In arbitrary dimension, we consider the semi-discrete elliptic operator $- \d_t^2 + \Am$, where $\Am$ is a finite difference approximation of the operator $-\nabla_x (\Gamma(x) \nabla_x)$. For this operator we derive a global Carleman estimate, in which the usual large parameter is connected to the discretization step-size. We address discretizations on some families of smoothly varying meshes. We present consequences of this estimate such as a partial spectral inequality of the form of that proven by G.~Lebeau and L.~Robbiano for $A^m$ and a null controllability result for the parabolic operator $\partial_t + A^m$, for the lower part of the spectrum of $A^m$. With the control function that we construct (whose norm is uniformly bounded) we prove that the $L^2$-norm of the final state converges to zero exponentially, as the step-size of the discretization goes to zero. A relaxed observability estimate is then deduced.