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Iterative Methods for Elliptic Finite Element Equations on General Meshes

  • R. A. Nicolaides
  • Shenaz Choudhury
Part of the ICASE/NASA LaRC Series book series (ICASE/NASA)

Abstract

It is fair to say that the development of iterative solution techniques for all kinds of discretized partial differential equations remains a vigorous branch of numerical analysis. Perhaps the greater part of the effort has gone into multigrid algorithms, the next most common topic being preconditioning methods. Traditionally applied to elliptic problems, multigrid methods have also recently been successfully applied to solving the hyperbolic equations of gas dynamics (see [23] for a survey).

Keywords

Iterative Method Conjugate Gradient Conjugate Gradient Method Multigrid Method Finite Element Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    O. Axelsson and V.A. Barker, Finite Element Solution of Boundary Value Problems, Academic Press, Orlando, 1984.MATHGoogle Scholar
  2. 2.
    M. Bercovier and A. Rosenthal, “Using the conjugate gradient method with preconditioning for solving FEM approximations of elasticity problems,” Engrg. Comput., 3(1986), 77.CrossRefGoogle Scholar
  3. 3.
    P.E. Bjorstad and O.B. Widlund, “Iterative methods for the solution of elliptic problems on regions partitioned into substructures,” Technical Report 136, Courant Institute of Mathematical Sciences, New York University, 1984.Google Scholar
  4. 4.
    J.H. Bramble, J.E. Pasciak, and A.H. Schatz, “The construction of preconditioners for elliptic problems by substructuring,” Math. Comp., 47(1986), 103.CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    N.I. Buleev, “A numerical method for the solution of two-dimensional and three-dimensional equations of diffusion,” Mat. Sb., 51(1960), 227.Google Scholar
  6. 6.
    T.F. Chan and D.C. Resasco, “A survey of preconditioners for domain decomposition,” Research Report YALEU/DCS/RR-414, Department of Computer Science, Yale University, New Haven, 1985.Google Scholar
  7. 7.
    R. Chandra, “Conjugate gradient methods for partial differential equations,” Ph.D. dissertation, Department of Computer Science, Yale University, New Haven, 1978.Google Scholar
  8. 8.
    P. Concus, G.H. Golub, and G. Meurant, “Block preconditioning for the conjugate gradient method,” SI AM J. Sci. Statist. Comput., 6(1985), 220.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    P. Concus, G.H. Golub, and D.P. O’DLeary, “A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations,” Sparse Matrix Computations(J.R. Bunch and D.J. Rose, eds.), Academic Press, New York, 1976, p. 309.Google Scholar
  10. 10.
    M. Dryja, “A capacitance matrix method for the Dirichlet problem on polygonal region,” Numer. Math., 39(1982), 51.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    T. Dupont, R.P. Kendall, and H.H. Rachford, Jr., “An approximate factorization procedure for solving self-adjoint elliptic difference equations” SIAM J. Numer. Anal, 5(1968), 559.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    T. Dupont, “A factorization procedure for the solution of elliptic difference equations,” SIAM J. Numer. Anal, 5(1968), 753.CrossRefMathSciNetGoogle Scholar
  13. 13.
    H.C. Elman, “Iterative methods for large, sparse, nonsymmetric systems of linear equations,” Ph.D. dissertation, Department of Computer Science, Yale University, New Haven, 1982.Google Scholar
  14. 14.
    G.E. Forsythe and W.R. Wasow, Finite-Difference Methods for Partial Differential Equations, Wiley, New York, 1960.MATHGoogle Scholar
  15. 15.
    S.P. Frankel, “Convergence rates of iterative treatments of partial differential equations,” Math. Tables Aids Comput., 4(1950), 65.CrossRefMathSciNetGoogle Scholar
  16. 16.
    G.H. Golub and D. Mayers, “The use of pre-conditioning over irregular regions,” Lecture at the 6th International Conference on Computing Methods in Applied Sciences and Engineering (Versailles, 1983 ).Google Scholar
  17. 17.
    M.D. Gunzburger and R.A. Nicolaides, “On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations,” Appl Numer. Math., 2(1986).Google Scholar
  18. 18.
    I. Gustafsson, “A class of first-order factorization methods,” BIT, 18(1978), 142.CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    I. Gustafsson, “Modified incomplete Cholesky methods,” Preconditioning Methods: Analysis and Applications(D.J. Evans, ed.), Gordon and Breach, New York, 1983, p. 265.Google Scholar
  20. 20.
    W. Hackbusch, Multi-Grid Methods and Applications, Springer-Verlag, Berlin, 1985.CrossRefMATHGoogle Scholar
  21. 21.
    L.A. Hageman and D.M. Young, Applied Iterative Methods, Academic Press, New York, 1981.MATHGoogle Scholar
  22. 22.
    T.J.R. Hughes, I. Levit, and J. Winget, “Element-by-element implicit algorithms for heat conduction,” J. Engrg. Mech., 109(1983), 576.CrossRefGoogle Scholar
  23. 23.
    A. Jameson, “Multigrid algorithms for compressible flow calculations,” MAE Report 1743, Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 1985.Google Scholar
  24. 24.
    D.S. Kershaw, “The incomplete Cholesky conjugate gradient method for the iterative solution of systems of linear equations,” J. Comput. Phys., 26(1978), 43.CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    R. Löhner and K. Morgan, “An unstructured multigrid method for elliptic problems,” Internat. J. Numer. Methods Engrg. 24(1987), 101.CrossRefMATHGoogle Scholar
  26. 26.
    D.G. Luenberger, Linear and Nonlinear Programming, 2nd ed., Addison-Wesley, Reading, MA, 1984.MATHGoogle Scholar
  27. 27.
    T.A. Manteuffel, “An incomplete factorization technique for positive definite linear systems,” Math. Comp., 34(1980), 473.CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    J.A. Meijerink and H.A. van der Vorst, “An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix,” Math. Comp., 31(1977), 148.MATHMathSciNetGoogle Scholar
  29. 29.
    J.A. Meijerink and H.A. van der Vorst, “Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems,” J. Comput. Phys., 44(1981), 134.CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    R.A. Nicolaides, “On the l2 convergence of an algorithm for solving finite element equations,” Math. Comp., 31(1977), 892.MATHMathSciNetGoogle Scholar
  31. 31.
    R.A. Nicolaides, “On some theoretical and practical aspects of multigrid methods,” Math. Comp., 33(1979), 933.CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    R.A. Nicolaides, “Deflation of conjugate gradients with applications to boundary value problems,” SIAM J. Numer. Anal., 24(1987), 355.CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    B. Nour-Omid and B.N. Parlett, “Element preconditioning using splitting techniques,” SIAM J. Sci. Statist. Comput., 6(1985), 761.CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    L.F. Richardson, “The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam,” Philos. Trans. Roy. Soc. London Ser. A, 210(1910), 307, and Proc. Roy. Soc. London Ser. A, 83(1910), 335.CrossRefMATHGoogle Scholar
  35. 35.
    J. Ruge and K. Stüben, “Efficient solution of finite difference and finite element equations by algebraic multigrid,” in Proceedings of the Multigrid Conference (Bristol, 1983 ).Google Scholar
  36. 36.
    H. Rutishauser, “Theory of gradient methods,” in Refined Iterative Methods for Computation of the Solution and the Eigenvalues of Self-Adjoint Boundary Value Problems, (by M. Engeli, T. Ginsburg, H. Rutishauser, and E. Stiefel), Birkhéuser, Basel, 1959, p. 24.CrossRefGoogle Scholar
  37. 37.
    H.L. Stone, “Iterative solution of implicit approximations of multidimensional partial differential equations,” SIAM J. Numer. Anal., 5(1968), 530.CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962.Google Scholar
  39. 39.
    R.S. Varga, “Factorization and normalized iterative methods,” in Boundary Value Problems in Differential Equations(R.E. Langer, ed.), University of Wisconsin Press, Madison, 1960, p. 121.Google Scholar
  40. 40.
    D.M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York  1988

Authors and Affiliations

  • R. A. Nicolaides
  • Shenaz Choudhury

There are no affiliations available

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