Discontinuous finite volume element method for a coupled Navier-Stokes-Cahn-Hilliard phase field model

In this paper, we propose a discontinuous finite volume element method to solve a phase field model for two immiscible incompressible fluids. In this finite volume element scheme, discontinuous linear finite element basis functions are used to approximate the velocity, phase function, and chemical potential while piecewise constants are used to approximate the pressure. This numerical method is efficient, optimally convergent, conserving the mass, convenient to implement, flexible for mesh refinement, and easy to handle complex geometries with different types of boundary conditions. We rigorously prove the mass conservation property and the discrete energy dissipation for the proposed fully discrete discontinuous finite volume element scheme. Using numerical tests, we verify the accuracy, confirm the mass conservation and the energy law, test the influence of surface tension and small density variations, and simulate the driven cavity, the Rayleigh-Taylor instability.