# Isogeometric Schwarz Preconditioners with Generalized B-Splines for the Biharmonic Problem +
## 5 Numerical Results
### Two-Dimensional Tests: OAS(2) Scalability in \(N\) and Optimality in \(H/\gamma\)
\begin{table}
\begin{tabular}{r r r r r r r r r r r r r} \hline \hline  & & \multicolumn{6}{c}{OAS prec. with \(p=2,k=1,r=0\), 2D deformed domain} \\ \hline  & & \multicolumn{2}{c}{\(1/h=8\)} & \multicolumn{2}{c}{\(1/h=16\)} & \multicolumn{2}{c}{\(1/h=32\)} & \multicolumn{2}{c}{\(1/h=64\)} & \multicolumn{2}{c}{\(1/h=128\)} & \multicolumn{2}{c}{\(1/h=256\)} \\  & \multicolumn{2}{c}{\(N\)} & \multicolumn{2}{c}{\(\kappa_{2}\)} & \multicolumn{2}{c}{it.} & \multicolumn{2}{c}{\(\kappa_{2}\)} & \multicolumn{2}{c}{it.} & \multicolumn{2}{c}{\(\kappa_{2}\)} & \multicolumn{2}{c}{it.} & \multicolumn{2}{c}{\(\kappa_{2}\)} & \multicolumn{2}{c}{it.} & \multicolumn{2}{c}{\(\kappa_{2}\)} & \multicolumn{2}{c}{it.} \\ \hline non-prec. & \(1.09\times 10^{2}\) & 19 & \(2.02\times 10^{3}\) & 105 & \(3.53\times 10^{4}\) & 460 & \(5.96\times 10^{5}\) & 1950 & \(9.86\times 10^{6}\) & 8175 & \(1.61\times 10^{8}\) & 33,502 \\ \hline \multirow{4}{*}{2-level OAS} & \(2\times 2\) & 8.21 & 11 & \(3.85\times 10^{1}\) & 18 & \(2.56\times 10^{2}\) & 33 & \(1.97\times 10^{3}\) & 62 & \(1.57\times 10^{4}\) & 125 & \(1.26\times 10^{5}\) & 257 \\  & \(4\times 4\) & & & \(2.28\times 10^{1}\) & 24 & \(1.56\times 10^{2}\) & 50 & \(1.22\times 10^{3}\) & 123 & \(9.84\times 10^{3}\) & 315 & \(7.96\times 10^{4}\) & 723 \\  & \(8\times 8\) & & & & & \(4.05\times 10^{1}\) & 33 & \(2.97\times 10^{2}\) & 79 & \(2.41\times 10^{3}\) & 228 & \(1.97\times 10^{4}\) & 645 \\  & \(16\times 16\) & & & & & & \(5.49\times 10^{1}\) & 41 & \(4.07\times 10^{2}\) & 118 & \(3.34\times 10^{3}\) & 330 \\  & \(32\times 32\) & & & & & & & \(6.75\times 10^{1}\) & 48 & \(5.02\times 10^{2}\) & 133 \\  & \(64\times 64\) & & & & & & & & & \(7.67\times 10^{1}\) & 53 \\ \hline \multicolumn{11}{c}{OAS prec. with \(p=3,k=2,r=0\), 2D deformed domain} \\ \hline non-prec. & \(5.17\times 10^{1}\) & 22 & \(8.20\times 10^{2}\) & 78 & \(1.33\times 10^{4}\) & 286 & \(2.16\times 10^{5}\) & 1159 & \(3.49\times 10^{6}\) & 4854 & \(5.61\times 10^{7}\) & 19,713 \\ \hline \multirow{4}{*}{2-level OAS} & \(2\times 2\) & 7.59 & 13 & \(3.06\times 10^{1}\) & 18 & \(1.85\times 10^{2}\) & 33 & \(1.36\times 10^{3}\) & 65 & \(1.07\times 10^{4}\) & 138 & \(8.54\times 10^{4}\) & 313 \\  & \(4\times 4\) & & & \(1.83\times 10^{1}\) & 23 & \(9.86\times 10^{1}\) & 42 & \(6.97\times 10^{2}\) & 95 & \(5.41\times 10^{3}\) & 255 & \(4.30\times 10^{4}\) & 707 \\  & \(8\times 8\) & & & & & \(4.11\times 10^{1}\) & 32 & \(2.49\times 10^{2}\) & 69 & \(1.95\times 10^{3}\) & 177 & \(1.54\times 10^{4}\) & 509 \\  & \(16\times 16\) & & & & & & & \(7.91\times 10^{1}\) & 41 & \(3.66\times 10^{2}\) & 80 & \(4.08\times 10^{3}\) & 258 \\  & \(32\times 32\) & & & & & & & & \(7.73\times 10^{1}\) & 42 & \(5.15\times 10^{2}\) & 91 \\  & \(64\times 64\) & & & & & & & & & \(8.50\times 10^{1}\) & 43 \\ \hline \multicolumn{11}{c}{OAS prec. with \(p=4,k=3,r=0\), 2D reference domain (unit square)} \\ \hline non-prec. & \(2.77\times 10^{1}\) & 10 & \(3.81\times 10^{2}\) & 27 & \(5.88\times 10^{3}\) & 66 & \(9.32\times 10^{4}\) & 211 & \(1.49\times 10^{6}\) & 772 & \(2.38\times 10^{7}\) & 6202 \\ \hline \multirow{4}{*}{2-level OAS} & \(2\times 2\) & 4.13 & 8 & 6.76 & 11 & \(1.70\times 10^{1}\) & 14 & \(8.19\times 10^{1}\) & 21 & \(5.52\times 10^{2}\) & 36 & \(4.21\times 10^{3}\) & 116 \\  & \(4\times 4\) & & & 5.30 & 16 & \(1.05\times 10^{1}\) & 18 & \(3.62\times 10^{1}\) & 28 & \(2.02\times 10^{2}\) & 49 & \(1.45\times 10^{3}\) & 123 \\  & \(8\times 8\) & & & & & 6.25 & 16 & \(1.73\times 10^{1}\) & 24 & \(7.60\times 10^{1}\) & 44 & \(4.71\times 10^{2}\) & 97 \\  & \(16\times 16\) & & & & & & & 6.56 & 16 & \(2.27\times 10^{1}\) & 26 & \(1.18\times 10^{2}\) & 55 \\  & \(32\times 32\) & & & & & & & & 6.38 & 16 & \(2.40\times 10^{1}\) & 27 \\  & \(64\times 64\) & & & & & & & & & 6.26 & 18 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: OAS preconditioner in 2D deformed and reference domains: condition number \(\kappa_{2}(T_{OAS})\) and iteration counts it.