Weighted isogeometric collocation based on Spline Projectors

Abstract

In this paper, we propose a novel version of weighted isogeometric collocation that is especially suited for adaptive THB-spline refinement.It is well known that the choice of the collocation nodes is crucial to ensure stability and good approximation properties, especially when adaptive refinement is performed. In order to address this issue, we make use of a particular class of locally supported quasi-interpolant schemes to propose the new method of Weighted Isogeometric Collocation based on Spline Projectors (WICSP).We show that WICSP performs well in the case of tensor-product spline discretizations, both with respect to the rate of convergence and computational complexity. In particular, we observe experimentally an optimal rate of convergence for odd degree basis functions and obtain a dimension-independent computational complexity O(np) for matrix-free applications, similar to other approaches such as weighted quadrature (Calabrò et al., 2017) and clustered collocation (Montardini et al., 2017).We explore how these results extend to the case of adaptively refined THB-spline discretizations. We observe that WICSP is compatible with THB-spline refinement, exhibiting good accuracy and low computational costs. In fact, we get a complexity of O(np2d) for matrix assembly and O(npd) for the matrix-free case. This compares well with the available methods for isogeometric THB-spline discretizations.