Superlinear Preconditioners for Finite Differences Linear Systems

Serra‐Capizzano, Stefano; Possio, Cristina Tablino

doi:10.1137/s0895479802416058

Cited by 7 publications

(23 citation statements)

References 17 publications

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“…Concerning the case of high order Finite Differences discretizations and Finite Elements approximation, we recall that in [30,34,35,33] we derived asymptotic expansions concerning the preconditioned matrices P −1 n A n in terms of related Toeplitz structures. Moreover, it was proved that the sequence {P −1 n A n } n is clustered at unity and is spectrally bounded if a(x) is regular enough and positive.…”

Section: Higher Order Fd Discretizations and Fem Methodsmentioning

confidence: 99%

“…Moreover, T N is the Toeplitz matrix of size N generated by 2 − 2 cos(z 1 ), i.e., the usual one dimensional discrete Laplacian, S N is h times the Toeplitz matrix of size N generated by pi sin(z 1 ), and the global dimension n of the linear system is given by N 3 . Therefore, the Hermitian part of A n (Q) is exactly the discretization of the diffusion terms, i.e., If we consider the same discretization scheme over the domain then, following the results in [35], there exists a matrix such that (16) and therefore…”

Section: The Case Of a = 1 And Constant P(x)mentioning

confidence: 99%

“…If we consider the same discretization scheme over the domain then, following the results in [35], there exists a matrix depending only on and Q (the same as in Subsection 3.2) such that…”

Section: Re(a N (A Q)) = N (A Q)mentioning

confidence: 99%

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

et al. 2004

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Summary.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  

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Section: Higher Order Fd Discretizations and Fem Methodsmentioning

confidence: 99%

Section: The Case Of a = 1 And Constant P(x)mentioning

confidence: 99%

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

et al. 2004

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“…Moreover, in [15] the independence of preconditioned iterations from the mesh was observed. The eigenvalue distribution for the diffusive part of the latter problem was investigated in [23,26,25].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we focus our attention on the case when q is nonzero and Ω is a connected finite union of d-dimensional rectangles (a plurirectangle) so that A(1, 0) (and consequently the whole preconditioner P (a)) is symmetric and positive definite as proven in [26]. In particular, the authors of [23,26] found that, if a(x) is positive and regular enough and q(x) ≡ 0, then the preconditioned sequence shows a proper eigenvalue clustering at the unity (for the notion of proper eigenvalue and singular value clustering, see Definition 2.2), and we prove here that the same holds true in the complex field for problem (1.1) as well.…”

Section: Introductionmentioning

confidence: 99%

Spectral Analysis of a Preconditioned Iterative Method for the Convection‐Diffusion Equation

Bertaccini¹,

Golub²,

Serra‐Capizzano³

2007

SIAM J. Matrix Anal. & Appl.

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Abstract. The convergence features of a preconditioned algorithm for the convection-diffusion equation based on its diffusion part are considered. Analyses of the distribution of the eigenvalues of the preconditioned matrix in arbitrary dimensions and of the fundamental parameters of convergence are provided, showing the existence of a proper cluster of eigenvalues. The structure of the cluster is not influenced by the discretization. An upper bound on the condition number of the eigenvector matrix under some assumptions is provided as well. The overall cost of the algorithm is O(n), where n is the size of the underlying matrices.

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Quasi-optimal preconditioners for finite element approximations of diffusion dominated convection-diffusion equations on (nearly) equilateral triangle meshes

Russo

Serra‐Capizzano

Possio

2014

Numer. Linear Algebra Appl.

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The paper is devoted to the spectral analysis of effective preconditioners for linear systems obtained via a finite element approximation to diffusion-dominated convection-diffusion equations. We consider a model setting in which the structured finite element partition is made by equilateral triangles. Under such assumptions, if the problem is coercive and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequences exhibit a strong eigenvalue clustering at unity, the preconditioning matrix sequence and the original matrix sequence are spectrally equivalent, and under the constant coefficients assumption, the eigenvector matrices have a mild conditioning. The obtained results allow to prove the conjugate gradient optimality and the generalized minimal residual quasi-optimality in the case of structured uniform meshes. The interest of such a study relies on the observation that automatic grid generators tend to construct equilateral triangles when the mesh is fine enough. Numerical tests, both on the model setting and in the non-structured case, show the effectiveness of the proposal and the correctness of the theoretical findings

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Superlinear Preconditioners for Finite Differences Linear Systems

Cited by 7 publications

References 17 publications

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

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

Spectral Analysis of a Preconditioned Iterative Method for the Convection‐Diffusion Equation

Quasi-optimal preconditioners for finite element approximations of diffusion dominated convection-diffusion equations on (nearly) equilateral triangle meshes

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