Abstract
We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open curves we propose natural boundary conditions that respect the appropriate gradient flow structure. Based on suitable weak formulations we introduce finite element approximations using piecewise linear elements. For some of the schemes a stability result can be shown. The derived schemes can be employed in very different contexts. For example, we apply the schemes to the Angenent metric in order to numerically compute rotationally symmetric self-shrinkers for the mean curvature flow. Furthermore, we utilise the schemes to compute geodesics that are relevant for optimal interface profiles in multi-component phase field models.
Highlights
In this paper we consider numerical approximations for gradient flows of curves evolving in Riemannian manifolds that are conformally flat
We allow both closed and open curves, where in the latter case appropriate boundary conditions need be considered in order to respect the required gradient flow structure
We define the Riemannian manifold with the help of its metric tensor as follows
Summary
In this paper we consider numerical approximations for gradient flows of curves evolving in Riemannian manifolds that are conformally flat. We are not aware of existing work on boundary value problems for elastic curves in Riemannian manifolds, but note that the case of a Euclidean ambient space has been considered in e.g. Barrett, have considered the evolution of closed curves in Riemannian manifolds that are conformally equivalent to the Euclidean plane in the recent works [11,12] Building on these works, in this paper we derive boundary value problems for curvature flow and elastic flow in such manifolds, and we believe that for elastic flow the obtained formulations are new in the literature. Usefulness of curvature flow and elastic flow in computing geodesics in Riemannian manifolds, both in the case of closed curves, and in the case of curves with boundary conditions We end this introduction with the presentation of some example metrics for (1.1).
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