## FAST NONSYMMETRIC ITERATIONS AND PRECONDITIONING FOR NAVIER-STOKES EQUATIONS*

### BibTeX

@MISC{Elmant_fastnonsymmetric,

author = {Howard Elmant and David Silvester},

title = {FAST NONSYMMETRIC ITERATIONS AND PRECONDITIONING FOR NAVIER-STOKES EQUATIONS*},

year = {}

}

### OpenURL

### Abstract

Abstract. Discretization and linearization of the steady-state Navier-Stokes equations gives rise to a nonsymmetric indefinite linear system of equations. In this paper, we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization. We confirm and supplement these analytic results with a series of numerical experiments indicating that Krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter. In addition, we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner. Key words. Navier-Stokes, iterative methods, preconditioners, Krylov subspace AMS subject classifications. 65F10, 65N12, 65N22, 65M60 1. Introduction. Consider the steady-state Navier-Stokes problem: given data f, find the velocity u and pressure p satisfying (1.1) 1-v V2u + u(div u) + u. Vu + grad p f

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Citation Context ...esults indicate that the lower bound is also tight. We test the preconditioners here with two Krylov subspace methods for solving nonsymmetric systems: the generalized minimum residual method (GMRES) =-=[15]-=- and a simple implementation of the quasi-minimum residual method (QMR) [8] based on coupled two-term recurrences without look-ahead. GMRES demonstrates the performance of the preconditioners with the... |

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(Show Context)
Citation Context ... Reh h llwll/v 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 =-=[12]-=-. In ourPRECONDITIONING FOR NAVIER-STOKES EQUATIONS 35 experiments, we demonstrate the effectiveness of the ideas using both a standard Galerkin discretization on a set of quasi-uniform grids and a s... |

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Citation Context ...s number Reh h llwll/v 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 [12]. In ourPRECONDITIONING FOR NAVIER-STOKES EQUATIONS 35 experiments, we demonstrate the effectiveness of the ideas using both a standard Galerkin discretization on a set of quasi-uniform grid... |

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Citation Context ...re real and nonnegative. Moreover, since N and A are first-order and second-order operators, respectively, the eigenvalues of are uniformly bounded in modulus by a constant 3 that is independent of h =-=[5]-=-. Therefore, the spectrum of I /2 is contained in the interval [1, 1 + 32/v2], or, equivalently, v 2 (p, Cp) < <1. 32 + v2 (p, Sp) Combining this with (1.4) and (2.9) gives ’21)2 (p, Cp) t 2 - -Jr" 13... |

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Citation Context ...pproximating matrix is/ F (I (H- 1R)k)- 1. 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 ft. A slightly more sophisticated idea is to use an alternating line relaxation. ... |

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Citation Context ...ring 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 18], =-=[22]-=- for further details.) The horizontal line Gauss-Seidel method for Fw v performs the iteration 1/3 0 --0, W + 11)i "JI- H v Fw For k steps of this iteration, the approximating matrix is/ F (I (H- 1R)k... |

1 |
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Citation Context ... of the even iterates stagnate. A rigorous explanation for this behavior can be given by relating the optimal polynomials implicitly generated by GMRES. In particular, it follows from the analysis in =-=[6]-=- that for a particular starting guess (not zero), the th GMRES polynomial for the triangular case is identical to the (2i 1)st GMRES polynomial for the diagonal case. Tables 4 and 5 show analogous ite... |

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Citation Context ...0(382) 229 Remark 3. In addition to the implementation of QMR with a coupled two-term recurrence (QMR2) discussed above, we tested a version without look-ahead based on a three-term recurrence (QMR3) =-=[7]-=-, and the definitive (Fortran) implementation of a two-term QMR with look-ahead (QMRz) from the QMRPAK directory in Netlib. For the preconditioners discussed above, the performances of the three varia... |

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Citation Context ...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], [10, pp. l Off], and =-=[19]-=-); i.e., defining a mesh parameter h, a velocity space Vh, ,and a pressure space Ph, there exist constants V, F, independent of h, such that (1..4) y2 < (P, BA-1Btp) < i-,2 YP Ph. (P, Qp) Here, Q is t... |

1 |
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(Show Context)
Citation Context ... 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 14], 16], =-=[20]-=- for the discrete Stokes equations, where the coefficient matrix has the form Consider the preconditioner for (2.1). The eigenvalues of the preconditioned operator are then given by the solution to th... |

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