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 when using mixed finite element methods. Our target application is simulation within the Proteus toolkit; in particular, we will give results for a dynamic dam-break problem in 2D. We focus on a preconditioner motivated by approximate commutators which has proved effective, displaying mesh-independent convergence for the constant coefficient single-phase Navier–Stokes equations. This approach is known as the “pressure convection–diffusion” (PCD) preconditioner [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]. However, the original technique fails to give comparable performance in its given form when applied to variable coefficient Navier–Stokes systems such as those arising in two-phase flow models. Here we develop a generalisation of this preconditioner appropriate for two-phase flow, requiring a new form for PCD. We omit considerations of boundary conditions to focus on the key features of two-phase flow. Before considering our target application, we present numerical results within the controlled setting of a simplified problem using a variety of different mixed elements. We compare these results with those for a straightforward extension to another commutator-based method known as the “least-squares commutator” (LSC) preconditioner, a technique also discussed in the aforementioned reference. We demonstrate that favourable properties of the original PCD and LSC preconditioners (without boundary adjustments) are retained with the new preconditioners in the two-phase situation.