Monolithic and Block Overlapping Schwarz Preconditioners for the Incompressible Navier--Stokes Equations

Monolithic preconditioners applied to the linear systems arising during the solution of the discretized incompressible Navier--Stokes equations are typically more robust than preconditioners based on incomplete block factorizations. Lower number of iterations and a reduced sensitivity to parameters like velocity and viscosity can significantly outweigh the additional cost for their setup. Different monolithic preconditioning techniques are introduced and compared to a selection of block preconditioners. In particular, two-level additive overlapping Schwarz methods (OSM) are used to set up monolithic preconditioners and to approximate the inverses arising in the block preconditioners. GDSW-type (Generalized Dryja--Smith--Widlund) coarse spaces are used for the second level. These highly scalable, parallel preconditioners have been implemented in the solver framework \texttt{FROSch} (Fast and Robust Overlapping Schwarz), which is part of the software library \texttt{Trilinos}. The new GDSW-type coarse space GDSW\expStar{} is introduced; combining it with other techniques results in a robust algorithm. The block preconditioners PCD (Pressure Convection--Diffusion), SIMPLE (Semi-Implicit Method for Pressure Linked Equations), and LSC (Least-Squares Commutator) are considered to various degrees. The OSM for the monolithic as well as the block approach allows the optimized combination of different coarse spaces for the velocity and pressure component, enabling the use of tailored coarse spaces. The numerical and parallel performance of the different preconditioning methods for finite element discretizations of stationary as well as time-dependent incompressible fluid flow problems is investigated and compared. Their robustness is analyzed for a range of Reynolds and Courant-Friedrichs-Lewy (CFL) numbers with respect to a realistic problem setting.