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

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 (0000). 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 (0000). 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.