Abstract
In this paper, we identify families of quadrature rules that are exact for sufficiently smooth spline spaces on uniformly refined triangles in R 2 \mathbb {R}^2 . Given any symmetric quadrature rule on a triangle T T that is exact for polynomials of a specific degree d d , we investigate if it remains exact for sufficiently smooth splines of the same degree d d defined on the Clough–Tocher 3-split or the (uniform) Powell–Sabin 6-split of T T . We show that this is always true for C 2 r − 1 C^{2r-1} splines having degree d = 3 r d=3r on the former split or d = 2 r d=2r on the latter split, for any positive integer r r . Our analysis is based on the representation of the considered spline spaces in terms of suitable simplex splines.
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