Optimal convergence of IgA collocation methods

Isogeometric collocation discretizes the strong form of a PDE on smooth spline spaces and therefore avoids element integration, but its spatial accuracy is highly sensitive to the placement of the collocation nodes. For this reason methods are tested on linear elliptic problems in order to verify convergence properties. Classical choices such as Greville abscissae may yield suboptimal convergence in both H1 and L2 norms for several spline degrees and problem settings. Node sets derived from (estimated) superconvergent (Cauchy–Galerkin) points – e.g. alternating subsets, clustered variants, or least–squares sets – frequently improve the observed L2 behaviour and in favourable cases approach the Galerkin benchmark, though this is not universal across all degrees, boundary conditions, and PDE types. In this paper we find for the first choices of points that recover optimal convergence for polynomial degrees p=4,6,8. The construction is made in order to recover the symmetry inside every knot span also for even degrees, as done in the above mentioned methods. Although the exact reason for this behaviour could not be clearly identified, the numerical evidence suggests that restoring local symmetry recovers the optimal rate. Unfortunately, as most of the previously proposed methods, this results in a collocation system with more equations than degrees of freedom number of degrees of freedom, thus the overall system is solved in a least–square sense.