Gaussian quadrature for C1 cubic Clough–Tocher macro-triangles

Kosinka, J. & Bartoň, M., 1-May-2019, In : Journal of Computational and Applied Mathematics. 351, p. 6-13 8 p.

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  • Gaussian quadrature for C 1 cubic Clough-Tocher macro-triangles

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  • Gaussianquadraturefor C 1 cubicClough–Tocher macro-triangles

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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.

Original languageEnglish
Pages (from-to)6-13
Number of pages8
JournalJournal of Computational and Applied Mathematics
Publication statusPublished - 1-May-2019


  • Clough–Tocher spline space, Gaussian quadrature rules, Numerical integration

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