# Scalable Block Preconditioners for Linearized Navier-Stokes Equations at High Reynolds Number
## 7 Numerical Results
### Handling the Low Viscosity Case. Two-Direction GS/BFBt-c  
The next test is performed using the 2 dir GS smoother and the BFBt-c preconditioner. The Multigrid for the \((1,1)\) block is now \(2W(2,2)\). In Figure 8 (left), we compare the spectrum of BFBt and BFBt-c, while, in Figure 8 (right), we show the spectrum of the \((1,1)\) block and the one of the BFBt-c preconditioned Schur complement. The plot reveals the clear superiority of the BFBt-c preconditioner, which provides a better clustering of the eigenvalues. To better follow the circulant flow with so high Reynolds number, we choose to select as the coarsest mesh M20 instead of M10. As a first consequence, the cost per iteration is expected to increase, mainly for the smallest problems. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Matrix**} & \multicolumn{3}{c}{\(\
u=0.1\)} & \multicolumn{3}{c}{\(\
u=0.01\)} & \multicolumn{3}{c}{\(\
u=0.005\)} \\ \cline{2-10}  & **iter** & **time** & \(\mathtt{tres}\) & **iter** & **time** & \(\mathtt{tres}\) & **iter** & **time** & \(\mathtt{tres}\) \\ \hline M20 & 45 & 1.76 & 2.40E-3 & 59 & 2.29 & 3.33E-3 & 99 & 3.83 & 3.56E-2 \\ M40 & 49 & 3.49 & 3.30E-3 & 65 & 4.53 & 3.90E-3 & 215 & 15.77 & 4.06E-2 \\ M80 & 53 & 8.76 & 2.20E-3 & 69 & 11.46 & 3.50E-3 & 232 & 39.45 & 1.22E-2 \\ M160 & 55 & 38.46 & 1.11E-3 & 71 & 50.04 & 2.10E-3 & 113 & 82.38 & 4.60E-3 \\ M320 & 55 & 162.95 & 4.69E-4 & 73 & 215.95 & 1.10E-3 & 121 & 420.06 & 1.50E-3 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results using [PERSON] and BFBt-c. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Matrix**} & \multicolumn{3}{c}{\(\
u=0.1\)} & \multicolumn{3}{c}{\(\
u=0.01\)} & \multicolumn{3}{c}{\(\
u=0.005\)} \\ \cline{2-10}  & **iter** & **time** & \(\mathtt{tres}\) & **iter** & **time** & \(\mathtt{tres}\) & **iter** & **time** & \(\mathtt{tres}\) \\ \hline M20 & 43 & 1.70 & 2.00E-4 & 56 & 2.21 & 8.85E-4 & 55 & 2.18 & 1.78E-2 \\ M40 & 47 & 3.49 & 1.32E-4 & 60 & 4.47 & 6.75E-4 & 78 & 5.75 & 1.25E-2 \\ M80 & 50 & 8.87 & 3.43E-5 & 63 & 11.20 & 1.66E-4 & 91 & 16.23 & 1.60E-3 \\ M160 & 52 & 39.91 & 3.49E-5 & 62 & 47.67 & 1.32E-5 & 103 & 81.14 & 5.58E-4 \\ M320 & 53 & 171.20 & 9.57E-5 & 65 & 211.05 & 3.71E-5 & 111 & 370.35 & 2.74E-4 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Results using sGS and BFBt-c.  
Table 10 shows the results while using this combination. The preconditioners are scalable for all viscosities, confirming the promising impression that we got from the spectral information. However, the true residual of the final solution is a couple of orders of magnitude larger than the original tolerance of GMRES. To try saving some of the added computational time, we employed strategy _split-GS_, which is supposed to be cheaper than _2 dir-GS_. We show the results in Table 11: when compared with Table 10, these results indeed show a lower computational time, revealing, as expected, that _split-GS_ is a cheap variant of _2 dir-GS_. However, it also introduces some instabilities that prevent the reduction of the true residual in some cases. If we compare Tables 8-11, we can see that, for high viscosities, the results are scalable using any method. In this situation, the most efficient approach in terms of computational time is the one that involves the Jacobi smoother, since the convection is not sufficiently strong to justify the use of a complicated flow-following technique. However, when it comes to low viscosities, the Gauss-Seidel smoother with proper ordering is the only strategy that manages to produce scalable results. We also report the results while using a Block Triangular Preconditioner, as shown in Table 12. The higher number of iterations needed is balanced by the reduced cost per iteration, producing a final computational time that is better than in the previous case. However, the true residual gets even higher in some cases.