PhD Defense: Local Projectors for (Truncated) Hierarchical B-splines and their Application to Weighted Isogeometric Collocation

Local Projectors for (Truncated) Hierarchical B-splines and their Application to Weighted Isogeometric Collocation

This thesis is divided in two parts. The first part is devoted to techniques for adaptive spline projection via quasi-interpolation, enabling the ecient approximation of given functions. We employ local least-squares fitting in restricted hierarchical spline spaces to establish novel projection operators for hierarchical splines of degree p. This leads to efficient spline projectors that require O(pd) floating point operations and O(1) evaluations of the given function per degree of freedom, while providing essentially the same accuracy as global approximation. Our spline projectors are based on a unifying framework for quasi-interpolation in hierarchical spline spaces. We present a detailed comparison with the scheme of [1].

In the second part, we propose a novel version of weighted isogeometric collocation that is especially suited for adaptive THB-spline renement. It is well known that the choice of the collocation nodes is crucial to ensure stability and good approximation properties, especially when adaptive renement 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 ofWeighted 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 [2] and clustered collocation [3].

We explore how these results extend to the case of adaptively refined THB-spline discretizations. We observe that WICSP is compatible with THB-spline refinement, 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.