Title: Non-singular Green's functions for the unbounded Poisson equation in one, two and three dimensions,
Abstract: In this paper, we derive the non-singular Green's functions for the unbounded
Poisson equation in two and three dimensions using a spectral approach to
regularize the homogeneous equation. The resulting 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 regularized Green's function, as this is useful in applications where
the Poisson equation represents potential functions of a vector field.