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• 1
Book
Oxford [u.a.] :Oxford Univ. Press,
Title: Finite elements and fast iterative solvers : with applications in incompressible fluid dynamics
Author: Elman, Howard C.
Edition: 2. ed.
Publisher: Oxford [u.a.] :Oxford Univ. Press,
Year of publication: 2014
Pages: XIV, 479 S. : , graph. Darst.
Series Statement: Numerical mathematics and scientific computation
ISBN: 978-0-19-967879-2 , 978-0-19-967880-8
Type of Medium: Book
Language: English
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• 2
Electronic Resource
Springer
Numerische Mathematik 76 (1997), S. 209-230
ISSN: 0945-3245
Keywords: Mathematics Subject Classification (1991): 65F10, 65G99, 65L10, 65L12, 65N22
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary. The one-dimensional discrete Poisson equation on a uniform grid with $n$ points produces a linear system of equations with a symmetric, positive-definite coefficient matrix. Hence, the conjugate gradient method can be used, and standard analysis gives an upper bound of $O(n$ ) on the number of iterations required for convergence. This paper introduces a systematically defined set of solutions dependent on a parameter $\beta$ , and for several values of $\beta$ , presents exact analytic expressions for the number of steps $k(\beta,\tau,n$ ) needed to achieve accuracy $\tau$ . The asymptotic behavior of these expressions has the form $O(n^{\alpha$ )} as $n \rightarrow \infty$ and $O(\tau^{\gamma$ )} as $\tau \rightarrow 0$ . In particular, two choices of $\beta$ corresponding to nonsmooth solutions give $\alpha = 0$ , i.e., iteration counts independent of $n$ ; this is in contrast to the standard bounds. The standard asymptotic convergence behavior, $\alpha = 1$ , is seen for a relatively smooth solution. Numerical examples illustrate and supplement the analysis.
Type of Medium: Electronic Resource
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• 3
Electronic Resource
Springer
Numerische Mathematik 67 (1994), S. 177-190
ISSN: 0945-3245
Keywords: Mathematics Subject Classification (1991): 65F10
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary. We derive analytic bounds on the convergence factors associated with block relaxation methods for solving the discrete two-dimensional convection-diffusion equation. The analysis applies to the reduced systems derived when one step of block Gaussian elimination is performed on red-black ordered two-cyclic discretizations. We consider the case where centered finite difference discretization is used and one cell Reynolds number is less than one in absolute value and the other is greater than one. It is shown that line ordered relaxation exhibits very fast rates of convergence.
Type of Medium: Electronic Resource
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• 4
Electronic Resource
Springer
Numerische Mathematik 83 (1999), S. 231-257
ISSN: 0945-3245
Keywords: Mathematics Subject Classification (1991):65N22
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary. In this work we calculate the eigenvalues obtained by preconditioning the discrete Helmholtz operator with Sommerfeld-like boundary conditions on a rectilinear domain, by a related operator with boundary conditions that permit the use of fast solvers. The main innovation is that the eigenvalues for two and three-dimensional domains can be calculated exactly by solving a set of one-dimensional eigenvalue problems. This permits analysis of quite large problems. For grids fine enough to resolve the solution for a given wave number, preconditioning using Neumann boundary conditions yields eigenvalues that are uniformly bounded, located in the first quadrant, and outside the unit circle. In contrast, Dirichlet boundary conditions yield eigenvalues that approach zero as the product of wave number with the mesh size is decreased. These eigenvalue properties yield the first insight into the behavior of iterative methods such as GMRES applied to these preconditioned problems.
Type of Medium: Electronic Resource
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• 5
Electronic Resource
Springer
BIT 29 (1989), S. 890-915
ISSN: 1572-9125
Keywords: Primary: 65F10, 65N20 ; Secondary: 15A06 ; Linear systems ; iterative methods ; preconditioners ; incomplete factorizations ; non-self-adjoint ; convection-diffusion
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Two classes of incomplete factorization preconditioners are considered for nonsymmetric linear systems arising from second order finite difference discretizations of non-self-adjoint elliptic partial differential equations. Analytic and experimental results show that relaxed incomplete factorization methods exhibit numerical instabilities of the type observed with other incomplete factorizations, and the effects of instability are characterized in terms of the relaxation parameter. Several stabilized incomplete factorizations are introduced that are designed to avoid numerically unstable computations. In experiments with two-dimensional problems with variable coefficients and on nonuniform meshes, the stabilized methods are shown to be much more robust than standard incomplete factorizations.
Type of Medium: Electronic Resource
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• 6
Electronic Resource
Chichester : Wiley-Blackwell
International Journal for Numerical Methods in Fluids 22 (1996), S. 755-770
ISSN: 0271-2091
Keywords: Stokes ; multigrid ; Krylov subspace ; conjugate gradient ; conjugate residual ; Uzawa ; Engineering ; Engineering General
Source: Wiley InterScience Backfile Collection 1832-2000
Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
Notes: Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizations a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the discretization. In this paper we compare the performance of four such methods, namely variants of the Uzawa, preconditioned conjugate gradient, preconditioned conjugate residual and multigrid methods, for solving several two-dimensional model problems. The results indicate that multigrid with smoothing based on incomplete factorization is more efficient than the other methods, but typically by no more than a factor of two. The conjugate residual method has the advantage of being independent of iteration parameters.