Interpolation by spline spaces on classes of triangulations

We describe a general method for constructing triangulations Δ which are suitable for interpolation by Sqr(Δ), r=1,2, where Sqr(Δ) denotes the space of splines of degree q and smoothness r. The triangulations Δ are obtained inductively by adding a subtriangulation of locally chosen scattered points in each step. By using Bézier–Bernstein techniques, we determine the dimension and construct Lagrange and Hermite interpolation sets for Sqr(Δ), r=1,2. The Hermite interpolation sets are obtained as limits of the Lagrange interpolation sets. The interpolating splines can be computed locally by passing from triangle to triangle. Several numerical results on interpolation of functions and scattered data are given.