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 tot al degree less than or equal to k on Delta. In this paper, we construc t a local basis of some subspace of the space S-k(r)(Delta), where k g reater than or equal to 3r + 2, that can be used to provide the highes t order of approximation, with the property that the approximation con stant of this order is independent of the geometry of Delta with the e xception of the smallest angle in the partition. This result is obtain ed by means of a careful, choice of locally supported basis functions which, however, require a very technical proof to justify their stabil ity in optimal-order approximation. A new formulation of smoothness co nditions for piecewise polynomials in terms of their B-net representat ions is derived for this purpose.