Quadrature rules for C1 quadratic spline finite elements on the Powell–Sabin 12-split

We present quadrature rules for the space of C1 quadratic splines on triangulations refined according to the Powell–Sabin (PS) 12-split. Focusing on a single triangle, we first provide a symmetric 4-node quadrature rule with positive weights, which is exact on the considered spline space. For its construction we make use of a local simplex spline basis. The rule is shown to be optimal, with the minimum number of nodes. Next, in view of using C1 quadratic splines on triangulations in Galerkin finite element discretizations of differential problems, we design a collection of weighted quadrature rules that are helpful for an efficient formation of the linear systems arising in such discretizations. Lastly, we provide numerical comparisons with elementwise Gaussian rules and we illustrate the performance of the presented quadrature rules in the context of C1 quadratic spline finite elements on triangulations.