Paper · Oct 9, 2021

A novel finite element approximation of anisotropic curve shortening flow

Klaus Deckelnick, Robert Nürnberg

math.NAWider flowsarXiv:2110.04605v2Interfaces Free Bound. 25 (2023) 671--708
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Abstract

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 e.g.\ 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 –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.

Comments: 32 pages, 12 figures

Curve shorteningOther curvature flowsAnisotropic & nonlocal flowsStability & uniquenessHarmonic maps & DeTurck trickNumerics & visualization

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Discrete hyperbolic curvature flow in the plane (Klaus Deckelnick and Robert Nürnberg, 2022)1Deckelnick 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)7Peachey 2022 Convex Ancient Solutions to Anisotropic Curve Shortening Flow (Theodora Bourni and Benjamin Richards, 2023)4Bourni 2023 Stability analysis for the anisotropic curve shortening flow of planar networks (Michael Gößwein, Matteo Novaga and Paola Pozzi, 2023)3Gößwein 2023 The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature (Patrick Guidotti, 2023)5Guidotti 2023 Which shapes can appear in a Curve Shortening Flow Singularity? (Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene et al., 2024)6Angenent 2024 Curve shortening flow with an ambient force field (Sam Cuthbertson, Glen Wheeler and Valentina-Mira Wheeler, 2024)8Cuthbertson 2024 Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick (Klaus Deckelnick and Vanessa Styles, 2026)2Deckelnick 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Discrete hyperbolic curvature flow in the plane

    Klaus Deckelnick and Robert NürnbergWider flows PDF

    Topics Numerics & visualization, Curve shortening, Other curvature flows · Same authors Klaus Deckelnick, Robert Nürnberg · Shared terms “error bound”, “semidiscrete finite”, “numerical simulations”

  2. Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick

    Klaus Deckelnick and Vanessa StylesWider flows PDF

    Topics Numerics & visualization, Harmonic maps & DeTurck trick · Same author Klaus Deckelnick · Shared terms “finite element”, “error bound”, “semidiscrete”

  3. Stability analysis for the anisotropic curve shortening flow of planar networks

    Michael Gößwein, Matteo Novaga and Paola PozziWider flows PDF

    Topics Anisotropic & nonlocal flows, Curve shortening, Stability & uniqueness

  4. Convex Ancient Solutions to Anisotropic Curve Shortening Flow

    Theodora Bourni and Benjamin RichardsWider flows PDF

    Topics Curve shortening · Shared terms “anisotropic curve”

  5. The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature

    Patrick GuidottiWider flows PDF

    Topics Numerics & visualization, Curve shortening · Shared terms “numerical”, “experiments”

  6. Which shapes can appear in a Curve Shortening Flow Singularity?

    Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene et al.Wider flows PDF

    Topics Numerics & visualization, Curve shortening · Shared terms “numerical simulations”

  7. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows PDF

    Topics Curve shortening, Stability & uniqueness

  8. Curve shortening flow with an ambient force field

    Sam Cuthbertson, Glen Wheeler and Valentina-Mira WheelerWider flows PDF

    Topics Curve shortening · Shared terms “anisotropic curve”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  3. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.