# Scalable Block Preconditioners for Linearized Navier-Stokes Equations at High Reynolds Number
## 7 Numerical Results
### Multigrid and BFBt
#### 7.4.3 Two-Direction GS/BFBt  
The next combination uses a Gauss-Seidel smoother applied using two patterns that try to follow the convective flow. We also employed a W cycle, which means that the Multigrid used is now \(2W(1,1)\). Table 6 shows the spectral properties. The results are promising, at least for the Multigrid preconditioner: the smallest eigenvalue \(\alpha_{F}\) is very close to 1 and it does not suffer from refinement of the mesh or reduction in the viscosity. The spectral radius \(\beta_{F}\) is slightly above 1 and again does not deteriorate too much changing the properties of the problem: there is a little growth when the viscosity is reduced and there is a decrease when the mesh is refined (this can be explained noticing that the convective phenomena are better represented when the matrix is larger, therefore improving the efficiency of this Multigrid scheme that is based on these phenomena). Figure 7 shows the spectrum of the preconditioned \((1,1)\) block while using the three previous smoothers. It is clearly visible how only the last choice is able to cluster the eigenvalues around one. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Matrix** & **Viscosity** & \(\alpha_{F}\) & \(\beta_{F}\) & \(\alpha_{S}\) & \(\beta_{S}\) \\ \hline \multirow{3}{*}{M40} & 0.1 & 1.0000 & 1.002 & 0.6966 & 25.44 \\  & 0.01 & 0.9784 & 1.006 & 0.4527 & 21.83 \\  & 0.005 & 0.6591 & 1.422 & 0.2522 & 25.15 \\ \hline \multirow{3}{*}{M80} & 0.1 & 1.0000 & 1.003 & 0.2159 & 26.47 \\  & 0.01 & 0.9984 & 1.015 & 0.1290 & 25.82 \\ \cline{1-1}  & 0.005 & \(<\)0 & 19.35 & 0.0349 & 27.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Spectral properties of the preconditioned matrices while using sGS and BFBt. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Matrix** & \(\
u\) = 0.1 & \(\
u\) = 0.01 & \(\
u\) = 0.005 \\ \hline M40 & 67 & 110 & \(\dagger\) \\ M80 & 97 & 195 & \(\dagger\) \\ M160 & 141 & 336 & \(\dagger\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results using Jacobi and BFBt. The \(\dagger\) symbol means no convergence within 400 iterations. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Matrix** & \(\
u\) = 0.1 & \(\
u\) = 0.01 & \(\
u\) = 0.005 \\ \hline M40 & 64 & 107 & \(\dagger\) \\ M80 & 87 & 183 & \(\dagger\) \\ M160 & 135 & 334 & \(\dagger\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results using sGS and BFBt. The \(\dagger\) symbol means no convergence within 400 iterations. Table 7 displays the results with this combination: they are again not scalable, but the behavior for the lowest viscosity is improved, thanks to the more efficient smoother. In Table 7, we also show the effect of the relaxation parameter \(\omega\), as proposed in [7], and briefly described at the end of Section 4, with the aim of improving the condition number of the preconditioned matrices. Setting \(\omega<1\) provides a slight decrease in the number of iterations, and this is due to the spectra of the preconditioned blocks, which partially overlap. Because of these (not completely satisfactory) preliminary results, the relaxation parameter will not be employed in the next results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Matrix** & **Viscosity** & \(\mathbf{\alpha}_{F}\) & \(\mathbf{\beta}_{F}\) & \(\mathbf{\alpha}_{S}\) & \(\mathbf{\beta}_{S}\) \\ \hline  & 0.1 & 0.9907 & 1.022 & 0.6973 & 25.46 \\ M40 & 0.01 & 0.9912 & 1.133 & 0.4899 & 21.84 \\  & 0.005 & 0.9153 & 1.291 & 0.1962 & 25.20 \\ \hline  & 0.1 & 0.9907 & 1.027 & 0.2159 & 26.55 \\ M80 & 0.01 & 0.9889 & 1.032 & 0.1242 & 25.83 \\  & 0.005 & 0.9619 & 1.130 & 0.0350 & 25.25 \\ \hline  & 0.1 & 0.9908 & 1.036 & 0.0554 & 27.20 \\ M160 & 0.01 & 0.9874 & 1.036 & 0.0315 & 26.68 \\  & 0.005 & 0.9859 & 1.050 & 0.0283 & 26.46 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Spectral properties of the preconditioned matrices using 2 dir GS and BFBt.