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- W2886172065 abstract "The quadrature rule of Hammer and Stroud (1956) for cubic polynomials has been shown to be exact for a larger space of functions, namely the C1 cubic Clough–Tocher spline space over a macro-triangle if and only if the split-point is the barycentre of the macro-triangle Kosinka and Bartoň (2018). We continue the study of quadrature rules for spline spaces over macro-triangles, now focusing on the case of C1 quadratic Powell–Sabin 6-split macro-triangles. We show that the 3-node Gaussian quadrature(s) for quadratics can be generalised to the C1 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 Kosinka and Bartoň (2018). 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 C1 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." @default.
- W2886172065 created "2018-08-22" @default.
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- W2886172065 date "2019-03-01" @default.
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- W2886172065 title "On numerical quadrature for <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=mml63 display=inline overflow=scroll altimg=si4.gif><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> quadratic Powell–Sabin 6-split macro-triangles" @default.
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- W2886172065 doi "https://doi.org/10.1016/j.cam.2018.07.051" @default.
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