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

TL;DR

This paper introduces a hybrid IEQ-FEM approach for the Cahn-Hilliard-Navier-Stokes equations that ensures unconditional energy stability, mass conservation, and improved computational efficiency through rigorous proofs and numerical validation.

Contribution

A novel hybrid IEQ-FEM method combining first- and second-order schemes for improved stability and efficiency in solving CHNS equations.

Findings

The hybrid IEQ-FEM guarantees unconditional energy stability.

Numerical experiments confirm high accuracy and efficiency.

The method conserves mass and dissipates energy as proven theoretically.

Abstract

We investigate two unconditionally energy stable invariant energy quadratization (IEQ) finite element methods (FEMs) [Chen et al. Numerical Algorithms, DOI: 10.1007/s11075-024-01910-z, 2024] for solving 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. \textcolor{black}{The auxiliary energy function introduced by the IEQ approach, modeling the square root of the nonlinear part of the energy, does not belong to the finite element space used for the spatial discretization.} 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 method.