We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${{\mathbb{R}}}^{d}$, $d\geq 2$. 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.

中文翻译：

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离散各向异性曲线在更高维数下缩短流动


我们通过 ${{\mathbb{R}}}^{d}$, $d\geq 2$ 中的各向异性曲线缩短流引入了参数曲线演化的新颖公式。重构取决于对参数化切向速度的适当操纵，从而产生严格的抛物线微分方程。此外，导出的方程是发散形式，从而产生了自然变分数值方法。对于基于分段线性元素的完全离散有限元近似，我们证明了最佳误差估计。数值模拟证实了理论结果并证明了该方法的实用性。