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.

翻译：摘要：我们提出了一种在${\mathbb R}^d$（$d\geq2$）中通过各向异性曲线缩短流演化参数化曲线的新公式。该重构依赖于对参数化切向速度的适当操作，从而得到严格抛物型微分方程。此外，推导出的方程呈散度形式，由此产生自然的变分数值方法。对于基于分段线性单元的全离散有限元逼近，我们证明了最优误差估计。数值模拟验证了理论结果并展示了该方法的实用性。