We study the role of preconditioning strategies recently developed for coercive problems in connection with a two-step iterative method, based on the Hermitian skew-Hermitian splitting (HSS) of the coefficient matrix, proposed by Bai, Golub and Ng for the solution of nonsymmetric linear systems whose real part is coercive. As a model problem we consider Finite Differences (FD) matrix sequences {A n (a,p)} n discretizing the elliptic (convection-diffusion) problem with Ω being a plurirectangle of R d with a(x) being a uniformly positive function and p(x) denoting the Reynolds function: here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes. More precisely, in connection with preconditioned HSS/GMRES like methods, we consider the preconditioning sequence {P n (a)} n , P n (a):= D n 1/2(a)A n (1,0) D n 1/2(a) where D n (a) is the suitably scaled main diagonal of A n (a,0). If a(x) is positive and regular enough, then the preconditioned sequence shows a strong clustering at unity so that the sequence {P n (a)} n turns out to be a superlinear preconditioning sequence for {A n (a,0)} n where A n (a,0) represents a good approximation of Re(A n (a,p)) namely the real part of A n (a,p). The computational interest is due to the fact that the preconditioned HSS method has a convergence behavior depending on the spectral properties of {P n -1(a)Re(A n (a,p))} n ≈ {P n -1(a)A n (a,0)} n : therefore the solution of a linear system with coefficient matrix A n (a,p) is reduced to computations involving diagonals and to the use of fast Poisson solvers for {A n (1,0)} n . Some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques.