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In this paper we present an adaptive variational procedure to solve a two-phase field model with variable density, viscosity. The model is a couple system that consists of incompressible Navier-Stokes equations and Allen-Cahn equation. In the scheme the projection method based on pressure is used to deal with the Navier-Stokes equations and stabilization approach is used for the Allen-Cahn equation. By some subtle explicit-implicit treatments, the proposed scheme is energy stable. In particular the adaptive refinement/coarsening is carried out by evaluating the residual error indicators on the error estimates of the Allen-Cahn equation. Some numerical examples are performed to show the accuracy and efficiency.

An adaptive variational procedure for a two-phase model with variable density and viscosity