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

Abstract

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 α= γ₀/ μ, where γ₀ is the surface tension coefficient and μ 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 μ and γ₀ values. Furthermore, the method exhibits superior mesh quality preservation compared with the BGN method, particularly in the small-time-step regime.

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BibTeX RIS

BibTeX

@misc{lei2026energystable,
  title = {{An energy--stable ALE--FEM with compensated tangential velocity for Navier--Stokes free--boundary flows}},
  author = {Nuo Lei and Weiying Zheng and Na Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09301},
  eprint = {2610.09301},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09301},
}

RIS

TY  - UNPB
TI  - An energy--stable ALE--FEM with compensated tangential velocity for Navier--Stokes free--boundary flows
AU  - Nuo Lei
AU  - Weiying Zheng
AU  - Na Zhu
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09301
UR  - https://arxiv.org/abs/2610.09301
AB  - 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 α= γ_0/ μ, where γ_0 is the surface tension coefficient and μ 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 μ and γ_0 values. Furthermore, the method exhibits superior mesh quality preservation compared with the BGN method, particularly in the small-time-step regime.
ER  -

Abstract

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