Discrete approximation of charge density Given the electric potential $\Phi(r)$ and the Poisson's equation: 
$$ \nabla^2 \Phi(r) = - 4\pi \rho(r)$$
Consider the 2-dimensional case and let's say that I want to discretize this using a square grid where at the center of the grid I have a point charge with charge $q$. What would be the expression (= a discrete approximation) for $\rho(r)$?
 A: Since you have a point charge there is nothing much you can do but to treat it as a delta function in the fashion of $q\delta(r-r_0)$. So in any other grid point you don't have any charge.
A: I) It is true that the electric charge density $\rho(\vec{r})$ of $N$ idealized point charges in $d$ dimensions is precisely given by $N$ $d$-dimensional delta functions.
II) However, here we assume that OP is really considering a macroscopic system that are well described by the classical continuum theory (and that the underlying (sub)atomic/molecular constituents are irrelevant). In the context of solving differential equations numerically, such as, e.g. Poisson equation, what is usually meant by a discretization is to replace a continuous space(time) region with a dense enough lattice on which the PDE can be numerically solved. To trust the discretization, the $\rho$ values of the lattice points should preferably be representative for the entire continuous charge density $\rho(\vec{r})$, and not delta function peaks.  
