Collocation Points for Optimal-Order Convergence in Isogeometric Analysis

  • Belardo, Maria Roberta (Scuola Superiore Meridionale)
  • Calabrò, Francesco (Università degli Studi di Napoli "Federico II)

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Isogeometric collocation (IgA-C) enforces PDEs in strong form at selected sites of smooth spline spaces, avoiding element integration and reducing assembly cost relative to Galerkin IgA. A persistent limitation is that accuracy is highly sensitive to the collocation nodes: standard choices such as Greville abscissae exhibit suboptimal convergence compared to the Galerkin reference, and the loss of optimal rates may depend on spline degree. Improving node selection is therefore central to making IgA-C a reliable high-order alternative. Several contributions have explored collocation point selection through its relationship with Galerkin discretizations and superconvergence. Variational collocation introduces Cauchy–Galerkin points, defined as zeros of the Galerkin residual, and proposes element-invariant approximations for their computation [2]. Related results show that collocation at Galerkin superconvergent locations can recover optimal L2 convergence for odd spline degrees in square collocation schemes [3]. Building on this context, recent results show that a simple structural feature, i.e. local symmetry of the interior sites within each knot span, is correlated with optimal-order convergence in global least-squares IgA-C [1]. For spline degrees p = 3–8, ensembles of elementwise symmetric perturbations of interior collocation points are tested in overdetermined least-squares solves of standard second-order elliptic benchmarks. For p >= 4, symmetric configurations systematically achieve Galerkin-like convergence, with L2 errors approaching order p + 1 and H1 errors approaching order p. Therefore, a new selection of collocation points is proposed for even degrees, mitigating the odd–even discrepancy observed with standard choices. Validation in 1D and in 2D second-order problems on mapped geometries confirms robustness across refinements and boundary treatments, providing a practical route to optimal-order accuracy in least-squares IgA-C. [1] M.R. Belardo, F. Calabrò, Optimal Convergence of IgA Collocation Methods, 2025, submitted. [2] H. Gomez, L. De Lorenzis, The variational collocation method, CMAME, 2016. [3] M. Montardini, G. Sangalli, L. Tamellini, Optimal-order isogeometric collocation at Galerkin superconvergent points, CMAME, 2017.