On regularizations of the delta distribution
Bamdad Hosseini, Nilima Nigam, John M. Stockie

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 to a singular term as a parameter (associated with the {support size} of ) shrinks to zero. We characterize this convergence in both the weak- 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…
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