  We develop a method for evaluation of A. Einstein's strength of systems of
partial differential and difference equations based on the computation of
Hilbert-type dimension polynomials of the associated differential and
difference field extensions. Also we present algorithms for such computations,
which are based on the Gr\"obner basis method adjusted for the modules over
rings of differential, difference and inversive difference operators. The
developed technique is applied to some fundamental systems of PDEs of
mathematical physics such as the diffusion equation, Maxwell equations and
equations for an electromagnetic field given by its potential. In each of these
cases we determine the strength of the original system of PDEs and the strength
of the corresponding systems of partial difference equations obtained by
forward and symmetric difference schemes. In particular, we obtain a method for
comparing two difference schemes from the point of view of their strength.
