Preconditioners for Two-Phase Incompressible Navier-Stokes Flow

We consider iterative methods for solving the linearised Navier-Stokes equations arising from two-phase flow problems and the efficient preconditioning of such systems. We focus on two preconditioners which have proved effective and display mesh-independent convergence for the constant coefficient Navier-Stokes equations. These two preconditioners are known as "pressure convection-diffusion" (PCD) and "least-squares commutator" (LSC) [H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, second ed., Oxford University Press, 2014, Chap. 9]. However, these techniques fail to give comparable performance in their given form when applied to variable coefficient Navier-Stokes systems such as those arising in two-phase flow models. Here we move towards developing generalisations of these preconditioners appropriate for two-phase flow; in particular, this requires a new form for PCD. We omit considerations of boundary conditions to focus on the key features of two-phase flow. Our numerical results demonstrate that the favourable properties of the original preconditioners (without boundary adjustments) are retained. Further, we test our two-phase PCD approach on a dynamic dam-break simulation using the Proteus toolkit.