NUMERICAL ANALYSIS FOR A LOCALLY DAMPED WAVE EQUATION

We consider a semi-discrete finite element formulation with artificial viscosity for the numerical approximation of a problem that models the damped vibrations of a string with fixed ends. The damping coefficient depends on the spatial variable and is effective only in a sub-interval of the domain. For this scheme, the energy of semi-discrete solutions decays exponentially and uniformly with respect to the mesh parameter to zero. We also introduce an implicit in time discretization. Error estimates for the semidiscrete and fully discrete schemes in the energy norm are provided and numerical experiments performed.