Effortless quasi-interpolation in hierarchical spaces

Abstract

We present a general and simple procedure to construct quasi-interpolants in hierarchical spaces, which are composed of a hierarchy of nested spaces. The hierarchical quasi-interpolants are described in terms of the truncated hierarchical basis. Once for each level in the hierarchy a quasi-interpolant is selected in the corresponding space, the hierarchical quasi-interpolants are obtained without any additional manipulation. The main properties of the quasi-interpolants selected at each level are preserved in the hierarchical construction. In particular, hierarchical local projectors are constructed, and the local approximation order of the underling hierarchical space is investigated. The presentation is detailed for the truncated hierarchical B-spline basis, and we discuss its extension to a more general framework.

Generalized spline spaces over T-meshes: Dimension formula and locally refined generalized B-splines

“…In particular, it has already been shown in the literature that the construction and properties of (truncated) hierarchical B-splines and analysis-suitable T-splines also extend to the generalized spline context, see [15,26,35] and [3,4], respectively.…”

“…In particular, GB-splines support (locally refined) hierarchical structures in the same way as polynomial B-splines, see [26] (and also [15,35]). T-spline structures based on GB-splines have been addressed in [3,4].…”

Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

“…Corollary 2 of Ref. 41. In particular, this implies Next, we present a concrete refinement algorithm to specify the setting of Sec.…”

“…Together with some ideas from Ref. 41, we use this observation to define a Scott-Zhang-type projector J • : L 2 (Ω) → X • which is locally L 2 -and H 1 -stable and has a first-order approximation property (Sec. 5.10).…”

Modified PHT-splines

“…which needs to solve C j globally. Similar to [5,35], an efficient method based on local geometric information can be provided as follows.…”