The linear, decoupled and fully discrete finite element methods for simplified two-phase ferrohydrodynamics model

In this paper, we consider numerical approximations of a phase field model for simplified two-phase ferrofluids. This model is a highly nonlinear and coupled multiphysics PDE system with Cahn-Hilliard equations, Navier-Stokes equations, magnetization equation and magnetostatic equation. By combining the artificial compressibility method for the Navier-Stokes equations, the convex splitting method or the stabilize explicit method for Cahn-Hilliard systems, the subtle implicit-explicit treatments and some extra stabilization terms for nonlinear coupling terms, we construct two linear, decoupled and fully discrete finite element methods to solve multiphysics system efficiently. The proposed schemes do not enforce any artificial boundary condition on the pressure. Furthermore, the energy stability and unique solvability are obtained for the proposed schemes. In order to accurately capture the diffuse interface, we apply the adaptive mesh strategy. Finally, a series of numerical experiments verify the theory and illustrate the efficiency and effectiveness of these methods.