Quadrature rules for splines of high smoothness on uniformly refined triangles

Eddargani, Salah; Manni, Carla; Speleers, Hendrik

doi:10.1090/mcom/4058

Quadrature rules for splines of high smoothness on uniformly refined triangles
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by Salah Eddargani, Carla Manni and Hendrik Speleers;

Math. Comp.

DOI: https://doi.org/10.1090/mcom/4058

Published electronically: February 3, 2025

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Abstract:

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

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