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

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.


Introduction
Green's functions are the preferred methods for solving linear differential equations in an unbounded domain, i.e. with free-space boundary conditions. The Green's function represents a homogeneous solution which is derived analytically, and then used to obtain the particular solution by a convolution with the right-hand-side of the Poisson equation.
The analytical Green's function of a continuous smooth field is singular at its origin. Applied in discretized numerical calculations, the singularity of the Green's function evidently causes a number of difficulties. In order to amend this, smoothing regularization techniques have been applied (e.g. [1,2,3]) which introduce a continuous and smooth field distribution around the discrete points and thus avoid the singularity of the Green's function. However, most regularization methods that have been applied are based on functions that only conserve a finite number of field moments, and are thus only accurate up to a finite order.
This work presents Green's functions for the unbounded Poisson equation using a spectral cut-off regularization to avoid the aforementioned singularity in the real space representation. Using a spectral regularization, we impose a cut-off scale at a given discretization length, after which we derive the Green's function analytically. As no explicit smoothing function is applied, the proposed Green's function obtains an optimal accuracy. Furthermore, the gradient vector of the Green's function is also derived. This function is useful in applications where the Poisson equation represents a vector or scalar potential function of a vector field as it enables a direct solution of the vector field.
The proposed Green's functions may be used directly in efficient computational methods such as the fast multipole method [4,5]. Using regularized Green's functions in a mesh based Poisson solver to improve the convergence order has recently been presented in [6,7,8]. An implementation of the Green's function proposed in this work has been shown to obtain an optimal accuracy at machine precision of the mesh based solver, provided that the right-hand-side function B(x) is sufficiently resolved by the mesh discretization [7,8].

Methodology
The Poisson equation in an unbounded domain is formally stated as: Here B(x) is a known bounded field and A(x) is the desired solution field. In many applications such as astrophysics, electrodynamics and vortex dynamics, the vector field to be solved v(x) is described by potential functions using the Helmholtz decomposition: The fundamental operations describing the conservation of the flux and circulation of the vector field v is (2) it follows that these may be expressed by the potential functions as: Both potentials may thus be obtained by solving a Poisson equation. By utilizing the linearity of the Poisson equation, and considering the equations by their Fourier transform, we obtain the algebraic equations: Here · denotes a Fourier transformed variable, ι the imaginary unit, k is the angular wave-number of the Fourier space, and k = |k| its modulus. G(k) is the Green's function in Fourier space which represents the homogeneous solution to the Poisson equation. We may furthermore obtain the vector field of Eq. (2) directly by incorporating the gradient operator into the Green's function as K(x) = ∇G(x) after which: As the Green's function in Fourier space (Eq. (4)) is in fact unbounded, an infinite domain in the Fourier space is needed in order to obtain the correct solution. However, for a discrete approximation, such as is used in numerical simulations, the discretization of the domain in the physical space results in a bounded domain in the Fourier space, represented by the set of wave-numbers k i = {−k s /2, ... , k s /2}. Here k s = 2π/h is the angular sampling wave-number corresponding to the discretization length h and k s /2 is the highest resolved angular wave-number due to the Nyquist-Shannon sampling theorem. The exact Green's function of Eq. (4) thus cannot not be represented in a discretized space, and using it will introduce errors in the calculation with a limited convergence rate of O(h 2 ) [9].
In order to amend this and derive a Green's function which is bounded by a maximum wavenumber, we regularize the Green's function using a radial cut-off function in Fourier space: This leads to a regularized Green's function and a corresponding homogeneous equation of: Using the radial cut-off function of Eq. (6), it is seen that the Green's function of the non-regularized equation (Eq. (4)) is unchanged in the range of wave-numbers 0 < k ≤ k s /2.

Results
By utilizing the radial symmetry of the Green's function, the regularization function in real space ζ(ρ) may be obtained by considering the unbounded Fourier transform in d-dimensional spherical/polar coordinates which is given by: where J ν denotes the Bessel function of the first kind and of order ν. Using an integral identity of the Bessel functions [10], we may satisfy Eq. (6) by using: Here we have shifted to normalized coordinates s = σk and ρ = r/σ where σ is determined by fulfilling s = 1 for k = k s /2: Considering Eqs. (8) and (9), we find that the regularization function in real space is given by: For the case of d = 2 this is identical to that presented in the references [1,3].
We may now obtain the corresponding Green's function by radial integration of the homogeneous equation (Eq. (7)) in the unbounded real space. For the two-dimensional case (d = 2), we obtain: where C is an arbitrary integration constant, which we may use to define a reference value. In this work we choose C to obtain the asymptotic behavior of the singular Greens function for large ρ whatever the value of σ: Here L is a chosen global reference length.
To evaluate the Green's function, Eq. (12) may be rewritten in terms of a series expansion: where γ = 0.5772156649 is Euler's constant. Unfortunately, this Green's function is computationally very expensive to evaluate as it contains large factorials to ensure an accurate estimate. The gradient vector of the Green's function, which is used to calculate the vector field directly in Eq. (5), is however much easier to evaluate and is given by: Here K(ρ) ≥ 0 and K(0) = 0 and e r = x/|x| is the radial normal vector. This result is identical to that presented in the references [1,3].
Using the same approach for the three-dimensional domain (d = 3), we obtain the Green's function: where Si(ρ) = ρ 0 sin(q) q dq is the sine integral function.
The gradient vector of the Green's function is then obtained as: where we also have that K(ρ) ≥ 0 and K(0) = 0. The derived non-singular Green's functions are compared to the singular Green's functions in Fig. 1. An implementation of the non-singular Green's functions into the Poisson solver presented in references [6,7,8], has shown to obtain an optimal accuracy at machine precision, given that the right-hand-side function B(x) is sufficiently resolved by the mesh discretization.

Conclusion
We have derived the non-singular Green's functions for the unbounded Poisson equation, in two and in three dimensions, using a spectral approach. By applying a radial cut-off function in Fourier space, the regularized Green's function is derived analytically in unbounded domains subject to a minimum resolved length scale (e.g. a mesh size h). As a sharp cut-off function is used, smoothing errors are avoided and the obtained regularized Green's function achieves an optimal accuracy for discretized fields.
The gradient vector of the Green's function was furthermore derived. The obtained Green's function for the two-dimensional case corresponds to that presented in references [1,3]. In the present work, we formally showed the spectral derivation of this function, and we extended it to three-dimensions.