A variational approach to the analysis of the continuous space-time FEM for the wave equation

TL;DR

This paper provides a comprehensive stability, convergence, and error analysis of a space-time finite element method for the wave equation, including new a posteriori error estimates validated by numerical experiments.

Contribution

It introduces a stability analysis without mesh restrictions, derives quasi-optimal error estimates, and develops a constant-free a posteriori error estimator for the wave equation.

Findings

Proves stability without mesh size restrictions.

Establishes quasi-optimal convergence rates.

Develops a reliable a posteriori error estimate.

Abstract

We present a stability and convergence analysis of the space-time continuous finite element method for the Hamiltonian formulation of the wave equation. More precisely, we prove a continuous dependence of the discrete solution on the data in a $C^0([0, T]; X)$-type energy norm, which does not require any restriction on the meshsize or the time steps. Such stability estimates are then used to derive a priori error estimates with quasi-optimal convergence rates, where a suitable treatment of possible nonhomogeneous Dirichlet boundary conditions is pivotal to avoid loss of accuracy. Moreover, based on the properties of a postprocessed approximation, we derive a constant-free, reliable a posteriori error estimate in the $C^0([0, T]; L^2(\Omega))$ norm for the semidiscrete-in-time formulation. Several numerical experiments are presented to validate our theoretical findings.