Uniform exponential stability of Galerkin approximations for damped wave systems
Submission Status: | under review |
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- 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 stabil-
ity 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 esti-
mates under minimal regularity assumptions. We also prove unconditional and uniform
exponential stability for the time discretization by certain one-step methods. The valid-
ity of the theoretical results as well as the necessity of some of the conditions required
for our analysis are demonstrated in numerical tests