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A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations

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Abstract

A class of modified block SSOR preconditioners is presented for solving symmetric positive definite systems of linear equations, which arise in the hierarchical basis finite element discretizations of the second order self‐adjoint elliptic boundary value problems. This class of methods is strongly related to two level methods, standard multigrid methods, and Jacobi‐like hierarchical basis methods. The optimal relaxation factors and optimal condition numbers are estimated in detail. Theoretical analyses show that these methods are very robust, and especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

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References

  1. O. Axelsson, A generalized SSOR method, BIT 12 (1972) 443-467.

    Article  MATH  MathSciNet  Google Scholar 

  2. O. Axelsson, Incomplete block matrix factorization preconditioning methods. The ultimate answer?, J. Comput. Appl. Math. 12/13 (1985) 3-18.

    Article  MathSciNet  Google Scholar 

  3. O. Axelsson, Iterative Solution Methods (Cambridge University Press, Cambridge, 1994).

    Google Scholar 

  4. O. Axelsson and V.A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation (Academic Press, New York, 1984).

    Google Scholar 

  5. O. Axelsson and B. Polman, On approximate factorization methods for block matrices suitable for vector and parallel processors, Linear Algebra Appl. 77 (1986) 3-26.

    Article  MATH  MathSciNet  Google Scholar 

  6. Z.-Z. Bai, The modified block SSOR preconditioners for symmetric positive definite linear systems, to appear (1997).

  7. R.E. Bank and T.F. Dupont, Analysis of a two-level scheme for solving finite element equations, Report CNA-159, Center for Numerical Analysis, University of Texas at Austin (1980).

  8. R.E. Bank and T.F. Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981) 35-51.

    Article  MATH  MathSciNet  Google Scholar 

  9. R.E. Bank, T.F. Dupont and H. Yserentant, The hierarchical basis multigrid method, Numer. Math. 52 (1988) 427-458.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Braess, The contraction number of a multigrid method for solving Poisson equation, Numer. Math. 37 (1981) 387-404.

    Article  MATH  MathSciNet  Google Scholar 

  11. C. Concus, G.H. Golub and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Statist. Comput. 6 (1985) 220-252.

    Article  MATH  MathSciNet  Google Scholar 

  12. G.H. Golub and C.F. Van Loan, Matrix Computation, 3rd ed. (Johns Hopkins University Press, Baltimore, MD, 1996).

    Google Scholar 

  13. I. Gustafsson, A class of first order factorization methods, BIT 18 (1978) 142-156.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. Hackbusch, Multigrid Methods and Applications (Springer, Berlin, 1985).

    Google Scholar 

  15. A. Hadjidimos, Accelerated overrelaxation method, Math. Comp. 32 (1978) 149-157.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Hadjidimos, On the generalization of the basic iterative methods for the solution of the linear systems, Internat. J. Comput. Math. 14 (1983) 355-369.

    MATH  MathSciNet  Google Scholar 

  17. A. Hadjidimos, A. Psimarni and A. Yeyios, On the convergence of some generalized iterative methods, Linear Algebra Appl. 75 (1986) 117-132.

    Article  MATH  MathSciNet  Google Scholar 

  18. L.A. Hageman and D.M. Young, Applied Iterative Methods (Academic Press, New York, 1981).

    Google Scholar 

  19. J.F. Maitre and F. Musy, The contraction number of a class of two-level methods; an exact evaluation for some finite element subspaces and model problems, in: Multigrid Methods, eds. W. Hackbusch and U. Trottenberg, Lecture Notes in Mathematics, Vol. 960 (Springer, Berlin, 1992).

    Google Scholar 

  20. M.G. Neytcheva, Arithmetic and communication complexity of preconditioning methods, Ph.D. thesis, Katholieke Universiteit, Nijmegen (September 1995).

    Google Scholar 

  21. Y. Saad, Iterative Methods for Sparse Linear Systems (PWS, Boston, 1996).

    Google Scholar 

  22. R.S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1962).

    Google Scholar 

  23. D.M. Young, Iterative Solution of Large Linear Systems (Academic Press, New York, 1971).

    Google Scholar 

  24. H. Yserentant, On the multi-level splitting finite element spaces, Numer. Math. 49 (1986) 379-412.

    Article  MATH  MathSciNet  Google Scholar 

  25. H. Yserentant, Hierarchical bases give conjugate gradient type methods a multigrid speed of convergence, Appl. Math. Comput. 19 (1986) 347-358.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100080, PR China

    Zhong‐Zhi Bai

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  1. Zhong‐Zhi Bai
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Bai, Z. A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations. Advances in Computational Mathematics 10, 169–186 (1999). https://doi.org/10.1023/A:1018974514896

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