  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.
