Advertisement

Mixed Defect Correction Iteration for the Solution of a Singular Perturbation Problem

  • P. W. Hemker
Part of the Computing Supplementum book series (COMPUTING, volume 5)

Abstract

We describe a discretization method (mixed defect correction) for the solution of a two-dimensional elliptic singular perturbation problem. The method is an iterative process in which two basic discretization schemes are used: one with and one without artificial diffusion. The resulting method is stable and yields a 2nd order accurate approximation in the smooth parts of the solution, without using any special directional bias in the discretization. The method works well also for problems with interior or boundary layers.

Keywords

Boundary Data Multigrid Method Smooth Part Discrete Operator Singular Perturbation Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Brandt, A., Dinar, N.: Multigrid solutions to elliptic flow problems. In: Numerical Methods for Partial Differential Equations (Parter, S. V., ed.), pp. 53–147. New York: Academic Press 1979.Google Scholar
  2. [2]
    Brandt, A.: Guide to multigrid development. In: Multigrid Methods (Hackbusch, W., Trottenberg, U., eds.), pp. 220–312. (Lecture Notes in Mathematics, Vol. 960.) Berlin-Heidelberg-New York: Springer 1982.CrossRefGoogle Scholar
  3. [3]
    Hackbusch, W.: Bemerkungen zur iterierten Defektkorrektur und zu ihrer Kombination mit Mehrgitterverfahren. Rev. Roum. Math. Pures Appl. 26, 1319–1329 (1981).MathSciNetMATHGoogle Scholar
  4. [4]
    Hackbusch, W.: On multigrid iterations with defect correction. In: Multigrid Methods (Hackbusch, W., Trottenberg, U., eds.), pp. 461–473. (Lecture Notes in Mathematics, Vol. 960.) Berlin-Heidelberg-New York: Springer 1982.CrossRefGoogle Scholar
  5. [5]
    Hemker, P. W.: Fourier analysis of gridfunctions, prolongations and restrictions. Report NW 93/80, Mathematisch Centrum, Amsterdam, 1980.MATHGoogle Scholar
  6. [6]
    Hemker, P. W.: An accurate method without directional bias for the numerical solution of a 2 D elliptic singular perturbation problem. In: Theory and Applications of Singular Perturbations (Eckhaus, W., de Jager, E. M., eds.), pp. 192–206. (Lecture Notes in Mathematics, Vol. 942.) Berlin-Heidelberg-New York: Springer 1982.CrossRefGoogle Scholar
  7. [7]
    Hemker, P. W.: Mixed defect correction iteration for the accurate solution of the convection diffusion equation. In: Multigrid Methods (Hackbusch, W., Trottenberg, U., eds.), pp. 485–501. (Lecture Notes in Mathematics, Vol. 960.) Berlin-Heidelberg-New York: Springer 1982.CrossRefGoogle Scholar
  8. [8]
    Hemker, P. W., de Zeeuw, P. M.: Defect correction for the solution of a singular perturbation problem. In: Scientific Computing (Stepleman, R. S., ed.), North-Holland 1983.Google Scholar
  9. [9]
    Miller, J. J. H. (ed.): Boundary and Interior Layers — Computational and Asymptotic Methods. Dublin: Boole Press 1980.MATHGoogle Scholar
  10. [10]
    Hughes, T. J. R., Brooks, A.: A multidimensional upwind scheme with no crosswind diffusion. In: Finite Element Methods for Convection Dominated Flows (Hughes, T. J. R., ed.). (AMD, Vol. 34.) The American Society of Mechanical Engineers 1979.Google Scholar
  11. [11]
    Stetter, H.: The defect correction principle and discretization methods. Num. Math. 29, 425–443 (1978).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • P. W. Hemker
    • 1
  1. 1.Department of Numerical MathematicsCentre for Mathematics and Computer ScienceAmsterdamThe Netherlands

Personalised recommendations