  The aim of this paper is twofold. First, we prove $L^p$ estimates for a
regularized Green's function in three dimensions. We then establish new
estimates for the discrete Green's function and obtain some positivity results.
In particular, we prove that the discrete Green's functions with singularity in
the interior of the domain cannot be bounded uniformly with respect of the mesh
parameter $h$. Actually, we show that at the singularity the discrete Green's
function is of order $h^{-1}$, which is consistent with the behavior of the
continuous Green's function. In addition, we also show that the discrete
Green's function is positive and decays exponentially away from the
singularity. We also provide numerically persistent negative values of the
discrete Green's function on Delaunay meshes which then implies a discrete
Harnack inequality cannot be established for unstructured finite element
discretizations.
