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, for instance, 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 H1-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.
1
Otto-von-Guericke-Universität Magdeburg, Germany
2
University of Trento, Italy
Klaus Deckelnick; Robert Nürnberg. A novel finite element approximation of anisotropic curve shortening flow. Interfaces and free boundaries, Tome 25 (2023) no. 4, pp. 671-708. doi: 10.4171/ifb/500
@article{10_4171_ifb_500,
author = {Klaus Deckelnick and Robert N\"urnberg},
title = {A novel finite element approximation of anisotropic curve shortening flow},
journal = {Interfaces and free boundaries},
pages = {671--708},
year = {2023},
volume = {25},
number = {4},
doi = {10.4171/ifb/500},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/500/}
}
TY - JOUR
AU - Klaus Deckelnick
AU - Robert Nürnberg
TI - A novel finite element approximation of anisotropic curve shortening flow
JO - Interfaces and free boundaries
PY - 2023
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UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/500/
DO - 10.4171/ifb/500
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