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arXiv · 1609.01971

Optimal-order isogeometric collocation at Galerkin superconvergent points

Monica Montardini·Giancarlo Sangalli·Lorenzo Tamellini
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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 [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.

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Monica Montardini, Giancarlo Sangalli, Lorenzo Tamellini. 2016-09-07. Optimal-order isogeometric collocation at Galerkin superconvergent points. https://doi.org/10.1016/j.cma.2016.09.043

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