Optimal-order isogeometric collocation at Galerkin superconvergent points
For researchers in isogeometric analysis, this work offers an incremental improvement in convergence order for collocation methods, but lacks rigorous justification.
This paper investigates an isogeometric collocation method using Galerkin superconvergent points, achieving optimal L2-convergence for odd-degree B-splines/NURBS approximations, improving upon prior work. However, the method's robustness and mathematical foundation remain unclear.
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 [1] and the variational collocation method presented in [2]. The focus is on smoothest B-splines/NURBS approximations, i.e, having global $C^{p-1}$ continuity for polynomial degree $p$. Within the framework of [2], we select as collocation points a subset of those considered in [1], which are related to the Galerkin superconvergence theory. With our choice, that features local symmetry of the collocation stencil, we improve the convergence behaviour with respect to [2], achieving optimal $L^2$-convergence for odd degree B-splines/NURBS approximations. The same optimal order of convergence is seen in [1], 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.