Advertisement

Richardson extrapolation based on superconvergent Nyström and degenerate kernel methods

  • C. Allouch
  • D. Sbibih
  • M. TahrichiEmail author
Article
  • 4 Downloads

Abstract

For computing the approximated solution of a second kind integral equation with a smooth kernel, we investigate in this paper the Richardson extrapolation using superconvergent Nyström and degenerate kernel methods based on interpolatory projection onto the space of (discontinuous) piecewise polynomials of degree \(\le r - 1.\) We obtain asymptotic series expansions for the approximate solutions and we show that the order of convergence 4r in the interpolation at Gauss points can be improved to \(4r+2\). We illustrate the improvement of the order of convergence by numerical experiments.

Keywords

Extrapolation Integral equation Interpolation Gauss points 

Mathematics Subject Classification

45B05 (Fredholm integral equation) 

Notes

References

  1. 1.
    Allouch, C., Sablonnire, P., Sbibih, D.: Superconvergent Nyström and degenerate kernel methods for solving multivariable integral equations of the second kind. J. Comput. Appl. Math. 236, 449–512 (2012)CrossRefGoogle Scholar
  2. 2.
    Allouch, C., Sablonnire, P., Sbibih, D., Tahrichi, M.: Superconvergent Nyström and degenerate kernel methods for eigenvalue problems. Appl. Math. Comput. 217, 7851–7866 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Allouch, C., Sbibih, D., Tahrichi, M.: Superconvergent Nyström and degenerate kernel methods for Hammerstein integral equations. J. Comput. Appl. Math. 258, 30–41 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Allouch, C., Sablonnière, P., Sbibih, D., Tahrichi, M.: Superconvergent Nyström and degenerate kernel methods for the numerical solution of integral equations of the second kind. J. Integral Equ. Appl. 24, 463–485 (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  6. 6.
    Atkinson, K., Graham, I., Sloan, I.: Piecewise continuous collocation for integral equations. SIAM J. Numer. Anal. 20, 172–186 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chandler, G.A.: Superconvergence of numerical solutions of second kind integral equations. PhD thesis, Australian National University (1979)Google Scholar
  8. 8.
    Chatelin, F.: Spectral Approximation of Linear Operators. Academic, New York (1983)zbMATHGoogle Scholar
  9. 9.
    de Boor, C., Swartz, B.: Collocation at Gaussian points. SIAM J. Numer. Anal. 10, 582–606 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lin, Q., Liu, J.: Extrapolation method for Fredholm integral equations with non-smooth kernels. Numer. Math. 35, 459–464 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lin, Q., Sloan, I.H., Xie, R.: Extrapolation of the iterated-collocation method for integral equations of the second kind. SIAM J. Numer. Anal 27(6), 1535–1541 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kulkarni, R.P.: A superconvergence result for solutions of compact operator equations. Bull. Aust. Math. Soc. 68, 517–528 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kulkarni, R.P., Grammont, L.: Extrapolation using a modified projection method. Numer. Funct. Anal. Optim. 30(11–12), 1339–1359 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    MacLean, W.: Asymptotic error expansions for numerical solutions of integral equations. IMA J. Numer. Anal. 9, 373–384 (1989)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Sloan, I.H.: Improvement by iteration for compact operator equations. Math. Comput. 30, 758–764 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.University Mohammed I, MFN, Team of Modelling and Scientific ComputingNadorMorocco
  2. 2.University Mohammed I, FSO, ANO LaboratoryOujdaMorocco

Personalised recommendations