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

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 $\lambda \ne -1$. The source can extend in all the Euclidean space ${\mathbf{R}}^3$, provided it decays at least as $r^{-3}$. A multi-domain approach is used, along with spherical coordinates $(r,\theta ,\varphi )$. In each domain, Chebyshev polynomials (in $r$ or $1/r$) and spherical harmonics (in $\theta$ and $\varphi$) 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 $l\ge k-3$ (scalar case) or $l\ge k-5$ (vectorial case).

Minor revisions. 31 pages, 13 figures, accepted for publication J. Comp. Phys

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• Quasiequilibrium sequences of synchronized and irrotational binary neutron stars in general relativity. I. Method and tests
• Einstein Constraints on Asymptotically Euclidean Manifolds

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