# Isogeometric Schwarz Preconditioners with Generalized B-Splines for the Biharmonic Problem +
## 2 GB-Splines: Definition and Basic Properties
### Univariate GB-Splines  
In order to follow the IGA paradigm, we consider \(n,p\in\mathbb{N}\) with \(p\geq 2\) and \(\Sigma=\{\xi_{1}\leq\xi_{2}\leq\ldots\leq\xi_{n+p}\leq\xi_{n+p+1}\}\) be a knot vector in the one-dimensional closed interval \([0,1]\) that is connected to the partition \(\Delta\) defined in (2) as follows:  
\[\Sigma=\{\xi_{1},\xi_{2},\ldots,\xi_{n+p+1}\}=\big{(}\overbrace{\eta_{0}, \ldots,\eta_{0}}^{\mu_{0}},\ldots,\overbrace{\eta_{\ell+1},\ldots,\eta_{\ell+ 1}}^{H_{\ell+1}}\big{)} \tag{7}\]  
for some positive integers \(\mu_{0},\cdots,\mu_{\ell+1}\) with \(\sum_{i=0}^{\ell+1}\mu_{i}=n+p+1\). Moreover, in what follows, \(\Sigma\) is assumed to be a \(p\)-open knot vector that interpolates at the ends of the one-dimensional parametric space \(\tilde{I}:=(0,1)\), that is,  
\[0=\xi_{1}=\cdots=\xi_{p+1}<\cdots<\xi_{n+1}=\cdots=\xi_{n+p+1}=1.\]  
Now, we are in a position to introduce GB-splines of degree \(p\) that are associated with a knot vector \(\Sigma\) and a generalized polynomial space \(\mathbb{G}\mathbb{P}_{p}^{U,V}(\Delta)\). Both of these components are connecting to a partition \(\Delta\) given in (2) (see [6; 7; 38; 42] and references therein). Definition 2: Given a partition \(\Delta\), let \(\mathbb{G}\mathbb{P}_{p}^{U,V}(\Delta)\) be a generalized polynomial space of degree \(p\geq 2\), and let \(\Sigma\) be a knot vector that is connected to \(\Delta\) as in (7). For any \(\xi_{j}<\xi_{j+1}\), let \(u_{j},v_{j}\) be the unique functions in \(\langle D^{p-1}U,D^{p-1}V\rangle\) on \(\{\xi_{j},\xi_{j+1}\}\) satisfying  
\[u_{j}(\xi_{j})=1,\quad u_{j}(\xi_{j+1})=0,\qquad v_{j}(\xi_{j})=0,\quad v_{j} (\xi_{j+1})=1.\]  
We set  
\[\widehat{N}_{i,1,\Sigma}^{U,V}(\zeta):= \begin{cases}v_{i}(\zeta),&\text{if } \xi_{i}\leq\zeta<\xi_{i+1},\\ u_{i+1}(\zeta),&\text{if }\xi_{i+1}\leq\zeta<\xi_{i+2},,\\ 0,&\text{otherwise},\end{cases}  \tag{8}\]  
and the \(j\)-th GB-spline \(\widehat{N}_{i,p,\Sigma}^{U,V}(\zeta)\) of degree \(p\) over \(\Sigma\) is defined recursively using the formula  
\[\widehat{N}_{j,p,\Sigma}^{U,V}(\zeta):=\int_{\xi_{j}}^{\zeta}\delta_{j,p-1, \Sigma}^{U,V}\widehat{N}_{j,p-1,\Sigma}^{U,V}(r)\;dr-\int_{\xi_{j+1}}^{\zeta} \delta_{j+1,p-1,\Sigma}^{U,V}\widehat{N}_{j+1,p-1,\Sigma}^{U,V}(r)\;dr, \tag{9}\]  
where  
\[\delta_  
It is noteworthy that the recursive formula is referred to as the integral form of the Cox-de Boor algorithm, which is utilized for constructing polynomial B-splines of degree \(p\) (see [43] for a complete description). The GB-spline space is defined as the space spanned by GB-splines, namely,  
\[\bar{\mathcal{S}}_{h}=\text{span}\{\bar{N}^{U,V}_{j,p,\Sigma}(\zeta):\;1\leq j \leq n\}. \tag{11}\]  
According to the definitions of \(U\) and \(V\) in (3), GB-splines \(\bar{N}^{U,V}_{j,p,\Sigma}(\zeta)\) with section spaces as given in (4)-(6) are commonly known as classical (polynomial) B-splines, exponential GB-splines and trigonometric GB-splines, respectively (see Figure 1 for some examples). The exponential GB-splines and trigonometric ones possess fundamental properties similar to those of the classical polynomial B-splines, see, e.g., [6; 7; 42] for their proofs. **Property 1**. : _The GB-splines \(\{\bar{N}^{U,V}_{j,p,\Sigma}(\zeta):\;j=1,\ldots,n\}\) of degree \(p\geq 2\) over the knot vector \(\Sigma\) are characterized by the following properties:_  
1. _Positivity:___\(\bar{N}^{U,V}_{j,p,\Sigma}(\zeta)>0,\quad\zeta\in(\bar{\zeta}_{j},\bar{\zeta} _{j+p+1}).\)__
2. _Compact support:_ _if_ \(\zeta\
otin[\zeta_{j},\zeta_{j+p+1}]\)_,_ \(\bar{N}^{U,V}_{j,p,\Sigma}(\zeta)=0.\)__
3. _Local partition of unity:___\(\sum_{j=m-p}^{m}\bar{N}^{U,V}_{j,p,\Sigma}(\zeta)=1,\quad\zeta\in[\zeta_{m}, \zeta_{m+1}).\)__
4. _Local linear independence:___\(\{\bar{N}^{U,V}_{j,p,\Sigma}(\zeta):\;j=m-p,\cdots,m\}\) _are linearly independent on_ \([\zeta_{m},\zeta_{m+1}].\)__
5.