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

摘要

We study the role of preconditioning strategies recently developedudfor coercive problems in connection with a two-step iterative method,udbased on the Hermitian skew-Hermitian splitting (HSS) of theudcoefficient matrix, proposed by Bai, Golub and Ng for the solutionudof nonsymmetric linear systems whose real part is coercive.udAs a model problem we consider Finite Differences (FD) matrix sequencesud${A_n(a,p)}_n$ discretizing the elliptic (convection-diffusion)udproblemud%udegin{equation}udlabel{eq:pde-abs}udleft{udegin{array}{l}ud%A_{a,p} u equivud-abla^T[a(x)abla u(x)]+sum_{j=1}^d {partialudover partial x_j}(p(x)u(x)) =f(x),quad quadudxin Omega, ud uddisplaystyleud{m Dirichlet BC},udend{array}udight.udend{equation}udwith $Omega$ being a plurirectangle of ${f R}^d$ with $a(x)$udbeing a uniformly positive function and $p(x)$ denoting theudReynolds function: here for plurirectangle we mean a connectedudunion of rectangles in $d$ dimensions with edges parallel to the axes.udMore precisely, in connection with preconditionedudHSS/GMRES like methods, we consider the preconditioning sequenceud${P_n(a)}_n$, $P_n(a):= D_n^{1/2}(a)A_n(1,0) D_n^{1/2}(a)$udwhere $ D_n(a)$ is the suitably scaled main diagonal ofud$A_n(a,0)$. If $a(x)$ is positive and regular enough, then theudpreconditioned sequence shows a strong clustering at unity so thatudthe sequence ${P_n(a)}_n$ turns out to be a superlinearudpreconditioning sequence for ${A_n(a,0)}_n$ where $A_n(a,0)$udrepresents a good approximation of ${m Re}(A_n(a,p))$ namely theudreal part of $A_n(a,p)$.ududThe computational interest is due to the fact that the preconditioned HSSudmethod has a convergence behavior depending on the spectral propertiesudof ${P_n^{-1}(a){m Re}(A_n(a,p))}_npproxud{P_n^{-1}(a)A_n(a,0)}_n$: therefore the solution of a linearudsystem with coefficient matrix $A_n(a,p)$ is reduced to computations involvinguddiagonals and to the use of fast Poisson solvers for ${A_n(1,0)}_n$.ududSome numerical experimentations confirm the optimality of the discussedudproposal and its superiority with respect to existing techniques.