Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds

TL;DR

This paper develops finite element schemes for approximating boundary value problems related to curvature and elastic flows in conformally flat Riemannian manifolds, with applications to self-shrinkers and geodesics.

Contribution

It introduces variational finite element methods with natural boundary conditions that preserve the gradient flow structure for curvature and elastic flows in 2D Riemannian manifolds.

Findings

Stable schemes for curvature and elastic flows are proposed.

Numerical computation of self-shrinkers in Angenent metric demonstrated.

Geodesics relevant to phase field models are computed.

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.