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A particular class of planar two-patch geometries, called bilinear-like G 2 two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like G 2 two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations.We investigate the space of C 2 -smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the C 2 -smoothness of an isogeometric function and the G 2 -smoothness of its graph surface (cf. [12,20]). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like G 2 two-patch parameterizations. Numerical results obtained by performing L 2 -approximation indicate that already the subspace possesses optimal approximation properties.

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We study the space of C 2 -smooth isogeometric functions on bilinearly parameterized multipatch domains Ω ⊂ R 2 , where the graph of each isogeometric function is a multi-patch spline surface of bidegree (d, d), d ∈ {5, 6}. The space is fully characterized by the equivalence of the C 2 -smoothness of an isogeometric function and the G 2 -smoothness of its graph surface (cf. [10,15]). This is the reason to call its functions C 2 -smooth geometrically continuous isogeometric functions.In particular, the dimension of this C 2 -smooth isogeometric space is investigated. The study is based on the decomposition of the space into three subspaces and is an extension of the work [14] to the multi-patch case. In addition, we present an algorithm for the construction of a basis, and use the resulting globally C 2 -smooth functions for numerical experiments, such as performing L 2 approximation and solving triharmonic equation, on bilinear multi-patch domains. The numerical results indicate optimal approximation order.

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We present an isogeometric framework based on collocation to construct a C 2 -smooth approximation of the solution of the Poisson's equation over planar bilinearly parameterized multi-patch domains. The construction of the used globally C 2 -smooth discretization space for the partial differential equation is simple and works uniformly for all possible multipatch configurations. The basis of the C 2 -smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches, edges and vertices of the multi-patch domain. For the selection of the collocation points, which is important for the stability and convergence of the collocation problem, two different choices are numerically investigated. The first approach employs the tensor-product Greville abscissae as collocation points, and shows for the multi-patch case the same convergence behavior as for the one-patch case [2], which is suboptimal in particular for odd spline degree. The second approach generalizes the concept of superconvergent points from the one-patch case (cf. [1,15,32]) to the multi-patch case. Again, these points possess better convergence properties than Greville abscissae in case of odd spline degree.While in case of a one-patch domain, the higher continuity of the functions can be easily guaranteed by using spline functions with the desired smoothness within the patch, the construction of C s -smooth (s ≥ 1) isogeometric spline spaces over multi-patch domains is challenging, and is the task of current research, see e.g. [6-9, 20-23, 28, 29, 33, 34, 40, 41] for s = 1 and [24][25][26][27]39] for s = 2. The design of C s -smooth multi-patch spline spaces is linked to the concept of geometric continuity of multi-patch surfaces, cf. [17,35], due to the fact that an isogeometric function is C s -smooth on a multi-patch domain if and only if its associated multi-patch graph surface is G s -smooth, cf. [16,28].So far, the problem of isogeometric collocation has been mostly explored on one-patch domains, see e.g. [1, 2, 5, 13-15, 30-32, 36, 38]. For analyzing the convergence behavior of a particular approach, the errors are generally computed with respect to L ∞ , W 1,∞ and W 2,∞ norm, or equivalently with respect to L 2 , H 1 and H 2 norm, respectively. The study of isogeometric collocation methods has started in [2] by using the Greville and Demko abscissae as collocation points. In comparison to the Galerkin approach, both choices show a suboptimal convergence behavior with respect to L 2 (L ∞ ) norm, namely of orders O(h p ) and O(h p−1 ) under h-refinement for even and odd spline degree p, respectively, and additionally a suboptimal convergence order O(h p−1 ) with respect to H 1 (W 1,∞ ) norm but just in case of odd spline degree p.In [1], an isogeometric collocation method has been presented, which is based on the computation and on the use of specific collocation points called superconvergent points. The proposed technique possesses only for even spline degree p in case of the L 2 ...

Abstract. In this paper, the geometric interpolation of planar data points and boundary tangent directions by a cubic G 2 Pythagorean-hodograph (PH) spline curve is studied. It is shown that such an interpolant exists under some natural assumptions on the data. The construction of the spline is based upon the solution of a tridiagonal system of nonlinear equations. The asymptotic approximation order 4 is confirmed.