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 thePᵣ × Pᵣ × 𝐏ᵣ₊₁ × Pᵣfinite element spaces, whereris the degree of the local polynomials, and derive the optimalL²error estimates for the phase-field variable, chemical potential, and pressure in the case ofr ≥ 1 , and for the velocity whenr ≥ 2 , without relying on the quasi-projection operator technique proposed in [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 optimalL²convergence rate for the velocity whenr=1 .