Gaussian quadrature for C1 cubic Clough–Tocher macro-triangles

Jiří Kosinka*, Michael Bartoň

*Bijbehorende auteur voor dit werk

Onderzoeksoutput: ArticleAcademicpeer review

6 Citaten (Scopus)
103 Downloads (Pure)


A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was designed by Hammer and Stroud (1956). The quadrature rule requires n + 2 quadrature points: the barycentre of the simplex and n + 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex. In the planar case, this particular rule belongs to a two-parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles. We prove that this rule is exact for a larger space, namely the C1 cubic Clough–Tocher spline space over macro-triangles if and only if the split-point is the barycentre. This results in a factor of three reduction in the number of quadrature points needed to integrate the Clough–Tocher spline space exactly.

Originele taal-2English
Pagina's (van-tot)6-13
Aantal pagina's8
TijdschriftJournal of Computational and Applied Mathematics
StatusPublished - 1-mei-2019

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