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Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems

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Part of the book series: Scientific Computation ((SCIENTCOMP))

Abstract

Large linear systems are solved for modeling many scientific and engineering applications. Often these systems result from a discretization of model equations using Finite Elements, Finite Volumes or Finite Differences. The systems tend to become very large for three dimensional problems. Some models involve both time and space as independent parameters and therefore it is necessary to solve such a linear system efficiently at all time-steps.

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References

  1. J. Bear, Dynamics of Fluids in Porous Meida, Elsevier, New York, 1972.

    Google Scholar 

  2. H. De Gersem, K. Hameyer, A deflated iterative solver for magnetostatic finite element models with large differences in permeability, Eur. Phys. J. Appl. Phys. 13 (2000), 45–49.

    Article  Google Scholar 

  3. G. De Josselin de Jong, Singularity distributions for the analysis of multiple flyuid flow in porous media, Journal of Geothermal Research 65 (1960), 3739–3758.

    ADS  Google Scholar 

  4. J. Frank, C. Vuik, On the construction of deflation-based preconditiones, SIAM J. Sci. Comput. 23 (2001), 442–562.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. H. Golub, C. F. van Loan, Matrix Computations, Third edition, The Johns Hopkins University Press, Baltimore, 1996.

    MATH  Google Scholar 

  6. C. B. Jenssen, P. Å. Weinerfelt, Coarse grid correction scheme for implicit multiblock Euler calculations, AIAA Journal 33 (1995), 1816–1821.

    Article  ADS  MATH  Google Scholar 

  7. E. F. Kaasschieter, Preconditioned conjugate gradients for solving singular systems, J. Comput. Appl. Math. 24 (1988), 265–275.

    Article  MathSciNet  MATH  Google Scholar 

  8. E. Kreyszig, Introductory Functional Analysis with Applications, Willey, New York, 1989.

    MATH  Google Scholar 

  9. L. W. Lake, Enhanced Oil Recovery, Pretience-Hall, Englewood Cliffs, 1989.

    Google Scholar 

  10. D. C. Lay, Linear Algebra and Its Applications, Addison-Wesley, Longman Scientific, Reading, Massachusetts, 1996.

    Google Scholar 

  11. A. Padiy, O. Axelsson, and B. Polman, Generalized augmented matrix preconditioning approach and its application to itergative solution of illconditioned algebraic systems, SIAM J. Matrix Anal. Appl. 22 (2000), 793–818.

    Article  MathSciNet  MATH  Google Scholar 

  12. E. Perchat, L. Fourment and T. Coupez, Parallel incomplete factorisations for generalised Stokes problems: application to hot metal forging simulation, Report, EPFL, Lausanne, 2001.

    Google Scholar 

  13. G. J. M. Pieters, Stability analysis for a saline layer formed by uniform using finite elements, Report RANA 01-07, Eindhoven University of Technology, Eindhoven,2001.

    Google Scholar 

  14. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Elsevier Science Publishers, Amsterdam, 1984.

    MATH  Google Scholar 

  15. A. van der Sluis, H. van der Vorst, The rate of convergence of conjugate gradients, Numer. Math. 48 (1986), 543–560.

    Article  MathSciNet  MATH  Google Scholar 

  16. C. Vuik, A. Segal, L. el Yaakoubi and E. Dufour, A comparison of various deflation vectors applied to elliptic problems with discontinuous coefficients, Appl. Numer. Math. 41 (2002), 219–233.

    Article  MathSciNet  MATH  Google Scholar 

  17. C. Vuik, A. Segal, and J. A. Meijerink, An efficient preconditioned DG method for the solution of a class of layered problems with extreme contasts in the coefficients, J. Comput. Phys. 152 (1999), 385–403.

    Article  ADS  MATH  Google Scholar 

  18. C. Vuik, A. Segal, J. A. Meijerink, and G. T. Wijma, The construction of projection vectors for a Deflated ICCG method applied to problems with extreme contrasts in the coefficients, J. Comput. Phys. 172 (2001), 426–450.

    Article  MathSciNet  ADS  MATH  Google Scholar 

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© 2004 Springer-Verlag Berlin Heidelberg

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Vermolen, F., Vuik, K., Segal, G. (2004). Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems. In: Křížek, M., Neittaanmäki, P., Korotov, S., Glowinski, R. (eds) Conjugate Gradient Algorithms and Finite Element Methods. Scientific Computation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18560-1_7

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  • DOI: https://doi.org/10.1007/978-3-642-18560-1_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62159-8

  • Online ISBN: 978-3-642-18560-1

  • eBook Packages: Springer Book Archive

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