Optimal-order isogeometric collocation at Galerkin superconvergent points

Abstract

In this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least-squares collocation method presented in Anitescu et al. (2015) and the variational collocation method presented in Gomez and De Lorenzis (2016). The focus is on smoothest B-splines/NURBS approximations, i.e, having global Cp−1 continuity for polynomial degree p. Within the framework of Gomez and De Lorenzis (2016), we select as collocation points a subset of those considered in Anitescu et al. (2015), which are related to the Galerkin superconvergence theory. With our choice, that features local symmetry of the collocation stencil, we improve the convergence behavior with respect to Gomez and De Lorenzis (2016), achieving optimal L2-convergence for odd degree B-splines/NURBS approximations. The same optimal order of convergence is seen in Anitescu et al. (2015), where, however a least-squares formulation is adopted. Further careful study is needed, since the robustness of the method and its mathematical foundation are still unclear.