@article {ipa|Optimal-order isogeometric collocation at Galerkin superconvergent points|2016, title = {Optimal-order isogeometric collocation at Galerkin superconvergent points}, year = {2016}, month = {September}, pages = {22 p.}, institution = {CNR-IMATI}, type = {Working paper}, address = {Pavia}, 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 [1] and the variational collocation method presented in [14]. The focus here is on smoothest B-splines/NURBS approximations, i.e, having global C p-1 continuity for polynomial degree p. In particular, we show that using as collocation points a suitable subset of those considered in [1] (which are related to the Galerkin superconvergence theory) it is possible to achieve optimal L2-convergence for odd degree B-splines/NURBS approximations with a pure collocation scheme, i.e., considering as many collocation points as degrees-of-freedom. The method in [1], instead, is based on a leastsquares formulation due to the fact that the set of collocation points outnumbers the degrees-of-freedom to be computed. We especially highlight that we obtain fourth-order convergence for the L2 and L$\infty$ norm of the error when considering cubic basis functions. Further careful analysis is however needed, since the robustness of the method and its mathematical foundations are still unclear.

}, keywords = {Superconvergent points}, url = {http://irs.imati.cnr.it/reports/irs16-13}, author = {M. Montardini and G. Sangall and L. Tamellini} }