On regularizations of the delta distribution

TL;DR

This paper investigates how regularizations of the Dirac delta distribution converge in different mathematical topologies and introduces a framework to compare various regularization methods, with applications to elliptic and hyperbolic PDEs.

Contribution

It characterizes the convergence of delta regularizations in multiple topologies and develops a comprehensive framework for comparing different regularization techniques.

Findings

Convergence in weak-* topology and weighted Sobolev norms established.

Framework includes many existing regularization methods.

Analysis of solution convergence for PDEs with regularized sources.

Abstract

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.