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:我们从古典异位曲线中将DeTurck 的诡计从缩短流到厌食环境。 在这里, 允许厌食性能量密度依赖于空间, 允许对芬斯勒测量值进行解释, 从而在里曼尼多元体中产生大地测量曲线流。 假设密度是纯正和平稳的, 我们为厌食性曲线缩短流引入一种新型的弱化配方。 然后我们得出一个最佳的 $H$1$- error, 结合为连续的半分解定元素, 使用片态线性元素。 此外, 我们考虑一些完全实用的完全离散的计划, 并证明它们无条件的稳定性 。 最后, 我们提出数个数字模拟, 包括一些确认导出误差的趋同实验, 以及晶体曲线流和地标曲流的应用 。