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 ...
