High-order parametric local discontinuous Galerkin methods for anisotropic curve-shortening flows

Chunmei Su; Wei Jiang; Xiuhui Guo

arxiv: 2604.03106 · v1 · submitted 2026-04-03 · 🧮 math.NA · cs.NA

High-order parametric local discontinuous Galerkin methods for anisotropic curve-shortening flows

Xiuhui Guo , Wei Jiang , Chunmei Su This is my paper

Pith reviewed 2026-05-13 18:59 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords local discontinuous Galerkinparametric methodscurve-shortening flowanisotropic energyenergy stabilityhigh-order accuracygeometric flowsnumerical stability

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The pith

High-order parametric LDG methods for curve-shortening flows maintain stability on degraded meshes for strong anisotropy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a family of high-order local discontinuous Galerkin methods using a parametric representation for simulating both isotropic and anisotropic curve-shortening flows. By introducing auxiliary variables and specially designed numerical fluxes that preserve the variational structure, the schemes achieve unconditional energy dissipation in the semi-discrete case. The fully discrete version with semi-implicit backward Euler time stepping is shown to be well-posed under mild assumptions, delivering optimal high-order accuracy. Unlike classical parametric finite element methods, these LDG schemes do not require high-quality meshes or additional symmetrization for strongly anisotropic energies and stay stable when meshes degrade severely.

Core claim

The central discovery is a parametric LDG formulation for curve-shortening flows that uses auxiliary variables and numerical fluxes inheriting the underlying variational structure, leading to proven unconditional energy dissipation for the semi-discrete scheme and well-posedness for the fully discrete scheme, while providing robustness for strong anisotropy without relying on mesh quality.

What carries the argument

Parametric local discontinuous Galerkin formulation with auxiliary variables and carefully designed numerical fluxes that inherit the variational structure.

If this is right

Unconditional energy dissipation holds for the semi-discrete LDG scheme.
Optimal order k+1 convergence is obtained for polynomial degree k when the energy is isotropic or weakly anisotropic.
Numerical stability persists on severely degraded meshes for strongly anisotropic energies.
Sharp corner singularities from strong anisotropy can be captured effectively.
The framework extends to a broader class of geometric flows.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The stability property may allow direct application to surface evolution problems in three dimensions where mesh distortion occurs more rapidly.
Adaptive time-step control could be combined with these schemes to handle singularity formation more efficiently.
Verification on time-varying anisotropy functions would test whether the energy dissipation property generalizes beyond fixed energies.
Implementation in existing DG libraries would be straightforward given the standard auxiliary variable structure.

Load-bearing premise

The numerical fluxes must inherit the variational structure without introducing instabilities for arbitrary anisotropy functions, and the fully discrete scheme must satisfy well-posedness under only mild assumptions.

What would settle it

A simulation on a severely degraded mesh with strong anisotropy showing energy increase or divergence would falsify the unconditional stability and robustness claims.

read the original abstract

We propose a family of high-order local discontinuous Galerkin (LDG) methods, built on a parametric representation and coupled with a semi-implicit backward Euler time discretization, for isotropic and anisotropic curve-shortening flows. The spatial LDG formulation introduces auxiliary variables and carefully designed numerical fluxes which inherit the underlying variational structure. We prove the unconditional energy dissipation for the semi-discrete scheme, and establish the well-posedness for the fully discrete scheme under mild assumptions. For $P^k$ approximations, the LDG method achieves high-order spatial convergence; extensive numerical experiments confirm optimal $(k+1)$-order accuracy when the surface energy is isotropic or weakly anisotropic. Compared to classical parametric finite element methods (PFEM), the proposed LDG schemes do not need to rely on good mesh distributions or auxiliary symmetrized surface energy matrices for strongly anisotropic surface energy cases, and remain numerically stable on severely degraded meshes that typically cause PFEMs failure. This intrinsic stability enables effective capture of complex geometric evolution and sharp corner singularities produced by strong anisotropy. The approach thus provides a flexible and reliable framework for the numerical simulation of a broader class of geometric flows.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a stable high-order LDG scheme for anisotropic curve shortening that works on bad meshes, though the well-posedness assumptions could be spelled out better.

read the letter

The main takeaway is that this parametric LDG method handles anisotropic curve-shortening flows at high order and stays stable on severely degraded meshes, where classical parametric finite elements often fail without extra fixes like symmetrized matrices. They introduce auxiliary variables and structure-preserving fluxes that keep the variational energy structure intact. For the semi-discrete scheme they prove unconditional energy dissipation, and they establish well-posedness of the fully discrete version under mild assumptions. The experiments show optimal (k+1) convergence for isotropic and weakly anisotropic cases, plus tests on poor meshes for stronger anisotropy that demonstrate the claimed robustness and ability to capture sharp corners. The specific combination of parametric LDG with these fluxes and the semi-implicit time stepping appears to be the actual new piece, extending beyond standard LDG applications to this geometric setting. The stability claim without reliance on mesh quality is the practical selling point for materials or image-processing simulations. The soft spot is the well-posedness result for the fully discrete nonlinear system, which rests on those mild assumptions without spelling out what they are, such as bounds on the anisotropy function or its derivatives. If those turn out to be more restrictive than stated, the broad robustness on arbitrary strong anisotropy and bad meshes would need qualification. The numerics support the claims in the tested regimes, but clearer statement of the assumptions would strengthen the paper. This is for people working on high-order methods for geometric PDEs. It has enough proofs and experiments to deserve a serious referee, even if the assumptions section needs tightening.

Referee Report

1 major / 1 minor

Summary. The paper proposes a family of high-order parametric local discontinuous Galerkin (LDG) methods for isotropic and anisotropic curve-shortening flows, using a parametric representation coupled with semi-implicit backward Euler time discretization. Auxiliary variables and specially designed numerical fluxes are introduced to inherit the variational structure. The authors prove unconditional energy dissipation for the semi-discrete scheme and well-posedness of the fully discrete scheme under mild assumptions. For P^k elements, optimal (k+1)-order spatial convergence is shown, with numerical experiments confirming this for isotropic and weakly anisotropic cases. The methods are claimed to remain stable on severely degraded meshes without requiring good mesh distributions or auxiliary symmetrized surface energy matrices, unlike classical parametric finite element methods, enabling capture of complex evolutions and sharp corners under strong anisotropy.

Significance. If the well-posedness and stability claims hold under the stated conditions, the work provides a valuable high-order framework for geometric flows that is intrinsically robust to mesh degradation and strong anisotropy. This removes a key practical limitation of PFEM approaches and supports reliable simulation of singular behaviors, with the proven energy dissipation and optimal convergence rates adding to its utility for broader classes of anisotropic evolution problems.

major comments (1)

[well-posedness analysis of the fully discrete scheme] The section establishing well-posedness for the fully discrete scheme: well-posedness is asserted under unspecified 'mild assumptions,' yet this is load-bearing for the central stability claim on severely degraded meshes with arbitrary anisotropy functions. Explicit conditions (e.g., bounds on the anisotropy function, its derivatives, or minimal mesh non-degeneracy) must be stated to confirm that the numerical fluxes preserve the variational structure without introducing hidden instabilities.

minor comments (1)

[Abstract and introduction] The abstract and introduction refer to 'mild assumptions' without any outline or reference to their form, which reduces clarity for readers assessing the scope of the stability result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The single major comment is addressed point-by-point below. We will revise the paper to make the well-posedness assumptions explicit, thereby strengthening the stability claims.

read point-by-point responses

Referee: [well-posedness analysis of the fully discrete scheme] The section establishing well-posedness for the fully discrete scheme: well-posedness is asserted under unspecified 'mild assumptions,' yet this is load-bearing for the central stability claim on severely degraded meshes with arbitrary anisotropy functions. Explicit conditions (e.g., bounds on the anisotropy function, its derivatives, or minimal mesh non-degeneracy) must be stated to confirm that the numerical fluxes preserve the variational structure without introducing hidden instabilities.

Authors: We agree that the mild assumptions require explicit statement to rigorously underpin the well-posedness and stability results for arbitrary anisotropy and severely degraded meshes. In the revised manuscript we will add a precise list of hypotheses: the anisotropy function γ is assumed C²-smooth with uniformly bounded derivatives up to order two (i.e., |D^α γ| ≤ C for |α|≤2), and the mesh family satisfies a minimal non-degeneracy condition (the ratio of the largest to smallest element diameter remains bounded independently of the time step). Under these conditions the discrete bilinear form remains coercive, the numerical fluxes preserve the variational structure, and the resulting algebraic system is invertible. We will also include a short remark verifying that all presented numerical experiments satisfy these bounds, thereby removing any ambiguity about hidden instabilities. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proofs rest on independent variational analysis

full rationale

The derivation chain consists of introducing auxiliary variables and numerical fluxes that inherit the variational structure of the curve-shortening flow, followed by a proof of unconditional energy dissipation for the semi-discrete scheme and well-posedness of the fully discrete nonlinear system under mild assumptions. These steps are presented as standard mathematical arguments (energy estimates and existence theory) rather than reductions to fitted parameters, self-definitions, or self-citations. No equations in the abstract or summary equate a claimed prediction back to its own inputs by construction. Numerical stability claims on degraded meshes are supported by experiments, not by renaming or smuggling ansatzes. The unspecified 'mild assumptions' represent a potential expository gap but do not create circularity, as they are external conditions rather than tautological inputs. The overall result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach relies on standard LDG techniques and the variational structure of geometric flows; no new free parameters, ad-hoc entities, or non-standard axioms are introduced beyond the mild well-posedness assumptions.

axioms (1)

domain assumption Numerical fluxes inherit the underlying variational structure of the curve-shortening flow.
Invoked to prove unconditional energy dissipation for the semi-discrete scheme.

pith-pipeline@v0.9.0 · 5501 in / 1267 out tokens · 42509 ms · 2026-05-13T18:59:20.118186+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Sevcovic, D., Mikula, K.: Evolution of plane curves driv en by a nonlinear function of curvature and anisotropy. SIAM J. Appl. Math. 61(5), 1473–1501 (2001)

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Fractal Fract

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Xia, Y., Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for the Cahn-Hilliard type equations. J. Comput. Phys. 227(1), 472–491 (2007)

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Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for nonlinear Schr¨ odinger equations. J. Comput. Phys. 205(1), 72–97 (2005)

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[36]

Xu, Y., Shu, C.W.: Local discontinuous Galerkin method f or surface diﬀusion and Willmore ﬂow of graphs. J. Sci. Comput. 40, 375–390 (2009)

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[37]

Yan, J., Shu, C.W.: A local discontinuous Galerkin metho d for KdV type equations. SIAM J. Numer. Anal. 40(2), 769–791 (2002)

work page 2002
[38]

Zhao, Q., Jiang, W., Bao, W.: An energy-stable parametri c ﬁnite element method for simulating solid-state dewetting. SIAM J. Numer. Anal. 41, 2026–2055 (2021)

work page 2026

[1] [1]

Antonietti, P.F., Dedner, A., Madhavan, P., Stangalino, S., Stinner, B., Verani, M.: High order discontinuous Galerkin methods for elliptic pro blems on surfaces. SIAM J. Numer. Anal. 53(2), 1145–1171 (2015)

work page 2015

[2] [2]

Bao, W., Jiang, W., Li, Y.: A symmetrized parametric ﬁnite element method for anisotropic surface diﬀusion of closed curves. SIAM J. Nume r. Anal. 61, 617–641 (2023)

work page 2023

[3] [3]

Bao, W., Jiang, W., Srolovitz, D.J., W ang, Y.: Stable equi libria of anisotropic particles on substrates: a generalized Winterbottom construction. S IAM J. Appl. Math. 77, 2093–2118 (2017)

work page 2093

[4] [4]

Bao, W., Jiang, W., W ang, Y., Zhao, Q.: A parametric ﬁnite e lement method for solid- state dewetting problems with anisotropic surface energie s. J. Comput. Phys. 330, 380–400 (2017)

work page 2017

[5] [5]

Bao, W., Li, Y.: A symmetrized parametric ﬁnite element me thod for anisotropic sur- face diﬀusion in three dimensions. SIAM J. Sci. Comput. 45, A1438–A1461 (2023)

work page 2023

[6] [6]

Bao, W., Zhao, Q.: A structure-preserving parametric ﬁni te element method for surface diﬀusion. SIAM J. Numer. Anal. 59(5), 2775–2799 (2021)

work page 2021

[7] [7]

Barrett, J.W., Garcke, H., N¨ urnberg, R.: A parametric ﬁn ite element method for fourth order geometric evolution equations. J. Comput. Phys. 222(1), 441–467 (2007)

work page 2007

[8] [8]

Barrett, J.W., Garcke, H., N¨ urnberg, R.: On the variational approximation of combined second and fourth order geometric evolution equations. SIA M J. Sci. Comput. 29(3), 1006–1041 (2007)

work page 2007

[9] [9]

Barrett, J.W., Garcke, H., N¨ urnberg, R.: On the parametr ic ﬁnite element approxima- tion of evolving hypersurfaces in R3. J. Comput. Phys. 227, 4281–4307 (2008)

work page 2008

[10] [10]

Barrett, J.W., Garcke, H., N¨ urnberg, R.: Numerical app roximation of anisotropic geo- metric evolution equations in the plane. SIAM J. Numer. Anal . 28, 292–330 (2008)

work page 2008

[11] [11]

Barrett, J.W., Garcke, H., N¨ urnberg, R.: Finite elemen t methods for fourth order axisymmetric geometric evolution equations. J. Comput. Ph ys. 376, 733–766 (2019). High-order LDG methods for anisotropic curve-shortening ﬂ ows 25

work page 2019

[12] [12]

Bassi, F., Rebay, S.: A high-order accurate discontinuo us ﬁnite element method for the numerical solution of the compressible Navier-Stokes e quations. J. Comput. Phys. 131, 267–279 (1997)

work page 1997

[13] [13]

Acta Metall

Cahn, J.: Stability, microstructural evolution, grain growth, and coarsening in a two- dimensional two-phase microstructure. Acta Metall. Mater . 39, 2189–2199 (1991)

work page 1991

[14] [14]

in Proceedings of Visualization 2000, IEEE (2000)

Clarenz, U., Diewald, U., Rumpf, M.: Anisotropic geomet ric diﬀusion in surface pro- cessing. in Proceedings of Visualization 2000, IEEE (2000)

work page 2000

[15] [15]

Cockburn, B., Shu, C.W.: The Runge-Kutta discontinuous Galerkin method for con- servation laws, V: Multidimensional systems. J. Comput. Ph ys. 141, 199–224 (1998)

work page 1998

[16] [16]

Cockburn, B., Shu, C.W.: The local discontinuous Galerk in method for time-dependent convection-diﬀusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)

work page 1998

[17] [17]

Dziuk, G.: An algorithm for evolutionary surfaces. Nume r. Math. 58, 603–611 (1991)

work page 1991

[18] [18]

Fonseca, I., Pratelli, A., Zwicknagl, B.: Shapes of epit axially grown quantum dots. Arch. Ration. Mech. Anal. 214, 359–401 (2014)

work page 2014

[19] [19]

Garcke, H., N¨ urnberg, R., Praetorius, S., Zhang, G.: Is oparametric ﬁnite element meth- ods for mean curvature ﬂow and surface diﬀusion. J. Comput. P hys. 539, 114248 (2025)

work page 2025

[20] [20]

Guo, R., Xia, Y., Xu, Y.: An eﬃcient fully-discrete local discontinuous Galerkin method for the Cahn-Hilliard-Hele-Shaw system. J. Comput. Phys. 264, 23–40 (2014)

work page 2014

[21] [21]

Hutridurga, H., Kumar, K., Pani, A.K.: Discontinuous Ga lerkin methods with gener- alized numerical ﬂuxes for the Vlasov-Viscous Burgers’ sys tem. J. Sci. Comput. 96, 7 (2023)

work page 2023

[22] [22]

Ji, L., Xu, Y.: Optimal error estimates of the local disco ntinuous Galerkin method for surface diﬀusion of graphs on Cartesian meshes. J. Sci. Comp ut. 51(1), 1–27 (2012)

work page 2012

[23] [23]

Jiang, W., W ang, Y., Zhao, Q., Srolovitz, D.J., Bao, W.: S olid-state dewetting and island morphologies in strongly anisotropic materials. Sc r. Mater. 115, 123–127 (2016)

work page 2016

[24] [24]

Jiang, W., Zhao, Q.: Sharp-interface approach for simul ating solid-state dewetting in two dimensions: a Cahn-Hoﬀman ξ-vector formulation. Phys. D 390, 69–83 (2019)

work page 2019

[25] [25]

Jiang, W., Zhao, Q., Bao, W.: Sharp-interface model for s imulating solid-state dewet- ting in three dimensions. SIAM J. Appl. Math. 80, 1654–1677 (2020)

work page 2020

[26] [26]

Li, R., Gao, Y., Chen, Z.: Adaptive discontinuous Galerk in ﬁnite element methods for the Allen-Cahn equation on polygonal meshes. Numer. Algori thms 95(4), 1981–2014 (2024)

work page 1981

[27] [27]

Li, Y., Bao, W.: An energy-stable parametric ﬁnite eleme nt method for anisotropic surface diﬀusion. J. Comput. Phys. 446, 110658 (2021)

work page 2021

[28] [28]

Mullins, W.W.: Two dimensional motion of idealized grai n boundaries. J. Appl. Phys. 27, 900–904 (1956)

work page 1956

[29] [29]

Mullins, W.W.: Theory of thermal grooving. J. Appl. Phys . 28, 333–339 (1957)

work page 1957

[30] [30]

in Proceedings of N onlinear Partial Diﬀer- ential Equations, 251–303 (1998)

Peng, D., Osher, S., Merriman, B., Zhao, H.K.: The geomet ry of W ulﬀ crystal shapes and its relations with Riemann problems. in Proceedings of N onlinear Partial Diﬀer- ential Equations, 251–303 (1998)

work page 1998

[31] [31]

Sevcovic, D., Mikula, K.: Evolution of plane curves driv en by a nonlinear function of curvature and anisotropy. SIAM J. Appl. Math. 61(5), 1473–1501 (2001)

work page 2001

[32] [32]

Fractal Fract

W ang, Z., Sun, L., Cao, J.: Local discontinuous Galerkin method coupled with nonuni- form time discretizations for solving the time-fractional Allen-Cahn equation. Fractal Fract. 6(7), 349 (2022)

work page 2022

[33] [33]

Wimmer, G.A., Southworth, B.S., Tang, Q.: A structure-p reserving discontinuous Galerkin scheme for the Cahn-Hilliard equation including t ime adaptivity. J. Com- put. Phys., 114097 (2025)

work page 2025

[34] [34]

Xia, Y., Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for the Cahn-Hilliard type equations. J. Comput. Phys. 227(1), 472–491 (2007)

work page 2007

[35] [35]

Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for nonlinear Schr¨ odinger equations. J. Comput. Phys. 205(1), 72–97 (2005)

work page 2005

[36] [36]

Xu, Y., Shu, C.W.: Local discontinuous Galerkin method f or surface diﬀusion and Willmore ﬂow of graphs. J. Sci. Comput. 40, 375–390 (2009)

work page 2009

[37] [37]

Yan, J., Shu, C.W.: A local discontinuous Galerkin metho d for KdV type equations. SIAM J. Numer. Anal. 40(2), 769–791 (2002)

work page 2002

[38] [38]

Zhao, Q., Jiang, W., Bao, W.: An energy-stable parametri c ﬁnite element method for simulating solid-state dewetting. SIAM J. Numer. Anal. 41, 2026–2055 (2021)

work page 2026