Paper · May 20, 2026

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

Klaus Deckelnick, Vanessa Styles

math.NAWider flowsarXiv:2605.21445
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Abstract

The paper is concerned with the error analysis of a numerical scheme for the approximation of parametric mean curvature flow. The scheme we study is based on a reparametrization using the DeTurck trick and was proposed by Elliott and Fritz in [15]. In the semidiscrete case, for a spatial discretization by finite elements of order we prove an optimal -error estimate for the position vector. We present numerical experiments that confirm this error bound and demonstrate that the scheme has good properties with respect to the distribution of mesh points as already observed in [15].

Mean curvature flowHarmonic maps & DeTurck trickNumerics & visualization

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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 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 The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature (Patrick Guidotti, 2023)7Guidotti 2023 Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus (Roman Bahadursingh, 2024)4Bahadursingh 2024 Self-shrinkers whose asymptotic cones fatten (Daniel Ketover, 2024)6Ketover 2024 Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group (Giovanna Citti, Nicolas Dirr, Federica Dragoni et al., 2024)8Citti 2024 Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds (Miroslav Kolar and Daniel Sevcovic, 2025)5Kolar 2025 A curvature flow approach to dorsal closure modelling (Shuhui He, Ben Whale, Glen Wheeler et al., 2025)3He 2025
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 PDF

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

  2. Discrete hyperbolic curvature flow in the plane

    Klaus Deckelnick and Robert NürnbergWider flows PDF

    Topics Numerics & visualization · Same author Klaus Deckelnick · Shared terms “error bound”, “semidiscrete”, “numerical”

  3. A curvature flow approach to dorsal closure modelling

    Shuhui He, Ben Whale, Glen Wheeler et al.Wider flows PDF

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

  4. Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus

    Roman BahadursinghWider flows PDF

    Topics Numerics & visualization · Shared terms “numerical”

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

    Miroslav Kolar and Daniel SevcovicWider flows PDF

    Topics Numerics & visualization · Shared terms “numerical experiments”, “position vector”, “approximation”

  6. Self-shrinkers whose asymptotic cones fatten

    Daniel KetoverWider flows PDF

    Topics Numerics & visualization · Shared terms “numerical experiments”

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

    Patrick GuidottiWider flows PDF

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

  8. Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group

    Giovanna Citti, Nicolas Dirr, Federica Dragoni et al.Wider flows PDF

    Topics Numerics & visualization · Shared terms “proposed”, “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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

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.