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

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 a stability result is then used to derive~\emph{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~\emph{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.