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

TL;DR

This paper introduces and compares monolithic and block overlapping Schwarz preconditioners for solving linear systems from incompressible Navier-Stokes equations, emphasizing scalability, robustness, and efficiency in various flow regimes.

Contribution

The paper develops new GDSW* coarse spaces and integrates them into overlapping Schwarz preconditioners, enhancing robustness and flexibility for fluid flow simulations.

Findings

Monolithic preconditioners outperform block preconditioners in robustness.

The GDSW* coarse space improves convergence and scalability.

Numerical tests show effectiveness across different Reynolds and CFL numbers.

Abstract

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 FROSch (Fast and Robust Overlapping Schwarz), which is part of the software library Trilinos. The new GDSW-type coarse space GDSW* 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 components, 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.