2019
DOI: 10.1016/j.cam.2018.07.051
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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 generalis… Show more

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Cited by 9 publications
(2 citation statements)
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“…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.…”
Section: D and Beyondmentioning
confidence: 99%
“…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.…”
Section: D and Beyondmentioning
confidence: 99%
“…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].…”
Section: Introductionmentioning
confidence: 99%