Anisotropic approximation on space-time domains

TL;DR

The paper develops an analytic framework for anisotropic approximation of time-dependent functions on space-time domains by fixed-order polynomials in time and space. It introduces temporal and spatial moduli of smoothness, proves Marchaud-type inequalities and time-dependent Jackson-type estimates, and defines anisotropic Besov spaces to capture separate time/space regularity. Whitney-type inequalities for these Besov spaces on space-time partitions yield local and global approximation bounds, along with embeddings that connect Besov regularity to Lebesgue control. The results are then used to derive direct estimates for discontinuous, adaptive space-time finite elements, providing constructive tools for AFEM in time-dependent PDEs with quantified approximation rates and complexity bounds.

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.

Figures (5)

• Figure 1: Illustration for $d=2$
• Figure 2: $Q_k$ and $\tilde{Q}_k$
• Figure 3: Illustration of subsequence $\tilde{Q}^{(j)}_k$ with $Q_k\subset \tilde{Q}^{(j)}_k$, $j=1,\dots, 4$.
• Figure 4: Illustration for $r=1$
• Figure :

Theorems & Definitions (64)

• Theorem 1.1: Jackson-type inequality
• Theorem 1.2: Whitney-type inequality
• Theorem 1.3: Embedding result
• Remark 1.4
• Theorem 1.5: Direct theorem
• Definition 1.6
• Remark 1.7
• Lemma 1.8
• proof
• Theorem 1.9