# Isogeometric Schwarz Preconditioners with Generalized B-Splines for the Biharmonic Problem +
## 2 GB-Splines: Definition and Basic Properties
### Multivariate GB-Splines  
Multidimensional GB-splines can be easily constructed using tensor products. For the sake of simplicity, we restrict our discussion to the bivariate case, noting that the construction in higher dimensions follows similarly.  
Let \(\widehat{\Omega}:=(0,1)\times(0,1)\) be the two-dimensional parametric space. With \(d=1,2\), assuming \(n_{d},\ell_{d}\in\mathbb{N}\) and the degree \(p_{d}\geq 2\), we consider the partitions \(\Delta_{d}=\{0=\eta_{d,0}<\dots<\eta_{d,d_{d}+1}=1\}\) and their corresponding \(p_{d}\)-open knot vector \(\Sigma_{d}=\{0=\xi_{d,1},\dots,\xi_{d,n_{d}+p_{d}+1}=1\}\). Additionally, section spaces \(\mathbb{G}\mathbb{P}_{p_{d}}^{U_{d},V_{d}}\) can be appropriately selected according to PDEs problem of interest. We first introduce the set of multi-indices as follows:  
* \(\mathbf{p}=(p_{1},p_{2})\);
* \(\mathbf{\xi}=(\xi_{1},\xi_{2})\);
* \(\mathbf{\Sigma}=\{\Sigma_{1},\Sigma_{2}\}\);
* \(\mathbf{U}=\{U_{1},U_{2}\}\) and \(\mathbf{V}=\{V_{1},V_{2}\}\);
* \(\mathbf{J}=\{\mathbf{j}=(j_{1},j_{2}):\,1\leq j_{d}\leq n_{d},\;d=1,2\}\).  
Then, we introduce the set of bivariate GB-splines  
\[\{\hat{N}_{j,p,\Sigma}^{\mathbf{U},\mathbf{V}}(\xi):=\hat{N}_{j_{1},p_{1},\Sigma_{1}}^{U_ {1},V_{1}}(\zeta_{1})\hat{N}_{j_{2},p_{2},\Sigma_{2}}^{U_{2},V_{2}}(\zeta_{2}),\quad\forall j\in J\}.\]  
The GB-spline space on \(\widehat{\Omega}\) is defined as the space spanned by bivariate GB-splines  
\[\widehat{S}_{h}:=\text{span}\{\hat{N}_{j,p,\Sigma}^{\mathbf{U},\mathbf{V}}(\xi),\ j\in J\}.\]  
In the context of IGA, the physical domain \(\Omega\) can be described by means of a global geometric map \(\mathbf{F}:\widehat{\Omega}\to\Omega\), which is given by  
\[\mathbf{F}(\hat{x})=\sum_{j\in J}\hat{N}_{j,p,\Sigma}^{\mathbf{U},\mathbf{V}}(\xi)\ C_{j}, \quad C_{j}\in\mathbb{R}^{d},\ \hat{x}\in\widehat{\Omega}\]  
where \(\mathbf{C}_{j}\) are the generalized control points. In accordance with the isoparametric concept, we define the space \(\widehat{V}_{h}\) in the parameter domain \(\widehat{\Omega}\) as follows:  
\[\widehat{V}_{h}:=\text{span}\{\hat{N}_{j,p,\Sigma}^{\mathbf{U},\mathbf{V}}(\xi),\ j=(j_{1},j_{2})\in J\quad\text{with}\ \ 3\leq j_{d}\leq n_{d}-2,\ d=1,2\}, \tag{16}\]  
and the space \(V_{h}\) in the physical domain \(\Omega\)  
\[V_{h}:=\text{span}\{N_{j,p,\Sigma}^{\mathbf{U},\mathbf{V}}(\xi):=\hat{N}_{j,p,\Sigma} ^{\mathbf{U},\mathbf{V}}(\xi)\circ\mathbf{F}^{-1},\ j=(j_{1},j_{2})\in J\quad\text{with }\ 3\leq j_{d}\leq n_{d}-2,\ d=1,2\}. \tag{17}\]  
where the homogeneous Dirichlet boundary condition that we consider leads to the elimination of the first and last two functions in each coordinate. For simplicity in the presentation, we consider only the homogeneous Dirichlet boundary conditions throughout this paper. However, it is worth noting that a similar approach can be extended to accommodate different types of boundary conditions (see Section 6 for a explanation about handling various boundary conditions).  
Observing that in multi-indices knot vector \(\mathbf{\Sigma}\) defined above, two knot vectors \(\Sigma_{1}=\{0=\xi_{1,1},\ldots,\xi_{1,n_{1}+p_{1}+1}=1\}\) and \(\Sigma_{2}=\{0=\xi_{2,1},\ldots,\xi_{2,n_{2}+p_{2}+1}=1\}\) result in a mesh of rectangular elements in the parametric space, those mesh elements are mapped via the geometric map \(\mathbf{F}\) to elements in the physical space. The physical mesh on \(\Omega\) is therefore  
\[\mathcal{T}_{h}=\{\mathbf{F}((\xi_{1,i},\xi_{1,i+1})\times(\xi_{2,j},\xi_{2,j+ 1}),\ \text{with}\ i=1,\ldots,n_{1}+p_{1},j=1,\ldots,n_{2}+p_{2}\},\]  
where the empty elements are excluded.  
The discrete formulation of the model problem (1) becomes then:  
Find \(u\in V_{h}\) such that  
\[a(u,v)=\int_{\Omega}fv\ dx,\qquad\forall v\in V_{h} \tag{18}\]  
with bilinear form \(a(u,v):=\int_{\Omega}\Delta u\Delta v\ dx\).