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  • 1
    Electronic Resource
    Electronic Resource
    A least-squares procedure for the solution of transport problems (1990)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 10 (1990), S. 623-636 
    Details
    ISSN: 0271-2091
    Keywords: Least-squares method ; Optimal control problem ; Conjugate gradient ; Advection-diffusion equation ; Perturbation series ; 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: In this paper a least-squares formulation associated with a conjugate gradient algorithm is proposed for the solution of transport problems. In this procedure the advection-diffusion equation is first discretized in time using an implicit scheme. At each time step the resulting partial differential equation is replaced by an optimal control problem. This minimization problem involves the minimization of a functional defined via a state equation. This functional is chosen in order to force the numerical solution of the advection-diffusion equation to be equal to the hyperbolic advective part of this equation. The effectiveness of the method is shown through a one-dimensional example involving advective and diffusive transport. No oscillation and high accuracy have been obtained for the entire range of Peclet numbers with a Courant number well in excess of unity.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650100603
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  • 2
    Electronic Resource
    Electronic Resource
    An iterative penalty method for the least squares solution of boundary value problems (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 13 (1997), S. 257-281 
    Details
    ISSN: 0749-159X
    Keywords: boundary value problems ; collocation least squares method ; augmented Lagrangian method ; Uzawa's algorithm ; preconditioned conjugate gradient ; Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This article is concerned with iterative techniques for linear systems of equations arising from a least squares formulation of boundary value problems. In its classical form, the solution of the least squares method is obtained by solving the traditional normal equation. However, for nonsmooth boundary conditions or in the case of refinement at a selected set of interior points, the matrix associated with the normal equation tends to be ill-conditioned. In this case, the least squares method may be formulated as a Powell multiplier method and the equations solved iteratively. Therein we use and compare two different iterative algorithms. The first algorithm is the preconditioned conjugate gradient method applied to the normal equation, while the second is a new algorithm based on the Powell method and formulated on the stabilized dual problem. The two algorithms are first compared on a one-dimensional problem with poorly conditioned matrices. Results show that, for such problems, the new algorithm gives more accurate results. The new algorithm is then applied to a two-dimensional steady state diffusion problem and a boundary layer problem. A comparison between the least squares method of Bramble and Schatz and the new algorithm demonstrates the ability of the new method to give highly accurate results on the boundary, or at a set of given interior collocation points without the deterioration of the condition number of the matrix. Conditions for convergence of the proposed algorithm are discussed. © 1997 John Wiley & Sons, Inc.
    Additional Material: 15 Ill.
    Type of Medium: Electronic Resource
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