We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics, giving rise to e.g.\ geodesic curvature flow in Riemannian manifolds. Assuming that the density is strictly convex and smooth, we introduce a novel weak formulation for anisotropic curve shortening flow. We then derive an optimal $H^1$--error bound for a continuous-in-time semidiscrete finite element approximation that uses piecewise linear elements. In addition, we consider some fully practical fully discrete schemes and prove their unconditional stability. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bound, as well as applications to crystalline curvature flow and geodesic curvature flow.

翻译：我们将经典的各向同性曲线缩短流中的德图尔克技巧扩展至各向异性情形。此处各向异性能量密度允许依赖于空间位置，这可在芬斯勒度量的框架下进行解释，例如引出了黎曼流形中的测地曲率流。假设密度函数严格凸且光滑，我们引入了一种针对各向异性曲线缩短流的新型弱形式。随后推导了基于分段线性元的连续时间半离散有限元逼近的最优$H^1$误差界。此外，我们考虑了若干完全实用的全离散格式并证明了其无条件稳定性。最后，通过数值模拟验证了所得误差界，并展示了该方法在晶化曲率流与测地曲率流中的应用。