Optimal error estimates of a SAV–FEM for the Cahn–Hilliard–Navier–Stokes model

We carry out the optimal error estimates of a scalar auxiliary variable (SAV) based the Euler semi-implicit finite element method for the Cahn–Hilliard–Navier–Stokes (CHNS) system in this paper. A SAV is introduced to reformulate the CHNS system into an equivalent system. We apply the Euler semi-implicit method and finite element method for the temporal and spatial discretizations to obtain a linear and unconditional energy stable scheme. The stability of numerical solutions under different norms and the optimal error estimates for our proposed numerical schemes are also presented. Finally, numerical examples are presented to demonstrate the accuracy of the proposed method, and the discrete energy is dissipated in numerical simulations.