The superconsistent collocation method, which is based on a
collocation grid different from the one used to represent the
solution, has proven to be very accurate in the resolution of
various functional equations. Excellent results can be also
obtained for what concerns preconditioning. Some analysis and
numerous experiments, regarding the use of finite-differences
preconditioners, for matrices arising from pseudospectral
approximations of advection-diffusion boundary value problems, are
presented and discussed, both in the case of Legendre and
Chebyshev representation nodes.