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  • 1
    Electronic Resource
    Electronic Resource
    Conjugate gradient methods and ILU preconditioning of non-symmetric matrix systems with arbitrary sparsity patterns (1989)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 9 (1989), S. 213-233 
    Details
    ISSN: 0271-2091
    Keywords: Preconditioning ; Conjugate gradients ; Non-symmetric matrices ; Finite elements ; Convective transport ; 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: Preconditioning techniques based on incomplete Gaussian elimination for large, sparse, non-symmetric matrix systems are described. A certain level of fill-in may be specified in the incomplete factorizations. All methods considered may be applied to matrices with arbitrary sparsity patterns, for instance those associated with the general preprocessor algorithms or adaptive mesh techniques. The preconditioners have been combined with five conjugate gradient-like methods and tested on finite element discretized scalar convection-diffusion equations in 2D and 3D. It is found from numerical experiments that an amount of fill-in corresponding to about 50% of the number of original non-zero matrix entries is the optimal choice for this class of preconditioners. The preconditioners show almost no sensitivity to grid distortion. In problems with significantly variable coefficients or anisotropy the preconditioners stabilize the basic iterative schemes in addition to reducing the computational work substantially, mostly by more than 90%. The modified preconditioning technique, where fill-in is added on the main diagonal, performs in general better than the standard incomplete LU factorization, but is inferior to the latter in 3D problems and for matrix systems with complicated sparsity patterns.
    Additional Material: 16 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650090207
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  • 2
    Electronic Resource
    Electronic Resource
    Implicit finite element methods for two-phase flow in oil reservoirs (1990)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 10 (1990), S. 651-681 
    Details
    ISSN: 0271-2091
    Keywords: Porous media ; Two-phase flow ; Oil recovery ; Finite elements ; Preconditioning ; Conjugate gradients ; 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: The equations governing immiscible, incompressible, two-phase, porous media flow are discretized by generalized streamline diffusion Petrov-Galerkin methods in space and by implicit differences in time. Systems of non-linear algebraic equations are solved by Newton-Raphson iteration employing ILU-preconditioned conjugate-gradient-like methods to the non-symmetric matrix system in each iteration. The resulting solution methods are robust, enable complex grids with irregular nodal orderings and allow capillary effects.Several numerical formulations are tested and compared for one-, two- and three-dimensional flow cases, with emphasis on problems involving saturation shocks, heterogeneous media and curved boundaries. For reservoirs consisting of multiple rock types with differing capillary pressure properties, it is shown that traditional Bubnov-Galerkin methods give poor results and the new Petrov-Galerkin formulations are required. Investigations regarding the behaviour of several preconditioned conjugate-gradient-like methods in these type of problems are also reported.
    Additional Material: 14 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650100605
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