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## Fast Nonsymmetric Iterations and Preconditioning for Navier-Stokes Equations (1994)

Venue: | SIAM J. Sci. Comput |

Citations: | 74 - 10 self |

### Citations

2076 | GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems.
- Saad, Schultz
- 1986
(Show Context)
Citation Context ...tioned linear systems, we consider two popular Krylov subspace iter10 ative methods for nonsymmetric systems: the restarted generalized minimum residual method GMRES(s), where s is the restart length =-=[14]-=-, and a simple implementation of the quasiminimum residual method (QMR) [7] based on coupled two-term recurrences without lookahead. We use s = 10; for this choice, the storage requirements of restart... |

1697 |
Mixed and Hybrid Finite Element Methods,
- Brezzi, Fortin
- 1991
(Show Context)
Citation Context ... insisting that its mean is zero. Working in a conventional mixed finite element framework, we will further assume that the underlying velocity and pressure approximations are (div-)stable (see e.g., =-=[2]-=-, p. 57, [9], pp. 10ff, [18]), i.e., defining a mesh parameter h, a velocity space V h and a pressure space P h , there exist constants fl, \Gamma, independent of h, such that (1.4) fl 2 (p; BA \Gamma... |

959 |
Introduction to Numerical Analysis
- Stoer, Bulirsch
- 1991
(Show Context)
Citation Context ... = B \Gamma F \Gamma1 +F \Gammat 2 \Delta B t denote the symmetric part of S, and R = B \Gamma F \Gamma1 \GammaF \Gammat 2 \Delta B t its skew-symmetric part, so that S = C+R. By Bendixson's Theorem (=-=[16]-=-, p. 418), any eigenvaluesof the problem (2.7) satisfies (2.8) min p (p; Cp) (p; 1 Qp)sRe()smax p (p; Cp) (p; 1 Qp) ; jIm()jsmax p j(p; Rp)j (p; 1 Qp) : To construct bounds on these Rayleigh quotients... |

913 |
Matrix Iterative Analysis.
- Varga
- 2000
(Show Context)
Citation Context ...ordering of the velocity grid. Thus, H is a block lower triangular matrix consisting of the block diagonal of F (a tridiagonal matrix) together with the strict block lower triangular part of F . (See =-=[17, 21]-=- for further details.) The horizontal line Gauss-Seidel method for Fw = v performs the iteration w 0 = 0; w i+1 = w i + H \Gamma1 (v \Gamma Fw i ): For k steps of this iteration, the approximating mat... |

548 |
Streamline upwind/Petrov-Galerkin formulations for convection dominated flow with particular emphasis on the incompressible Navier-Stokes equations.
- Brooks, Hughes
- 1982
(Show Context)
Citation Context ...ds number Re h = hkwk= is greater than unity. In such situations, the addition of streamwise diffusion to the discrete system is known to give added stability, both theoretically and numerically, see =-=[3]-=- and [11]. In our experiments, we demonstrate the effectiveness of the ideas using both a standard Galerkin discretization on a set of quasi-uniform grids, and a streamline-upwind scheme on a set of u... |

521 |
Numerical solution of partial differential equations by the finite element method,
- Johnson
- 1987
(Show Context)
Citation Context ...r Re h = hkwk= is greater than unity. In such situations, the addition of streamwise diffusion to the discrete system is known to give added stability, both theoretically and numerically, see [3] and =-=[11]-=-. In our experiments, we demonstrate the effectiveness of the ideas using both a standard Galerkin discretization on a set of quasi-uniform grids, and a streamline-upwind scheme on a set of uniform gr... |

460 |
Iterative Solution of Large Linear Systems
- Young
- 2003
(Show Context)
Citation Context ...ordering of the velocity grid. Thus, H is a block lower triangular matrix consisting of the block diagonal of F (a tridiagonal matrix) together with the strict block lower triangular part of F . (See =-=[17, 21]-=- for further details.) The horizontal line Gauss-Seidel method for Fw = v performs the iteration w 0 = 0; w i+1 = w i + H \Gamma1 (v \Gamma Fw i ): For k steps of this iteration, the approximating mat... |

395 | QMR: a quasi-minimal residual method for non-Hermitian linear systems
- Freund, Nachtigal
- 1991
(Show Context)
Citation Context ...ioner. Remark 3. In addition to the implementation of QMR with a coupled two-term recurrence (QMR 2 ) discussed above, we tested a version without look-ahead based on a three-term recurrence (QMR 3 ) =-=[6]-=-, and the definitive (Fortran) implementation of two-term QMR with lookahead (QMR 2 ) from the QMRPAK directory in Netlib. For these preconditioners, the performances of the three variants were virtua... |

196 |
Finite Element Methods for Viscous Incompressible flows.
- Gunzburger
- 1989
(Show Context)
Citation Context ...) operator. There are, however, a number of disadvantages to the time-dependent approach. Simple time discretization methods based on the l 2 --projection onto the discretely divergence-free subspace =-=[9]-=- have an O(h) CFL restriction on the time step, which impinges on efficiency. On the other hand, absolutely stable schemes like the method of backward characteristics are known to be sensitive to impl... |

72 |
Numerical Solution of Partial DiĀ®erential Equations by the Finite Element Method
- Johnson
- 1995
(Show Context)
Citation Context ...r Re h = hkwk= is greater than unity. In such situations, the addition of streamwise diusion to the discrete system is known to give added stability, both theoretically and numerically, see [3] and =-=[11]-=-. In our experiments, we demonstrate the eectiveness of the ideas using both a standard Galerkin discretization on a set of quasi-uniform grids, and a streamline-upwind scheme on a set of uniform gri... |

65 |
A preconditioned iterative method for saddle point problems.
- Rusten, Winter
- 1992
(Show Context)
Citation Context ...e matrices in question are implicitly parameterized by h. For simplicity, we state our results under the assumption that B of (1.3) has full rank. The first idea is derived from a method developed in =-=[13, 15, 19]-=- for the discrete Stokes equations, where the coefficient matrix has the form (2.1) ` A B t B O ' : Consider the preconditioner ` A 0 0 Q ' for (2.1). The eigenvalues of the preconditioned operator ar... |

62 |
Realistic eigenvalue bounds for the Galerkin mass matrix
- Wathen
- 1987
(Show Context)
Citation Context ...discretization. In this case, however, it is known that Q can be replaced by some approximationsQ without affecting asymptotic convergence properties; only the constants fl and \Gamma of (1.4) change =-=[20]-=-. In the experiments discussed in Sections 3 and 4, we replace Q with a diagonal matrix consisting of the main diagonal of Q. 3. Numerical results I: Exact convection-diffusion solves. In this section... |

41 |
Bounds for iterates, inverses, spectral variation and fields of values of nonnormal matrices;
- Henrici
- 1962
(Show Context)
Citation Context ...e presence of (V) in these bounds is an artifact of the proof of the Bauer-Fike theorem; there are more subtle analyses ([8], pp. 344ff), as well as bounds that do not require diagonalizable matrices =-=[10]-=-. We have observed that the eigenvalues of A T are insensitive to perturbations, and we believe that the presence of (V) is pessimistic. This supposition is supported by the experimental results descr... |

36 |
Fast iterative solution of stabilized Stokes systems part ii: using block preconditioners
- Silvester, Wathen
- 1994
(Show Context)
Citation Context ... discrete Stokes problems for it implies that, using a quasi-uniform mesh, the Schur complement BA \Gamma1 B t has condition number bounded independently of h. It is also known from our previous work =-=[15] that whens! 1, &quo-=-t;optimal " preconditioners for the Laplacian sub-blocks give rise to "optimal" preconditioners for the Stokes problem in the sense that the spectra of the underlying discrete operators... |

26 |
Preconditioning by fast direct methods for non self-adjoint nonseparable elliptic equations
- Elman, Schultz
- 1986
(Show Context)
Citation Context ...l and nonnegative. Moreover, since N and A are first-order and second-order operators, respectively, the eigenvalues of ~ N are uniformly bounded in modulus by a constant ffi that is independent of h =-=[5]-=-. Therefore, the spectrum of I \Gamma 1 2 ~ N 2 is contained in the interval \Theta 1; 1 + ffi 2 = 2 , or, equivalently, 2 ffi 2 +s2s(p; Cp) (p; 1sS1 p)s1: Combining this with (1.4) and (2.9) gives fl... |

25 |
Stability of the LagrangeāGalerkin method with non-exact integration,
- Morton, Priestley, et al.
- 1988
(Show Context)
Citation Context ...s. There are many ways to do this: one way is to make use of the "characteristics" associated with the hyperbolic part of the Navier-Stokes operator via a Lagrange-Galerkin approach (for exa=-=mple, see [12]-=-). The associated transpose-diffusion splitting leads to absolutely stable temporal discretizations so that large time steps can be taken. At each time step, a symmetric indefinite matrix system corre... |

22 |
Fast iterative solution of stabilized Stokes systems part i: using simple diagonal preconditioners
- Wathen, Silvester
- 1993
(Show Context)
Citation Context ...e matrices in question are implicitly parameterized by h. For simplicity, we state our results under the assumption that B of (1.3) has full rank. The first idea is derived from a method developed in =-=[13, 15, 19]-=- for the discrete Stokes equations, where the coefficient matrix has the form (2.1) ` A B t B O ' : Consider the preconditioner ` A 0 0 Q ' for (2.1). The eigenvalues of the preconditioned operator ar... |

13 |
Varga,Matrix Iterative Analysis,
- S
- 2000
(Show Context)
Citation Context ...ordering of the velocity grid. Thus, H is a block lower triangular matrix consisting of the block diagonal of F (a tridiagonal matrix) together with the strict block lower triangular part of F . (See =-=[17, 21]-=- for further details.) The horizontal line Gauss-Seidel method for Fw = v performs the iteration w 0 = 0; w i+1 = w i +H 1 (v Fw i ): For k steps of this iteration, the approximating matrix is ^ F ... |

9 |
Incomplete block-matrix factorization iterative methods for convection-diffusion problems
- Axelson, Eijkhout, et al.
- 1989
(Show Context)
Citation Context ... is only to demonstrate "proof-of-concept;" many other techniques for approximating the action of F \Gamma1 are possible, for both diffusion-dominated and convection-dominated flow. See, for=-= example, [1, 4, 13, 15, 19]-=-. In conclusion, in this paper we have shown that appropriately preconditioned Krylov subspace methods can be used to solve the discrete Oseen equations, a linearized version of the Navier-Stokes equa... |

4 |
Line iterative methods for cyclically reduced convectiondiusion problems
- Elman, Golub
- 1992
(Show Context)
Citation Context ... = F (I \Gamma (H \Gamma1 R) k ) \Gamma1 . It has been observed that the performance of relaxation methods of this type can be improved if the sweep direction follows the underlying direction of flow =-=[4]-=-. Our benchmark problem has a circular flow, so that no simple line relaxation can mimic the flow direction throughout \Omega\Gamma A slightly more sophisticated idea is to use an alternating line rel... |

4 |
urth, A combined conjugate gradient-multigrid algorithm for the numerical solution of the Stokes problem
- Verf
- 1984
(Show Context)
Citation Context ... zero. Working in a conventional mixed finite element framework, we will further assume that the underlying velocity and pressure approximations are (div-)stable (see e.g., [2], p. 57, [9], pp. 10ff, =-=[18]-=-), i.e., defining a mesh parameter h, a velocity space V h and a pressure space P h , there exist constants fl, \Gamma, independent of h, such that (1.4) fl 2 (p; BA \Gamma1 B t p) (p; Qp)s\Gamma 2 8p... |

3 |
Bounds for iterates, inverses, spectral variation, and of values of nonnormal matrices
- Henrici
- 1962
(Show Context)
Citation Context ...e presence of (V) in these bounds is an artifact of the proof of the Bauer-Fike theorem; there are more subtle analyses ([8], pp. 344), as well as bounds that do not require diagonalizable matrices =-=[10]-=-. We have observed that the eigenvalues of A T are insensitive to perturbations, and we believe that the presence of (V) is pessimistic. This supposition is supported by the experimental results desc... |