The construction of classical hierarchical B-splines can be suitably modified in order to define locally supported basis functions that form a partition of unity. We will show that this property can be obtained by reducing the support of basis functions defined on coarse grids, according to finer levels in the hierarchy of splines. This truncation not only decreases the overlapping of supports related to basis functions arising from different hierarchical levels, but it also improves the numerical properties of the corresponding hierarchical basis -which is denoted as truncated hierarchical B-spline (THB-spline) basis. Several computed examples will illustrate the adaptive approximation behavior obtained by using a refinement algorithm based on THB-splines.

The problem of constructing a normalized hierarchical basis for adaptively refined spline spaces is addressed. Multilevel representations are defined in terms of a hierarchy of basis functions, reflecting different levels of refinement. When the hierarchical model is constructed by considering an underlying sequence of bases Γℓ=0,…,N−1 with properties analogous to classical tensor-product B-splines, we can define a set of locally supported basis functions that form a partition of unity and possess the property of coefficient preservation, i.e., they preserve the coefficients of functions represented with respect to one of the bases Γℓ. Our construction relies on a certain truncation procedure, which eliminates the contributions of functions from finer levels in the hierarchy to coarser level ones. Consequently, the support of the original basis functions defined on coarse grids is possibly reduced according to finer levels in the hierarchy. This truncation mechanism not only decreases the overlapping of basis supports, but it also guarantees strong stability of the construction. In addition to presenting the theory for the general framework, we apply it to hierarchically refined tensor-product spline spaces, under certain reasonable assumptions on the given knot configuration

We consider fast solvers for large linear systems arising from the Galerkin approximation based on B-splines of classical ddimensional\ud elliptic problems, d ≥ 1, in the context of isogeometric analysis. Our ultimate goal is to design iterative algorithms\ud with the following two properties. First, their computational cost is optimal, that is linear with respect to the number of degrees of\ud freedom, i.e. the resulting matrix size. Second, they are totally robust, i.e., their convergence speed is substantially independent of\ud all the relevant parameters: in our case, these are the matrix size (related to the fineness parameter), the spline degree (associated to\ud the approximation order), and the dimensionality d of the problem.We review several methods like PCG, multigrid, multi-iterative\ud algorithms, and we show how their numerical behavior (in terms of convergence speed) can be understood through the notion of\ud spectral distribution, i.e. through a compact symbol which describes the global eigenvalue behavior of the considered stiffness\ud matrices. As a final step, we show how we can design an optimal and totally robust multi-iterative method, by taking into account\ud the analytic features of the symbol. A wide variety of numerical experiments, few open problems and perspectives are presented\ud and critically discussed.\ud ⃝c 2014 Elsevier B.V. All rights reserved

We develop a multi-degree polar spline framework with applications to both geometric modeling and isogeometric analysis. First, multi-degree splines are introduced as piecewise non-uniform rational B-splines (NURBS) of non-uniform or variable polynomial degree, and a simple algorithm for their construction is presented. Then, an extension to two-dimensional polar configurations is provided by means of a tensor-product construction with a collapsed edge. Suitable combinations of these basis functions yield C k smooth polar splines for any k ≥ 0. We show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality. Explicit constructions for k ∈ {0, 1, 2} are presented. Optimal approximation behavior is observed numerically, and examples of free-form design, smooth hole-filling, and high-order partial differential equations demonstrate the applicability of the developed framework.

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We consider the stiffness matrices coming from the Galerkin B-spline isogeometric analysis approximation of classical elliptic problems. By exploiting specific spectral properties compactly described by a symbol, we design efficient multigrid methods for the fast solution of the related linear systems. We prove the optimality of the two-grid methods (in the sense that their convergence rate is independent of the matrix size) for spline degrees up to 3, both in the 1D and 2D case. Despite the theoretical optimality, the convergence rate of the two-grid methods with classical stationary smoothers worsens exponentially when the spline degrees increase. With the aid of the symbol, we provide a theoretical explanation of this exponential worsening and by a proper factorization of the symbol we provide a preconditioned conjugate gradient 'smoother', in the spirit of the multi-iterative strategy, that allows us to obtain a good convergence rate independent both of the matrix size and of the spline degrees. A selected set of numerical experiments confirms the effectiveness of our proposal and the numerical optimality with a uniformly high convergence rate, also for the V-cycle multigrid method and large spline degrees.

We present a general and simple procedure to construct quasi-interpolants in hierarchical spaces, which are composed of a hierarchy of nested spaces. The hierarchical quasi-interpolants are described in terms of the truncated hierarchical basis. Once for each level in the hierarchy a quasi-interpolant is selected in the corresponding space, the hierarchical quasi-interpolants are obtained without any additional manipulation. The main properties of the quasi-interpolants selected at each level are preserved in the hierarchical construction. In particular, hierarchical local projectors are constructed, and the local approximation order of the underling hierarchical space is investigated. The presentation is detailed for the truncated hierarchical B-spline basis, and we discuss its extension to a more general framework.