Advertisement

The Defect Correction Approach

  • K. Böhmer
  • P. W. Hemker
  • H. J. Stetter
Part of the Computing Supplementum book series (COMPUTING, volume 5)

Abstract

This is an introductory survey of the defect correction approach which may serve as a unifying frame of reference for the subsequent papers on special subjects.

Keywords

Asymptotic Expansion Discretization Method Multigrid Method Discretization Error Approximate Inverse 
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]
    Allgower, E. L., Böhmer, K.: A mesh independence principle for operator equations and their discretizations. GMD Report, Bonn, 1984.Google Scholar
  2. [2]
    Allgower, E. L., Böhmer, K., McCormick, S. F.: Discrete defect corrections: the basic ideas. ZAMM 62, 371–377 (1982).CrossRefGoogle Scholar
  3. [3]
    Allgower, E. L., McCormick, S. F.: Newton’s method with mesh refinements for numerical solution of nonlinear two-point boundary value problems. Numer. Math. 29, 237–260 (1978).MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Atkinson, K. E.: Iterative variants of the Nyström method for the numerical solution of integral equations. Numer. Math. 22, 17–33 (1973).MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Bank, R. E., Sherman, A. H.: An adaptive multi-level method for elliptic boundary value problems. Computing 26, 91–105 (1982).MathSciNetCrossRefGoogle Scholar
  6. [6]
    Barkai, D., Brandt, A.: Vectorized multigrid Poisson solver for the CDC CYBER 205. Appl. Math. and Computation 48, 215–227 (1983).CrossRefGoogle Scholar
  7. [7]
    Björck, A., Pereyra, V.: Solution of Vandermonde systems of equations. Math. Comp. 24, 893–903 (1970).MathSciNetCrossRefGoogle Scholar
  8. [8]
    Böhmer, K.: Defect corrections via neighbouring problems. I. General theory., MRC Report, University of Wisconsin-Madison No. 1750 (1977).Google Scholar
  9. [9]
    Böhmer, K.: Discrete Newton methods and iterated defect corrections, I. General theory, II. Initial and boundary value problems in ordinary differential equations. Berichte Nr. 10, 11, Universität Karlsruhe, Fakultät für Mathematik (1978).Google Scholar
  10. [10]
    Böhmer, K.: Discrete Newton methods and iterated defect corrections. Numer. Math. 37, 167–192 (1981).MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Böhmer, K.: Asymptotic expansion for the discretization error in linear elliptic boundary value problems on general regions. Math. Z. 177, 235–255 (1981).MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Brakhage, H.: Über die numerische Behandlung von Integralgleichungen nach der Quadraturformelmethode. Numer. Math. 2, 183–196 (1960).MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comp. 31, 333–390 (1977).MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Brandt, A., Dinar, N: Multigrid solutions to elliptic flow problems. In: Numerical Methods for Partial Differential Equations (Parter, S., ed.), pp. 53–147. Academic Press 1979.Google Scholar
  15. [15]
    Bulirsch, R., Stoer, J.: Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math. 8, 1–13 (1966).MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Dendy, J. E., jr.: Black box multigrid. J. Comp. Phys. 48, 366–386 (1982).MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Fox, L.: Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations. Proc. Roy. Soc. London A 190, 31–59 (1947).Google Scholar
  18. [18]
    Fox, L.: The solution by relaxation methods of ordinary differential equations. Proc. Cambridge Phil. Soc. 45, 50–68 (1949).MATHCrossRefGoogle Scholar
  19. [19]
    Fox, L., ed.: Numerical Solution of Ordinary and Partial Differential Equations. Oxford: Pergamon Press 1962.MATHGoogle Scholar
  20. [20]
    Fox, L.: The Numerical Solution of Two-point Boundary Value Problems in Ordinary Differential Equations. Oxford: University Press 1957.Google Scholar
  21. [21]
    Foerster, H., Witsch, K.: Multigrid software for the solution of elliptic problems on rectangular domains: MGOO. In: Multigrid Methods (Hackbusch, W., Trottenberg, U., eds.), pp. 427–460. (Lecture Notes in Mathematics, Vol. 960.) Berlin-Heidelberg-New York: Springer 1982.CrossRefGoogle Scholar
  22. [22]
    Frank, R.: The method of iterated defect-correction and its application to two-point boundary value problems. Numer. Math. 25, 409–419 (1976).MATHCrossRefGoogle Scholar
  23. [23]
    Frank, R., Hertling, J., Ueberhuber, C. W.: An extension of the applicability of iterated deferred corrections. Math. Comp. 31, 907–915 (1977).MathSciNetMATHGoogle Scholar
  24. [24]
    Frank, R., Ueberhuber, C. W.: Iterated defect correction for differential equations, part I: Theoretical results. Computing 20, 207–228 (1978).MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    Frank, R., Ueberhuber, C. W.: Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations. BIT 17, 46–159 (1977).MathSciNetCrossRefGoogle Scholar
  26. [26]
    Gragg, W. B.: Repeated extrapolation to the limit in the numerical solution of ordinary differential equations. Thesis, UCLA, 1963.Google Scholar
  27. [27]
    Gragg, W. B.: On extrapolation algorithms for ordinary initial value problems. SIAM J. Num. Anal. 2, 384–403 (1965).MathSciNetGoogle Scholar
  28. [28]
    Hackbusch, W.: Error analysis of the nonlinear multigrid method of the second kind. Apl. Mat. 26, 18–29 (1981).MathSciNetMATHGoogle Scholar
  29. [29]
    Hackbusch, W.: Bemerkungen zur iterierten Defektkorrektur und zu ihrer Kombination mit Mehrgitterverfahren. Report 79-13, Math. Institut Universität Köln (1979); Rev. Roum. Math. Pures Appl. 26, 1319–1329 (1981).MathSciNetMATHGoogle Scholar
  30. [30]
    Hackbusch, W.: Die schnelle Auflösung der Fredholmschen Integralgleichung zweiter Art. Beiträge Numer. Math. 9, 47–62 (1981).Google Scholar
  31. [31]
    Hackbusch, W.: On the regularity of difference schemes — Part II: Regularity estimates for linear and nonlinear problems. Ark. Mat. 21, 3–28 (1982).MathSciNetCrossRefGoogle Scholar
  32. [32]
    Hairer, E.: On the order of iterated defect corrections. Numer. Math. 29, 409–424 (1978).MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Hemker, P. W., Schippers, H.: Multiple grid methods for the solution of Fredholm equations of the second kind. Math. Comp. 36, 215–232 (1981).MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Hemker, P. W., Wesseling, P., De Zeeuw, P. M.: A portable vector code for autonomous multigrid modules. In: PDE Software: Modules, Interfaces and Systems (Engquist, B., ed.), pp. 29–40. Procs. IFIP WG 2.5 Working Conference, North-Holland, 1984.Google Scholar
  35. [34a]
    Lin Qun, Zhu Qiding: Asymptotic expansions for the derivative of finite elements, J. Comp. Math. 2 (1984, to appear).Google Scholar
  36. [35]
    Lindberg, B.: Error estimation and iterative improvement for the numerical solution of operator equations. Report UIUCDS-R-76-820 (1976).Google Scholar
  37. [36]
    Lindberg, B.: Error estimation and iterative improvement for discretization algorithms.BIT 20, 486–500 (1980).MathSciNetMATHCrossRefGoogle Scholar
  38. [37]
    Marchuk, G. I., Shaidurov, V. V.: Difference Methods and Their Extrapolations. Berlin-Heidelberg-New York: Springer 1983.MATHGoogle Scholar
  39. [38]
    Munz, H.: Uniform expansions for a class of finite difference schemes for elliptic boundary value problems. Math. Comp. 36, 155–170 (1981).MathSciNetMATHCrossRefGoogle Scholar
  40. [39]
    Novak, Z., Wesseling, P.: Multigrid acceleration of an iterative method with application to transonic potential flow. (To appear.)Google Scholar
  41. [40]
    Ortega, J. M., Rheinboldt, C. W.: On discretization and differentiation of operators with applications to Newton’s method. SIAM J. Numer. Anal. 3, 143–156 (1966).MathSciNetMATHCrossRefGoogle Scholar
  42. [41]
    Pereyra, V.: The difference correction method for nonlinear two-point boundary value problems. — Techn. Rep. CS 18, Comp. Sc. Dept., Stanford Univ., California, 1965.Google Scholar
  43. [42]
    Pereyra, V.: Accelerating the convergence of discretization algorithms. SIAM J. Numer. Anal. 4, 508–533 (1967).MathSciNetMATHCrossRefGoogle Scholar
  44. [43]
    Pereyra, V.: Iterated deferred corrections for nonlinear operator equations. Numer. Math. 10, 316–323 (1967).MathSciNetMATHCrossRefGoogle Scholar
  45. [44]
    Pereyra, V.: Iterated deferred corrections for nonlinear boundary value problems. Numer. Math. 11, 111–125 (1968).MathSciNetMATHCrossRefGoogle Scholar
  46. [45]
    Pereyra, V.: Highly accurate numerical solution of quasilinear elliptic boundary-value problems in n dimensions. Math. Comp. 24, 111–783 (1970).Google Scholar
  47. [46]
    Pereyra, V., Proskurowski, W., Widlund, O.: High order fast Laplace solvers for the Dirichlet problem on general regions. Math. Comp. 31, 1–16 (1977).MathSciNetMATHCrossRefGoogle Scholar
  48. [47]
    Skeel, R. D.: A theoretical framework for proving accuracy results for deferred corrections. SIAM J. Numer. Anal. 19, 171–196 (1981).MathSciNetCrossRefGoogle Scholar
  49. [48]
    Stetter, H. J.: Asymptotic expansions for the error of discretization algorithms for nonlinear functional equations. Numer. Math. 7, 18–31 (1965).MathSciNetMATHCrossRefGoogle Scholar
  50. [49]
    Stetter, H. J.: Analysis of Discretization Methods for Ordinary Differential Equations. Berlin-Heidelberg-New York: Springer 1973.MATHGoogle Scholar
  51. [50]
    Stetter, H. J.: Economical global error estimation. In: Stiff Differential Systems (Willoughby, R. A., ed.), pp. 245–258. New York-London: Plenum Press 1974.Google Scholar
  52. [51]
    Stetter, H. J.: The defect correction principle and discretization methods. Numer. Math. 29, 425–443 (1978).MathSciNetMATHCrossRefGoogle Scholar
  53. [52]
    Ueberhuber, C. W.: Implementation of defect correction methods for stiff differential equations. Computing 23, 205–232 (1979).MathSciNetMATHCrossRefGoogle Scholar
  54. [53]
    Wesseling, P.: A robust and efficient multigrid method. In: Multigrid Methods (Hackbusch, W., Trottenberg, U., eds.), pp. 613–630. (Lecture Notes in Mathematics, Vol. 960.) Berlin-Heidelberg-New York: Springer 1982.Google Scholar
  55. [54]
    Zadunaisky, P. E.: A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations. In: The Theory of Orbits in the Solar System and in Stellar Systems. Proc. of Intern. Astronomical Union, Symp. 25, Thessaloniki (Contopoulos, G., ed.). 1964.Google Scholar
  56. [55]
    Zadunaisky, P. E.: On the estimation of errors propagated in the numerical integration of ordinary differential equations. Numer. Math. 27, 21–40 (1976).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • K. Böhmer
    • 1
  • P. W. Hemker
    • 2
  • H. J. Stetter
    • 3
  1. 1.Fachbereich MathematikPhilipps-UniversitätMarburgFederal Republic of Germany
  2. 2.Department of Numerical MathematicsCentre for Mathematics and Computer ScienceAmsterdamThe Netherlands
  3. 3.Institut für Angewandte und Numerische MathematikTechnische Universität WienWienAustria

Personalised recommendations