# Scalable Block Preconditioners for Linearized Navier-Stokes Equations at High Reynolds Number
## 8 Conclusions  
The algebraic solution of the discretized and linearized Navier Stokes equations, in case of high Reynolds number, is still a challenging numerical linear algebra problem. Our contribution is to give an overview of the main multigrid approaches to precondition the advection-diffusion block and to approximate the Schur complement matrix. We have also proposed some variants that can improve efficiency and scalability of the block preconditioner in the case of low viscosity value. The use of such sophisticated multigrid variants leads to solving the finest Navier-Stokes discretized system with almost one-million unknowns and 17 million nonzeros in a few minutes, even for the smallest (\(\
u=10^{-3}\)) value of the viscosity. We have also shown that the use of low-rank updates may constitute a viable improvement of a given block preconditioner when the number of the eigenvalues of the preconditioned \((1,1)\) block, which are far away from 1 (outliers) is small and roughly independent of \(h\).  
**Author Contributions:** Writing--original draft preparation, F.Z. and L.B.; writing--review and editing, F.Z. and L.B. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research has been funded by the project granted by the Fondazione Cassa di Risparmio di Padova e Rovigo (CARIPARO): _Matrix-Free Preconditioners for Large-Scale Convex Constrained Optimization Problems (PRECOP)_.  
**Acknowledgments:** We are indebted to [PERSON] for providing the initial version of the multigrid code and for fruitful discussions.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**ICP**} & \multicolumn{3}{c}{**BTP**} \\
**Matrix** & **Iter** & **Time** & **tres** & **Iter** & **Time** & **tres** \\ \hline M40 & 71 & 22.55 & 3.24E-4 & 84 & 21.20 & 2.00E-4 \\ M80 & 72 & 48.26 & 6.64E-5 & 86 & 43.73 & 7.56E-4 \\ M160 & 75 & 123.80 & 5.94E-5 & 83 & 108.82 & 1.84E-4 \\ M320 & 78 & 485.84 & 1.32E-5 & 82 & 422.72 & 2.77E-5 \\ \hline \hline \end{tabular}
\end{table}
Table 15: Results using right-preconditioned GMRES with split-GS for \(\
u=0.001\) with ICP (left) and BTP (left) and BTP (right). The coarsest mesh is M20.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Inexact Constraint Preconditioner**} & \multicolumn{3}{c}{**Block Triangular Preconditioner**} \\  & & \multicolumn{3}{c}{**Time**} & \multicolumn{3}{c}{**Time**} & \multicolumn{3}{c}{**Time**} \\
**Matrix** & **Iter** & **prep** & **GMRES** & **Total** & **tres** & **Iter** & **prep** & **GMRES** & **Total** & **tres** \\ \hline M20 & 115 & 0.51 & 4.85 & 5.36 & 6.60E-3 & 93 & 0.55 & 5.23 & 5.78 & 9.80E-3 \\ M40 & 86 & 1.38 & 9.29 & 10.67 & 2.40E-3 & 106 & 1.27 & 8.20 & 9.47 & 3.85E-4 \\ M80 & 92 & 4.15 & 31.20 & 35.35 & 3.32E-4 & 112 & 3.57 & 25.58 & 29.15 & 4.27E-4 \\ M160 & 98 & 11.87 & 127.11 & 138.98 & 4.75E-4 & 113 & 11.92 & 96.51 & 108.43 & 2.00E-4 \\ M320 & 118 & 59.26 & 658.55 & 707.81 & 5.90E-5 & 117 & 48.97 & 488.38 & 537.35 & 5.89E-5 \\ \hline \hline \end{tabular}
\end{table}
Table 14: Results using split-GS and low-rank update for \(\
u=0.001\) with ICP (left) and BTP (right). The coarsest mesh is M10.