STABILITY OF OPTIMAL-ORDER APPROXIMATION BY BIVARIATE SPLINES OVER ARBITRARY TRIANGULATIONS

Let Delta be a triangulation of some polygonal domain in R(2) and S-k(r)(Delta), the space of all bivariate C-r piecewise polynomials of total degree less than or equal to k on Delta. In this paper, we construct a local basis of some subspace of the space S-k(r)(Delta), where k greater than or equal to 3r + 2, that can be used to provide the highest order of approximation, with the property that the approximation constant of this order is independent of the geometry of Delta with the exception of the smallest angle in the partition. This result is obtained by means of a careful, choice of locally supported basis functions which, however, require a very technical proof to justify their stability in optimal-order approximation. A new formulation of smoothness conditions for piecewise polynomials in terms of their B-net representations is derived for this purpose.