Anisotropic approximation on space–time domains

We investigate anisotropic (piecewise) polynomial approximation of functions in Lebesgue spaces as well as anisotropic Besov spaces. For this purpose we study temporal and spatial moduli of smoothness and their properties. In particular, we prove Jackson- and Whitney-type inequalities on Lipschitz cylinders, i.e., space–time domains $I\times D$ with a finite interval $I$ and a bounded Lipschitz domain $D\subset R^d$, $d\in N$. As an application, we prove a direct estimate result for adaptive space–time finite element approximation in the discontinuous setting.