An Analysis of Block Successive Overrelaxation for a Class of Matrices with Complex Spectra

Chin, R. C. Y.; Manteuffel, Thomas A.

doi:10.1137/0725036

Cited by 30 publications

(14 citation statements)

References 15 publications

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“…These bounds also follow from the more general analysis given in [4]. for the rowwise and columnwise orderings, respectively.…”

Section: A One-dimensional Examplementioning

confidence: 79%

“…As a result, CG is not always viable in floating-point arithmetic. Finally, we note that for the full system, it is shown in [4] that SOR iteration may be convergent for values of y and ô where the block Jacobi method is divergent, as well as for values where our analysis does not apply.…”

Section: Comparison Of Boundsmentioning

confidence: 99%

“…For the full system (1.2), line methods of this type are known to be effective in the self-adjoint case, see e.g. [14,20,21], and they have also been applied successfully to non-self-adjoint problems [4,5]. Our analysis applies to finite difference discretizations of a constant-coefficient version of ( 1.1 ) with Dirichlet boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

“…In §3, we describe the discrete constant-coefficient two-dimensional convection-diffusion equation, and we present a convergence analysis of a block Jacobi method for solving the full system (1.2). This analysis is closely related to that of [4], which applies in a somewhat more general setting. In §4, we present the convergence analysis for the reduced system.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Iterative methods for cyclically reduced nonselfadjoint linear systems

Elman¹,

Golub²

1990

Math. Comp.

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Abstract. We study iterative methods for solving linear systems of the type arising from two-cyclic discretizations of non-self-adjoint two-dimensional elliptic partial differential equations. A prototype is the convection-diffusion equation. The methods consist of applying one step of cyclic reduction, resulting in a "reduced system" of half the order of the original discrete problem, combined with a reordering and a block iterative technique for solving the reduced system. For constant-coefficient problems, we present analytic bounds on the spectral radii of the iteration matrices in terms of cell Reynolds numbers that show the methods to be rapidly convergent. In addition, we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the "unreduced" system.

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“…These bounds also follow from the more general analysis given in [4]. for the rowwise and columnwise orderings, respectively.…”

Section: A One-dimensional Examplementioning

confidence: 79%

Section: Comparison Of Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Iterative methods for cyclically reduced nonselfadjoint linear systems

Elman¹,

Golub²

1990

Math. Comp.

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show abstract

“…The results express convergence behavior in terms of discrete cell Reynolds numbers oh/2, xh/2, and they are confirmed and supplemented by numerical experiments. The analytic and experimental results (as well as those of [2] and [15]) show that the nonsymmetric discrete problems arising from ( 1.4) are in some ways easier to solve than the symmetric ones.…”

Section: Discussionmentioning

confidence: 98%

Iterative methods for cyclically reduced nonselfadjoint linear systems. II

Elman¹,

Golub²

1991

Math. Comp.

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Abstract.We perform an analytic and experimental study of line iterative methods for solving linear systems arising from finite difference discretizations of non-self-adjoint elliptic partial differential equations on two-dimensional domains. The methods consist of performing one step of cyclic reduction, followed by solution of the resulting reduced system by line relaxation. We augment previous analyses of one-line methods, and we derive a new convergence analysis for two-line methods, showing that both classes of methods are highly effective for solving the convection-diffusion equation. In addition, we compare the experimental performance of several variants of these methods, and we show that the methods can be implemented efficiently on parallel architectures.

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On preconditioned Krylov subspace methods for discrete convection-diffusion problems

Chernesky

1997

Numer. Methods Partial Differential Eq.

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We study nonstationary iterative methods for solving preconditioned systems arising from discretizations of the convection-diffusion equation. The preconditioners arise from Gauss-Seidel methods applied to the original system. It is shown that the performance of the iterative solvers is affected by the relationship of the ordering of the underlying grid and the direction of the flow associated with the differential operator. Specifically, only those orderings that follow the flow give fast iterative solvers.

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An Analysis of Block Successive Overrelaxation for a Class of Matrices with Complex Spectra

Cited by 30 publications

References 15 publications

Iterative methods for cyclically reduced nonselfadjoint linear systems

Iterative methods for cyclically reduced nonselfadjoint linear systems

Iterative methods for cyclically reduced nonselfadjoint linear systems. II

On preconditioned Krylov subspace methods for discrete convection-diffusion problems

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