  A two-step preconditioned iterative method based on the
Hermitian/Skew-Hermitian splitting is applied to the solution of nonsymmetric
linear systems arising from the Finite Element approximation of
convection-diffusion equations. The theoretical spectral analysis focuses on
the case of matrix sequences related to FE approximations on uniform structured
meshes, by referring to spectral tools derived from Toeplitz theory. In such a
setting, if the problem is coercive, and the diffusive and convective
coefficients are regular enough, then the proposed preconditioned matrix
sequence shows a strong clustering at unity, i.e., a superlinear
preconditioning sequence is obtained. Under the same assumptions, the
optimality of the PHSS method is proved and some numerical experiments confirm
the theoretical results. Tests on unstructured meshes are also presented,
showing the some convergence behavior.
