# Non-singular Green's functions for the unbounded Poisson equation in one, two and three dimensions

## TL;DR

This paper derives non-singular Green's functions for the unbounded Poisson equation in 1D, 2D, and 3D using a Fourier domain cutoff, which are useful for applications with a minimum length scale.

## Contribution

It introduces a novel method to obtain non-singular Green's functions for the Poisson equation across multiple dimensions using a Fourier cutoff approach.

## Key findings

- Derived explicit non-singular Green's functions for 1D, 2D, and 3D.
- Provided the gradient of the non-singular Green's functions for vector field applications.
- Applicable to problems with a minimum resolved length scale, such as mesh-based simulations.

## Abstract

This paper is a revised version of the original paper of same title--published in Applied Mathematics Letters 89--containing some corrections and clarifications to the original text. We derive non-singular Green's functions for the unbounded Poisson equation in one, two and three dimensions, using a cut-off function in the Fourier domain to impose a smallest length scale when deriving the Green's function. The resulting non-singular Green's functions are relevant to applications which are restricted to a minimum resolved length scale (e.g. a mesh size h) and thus cannot handle the singular Green's function of the continuous Poisson equation. We furthermore derive the gradient vector of the non-singular Green's function, as this is useful in applications where the Poisson equation represents potential functions of a vector field.