Efficient numerical solution of Cahn-Hilliard-Navier-Stokes fluids in 2D

A finite element discretization of the variable density Cahn-Hilliard-Navier-Stokes system is presented. This system is then decoupled with the Cahn-Hilliard part solved via the nonlinear multigrid method proposed by the authors in [J. Comput. Phys., 212 (2006), pp. 288-304] and the Navier-Stokes part via a multigrid preconditioned GMRES scheme. This new solver is examined numerically and is shown to be efficient with respect to mesh refinement and robust with respect to the problem parameters.