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 $\{A_n(a,p)\}_n$ discretizing the elliptic (convection-diffusion) problem % \begin{equation} \label{eq:pde-abs} \left\{ \begin{array}{l} %A_{a,p} u \equiv -\nabla^T[a(x)\nabla u(x)]+\sum_{j=1}^d {\partial \over \partial x_j}(p(x)u(x)) =f(x),\quad \quad x\in \Omega, \\ \ \\ \displaystyle {\rm Dirichlet\ BC}, \end{array} \right. \end{equation} with $\Omega$ being a plurirectangle of ${\bf R}^d$ 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 ${\rm Re}(A_n(a,p))$ 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 $\{P_n^{-1}(a){\rm Re}(A_n(a,p))\}_n\approx \{P_n^{-1}(a)A_n(a,0)\}_n$: 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.