Approximation classes for the anisotropic space-time finite element method. An almost characterization

We study the approximation of $L_p$-functions, $p\in (0,\mathrm{\infty }]$, on cylindrical space-time domains ${\mathrm{\Omega }}_T:=[0,T]\times \mathrm{\Omega }$, $0<T<\mathrm{\infty }$, $\mathrm{\Omega }\subset {\mathbb{R}}^d$ Lipschitz, $d\in \mathbb{N}$, with respect to continuous anisotropic space-time finite elements on prismatic meshes. In particular, we propose a suitable refinement technique which creates (locally refined) prismatic meshes with sufficient smoothness and the desired anisotropy, and prove complexity estimates. Furthermore, we define a (quasi-)interpolation operator on this type of meshes and use it to characterize the corresponding approximation classes by showing direct and inverse estimates in terms of anisotropic Besov norms.