Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-29T03:56:12.086Z Has data issue: false hasContentIssue false
  • English
  • Français

A superconvergence result for solutions of compact operator equations

Published online by Cambridge University Press:  17 April 2009

Rekha P. Kulkarni
Rekha P. Kulkarni
Affiliation:
Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India, e-mail: rpk@math.iitb.ac.in
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Over the last 20 years, since the publication of Sloan's paper on the improvement by the iteration technique, various approaches have been proposed for post-processing the Galerkin solution of multi-dimensional second kind Fredholm Integral equation. These methods include the iterated Galerkin method proposed by Sloan, the Kantorovich method and the iterated Kantorovich method. Recently, Lin, Zhang and Yan have proposed interpolation as an alternative to the iteration technique. For an integral operator, with a smooth kernel using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree ≤ r − 1, previous authors have established an order r convergence for the Galerkin solution and 2r for the iterated Galerkin solution. Equivalent results have also been established for the interpolator projection at Gauss points and some interpolation post-processing technique. In this paper, a method is introduced and shown to have convergence of order 4r. The size of the system of equations that must be solved, in implementing this method, remains the same as for the Galerkin method.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 68 , Issue 3 , December 2003 , pp. 517 - 528
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Chandler, G.A., Superconvergence of numerical solutions of second kind integral equations, Ph.D. Thesis (Australian National University, ACT, Australia, 1979).Google Scholar
[2]Chatelin, F., Spectral approximation of linear operators (Academic Press, New York, 1983).Google Scholar
[3]Chatelin, F. and Lebbar, R., ‘The iterated projection solution for the Fredholm integral equation of second kind’, J. Austral. Math. Soc. Ser. B 22 (1981), 439–451.CrossRefGoogle Scholar
[4]de Boor, C., ‘A bound on the L ∞ norm of L 2-approximation by splines in terms of a global mesh ratio’, Maths. Comput. 30 (1976), 765–771.Google Scholar
[5]de Boor, C. and Swartz, B., ‘Collocation at Gaussin points’, SIAM J. Numer. Anal. 10 (1973), 582–606.Google Scholar
[6]Douglas, J. Jr., Dupont, T. and Wahlbin, L., ‘Optimal L ∞ error estimates for Galerkin approximations to solutions of two point boundary value problems’, Math. Comp. 29 (1975), 475–483.Google Scholar
[7]Graham, I.G., Joe, S. and Sloan, I.H., ‘Iterated Galerkin versus iterated collocation for integral equations of the second kind’, IMA J. Numer. Anal. 5 (1985), 355–369.Google Scholar
[8]Hu, Q., ‘Interpolation correction for collocation solutions of Fredholm integro-differential equations’, Math. Comp. 67 (1998), 987–999.CrossRefGoogle Scholar
[9]Kulkarni, R.P., ‘A New Superconvergent projection method for approximate solutions of eigenvalue problems’, Numer. Funct. Anal. Optim. 24 (2003), 75–84.Google Scholar
[10]Lin, Q., Zhang, S. and Yan, N., ‘An acceleration method for integral equations by using interpolation post-processing’, Adv. Comput. Math. 9 (1998), 117–129.CrossRefGoogle Scholar
[11]Richter, G.R., ‘Superconvergence of piecewise polynomial Galerkin approximations for Fredholm integral equations of the second kind’, Num. Math. 31 (1978), 63–70.Google Scholar
[12]Schock, E., ‘Galerkin like methods for equations of the second kind’, J. Integral Equations Appl. 4 (1982), 361–364.Google Scholar
[13]Sloan, I.H., ‘Improvement by iteration for compact operator equations’, Math. Comp. 30 (1976), 758–764.CrossRefGoogle Scholar
[14]Sloan, I.H., ‘Four variants of the Gałerkin method for Integral equations of the second kind’, IMA J. Numer. Anal. 4 (1984), 9–17.CrossRefGoogle Scholar
[15]Spence, A. and Thomas, K.S., ‘On superconvergence properties of Galerkin's method for compact operator equations’, IMA J. Numer. Anal. 3 (1983), 253–271.CrossRefGoogle Scholar