A novel finite element approximation of anisotropic curve shortening flow

TL;DR

This paper extends the anisotropic curve shortening flow to include space-dependent energy densities, introduces a new weak formulation, and provides error bounds and stable numerical schemes with applications to geometric flows.

Contribution

It develops a novel weak formulation for anisotropic curve shortening flow with space-dependent densities and establishes error bounds and stable finite element schemes.

Findings

Optimal $H^1$--error bound for semidiscrete approximation

Unconditional stability of fully discrete schemes

Numerical simulations confirming theoretical results

Abstract

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.