A two-grid mixed finite element method of a phase field model for two-phase incompressible flows

This paper develops a fully discrete two-grid mixed finite element scheme of a phase field model for the two immiscible and incompressible flows. In the model, the Navier-Stokes equations and Allen-Cahn equation are coupled by a phase function in which an extra phase induced stress term is added in the Navier-Stokes equations and a fluid induced transport term is added in the Allen-Cahn equation. We use the two-grid method to overcome the computational difficulties arising from the strong nonlinear coupling system. The stability and convergence of the two-grid method are obtained under the proper assumption of time discretization size k, space discretization size h and interface thickness ε. The analysis shows that the two-grid methods can get the same convergence as the classical one-grid methods provided that we choose proper coarse and fine mesh sizes. However, the two-grid methods can save much computational time compared with the one-grid methods. Some numerical experiments are provided to confirm the theoretical analysis.