  We consider the numerical approximation of linear damped wave systems by
Galerkin approximations in space and appropriate time-stepping schemes. Based
on a dissipation estimate for a modified energy, we prove exponential decay of
the physical energy on the continuous level provided that the damping is
effective everywhere in the domain. The methods of proof allow us to analyze
also a class of Galerkin approximations based on a mixed variational
formulation of the problem. Uniform exponential stability can be guaranteed for
these approximations under a general compatibility condition on the
discretization spaces. As a particular example, we discuss the discretization
by mixed finite element methods for which we obtain convergence and uniform
error estimates under minimal regularity assumptions. We also prove
unconditional and uniform exponential stability for the time discretization by
certain one-step methods. The validity of the theoretical results as well as
the necessity of some of the conditions required for our analysis are
demonstrated in numerical tests.
