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Optimal convergence analysis of fully discrete SAVs-FEM for the Cahn-Hilliard-Navier-Stokes equations

Haijun Gao, Xi Li, Cheng Wang, Minfu Feng

TL;DR

This paper develops a fully discrete, linear, and unconditionally energy-stable SAV-FEM scheme for the CHNS system in a 2D bounded domain. It blends two scalar auxiliary variables with a pressure-correction projection and Taylor-Hood–type finite elements, achieving optimal $L^2$ error estimates for $(\phi,\mu,p)$ when $r\ge1$ and for the velocity when $r\ge2$, without relying on quasi-projection techniques. Theoretical analysis establishes unconditional energy stability and sharp convergence rates, while numerical experiments confirm the predicted orders and demonstrate stable coarsening dynamics and shape relaxation under rotational boundary conditions. The results indicate a practically effective, decoupled, and energy-stable framework for simulating two-phase incompressible flows modeled by CHNS.

Abstract

We construct a fully discrete numerical scheme that is linear, decoupled, and unconditionally energy stable, and analyze its optimal error estimates for the Cahn-Hilliard-Navier-Stokes equations. For time discretization, we employ the two scalar auxiliary variables (SAVs) and the pressure-correction projection method. For spatial discretization, we choose the $P_r \times P_r \times \mathbf{P}_{r+1} \times P_r$ finite element spaces, where $r$ is the degree of the local polynomials, and derive the optimal $L^2$ error estimates for the phase-field variable, chemical potential, and pressure in the case of $r \geq 1$, and for the velocity when $r \geq 2$, without relying on the quasi-projection operator technique proposed in \textit{[Cai et al. SIAM J Numer Anal, 2023]}. Numerical experiments validate the theoretical results, confirming the unconditional energy stability and optimal convergence rates of the proposed scheme. Additionally, we numerically demonstrate the optimal $L^2$ convergence rate for the velocity when $r=1$.

Optimal convergence analysis of fully discrete SAVs-FEM for the Cahn-Hilliard-Navier-Stokes equations

TL;DR

This paper develops a fully discrete, linear, and unconditionally energy-stable SAV-FEM scheme for the CHNS system in a 2D bounded domain. It blends two scalar auxiliary variables with a pressure-correction projection and Taylor-Hood–type finite elements, achieving optimal error estimates for when and for the velocity when , without relying on quasi-projection techniques. Theoretical analysis establishes unconditional energy stability and sharp convergence rates, while numerical experiments confirm the predicted orders and demonstrate stable coarsening dynamics and shape relaxation under rotational boundary conditions. The results indicate a practically effective, decoupled, and energy-stable framework for simulating two-phase incompressible flows modeled by CHNS.

Abstract

We construct a fully discrete numerical scheme that is linear, decoupled, and unconditionally energy stable, and analyze its optimal error estimates for the Cahn-Hilliard-Navier-Stokes equations. For time discretization, we employ the two scalar auxiliary variables (SAVs) and the pressure-correction projection method. For spatial discretization, we choose the finite element spaces, where is the degree of the local polynomials, and derive the optimal error estimates for the phase-field variable, chemical potential, and pressure in the case of , and for the velocity when , without relying on the quasi-projection operator technique proposed in \textit{[Cai et al. SIAM J Numer Anal, 2023]}. Numerical experiments validate the theoretical results, confirming the unconditional energy stability and optimal convergence rates of the proposed scheme. Additionally, we numerically demonstrate the optimal convergence rate for the velocity when .

Paper Structure

This paper contains 14 sections, 15 theorems, 171 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Numerical scheme
  3. Notations and preliminaries
  4. Fully discretization scheme
  5. Discrete scheme
  6. Energy stability
  7. Error analysis
  8. Convergence analysis
  9. Numerical test
  10. Convergence tests
  11. Coarsening dynamics
  12. Shape Relaxation with Rotational Boundary Condition
  13. Verification of unconditional stability
  14. Conclusion

Key Result

Lemma 2.1

For all $0 \leq n \leq m$, let $a_n, b_n, c_n, d_n, \tau, C \geq 0$ such that suppose that $\tau d_n<1$, for all $n$, and set $\sigma_n=(1-\tau d_n)^{-1}$. Then, it holds

Figures (3)

  • Figure 1: Snapshots of phase function at difference times from left to right row by row with $t=0.001, 0.05, 0.1, 0.15, 0.3, 1, 3, 5$, respectively.
  • Figure 2: Snapshots of phase function at difference times from left to right row by row with $t=0, 0.01, 0.05, 0.08, 0.1, 0.2, 0.3, 0.5$, respectively.
  • Figure 3: Evolution of the energy with different time-step $\tau$.

Theorems & Definitions (25)

  • Lemma 2.1: 1990_Heywood_Finite_element_approximation_of_the_nonstationary_Navier_Stokes_problem_IV_Error_analysis_for_second_order_time_discretization
  • Lemma 2.2: 2015_HeYinnian_Unconditional_convergence_of_the_Euler_semi_implicit_scheme_for_the_three_dimensional_incompressible_MHD_equations
  • Lemma 2.3: 2019_YangXiaofeng_Convergence_analysis_of_an_unconditionally_energy_stable_projection_scheme_for_magneto_hydrodynamic_equations2024_YiNianyu_Convergence_analysis_of_a_decoupled_pressure_correction_SAV_FEM_for_the_Cahn_Hilliard_Navier_Stokes_model
  • Lemma 2.4: 2020_Chen_Hongtao_Optimal_error_estimates_for_the_scalar_auxiliary_variable_finite_element_schemes_for_gradient_flows2024_YiNianyu_Convergence_analysis_of_a_decoupled_pressure_correction_SAV_FEM_for_the_Cahn_Hilliard_Navier_Stokes_model
  • Theorem 2.1
  • proof
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • ...and 15 more