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An energy--stable ALE--FEM with compensated tangential velocity for Navier--Stokes free--boundary flows

Nuo Lei, Weiying Zheng, Na Zhu

arXiv:2610.09301v1math.NAcs.NA

TL;DR

Surface-tension-driven Navier–Stokes free-boundary flows require energy control while maintaining a usable moving mesh. The paper proposes an energy-stable ALE finite element method with compensated tangential velocity, and reports energy decay, mesh quality, and robustness across μ and γ0 values, including in the small-time-step regime.

  • Problem

    For surface-tension-driven free-boundary flows, energy-diminishing discretizations are important, while ALE mesh quality depends on tangential motion that BGN may not control well as Δt → 0.

    Source: §1 · pp. 1–2

  • Method

    The method uses a compensated tangential velocity based on the discrete tangential curvature residual, with relaxation scaled by α = γ0/μ.

    Source: §1 · pp. 2–3, §6 · p. 18

  • Results

    The fully discrete scheme is unconditionally energy-stable, recovers its semi-discrete counterpart as Δt → 0, and preserves energy decay and high-quality meshes across variations in μ and γ0.

    Source: §1 · p. 3, §6 · p. 18

  • Takeaways & Limitations

    The reported CTV-ALE method maintains effective tangential redistribution and high-quality interface and bulk meshes, including in the small-Δt regime where BGN degrades.

    Source: §1 · p. 3

    The relaxation parameter ε is assumed sufficiently small and independent of the physical parameters.

    Source: §1 · p. 3

Abstract

from arXiv · show

We propose an energy-stable arbitrary Lagrangian-Eulerian (ALE) finite element formulation for incompressible Navier-Stokes flows with surface-tension-driven free boundaries. The key ingredient is a multiplier-free compensated tangential velocity formulation, which redistributes boundary nodes via the discrete tangential curvature residual. The relaxation parameter governing the artificial tangential velocity is scaled according to the viscous-capillary balance, yielding the natural choice $\alpha= \gamma_0/ \mu $, where $\gamma_0$ is the surface tension coefficient and $\mu$ is the dynamic viscosity. We prove that the fully discrete scheme is well-posed, energy-dissipative, and temporally consistent with its semi-discrete counterpart. Numerical results confirm the robustness of the proposed method across a range of $\mu$ and $\gamma_0$ values. Furthermore, the method exhibits superior mesh quality preservation compared with the BGN method, particularly in the small-time-step regime.

1. Introduction

The introduction presents an energy-stable ALE finite element approach for surface-tension-driven Navier–Stokes free-boundary flows. Its compensated tangential velocity addresses mesh redistribution while supporting the discrete energy behavior and mesh quality reported for the method.

1. Introduction

  • Surface tension couples curvature-dependent interfacial forces to incompressible Navier–Stokes flow, whose kinetic and interfacial energies decrease through viscous dissipation.Discrete energy-dissipation laws provide a priori bounds and prevent artificial energy production from discretization.Source: §1 · p. 1
  • The kinematic condition fixes normal mesh motion, while tangential motion can redistribute boundary nodes; BGN redistribution may deteriorate as Δt → 0.ALE permits the bulk mesh velocity to be chosen independently of the fluid velocity.Source: §1 · pp. 1–2
  • Earlier MDR-type approaches target mesh redistribution while preserving prescribed normal motion, but energy-stable time discretization does not follow automatically from their continuous principles.Related work developed energy-stable MDR schemes for geometric curvature flows and a dual formulation using an auxiliary multiplier.Source: §1 · p. 2
  • The method sets tangential ALE motion from the discrete tangential curvature residual, complementing the fluid-determined normal motion.The relaxation parameter ε is sufficiently small and independent of physical parameters; the discrete tangential velocity can remain nonzero even when ε = 0.Source: §1 · pp. 2–3
  • The fully discrete CTV-ALE scheme is unconditionally energy-stable, recovers the semi-discrete scheme as Δt → 0, and maintains energy decay and high-quality meshes across variations in μ and γ0.The introduction also reports that it overcomes BGN mesh-quality degradation in the small-Δt regime.Source: §1 · p. 3

2. A free-boundary problem for the Navier-Stokes equations

The section formulates a surface-tension-driven Navier–Stokes free-boundary problem in ALE form and closes it with a compensated tangential-velocity formulation. It presents an energy-dissipation result for that formulation.

2.1. The model problem.

  • The model couples incompressible Navier–Stokes flow to a moving free boundary whose stress condition involves surface tension and mean curvature.The stress is σ = 2µD(u) − pI, and the surface tension coefficient is γ0.Source: §2.1 · pp. 3–4

2.2. The ALE formulation.

  • The ALE velocity moves the domain, while its surface trace is extended into the bulk by a minimal-deformation-rate problem.The ALE time derivative includes mesh advection, and the resulting fluid equation uses relative velocity u − w.Source: §2.2 · p. 4

2.3. The CTV formulation.

  • The continuous tangential-curvature term vanishes after integration by parts, but its discrete counterpart is nonzero and contributes to numerical stability.With ε = 0, the formulation gives purely normal boundary motion.Source: §2.3 · pp. 4–5
  • The surface velocity combines the fluid's normal velocity with a tangential component defined by the CTV surface variational problem.The formulation sets α = γ0/µ and permits a small relaxation parameter ε, including ε = 0.Source: §2.3 · pp. 4–5

2.4. Total energy-dissipation.

  • For zero forcing and ε = 0, the total energy is dissipative; the energy estimate applies for ε below a geometry-dependent threshold.The stated energy consists of kinetic energy and surface energy.Source: §2.4 · p. 5

3. Semi-discrete ALE finite element method

The semi-discrete method evolves an isoparametric finite-element mesh while retaining its connectivity and moving its geometry nodes. Its energy estimate inherits a dissipation property under stated conditions.

3.1. Finite element meshes.

  • The initial domain uses a shape-regular, quasi-uniform isoparametric triangulation fitted to the interface, with geometric approximation accuracy O(h^(k+1)).The mesh connectivity is held fixed during evolution while geometry nodes move.Source: §3.1 · p. 7

3.2. Finite element spaces.

  • The method defines polynomial finite-element spaces on the volume and boundary meshes and projects the discontinuous face normal into a boundary space.The text assumes polynomial degree k ≥ 2 and uses the projected normal to define the tangential space and projection.Source: §3.2 · pp. 7–8

3.3. Semi-discrete schemes.

  • The semi-discrete scheme couples the fluid, curvature, tangential velocity, and bulk mesh velocity on the evolving mesh.For zero forcing and ε = 0, its total energy is dissipative under the theorem's stated small-ε condition.Source: §3.3 · p. 8

4. Fully discrete scheme

The fully discrete method advances the coupled ALE finite-element system in time and establishes energy control and temporal consistency. The temporal limit is shown to recover the semi-discrete solution.

4.1. The CTV scheme.

  • The fully discrete CTV scheme is linear at each step because the quantities used to form it are calculated at the preceding time level.It is posed on a uniform time partition and coupled with a one-step ALE mapping.Source: §4 · p. 9, §4.1 · pp. 9–10

4.2. Energy dissipation.

  • For any ℓ ≥ 1, if ε ≤ ε2 and 2(1 + µ)Δt < 1, problem (4.1) has a unique solution at each time step.The stated assumptions allow ε = 0.Source: §4.2 · pp. 10–11

4.3. Temporal consistency.

  • The CTV scheme is temporally consistent: as Δt tends to zero, its solution converges to a weak solution of the semi-discrete scheme.The convergence result applies for ε within the bounds specified by the semi-discrete and fully discrete energy theorems.Source: §4.3 · pp. 11–12

5. Numerical tests

The tests assess accuracy and parameter robustness, compare CTV and BGN mesh behavior, and examine three-dimensional capillary dynamics. Across these tests, they report second-order spatial convergence, energy decay, and mesh-quality behavior under varied conditions.

5.1. Accuracy test.

  • Both self-convergence errors exhibit asymptotically second-order convergence, consistent with the expected optimal spatial accuracy.The experiment compares solutions across successively refined meshes after pulling them back to the initial domain.Source: §5.1 · p. 14, Table 1 · p. 14

5.2. Robustness with respect to ε.

  • For ε = 1, 10^-4, 10^-8, and 0, total energy decreases monotonically and σm_max remains stable as ε approaches zero.The comparison uses a fixed time step Δt = 10^-2.Source: §5.2 · p. 14, Figure 2 · p. 15

5.3. Six-leaf clover.

  • At Δt = 10^-4 and μ = 10^-3, BGN meshes tangle and terminate computation, while CTV produces shape-regular meshes and supports long-time evolution.Figure 3 shows the meshes at several times.Source: §5.3 · p. 15, Figure 3 · p. 16
  • At μ = 10^-2, CTV retains good mesh quality and remains stable to t = 1, while illegal concave BGN elements terminate computation after t = 0.3.The CTV mesh-quality indicator σm_max remains around 10 throughout the simulation.Source: §5.3 · p. 15, Figure 4 · p. 16, Figure 5 · p. 17

5.4. Three-dimensional damping of circularity.

  • In the damping test, circularity oscillates, and higher surface-tension coefficient γ0 produces higher oscillation frequency; computed mean frequency f closely follows f ∼ γ0^1/2.The tested γ0 values were 1, 2, 4, 6, 8, and 10; the scaling is consistent with capillary oscillation behavior reported in the literature.Source: §5.4 · pp. 15–16

5.5. Three-dimensional flow in a popcorn-shaped domain.

  • In the popcorn-shaped 3D droplet test, meshes retain good quality at the reported stages despite the initial interface’s nonuniform curvature.The test uses μ = 0.01, γ0 = 1, and Δt = 10^-3.Source: §5.5 · pp. 17–18, Figure 8 · p. 18

6. Conclusion

The conclusion presents an energy-stable ALE–FEM for surface-tension-driven free-boundary flows, with a compensated tangential velocity formulation. It reports well-posedness, energy-dissipation estimates, and consistency with the semi-discrete formulation, while identifying convergence results and error estimates as future work.

6. Conclusion

  • The fully discrete scheme is well-posed and energy-dissipative, with energy-dissipation estimates also established for the semi-discrete formulation.Source: §6 · p. 18
  • The surface mesh velocity follows fluid motion normally and uses compensated tangential velocity for redistribution, with α = γ0/µ and an MDR bulk extension.Source: §6 · p. 18
  • As ∆t → 0, the fully discrete CTV formulation consistently recovers its semi-discrete counterpart.Source: §6 · p. 18
  • Establishing rigorous convergence results and error estimates for ALE finite element methods in free-boundary and two-phase problems is identified as future work.Source: §6 · p. 18
  • Figure 9 shows mesh velocity fields at different time instants.Source: Figure 9 · p. 19
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