A Multidomain Spectral Method for Scalar and Vectorial Poisson Equations with Noncompact 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 ΔN→+λ∇→(∇→ċN→=S→ with λ≠−1. The source can extend in all the Euclidean space R3, provided it decays at least as r−3. A multidomain approach is used, along with spherical coordinates (r, θ, φ). In each domain, Chebyshev polynomials (in r or 1/r) and spherical harmonics (in θ and φ) 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 evanescents; i.e., it decreases as exp(−N), if the source does not contain any spherical harmonics of index l≥k−3 (scalar case) or l≥k−5 (vectorial case).