Numerical approximation of curve evolutions in Riemannian manifolds

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Title: Numerical approximation of curve evolutions in Riemannian manifolds
Authors: Barrett, JW
Garcke, H
Nürnberg, R
Item Type: Working Paper
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 ${\mathbb R}^d$, $d\geq 3$. 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.
Copyright Statement: © 2018 The Author(s).
Keywords: math.NA
65M60, 53C44, 53A30, 35K55
Notes: 49 pages, 15 figures
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences

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