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A minimizing-movement framework for parametric finite element approximation of anisotropic curve shortening flows

Wei Jiang, Chunmei Su, Ganghui Zhang

arXiv:2610.36753v1math.NAcs.NA

TL;DR

Anisotropic curve-shortening schemes face stability conditions tied to the energy density, while strongly anisotropic cases also pose substantial analysis and numerical challenges. This paper introduces a minimizing-movement PFEM framework with modified energies and convex splitting; its schemes have unique one-step solutions and dissipate energy under stated conditions, with energy decrease also observed in a strongly anisotropic computation.

  • Problem

    Existing one-step energy estimates depend on pointwise conditions involving the anisotropy, while strongly anisotropic cases make accurate, energy-stable approximation more challenging.

    Source: §1 · p. 2

  • Method

    The framework pairs a weighted L2 penalty with anisotropic curve energy and uses convex-implicit, concave-explicit splitting to construct schemes for positive C2 surface densities.

    Source: §1 · p. 3

  • Results

    For positive C2 surface densities, the unconstrained scheme has a unique solution for arbitrary time-step size under elementwise non-degeneration and satisfies energy dissipation; a strongly anisotropic computation also records decreasing energy.

    Source: §4.3 · p. 17, §5 · p. 21

  • Takeaways & Limitations

    The framework extends the stability guarantee to nonsymmetric and strongly anisotropic densities that may violate 3γ(N) > γ(−N).

    Source: §1 · p. 2

    The analysis does not cover the cited weakly anisotropic density for 1 < r < 2 because it is not C2 at coordinate-axis orientations, though its discrete formulation can be implemented numerically.

    Source: §2.1 · p. 6

Abstract

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We propose a minimizing-movement framework for constructing parametric finite element methods (PFEMs) for anisotropic curve shortening flows. The cornerstone is an admissible modified energy: a smooth, convex, and iteration-dependent surrogate that majorizes the original anisotropic energy and agrees with it at the current iterate. For any fixed finite element degree $n\geq1$ and any positive surface energy density $\gamma\in C^2(\mathbb{S}^1)$, the unconstrained one-step problem has a unique solution for any time-step size whenever the parametric derivative of the current discrete curve does not vanish on any mesh element. It also satisfies a one-step discrete energy dissipation inequality. It therefore preserves the variational structure of the underlying $L^2$-gradient flow. The framework recovers a variational interpretation of Dziuk's isotropic scheme and applies directly to the weak anisotropies. For strong anisotropies, a convex splitting removes the weak-anisotropy restriction on the original density. A scalar constraint further extends the framework to area-preserving flows. Numerical experiments exhibit approximately the expected convergence for smooth isotropic and weakly anisotropic examples. They also confirm energy dissipation and area preservation. In the strongly anisotropic regime, the computations produce sharp facet-like profiles but reveal mesh-scale oscillations under the grid refinement.

1. Introduction

Existing anisotropic PFEM energy estimates impose conditions on the surface density, while strongly anisotropic flows pose additional numerical challenges. This work develops a minimizing-movement framework with modified energies and convex splitting to obtain broader energy stability and a uniquely solvable time step.

1. Introduction

  • Strong anisotropy can destroy parabolicity in some orientations, and its Wulff shapes may have facets and corners, complicating analysis and numerical approximation.The paper uses convex splitting, treating the convex contribution implicitly and the concave contribution explicitly.Source: §1 · pp. 2–3
  • Earlier anisotropic surface-energy-matrix PFEM estimates still require pointwise conditions on the anisotropy, motivating a framework with a broader stability guarantee.The proposed analysis includes nonsymmetric and strongly anisotropic densities that may violate 3γ(N) > γ(−N).Source: §1 · p. 2
  • A weighted L2 penalty on the previous discrete curve makes the minimizing-movement formulation compatible with parametric finite elements without a bulk distance functional.The discrete curves lie in a continuous periodic piecewise-polynomial finite element space of fixed degree n ≥1.Source: §1 · p. 3
  • The smooth, convex, iterate-dependent modified energy yields a strictly convex time-step problem, a unique update, and dissipation of the true anisotropic energy.It is consistent with the original energy at the current iterate and majorizes that energy.Source: §1 · p. 3
  • Convex splitting combined with an admissible modified energy extends the PFEM’s one-step energy estimate to every positive γ ∈C2(S1), without symmetry or the condition 3γ(N) > γ(−N).The paper identifies this as the first fully discrete PFEM with a one-step energy estimate for this entire class of surface energy densities.Source: §1 · pp. 3–4
  • The framework extends to area-preserving curve shortening through a Lagrange multiplier and recovers Dziuk’s isotropic PFEM as the Euler–Lagrange equation of one minimizing-movement step.This recovery provides an incremental variational interpretation of Dziuk’s scheme.Source: §1 · p. 4

2. A minimizing-movement framework for CSF

This section develops a minimizing-movement framework for curve shortening flows, establishing unique one-step solvability and a discrete energy estimate under a nonvanishing-derivative condition on each element. It applies to isotropic and weakly anisotropic flows and extends to strongly anisotropic densities through convex splitting.

2.1. An abstract minimizing-movement framework

  • For any τ > 0, an admissible modified-energy step has a unique minimizer and satisfies a one-step energy estimate if the current discrete parametrization is elementwise nondegenerate.The modified energy is a convex, continuously differentiable majorizer that agrees with the original energy at the current iterate. The result does not guarantee that the next parametrization remains elementwise nondegenerate.Source: §2.1 · p. 7

Definition 2.5. For X, Xk

  • The isotropic modified energy is admissible, yielding a unique solution characterized by its Euler–Lagrange equation and a one-step energy estimate for every τ > 0.The energy estimate is a one-step length estimate.Source: Definition 2.5. For X, Xk · p. 8

2.2. Isotropic case: recovery of Dziuk’s scheme

  • For n = 1, the isotropic construction recovers Dziuk’s consistent-mass linearly implicit polygonal scheme; for n > 1, it gives a conforming piecewise-polynomial Galerkin extension.Both are characterized as minimizing-movement steps and have a direct variational proof of the one-step length estimate.Source: Definition 2.5. For X, Xk · p. 8

2.3. Weakly anisotropic case

  • For positive γ ∈ C2(S1) satisfying weak anisotropy, the positively one-homogeneous density bγ is convex, enabling the weakly anisotropic construction.Densities that are weakly anisotropic but not C2 at coordinate-axis orientations, such as ℓr anisotropies with 1 < r < 2, lie beyond the analysis, though the discrete formulation can be examined numerically.Source: §2.1 · p. 6, §2.3 · p. 9

Definition 2.10. For X, Xk

  • For weakly anisotropic γ, the constructed modified energy satisfies the admissibility conditions, including convexity and continuous differentiability on the trial space.The construction’s differentiability conclusion also applies to trial functions with collapsed edges.Source: Definition 2.10. For X, Xk · p. 10
  • For weakly anisotropy, the one-step problem has a unique solution and satisfies a one-step energy estimate for every τ > 0, provided the current curve derivative is nonzero on each element.The uniqueness and estimate are stated under the weak-anisotropy assumption.Source: Definition 2.10. For X, Xk · p. 10

2.4. Strongly anisotropic case via convex splitting

  • Convex splitting writes γ = γc − γe with positive weakly anisotropic components, restoring the convexity needed to apply the modified-energy framework to strongly anisotropic densities.The explicit contribution from γe does not affect the minimizer; it is included to ensure the complete energy satisfies condition (M1).Source: §2.4 · pp. 10–11
  • For a convex splitting and a current curve derivative nonzero on every element, the strongly anisotropic one-step problem has a unique solution and satisfies an energy estimate.The theorem establishes the strongly anisotropic minimizing-movement step under these conditions.Source: §2.4 · p. 12

3. Extension to area-preserving curve shortening flow

A single scalar area constraint extends the modified-energy minimizing-movement framework to isotropic, weakly anisotropic, and strongly anisotropic area-preserving curve shortening flows. Every global minimizer preserves signed area and satisfies a discrete energy dissipation inequality.

3. Extension to area-preserving curve shortening flow

  • A single scalar area constraint extends the modified-energy framework to area-preserving flows across isotropic, weakly anisotropic, and strongly anisotropic regimes.The construction retains the appropriate modified energy for each anisotropy type.Source: §3 · p. 12
  • For strongly anisotropic densities, a constant convex splitting can be used to select the modified energy.Source: §3 · p. 12
  • Every global minimizer preserves signed area and satisfies the discrete energy dissipation inequality.The theorem also establishes constraint qualification and a corresponding Lagrange multiplier.Source: §3 · p. 13
  • Taking γ ≡ 1 recovers the area-preserving isotropic minimizing-movement scheme, with the energy estimate becoming a length dissipation inequality.Source: §3 · p. 14

4. Numerical results

The experiments assess coupled convergence, strong-anisotropy mesh behavior, and structure-preserving properties. Smooth isotropic and weakly anisotropic tests show near-second-order coupled decay, while strong-anisotropy refinement exposes mesh-scale oscillations.

4. Numerical results

  • The experiments examine coupled convergence, mesh-dependent oscillations, energy decay, area preservation, mesh quality, and solver performance using Newton's method with previous-step initialization and a residual tolerance of 10^-12.Computations use continuous piecewise-linear finite elements (n = 1).Source: §4 · p. 14

4.1. Convergence tests for isotropic and weakly anisotropic flows

  • The convergence tests use τ = h^2 and compare successive mesh levels by manifold distance, so the resulting order reflects coupled space–time error rather than separate spatial and temporal orders.An order near two is consistent with the expected combined error under this time-step coupling.Source: §4.1 · pp. 14–15
  • Across isotropic and three weakly anisotropic densities, CSF and AP-CSF adjacent-grid errors show approximately second-order decay over the displayed mesh range.The AP-CSF tests show slopes close to two, so the scalar Lagrange multiplier does not reduce the observed coupled convergence order.Source: §4.1 · p. 15, Figure 1 · p. 15, Figure 2 · p. 16

4.2. Oscillations in the strongly anisotropic case

  • Strong-anisotropy refinement shows coarse-pair orders near two, followed by nonmonotone errors and negative orders; the full sequences do not demonstrate mesh convergence.Table 1 reports adjacent-grid manifold distances and observed orders at T = 0.03 for the strongly anisotropic 2-fold density.Source: §4.2 · p. 16, Table 1 · p. 17
  • Increasing the splitting constant delays, but does not eliminate, the deterioration in adjacent-grid agreement.Deterioration begins at mesh pair (128, 256) for C = 0.2 and 1.0, and at (256, 512) for C = 10.0.Source: §4.2 · pp. 16–17
  • Figure 3 associates the loss of adjacent-grid agreement with sharp, mesh-scale oscillations, but the experiments do not identify their mechanism or establish convergence to a strongly anisotropic continuum flow.The splitting constant is not a physical isotropic shortening coefficient; for C ≥ Cmin, the one-step energy estimate remains valid for a global minimizer.Source: §4.2 · p. 17, Figure 3 · p. 18

4.3. Numerical assessment of the structure-preserving properties

  • Weakly anisotropic examples show monotonically decreasing normalized energy and moderate mesh ratios; the strongly anisotropic example develops facets and a markedly increasing mesh ratio while its energy decreases.Newton's method meets the prescribed residual tolerance at every reported step.Source: §4.3 · p. 17, Figure 4 · p. 19
  • For the strongly anisotropic flower, petals contract before sharp facets emerge, and the curve continues shrinking while discrete energy decreases throughout.Newton's method converges in a small number of iterations at each time step.Source: §4.3 · p. 17, Figure 5 · p. 20

5. Conclusion

The framework provides a variational PFEM for isotropic and anisotropic curve shortening, with energy dissipation and a conditional uniqueness guarantee for unconstrained steps. Experiments show convergence in smooth isotropic and weakly anisotropic cases, while strong anisotropies exhibit facet-like profiles and refinement oscillations.

5. Conclusion

  • Experiments show expected convergence for smooth isotropic and weakly anisotropic examples, but strong-anisotropy refinements may develop mesh-scale oscillations and cusps.The computations also confirm favorable mesh quality, energy decay, area preservation, and facet-like profiles in strongly anisotropic cases.Source: §5 · p. 21
  • For positive γ ∈ C^2(S^1), each unconstrained step has a unique solution for any time-step size if the current curve derivative is nonzero on every mesh element.The proposed schemes satisfy energy dissipation for isotropic, weakly anisotropic, and strongly anisotropic densities.Source: §5 · p. 21
  • The framework recovers Dziuk’s isotropic scheme, applies directly to weak anisotropies, and handles strong anisotropies through convex splitting.Source: §5 · p. 21
  • With one scalar constraint, each area-preserving step has a global minimizer that exactly preserves signed area and dissipates energy, though uniqueness is not guaranteed.Source: §5 · p. 21
  • Future work will extend the framework to surface diffusion and geometric surface flows in R^3.Source: §5 · p. 21
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