Bivariate spline interpolation with optimal approximation order

Let Delta be a triangulation of some polygonal domain Omega subset of R-2 a nd let S-q(r)(Delta) denote the space of all bivariate polynomial splines o f smoothness r and degree q with respect to Delta. We develop thr first Her mite-type interpolation scheme for S-q(r)(Delta), q greater than or equal t o 3r + 2, whose approximation error is bounded above by Kh(q+1), where h is the maximal diameter of the triangles in Delta, and the constant K only de pends on the smallest angle of the triangulation and is independent of near -degenerate edges and near-singular vertices. Moreover, the fundamental fun ctions of our scheme are minimally supported and form a locally linearly in dependent basis for a superspline subspace of S-q(r)(Delta). This shows tha t the optimal approximation order can be achieved by using minimally suppor ted splines. Our method of proof is completely different from the quasi-int erpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].