Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Multiple grid methods for the solution of Fredholm integral equations of the second kind

Authors: P. W. Hemker and H. Schippers
Journal: Math. Comp. 36 (1981), 215-232
MSC: Primary 65R20; Secondary 45L10
MathSciNet review: 595054
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper multiple grid methods are applied for the fast solution of the large nonsparse systems of equations that arise from the discretization of Fredholm integral equations of the second kind. Various multiple grid schemes, both with Nyström and with direct interpolation, are considered. For these iterative methods, the rates of convergence are derived using the collectively compact operator theory by Anselone and Atkinson. Estimates for the asymptotic computational complexity are given, which show that the multiple grid schemes result in $ \mathcal{O}({N^2})$ arithmetic operations.

References [Enhancements On Off] (What's this?)

  • [1] P. M. Anselone, Collectively Compact Operator Approximation Theory, Prentice-Hall, Englewood Cliffs, N.J., 1971. MR 0443383 (56:1753)
  • [2] K. E. Atkinson, "Iterative variants of the Nyström method for the numerical solution of integral equations," Numer. Math., v. 22, 1973, pp. 17-31. MR 0337038 (49:1811)
  • [3] K. E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, Pa., 1976. MR 0483585 (58:3577)
  • [4] H. Brakhage, "Über die numerische Behandlung von Integralgleichungen nach der Quadraturformelmethode," Numer. Math., v. 2, 1960, pp. 183-196. MR 0129147 (23:B2184)
  • [5] A. Brandt, "Multi-level adaptive solutions to boundary-value problems," Math. Comp., v. 31, 1977, pp. 333-390. MR 0431719 (55:4714)
  • [6] A. Brandt, "Multi-level adaptive techniques for singular perturbation problems," in Numerical Analysis of Singular Perturbation Problems (P. W. Hemker and J. J. H. Miller, Eds.), Academic Press, London, 1979. MR 556515 (81d:65057)
  • [7] W. Hackbusch, Die schnelle Auflösung der Fredholmschen Integralgleichung zweiter Art, Report 78-4, Universität zu Köln, 1978.
  • [8] W. Hackbusch, An Error Analysis of the Nonlinear Multi-Grid Method of Second Kind, Report 78-15, Universität zu Köln, 1978.
  • [9] P. M. Prenter, "A collocation method for the numerical solution of integral equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 570-581. MR 0327064 (48:5406)
  • [10] H. Schippers, Multi-Grid Techniques for the Solution of Fredholm Integral Equations of the Second Kind, Colloq. Numerical Treatment of Integral Equations, MC-Syllabus 41, Mathematisch Centrum, Amsterdam, 1979. MR 570811 (81g:65183)
  • [11] H. J. Stetter, "The defect correction principle and discretization methods," Numer. Math., v. 29, 1978, pp. 425-443. MR 0474803 (57:14436)
  • [12] P. Wesseling & P. Sonneveld, "Numerical experiments with a multiple grid and preconditioned Lanczos type method," Proc. IUTAM-Symposium on Approximation Methods for Navier-Stokes Problems, Paderborn, Sept. 1979, Lecture Notes in Math., 771, Springer. MR 566020 (81b:65117)
  • [13] P. Wesseling, "The rate of convergence of a multiple grid method," to appear in: Proc. Dundee Biennial Conference on Numerical Analysis, June 1979, Lecture Notes in Math., Springer. (Available as Report NA-30, Delft University of Technology.) MR 569468 (81j:65120)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65R20, 45L10

Retrieve articles in all journals with MSC: 65R20, 45L10

Additional Information

Keywords: Fredholm integral equations of the second kind, multiple grid methods
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society