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

TL;DR

This paper develops a unified stability and error analysis framework for Galerkin-type time discretizations applicable to both parabolic and hyperbolic problems, emphasizing robustness and reduced regularity requirements.

Contribution

It introduces a novel approach that avoids Grönwall estimates, broadening the applicability and stability analysis of space-time Galerkin methods for various PDEs.

Findings

Establishes stability without Grönwall estimates.

Provides a priori error bounds under minimal regularity.

Applicable to arbitrary approximation degrees.

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 Gr\"onwall estimates, applicability to arbitrary approximation degrees, reduced regularity assumptions, and robustness with respect to the model parameters.