On the exponential stability of uniformly damped wave equations and their structure-preserving discretization

Abstract

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 mild solutions using Lyapunov-type arguments. For the formulation of our results, we use the language of Hilbert complexes which provides all the tools required for our analysis and is also general enough to cover a number of interesting examples. Some of these are briefly discussed in the course of the manuscript. The functional analytic setting and the main arguments in our proofs are chosen such that they transfer almost verbatim to the discrete setting. We thus obtain corresponding decay results for numerical approximations of a variety of problems obtained by compatible discretization strategies which can be seen as our main contribution.