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

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 ∆ ~ N + λ~ ∇(~ ∇ · ~ N) = ~ S with λ 6= −1. The source can extend in all the Euclidean space R, provided it decays at least as r−3 (scalar case) or r−4 (vectorial case). A multi-domain 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 evanescent, i.e. 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).