An energy stable $C^0$ finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density

Huaxiong Huang; Lingyue Shen; Ping Lin; Shixin Xu; Zilong Song

REVIEW 3 major objections 4 minor 65 references

An energy stable $C^0$ finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper develops a C0 finite element scheme for a quasi-incompressible phase-field model of moving contact lines and argues that the scheme conserves the mass of each phase and dissipates the discrete total energy on every time step.

desk verdict A credible variational model and C0 FEM for variable-density moving contact lines, but the advertised exact mass conservation is not proven as written; energy stability likely survives as an inequality. read the letter →

arxiv 1908.03681 v2 pith:VM53F6HE submitted 2019-08-10 math.NA cs.NA

Lingyue Shen , Huaxiong Huang , Ping Lin , Zilong Song , Shixin Xu This is my paper · ORCID

classification math.NAcs.NA MSC 65M6065N3076D0576T99

keywords energystabilitymovingcontactlinequasi-incompressibleNavier-Stokes-Cahn-HilliardvariabledensityC0finiteelementphase-fieldmethodmassconservationlargeratio

topics Navier-Stokes Regularity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper develops a fully discrete $C^0$ finite element scheme for a quasi-incompressible Navier-Stokes-Cahn-Hilliard system with moving contact lines and variable density, and argues that the scheme conserves the mass of each phase exactly and dissipates a discrete total energy on every time step. The model itself is derived from the energy variational method with a mass-averaged velocity, and the boundary condition is a density-extended General Navier-Stokes Boundary Condition in which the density ratio appears explicitly in the contact-line dynamics. If the claims are correct, the scheme gives a practical numerical route to wetting and spreading problems with large density contrasts, and it does so with equal-order $P1$ and $P2$ elements, sidestepping the inf-sup condition. Convergence tests report second-order in $L^2$ for $P1$ and third-order for $P2$ elements at both low and high density ratios.

What carries the argument

The machinery is the discrete energy identity, built on the discrete total energy $E_h^{n,\mathrm{tot}} = \int_\Omega \frac12 \rho_h^n |u_h^n|^2\,dx + \frac1\beta \int_\Omega \rho_h^n(\frac1\epsilon G(c_h^n)+\frac\epsilon2|\nabla c_h^n|^2)\,dx + \frac{\alpha_w}{\beta}\int_{\partial\Omega_w} f_w(c_h^n)\,dS$. The proof tests the four weak equations (64) with the special choices $\psi_h = \frac{\Delta t}{\beta}\bar\mu_h^{n+1}$, $\chi_h = \frac{1}{\beta}(c_h^{n+1}-c_h^n)$, $v_h=\Delta t u_h^{n+1}$, and $q_h = \frac{\Delta t}{\beta}\bar p_h^{n+1}$. The discrete chain-rule identities $G(c^{n+1})-G(c^n) = g(c^{n+1},c^n)(c^{n+1}-c^n)$ and $\rho^{n+1}-\rho^n = -\alpha\rho^{n+1}\rho^n(c^{n+1}-c^n)$ convert the chemical-potential products into exact differences of the mixing energy, while the cancellation between the pressure term in the momentum equation and the quasi-incompressibility equation removes the pressure. The same algebraic structure gives the componentwise mass conservation in Lemma 3.3 by taking the phase equation and the quasi-incompressibility equation together with special test functions.

What would settle it

Take one step of the proposed scheme on the Couette test at low density ratio, with a fixed wall, and evaluate both sides of Eq. (70); if the left-hand side is less negative than the right-hand side by approximately $\frac12\int_\Omega \rho^n |u^{n+1}_h-u^n_h|^2\,dx$, then the exact equality in the theorem fails.

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Extended reading notes

Core claim

The central claim is that a natural fully discrete finite element discretization of the quasi-incompressible NSCH system preserves the physics that matter in phase-field wetting simulations: the total mass and the mass of each component are conserved, and the discrete total energy satisfies $E_h^{n+1,\mathrm{tot}} - E_h^{n,\mathrm{tot}} = -\Delta t D_h^{n+1} - \Delta t \int_{\partial\Omega_w} (1/l_s Re) u_s^{n+1}\cdot u_w dS$, where $D_h^{n+1}$ is a non-negative sum of bulk viscous, compressible, chemical-potential-gradient, wall mobility, and slip dissipation terms. With a fixed wall the right-hand side is non-positive, so the scheme is energy stable at the fully discrete level. The construction is made possible by the quasi-incompressibility constraint $\nabla\cdot u = \alpha\nabla\cdot(M\nabla\tilde\mu)$ with $\alpha=(\rho_2-\rho_1)/(\rho_1\rho_2)$, whose pressure term $\alpha p$ in the chemical potential acts as a pressure stabilization and removes the need for inf-sup compatible velocity-pressure spaces. Numerical simulations of Couette flow, droplets in shear flow, and a rising bubble with density ratio $1000{:}1$ are presented as evidence that the scheme is mass-conservative, energy-dissipative, and second or third-order convergent.

Load-bearing premise

The discrete energy law is proved as an exact equality, but the proof of the kinetic-energy update in Eq. (59) omits the non-negative term $\frac12\int_\Omega \rho^n |u^{n+1}-u^n|^2\,dx$, so the equality as stated is not rigorously established unless that term cancels.

Editorial extensions

If this is right

For fixed walls, every computed time step satisfies $E_h^{n+1,\mathrm{tot}}\le E_h^{n,\mathrm{tot}}$, giving a nonlinear stability guarantee that prevents unbounded energy growth in long wetting and spreading runs.
The componentwise mass conservation means each phase's total mass stays fixed at machine precision across the interface, avoiding the spurious mass loss that can distort phase-field interface motion.
Because the quasi-incompressible $\Delta p$ term plays the role of pressure stabilization, equal-order $P1$ and $P2$ velocity-pressure spaces are sufficient, which simplifies code design relative to schemes requiring inf-sup stable pairs.
The density-extended GNBC makes the contact-line slip and phase-field boundary flux depend on $\alpha=(\rho_2-\rho_1)/(\rho_1\rho_2)$, so simulations with different density contrasts but identical surface tensions are expected to exhibit different contact-line speeds.
The reported convergence rates imply the scheme is usable for quantitative interface tracking: $P1$ gives roughly second order and $P2$ roughly third order in $L^2$ for velocity and phase variable in the tested Couette setup.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A consequence the paper leaves implicit: the density-dependent boundary term $L(c)=\epsilon\rho\partial_n c+\alpha_w df_w/dc$ suggests a testable prediction, namely that contact-line dynamics should change with density ratio even when surface energy parameters are held fixed.
The scheme's pressure-stabilization mechanism resembles pseudo-compressibility methods, so a natural test is whether the same $C^0$ formulation extends to three-dimensional moving-contact-line problems and to surfactant-laden variants of the quasi-incompressible NSCH system; the paper does not present such extensions.
If the missing non-negative kinetic-energy term $\frac12\int_\Omega \rho^n |u^{n+1}-u^n|^2\,dx$ is restored in the proof, the asserted discrete energy identity becomes a dissipation inequality, so Theorem 3.2 and the inherited fully discrete Theorem 3.4 are most safely read as an energy inequality rather than an exact equality.
The fully discrete theorem is stated without an independent proof; its proof is deferred to following Theorem 3.2, so any gap in the time-discrete energy identity propagates directly to the finite element energy law.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Short version: this is a serious paper, not a desk reject, but the headline claims outrun the proofs. The genuinely new piece is the thermodynamically consistent q-NSCH model with mass-averaged velocity and a GNBC, plus a C0 finite element scheme that avoids the inf-sup condition thanks to the Δp term. The variational derivation in Section 2 is coherent, and the simulations with density ratio 100:1 are practical evidence that the scheme is implementable. Credit is also due for not fitting parameters to a manufactured benchmark; the reference comparisons are standard.

The soft spot is the fully discrete analysis, and it is load-bearing. Lemma 3.3 chooses ψ_h = q_h = ρ_h^{n+1} = ρ(c_h^{n+1}) as test functions in (64a) and (64d). But ρ(·) is a rational function, so for P1/P2 c_h^n it is not in H_h; the Galerkin equations simply do not apply to that choice. There is also a sign inconsistency in the same step: substituting into (64d) gives −(1/α)∫∇ρ·u = −∫M(∇μ̄+α∇p̄)·∇ρ, while the proof writes a plus sign on the right. So exact phase-mass conservation, one of the two advertised properties, is unsupported as written. This may be repairable—project ρ into H_h and prove an approximate conservation law—but the exact statement in Lemma 3.3 is not established.

The energy law has a smaller wound. In the time-discrete proof, Eq. (59) states an equality for the kinetic-energy change without the nonnegative term (1/2)∫ρ^n|u^{n+1}−u^n|^2. That turns the equality into an inequality, which is enough for energy stability. So Theorem 3.2 likely survives as a stability bound, but not with the stated exact form. The fully discrete Theorem 3.4 is one line saying “follow the proof of Theorem 3.2,” so it inherits the issue and does not independently fix Lemma 3.3.

The convergence section also overclaims. Only two refinement intervals are used for the rates, and several first-step rates are well below the stated second/third order. That is a numerical-report weakness, not a fatal one.

Citation pattern is fine; prior q-NSCH and FEM work is properly acknowledged. This paper is for people building phase-field FEMs for moving contact lines, especially with large density ratios. It does not close the analysis gap, but it gives a usable starting point. I would send it to referees with the instruction to focus on Lemma 3.3; the authors need to fix the test-function argument or soften the claim. With that repaired, I would be comfortable with publication.

Referee Report

3 major / 4 minor

Summary. The paper develops a quasi-incompressible phase-field model for two-phase flows with moving contact lines and variable density, derived from an energy variational approach with a generalized Navier boundary condition. The authors then propose a C0 finite element discretization of the reformulated system and claim two headline properties: exact conservation of the mass of each phase (Lemma 3.3) and a fully discrete energy law with non-positive dissipation (Theorems 3.2 and 3.4). Numerical experiments cover Couette-flow convergence studies for low and high density ratios, moving droplets in shear flow, and a rising bubble with density ratio 0.001:1. The abstract further claims second-order convergence for P1 and third-order for P2 elements in the L2 norm.

Significance. If the two headline properties were rigorously established, this would be a useful contribution: a mass-conservative, energy-stable C0 finite element scheme for a thermodynamically consistent quasi-incompressible Navier-Stokes-Cahn-Hilliard system would extend the available toolbox for moving-contact-line simulations to large density ratios. The model derivation is self-consistent and first-principles, and the pressure-stabilized formulation that avoids the inf-sup condition is a genuine practical benefit. The numerical experiments target large density ratios and include diagnostic checks of mass, energy, and the quasi-incompressibility condition, which are appropriate validation targets. However, the advertised properties are currently not proven: the mass-conservation lemma is based on an invalid choice of test functions, and the discrete energy theorem omits a non-negative kinetic term. The significance is therefore conditional on the authors repairing these proofs or modifying the claims.

major comments (3)

[Section 3.2, Lemma 3.3] The proof of exact phase-mass conservation is not valid for the stated Galerkin scheme. Setting ψ_h = q_h = ρ_h^{n+1} in (64a) and (64d) is not an admissible choice because ρ_h^{n+1} = ρ1ρ2/(ρ2 c_h^{n+1}+ρ1(1-c_h^{n+1})) is a rational function of c_h^{n+1}, and for P1/P2 elements it is not a piecewise polynomial and hence not an element of the finite-dimensional test space H_h; equations (64) hold only for test functions in H_h. Moreover, substituting q_h = ρ_h^{n+1} into (64d) gives −∫∇ρ_h^{n+1}·u_h^{n+1} dx = −α∫M(∇μ̄_h^{n+1}+α∇p̄_h^{n+1})·∇ρ_h^{n+1} dx, so after division the displayed identity in the proof has the opposite sign. The claimed cancellation of the flux terms therefore does not follow, and the exact conservation identities (65a)-(65b) are unsupported.
[Section 3.1, Theorem 3.2 and Eq. (59)] Equation (59) omits the non-negative term (1/2)∫_Ω ρ^n |u^{n+1}-u^n|^2 dx that arises when the discrete momentum equation is multiplied by Δt u^{n+1}; the standard identity ∫ρ^n(u^{n+1}-u^n)·u^{n+1} = (1/2)∫ρ^n(|u^{n+1}|^2-|u^n|^2+|u^{n+1}-u^n|^2) shows that this term cannot cancel with the skew-symmetric terms in (49c). Consequently the discrete energy law (55) is not established as an equality; at best the calculation gives the inequality E^{n+1,tot}-E^{n,tot} ≤ −(non-negative dissipation). Since Theorem 3.4 is proved only by reference to the proof of Theorem 3.2, the fully discrete energy law (70) inherits the same gap. The energy-stability conclusion likely survives, but the theorem as stated is not rigorously established.
[Section 4.1, Tables 1 and 2] The convergence study does not provide enough evidence for the abstract's claim of second-order (P1) and third-order (P2) convergence in the L2 norm. Each error column reports only two observed rates, and the rates are not stable across the two intervals: for example, the P1 low-density-ratio Err(uy) rates are 1.46 and 2.78, and the P1 high-density-ratio Err(c) rates are 1.12 and 2.52. With only two refinement intervals no asymptotic regime is exhibited, and several rates fall well below the claimed orders. The authors should add at least one more refinement level, report all observed rates in both tables, and either substantiate or temper the convergence claim.

minor comments (4)

[Eq. (49c)] The term written as "+ 1/2(ρ^{n+1}-ρ^n/Δt + ∇·(ρ^n u^n)))" has a parenthesis mismatch and does not indicate what is multiplied; it should read something like "+ (1/2)((ρ^{n+1}-ρ^n)/Δt + ∇·(ρ^n u^n)) u^{n+1}".
[Section 4.3 and Fig. 5] The text in Section 4.3 uses "quasi-impressible" where "quasi-incompressible" is meant, and the caption of Fig. 5 says "L2 norm of u" while the text and the figure label refer to the L2 norm of ∇·u.
[Section 4.1, Tables 1 and 2] The convergence tables report errors only at time t = 0.2 and only for the spatial discretization; the manuscript states "convergence rate in the sense of L2 norm" without specifying that no temporal convergence study is presented, so the claimed orders should be attributed to the spatial discretization under the chosen time step.
[Throughout] There are several typos and formatting artifacts, including "immersible" for "immiscible" in the Introduction, "Appendix Appendix A" for "Appendix A", and mismatched parentheses in Eq. (64c). A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the scheme's claimed mass conservation and energy stability are derived from the discrete variational equations and tested against reference solutions; no claim reduces to an input or to a self-citation.

full rationale

The paper's main claims are a fully discrete C0 finite element scheme (Eq. 64) with exact phase-mass conservation (Lemma 3.3) and a discrete energy law (Theorems 3.2 and 3.4). The derivation chain is variational: given the total energy (13) and dissipation (15), the constitutive relations (23) are obtained by enforcing dE/dt = -Delta; the subsequent theorems verify algebraic consequences of those relations. This is not circular because the numerical scheme is not obtained by assuming its own stability or conservation properties. Self-citations ([13,42,55], sharing co-author P. Lin) are used for design guidance and algebraic identities, but Lemma 3.1's identities are elementary algebra and the fully discrete FE analysis is carried out in the paper, so the citations are not load-bearing. Numerical convergence tests compare against finer-mesh reference solutions without fitting, so the observed rates are independent evidence. The proof gaps flagged by a strict correctness review (Lemma 3.3 chooses test functions rho_h^{n+1} and rho_h^{n+1}c_h^{n+1} that need not lie in H_h; Eq. (59) drops the non-negative kinetic term (1/2)∫ rho_n |u^{n+1}-u^n|^2; Theorem 2.3 states 'The proof is similar as Theorem 2.2. Here we omit the details') are mathematical validity issues, not circular reductions. None of the paper's predictions is, by construction, equal to an input, and no load-bearing result is imported solely from a self-citation chain.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the chosen mixture rule and energy/dissipation functionals; these are modeling choices, not fitted to data. No free parameters are fitted in the derivation or the numerical tests, and no new physical entities are introduced.

assumptions (6)

domain assumption Density depends on concentration only via 1/ρ = c/ρ1 + (1-c)/ρ2 (Eq. 2)
This mixture rule is the basis for the quasi-incompressibility condition (Eq. 9) and the entire model.
domain assumption The total energy is the sum of kinetic, bulk mixing, and wall terms with the given G(c) and f_w(c) (Eqs. 13-14)
The variational derivation compares this energy with the dissipation functional.
domain assumption The dissipation functional has the prescribed form (Eq. 15) with constant mobilities and linear flux laws
Onsager's principle is used to identify fluxes via dE/dt = -Δ.
domain assumption The Stokes hypothesis λ = -2η/3 is adopted (Section 2.3)
Simplifies the dimensionless system; commonly used for gases and assumed without discussion.
standard math Solutions are assumed smooth for the energy laws (Theorems 2.2 and 2.3)
The proofs use integration by parts and Reynolds transport, which require smoothness.
domain assumption For the finite element analysis, ρ^n ∈ L∞(Ω) and positive [55]
Used in the weak formulation and stability proof, quoted from prior work.

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Cite this review

Pith. "Pith review of An energy stable $C^0$ finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density." pith.science (2026). https://pith.science/paper/VM53F6HE

@misc{pith2026190803681,
  author       = {Pith},
  title        = {Pith review of: An energy stable $C^0$ finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VM53F6HE}},
  note         = {Machine review of arXiv:1908.03681}
}

read the original abstract

In this paper, we focus on modeling and simulation of two-phase flow with moving contact lines and variable density. A thermodynamically consistent phase-field model with General Navier Boundary Condition is developed based on the concept of quasi-incompressibility and the energy variational method. Then a mass conserving and energy stable C0 finite element scheme is developed to solve the PDE system. Various numerical simulation results show that the proposed schemes are mass conservative, energy stable and the 2nd order for P1 element and 3rd order for P2 element convergence rate in the sense of L2 norm.

Figures

Figures reproduced from arXiv: 1908.03681 by the authors.

**Figure 2.** Initial condition of phase 1 concentration [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** The interface and velocity profile at T = 0.4 large density ratio ρ1 = 0.1, ρ2 = 10. The fluid velocities on the wall are shown in Fig.4. It shows that for both low and high density ratio case, P1 element could yield consistent contact velocity with P2 element. In Fig.5, we check the L 2 norm of ∇ · u with different . The results confirm that as decreases, the solution converges to the sharp interface incompressi… view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Velocity on wall around equilibrium state.Left: low density ratio [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

**Figure 5.** Figure 5: L 2 norm of ∇ · u with different . over time for both methods and two density ratios, indicating that our schemes are energy stable. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

**Figure 6.** Figure 6: Mass conservation for each component. Left: low density ratio; Right: high density ratio. [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

**Figure 7.** Figure 7: Total energy as a function of time. Left: lower density ratio; Right: high density ratio. [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

**Figure 8.** Figure 8: Moving droplet in shear flow with acute static contact angle [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]

**Figure 9.** Figure 9: Moving droplet in shear flow with acute static contact angle [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

**Figure 10.** Figure 10: Moving droplet in shear flow with obtuse static contact angle [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]

**Figure 11.** Figure 11: Moving droplet in shear flow: dynamics of the distance between two contact points. [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]

**Figure 12.** Figure 12: Rising Bubble interface with velocity filed (first and third rows) and [PITH_FULL_IMAGE:figures/full_fig_p027_12.png]

**Figure 13.** Figure 13: Rising Bubble interface with velocity filed (first and third rows) and [PITH_FULL_IMAGE:figures/full_fig_p028_13.png]

**Figure 14.** Figure 14: Rising velocity. Solid lines are rising velocity and vertical dash lines are time when bubble break [PITH_FULL_IMAGE:figures/full_fig_p029_14.png]

Discussion (0). Continue with ORCID to comment.

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