Locally Linearly Independent Basis forC 1 Bivariate Splines of Degree q 5

We construct a locally linearly independent basis for the space S 1 q (() (q 5). Bases with this property were available only for some subspaces of smooth bivariate splines. x1. Introduction Let IR 2 be a simply connected polygonal domain, and let denote a triangulation of consisting of N triangles, V vertices and E edges. Given 0 r < q, consider the linear space of bivariate polynomial splines of degree q and smoothness r, S r q (() := fs 2 C r (() : s T 2 q for all triangles T 2 g; where q := spanfx i y j : i 0; j 0; i + j qg is the space of bivariate polynomials of total degree q. To simplify notation, we set d q := dim q = (q + 1)(q + 2) 2 ; q = 0; 1; : : : : The question of identifying the dimension of S r q (() was rst considered by Strang 14]. Morgan & Scott 11] showed that dimS 1 q ((1) where V I and E I denote the number of interior vertices and interior edges respectively, and is the number of singular vertices of , i.e., those interior vertices for which the adjacent edges of each attached edge are collinear, so ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.