# Isogeometric Schwarz Preconditioners with Generalized B-Splines for the Biharmonic Problem +
## 3 The Overlapping Schwarz Preconditioners
### Subdomains and Subspace Decomposition
Therefore, except for the boundary subdomains, the local subspaces and the coarse space are expressed as follows:  
\[\widehat{V}_{k}:=\operatorname{span}\{\widehat{N}_{j,p,\Sigma}^{U,V}(\xi),\,s _{k}-r\leq j\leq s_{k+1}+r\}\qquad k=1,2,..,N, \tag{19}\]3 \[\widehat{V}_{k}:=\operatorname{span}\{\widehat{N}_{j,p,\Sigma}^{U,V}(\xi),\,s _{k}-r\leq j\leq s_{k+1}+r\}\qquad k=1,2,..,N, \tag{19}\]4  
The subdomains in physical space correspond to the images of the subdomains in parameter space:  
\[\widehat{V}_{k}:=\operatorname{span}\{\widehat{N}_{j,p,\Sigma}^{U,V}(\xi),\,s _{k}-r\leq j\leq s_{k+1}+r\}\qquad k=1,2,..,N, \tag{19}\]5  
In order to estimate the condition number, we require the following two assumptions concerning the mesh and subdivision:  
**Assumption 1**. : _The parametric mesh in each extended subdomain \(\widehat{\Omega}_{kl}^{\prime\prime}\) has the property of quasi-uniformity, which implies that there exists a positive real number \(h\) such that \(h\approx\text{diam}\big{(}\widehat{\Omega}_{kl}^{\prime\prime}\big{)}\) for all subdomains \(\widehat{\Omega}_{kl}^{\prime\prime}\)._  
**Assumption 2**. : _The overlap index \(r\) remains bounded, independently of both the mesh size and the polynomial degree._  
It is worth noting that our analysis relies on the local quasi-uniformity assumption regarding the knot vector, which is a conventional assumption in both mathematical isogeometric literature (see, e.g., [17]) and finite element literature (see, e.g., [47]). Assumption 2 allows us to consider the case of practical significance, namely, when small overlap leads to optimal or close-to-optimal condition numbers, see, e.g., [48]. The discrete GB-spline space \(V_{h}\) can be expressed as a sum of coarse and local spaces using the local and coarse embedding operators \(\mathbf{I}_{kl}:V_{kl}\to V_{h}\), \(k=1,..,N\), \(l=1,..,M\) and \(\mathbf{I}_{0}:V_{0}\to V_{h}\). Specifically, the decomposition is given by:  
\[\widehat{V}_{k}:=\operatorname{span}\{\widehat{N}_{j,p,\Sigma}^{U,V}(\xi),\,s _{k}-r\leq j\leq s_{k+1}+r\}\qquad k=1,2,..,N, \tag{19}\]6  
Additionally, we define the local projections \(\widetilde{\mathbf{T}}_{kl}:V_{h}\to V_{kl}\) and the coarse projection \(\widetilde{\mathbf{T}}_{0}:V_{h}\to V_{0}\) as follows:  
\[\widehat{V}_{k}:=\operatorname{span}\{\widehat{N}_{j,p,\Sigma}^{U,V}(\xi),\,s _{k}-r\leq j\leq s_{k+1}+r\}\qquad k=1,2,..,N, \tag{19}\]7  
and \(\mathbf{T}_{kl}=\mathbf{I}_{kl}\widetilde{\mathbf{T}}_{kl}\) and \(\mathbf{T}_{0}=\mathbf{I}_{0}\widetilde{\mathbf{T}}_{0}\). Then, our proposed OAS operator \(\mathbf{T}_{OAS}\) can be viewed as either _the one-level version with only local problems_ or _the two-level version with both coarse and local problems_:  
\[\widehat{V}_{k}:=\operatorname{span}\{\widehat{N}_{j,p,\Sigma}^{U,V}(\xi),\,s _{k}-r\leq j\leq s_{k+1}+r\}\qquad k=1,2,..,N, \tag{19}\]8  
The matrix form of the OAS operator \(\mathbf{T}_{OAS}\) can be written as  
\[\widehat{V}_{k}:=\operatorname{span}\{\widehat{N}_{j,p,\Sigma}^{U,V}(\xi),\,s _{k}-r\leq j\leq s_{k+1}+r\}\qquad k=1,2,..,N, \tag{19}\]9  
where \(\mathcal{A}\) is the stiffness matrix associated with the discrete problem (18) and the respective OAS preconditioners \(\mathbf{B}_{OAS}\) are  
\[ \begin{split}\mathbf{B}_{OAS}&=\mathbf{B}_{OAS(1)} :=\sum_{k=1}^{N}\sum_{l=1}^{M}R_{kl}^{T}\,A_{kl}^{-1}\,R_{kl},\\ \mathbf{B}_{OAS}&=\mathbf{B}_{OAS(2)}& :=R_{0}^{T}\,A_{0}^{-1}\,R_{0}+\sum_{k=1}^{N}\sum_{l=1}^{M}R_{kl}^{T}\,A_{kl} ^{-1}\,R_{kl}.\end{split}  \tag{24}\]  
Here, local restriction matrices \(R_{kl}:V_{h}\to V_{kl}\) are defined as the transpose of the natural embedding matrices \(R_{kl}^{T}:V_{kl}\to V_{h}\), \(A_{kl}\) are the local stiffness matrices restricted to the subspace \(V_{kl}\), the matrix \(R_{0}^{T}\) corresponds to the interpolation from the coarse space \(V_{0}\) to the GB-spline space \(V_{h}\) and \(A_{0}\) is the coarse stiffness matrix associated with the coarse space \(V_{0}\).