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The iterated projection solution for the Fredholm integral equation of second kind

Published online by Cambridge University Press:  17 February 2009

Françoise Chatelin  and
Rachid Lebbar
Rachid Lebbar
Affiliation:
Université de Grenoble, Mathématiques Appliquées, IMAG, BP 53X, 38041 GRENOBLE-CEDEX, France
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Abstract

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We are concerned with the solution of the second kind Fredholm equation (and eigenvalue problem) by a projection method, where the projection is either an orthogonal projection on a set of piecewise polynomials or an interpolatory projection at the Gauss points of subintervals.

We study these cases of superconvergence of the Sloan iterated solution: global superconvergence for a smooth kernel, and superconvergence at the partition points for a kernel of “Green's function” type. The mathematical analysis applies for the solution of the inhomogeneous equation as well as for an eigenvector.

Type
Research Article
Information
The ANZIAM Journal , Volume 22 , Issue 4 , April 1981 , pp. 439 - 451
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]de Boor, C. and Swartz, B., “Collocation at Gaussian points”, SIAM J. Num. Anal. 10 (1973), 582–606.CrossRefGoogle Scholar
[2]de Boor, C. and Swartz, B., “Collocation approximation to eigenvalues of an ordinary differential equation: the principle of the thing”, Math. Comp. 35 (1980), 679–694.CrossRefGoogle Scholar
[3]de Boor, C. and Swartz, B., “Collocation approximation to eigenvalues of an ordinary differential equation: numerical illustrations”, Math. Comp. 36 (1981).CrossRefGoogle Scholar
[4]de Boor, C. and Swartz, B., “Local piecewise polynomial projection methods for an ode which give high-order convergence at knots”, Math. Comp. 36 (1981).CrossRefGoogle Scholar
[5]Chandler, G. A., “Global superconvergeuce of iterated Galerkin solutions for second kind integral equations”, Australian National University Technical Report (1978).Google Scholar
[6]Chandler, G. A., Superconvergence of numerical solutions of second kind integral equations (Ph. D. Thesis, Australian National University, 1979).CrossRefGoogle Scholar
[7]Chatelin, F., “Numerical computation of the eigenelements of linear integral operators by iterations”, SIAM J. Num. Anal. 15 (1978), 1112–1124.CrossRefGoogle Scholar
[8]Chatelin, F., “Sur les homes d'erreur a posteriori pour les éléments propres d'opérateurs linéaires”, Numer. Math. 32 (1979), 233–246.CrossRefGoogle Scholar
[9]Chatelin, F., Linear, spectral approximation in Banach spaces (Academic Press, New-York, to appear).Google Scholar
[10]Lebbar, R., “Méthode de Galerkin iterée pour l'équation intégrale de Fredholm de 2e espèce” (Rapport D. E. A., Université de Grenoble, 1980).Google Scholar
[11]Qun, Lin and Jiaquan, Liu, “Extrapolation methods for Fredholm integral equations with non-smooth kernels” Numer. Math. 35 (1980), 459–464.CrossRefGoogle Scholar
[12]Richter, G. R., “Superconvergence of piecewise polynomial Galerkin approximations for Fredholm integral equations of the 2nd kind”, Numer. Math. 31 (1978), 63–70.CrossRefGoogle Scholar
[13]Sloan, I. H., “Error analysis for a class of degenerate kernel methods”, Numer. Math. 25 (1976), 231–238.CrossRefGoogle Scholar
[14]Sloan, I. H., “Iterated Galerkin method for eigenvalue problems”, SIAM J. Num. Anal. 13 (1976), 753–760.CrossRefGoogle Scholar
[15]Sloan, I. H., “A review of numerical methods for Fredholm equations of the 2nd kind”, in the Application and numerical solution of integral equations (eds. Anderssen, , de Hoog, , Lukas, Sijthoff & Noordhoff, Leyden, 1980).Google Scholar
[16]Sloan, I. H. and Burn, B. J., “Collocation with polynomials for integral equations of the 2nd kind: a new approach to the theory”, Journ. Integral Equ. 1 (1979), 77–94.Google Scholar
[17]Sloan, I. H., Noussair, E. and Burn, B. J., “Projection method for equations of the 2nd kind” J. Math. Anal. Appl. 69 (1979), 84–103.CrossRefGoogle Scholar
[18]Wilkinson, J. H., The algebraic eigenvalue problem (Clarendon Press, Oxford, 1965).Google Scholar