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

Galerkin-type time discretizations are variational time-stepping schemes in which the test and trial spaces consist of piecewise polynomials in time. This paper reviews the schemes from this class available in the literature for parabolic and hyperbolic problems. In particular, we focus on the historical difficulties in their stability and \emph{a priori} error analysis that have limited their applicability, especially for wave propagation problems. We discuss how the limitations of standard energy arguments are behind these theoretical gaps, and present a unified framework that yields stability bounds and \emph{a priori} error estimates in $L^{\infty}(0, T; \mathcal{X})$ norms. Crucially, the stability analysis relies on a suitable choice of nonstandard test functions. This approach is characterized by the absence of Grönwall estimates, applicability to arbitrary approximation degrees, reduced regularity assumptions, and robustness with respect to model parameters.