Stability of optimal-order approximation by bivariate splines over arbitrary triangulations

Abstract:

Let $\Delta$ be a triangulation of some polygonal domain in ${\mathbb {R}^2}$ and $S_k^r(\Delta )$, the space of all bivariate ${C^r}$ piecewise polynomials of total degree $\leqslant k$ on $\Delta$. In this paper, we construct a local basis of some subspace of the space $S_k^r(\Delta )$, where $k \geqslant 3r + 2$, 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 ${\text {B}}$-net representations is derived for this purpose.