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Divergence-free and mass-conservative virtual element methods for the Navier-Stokes-Cahn-Hilliard system

Alberth Silgado, Giuseppe Vacca

TL;DR

This work develops high-order divergence-free and $C^1$-conforming virtual element discretizations for the time-dependent Navier–Stokes–Cahn–Hilliard system, emphasizing a variational formulation that uses only velocity, pressure, and phase-field variables. A novel skew-symmetric convective form for the Cahn–Hilliard equation enables discrete mass conservation and energy-type bounds at semi- and fully-discrete levels, while projection-based Ritz operators yield optimal error estimates. The approach leverages polygonal meshes and space-time discretizations to achieve exact incompressibility, mass conservation, and high-order accuracy without auxiliary variables. Numerical experiments on diverse meshes confirm the predicted convergence rates and the mass-conserving property, highlighting robustness and applicability to complex two-phase flows. The framework lays groundwork for future energy-law preserving schemes on general meshes and for further extensions to fully discrete energy-dissipative formulations.

Abstract

In this work, we design and analyze semi/fully-discrete virtual element approximations for the time-dependent Navier--Stokes-Cahn--Hilliard equations, modeling the dynamics of two-phase incompressible fluid flows with diffuse interfaces. A new variational formulation is derived involving solely the velocity, pressure, and phase field, together with corresponding a priori energy estimates. The spatial discretization is based on the coupling divergence-free and $C^1$-conforming elements of high-order, while the time discretization employs a classical backward Euler scheme. By introducing a novel skew-symmetric trilinear form to discretize the convective term in the Cahn--Hilliard equation, we propose discrete schemes that satisfy mass conservation and energy bounds. Moreover, optimal error estimates are provided for both formulations. Finally, two numerical experiments are presented to support our theoretical findings and to illustrate the good performance of the proposed schemes for different polynomial degrees and polygonal meshes.

Divergence-free and mass-conservative virtual element methods for the Navier-Stokes-Cahn-Hilliard system

TL;DR

This work develops high-order divergence-free and -conforming virtual element discretizations for the time-dependent Navier–Stokes–Cahn–Hilliard system, emphasizing a variational formulation that uses only velocity, pressure, and phase-field variables. A novel skew-symmetric convective form for the Cahn–Hilliard equation enables discrete mass conservation and energy-type bounds at semi- and fully-discrete levels, while projection-based Ritz operators yield optimal error estimates. The approach leverages polygonal meshes and space-time discretizations to achieve exact incompressibility, mass conservation, and high-order accuracy without auxiliary variables. Numerical experiments on diverse meshes confirm the predicted convergence rates and the mass-conserving property, highlighting robustness and applicability to complex two-phase flows. The framework lays groundwork for future energy-law preserving schemes on general meshes and for further extensions to fully discrete energy-dissipative formulations.

Abstract

In this work, we design and analyze semi/fully-discrete virtual element approximations for the time-dependent Navier--Stokes-Cahn--Hilliard equations, modeling the dynamics of two-phase incompressible fluid flows with diffuse interfaces. A new variational formulation is derived involving solely the velocity, pressure, and phase field, together with corresponding a priori energy estimates. The spatial discretization is based on the coupling divergence-free and -conforming elements of high-order, while the time discretization employs a classical backward Euler scheme. By introducing a novel skew-symmetric trilinear form to discretize the convective term in the Cahn--Hilliard equation, we propose discrete schemes that satisfy mass conservation and energy bounds. Moreover, optimal error estimates are provided for both formulations. Finally, two numerical experiments are presented to support our theoretical findings and to illustrate the good performance of the proposed schemes for different polynomial degrees and polygonal meshes.
Paper Structure (32 sections, 20 theorems, 161 equations, 5 figures)

This paper contains 32 sections, 20 theorems, 161 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Scope and contributions.
  3. Notation.
  4. Outline.
  5. The model problem and continuous weak formulation
  6. A priori energy estimates
  7. Conforming virtual element approximation
  8. Polygonal decomposition and polynomial projections
  9. Virtual element spaces for the velocities and pressures
  10. Virtual element space for the phase variable
  11. Virtual element forms
  12. Stabilizing forms
  13. Discrete first-order forms
  14. Discrete second- and fourth-order forms:
  15. Discrete convective forms:
  16. ...and 17 more sections

Key Result

Proposition 2.1

Suppose that the initial values $(\boldsymbol{u}_0, \varphi_0)$ in initial:data satisfy $\mathcal{J}_{\varepsilon,\lambda}(\boldsymbol{u}_0,\varphi_0) < \infty$. Then, every regular solution $(\boldsymbol{u}, \varphi)$ of problem Mmodel_a satisfies the following estimates: where $C_{{\rm stab}}$ is a positive constant depending on the data and physical parameters.

Figures (5)

  • Figure 1: Test 1. Sample of the adopted meshes.
  • Figure 2: Test 1. Convergence histories of the VEM errors \ref{['compu:errors']} at final time $T=1$. The left panel reports the results for $(k,\ell)=(1,2)$, whereas the right panel corresponds to $(k,\ell)=(2,3)$.
  • Figure 3: Test 2. Snapshots of the numerical phase solutions $\varphi^n_h$ at different time instants $t_n$, using the fully-discrete VE scheme \ref{['fully:disc:schm']}, $\nu=1$, $\lambda=\gamma=0.1$, $\varepsilon=2 \times 10^{-2}$, and $(k,\ell)=(1,2)$.
  • Figure 4: Test 2. Snapshots and streamlines of the numerical velocity solutions $\boldsymbol{u}^n_h$ at different time instants, using the fully-discrete VE scheme \ref{['fully:disc:schm']}, $\nu=1$, $\lambda=\gamma=0.1$, and $\varepsilon=2 \times 10^{-2}$, $(k,\ell)=(1,2)$.
  • Figure 5: Test 2. Snapshots of the numerical pseudo-pressure solution $\widehat{p}^n_h$ at different time instants, using the fully-discrete VE scheme \ref{['fully:disc:schm']}, $\nu=1$, $\lambda=\gamma=0.1$, and $\varepsilon=2 \times 10^{-2}$, $(k,\ell)=(1,2)$.

Theorems & Definitions (26)

  • Proposition 2.1: Continuous conservation laws
  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1: Bramble-Hilbert
  • Proposition 3.2: Properties of the discrete forms
  • Remark 3.1
  • Lemma 4.1: Grönwall’s Lemma
  • Theorem 4.1: Semi-discrete conservation law and energy stability
  • Lemma 4.2: Discrete Grönwall inequality
  • Theorem 4.2: Fully-discrete conservation law and energy stability
  • ...and 16 more