Abstract
Abstract In this paper, we introduce the operator dependent range-separated (RS) tensor approximation of the discretized Dirac delta function (distribution) in R d . It is constructed by application of the elliptic operator to the RS tensor representation of the associated Green kernel discretized on the d-dimensional Cartesian grid. The proposed local-global decomposition of the Dirac delta can be applied for solving the potential equations in a non-homogeneous medium when the density in the right-hand side is given by a large sum of pointwise singular charges. As an example of applications, we describe the regularization scheme for solving the Poisson-Boltzmann equation that models the electrostatics in bio-molecules. We show how the idea of the operator dependent RS tensor decomposition of the Dirac delta can be generalized to the closely related problem on range-separated tensor representation of the elliptic resolvent. This approach paves the way for application of tensor numerical methods to elliptic problems with non-regular data. Numerical tests confirm the expected localization properties of the RS tensor approximation of the Dirac delta represented on a tensor grid in 3D.
Published Version
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