Approximation by piecewise constants on convex partitions (Q765690)

Let \(\Omega\subset\mathbb{R}^d\), \(d\in \mathbb{N}\), be a bounded domain and let \(\Delta\) be a convex partition of \(\Omega\). Denote by \(|\Delta|\) the number of convex cells in \(\Delta\). For \(\Omega := (0,1)^d\), the author shows that the order of approximation of an \(f\in W^2_p (\Omega)\) by piecewise constants can be improved to \(\mathcal{O}(|\Delta|^{-2/(d+1)})\) by using suitable anisotropic convex partitions obtained from a simple algorithm. Furthermore, he proves that the saturation order of piecewise constant approximation in the \(L_\infty\)-norm on convex partitions is \(|\Delta|^{-2/(d+1)}\). In addition, it is shown that the saturation order for linear approximation on convex partitions is \(|\Delta|^{-2/d}\), which is the same as for isotropic partitions.