  In this article we consider regularizations of the Dirac delta distribution
with applications to prototypical elliptic and hyperbolic partial differential
equations (PDEs). We study the convergence of a sequence of distributions
$\mathcal{S}_H$ to a singular term $\mathcal{S}$ as a parameter $H$ (associated
with the {support size} of $\mathcal{S}_H$) shrinks to zero. We characterize
this convergence in both the weak-$\ast$ topology of distributions, as well as
in a weighted Sobolev norm. These notions motivate a framework for constructing
regularizations of the delta distribution that includes a large class of
existing methods in the literature. This framework allows different
regularizations to be compared. The convergence of solutions of PDEs with these
regularized source terms is then studied in various topologies such as
pointwise convergence on a deleted neighborhood and weighted Sobolev norms. We
also examine the lack of symmetry in tensor product regularizations and effects
of dissipative error in hyperbolic problems.
