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

TL;DR

This paper develops and analyzes a fully discrete, energy-stable numerical scheme for the coupled Cahn-Hilliard-Navier-Stokes equations, achieving optimal error estimates and validating them through numerical experiments.

Contribution

It introduces a linear, decoupled, unconditionally energy-stable scheme using SAVs and finite element spaces, with optimal error analysis without relying on quasi-projection techniques.

Findings

The scheme is unconditionally energy stable.

Optimal L2 error estimates are derived for key variables.

Numerical experiments confirm theoretical convergence rates.

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 $r=1$.