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

TL;DR

This paper characterizes approximation classes for anisotropic space-time finite element methods on prismatic meshes, introducing a refinement technique and interpolation operators to analyze approximation properties in anisotropic Besov norms.

Contribution

It provides a new refinement technique and a characterization of approximation classes for anisotropic space-time finite element methods using Besov norms.

Findings

Proposed a refinement technique for prismatic meshes with anisotropy.

Defined a (quasi-)interpolation operator for these meshes.

Characterized approximation classes via anisotropic Besov norms.

Abstract

We study the approximation of $L_p$-functions, $p\in (0,\infty]$, on cylindrical space-time domains $\Omega_T:=[0,T]\times \Omega$, $0<T<\infty$, $\Omega\subset \R^d$ 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.