Adaptive version of Simpler GMRES

Jiránek, Pavel; Rozložńık, Miroslav

doi:10.1007/s11075-009-9311-2

Cited by 16 publications

(38 citation statements)

References 16 publications

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“…Of course, this strategy means that a flexible GMRES (FGMRES) algorithm with right preconditioning is used instead of GMRES as the outer solver [37]. The flexible GMRES implementation used is the one described in [31] based on the simple GMRES algorithm in [38]. The tolerance of the inner iterations is denoted by tol in .…”

Section: The Use Of Inexact Solversmentioning

confidence: 99%

A Relaxed Dimensional Factorization preconditioner for the incompressible Navier–Stokes equations

Benzi

Niu

et al. 2011

Journal of Computational Physics

102

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a b s t r a c tIn this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo [7]. Numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the RDF preconditioner with respect to both mesh size and viscosity.

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Section: The Use Of Inexact Solversmentioning

confidence: 99%

A Relaxed Dimensional Factorization preconditioner for the incompressible Navier–Stokes equations

Benzi

Niu

et al. 2011

Journal of Computational Physics

102

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“…For the latter we use the implementation based on the simpler GMRES algorithm described in [19]. Here, to solve the linear systems with the velocity-pressure equation and the vorticityhelicity equation, instead of applying one action of the AL-type preconditioners, a few inner GMRES iterations with corresponding preconditioners are used.…”

Section: Preconditioners and Solvers For Steady Problemsmentioning

confidence: 99%

Assessment of a vorticity based solver for the Navier–Stokes equations

Benzi

Olshanskii

Rebholz

et al. 2012

Computer Methods in Applied Mechanics and Engineering

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a b s t r a c tWe investigate numerically a recently proposed vorticity based formulation of the incompressible Navier-Stokes equations. The formulation couples a velocity-pressure system with a vorticity-helicity system, and is intended to provide a numerical scheme with enhanced accuracy and superior conservation properties. For a few benchmark problems, we study the performance of a finite element method for this formulation and compare it with the commonly used velocity-pressure based finite element method. It is shown that both steady and unsteady discrete problems in the new formulation admit simple decoupling strategies followed by the application of iterative solves to auxiliary subproblems. Further, we compare several iterative strategies to solve the discrete problems and study the interplay between the choice of stabilization parameters in the finite element method and the efficiency of linear algebra solvers.

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“…By Lemma , the block columns in

Z_{m}

are linear independent, and it can be regarded as the basis of

K_{m} false(A, R_{0} false)

. Lemma also allows

κ_{F} ({scriptZ}_{m}) = κ_{2} ({normalΓ}_{m} L_{m}^{- 1} {normalΩ}_{m}^{- 1}),

from which the assertion is true by lemma 2.4 in the work of JR …”

Section: Glsgmres Methodsmentioning

confidence: 94%

“…Proof Note that

V_{m}

is well defined under the assumption

R_{0} \notin A K_{m - 1} false(A, R_{0} false)

. From we have

Z_{m}^{S} = [{\overset{R}{true}}_{m - 1}, V_{1}, …, V_{m - 1}] * Y_{m},

where, by , the block columns of

false[{\overset{R}{true}}_{m - 1}, V_{1}, …, V_{m - 1} false]

are F ‐orthogonal and Y m takes the form

Y_{m} = [\begin{matrix} ρ_{m - 1} false/ ρ_{0} \\ ξ_{1} false/ ρ_{0} & 1 \\ ⋮ & ⋱ \\ ξ_{m - 1} false/ ρ_{0} & 1 \end{matrix}] \in {double-struckR}^{m \times m} .

Then, by Lemma ,

κ_{F} false({scriptZ}_{m}^{normalS} false) = κ_{2} false(Y_{m} false)

, and the lower and upper bounds in follow straightforward from lemma 2.3 in the work of JR …”

Section: Glsgmres Methodsmentioning

confidence: 97%

“…When s = 1, it reduces to the standard simpler GMRES and its variant method using the normalized residual as the basis was proven to be conditionally stable . Further investigation on an adaptive version was done in the work of Jiránek and Rozlo

\overset{z}{true}

nĺk (JR) and Jing et al For s > 1, the idea in the works of JR was generalized to an (adaptive) simpler block GMRES method . However, for the GlSGMRES method, stability keeps unevaluated due to the absence of the definition of the condition number for the global method.…”

Section: Introductionmentioning

confidence: 99%

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Making global simpler GMRES more stable

Liu

Shen

Jia

2018

Numerical Linear Algebra App

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Summary The global GMRES method is well known for the solution of nonsymmetric linear systems with multiple right hands. In this paper, the condition number for evaluating the stability of the global simpler GMRES method is defined. With this condition number, it is shown that Zong et al.'s global simpler method with a simple basis of the Krylov matrix subspace might lead to instability when there is a large decrease in the residual norm. An adaptive global simpler method is then presented to avoid the potential instability. For this adaptive method, theoretical results deliver the lower and upper bounds for the condition number of the basis, and when the linear system is of single right hand, the estimate is proven to be tighter than Jiránek et al.'s upper bound by numerical tests. Numerical results also show that the adaptive strategy leads to a well‐conditioned basis, and in most cases, the global simpler GMRES with normalized residuals as the basis can also be reliable.

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Adaptive version of Simpler GMRES

Cited by 16 publications

References 16 publications

A Relaxed Dimensional Factorization preconditioner for the incompressible Navier–Stokes equations

A Relaxed Dimensional Factorization preconditioner for the incompressible Navier–Stokes equations

Assessment of a vorticity based solver for the Navier–Stokes equations

Making global simpler GMRES more stable

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