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 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.

💡 Research Summary

This paper develops a comprehensive theory of anisotropic polynomial approximation on space‑time cylinders I × D, where I is a finite time interval and D ⊂ ℝᵈ is a bounded Lipschitz domain. The authors introduce separate temporal and spatial moduli of smoothness, ω_{r₁,t} and ω_{r₂,x}, defined via higher‑order differences, and study their fundamental properties, including Marchaud‑type inequalities that hold for the full range 0 < p ≤ ∞.

Using these tools, they prove a Jackson‑type inequality (Theorem 1.1): for any f ∈ Lᵖ(I × D) there exists an anisotropic polynomial P of temporal degree r₁ − 1 and spatial multi‑degree |α| < r₂ such that the Lᵖ‑error is bounded by a constant (depending only on dimension, the degrees, and the Lipschitz characteristics of D) times the sum of the temporal and spatial moduli evaluated at the size of I and the diameter of D, respectively. This result extends classical Jackson estimates to the anisotropic, space‑time setting and to the quasi‑Banach regime p < 1.

The paper then defines anisotropic Besov spaces B_{s₁,s₂}^{q,q}(I × D) via the same moduli and shows that the Besov seminorm is equivalent to a suitable combination of temporal and spatial smoothness. With this framework, a Whitney‑type inequality (Theorem 1.2) is established: on each prism J × S of a space‑time partition 𝒫, there exists a local anisotropic polynomial whose Lᵖ‑error is controlled by the volume of the prism raised to the exponent α = 1/s₁ + d/s₂ − 1/q + 1/p multiplied by the local Besov seminorm. The constants involve geometric parameters κ_𝒫 and a(𝒫) that measure the anisotropy and shape regularity of the partition.

These approximation results are then applied to discontinuous adaptive space‑time finite element methods. For a given initial tensor‑product partition 𝒫₀, the authors construct a refined partition 𝒫 using an “Atomic Split” refinement algorithm. They prove a direct theorem (Theorem 1.5) stating that the number of added elements grows at most like ε^{-(1/s₁ + d/s₂)} to achieve a prescribed accuracy ε, and that there exists a discontinuous finite‑element function F in the space V_{r₁,r₂}^{DC}(𝒫) such that ‖f − F‖{Lᵖ} ≤ C₂ ε |f|{B_{s₁,s₂}^{q,q}}. The constants C₁ and C₂ depend only on dimension, smoothness parameters, the initial partition’s shape regularity, and the domain’s Lipschitz constants.

Technical contributions include: (i) a careful treatment of moduli of smoothness for 0 < p < 1, (ii) the introduction of LipProp(D) to capture the influence of non‑convex Lipschitz boundaries on approximation constants, (iii) the construction of anisotropic polynomial spaces that allow independent control of temporal and spatial degrees, and (iv) detailed geometric analysis of space‑time partitions, including the definition of κ_𝒫 (minimal head angle) and a(𝒫) (anisotropy measure).

The paper also provides auxiliary lemmas on convergence of sequences of Lipschitz domains, Marcinkiewicz‑type inequalities, and the relationship between Besov seminorms and moduli. An embedding theorem (Theorem 1.3) shows that the anisotropic Besov space continuously embeds into Lᵖ with a constant depending on the same geometric parameters.

In conclusion, the authors deliver a rigorous, anisotropic approximation framework that bridges the gap between classical isotropic theory and the needs of adaptive space‑time finite element methods. Their results lay the groundwork for further developments such as inverse theorems, extensions to continuous finite elements, and applications to nonlinear time‑dependent PDEs.