Semi-implicit, unconditionally energy stable, stabilized finite element method based on multiscale enrichment for the Cahn-Hilliard-Navier-Stokes phase-field model

In this paper, we establish a novel fully discrete semi-implicit stabilized finite element method for the Cahn-Hilliard-Navier-Stokes phase-field model by using the lowest equal-order (P-1/P-1/P-1/P-1) finite element pair, which consists of the stabilized finite element method based on multiscale enrichment for the spatial discretization and the first order semi-implicit scheme combined with convex splitting approximation for the temporal discretization. We prove that the fully discrete scheme is unconditional energy stable and mass conservative. We also carry out optimal error estimates both in time and space for the phase function, chemical potential and velocity in the appropriate norms. Finally, several numerical experiments are presented to confirm the theoretical results and the efficiency of the proposed scheme.