On numerical quadrature for C1 quadratic Powell–Sabin 6-split macro-triangles

Abstract

The quadrature rule of Hammer and Stroud [16] for cubic polynomials has been shown to be exact for a larger space of functions, namely the C 1 cubic Clough-Tocher spline space over a macro-triangle if and only if the split-point is the barycentre of the macro-triangle [21]. We continue the study of quadrature rules for spline spaces over macro-triangles, now focusing on the case of C 1 quadratic Powell-Sabin 6-split macro-triangles. We show that the 3-node Gaussian quadrature(s) for quadratics can be generalised to the C 1 quadratic Powell-Sabin 6-split spline space over a macro-triangle for a two-parameter family of inner split-points, not just the barycentre as in [21]. The choice of the inner split-point uniquely determines the positions of the edge split-points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature. Consequently, the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three.For the inner split-point at the barycentre, we prove that the two 3-node quadratic polynomial quadratures of Hammer and Stroud exactly integrate also the C 1 quadratic Powell-Sabin spline space if and only if the edge split-points are at their respective edge midpoints. For other positions of the inner and edge split-points we provide numerical examples showing that three nodes suffice to integrate the space exactly, but a full classification and a closed-form solution in the generic case remain elusive.

Gaussian quadrature for C1 cubic Clough–Tocher macro-triangles

“…For other spline spaces over planar triangulations, one may ask a similar question as we posed here, namely if a certain polynomial bivariate quadrature rule integrates a larger space, and if so, under what conditions. In the case of C 1 quadratic Powell-Sabin 6-split macro-triangles, the underlying spline space admits more degrees of freedom and consequently the 3-node Gaussian quadrature(s) for quadratics can be generalised to the C 1 quadratic Powell-Sabin 6-split spline space over a macro-triangle for a two-parameter family of inner split-points [4], not just the barycentre as it is in the case of the Clough-Tocher spline space considered here.…”

A geometric characterization of Powell-Sabin triangulations allowing the construction of C2 quartic splines

“…In [6], Powell and Sabin introduced a new refinement with the specific objective of contour plotting, managing to define a 𝐶 1 piecewise quadratic function from the values at the nodes of the function to be approximated and its gradient. Since then, interest in 𝐶 1 quadratic Powell-Sabin (PS-) splines has been maintained: for instance, in [7], the construction of normalized B-spline bases was addressed; in [8], differ-ential and discrete quasi-interpolants were defined; and in [9] Gaussian quadrature was studied. Blossoming was also used to build 𝐶 1 quadratic quasi-interpolants on PS-partitions [10].…”