## Preconditioners For Saddle Point Problems Arising In Computational Fluid Dynamics (2002)

Venue: | Appl. Numer. Math |

Citations: | 16 - 1 self |

### BibTeX

@ARTICLE{Elman02preconditionersfor,

author = {Howard C. Elman},

title = {Preconditioners For Saddle Point Problems Arising In Computational Fluid Dynamics},

journal = {Appl. Numer. Math},

year = {2002},

volume = {43},

pages = {75--89}

}

### Years of Citing Articles

### OpenURL

### Abstract

Discretization and linearization of the incompressible Navier-Stokes equations leads to linear algebraic systems in which the coefficient matrix has the form of a saddle point problem F B T B 0 u p = f g : (0.1) In this paper, we describe the development of efficient and general iterative solution algorithms for this class of problems. We review the case where (0.1) arises from the steady-state Stokes equations and show that solution methods such as the Uzawa algorithm lead naturally to a focus on the Schur complement operator BF 1 B T together with efficient strategies of applying the action of F 1 to a vector. We then discuss the advantages of explicitly working with the coupled form of the block system (0.1). Using this point of view, we describe some new algorithms derived by developing efficient methods for the Schur complement systems arising from the Navier-Stokes equations, and we demonstrate their effectiveness for solving both steady-state and evolutionary problems.

### Citations

1518 |
Iterative methods for sparse linear systems
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Citation Context ...-denite systems of equations, it is known that Richardson iteration is slower to converge than the conjugate gradient method (CG). The convergence factor for CG is bounded by = ( p 1)=( p + 1) [26]=-=-=-. 3. The action of the inverse of A is potentially costly. This is clearly the dominant cost of the algorithm. Although fast solvers for the Poisson equation are available, it would be desirable to av... |

1323 |
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Citation Context ... Discretization leads to a linear system of the form F B T B 0 u p = f 0 (3.2) to be solved at each step. Our solution strategy for this is to use Krylov subspace methods such as the GMRES [27], QMR [14] or BiCGSTAB(L) [32] algorithms, in combination with preconditioning. The latter is the critical component needed for rapid convergence. The discussion of the previous section leads to the i... |

337 | AMR: A quasi-minimal residual method for non-hermitian linear systems
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Citation Context ...ation leads to a linear system of the form F B T B 0 u p = f 0 (3.2) to be solved at each step. Our solution strategy for this is to use Krylov subspace methods such as the GMRES [27], QMR [14] or BiCGSTAB(L) [32] algorithms, in combination with preconditioning. The latter is the critical component needed for rapid convergence. The discussion of the previous section leads to the idea that a... |

305 |
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(Show Context)
Citation Context ...ill assume throughout that the spatial discretization is div-stable; we have in mind low ordersnite element discretizations that satisfy an inf-sup condition, or the MACsnite dierence discretization [=-=15, 17, 21]-=-. The solution methods considered are generalizations of techniques developed originally for the steady-state Stokes equations, and in Section 2 we outline the derivation of our point of view as it ev... |

130 |
Finite Element Approximation of the Navier-Stokes Equations
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(Show Context)
Citation Context ...ill assume throughout that the spatial discretization is div-stable; we have in mind low ordersnite element discretizations that satisfy an inf-sup condition, or the MACsnite dierence discretization [=-=15, 17, 21]-=-. The solution methods considered are generalizations of techniques developed originally for the steady-state Stokes equations, and in Section 2 we outline the derivation of our point of view as it ev... |

99 |
Inexact and preconditioned Uzawa algorithms for saddle-point problems
- Elman, Golub
- 1994
(Show Context)
Citation Context ...that (BA 1 B T ) is independent of h. This discussion also suggests that preconditioning by the mass matrix or some spectrally equivalent approximationsQMp may be benecial, which is indeed the case [11]. The preconditioned Uzawa algorithm updates the pressures as p k+1 = p k +sQ 1 Mp Bu k+1 : Ifsis dened as in (2.4) where min and max now represent extrema of the Rayleigh quotient (q;BA 1 B T q)... |

99 | Incompressible flow and the finite element method - Gresho, Sani - 1998 |

70 | A.: Analysis of the inexact uzawa algorithm for saddle point problems
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- 1997
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Citation Context ...s from the coupling between the pressure and velocity, itssrst step can be replaced by an approximate computation of the action of A 1 to produce an algorithm with similar convergence characteristics =-=[4, 11, 41]-=-. It is not straightforward to automate this process, however, since the inner iteration for the Poisson equation requires a stopping criterion. Thus, neither of these strategies completely resolves t... |

66 | Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations
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Citation Context ... systems, and for (3.2) we prefer a block-triangular preconditioner Q = Q F B T 0 Q S : (3.3) Iteration with this choice requires approximately half the steps needed with a block-diagonal version [8]. Applying the preconditioner, i.e., computing w s = Q F B T 0 Q S 1 v q 6 for given v, q entails solving the systems Q S s = q; Q F w = v B T s: (3.4) The only cost not incurred by the bl... |

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Citation Context ...ral values of , P 2 -P1 discretization, h = 1=64. then appear to be independent of the value of the viscosity parameter. These results are consistent with those of extensive experiments described in [=-=13, 18, 28-=-]. It is also shown empirically in [13] that the eigenvalues of the preconditioned matrix AQ 1 are clustered in a region that does not depend on , except for a small number of outliers. The number of ... |

58 |
A preconditioning iterative method for saddle point problems
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Citation Context ...stopping criterion. Thus, neither of these strategies completely resolves the drawbacks listed above, and we have come to prefer an alternative approach, developed independently by Rusten and Winther =-=[25-=-] and Silvester and Wathen [29, 39], which treats the saddle point problem (2.1) directly. The system (2.1) is symmetric indenite, so that the MINRES Krylov subspace method [22] is applicable. When th... |

54 | Efficient preconditioning of the linearized Navier–Stokes equations for incompressible flow
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Citation Context ...ionary problems. This idea was originally developed by Kay and Loghin [18] using the structure of the Green's function for the operator of (3.1). The approach presented here follows Silvester et. al. =-=-=-[28]. We start with (3.1) and forsxed m let w = u (m 1) denote the lagged convection coecient and +w r the resulting convection-diusion operator. Let us suppose that there is an analogous operator ( ... |

39 | Iterative techniques for time dependent Stokes problems
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- 1997
(Show Context)
Citation Context ...oice which is quasioptimal with respect to the energy norm. Second, although we have 5 considered only steady problems here, the same point of view can be adapted to the evolutionary Stokes equations =-=[3, 6]-=-. In this case, the matrix A consists of a linear combination of a velocity mass matrix and a discrete Laplace operator. Good preconditioners for the Schur complement operator require an approximate P... |

32 | Multigrid and Krylov subspace methods for the discrete Stokes equations
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- 1996
(Show Context)
Citation Context ... 2 for 1 , 2 independent of the mesh size, then so are the intervals of (2.6), as well as the convergence factor for MINRES. Detailed discussions of the eectiveness of this approach are given in [10=-=, 25, 29]-=-. The key point is that it achieves optimal convergence rates automatically, without a need for exact computation of the action of A 1 , or estimates of any parameters, or stopping criteria associated... |

30 |
Fast Iterative Solution of Stabilized Stokes Systems
- Silvester, Wathen
- 1994
(Show Context)
Citation Context ...her of these strategies completely resolves the drawbacks listed above, and we have come to prefer an alternative approach, developed independently by Rusten and Winther [25] and Silvester and Wathen =-=[29, 39-=-], which treats the saddle point problem (2.1) directly. The system (2.1) is symmetric indenite, so that the MINRES Krylov subspace method [22] is applicable. When this method is applied to a system A... |

25 |
Solution of sparse inde systems of linear equations
- Paige, Saunders
- 1975
(Show Context)
Citation Context ... by Rusten and Winther [25] and Silvester and Wathen [29, 39], which treats the saddle point problem (2.1) directly. The system (2.1) is symmetric indenite, so that the MINRES Krylov subspace method [22] is applicable. When this method is applied to a system Ax = b, the residual r k = b Ax k of the kth iterate satises kr k k min p k (0)2 k max 2(A) jp k ()j kr 0 k; (2.5) where k denotes the ... |

22 |
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- Bey, Wittum
- 1997
(Show Context)
Citation Context ...uation is a more dicult problem than the Poisson equation (in particular, the analysis of solution algorithms is far less well developed), there are eective solvers available for it, see for example [=-=2, 9, 24, 42]-=-. The Schur complement system is less straightforward. An operator Q S that is spectrally equivalent to the pressure mass matrix, as discussed above, is easy to implement and has also been shown to le... |

22 |
Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems
- Elman
- 1989
(Show Context)
Citation Context ...uation is a more dicult problem than the Poisson equation (in particular, the analysis of solution algorithms is far less well developed), there are eective solvers available for it, see for example [=-=2, 9, 24, 42]-=-. The Schur complement system is less straightforward. An operator Q S that is spectrally equivalent to the pressure mass matrix, as discussed above, is easy to implement and has also been shown to le... |

19 |
Block triangular preconditioners for nonsymmetric saddle point problems: field-of-values analysis
- KLAWONN, STARKE
- 1999
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Citation Context ...t is spectrally equivalent to the pressure mass matrix, as discussed above, is easy to implement and has also been shown to lead to (essentially) mesh independent rates of convergence for (3.1){(3.2) =-=[8, 19-=-]. However, performance deteriorates if the Reynolds number becomes large, i.e., if the viscosity is small. To remedy this diculty, we consider an alternative approach for constructing a precondition... |

14 |
Analysis and convergence of the MAC scheme l
- Nicolaides
- 1992
(Show Context)
Citation Context ...ill assume throughout that the spatial discretization is div-stable; we have in mind low ordersnite element discretizations that satisfy an inf-sup condition, or the MACsnite dierence discretization [=-=15, 17, 21]-=-. The solution methods considered are generalizations of techniques developed originally for the steady-state Stokes equations, and in Section 2 we outline the derivation of our point of view as it ev... |

9 | Analysis of Preconditioned Picard Iterations for the Navier-Stokes Equations
- Loghin
- 2001
(Show Context)
Citation Context ...rends for both \exact" and approximate versions of the preconditioner [12]. There has been a limited amount of analysis of this preconditioning strategy that provides insight into convergence. Lo=-=ghin [20]-=- has shown that the eigenvalues of the preconditioned linear systems are contained in a set that is independent of the mesh size. This does notsrmly establish that convergence rates are also independe... |

8 |
A Green’s function preconditioner for the steady-state Navier– Stokes equations, Rep NA-99/06
- Kay, Loghin
- 1999
(Show Context)
Citation Context ...roach for constructing a preconditioner, which leads to a methodology that adapts in a natural way to both steady-state and evolutionary problems. This idea was originally developed by Kay and Loghin =-=[18]-=- using the structure of the Green's function for the operator of (3.1). The approach presented here follows Silvester et. al. [28]. We start with (3.1) and forsxed m let w = u (m 1) denote the lagged ... |

8 | Convergence analysis of a multigrid method for convection-diffusion equations
- Reusken
(Show Context)
Citation Context ...uation is a more dicult problem than the Poisson equation (in particular, the analysis of solution algorithms is far less well developed), there are eective solvers available for it, see for example [=-=2, 9, 24, 42]-=-. The Schur complement system is less straightforward. An operator Q S that is spectrally equivalent to the pressure mass matrix, as discussed above, is easy to implement and has also been shown to le... |

8 |
Fast and robust solvers for time-discretised incompressible Navier-Stokes equations
- Silvester, Wathen
- 1996
(Show Context)
Citation Context ...d k ; M p d k )] 1=2 : But if QA and QMp are spectrally equivalent to the discrete Laplacian and mass matrix, respectively, then the residual norm of (2.8) is spectrally equivalent to the energy norm =-=[30]-=-. Consequently, the quantity minimized by preconditioned MINRES is a natural choice which is quasioptimal with respect to the energy norm. Second, although we have 5 considered only steady problems he... |

5 |
Some fast 3D element solvers for the generalized Stokes problem
- Cahouet, Chabard
- 1988
(Show Context)
Citation Context ...oice which is quasioptimal with respect to the energy norm. Second, although we have 5 considered only steady problems here, the same point of view can be adapted to the evolutionary Stokes equations =-=[3, 6]-=-. In this case, the matrix A consists of a linear combination of a velocity mass matrix and a discrete Laplace operator. Good preconditioners for the Schur complement operator require an approximate P... |

2 |
Multigrid solutions to elliptic problems
- Brandt, Dinar
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(Show Context)
Citation Context ...le of multigrid methods for saddle point problems takes the form of so-called \distributive iterations," in which the MG smoothing iteration is applied to a system obtained from a change of varia=-=bles [-=-5, 35, 43, 42]. This approach shares with (3.6) the use of a discrete convection-diusion operator F p . Consider the transformation F B T B 0 I B T 0 F p ^ u ^ p = f 0 ; u p = I B T 0 F p ^ ... |

2 | On some element methods for the numerical simulation of incompressible viscous - Dean, Glowinski - 1993 |

1 |
A multigrid version of a simple element method for the Stokes problem
- Pitkaranta, Saarinen
- 1985
(Show Context)
Citation Context ...s consist of a convection-diusion operator and a scaled discrete Laplacian. Smoothers for (5.1) are derived from smoothers for these individual blocks: see the references above for details. See also [=-=23, 38]-=- for other multigrid methods derived from the squared system associated with (3.2). Thus, we see that multigrid methods share many characteristics of the preconditioning approach considered here. We h... |