A multi-domain spectral method for scalar and vectorial Poisson equations with non-compact sources

TL;DR

This paper introduces a spectral multi-domain method using spherical coordinates and harmonic expansions to solve scalar and vectorial Poisson equations with non-compact sources in three-dimensional space, achieving high accuracy and exponential convergence.

Contribution

It develops a novel multi-domain spectral approach for elliptic equations with non-compact sources, extending the applicability in general relativity computations.

Findings

Error decreases as N^{-2(k-2)} for sources decaying as r^{-k}.

Error is exponential if source lacks high-order spherical harmonics.

Method effectively handles sources extending over all of R^3.

Abstract

We present a spectral method for solving elliptic equations which arise in general relativity, namely three-dimensional scalar Poisson equations, as well as generalized vectorial Poisson equations of the type $\Delta \vec{N} + \lambda \vec{\nabla}(\vec{\nabla}\cdot \vec{N}) = \vec{S}$ with $\lambda \not= -1$. The source can extend in all the Euclidean space ${\bf R}^3$, provided it decays at least as $r^{-3}$. A multi-domain approach is used, along with spherical coordinates $(r,\theta,\phi)$. In each domain, Chebyshev polynomials (in $r$ or $1/r$) and spherical harmonics (in $\theta$ and $\phi$) expansions are used. If the source decays as $r^{-k}$ the error of the numerical solution is shown to decrease at least as $N^{-2(k-2)}$, where $N$ is the number of Chebyshev coefficients. The error is even evanescent, i.e. decreases as $\exp(-N)$, if the source does not contain any spherical harmonics of index $l\geq k -3$ (scalar case) or $l\geq k-5$ (vectorial case).