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On a modified Hilbert transformation, the discrete inf-sup condition, and error estimates

Richard Löscher, Olaf Steinbach, Marco Zank

TL;DR

This work analyzes a modified Hilbert transformation ${\mathcal{H}}_T$ on a finite time interval to accompany space-time finite element discretizations of time-dependent PDEs. It establishes a mesh-dependent discrete inf-sup stability for the projection based on ${\mathcal{H}}_T$, with the stability constant scaling linearly with the mesh size $h$ and a Fourier-coefficient characterization of the stability. The paper derives Céa-type and interpolation-based error estimates for the projection error relative to the $L^2$ projection, proving optimal convergence under additional regularity such as $u\in H^2(0,T)$ and $\partial_t u(0)=0$, while explaining slower convergence in singular cases. Numerical experiments support the theory, showing near-optimal convergence in most scenarios and clarifying the influence of initial-time singularities; the results have direct implications for stable space-time FEMs for parabolic and hyperbolic evolution equations and point to promising extensions to higher-order spaces and wave-equation analysis.

Abstract

In this paper, we analyze the discrete inf-sup condition and related error estimates for a modified Hilbert transformation as used in the space-time discretization of time-dependent partial differential equations. It turns out that the stability constant depends linearly on the finite element mesh parameter, but in most cases, we can show optimal convergence. We present a series of numerical experiments which illustrate the theoretical findings.

On a modified Hilbert transformation, the discrete inf-sup condition, and error estimates

TL;DR

This work analyzes a modified Hilbert transformation on a finite time interval to accompany space-time finite element discretizations of time-dependent PDEs. It establishes a mesh-dependent discrete inf-sup stability for the projection based on , with the stability constant scaling linearly with the mesh size and a Fourier-coefficient characterization of the stability. The paper derives Céa-type and interpolation-based error estimates for the projection error relative to the projection, proving optimal convergence under additional regularity such as and , while explaining slower convergence in singular cases. Numerical experiments support the theory, showing near-optimal convergence in most scenarios and clarifying the influence of initial-time singularities; the results have direct implications for stable space-time FEMs for parabolic and hyperbolic evolution equations and point to promising extensions to higher-order spaces and wave-equation analysis.

Abstract

In this paper, we analyze the discrete inf-sup condition and related error estimates for a modified Hilbert transformation as used in the space-time discretization of time-dependent partial differential equations. It turns out that the stability constant depends linearly on the finite element mesh parameter, but in most cases, we can show optimal convergence. We present a series of numerical experiments which illustrate the theoretical findings.
Paper Structure (6 sections, 14 theorems, 176 equations, 2 figures, 8 tables)

This paper contains 6 sections, 14 theorems, 176 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. A modified Hilbert transformation
  3. A modified Hilbert transformation based projection
  4. Discrete inf-sup stability condition in $S_h^0(0,T)$
  5. Error estimates for the projection in $S_h^0(0,T)$
  6. Conclusions

Key Result

Lemma 2.2

For $0 \neq v \in H^s_{0,}(0,T) = [H^1_{0,}(0,T),L^2(0,T)]_s$ with $s \in (0,1]$, the inequality holds true.

Figures (2)

  • Figure 1: Functions $u_{h,\text{min}}$ realizing \ref{['inf sup L2 diskret']} with the smallest inf-sup constant $c_S$ for $N=32$ elements for $T=2$.
  • Figure 2: Justifying estimate \ref{['AbschFx2']}.

Theorems & Definitions (18)

  • Remark 2.1
  • Lemma 2.2
  • Remark 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Corollary 4.4
  • Lemma 4.5
  • Theorem 4.6
  • Corollary 4.7
  • ...and 8 more