A complete error analysis is performed for the spectral-Galerkin
approximation of a model Helmholtz equation with high wave numbers.
The analysis presented in this paper does not rely on the explicit
knowledge of continuous/discrete Green's functions and does not
require any mesh condition to be satisfied. Furthermore, new error
estimates are also established for multi-dimensional radial and
spherical symmetric domains. Illustrative numerical results in
agreement with the theoretical analysis are presented.