The Finite Element Discretization for Stream-Function Problems on Multiply Connected Domains

Gijzen, Martin B. van; Vreugdenhil, C. B.; Öksüzoǧlu, Hakan

doi:10.1006/jcph.1998.5886

Cited by 8 publications

(12 citation statements)

References 12 publications

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“…We discuss several choices for the outer iteration in Section 4. Our observations are illustrated for a Schur complement problem that stems from an ocean circulation model for steady barotropic flow as described in [25].…”

Section: Introductionmentioning

confidence: 99%

Relaxation strategies for nested Krylov methods

Eshof

Sleijpen

Gijzen

2005

Journal of Computational and Applied Mathematics

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There are classes of linear problems for which the matrix-vector product is a time consuming operation because an expensive approximation method is required to compute it to a given accuracy. In recent years different authors have investigated the use of, what is called, relaxation strategies for various Krylov subspace methods. These relaxation strategies aim to minimize the amount of work that is spent in the computation of the matrix-vector product without compromising the accuracy of the method or the convergence speed too much. In order to achieve this goal, the accuracy of the matrix-vector product is decreased when the iterative process comes closer to the solution. In this paper we show that a further significant reduction in computing time can be obtained by combining a relaxation strategy with the nesting of inexact Krylov methods. Flexible Krylov subspace methods allow variable preconditioning and therefore can be used in the outer most loop of our overall method. We analyze for several flexible Krylov methods strategies for controlling the accuracy of both the inexact matrix-vector products and of the inner iterations. The results of our analysis will be illustrated with an example that models global ocean circulation.

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Section: Introductionmentioning

confidence: 99%

Relaxation strategies for nested Krylov methods

Eshof

Sleijpen

Gijzen

2005

Journal of Computational and Applied Mathematics

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“…A preconditioned 2D elliptic surface problem. In [27], a finite element discretization of an elliptic PDE model for the ocean of planet earth is presented. This discretization leads to one linear system of N = 169,850 equations for each month with a nonchanging mildly nonsymmetric system matrix and changing right-handside vectors.…”

Section: 2mentioning

confidence: 99%

Mstab: Stabilized Induced Dimension Reduction for Krylov Subspace Recycling

Neuenhofen¹,

Greif²

2018

SIAM J. Sci. Comput.

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We introduce Mstab, a Krylov subspace recycling method for the iterative solution of sequences of linear systems, where the system matrix is fixed and is large, sparse, and nonsymmetric, and the right-hand-side vectors are available in sequence. Mstab utilizes the short-recurrence principle of induced dimension reduction-type methods, adapted to solve sequences of linear systems. Using IDRstab for solving the linear system with the first right-hand side, the proposed method then recycles the Petrov space constructed throughout the solution of that system, generating a larger initial space for subsequent linear systems. The richer space potentially produces a rapidly convergent scheme. Numerical experiments demonstrate that Mstab often enters the superlinear convergence regime faster than other Krylov-type recycling methods.1. Introduction. We consider iterative methods for the solution of sequences of large sparse nonsymmetric linear systemswith fixed nonsingular A ∈ C N ×N , where the right-hand sides b (ι) ∈ C N are provided in sequence. Such situations occur, for example, when applying an implicit time stepping scheme to numerically solve a transient partial differential equation (PDE). Relevant applications are, e.g., topology optimization [9], model reduction [6], structural dynamics [18], quantum chromodynamics [7], electrical circuit analysis [29], fluid dynamics [15], and optical tomography [13]. In all the aforementioned references a technique called Krylov subspace recycling (KSSR) is used.It is useful to start our discussion by establishing the notation and a few basic principles for solving a single linear system, Ax = b. Suppose x 0 is an initial guess of the solution, and let r 0 = b − Ax 0 be the initial residual. We define, as usual, the Krylov subspace of degree k ∈ N associated with A and r 0 as K k (A; r 0 ) := span{r 0 , Ar 0 , . . . , A k−1 r 0 }.Standard Krylov subspace methods solve a single linear system by searching in the kth iteration an approximate solution x k ∈ x 0 + K k (A; r 0 ) such that the residual r k *

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“…The problem denoted M4D2 arising in computational chemistry is proposed by Sherry Li from NERSC in [1]. The problems denoted STOMMEL1 and STOMMEL2 arising in ocean modeling are proposed by Martin van Gijzen from Delft University in [34]. The other linear systems are extracted from Tim Davis' matrix collection at the University of Florida [7].…”

Section: Numerical Experimentsmentioning

confidence: 99%

The BiCOR and CORS Iterative Algorithms for Solving Nonsymmetric Linear Systems

Carpentieri¹,

Jing²,

Huang³

2011

SIAM J. Sci. Comput.

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We present two iterative algorithms for solving real nonsymmetric and complex non-Hermitian linear systems of equations and that were developed from variants of the nonsymmetric Lanczos method. In this paper, we give the theoretical background of the two iterative methods and discuss their main computational aspects. Using a large number of numerical experiments, we analyze their convergence properties, and we also compare them with other popular nonsymmetric iterative solvers in use today.

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The Finite Element Discretization for Stream-Function Problems on Multiply Connected Domains

Cited by 8 publications

References 12 publications

Relaxation strategies for nested Krylov methods

Relaxation strategies for nested Krylov methods

Mstab: Stabilized Induced Dimension Reduction for Krylov Subspace Recycling

The BiCOR and CORS Iterative Algorithms for Solving Nonsymmetric Linear Systems

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