Uniform exponential stability of Galerkin approximations for damped wave systems

TL;DR

This paper proves uniform exponential stability for Galerkin approximations of damped wave systems, ensuring energy decay and convergence under minimal regularity and compatibility conditions, supported by numerical tests.

Contribution

It establishes uniform exponential stability for Galerkin and mixed finite element approximations of damped wave systems, including time discretizations, under broad conditions.

Findings

Exponential decay of energy in Galerkin approximations

Convergence and error estimates for mixed finite element methods

Unconditional stability of certain time-stepping schemes

Abstract

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.