Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh

Abstract:

Let $\Delta$ be the mesh in the plane obtained from a uniform square mesh by drawing in the north-east diagonal in each square. Let $\pi _{k,\Delta }^\rho$ be the space of bivariate piecewise polynomial functions in ${C^\rho }$, of total degree $\leq k$, on the mesh $\Delta$. Let $m(k,\rho )$ denote the approximation order of $\pi _{k,\Delta }^\rho$. In this paper, an upper bound for $m(k,\rho )$ is given. In the space $3 \leq 2k - 3\rho \leq 7$, the exact values of $m(k,\rho )$ are obtained: \[ \begin {array}{*{20}{c}} {m(k,\rho ) = 2k - 2\rho - 1} \hfill & {{\text {for}}\;2k - 3\rho = 3\;{\text {or}}\;4,} \hfill \\ {m(k,\rho ) = 2k - 2\rho - 2} \hfill & {{\text {for}}\;2k - 3\rho = 5,6\;{\text {or}}\;7.} \hfill \\ \end {array} \] In particular, this result answers negatively a conjecture of de Boor and Höllig.