisotropic curve shortening flow in higher codimension

TL;DR

This work develops a mathematically rigorous and computationally practical framework for anisotropic curve shortening flow of curves in arbitrary codimension. By applying a DeTurck-type reformulation, the authors obtain a strictly parabolic, divergence-form PDE that admits a variational finite element discretization. They prove unconditional stability for the fully discrete scheme and establish optimal error bounds: a in L2-norm of order $h^2$ in the H1 estimate and order $h^4$ in L2, under a mild coupling between space and time discretization. Numerical experiments in plane and in 3D validate convergence rates and demonstrate robust behavior across isotropic and anisotropic, including crystalline-like, anisotropies, while also illustrating how tangential motion yields well-distributed mesh points and the method’s ability to handle singularities up to the limits of the theory.

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.

Figures (11)

• Figure 1: Plots of the ratio ${\mathfrak r}^m$ for \ref{['eq:fea']} over time. Left: \ref{['eq:fea']}, middle: Dziuk99, right: triplejANI.
• Figure 2: Anisotropic curvature flow for an ellipse for the anisotropy \ref{['eq:gammaD99b']} with $(k,{\widehat{\delta}})=(3, 0.124)$. Solution at times $t=0,0.05,\ldots,0.25$ on the left. We also show plots of the discrete energy $E_\phi(x_h^m)$ (middle) and of the ratio ${\mathfrak r}^m$ (right) over time.
• Figure 3: Anisotropic curvature flow for a spiral for the anisotropy \ref{['eq:bgnL2']}. Solution at times $t=0,0.01,0.015,0.019$.
• Figure 4: Isotropic curve shortening flow for the trefoil knot \ref{['eq:trefoil']}. On the left, $x_h^m$ at times $t=0,0.5,\ldots,2,T=2.45$, with $x_h^M$ on the right. Below we show plots of $E_\phi(x_h^m)$, ${\mathfrak r}^m$ and $1/K^m_\infty$ over time.
• Figure 5: Isotropic curve shortening flow for the two interlocked rings \ref{['eq:irings']}. On the left, $x_h^m$ at times $t=0,0.25,\ldots,2,T=2.1$, in the middle $x^M_h$, with $x_h^m$ at time $t=0.5$ on the right. Below we also show plots of $E_\phi(x_h^m)$, ${\mathfrak r}^m$ and $1/K^m_\infty$ over time.

Theorems & Definitions (8)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof