# Isogeometric Schwarz Preconditioners with Generalized B-Splines for the Biharmonic Problem +
## 2 GB-Splines: Definition and Basic Properties
### Univariate GB-Splines
_Smoothness:_ _each_ \(\bar{N}^{U,V}_{j,p,\Sigma}\) _has a continuous derivative of order_ \(p-m_{i}\) _at the knot_ \(\xi_{i}\)_, where_ \(m_{i}\) _is the multiplicity of_ \(\xi_{i}\) _in the knot vector_ \(\{\xi_{j},\ldots,\xi_{j+p+1}\}\)_._
6. _Differentiation:_ _The derivative of GB-splines is represented in terms of two consecutive GB-splines of a lower degree as_ \[\frac{d}{d\zeta}\bar{N}^{U,V}_{j,p,\Sigma}(\zeta)=\delta^{U,V}_{j,p-1,\Sigma} \bar{N}^{U,V}_{j,p-1,\Sigma}(\zeta)-\delta^{U,V}_{j+1,p-1,\Sigma}\bar{N}^{U,V }_{j+1,p-1,\Sigma}(\zeta).\]  
**Remark 1**. : _A knot insertion procedure is available for GB-splines (see, e.g., [42; 44; 45] for more details) and thus, in an analogous way, the h-refinement obtained by knot insertion in NURRSB-based IGA can be extended to IGA GB-splines. In the present paper, our theoretical work is only an h-analysis and does not cover the dependence of convergence rate on \(p\)._  
Figure 1: On the knot vector \(\Sigma=\{0,0,0,1/5,2/5,3/5,4/5,1,1,1\}\), a comparison between (**a**) quadratic polynomial B-splines and (**b**) exponential GB-splines of degree 2, locally spanned by the functions \(\{1,U(\zeta),V(\zeta)\}\) where \(U(\zeta)=\exp(\alpha_{i}\zeta)\) and \(V(\zeta)=\exp(-\alpha_{i}\zeta)\) with \(\alpha_{i}=10\). It was demonstrated in [38] that GB-splines are regarded as a special instance of normalized Tchebycheff B-splines, ensuring that GB-splines possess all the properties as detailed in [46], Section 9. Of these, the most significant property for our \(h\)-analysis can be stated as follows:  
**Theorem 1**. : _GB-splines are normalized Tchebycheffian B-splines._  
Proof. : The proof can be obtained in a similar way to ([38], Theorem 11). For the sake of completeness, we state some important lemmas from [46], without providing their proofs, which will play a fundamental role in demonstrating the approximation and stability of our proposed preconditioners (see Lemmas 1 and 2 below). We will make use of the dual functionals \(\lambda_{i,p,\Sigma}^{U,V}\) (defined in ([46], Theorem 9.26)) that satisfy  
\[\lambda_{i,p,\Sigma}^{U,V}(\widehat{N}_{j,p,\Sigma}^{U,V})=\delta_{ij},\quad 1 \leq i,j\leq n, \tag{12}\]  
where \(\delta_{ij}\) is the Kronecker delta. Furthermore, a projector on \(\widehat{\mathcal{S}}_{h}\) is introduced in [46], Theorem 9.37, denoting with  
\[\widehat{\Pi}_{\widehat{\mathcal{S}}_{h}}v:=\sum_{j=1}^{n}(\lambda_{j,p, \Sigma}^{U,V}\,v)\widehat{N}_{j,p,\Sigma^{\prime}}^{U,V},\qquad\forall v\in L ^{2}(0,1). \tag{13}\]  
Now, we state a useful estimate of the functional \(\lambda_{i,p,\Sigma}^{U,V}\),along with the stability and approximation properties of the quasi-interpolant \(\widehat{\Pi}_{\widehat{\mathcal{S}}_{h}}\) (see ([46], Theorems 9.26 and 9.37) for their complete proofs). **Lemma 1**. : _If \(f\in L^{q}(\xi_{i},\xi_{i+p+1})\), with \(1\leq q\leq+\infty\), then_  
\[|\lambda_{i,p,\Sigma}^{U,V}(f)|\lesssim|\xi_{i+p+1}-\xi_{i}|^{-1/q}\|f\|_{L^{q} (\xi_{i}\xi_{i+p+1})}. \tag{14}\]  
**Lemma 2**. : _If the mesh on the parametric space \(\widehat{I}\) satisfies the quasi-uniformity condition (defined later), then we have_  
\[\|\hat{\sigma}-\widehat{\Pi}_{\widehat{\mathcal{S}}_{h}}\hat{\sigma}\|_{L^{2} (\widehat{I})}+h|\hat{\sigma}-\widehat{\Pi}_{\widehat{\mathcal{S}}_{h}}\hat{ \sigma}|_{H^{1}(\widehat{I})}+h^{2}|\widehat{\Pi}_{\widehat{\mathcal{S}}_{h}} \hat{\sigma}|_{H^{2}(\widehat{I})}\lesssim h^{2}|\hat{\sigma}|_{H^{2}( \widehat{I})},\;\forall\theta\in\widehat{\mathcal{S}}_{h}, \tag{15}\]  
_where \(h\) denotes the global mesh size of the mesh on the parametric space \(\widehat{I}\)._