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

Pedro Morin, Cornelia Schneider, Nick Schneider

Abstract

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

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

Abstract

We study the approximation of -functions, , on cylindrical space-time domains , , Lipschitz, , 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.
Paper Structure (20 sections, 44 theorems, 182 equations, 13 figures, 3 algorithms)

This paper contains 20 sections, 44 theorems, 182 equations, 13 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Preliminaries
  4. General setting
  5. Temporal and spatial moduli of smoothness for anisotropic Besov spaces
  6. Approximation classes
  7. Main results
  8. Space-time partition and mesh refinement
  9. Atomic refinement of prisms
  10. Refinement of patches preserving mesh regularity
  11. Algorithmic complexity
  12. Support of basis functions and quasi-interpolation
  13. An investigation of the mesh geometry of hanging nodes
  14. The (quasi-)interpolation operator
  15. Almost characterization of approximation classes with respect to space-time finite elements
  16. ...and 5 more sections

Key Result

Lemma 3.2

Let $\genfrac{\{}{\}}{0pt}{0}{I}{S}$ be a $\genfrac{\{}{\}}{0pt}{0}{1}{d}$-dimensional simplex derived from an element $\genfrac{\{}{\}}{0pt}{0}{I_0}{S_0}\in\genfrac{\{}{\}}{0pt}{0}{\mathcal{I}_0}{\mathcal{T}_0}$ by finitely many applications of the method $\genfrac{\{}{\}}{0pt}{0}{\textup{BISECT}(1 where we have used

Figures (13)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 8 more figures

Theorems & Definitions (122)

  • Remark 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Lemma 3.2
  • Remark 3.3
  • proof
  • Corollary 3.4
  • proof
  • ...and 112 more