Anisotropic approximation on space-time domains

TL;DR

This paper develops anisotropic polynomial approximation methods for functions in Lebesgue and Besov spaces on space-time domains, establishing key inequalities and applying them to adaptive finite element methods.

Contribution

It introduces new Jackson- and Whitney-type inequalities for anisotropic approximation on Lipschitz cylinders, advancing the theoretical foundation for adaptive space-time finite element methods.

Findings

Proved Jackson- and Whitney-type inequalities on Lipschitz cylinders.

Established properties of temporal and spatial moduli of smoothness.

Provided a direct estimate for adaptive space-time finite element approximation.

Abstract

We investigate anisotropic (piecewise) polynomial approximation of functions in Lebesgue spaces as well as anisotropic Besov spaces. For this purpose we study temporal and spacial 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.