# Isogeometric Schwarz Preconditioners with Generalized B-Splines for the Biharmonic Problem +
## 3 The Overlapping Schwarz Preconditioners
### Subdomains and Subspace Decomposition  
The subdomains and subspace decomposition are first built for the IGA GB-splines space in one-dimensional parameter space and then are extended to the physical domain of interest through a geometrical mapping \(\mathbf{F}\). Given a partition \(\Delta\) to which the knot vector \(\Sigma=\{\xi_{1}=0,\ldots,\xi_{n+p+1}=1\}\) is connected (as described in the previous section), we can choose a subset \(\{0=\xi_{i_{1}}<\xi_{i_{2}}<\cdots<\xi_{i_{N}}<\xi_{i_{N+1}}=1\}\) consisting of interface knots in the closure of the parametric space \(\overline{\left(\widehat{I}\right)}=[0,1]\). We can obtain a decomposition of \(\overline{\left(\widehat{I}\right)}\) by selecting a suitable subset of interface knots, that is,  
\[\overline{\left(\widehat{I}\right)}=[0,1]=\overline{\left(\bigcup_{k=1,..,N} \widehat{I}_{k}\right)},\ \ \text{where}\ \ \widehat{I}_{k}=(\xi_{i_{k}},\xi_{i_{k+1}}),\]  
into \(N\) sub-intervals \(\widehat{I}_{k}\) whose diameter \(\operatorname{diam}(\widehat{I}_{k}),1\leq k\leq N\), is assumed to have a similar characteristic diameter \(H\). With each interface knot \(\xi_{i_{k}},2\leq k\leq N\), we are able to associate an index \(s_{k}\in\{3,4,\ldots,n-2\}\), which is strictly increasing with respect to \(k\) and satisfies the inequality \(s_{k}<i_{k}<s_{k+}+p+1\). From this choice of \(s_{k}\) and Property 1, it follows that the GB-spline function \(\widehat{N}_{s_{k},p,\Sigma}^{U,V}\) has support that intersects both \(\widehat{I}_{k-1}\) and \(\widehat{I}_{k}\). Moreover, the construction of the basis function \(\widehat{N}_{s_{k},p,\Sigma}^{U,V}\) guarantees that there exists at least one such \(s_{k}\), even if it is not unique. An overlapping decomposition of \(\widehat{I}\) can then be formulated as follows. The overlap index, denoted by \(r\in\mathbb{N}\), represents the number of GB basis functions shared by adjacent subdomains that are defined as  
\[\widehat{V}_{k}:=\operatorname{span}\{\widehat{N}_{j,p,\Sigma}^{U,V}(\xi),\,s _{k}-r\leq j\leq s_{k+1}+r\}\qquad k=1,2,..,N, \tag{19}\]  
with the exception that \(3\leq j\leq s_{2}+r\) for the space \(\widehat{V}_{1}\) and \(s_{N}-r\leq j\leq n-2\) for the space \(\widehat{V}_{N}\). These subspaces constitute an overlapping decomposition of the GB-spline space \(\widehat{V}_{h}\) on one-dimensional parameter domain \(\widehat{I}\). For example, for the cubic degree case with maximum continuity \(C^{2}\), \(r=0\) represents the minimal overlap where only one common GB basis function is shared between adjacent subspaces, while more general \(2r+1\) indicates the number of GB basis functions in common among neighboring local subspaces. Moreover, it is necessary to introduce the overlap parameter (that will be used in the following analysis)  
\[\gamma:=h(2r+2), \tag{20}\]  
which is connected to the width \(\delta\) of the overlapping region in OAS approach for classical FEM. Additionally, the definitions of the extended subdomains \(\widehat{I}_{k}^{\prime}\) and the further extended subdomains \(\widehat{I}_{k}^{\prime\prime}\) are provided as follows:  
\[\widehat{I}_{k}^{\prime}=\bigcup_{\widehat{N}_{j,p,\Sigma}^{U,V}\in\widehat{ V}_{k}}\operatorname{supp}(\widehat{N}_{j,p,\Sigma}^{U,V})=(\xi_{s_{k}-r},\xi_{s_{k+1}+ r+p+1}), \tag{21}\]  
and  
\[\widehat{I}_{k}^{\prime\prime}=\bigcup_{\operatorname{supp}(\widehat{N}_{j, p,\Sigma}^{U,V})\cap\widehat{I}_{k}^{\prime}\
eq\emptyset}\operatorname{supp}( \widehat{N}_{j,p,\Sigma}^{U,V}), \tag{22}\]  
respectively, except for \(\widehat{I}_{1}^{\prime}\), \(\widehat{I}_{N}^{\prime}\), \(\widehat{I}_{1}^{\prime\prime}\) and \(\widehat{I}_{N}^{\prime\prime}\).