Quadrature rules for splines of high smoothness on uniformly refined triangles

TL;DR

This paper develops quadrature rules that are exact for high-smoothness spline spaces on refined triangles, expanding their applicability to complex spline functions used in geometric modeling and numerical analysis.

Contribution

It demonstrates that existing symmetric quadrature rules for polynomials remain exact for certain high-smoothness spline spaces on refined triangles, with explicit conditions based on spline degree and smoothness.

Findings

Quadrature rules are exact for $C^{2r-1}$ splines of degree $3r$ on Clough-Tocher splits.

Quadrature rules are exact for $C^{2r-1}$ splines of degree $2r$ on Powell-Sabin splits.

The analysis uses representation of spline spaces via simplex splines.

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.