Optimal approximation order of piecewise constants on convex partitions (Q2307485)

Let \(W_q^2(\Omega)\) be the Sobolev space on a cube \(\Omega\subset \mathbb{R}^d\). For a partition \(\Delta\) of \(\Omega\), let us denote by \(S_0(\Delta)\) the space of all piecewise constant functions \(s\,:\ \Omega\to\mathbb{R}\) that are constant on every cell \(\omega\in\Delta\). Furthermore, the set of all convex partitions of \(\Omega\) comprising at most \(N\) cell is denoted by \(\mathfrak{D}_N\). It is shown that if \[ \frac{2}{d+1} + \frac{1}{p} - \frac{1}{q}\geq 0, \quad \ 1\leq p \leq \infty, \ 1\leq q < \infty, \] then, for all \(f\in W_q^2(\Omega)\), \[ E_N(f)_p := \inf_{\Delta\in \mathfrak{D}_N}\inf_{s\in S_0(\Delta)}\|f - s\|_{L_p(\Omega)}\leq CN^{-2/(d+1)}\|f\|_{W_q^2(\Omega)}, \] where \(C\) depends only on \(d, p, q\). It is noted that the approximation order \(O(N^{-2/(d+1)})\) is achieved on a polyhedral partition obtained by anisotropic refinement of an adaptive dyadic partition. Additionally, some new estimates of the approximation order from the above and below are given for Sobolev-Slobodeckij spaces \(W_q^r(\Omega)\) embedded in \(L_p(\Omega)\).