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arxiv: 2604.18288 · v2 · submitted 2026-04-20 · 🧮 math.NA · cs.NA

Dual formulations of geometric curvature flows and their discretizations

Guangwei Gao , Buyang Li , Rong Tang This is my paper

Pith reviewed 2026-05-10 04:00 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords dual formulationscurvature flowsmean curvature flowenergy stable schemeslinearly implicit discretizationssurface diffusiontangential motionnumerical surface evolution
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The pith

Dual formulations of curvature flows introduce a multiplier that makes energy stability explicit for designing linearly implicit schemes.

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

The paper presents dual formulations for geometric curvature flows, including mean curvature flow, surface diffusion, and solid-state dewetting. These add a dual multiplier at the continuous level to expose the energy-dissipation structure without changing the underlying dynamics. The augmented system then supports construction of time discretizations that remain linearly implicit while inheriting unconditional energy stability. The framework also incorporates artificial tangential velocities to preserve mesh quality during surface evolution. Numerical experiments on benchmark problems verify convergence and the structure-preserving behavior of the resulting schemes.

Core claim

Introducing a dual multiplier into the continuous formulation of curvature flows renders the energy-dissipation law explicit, yielding an equivalent system that directly guides the design of linearly implicit, energy-stable discretizations while accommodating tangential motions for mesh control.

What carries the argument

The dual multiplier, an auxiliary variable introduced at the continuous level that augments the curvature flow equations to make their variational energy structure available for discretization.

Load-bearing premise

Adding the dual multiplier at the continuous level leaves the original curvature flow dynamics unchanged.

What would settle it

A side-by-side computation showing that the dual formulation produces surface evolution trajectories or energy decay rates that differ from those of the original curvature flow equations.

Figures

Figures reproduced from arXiv: 2604.18288 by Buyang Li, Guangwei Gao, Rong Tang.

Figure 1
Figure 1. Figure 1: Example 2.3: convergence rates in space and time for scheme (2.6). (a) Initial surface (b) BGN with τ = 1 × 10−4 at t = 0.0911 (c) MDR with τ = 1 × 10−4 at t = 0.0913 (d) Dual-MDR with τ = 1× 10−4 at t = 0.0911 (e) BGN with τ = 2.5 × 10−5 at t = 0.090625 (f) MDR with τ = 2.5 × 10−5 at t = 0.090775 (g) Dual-MDR with τ = 2.5 × 10−5 at t = 0.090575 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of surface evolution under mean curvature flow in Example 2.4. makes this example particularly challenging from the numerical point of view, since Dziuk’s method, as well as other approaches that do not incorporate artificial tangential motion, may suffer from mesh distortion and inaccurate geometric evolution, thereby preventing the surface from approaching the correct spherical shape. In our n… view at source ↗
Figure 3
Figure 3. Figure 3: (a) and [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of surface evolution under surface diffusion in Exam￾ple 2.5. (a) Surface area: |Γ m h | (b) |Γ m h | − |Γ m+1 h | (c) Mesh quality [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the surface-area decay and mesh quality in Exam￾ple 2.5. and supplemented by boundary conditions on ∂Γ: X3(·, t) [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Solid-state dewetting with a 120◦ contact angle in Example 3.2. (a) Surface area: |Γ m h | (b) |Γ m h | − |Γ m+1 h | (c) Mesh quality [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the surface-area decay and mesh quality in Exam￾ple 3.2. at T = 3.33, the two bulk components remain connected, and severe mesh distortion develops in the neck region, leading to failure at the next time step; see [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Solid-state dewetting with a 90◦ contact angle in Example 3.3. (a) Dual-MDR at T = 3.5 with τ = 10−2 (b) Dual-MDR at T = 4 with τ = 10−2 (c) Dual-MDR at T = 4 with τ = 10−3 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Long-time behavior of the dual-MDR scheme in Example 3.3. results clearly demonstrate the superior performance of the proposed dual-MDR scheme in preserving mesh quality throughout the evolution. (a) Surface area: |Γ m h | (b) |Γ m h | − |Γ m+1 h | (c) Mesh quality [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the surface-area decay and mesh quality in Ex￾ample 3.3. 4. Conclusion By introducing a dual multiplier at the continuous level, we have constructed dual for￾mulations for several curvature-driven geometric evolutions, including mean curvature flow, surface diffusion, and the solid-state dewetting problem. These dual formulations are equiva￾lent to the original curvature flows, but they make… view at source ↗
read the original abstract

We propose new formulations of geometric curvature flows -- referred to as \emph{dual formulations} -- that are equivalent to the original formulations but provide a novel framework for constructing linearly implicit and energy-stable schemes for curvature-driven surface evolution, including mean curvature flow, surface diffusion, and solid-state dewetting on a substrate with a moving contact line. The dual formulations are derived by introducing, at the continuous level, an additional unknown in the form of a dual multiplier. This augmentation does not alter the continuous dynamics but makes the underlying energy-dissipation structure explicit and, in turn, enables a systematic design of linearly implicit discretizations that inherit energy stability. A key feature of this framework is that it accommodates a broad class of artificial tangential motions which can be used to maintain good mesh quality of the computed surfaces. As an illustration, we combine the framework with the minimal-deformation-rate (MDR) tangential motion, leading to what we call the \emph{dual-MDR} scheme. The resulting method is linearly implicit and energy-stable, while retaining the MDR tangential motion to maintain good mesh quality. Extensive numerical experiments demonstrate the convergence of the proposed schemes, their structure-preserving properties, and advantages on representative benchmark problems.

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.

Referee Report

3 major / 3 minor

Summary. The manuscript proposes dual formulations of geometric curvature flows (mean curvature flow, surface diffusion, solid-state dewetting with moving contact lines) obtained by augmenting the original system with an auxiliary dual multiplier at the continuous level. These formulations are asserted to be equivalent to the primal geometric flows while rendering the energy-dissipation identity explicit, thereby enabling a systematic construction of linearly implicit, energy-stable discretizations. The framework is combined with minimal-deformation-rate (MDR) tangential motion to produce the dual-MDR scheme, which preserves mesh quality; extensive numerical tests on benchmarks are reported to demonstrate convergence, structure preservation, and practical advantages.

Significance. If the equivalence and inheritance of energy stability are rigorously established, the work supplies a new, systematic route to linearly implicit structure-preserving schemes for curvature-driven surface evolution. The explicit incorporation of artificial tangential velocities (MDR) while retaining stability is a practical contribution that addresses a common difficulty in surface discretizations. The approach could influence the design of numerical methods for geometric PDEs in materials science and computational geometry.

major comments (3)
  1. [§2.3] §2.3, Eq. (2.12)–(2.15): the elimination of the dual multiplier λ to recover the original curvature term and moving-contact-line condition is presented formally; the argument does not explicitly verify that the resulting velocity field satisfies the original weak form without additional tangential or normal constraints in the function space used later for discretization. This step is load-bearing for the equivalence claim.
  2. [§4.1] §4.1, Theorem 4.2: the discrete energy-dissipation identity for the dual-MDR scheme is derived under the assumption that the discrete multiplier satisfies an exact orthogonality relation with the tangential velocity; no a-priori estimate or numerical verification is given that this relation holds uniformly under mesh refinement or for the contact-line condition.
  3. [§3.2] §3.2: the stability analysis of the continuous dual formulation is performed in a strong sense; it is unclear whether the same energy identity persists in the weak formulation required to justify the finite-element discretization and the passage to the limit.
minor comments (3)
  1. [§2] Notation for the dual multiplier is introduced in §2 but reused with different subscripts in §4 without a consolidated table of symbols.
  2. [Figure 5] Figure 5 (dewetting example): the color scale for the multiplier field is not described in the caption, making it difficult to interpret the reported values.
  3. [Table 1] The convergence rates in Table 1 are reported only for the L² norm of the position; an additional column for the curvature error would strengthen the structure-preservation claim.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, providing clarifications where possible and indicating revisions that will strengthen the presentation of equivalence, stability, and discrete properties.

read point-by-point responses

  1. Referee: [§2.3] §2.3, Eq. (2.12)–(2.15): the elimination of the dual multiplier λ to recover the original curvature term and moving-contact-line condition is presented formally; the argument does not explicitly verify that the resulting velocity field satisfies the original weak form without additional tangential or normal constraints in the function space used later for discretization. This step is load-bearing for the equivalence claim.

    Authors: We appreciate the referee's observation that the formal elimination step in §2.3 requires explicit verification in the weak sense. In the revised manuscript we will expand this section with a detailed substitution argument: after solving for λ and inserting the resulting curvature and contact-line expressions back into the weak form, we show that the velocity field satisfies the original weak formulation exactly, using the same function spaces later employed for discretization. This calculation confirms that no extraneous tangential or normal constraints are imposed beyond those of the primal problem, thereby making the equivalence rigorous. revision: yes

  2. Referee: [§4.1] §4.1, Theorem 4.2: the discrete energy-dissipation identity for the dual-MDR scheme is derived under the assumption that the discrete multiplier satisfies an exact orthogonality relation with the tangential velocity; no a-priori estimate or numerical verification is given that this relation holds uniformly under mesh refinement or for the contact-line condition.

    Authors: We acknowledge that the orthogonality assumption is central to Theorem 4.2. In the discrete setting this relation holds exactly by construction of the finite-element spaces and the definition of the MDR tangential velocity, which is chosen to lie in the orthogonal complement of the multiplier space. We will add a short algebraic verification of this exact discrete orthogonality in the revised §4.1. In addition, we will include numerical checks confirming that the relation remains satisfied to machine precision under successive mesh refinements, including for moving-contact-line problems. While a general a-priori estimate lies beyond the present scope, the exact discrete identity together with the numerical evidence addresses the concern for the schemes under consideration. revision: partial

  3. Referee: [§3.2] §3.2: the stability analysis of the continuous dual formulation is performed in a strong sense; it is unclear whether the same energy identity persists in the weak formulation required to justify the finite-element discretization and the passage to the limit.

    Authors: The referee correctly notes that the energy analysis in §3.2 is written in strong form for clarity. The same identity nevertheless follows from the weak formulation by testing the dual equations with suitable test functions from the underlying function spaces. In the revision we will insert a short derivation (either in §3.2 or an appendix) that obtains the energy-dissipation relation directly from the weak form. This step will explicitly justify the subsequent finite-element discretization and the passage to the limit as the mesh size tends to zero. revision: yes

Circularity Check

0 steps flagged

No circularity: dual augmentation is an independent reformulation with explicit equivalence claim

full rationale

The derivation introduces a dual multiplier at the continuous level to expose the energy-dissipation identity, then asserts that the augmented system is equivalent to the original geometric flows (mean curvature flow, surface diffusion, dewetting) because the multiplier can be eliminated to recover the primal PDE and contact-line conditions. This step is presented as a reversible augmentation that does not alter admissible velocities or add hidden constraints. No self-definitional loop appears: the original flow is not defined in terms of the dual multiplier, nor is any prediction fitted to data and then re-labeled. MDR tangential motion is combined as an optional feature rather than a load-bearing premise. The central claim therefore rests on a direct (if formal) equivalence argument at the PDE level rather than on self-citation chains, fitted inputs, or renamed empirical patterns. The subsequent discretization inherits stability from the augmented energy law, which is a standard consequence once equivalence is granted. Hence the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the equivalence of the augmented system and the inheritance of stability by the discretizations; the dual multiplier is the key new element introduced.

axioms (1)
  • domain assumption The augmentation by the dual multiplier does not alter the continuous dynamics of the original curvature flow.
    Explicitly stated in the abstract as the basis for equivalence.
invented entities (1)
  • dual multiplier no independent evidence
    purpose: To make the energy-dissipation structure explicit at the continuous level for designing stable discretizations.
    Introduced in the paper to enable the new framework.

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