Instability in the Dimension of Spaces of Bivariate Piecewise Polynomials of Degree $2r$ and Smoothness Order <i>r</i>

Abstract

The spaces of bivariate splines on a particular triangulation are considered. The dimension of the space of splines with degree 2 and with smoothness order 1 on this triangulation has long been known to depend on the geometry of the triangulation. This result is extended to the space of splines of degree $2r$ and smoothness order r. The dimension of the spline space is shown to have a similar dependence on geometry for all values of $r \geqq 1$.