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 -\delta^T [a(x)\delta u(x)] + Sigma_(j=1)^(d) \partial /\partial x(j) (p(x) u(x)) = f (x), x. is an element of Omega, Dirichlet BC, with Omega 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) approximate to {P_n^(-1)_n(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.

Bertaccini, D., Golub, G., Serra Capizzano, S., TABLINO POSSIO, C. (2005). Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation. NUMERISCHE MATHEMATIK, 99(3), 441-484 [10.1007/s00211-004-0574-1].

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

TABLINO POSSIO, CRISTINA
2005

Abstract

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 -\delta^T [a(x)\delta u(x)] + Sigma_(j=1)^(d) \partial /\partial x(j) (p(x) u(x)) = f (x), x. is an element of Omega, Dirichlet BC, with Omega 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) approximate to {P_n^(-1)_n(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.
Articolo in rivista - Articolo scientifico
Matrix sequences, clustering, preconditioning, splitting iterations
English
2005
99
3
441
484
none
Bertaccini, D., Golub, G., Serra Capizzano, S., TABLINO POSSIO, C. (2005). Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation. NUMERISCHE MATHEMATIK, 99(3), 441-484 [10.1007/s00211-004-0574-1].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/216
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