Numerical approximation of curve evolutions in Riemannian manifolds

Abstract

We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk, the elliptic plane as well as any conformal parameterization of a two-dimensional surface in ${\mathbb{R}}^d$, $d\ge 3$. In these spaces we introduce stable numerical schemes for curvature flow and curve diffusion, and we also formulate a scheme for elastic flow. Variants of the schemes can also be applied to geometric evolution equations for axisymmetric hypersurfaces in ${\mathbb{R}}^d$. Some of the schemes have very good properties with respect to the distribution of mesh points, which is demonstrated with the help of several numerical computations.

49 pages, 15 figures

References in corpus (3)

• Finite element methods for fourth order axisymmetric geometric evolution equations
• Variational discretization of axisymmetric curvature flows
• The elastic flow of curves in the hyperbolic plane

Cited by in corpus (5)

• Stable approximations for axisymmetric Willmore flow for closed and open surfaces
• A novel finite element approximation of anisotropic curve shortening flow
• Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds
• Structure preserving discretisations of gradient flows for axisymmetric two-phase biomembranes
• An unconditionally stable finite element scheme for anisotropic curve shortening flow