Range-separated tensor decomposition of the discretized Dirac delta and elliptic operator inverse

Abstract

In this paper, we introduce the operator dependent range-separated (RS) tensor approximation of the discretized Dirac delta function (distribution) in Rd. 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.