# Isogeometric Schwarz Preconditioners with Generalized B-Splines for the Biharmonic Problem +
## 6 Conclusions  
In this paper, we study overlapping additive Schwarz (OAS) preconditioners for the biharmonic problem, which fits very naturally in the context of IGA due to the \(C^{1}\) continuity requirement associated to a conforming discrete space. More specifically, we consider the problem of solving the linear systems that arise from isogeometric discretization based on GB-splines of the biharmonic Dirichlet problem. We have developed a theoretical analysis on condition number bounds for the preconditioned system in terms of the involved mesh sizes and overlap parameter. A set of numerical tests have confirmed the theoretical predictions and investigated the performance of our preconditioners with respect to various discretization parameters. In the current paper, we have focused only on the biharmonic problem with homogeneous Dirichlet boundary conditions on the reference patch. We believe that our preconditioning techniques can be applied to the biharmonic problems either with mixed \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{\(k=p-1\), 2D Deformed Domain} & \multicolumn{4}{c}{\(k=p-1\), Unit Square} \\ \cline{2-10} \(p\) & \multicolumn{2}{c}{**Non-Prec.**} & \multicolumn{2}{c}{**2-Level OAS**} & \multicolumn{2}{c}{**Non-Prec.**} & \multicolumn{2}{c}{**2-Level OAS**} \\  & & \multicolumn{2}{c}{\(r=0\)} & \(r=2\) & \(r=4\) & \(r=p\) & & \(r=0\) & \(r=2\) & \(r=4\) & \(r=p\) \\ \hline
2 & \(5.96\times 10^{5}\) & \(1.97\times 10^{3}\) & \(2.43\times 10^{2}\) & \(7.70\times 10^{1}\) & \(2.43\times 10^{2}\) & \(2.07\times 10^{5}\) & \(1.70\times 10^{3}\) & \(2.18\times 10^{2}\) & \(7.14\times 10^{1}\) & \(2.18\times 10^{2}\) \\
3 & \(2.16\times 10^{5}\) & \(1.36\times 10^{3}\) & \(1.34\times 10^{2}\) & \(4.22\times 10^{1}\) & \(6.95\times 10^{1}\) & \(7.08\times 10^{4}\) & \(1.81\times 10^{2}\) & \(2.34\times 10^{1}\) & \(9.83\) & \(1.41\times 10^{1}\) \\
4 & \(2.88\times 10^{5}\) & \(5.03\times 10^{2}\) & \(8.58\times 10^{1}\) & \(3.22\times 10^{1}\) & \(3.22\times 10^{1}\) & \(9.32\times 10^{4}\) & \(8.19\times 10^{1}\) & \(1.91\times 10^{1}\) & \(1.00\times 10^{1}\) & \(1.00\times 10^{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Two-level OAS preconditioner on both the quarter ring domain and the reference unit square: condition number \(\kappa_{2}(T_{OAS})\) as a function of the spline polynomial degree \(p\) for maximal regularity \(k=p-1\) with different levels of overlap from symmetric minimal (\(r=0\)) to symmetric generous (\(r=p\)). Fixed \(1/h=64\), \(N=2\times 2\), \(H/h=32\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Two-Level OAS, 3D Cubic Domain**} \\ \hline  & & \multicolumn{2}{c}{\(r=0\)} & \multicolumn{2}{c}{\(r=1\)} \\  & \(N\) & \(\kappa_{2}=\lambda_{\text{max}}/\lambda_{\text{min}}\) & **it.** & \(\kappa_{2}=\lambda_{\text{max}}/\lambda_{\text{min}}\) & **it.** \\ \hline  & \(2\times 2\times 2\) & \(11.06=8.00/0.72\) & 9 & \(7.66=8.00/1.04\) & 7 \\  & \(3\times 3\times 3\) & \(22.30=8.10/0.36\) & 18 & \(9.43=9.43/1.00\) & 18 \\ \(p=2\) & \(4\times 4\times 4\) & \(25.86=8.09/0.31\) & 22 & \(14.28=11.52/0.81\) & 22 \\ \(k=1\) & \(5\times 5\times 5\) & \(24.54=8.08/0.33\) & 26 & \(17.08=11.25/0.66\) & 23 \\  & \(6\times 6\times 6\) & \(23.24=8.07/0.35\) & 28 & \(17.95=11.45/0.64\) & 25 \\  & \(7\times 7\times 7\) & \(22.52=8.07/0.36\) & 29 & \(17.28=11.38/0.66\) & 26 \\ \hline  & \(2\times 2\times 2\) & \(13.20=8.05/0.61\) & 14 & \(8.86=8.30/0.94\) & 13 \\ \(p=3\) & \(3\times 3\times 3\) & \(14.09=8.01/0.57\) & 18 & \(9.37=8.32/0.89\) & 20 \\ \(k=2\) & \(4\times 4\times 4\) & \(14.91=8.09/0.54\) & 22 & \(10.00=8.76/0.88\) & 22 \\  & \(5\times 5\times 5\) & \(15.85=8.03/0.51\) & 23 & \(10.08=8.77/0.87\) & 23 \\  & \(6\times 6\times 6\) & \(16.84=8.09/0.48\) & 24 & \(10.06=8.79/0.87\) & 23 \\ \hline \hline \end{tabular}
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