Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation

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 {An (a, p)}n discretizing the elliptic (convection-diffusion) problem (Equation Presented) (1){-∇ (D • Dirichlet BC, with Ω being a plurirectangle of Rd 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 {Pn(a)}n, Pn(a) := Dn1/2(a)An(1, 0)Dn1/2 (a) where D n(a) is the suitably scaled main diagonal of An(a, 0). If a(x) is positive and regular enough, then the preconditioned sequence shows a strong clustering at unity so that the sequence {Pn(a)}n turns out to be a superlinear preconditioning sequence for {An(a, 0)}n where An(a, 0) represents a good approximation of Re(An(a, p)) namely the real part of An(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(An(a, p))}n ≈ {P n-1(a)An(a, 0)}n: therefore the solution of a linear system with coefficient matrix An(a, p) is reduced to computations involving diagonals and to the use of fast Poisson solvers for {An(1, 0)}n. Some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques.