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On regularizations of the Dirac delta distribution. (English) Zbl 1349.35008
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.

MSC:
35A35 Theoretical approximation in context of PDEs
35A08 Fundamental solutions to PDEs
Software:
Chebfun; Matlab
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