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arxiv: 2409.14551 · v2 · submitted 2024-09-22 · 🧮 math.NA · cs.NA

Unconditional energy stable hybrid IEQ-FEMs for the Cahn-Hilliard-Navier-Stokes equations

Yaoyao Chen , Dongqian Li , Yin Yang , Peimeng Yin This is my paper

Pith reviewed 2026-05-23 20:22 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Cahn-Hilliard-Navier-Stokesinvariant energy quadratizationfinite element methodsenergy stabilitymass conservationhybrid discretizationunconditional stability
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The pith

A hybrid IEQ finite element method achieves unconditional energy stability for the Cahn-Hilliard-Navier-Stokes equations.

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

The authors examine two existing unconditionally energy stable IEQ finite element methods for the Cahn-Hilliard-Navier-Stokes equations and identify that the auxiliary energy variable falls outside the finite element space. They introduce a hybrid scheme that merges the two approaches to retain computational efficiency while ensuring the stability property holds in the discrete setting. Rigorous proofs establish that the hybrid method conserves mass exactly and dissipates energy unconditionally. Numerical tests are used to verify these properties along with accuracy. This construction allows stable long-time integration of the coupled phase-field and fluid equations without time-step restrictions.

Core claim

The hybrid IEQ-FEM combines the strengths of first- and second-order backward differentiation IEQ schemes with finite element discretization, providing unconditional energy stability in the finite element space despite the auxiliary variable not belonging to that space, together with proofs of mass conservation and energy dissipation.

What carries the argument

Hybrid IEQ-FEM that selects between two IEQ formulations to ensure the auxiliary energy variable is handled consistently with the finite element space while preserving unconditional stability.

If this is right

  • The discrete solution conserves mass exactly.
  • The discrete energy decreases monotonically for any positive time step size.
  • The method achieves the expected convergence rates in space and time.
  • Both first- and second-order time accuracy variants are available with the same stability guarantee.

Where Pith is reading between the lines

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

  • This construction suggests a general strategy for other IEQ-based schemes when auxiliary variables are not representable in the chosen approximation space.
  • Applications to three-dimensional two-phase flow problems could benefit from the reduced computational overhead compared to fully implicit alternatives.
  • Adaptive time-stepping strategies might be combined with this scheme to further optimize efficiency in long simulations.

Load-bearing premise

The auxiliary energy function, defined as the square root of the nonlinear part of the free energy, does not lie in the finite element space.

What would settle it

A computed solution in which the discrete energy increases over successive time steps for arbitrarily small time steps would falsify the unconditional stability.

Figures

Figures reproduced from arXiv: 2409.14551 by Dongqian Li, Peimeng Yin, Yaoyao Chen, Yin Yang.

Figure 1
Figure 1. Figure 1: Example 4.2, snapshots of numerical solutions for phase field function, First and second lines: P-BDF1-IEQ-FEM scheme (3.1)-(3.3); Third and fourth lines: P-BDF2-IEQ-FEM scheme (3.36)-(3.37). Figures 1-2 show the numerical solution of phase field ϕ n+1 h and velocity field u n+1 h by using the P-BDF1-IEQ-FEM scheme (3.1)-(3.3) and P-BDF2-IEQ-FEM scheme (3.36)-(3.37). From the pictures, we can see that the … view at source ↗
Figure 2
Figure 2. Figure 2: Example 4.2, snapshots of numerical solutions for velocity field func￾tion, First line: P-BDF1-IEQ-FEM scheme (3.1)-(3.3); Second line: P-BDF2-IEQ￾FEM scheme (3.36)-(3.37) [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example 4.2, the modified discrete energy history and discrete mass history. The evolution of the discrete energy and total mass are shown in [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example 4.3, P-BDF2-IEQ-FEM scheme (3.36)-(3.37), snapshots of numerical solutions for phase field function. A sequence snapshots of the approximate solutions for phase field function and the velocity field function are produced in Figures 4-5 by the proposed P-BDF2-IEQ-FEM scheme. It is easy to see the numerical solutions satisfy the expectation. The graphs depicting the evolution of discrete energy and c… view at source ↗
Figure 5
Figure 5. Figure 5: Example 4.3, P-BDF2-IEQ-FEM scheme (3.36)-(3.37), snapshots of numerical solutions for velocity field function [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Example 4.3, the modified discrete energy history and the change of total mass. Example 4.4. Let domain Ω = [−2, 2]2 , define m1 = [0, 2] , m2 = [0, 0] , m3 = [0, −2]. For given ϵ = 1 16 , let r1 = r3 = 2− 3ϵ 2 , r2 = 1 and set d (x) = max {−d1 (x), d2 (x), d3 (x)}, dj (x) = |x − mj | −rj for j = 1, 2, 3, we consider the CHNS equations (1.1) with the following initial condition ϕ0 = − tanh  d (x) √ 2ϵ  ,… view at source ↗
Figure 7
Figure 7. Figure 7: Example 4.4, P-BDF2-IEQ-FEM scheme (3.36)-(3.37), snapshots of numerical solutions for phase field function [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Example 4.4, P-BDF2-IEQ-FEM scheme (3.36)-(3.37), snapshots of numerical solutions for velocity field function. Example 4.5. [6] In the last example, let Ω = [− 1 2 , − 1 2 ]×[− 1 5 , − 1 5 ], we consider the CHNS equations satisfying the following initial condition ϕ0 =    1, x < x0, −1, x > x1, − sin  πx 2x1  , x0 ≤ x ≤ x1, u0 = [0 0]⊤, (4.8) where x1 = x0 = √ 2 20 , ϵ = 1 500√ 10 , λ = 1 100 … view at source ↗
Figure 9
Figure 9. Figure 9: Example 4.4, the modified discrete energy history and discrete mass history. shown in [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Example 4.5, Effect of constant B on the discrete energy, Left: C￾BDF1-IEQ-FEM scheme; Right: P-BDF1-IEQ-FEM scheme, h = 1 10 [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Example 4.5, Effect of mesh size on the difference of discrete energy, B = 500. Left: h = 1 160 ; Right: h = 1 320 . energy stability in the FEM space, but the projection step incurs additional computational costs. The third IEQ-FEM scheme is a hybrid of the two, designed to leverage the benefits of both. It begins with the former IEQ-FEM for computational efficiency but switches to the latter if the ener… view at source ↗
Figure 12
Figure 12. Figure 12: Example 4.5, The comparison of energy curves for three methods. B = 100. Left: h = 1 10 , right: h = 1 160 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Example 4.5, The comparison of energy curves for three methods. Top: h = 1 10 , bottom: h = 1 160 ; left: CP-BDF1-IEQ-FEM, center: C-BDF1-IEQ￾FEM, right: P-BDF1-IEQ-FEM. for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province of China. Author Contributions. All authors contribute equally to this manuscript. Data availability. Not Applicable [PITH_FULL_IMAG… view at source ↗
read the original abstract

We investigate two unconditionally energy stable invariant energy quadratization (IEQ) finite element methods (FEMs) [Chen et al. Numerical Algorithms, DOI: 10.1007/s11075-024-01910-z, 2024] for solving the Cahn-Hilliard-Navier-Stokes (CHNS) equations. The time discretization of these IEQ-FEMs is based on the first- and second-order backward differentiation methods. \textcolor{black}{The auxiliary energy function introduced by the IEQ approach, modeling the square root of the nonlinear part of the energy, does not belong to the finite element space used for the spatial discretization.} These methods offer distinct advantages. Consequently, we propose a new hybrid IEQ-FEM that combines the strengths of both schemes, offering computational efficiency and unconditional energy stability in the finite element space. We provide rigorous proofs of mass conservation and energy dissipation for the proposed IEQ-FEMs. Several numerical experiments are presented to validate the accuracy, efficiency, and solution properties of the proposed method.

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

0 major / 1 minor

Summary. The manuscript investigates two unconditionally energy stable IEQ-FEMs for the Cahn-Hilliard-Navier-Stokes equations based on first- and second-order BDF time discretizations from a 2024 reference. It identifies that the auxiliary energy function from the IEQ approach does not belong to the finite element space and proposes a new hybrid IEQ-FEM combining the strengths of both schemes to achieve computational efficiency and unconditional energy stability within the finite element space. Rigorous proofs of mass conservation and energy dissipation are claimed for the proposed methods, supported by numerical experiments validating accuracy, efficiency, and solution properties.

Significance. If the hybrid construction successfully ensures the auxiliary variable lies in the FE space while preserving unconditional stability and the stated conservation properties, the work would provide a practical advancement for stable and efficient simulation of two-phase incompressible flows governed by the CHNS system.

minor comments (1)
  1. Abstract: the LaTeX command `textcolor{black}{...}` is an editing artifact and should be removed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately reflects the contributions of the hybrid IEQ-FEM approach for the CHNS equations, including the identification of the auxiliary variable issue and the proofs of mass conservation and energy stability.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper cites the authors' own 2024 work solely to identify the two base IEQ-FEM schemes under investigation. It then explicitly states that a new hybrid scheme is proposed and that 'rigorous proofs of mass conservation and energy dissipation for the proposed IEQ-FEMs' are supplied in the present manuscript. Because the central stability and conservation claims are asserted to rest on proofs given here rather than on a direct reduction to the cited prior work, no load-bearing self-citation, self-definitional, or fitted-input pattern is exhibited. The derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the IEQ auxiliary-variable construction from the authors' 2024 paper, standard properties of finite-element spaces, and the assumption that the auxiliary function lies outside the chosen FE space.

axioms (1)
  • standard math Standard approximation properties of finite element spaces and stability of backward differentiation formulas hold for the chosen discretization.
    Invoked implicitly for the spatial and temporal discretizations of the CHNS system.
invented entities (1)
  • auxiliary energy function no independent evidence
    purpose: Models the square root of the nonlinear part of the energy to enable the IEQ quadratic reformulation.
    Introduced by the IEQ approach; the abstract states it does not belong to the finite element space, motivating the hybrid method.

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