Skip to main content
SpringerLink
Log in

A circulant preconditioner for domain decomposition algorithm for the solution of the elliptic problems

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

A preconditioned conjugate gradient (PCG)-based domain decomposition method was given in [11] and [12] for the solution of linear equations arising in the finite element method applied to the elliptic Neumann problem. The novelty of the proposed algorithm was that the recommended preconditioner was constructed by using symmetric-cyclic matrix. But we could give only the definitions of the entries of this cyclic matrix. Here we give a short description of this algorithm, the method of calculation of matrix entries and the results of calculation. The numerical experiments presented show, that this construction of precondition in the practice works well.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. O. Axelsson andV. A. Barker, Finite Element Solution of Boundary Value problems,Theory and Applications, New York, Academic Press, 1984.

    Google Scholar 

  2. J. H. Bramble, J. E. Pasciak andA. H. Schatz, An Iterative Method for Elliptic Problems on Region Partitioned into Sabstructures,Math. Comp. Vol.46 No. 174 (1986), 361–369.

    Google Scholar 

  3. J. H. Bramble, J. E. Pasciak andA. H. Schatz, The Construction of Preconditioners for Elliptic Problems by Substructuring I,Math. of Comp. 47 No. 175 (1986), 103–134.

    Google Scholar 

  4. J. H. Bramble, J. E. Pasciak andA. H. Schatz, The Construction of Preconditioners for Elliptic Problems by Substructuring II,Math. of Comp. 49 No. 184 (1986), 1–6.

    Google Scholar 

  5. J. H. Bramble, J. E. Pasciak andA. H. Schatz, The Construction of Preconditioners for Elliptic Problems by Substructuring III,Math. of Comp. 51 No. 184 (1988), 415–430.

    Google Scholar 

  6. J. H. Bramble, J. E. Pasciak andA. H. Schatz, The Construction of Preconditioners for Elliptic Problems by Substructuring IV,Math. of Comp. 53 No. 187 (1989), 1–24.

    Google Scholar 

  7. BruceW. Char, KeithO. Geddes, GastonH. Gonnet, BentonL. Leong, MichaelB. Monagan and StephenM. Watt,Maple V Language Reference Manual Springer-Verlag 1991.

  8. R. H. Chan, Toeplitz Preconditioners for Toeplitz Systems with Nonnegative Generating Functions,IMA Journal of Numerical Analysis 11 (1991), 333–345.

    Google Scholar 

  9. R. H. Chan andT. F. Chan, Circulant Preconditioners for Elliptic Problems, to appear inJournal of Numerical Linear Algebra with Application.

  10. M. Dryja, A Finite Element Capacitance Method For Elliptic Problems on Region Partitioned into Subregions,Numer. Math. 44 (1984), 153–168.

    Google Scholar 

  11. B. Kiss andG. Molnárka, A preconditioned Domain Decomposition Algorithm for the Solution of the Elliptic Neumann Problem, In.:Parallel Algorithm for Differential Equations, ed. by W. Hackbush, Proc. of 6. GAMM-seminar, Kiel, January, (19–21), 1990. 119–129.

  12. B. Kiss andG. Molnárka, A preconditioned Domain Decomposition Algorithm for the Algorithm for the Elliptic Neumann Problems,Periodica Mathematica Hungarica Vol.24(3), (1992), 151–165.

    Google Scholar 

  13. J. L. Lions andE. Maggenes,Problèmes aux Limites non Homogènes et Applications, Dunod, Paris, 1968.

    Google Scholar 

  14. A. M. Matsokin andS. V. Nepomnyaschikh, A Schwarz Alternating Methods in Subspace,Soviet Mathematics,29(10) (1985), 78–84.

    Google Scholar 

  15. C. Moler J. Little andS. Bangert,PRO-MATLAB for Sun Workstations, Vers 3.2 SUN. August, 17.1987. The Math Works Inc.

  16. P. A. Raviart andJ. M. Thomas,Introduction à l'Analyse Numérique des Équations aux Derivées Partielles, Massson, Paris, 1983.

    Google Scholar 

  17. G. Strang, A Proposal for Toeplitz Matrix Calculations,Studies in Appl. Math. 74 (1986), 171–176.

    Google Scholar 

  18. O. Widlund,An Extension Theorem for Finite Element Spaces with Three Applications, Tehnical Report 233, Dept. of Computer Science, Conrant Inst. 1986.

  19. O. Widlund, Iterative Substructuring Methods,Algorithm and theory for Elliptic Problems in the Plane Technical Report 287, Dept. of Computer Science, Courant Institute, 1987.

  20. S. Wolfram,MATHEMATICA, A System for Doing Mathematics by Computer, Addison-Wesely P.C. Inc. 1988.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, István Széchenyi College, Hedervári u. 3, H-9026, Győr, Hungary

    B. Kiss & G. Molnárka

  2. Department of Numerical Analysis, Loránd Eőtvős University, Múzeum KRT 6-8, H-1088, Budapest, Hungary

    N. A. A. Rahman

Authors
  1. B. Kiss
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Molnárka
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. A. A. Rahman
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The authors was supported in part by TEMPUS JEP-0102-92/1 and Hungarian National Sciencific Foundation OTKA No.: 2150.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kiss, B., Molnárka, G. & Rahman, N.A.A. A circulant preconditioner for domain decomposition algorithm for the solution of the elliptic problems. Period Math Hung 29, 67–80 (1994). https://doi.org/10.1007/BF01876204

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01876204

Mathematics subject classification numbers, 1991

Key words and phrases

Search

Navigation