Iterative linear Solvers for High-Order Discretizations of Multiphase Flows

We are presenting multigrid-multilevel methods for a high-order multiphase flow solver, comparing different approaches for smothers.

The foundation for this work are sharp interface discretizations for multi-phase flows. In particular, an extended discontinuous Galerkin (XDG, extended DG, also unfitted DG, UDG) method is employed. Here, the fluid interface is embedded within a Cartesian background mesh. The discontinuous finite elements, defined on the background mesh are extended in a fashion so that they are capable of approximating singularities (e.g. jumps and kinks) in the pressure and the velocity field with a high order of accuracy.

Unfortunately, the linear systems arising from such systems are rather difficult to solve, in comparison to single-phase flows, for various reasons. First, due to the elliptic nature of surface-tension driven problems splitting schemes such as the projection method do not work well. Second, the viscosity operator couples all velocity components at the interface (while for constant viscosity, the operator is block-diagonal), therefore all momentum components must be solved simultaneously.

In the presentation, we give a comparison of p- and h-Multigrid approaches, applying different smoothers. This includes rather simple Block-Jacobi smoothers as well as more sophisticated Schwarz and ILU-approaches. The discussion will include both, investigation of the algorithmic performance as well as scalability for high-performance computing.