On regularizations of the Dirac delta distribution

In this article we consider regularizations of the Dirac delta distribution with applications to prototypical elliptic and hyperbolic partial di↵erential equations (PDEs). We study the convergence of a sequence of distributions SH to a singular term S as a parameter H (associated with the support size of SH) 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 di↵erent 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 e↵ects of dissipative error in hyperbolic problems.

• Type: Journal
• Year: 2016
• Where: JCPHY
• Authors: Bamdad Hosseini, Nilima Nigam, John M. Stockie
• Keywords: JCPHY 2016