Optimal-order isogeometric collocation at Galerkin superconvergent points

TL;DR

This paper explores an isogeometric collocation method that achieves optimal convergence rates at Galerkin superconvergent points, especially for odd-degree B-splines/NURBS, improving upon previous methods.

Contribution

It introduces a new collocation point selection based on Galerkin superconvergence theory, enhancing convergence behavior and achieving optimal $L^2$-convergence for smooth B-spline/NURBS approximations.

Findings

Achieves optimal $L^2$-convergence for odd-degree B-splines/NURBS.

Improves convergence behavior over previous methods.

Uses a symmetric collocation stencil based on 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 [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.