An isogeometric collocation method using superconvergent points

• We propose an isogeometric collocation method with improved approximation properties.
• The locations of the collocation points are derived from the superconvergence theory.
• The proposed method achieves optimal convergence rates in the first derivative norms.
• We include a detailed comparison with standard collocation and the Galerkin method.

We develop an IGA collocation method modified by collocating at points other than the standard Greville abscissae. The method is related to orthogonal collocation used for solving differential equations and to the superconvergence theory, therefore we refer to this method as “super-collocation” (IGA-SC). By carefully choosing the collocation points, it can be seen that the IGA-SC converges in the first derivative (energy) norms at rates similar to that of the Galerkin solution. This is different from the collocation at Greville abscissae (IGA-C), where the convergence in energy norm for odd polynomial degrees is typically suboptimal. The method is tested on 1D, 2D and 3D numerical examples, in which it is compared to IGA-C and Galerkin’s method (IGA-G). The comparison includes a detailed cost vs. accuracy analysis, which shows an improved efficiency of the proposed method in particular for odd polynomial degrees.