# Isogeometric Schwarz Preconditioners with Generalized B-Splines for the Biharmonic Problem +
## 1 Introduction  
Isogeometric analysis (IGA) is a novel numerical approach for solving partial differential equations (PDEs), introduced by [PERSON] et al. in [1], which forms a link between Computer-Aided Design (CAD) and Finite Element Analysis (FEA) technologies (see also the book [2] and the references therein). CAD systems widely employ Non-Uniform Rational B-Splines (NURBS) to define complex geometries [3]. In the IGA approach which follows an isoparametric paradigm, the NURBS basis functions that represent the CAD geometry are directly utilized in solving PDEs. While NURBS are capable of exactly representing a wider class of geometries such as conic sections, transcendental curves such as cycloid or helices are excluded. In addition, the rational NURBS model has limits in exact differentiation and integration. To avoid these disadvantages, GB-splines describing both conic sections and transcendental geometries were introduced as a tool in the IGA framework and have further investigated in diverse fields (see, e.g., [4, 5, 6, 7, 8, 9, 10]). In the present paper, our model problem is the following biharmonic Dirichlet problem, e.g., arising in the context of thin plate theories of elasticity and the Stokes problem in stream function and vorticity formulation [11, 12]:  
\[\Delta^{2}u=f\ \text{in}\ \Omega,\quad u=\frac{\partial u}{\partial\textbf{n}}=0 \ \text{on}\ \partial\Omega, \tag{1}\]  
where \(\Omega\) is a bounded and connected CAD domain in \(\mathbb{R}^{d}\), **n** is the unit outward normal on the boundary \(\partial\Omega\) (see [13] (Ch. 5.9) for a detailed description). For fourth-order problems, the conforming finite element space necessitates a finite-dimensional subspace of the Sobolev space \(H^{2}(\Omega)\), implying that its basis functions belong to \(C^{1}(\Omega)\). Due to the complexity of \(C^{1}\)-continuous piecewise polynomial elements, \(H^{2}\)-conforming elements are less used in practice for solving biharmonic equations. The nonconforming methodcircumvents the complicated construction of \(C^{1}\) elements but its convergence depends on the elaborate design of the finite element space (see, for example, [13; 14]). In the IGA setting, spline basis functions of degree \(p\) with high smoothness up to \(C^{p-1}\) are useful when dealing with higher-order PDEs, e.g., the biharmonic equation that we consider. Additionally, thanks to [15], the biharmonic problem can be approximated using a mixed finite element method, which is beyond the scope of this paper. In this paper, particular attention is paid to the biharmonic primal formulation. The linear systems arising from the biharmonic Dirichlet problem discretized with IGA GB-splines are highly ill-conditioned as the mesh size \(h\) decreases and polynomial degree \(p\) increases. Thus, the development of effective preconditioners for IGA GB-splines is a challenging issue (see [16], particularly for the scalar elliptic case). To our knowledge, overlapping additive Schwarz (OAS) preconditioners for IGA approximations of PDEs by GB-splines were first presented in [16]. The novelty of this paper is that in IGA methodology using GB-splines as the basis for analysis, overlapping Schwarz preconditioners can be extended to the discretization of the biharmonic equation. Theoretical analysis for our two-level overlapping preconditioner shows that the resulting preconditioned biharmonic operator satisfies a condition number bound that is scalable for an increasing number of subdomains \(N\) and is bounded by a \(\left(1+\frac{H^{3}}{\gamma^{3}}\right)\)-term, where \(H\) and \(\gamma\), both of which will be defined later, are the characteristic subdomain size and an overlap parameter related to the size of the overlapping region between adjacent subdomains, respectively.