On numerical quadrature for \(C^1\) quadratic Powell-Sabin 6-split macro-triangles
In this paper the authors study Gaussian quadrature rules for polynomial spline spaces over triangulations, in particular for the case of $C^1$ quadratic Powell-Sabin 6-split macro-triangles, often used in interpolation problems over triangulations and in isogeometric analysis. After an introduction on Bézier triangles, Powell-Sabin 6-split and Hammer-Stroud quadratures for quadratic triangular polynomials is presented. Such quadratures ensure that the number of nodes is reduced from twelve or eighteen to three (optimal number). The construction of three new spline basis functions complements the basis of bivariate quadratic polynomials on a triangle to form a basis over the Powell-Sabin macro-element. The authors also generalize the problem to the non-symmetric case, i.e., to quadrature rules for the Powell-Sabin splines with general split-points. They obtain numerical evidence that three nodes suffice also in the general situation, but a rigorous proof of this fact is yet to be shown.
scientific article; zbMATH DE number 7006426
scientific article