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 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.

💡 Research Summary

This paper investigates preconditioning strategies for the large saddle‑point linear systems that arise from the discretization of the incompressible Navier‑Stokes equations. Two families of preconditioners are examined: (1) monolithic (or “single‑block”) preconditioners that treat the entire coupled velocity‑pressure matrix as one block, and (2) traditional block‑factorization preconditioners that approximate the Schur complement. The authors employ a two‑level additive overlapping Schwarz method (OSM) as the underlying domain‑decomposition framework for both families. In the first level, local subdomain solves are performed using inexpensive solvers (ILU or algebraic multigrid). In the second level, coarse‑grid corrections are constructed from generalized Dryja‑Smith‑Widlund (GDSW) type coarse spaces. Three variants are considered: the classical GDSW, the enriched RGDSW, and a newly introduced GDSW* coarse space. GDSW* provides a richer set of basis functions for the velocity component while allowing the pressure component to be handled by RGDSW, thereby achieving a tailored coarse representation for each field.

For the block‑factorization approach, three well‑known schemes are implemented: Pressure Convection‑Diffusion (PCD), SIMPLEC (a semi‑implicit pressure‑linked equations variant), and Least‑Squares Commutator (LSC). Each scheme requires an approximation of the inverse of the velocity block (F⁻¹) and, in the case of PCD and LSC, an approximation of the Schur complement S ≈ −B F⁻¹ Bᵀ. These inverses are also approximated by the two‑level OSM, ensuring a fair comparison between the monolithic and block strategies.

The numerical experiments cover a broad spectrum of discretizations (P2‑P1, Q2‑Q1, Q2‑P1‑disc., as well as stabilized P1‑P1 and Q1‑Q1), mesh types (structured and unstructured), and problem geometries (a backward‑facing step and a realistic arterial network). Reynolds numbers range from 200 to 3 200 and Courant–Friedrichs–Lewy (CFL) numbers from 0.5 to 30, thereby testing the robustness of the solvers under both low‑ and high‑inertia regimes and under aggressive time stepping. The linear systems are solved with GMRES, and the overall nonlinear problem is tackled by an inexact Newton–Krylov method.

Key findings include:

• The monolithic preconditioner consistently yields fewer GMRES iterations than the block preconditioners, especially at high Reynolds and high CFL numbers where the block approaches struggle to capture dominant convective effects.
• Although the setup phase of the monolithic preconditioner is about 1.3–1.7 times more expensive than that of the block methods (due to the construction of a larger coarse problem), the reduced iteration count more than compensates, leading to overall runtime reductions of 10–40 % in most test cases.
• The new GDSW* + RGDSW combination, which applies different coarse spaces to velocity and pressure, improves convergence by roughly 20–30 % compared with using a single GDSW coarse space for both fields.
• Parallel scalability is excellent: weak‑scalability tests up to 4096 cores maintain parallel efficiencies between 0.85 and 0.90, and strong‑scalability tests on a fixed problem size show near‑linear speedup up to several thousand cores.
• Among the block preconditioners, PCD and LSC perform adequately at low Reynolds numbers but deteriorate as convection dominates; SIMPLEC, relying on a diagonal approximation of the Schur complement, becomes unstable for large CFL numbers.

The implementation leverages the Trilinos ecosystem: FROSch provides the two‑level overlapping Schwarz framework and the various GDSW‑type coarse spaces; Teko supplies ready‑made block preconditioners; and FEDDLib handles mesh generation, finite‑element assembly, and the interface to Trilinos. Coarse‑grid solves are performed with a direct solver (e.g., MUMPS), and symbolic factorizations are reused across Newton iterations to amortize the setup cost.

In conclusion, the study demonstrates that a carefully designed monolithic preconditioner, equipped with a flexible, field‑specific coarse‑space strategy, outperforms traditional block‑factorization approaches in terms of robustness, iteration count, and overall computational efficiency for incompressible Navier‑Stokes simulations. The authors suggest future work on extending the methodology to non‑Newtonian fluids, fluid‑structure interaction, adaptive coarse‑space enrichment, and GPU‑accelerated local solvers, which could further broaden the applicability of the proposed techniques.