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Preconditioning indefinite systems arising from mixed finite element discretization of second-order elliptic problems

Submitted Papers

Part of the Lecture Notes in Mathematics book series (LNM,volume 1457)

Abstract

We discuss certain preconditioning techniques for solving indefinite linear systems of equations arising from mixed finite element discretizations of elliptic equations of second order. The techniques are based on various approximations of the mass matrix, say, by simply lumping it to be diagonal or by constructing a diagonal matrix assembled of properly scaled lumped element mass matrices. We outline two possible alternatives for preconditioning. One can precondition the original (indefinite) system by some indefinite matrix and hence use either a stationary iterative method or a generalized conjugate gradient type method. Alternatively as in the particular case of rectangular Raviart-Thomas elements, which we consider, one can perform iterations in a subspace, eliminating the velocity unknowns and then considering the corresponding reduced system which is elliptic. So we can use the ordinary preconditioned conjugate gradient method and any known preconditioner (of optimal order, for example, like the multigrid method) for the corresponding finite element discretization of the elliptic problem. Numerical experiments for some of the proposed iterative methods are presented.

Keywords

  • indefinite system
  • preconditioning
  • iterations in subspace
  • conjugate gradients
  • mixed finite elements
  • second order elliptic problems
  • Subject Classifications
  • AMS(MOS) 65F10
  • 65N20
  • 65N30

On leave from Center for Informatics and Computer Technology, Bulgarian Academy of Sciences, G. Bontchev str., bl. 25-A, 1113 Sofia, Bulgaria

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References

  1. O. Axelsson, A survey of vectorizable preconditioning methods for large scale finite element matrix problems, Colloquium Topics in Applied Numerical Analysis, J.G. Verwer, ed., CWI Syllabus 4, Math. Center, Amsterdam, 1984.

    Google Scholar 

  2. O. Axelsson. A generalized conjugate gradient, least square method, Numer. Math., 51 (1987), 209–227.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. O. Axelsson, P.S. Vassilevski. Algebraic multilevel preconditioning methods, I, Numer. Math. 56 (1989), 157–177.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. O. Axelsson, P.S. Vassilevski. A black box generalized conjugate gradient solver with inner iterations and variable step preconditioning, Report 9833, Department of Mathematics, University of Nijmegen, The Netherlands, 1989.

    MATH  Google Scholar 

  5. R.E. Bank, B.D. Welfert and H. Yserentant. A class of iterative methods for solving saddle point problems, Numer. Math., 56 (1990), 645–666.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. J.H. Bramble and J.E. Pasciak. A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comp., 181 (1988), 1–17.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. J.H. Bramble, J.E. Pasciak and J. Xu. The analysis of multigrid algorithms with non-nested spaces or non-inherited quadratic forms, Math. Comp. (to appear).

    Google Scholar 

  8. R.E. Ewing and M.F. Wheeler, Computational aspects of mixed finite element methods, Numerical Methods for Scientific Computing, R.S. Stepleman, ed., Amsterdam, North Holland, 1983, 163–172.

    Google Scholar 

  9. R.S. Falk and J.E. Osborn. Error estimates for mixed methods, RAIRO Numer. Anal. 14 (1980), 249–277.

    MathSciNet  MATH  Google Scholar 

  10. V. Girault and P.A. Raviart. Finite element approximation of the Navier-Stokes equations, Lecture Notes in Math. 749, Springer-Verlag, New York, 1981.

    MATH  Google Scholar 

  11. P. Lu, M.B. Allen and R.E. Ewing. A semi-implicit iteration scheme using the multigrid method for mixed finite element equations, (submitted).

    Google Scholar 

  12. P.A. Raviart and J.M. Thomas. A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, 606, Galligani and E. Magenes, eds., Springer-Verlag, Berlin, 1977, 292–315.

    CrossRef  Google Scholar 

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© 1990 Springer-Verlag

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Ewing, R.E., Lazarov, R.D., Lu, P., Vassilevski, P.S. (1990). Preconditioning indefinite systems arising from mixed finite element discretization of second-order elliptic problems. In: Axelsson, O., Kolotilina, L.Y. (eds) Preconditioned Conjugate Gradient Methods. Lecture Notes in Mathematics, vol 1457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090900

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  • DOI: https://doi.org/10.1007/BFb0090900

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53515-7

  • Online ISBN: 978-3-540-46746-5

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