  High resolution dark matter only simulations provide a realistic and fully
general means to study the theoretical predictions of cosmological structure
formation models for gravitational lensing. Due to the finite number of
particles, the density field only becomes smooth on scales beyond a few times
the local mean interparticle separation. This introduces noise on the
gravitational lensing properties such as the surface mass density, the
deflection angles and the magnification. At some small-scale mass limit, the
noise due to the discreteness of the N-body simulation becomes comparable to
the effects of physical substructures. We present analytic expressions to
quantify the Poisson noise and study its scaling with the particle number of
the simulation and the Lagrangian smoothing size. We use the Phoenix set of
simulations, currently the largest available dark matter simulations of
clusters to study the effect of limited numerical resolution and the
gravitational strong lensing effects of substructure. We quantify the smallest
resolved substructure, in the sense that the effect of the substructure on any
strong lensing property is significant compared to the noise, and we find that
the result is roughly independent of the strong lensing property. A simple
scaling relates the smallest resolved substructures in a simulation with the
resolution of the N-body simulation.
