Three recipes for quasi-interpolation with cubic Powell–Sabin splines

Abstract

We investigate the construction of bivariate quasi-interpolation methods based on C1 cubic Powell–Sabin B-spline representations. Rather than using a large set of functional data to specify all the parameters in such representations, we study how to reduce them by imposing different super-smoothness properties while retaining cubic precision. This results in three recipes, which are completely general in the sense that they can be implemented with any local cubic polynomial approximation scheme (or a mixture of them). More precisely, they embed C2 super-smoothness at the vertices and across the edges, C2 super-smoothness inside the macro-triangles, and smoothness of Clough–Tocher type, respectively. To demonstrate their usefulness, we derive four specific methods based on local Hermite and Lagrange interpolation. We conclude with a selection of numerical experiments.