This work is devoted to the
analysis of a viscous finite-difference space semi-discretization
of a locally damped wave equation in a regular 2-D domain. The
damping term is supported in a suitable subset of the domain, so
that the energy of solutions of the damped continuous wave
equation decays exponentially to zero as time goes to infinity.
Using discrete multiplier techniques, we prove that adding a
suitable vanishing numerical viscosity term leads to a uniform
(with respect to the mesh size) exponential decay of the energy
for the solutions of the numerical scheme. The numerical viscosity
term damps out the high frequency numerical spurious oscillations
while the convergence of the scheme towards the original damped
wave equation is kept, which guarantees that the low frequencies
are damped correctly. Numerical experiments are presented and
confirm these theoretical results. These results extend those by
Tcheugoué-Tébou and Zuazua [Numer. Math.95, 563–598 (2003)] where the 1-D case
was addressed as well the square domain in 2-D. The methods and
results in this paper extend to smooth domains in any space
dimension.