On Computing derivatives for C^1 interpolation schemes: an optimization

The application of Powell-Sabin's or Clough-Tocher's schemes to scattered data problems, as known requires the knowledge of the partial derivatives of first order at the vertices of an underlying triangulation. We study a local method for generating partial derivatives based on the minimization of the energy functional on the star of triangles sharing a node that we called a cell. The functional is associated to some piecewise polynomial function interpolating the points. The proposed method combines the global Method II by Renka and Cline (cf. [16, pp. 230-231]) with the variational approach suggested by Alfeld (cf. [2]) with care to efficiency in the computations. The locality together with some implementation strategies produces a method well suited for the treatment of a big amount of data. An improvement of the estimates is also proposed.