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Unconditional energy stable IEQ-FEMs for the Cahn-Hilliard-Navier-Stokes equations

Yaoyao Chen, Dongqian Li, Yin Yang, Peimeng Yin

Abstract

We propose several unconditionally energy stable invariant energy quadratization (IEQ) finite element methods (FEMs) to solve the Cahn-Hilliard-Navier-Stokes (CHNS) equations. The time discretization of these IEQ-FEMs is based on the first- and second-order backward differentiation methods. The intermediate function introduced by the IEQ approach is positioned in different function spaces: the continuous function space, and a combination of the continuous function and finite element spaces. These methods offer distinct advantages. Consequently, we propose a new hybrid IEQ-FEM that combines the strengths of both schemes, offering computational efficiency and unconditional energy stability in the finite element space. We provide rigorous proofs of mass conservation and energy dissipation for the proposed IEQ-FEMs. Several numerical experiments are presented to validate the accuracy, efficiency, and solution properties of the proposed IEQ-FEMs.

Unconditional energy stable IEQ-FEMs for the Cahn-Hilliard-Navier-Stokes equations

Abstract

We propose several unconditionally energy stable invariant energy quadratization (IEQ) finite element methods (FEMs) to solve the Cahn-Hilliard-Navier-Stokes (CHNS) equations. The time discretization of these IEQ-FEMs is based on the first- and second-order backward differentiation methods. The intermediate function introduced by the IEQ approach is positioned in different function spaces: the continuous function space, and a combination of the continuous function and finite element spaces. These methods offer distinct advantages. Consequently, we propose a new hybrid IEQ-FEM that combines the strengths of both schemes, offering computational efficiency and unconditional energy stability in the finite element space. We provide rigorous proofs of mass conservation and energy dissipation for the proposed IEQ-FEMs. Several numerical experiments are presented to validate the accuracy, efficiency, and solution properties of the proposed IEQ-FEMs.
Paper Structure (16 sections, 11 theorems, 96 equations, 13 figures, 5 tables, 1 algorithm)

This paper contains 16 sections, 11 theorems, 96 equations, 13 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. IEQ-FEMs for the CHNS Equations
  3. The semi-discrete FEM for the CHNS equations.
  4. The IEQ reformulation
  5. Fully discrete schemes
  6. First order fully discrete schemes
  7. P-BDF1-IEQ-FEM scheme
  8. C-BDF1-IEQ-FEM scheme
  9. CP-BDF1-IEQ-FEM scheme
  10. Second order fully discrete schemes
  11. P-BDF2-IEQ-FEM scheme
  12. C-BDF2-IEQ-FEM scheme
  13. Numerical examples
  14. Convergent rates
  15. Numerical solutions for the CHNS equations
  16. ...and 1 more sections

Key Result

Lemma 2.1

The semi-discrete finite element scheme eq2.1 conserves the total mass and the solution satisfies the energy dissipation law

Figures (13)

  • Figure 1: $\mathbf{Example\ \ref{['exam4']}}$, snapshots of numerical solutions for phase field function, First and second lines: P-BDF1-IEQ-FEM scheme \ref{['se3-1']}-\ref{['se3-3']}; Third and fourth lines: P-BDF2-IEQ-FEM scheme \ref{['se3-18']}-\ref{['se3-19']}.
  • Figure 2: $\mathbf{Example\ \ref{['exam4']}}$, snapshots of numerical solutions for velocity field function, First line: P-BDF1-IEQ-FEM scheme \ref{['se3-1']}-\ref{['se3-3']}; Second line: P-BDF2-IEQ-FEM scheme \ref{['se3-18']}-\ref{['se3-19']}.
  • Figure 3: $\mathbf{Example\ \ref{['exam4']}}$, the modified discrete energy history and discrete mass history.
  • Figure 4: $\mathbf{Example\ \ref{['exam2']}}$, P-BDF2-IEQ-FEM scheme \ref{['se3-18']}-\ref{['se3-19']}, snapshots of numerical solutions for phase field function.
  • Figure 5: $\mathbf{Example\ \ref{['exam2']}}$, P-BDF2-IEQ-FEM scheme \ref{['se3-18']}-\ref{['se3-19']}, snapshots of numerical solutions for velocity field function.
  • ...and 8 more figures

Theorems & Definitions (26)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 16 more