A regularization approach for solving Poisson's equation with singular charge sources and diffuse interfaces

TL;DR

This paper introduces a novel regularization method for solving Poisson's equation with singular charge sources and diffuse interfaces, enabling accurate numerical solutions where analytical methods are unavailable.

Contribution

It presents the first regularization approach that analytically represents Coulomb potential using Green's functions for complex diffuse interfaces.

Findings

Validated on spherical domain with diffuse interface

Compared with semi-analytical quasi-harmonic method

Demonstrated numerical effectiveness of the regularization

Abstract

Singular charge sources in terms of Dirac delta functions present a well-known numerical challenge for solving Poisson's equation. For a sharp interface between inhomogeneous media, singular charges could be analytically treated by fundamental solutions or regularization methods. However, no analytical treatment is known in the literature in case of a diffuse interface of complex shape. This letter reports the first such regularization method that represents the Coulomb potential component analytically by Green's functions to account for singular charges. The other component, i.e., the reaction field potential, then satisfies a regularized Poisson equation with a smooth source and the original elliptic operator. The regularized equation can then be simply solved by any numerical method. For a spherical domain with diffuse interface, the proposed regularization method is numerically validated and compared with a semi-analytical quasi-harmonic method.