In arbitrary dimension, we consider the semidiscrete elliptic operator -partial derivative(2)(t) + A(m), where A(m) is a finite-difference approximation of the operator -del(x)(Gamma(x)del(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 derivative(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.