Bivariate spline interpolation with optimal approximation order
Let \(S^r_q(\Delta)\) denote the space of bivariate polynomial splines of smoothness \(r\) and degree \(q\), with respect to a triangulation \(\Delta\) of a polygonal domain \(\Omega\subset\mathbb{R}^2\). The paper deals with some approximation properties of the superspline subspace \[ S^{r,\rho}_q(\Delta)= \{s\in S^r_q(\Delta)\mid s\in C^\rho(v),\;\forall\text{ vertices }v\text{ of }\Delta\}, \] where \(\rho= r+ [(r+ 1)/2]\) and \(q\geq 3r+2\). The main result of the paper establishes: A) the existence of a unique \(s_f\in S^{r,\rho}_q(\Delta)\) satisfying Hermite-type interpolation conditions for a given \(f\in C^{2r}(\Omega)\), B) the following estimate for \(f\in C^m(\Omega)\) \((m= 2r,\dots, q+1)\), and \(T\in\Delta\) \[ \|D^\alpha_x D^\beta_y(f- s_f)\|_{L_\infty(T)}\leq Kh^{m- \alpha-\beta}_T \max_{0\leq \mu\leq m}\|D^\mu_x D^{m- \mu}_y f\|_{C(T)} \] for all \(\alpha,\beta\geq 0\), \(\alpha+ \beta\leq m\), where \(h_T\) is the diameter of \(T\), and \(K= K(r,q,\theta_\Delta)\) (\(\theta_\Delta\) is the smallest angle in \(\Delta\)). The interpolation scheme is given in terms of a nodal basis \(\{s_i\}\) of \(S^{r,\rho}_q(\Delta)\), which is studied in the last section of the article. The authors point out that one of the several advantages of their approach, is that the estimation of the approximation error is completely local. It means that the estimate involves only the local Sobolev norm of \(f\) on the triangle \(T\) on which the error is measured.
scientific article; zbMATH DE number 1618395
scientific article