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In this article we consider regularizations of the Dirac delta distributionwith applications to prototypical elliptic and hyperbolic partial differentialequations (PDEs). We study the convergence of a sequence of distributions$\mathcal{S}_H$ to a singular term $\mathcal{S}$ as a parameter $H$ (associatedwith the {support size} of $\mathcal{S}_H$) shrinks to zero. We characterizethis convergence in both the weak-$\ast$ topology of distributions, as well asin a weighted Sobolev norm. These notions motivate a framework for constructingregularizations of the delta distribution that includes a large class ofexisting methods in the literature. This framework allows differentregularizations to be compared. The convergence of solutions of PDEs with theseregularized source terms is then studied in various topologies such aspointwise convergence on a deleted neighborhood and weighted Sobolev norms. Wealso examine the lack of symmetry in tensor product regularizations and effectsof dissipative error in hyperbolic problems.

On regularizations of the delta distribution