Paper · Dec 4, 2020

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

Harald Garcke, Robert Nürnberg

math.NAWider flowsarXiv:2012.02707v4
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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.

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Submitted
Dec 4, 2020
Last revised
Jul 20, 2021 (v4)
Subjects
math.NA, math.DG
Journal
Numer. Math. 149 (2021) 375--415
DOI
10.1007/s00211-021-01231-6

Comments: 42 pages, 21 figures

Mean curvature flowCurve shorteningOther curvature flowsSelf-shrinkers & translatorsSurfacesStability & uniquenessBoundaries & non-compactRotational symmetry & warped productsNumerics & visualizationPhase fields & multiphase flows

Its neighbourhood

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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 2016, 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 A survey of closed self-shrinkers with symmetry (Gregory Drugan, Hojoo Lee and Xuan Hien Nguyen, 2017)6Drugan 2017 Curve-shortening flow of open, elastic curves in ℝ² with repelling endpoints: A minimizing movement approach (Rufat Badal, 2019)3Badal 2019 Numerically computing the index of mean curvature flow self-shrinkers (Yakov Berchenko-Kogan, 2020)5Berchenko-Ko 2020 A novel finite element approximation of anisotropic curve shortening flow (Klaus Deckelnick and Robert Nürnberg, 2021)1Deckelnick 2021 Discrete hyperbolic curvature flow in the plane (Klaus Deckelnick and Robert Nürnberg, 2022)2Deckelnick 2022 Generalized Elastic Translating Solitons (Alvaro Pampano, 2022)4Pampano 2022 Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds (Miroslav Kolar and Daniel Sevcovic, 2025)7Kolar 2025 Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick (Klaus Deckelnick and Vanessa Styles, 2026)8Deckelnick 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A novel finite element approximation of anisotropic curve shortening flow

    Klaus Deckelnick and Robert NürnbergWider flows

    Topics Numerics & visualization, Curve shortening, Other curvature flows · Same author Robert Nürnberg · Shared terms “element approximation”, “finite element”, “linear elements”

  2. Discrete hyperbolic curvature flow in the plane

    Klaus Deckelnick and Robert NürnbergWider flows

    Topics Numerics & visualization, Curve shortening, Other curvature flows · Same author Robert Nürnberg · Shared terms “numerical”, “approximation”, “interface”

  3. Curve-shortening flow of open, elastic curves in with repelling endpoints: A minimizing movement approach

    Rufat BadalWider flows

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

  4. Generalized Elastic Translating Solitons

    Alvaro PampanoWider flows

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “variational”

  5. Numerically computing the index of mean curvature flow self-shrinkers

    Yakov Berchenko-KoganWider flows

    Topics Numerics & visualization, Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “symmetric self”, “numerically”, “Angenent”

  6. A survey of closed self-shrinkers with symmetry

    Gregory Drugan, Hojoo Lee and Xuan Hien NguyenWider flows

    Topics Numerics & visualization, Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “numerical”

  7. Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds

    Miroslav Kolar and Daniel SevcovicWider flows

    Topics Numerics & visualization, Surfaces, Stability & uniqueness · Shared terms “numerical approximation”

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

    Klaus Deckelnick and Vanessa StylesWider flows

    Topics Numerics & visualization · Shared terms “finite element”, “numerical”, “approximation”

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

In the guidebook

  1. 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.

  2. 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.

  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.