  We study damped wave propagation problems phrased as abstract evolution
equations in Hilbert spaces. Under some general assumptions, including a
natural compatibility condition for initial values, we establish exponential
decay estimates for all mild solutions using the language and tools of Hilbert
complexes. This framework turns out strong enough to conduct our analysis but
also general enough to include a number of interesting examples. Some of these
are briefly discussed. By a slight modification of the main arguments, we also
obtain corresponding decay results for numerical approximations obtained by
compatible discretization strategies.
