On regularizations of the Dirac delta distribution

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 S H to a singular term S as a parameter H (associated with the support size of S H ) 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 Sobolev norms. We also examine the lack of symmetry in tensor product regularizations and effects of dissipative error in hyperbolic problems.

Discretization of multipole sources in a finite difference setting for wave propagation problems

“…Theorem 1 and moment conditions given in equation 11 are extensions of what is presented in Hosseini et al (2016) for |s| = 0. Given equation 13, we refer to q as the singular source approximation order and η as being a q-order approximation of D s δ(x − x * ).…”

“…We use the qualifier "continuum" to differentiate at times between the discrete moment conditions. Recent work by Hosseini et al (2016) addresses the mode of convergence of f → f subject to regularized source terms satisfying the continuum moment conditions, mainly convergence in a weak- * topology (distribution sense) and in a weighted Sobelev norm. Both Tornberg and Engquist (2003) and Hosseini et al (2016) argue that f → f implies u → u as → 0, point-wise away from the source location, in particular for elliptic operators L. This argument hinges on the integral representation of elliptic operators and the smoothness of their kernel (i.e., Green's functions) away from source location.…”

Multiscale modeling of vascularized tissues via nonmatching immersed methods

“…For a discussion on the properties of possible Dirac delta approximations to use, we refer the reader to the excellent review paper [24]. The above formula can be straightforwardly generalized to the case of N vessels, of radii a i , i = 1, .…”

“…In the original Immersed Boundary Method [30] the singular source terms are formally written in terms of the Dirac delta distribution, and their discretization follows two possible routes: i) the Dirac delta distribution is approximated through a smooth function, or ii) the variational definition of the Dirac distribution is used directly in the Finite Element formulation of the problem. For finite difference schemes, the first solution is the only viable option, even though the use of smooth kernels may excessively smear the singularities, leading to large errors in the approximation [24]. In the context of finite elements, both solutions are possible.…”

A New Family of Regularized Kernels for the Harmonic Oscillator

“…Although the blob approximations converge to the unregularized system as → 0, blobs with small generate large variations in ∂G ∂r , leading to severe time-step restrictions when resolving the evolution dynamics. The quality of different delta distribution regularizations has recently been analyzed in a functional analysis setting [12]. Alternatively, one can (2) is integrated with a fourth-order symplectic integrator with ∆t = 3.91 × 10 −3 .…”