  It is by now well-known that one can recover a potential in the wave equation
from the knowledge of the initial waves, the boundary data and the flux on a
part of the boundary satisfying the Gamma-conditions of J.-L. Lions. We are
interested in proving that trying to fit the discrete fluxes, given by discrete
approximations of the wave equation, with the continuous one, one recovers, at
the limit, the potential of the continuous model. In order to do that, we shall
develop a Lax-type argument, usually used for convergence results of numerical
schemes, which states that consistency and uniform stability imply convergence.
In our case, the most difficult part of the analysis is the one corresponding
to the uniform stability, that we shall prove using new uniform discrete
Carleman estimates, where uniform means with respect to the discretization
parameter. We shall then deduce a convergence result for the discrete inverse
problems. Our analysis will be restricted to the 1-d case for space
semi-discrete wave equations discretized on a uniform mesh using a finite
differences approach.
