Hosseini, Bamdad; Nigam, Nilima; Stockie, John M. On regularizations of the Dirac delta distribution. (English) Zbl 1349.35008 J. Comput. Phys. 305, 423-447 (2016). Summary: 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-topology of distributions and 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. Cited in 18 Documents MSC: 35A35 Theoretical approximation in context of PDEs 35A08 Fundamental solutions to PDEs Keywords:Dirac delta function; singular source term; discrete delta function; approximation theory; weighted Sobolev spaces Software:Matlab; Chebfun PDFBibTeX XMLCite \textit{B. Hosseini} et al., J. Comput. Phys. 305, 423--447 (2016; Zbl 1349.35008) Full Text: DOI arXiv References: [1] Agnelli, J. P.; Garau, E. M.; Morin, P., A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces, ESAIM: Math. Model. Numer. Anal., 48, 1557-1581 (2014) · Zbl 1305.35026 [2] Aheizer, N. 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