We propose a new class of additive and multiplicative multilevel methods for isogeometric collocation methods based on generalized B-splines (GB-splines). To our knowledge, this represents the first systematic development of multilevel solvers for isogeometric collocation schemes based on GB-splines. We provide numerical evidence supporting the optimality of our proposed preconditioners, accelerated by GMRES, with respect to the number of levels. Furthermore, we conduct a comprehensive numerical study to assess the robustness of the proposed preconditioners with respect to isogeometric discretization parameters. Particular attention is devoted to investigating the effectiveness of the multilevel solvers in the presence of domain deformations.

We study overlapping additive Schwarz (OAS) preconditioners for the solution of elliptic boundary value problems discretized using isogeometric collocation methods based on generalized B-splines (GB-splines). Through a series of numerical experiments, we demonstrate the scalability of the proposed preconditioning strategy with respect to the number of subdomains, as well as its robustness with respect to the parameters of the isogeometric discretization.

We construct an overlapping additive Schwarz preconditioner for the biharmonic Dirichlet problems discretized by isogeometric analysis based on generalized B-splines (GB-splines) and analyze its optimal convergence rate bound that is cubic in the ratio between subdomains and overlap sizes. Our analysis is validated through a set of numerical experiments that illustrate good behavior of the proposed preconditioner with respect to the model parameters.

We present the construction of additive multilevel preconditioners, also known as BPX preconditioners, for the solution of the linear system arising in isogeometric adaptive schemes with (truncated) hierarchical B-splines. We show that the locality of hierarchical spline functions, naturally defined on a multilevel structure, can be suitably exploited to design and analyze efficient multilevel decompositions. By obtaining smaller subspaces with respect to standard tensor-product B-splines, the computational effort on each level is reduced. We prove that, for suitably graded hierarchical meshes, the condition number of the preconditioned system is bounded independently of the number of levels. A selection of numerical examples validates the theoretical results and the performance of the preconditioner.

Overlapping Additive Schwarz (OAS) preconditioners are here constructed for isogeometric collocation discretizations of the system of linear elasticity in both two and three space dimensions. Isogeometric collocation methods are recent variants of isogeometric analysis based on the numerical approximation of the strong form of partial differential equations at appropriate collocation points. Numerical results in two and three dimensions show that two-level OAS preconditioners are scalable in the number of subdomains N, quasi-optimal with respect to the mesh size h and optimal with respect to the spline polynomial degree p. Moreover, two-level OAS preconditioners are more robust than one-level OAS and non-preconditioned GMRES solvers when the material tends to the incompressible limit, as well as in the presence of strong deformation of the NURBS geometry.

The basis of T-splines are the point-based splines (PB splines) that are unstructured meshless splines. In this paper, we study associated PB splines with local knot vectors that are arbitrarily distributed in $$([0,1]\cap {\mathbb {Q}})^d, d=1,2$$ , where $${\mathbb {Q}}$$ is the set of rational numbers. We prove the linear independence of linear PB splines under a mild assumption that their central knots are all distinct. The linearly independent property is one of important prerequisites for isogeometric analysis. Moreover, we illustrate that the same assumption can not be extended to two-dimensional case, by giving a set of linearly dependent bilinear PB splines.

We propose and analyze optimal additive multilevel solvers for isogeometric discretizations of scalar elliptic problems for locally refined T-meshes. Applying the refinement strategy in Morgenstern and Peterseim (2015, Comput. Aided Geom. Design, 34, 50-66) we can guarantee that the obtained T-meshes have a multilevel structure, and that the associated T-splines are analysis-suitable, for which we can define a dual basis and a stable projector. Taking advantage of the multilevel structure, we develop two BPX preconditioners: the first on the basis of local smoothing only for the functions affected by a newly added edge by bisection, and the second smoothing for all the functions affected after adding all the edges of the same level. We prove that both methods have optimal complexity, and present several numerical experiments to confirm our theoretical results, and also to compare the practical performance of the proposed preconditioners.

We construct and analyze an overlapping Schwarz preconditioner forelliptic problems discretized with isogeometricanalysis. The preconditioner is based on partitioning the domainof the problem into overlapping subdomains, solving localisogeometric problems on these subdomains, and solving anadditional coarse isogeometric problem associated with thesubdomain mesh. We develop an $h$-analysis of the preconditioner,showing in particular that the resulting algorithm is scalable andits convergence rate depends linearly on the ratio betweensubdomain and „overlap sizes” for fixed polynomial degree $p$and regularity $k$ of the basis functions. Numerical results in two-and three-dimensional tests show the good convergence properties of thepreconditioner with respect to the isogeometric discretizationparameters $h, p, k$, number of subdomains $N$, overlap size, andalso jumps in the coefficients of the ellipticoperator.

Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with n≥2, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r modulo n. Similar expressions are also dealt with for the case of Lucas numbers. Symmetric and anti-symmetric properties of Fibonacci and Lucas numbers are used in the proofs.

We present optimal additive and multiplicative multilevel methods, such as BPX preconditioner and multigrid V-cycle, for the solution of linear systems arising from isogeometric collocation discretizations of second order elliptic problems. These resulting preconditioners, accelerated by GMRES, lead to optimal complexity for the number of levels, and illustrate their good performance with respect to the isogeometric discretization parameters such as the spline polynomial degree and regularity of the isogeometric basis functions, as well as with respect to domain deformations.

We present the construction of additive multilevel preconditioners, also known as BPX preconditioners, for 4 the solution of the linear system arising in isogeometric adaptive schemes with (truncated) hierarchical B-splines. We show 5 that the locality of hierarchical spline functions, naturally defined on a multilevel structure, can be suitably exploited to 6 design and analyze efficient multilevel decompositions. By obtaining smaller subspaces with respect to standard tensor7 product B-splines, the computational effort on each level is reduced. We prove that, for suitably graded hierarchical 8 meshes, the condition number of the preconditioned system is bounded independently of the number of levels. A selection 9 of numerical examples validates the theoretical results and the performance of the preconditioner. 10

The paper presents some properties of Generalized T-splines (GT-splines), which are crucial to their actual application. In particular, we construct a dual basis for a noteworthy class of GT-splines, which allows to show that, under suitable conditions, they form a partition of unity. Moreover, we study the approximation properties of the GT-spline space by constructing a class of quasi-interpolants which belong to it and are defined by giving a dual basis.

In this paper we consider spaces of bivariate splines of bi-degree (m, n) with maximal order of smoothness over domains associated to a two-dimensional grid. We define admissible classes of domains for which suitable combinatorial technique allows us to obtain the dimension of such spline spaces and the number of tensor-product B-splines acting effectively on these domains. Following the strategy introduced recently by Giannelli and Juettler, these results enable us to prove that under certain assumptions about the configuration of a hierarchical T-mesh the hierarchical B-splines form a basis of bivariate splines of bi-degree (m, n) with maximal order of smoothness over this hierarchical T-mesh. In addition, we derive a sufficient condition about the configuration of a hierarchical T-mesh that ensures a weighted partition of unity property for hierarchical B-splines with only positive weights.

The paper considers the extension of the T-spline approach to the Generalized B-splines (GB-splines), a relevant class of non-polynomial splines. The Generalized T-splines (GT-splines) are based both on the framework of classical polynomial T-splines and on the Trigonometric GT-splines (TGT-splines), a particular case of GT-splines. Our study of GT-splines introduces a class of T-meshes (named VMCR T-meshes) for which both the corresponding GT-splines and the corresponding polynomial T-splines are linearly independent. A practical characterization can be given for a sub-class of VMCR T-meshes, which we refer to as weakly dual-compatible T-meshes, which properly includes the class of dual-compatible (equivalently, analysis-suitable) T-meshes for an arbitrary (polynomial) order.

The system of linear elasticity for compressible composite materials is discretized with Isogeometric Analysis and the resulting discrete system is solved iteratively by PCG with an Overlapping Schwarz preconditioner, requiring the solution of local elasticity problems on overlapping subdomains and the solution of a coarse elasticity problem associated with the subdomain coarse mesh. The proposed preconditioner has an optimal convergence rate bound that is scalable in the number of subdomains and is linear in the ratio between subdomain and overlap sizes. This study also shows the preconditioner robustness with respect to the presence of discontinuous elastic coefficients in composite materials and to domain deformation.

Isogeometric collocation methods are very recent and promising numerical schemes that preserve the advantages of isogeometric analysis but often exhibit better performances than their Galerkin counterparts. In the present paper, an additive overlapping Schwarz method for isogeometric collocation discretizations is introduced and studied. The resulting preconditioner, accelerated by GMRES, is shown to be scalable with respect to the number of subdomains and very robust with respect to the isogeometric discretization parameters such as the mesh size and polynomial degree, as well as with respect to the presence of discontinuous elliptic coefficients and domain deformations.

In this paper we propose a strategy for generating consistent hierarchical T-meshes which allow local refinement and offer a way to obtain spline basis functions with highest order smoothness incrementally. We describe the required ordering of line-segments during refinement and the construction of spline basis functions. We give our strategy for generating consistent hierarchical T-meshes over any shape of a two-dimensional domain.

In this paper, we analyze the diversity order of the optimal transmit antenna selection (TAS) for spatial multiplexing (SM) systems with maximum-likelihood (ML) detection. By deriving the upper and lower bounds on the diversity order, we prove that the optimal TAS of selecting L antennas among M transmit antennas in the SM system with M transmit antennas and N receive antennas obtains a diversity order of N(M - L + 1), which coincides with the well-known results of the respective diversity orders MN and N for the special cases when L = 1 and L = M. Monte Carlo simulations confirm our theoretical result.

We prove that the dimension of trivariate tensor-product spline space of tri-degree (m,m,m) with maximal order of smoothness over a three-dimensional domain coincides with the number of tensor-product B-spline basis functions acting effectively on the domain considered. A domain is required to belong to a certain class. This enables us to show that, for a certain assumption about the configuration of a hierarchical mesh, hierarchical B-splines span the spline space.This paper presents an extension to three-dimensional hierarchical meshes of results proposed recently by Giannelli and Jüttler for two-dimensional hierarchical meshes.

The paper’s main aim consists of extending the T-spline approach to trigonometric generalized B-splines, a particularly relevant case of non-polynomial splines. Such goal can be achieved by a careful revision of some results concerning the basic properties of the trigonometric generalized B-splines and by a formalization of the concept of T-splines in the trigonometric setting. Moreover, fundamental for the use of this new tool is the study of the noteworthy case with constant frequencies and of the linear independence of the corresponding blending functions, which can be proved to be strongly linked to the linear independence of the polynomial blending functions associated to the same T-mesh.