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Optimal convergence of the arbitrary Lagrangian-Eulerian interface tracking method for two-phase Navier--Stokes flow without surface tension

Buyang Li, Shu Ma, Weifeng Qiu

TL;DR

This work develops and analyzes an arbitrary Lagrangian–Eulerian interface-tracking finite element method for sharp-interface two-phase Navier–Stokes flow without surface tension. The method tracks the interface with the fluid velocity while extending the bulk mesh harmonically, using high-order curved elements and Taylor–Hood spaces to achieve optimal $O(h^k)$ convergence in $L^ty(0,T; H^1(Omega))$ for $k\ge 2$, despite only piecewise smooth solutions within each phase. A key innovation is the matrix–vector formulation combined with an interpolated evolving mesh and a lifting operator, which enables robust consistency analysis and error estimates when the numerically tracked interface does not coincide with the exact interface. The main result is established via a homotopy between interpolated and actual domains and comprehensive energy estimates, with numerical experiments confirming the predicted convergence rates and illustrating dynamics such as center of mass, circularity, rise velocity, and energy for a bubble in gravity-driven flow without surface tension. The approach provides a rigorous foundation for high-order, stable ALE-FEMs in two-phase NS settings where the interface evolves dynamically and global solution regularity is limited by the moving boundary.

Abstract

Optimal-order convergence in the $H^1$ norm is proved for an arbitrary Lagrangian-Eulerian interface tracking finite element method for the sharp interface model of two-phase Navier-Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid's velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete arbitrary Lagrangian-Eulerian interface tracking finite element method is shown to be $O(h^k)$ in the $L^\infty(0, T; H^1(Ω))$ norm for the Taylor-Hood finite elements of degree $k \ge 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.

Optimal convergence of the arbitrary Lagrangian-Eulerian interface tracking method for two-phase Navier--Stokes flow without surface tension

TL;DR

This work develops and analyzes an arbitrary Lagrangian–Eulerian interface-tracking finite element method for sharp-interface two-phase Navier–Stokes flow without surface tension. The method tracks the interface with the fluid velocity while extending the bulk mesh harmonically, using high-order curved elements and Taylor–Hood spaces to achieve optimal convergence in for , despite only piecewise smooth solutions within each phase. A key innovation is the matrix–vector formulation combined with an interpolated evolving mesh and a lifting operator, which enables robust consistency analysis and error estimates when the numerically tracked interface does not coincide with the exact interface. The main result is established via a homotopy between interpolated and actual domains and comprehensive energy estimates, with numerical experiments confirming the predicted convergence rates and illustrating dynamics such as center of mass, circularity, rise velocity, and energy for a bubble in gravity-driven flow without surface tension. The approach provides a rigorous foundation for high-order, stable ALE-FEMs in two-phase NS settings where the interface evolves dynamically and global solution regularity is limited by the moving boundary.

Abstract

Optimal-order convergence in the norm is proved for an arbitrary Lagrangian-Eulerian interface tracking finite element method for the sharp interface model of two-phase Navier-Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid's velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete arbitrary Lagrangian-Eulerian interface tracking finite element method is shown to be in the norm for the Taylor-Hood finite elements of degree . This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.
Paper Structure (13 sections, 7 theorems, 128 equations, 5 figures, 1 table)

This paper contains 13 sections, 7 theorems, 128 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The ALE evolving finite element method
  3. The ALE reformulation of PDE
  4. The ALE mapping and ALE evolving FEM
  5. Interpolated evolving mesh and consistency analysis
  6. Interpolated ALE mapping and interpolated evolving mesh
  7. Lifting operator
  8. Weak formulation on the interpolated evolving mesh
  9. Matrix-vector form of the weak formulation
  10. Convergence of the ALE evolving FEM
  11. Comparison of integrals on two different domains
  12. The proof of Theorem \ref{['Main-THM']}
  13. Numerical results

Key Result

proposition 1

We assume that the flow map $\phi:\varOmega_\pm(0)\times[0,T]\rightarrow {\mathbb R}^d$ and its inverse $\phi(\cdot,t)^{-1}:\varOmega_\pm(t)\rightarrow \varOmega_\pm(0)$ are both sufficiently smooth so that the subdomains $\varOmega_\pm(t)$ are sufficiently smooth and the interpolated triangulations Then there exist positive constants $h_0$ such that for $h\leqslant h_0$, the following consistency

Figures (5)

  • Figure 1: Domain $\varOmega$ in the case $d = 2$.
  • Figure 2: Spatial discretization errors of the numerical solutions at $T = 0.1$.
  • Figure 4: Left: mesh $M_1$. Right: mesh $M_2$. Dark color: domain $\varOmega_{-}(0)$.
  • Figure 6: The number of remeshing steps with respect to time.
  • Figure 9: Benchmark quantities with respect to time

Theorems & Definitions (12)

  • remark 1
  • remark 2
  • remark 3
  • proposition 1: Consistency estimates
  • theorem 1: Convergence of the ALE evolving FEM
  • lemma 1
  • remark 4
  • lemma 2
  • lemma 3
  • remark 5: Norm equivalence
  • ...and 2 more