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Galerkin-type time discretizations for parabolic and hyperbolic problems: stability and a priori error analysis

Sergio Gómez

TL;DR

This work develops a unified variational framework to analyze Galerkin-type time discretizations for parabolic and hyperbolic problems, focusing on stability and a priori error estimates in $L^{\infty}(0,T;X)$. By employing nonstandard test functions and time-weighted techniques, it avoids Grönwall-type growth, accommodates arbitrary polynomial degrees, and remains robust with respect to parameter variations. The analysis encompasses DG and CG time discretizations across heat and wave equations, using a suite of time projections (Thomée, Aziz–Monk, Walkington) and a time reconstruction operator to derive stability and convergence results. It also addresses nonhomogeneous Dirichlet data, different wave formulations, and variants such as least-squares stabilization and exponential weights, providing a comprehensive, adaptable foundation for space–time discretizations of parabolic and hyperbolic problems. The results offer unconditional stability with optimal convergence under standard regularity and pave the way for nonlinear extensions and practical, efficient implementations in space–time finite element frameworks.

Abstract

We present a unified framework for the analysis of space-time methods based on Galerkin-type time discretizations for parabolic and hyperbolic problems. Crucially, the stability analysis relies on a suitable choice of test functions to establish the continuous dependence of the discrete solution on the data in $L^{\infty}(0, T; X)$ norms, which is then used to derive a priori error estimates. This approach closes the gap in the analysis of some methods in this class caused by the limitation of standard energy arguments, and is characterized by the absence of Grönwall estimates, applicability to arbitrary approximation degrees, reduced regularity assumptions, and robustness with respect to the model parameters.

Galerkin-type time discretizations for parabolic and hyperbolic problems: stability and a priori error analysis

TL;DR

This work develops a unified variational framework to analyze Galerkin-type time discretizations for parabolic and hyperbolic problems, focusing on stability and a priori error estimates in . By employing nonstandard test functions and time-weighted techniques, it avoids Grönwall-type growth, accommodates arbitrary polynomial degrees, and remains robust with respect to parameter variations. The analysis encompasses DG and CG time discretizations across heat and wave equations, using a suite of time projections (Thomée, Aziz–Monk, Walkington) and a time reconstruction operator to derive stability and convergence results. It also addresses nonhomogeneous Dirichlet data, different wave formulations, and variants such as least-squares stabilization and exponential weights, providing a comprehensive, adaptable foundation for space–time discretizations of parabolic and hyperbolic problems. The results offer unconditional stability with optimal convergence under standard regularity and pave the way for nonlinear extensions and practical, efficient implementations in space–time finite element frameworks.

Abstract

We present a unified framework for the analysis of space-time methods based on Galerkin-type time discretizations for parabolic and hyperbolic problems. Crucially, the stability analysis relies on a suitable choice of test functions to establish the continuous dependence of the discrete solution on the data in norms, which is then used to derive a priori error estimates. This approach closes the gap in the analysis of some methods in this class caused by the limitation of standard energy arguments, and is characterized by the absence of Grönwall estimates, applicability to arbitrary approximation degrees, reduced regularity assumptions, and robustness with respect to the model parameters.
Paper Structure (53 sections, 44 theorems, 263 equations, 2 tables)

This paper contains 53 sections, 44 theorems, 263 equations, 2 tables.

Table of Contents

  1. Keywords.
  2. Mathematics Subject Classification.
  3. Introduction
  4. Notation.
  5. Outline.
  6. Model problems
  7. The heat equation
  8. The wave equation
  9. Preliminaries
  10. Mesh notation and discrete spaces
  11. Space and time meshes.
  12. Discrete spaces.
  13. Time jumps.
  14. A polynomial inverse estimate
  15. Auxiliary weight functions
  16. ...and 38 more sections

Key Result

Lemma 3.1

For any $q \in \mathbb{N}$ and any Banach space $(Z, \|\cdot\|_{Z})$, there exists a positive constant $C_{\mathsf{inv}}$ depending only on $q$ such that, for $n = 1, \ldots, N$, it holds

Theorems & Definitions (105)

  • Remark 2.1: Robust continuous dependence on the data
  • Remark 2.2: Additional regularity on $f$
  • Remark 2.3: Space--time methods
  • Remark 2.4: Well-posedness of \ref{['eq:second-order-weak-formulation-wave']}
  • Remark 2.5: Nonhomogeneous Dirichlet boundary conditions
  • Remark 2.6: Space--time methods
  • Lemma 3.1: Inverse estimate
  • Remark 3.2: $q$-dependence of $C_{\mathsf{inv}}$
  • Lemma 3.3: Identities involving $\varphi_n$
  • proof
  • ...and 95 more