BIT Numerical Mathematics

, Volume 29, Issue 4, pp 748–768 | Cite as

Two-level hierarchically preconditioned conjugate gradient methods for solving linear elasticity finite element equations

  • M. Jung
  • U. Langer
  • U. Semmler
Preconditioned Conjugate Gradient Methods


In this paper we study and compare some preconditioned conjugate gradient methods for solving large-scale higher-order finite element schemes approximating two- and three-dimensional linear elasticity boundary value problems. The preconditioners discussed in this paper are derived from hierarchical splitting of the finite element space first proposed by O. Axelsson and I. Gustafsson. We especially focus our attention to the implicit construction of preconditioning operators by means of some fixpoint iteration process including multigrid techniques. Many numerical experiments confirm the efficiency of these preconditioners in comparison with classical direct methods most frequently used in practice up to now.

AMS (MOS) Subject Classifications

65F10 65N20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    O. Axelsson,On multigrid methods of the two-level type, In [20], pp. 352–367.Google Scholar
  2. [2]
    O. Axelsson and I. Gustafsson,Preconditioning and two-level multigrid methods of arbitrary degree of approximation, Math. Comp. 40 (1983), 219–242.Google Scholar
  3. [3]
    R. Bank and T. Dupont,Analysis of a two-level scheme for solving finite element equations. Report CNA-159, Centre for Numerical Analysis, The University of Texas at Austin, 1980.Google Scholar
  4. [4]
    D. Braess,The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation. In [20], pp. 368–386.Google Scholar
  5. [5]
    D. Braess,The contraction number of a multigrid method for solving the Poisson equation. Numer. Math. 37 (1981), 387–404.Google Scholar
  6. [6]
    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 [20], 535–544.Google Scholar
  7. [7]
    C. A. Thole,Beiträge zur Fourieranalysis von Mehrgittermethoden: V-cycle, ILU-Glättung, anisotrope Operatoren. Diplomarbeit, Institut für Angewandte Mathematik, Universität Bonn, 1983.Google Scholar
  8. [8]
    N. Schieweck,A multigrid convergence proof by a strengthened Cauchy inequality for symmetric elliptic boundary value problems. InSecond Multigrid Seminar, Garzau, November 5–8, 1985, Report R-MATH-08/86, AdW der DDR, Karl-Weierstraß-Institut für Mathematik, Berlin, 1986, pp. 49–62.Google Scholar
  9. [9]
    M. Jung, Konvergenzfaktoren von Mehrgitterverfahren für Probleme der ebenen, linearen Elastizitätstheorie, ZAMM 67 (1987), 165–173.Google Scholar
  10. [10]
    D. Braess,The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation. Math. Comp. 42 (1984), 505–519.Google Scholar
  11. [11]
    R. Verfürth,The contraction number of a multigrid method with mesh ratio 2 for solving Poisson's equation, Linear Algebra Appl. 60 (1984), 113–128.Google Scholar
  12. [12]
    H. Yserentant,On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986), 379–412.Google Scholar
  13. [13]
    O. Axelsson and P. S. Vassilevski,Algebraic multilevel preconditioning methods, I, Report 881, Department of Mathematics, Catholic University Nijmegen, The Netherlands, 1988.Google Scholar
  14. [14]
    V. G. Korneev and U. Langer,Approximate solution of plastic flow theory problems. Teubner-Texte zur Mathematik, Bd. 69, Teubner-Verlag, Leipzig, 1984.Google Scholar
  15. [15]
    M. Jung,Eine Einführung in die Theorie und Anwendung von Mehrgitterverfahren. Wissenschaftliche Schriftenreihe der TU Karl-Marx-Stadt, Karl-Marx-Stadt, 1989.Google Scholar
  16. [16]
    O. Axelsson and V. A. Barker,Finite Element Solution of Boundary Value Problems. Theory and Computation. Academic Press, New York, 1984.Google Scholar
  17. [17]
    A. A. Samarskij and E. S. Nikolaev,Methods for solving grid equations. Nauka, Moscow, 1978, (in Russian).Google Scholar
  18. [18]
    U. Langer and W. Queck,Preconditioned Uzawa-type iterative methods for solving mixed finite element equations: Theory — Applications — Software. Wissenschaftliche Schriftenreihe der TU Karl-Marx-Stadt, Karl-Marx-Stadt, 3/1987.Google Scholar
  19. [19]
    M. Jung, U. Langer, A. Meyer, W. Queck and M. Schneider,Multigrid preconditioners and their applications. InThird Multigrid Seminar, Biesenthal, May 2–6, 1988, Report R-MATH-89, AdW der DDR, Karl-Weierstraß-Institut für Mathematik, Berlin, 1989.Google Scholar
  20. [20]
    W. Hackbusch and U. Trottenberg (editors),Multigrid Methods, Proceedings of the Conference held at Köln-Porz, November 23–27, 1981, Lect. Notes in Mathe. 960, Springer-Verlag, Berlin, 1982.Google Scholar
  21. [21]
    T. A. Mannteuffel,An incomplete factorization technique for positive definite linear systems, Math. Comp. 34 (1980), 473–497.Google Scholar
  22. [22]
    I. Gustafsson,A class of first order factorization methods. Comp. Sc. 77.04R, Chalmers Univ. of Techn., Gothenburg 1977.Google Scholar
  23. [23]
    A. Meyer,On the O(h −1)-property of MAF-preconditioning for certain matrices arising in FEM- and FDM-discretization, Preprint Nr. 45, TU Karl-Marx-Stadt, Karl-Marx-Stadt, 1987.Google Scholar
  24. [24]
    Ph. G. Ciarlet and P. A. Raviart,Maximum principle and uniform convergence for the finite element method. Comp. Meth. Appl. Mech. Engin. 2 (1973), 17–31.Google Scholar
  25. [25]
    B. Heinrich,Finite difference methods on irregular networks. ISNM, vol. 82, Birkhäuser Verlag, Basel, 1987.Google Scholar
  26. [26]
    V. G. Korneev,Finite element schemes of higher order of accuracy, Izdatel'stvo LGU, Leningrad, 1977, (in Russian).Google Scholar
  27. [27]
    U. Semmler and J. Leopold,Dialogorientierter automatischer FEM-Vernetzungsgenerator und seine Anwendung im Programmsystem FEP AS, Preprint Nr. 41, TU Karl-Marx-Stadt, Karl-Marx-Stadt 1987.Google Scholar
  28. [28]
    U. Semmler,Vorkonditionierung für konjugierte Gradientenverfahren zur Lösung großdimensionierter FEM-Systeme. Preprint Nr. 92, TU Karl-Marx-Stadt, Karl-Marx-Stadt 1989.Google Scholar

Copyright information

© BIT Foundations 1989

Authors and Affiliations

  • M. Jung
    • 1
  • U. Langer
    • 1
  • U. Semmler
    • 1
  1. 1.Technical University of Karl-Marx-StadtGerman Democratic Republic

Personalised recommendations