# Multi-parameter error resolution for the collocation method of Volterra integral equations

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## Abstract

In this paper, a multi-parameter error resolution technique is introduced and applied to the collocation method for Volterra integral equations. By using this technique, an approximation of higher accuracy is obtained by using a multi-processor in parallel. Additionally, a correction scheme for approximation of higher accuracy and a global superconvergence result are presented.

### AMS subject classification

65R20### Key words

Collocation integral equations multi-parameter error resolution parallel algorithm## Preview

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### References

- 1.K. E. Atkinson,
*A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind*, Society for Industrial and Applied Mathematics, Philadelphia, 1976.MATHGoogle Scholar - 2.H. Brunner,
*Iterated collocation methods and their discretizations for Volterra integral equations*, SIAM J. Numer. Anal., 21 (1984), pp. 1132–1145.MathSciNetCrossRefMATHGoogle Scholar - 3.H. Brunner,
*Collocation methods for one-dimensional Fredholm and Volterra integral equations*, in The State of the Art in Numerical Analysis, A. Iserles and M. J. D. Powell, eds., Oxford University Press, Oxford, 1987, pp. 563–600.Google Scholar - 4.H. Brunner,
*Open problems in the discretization of Volterra integral equations*, Lecture Notes, Memorial University of Newfoundland, 1993.Google Scholar - 5.H. Brunner and P. J. van der Houwen,
*The Numerical Solution of Volterra Equations*, CWI Monograph 3, North-Holland, Amsterdam, 1986.MATHGoogle Scholar - 6.F. Chatelin and R. Lebbar,
*The iterated projection solution for the Fredholm integral equation of second kind*, J. Austral. Math. Soc. Ser. B, 22 (1981), pp. 439–451.MathSciNetMATHCrossRefGoogle Scholar - 7.I. G. Graham, S. Joe, and I. H. Sloan,
*Iterated Galerkin versus iterated-collocation for integral equations of the second kind*, IMA J. Numer. Anal., 5 (1985), pp. 355–369.MathSciNetMATHGoogle Scholar - 8.S. Joe,
*Collocation methods using piecewise polynomials for second kind integral equations*, J. Comput. Appl. Math., 12/13 (1985), pp. 391–400.MathSciNetCrossRefGoogle Scholar - 9.Q. Lin and J. Shi,
*Iterative corrections and a posteriori error estimate for integral equations*, J. Comput. Math., 11 (1993), pp. 297–300.MathSciNetMATHGoogle Scholar - 10.Q. Lin, I. H. Sloan, and R. Xie,
*Extrapolation of the iterated-collocation method for integral equations of the second kind*, SIAM J. Numer. Anal., 27 (1990), pp. 1535–1541.MathSciNetCrossRefMATHGoogle Scholar - 11.Q. Lin and A. Zhou,
*Defect correction for finite element gradient*, Systems Sci. Math. Sci., 5 (1992), 287–288.MathSciNetMATHGoogle Scholar - 12.W. McLean,
*Asymptotic error expansions for numerical solutions of integral equations*, IMA J. Numer. Anal., 9 (1989), pp. 373–384.MathSciNetMATHGoogle Scholar - 13.I. H. Sloan,
*Superconvergence*, in: Numerical Solution of Integral Equations, M. A. Golberg, ed., Plenum Press, New York, 1990, pp. 35–70.Google Scholar - 14.A. Zhou,
*Extrapolation for collocation method of the first kind Volterra integral equations*, Acta Math. Sci., 11 (1991), pp. 471–476.MATHGoogle Scholar - 15.A. Zhou, C. B. Liem, and T. M. Shih,
*A parallel algorithm based on multi-parameter asymptotic error expansion*, in Proc. of Conference on Scientific Computation, Hong Kong, March 17–19, 1994.Google Scholar - 16.A. Zhou, C. B. Liem, T. M. Shih, and T. Lü,
*A multi-parameter splitting extrapolation and a parallel algorithm*, Research Report IMS-61, Institute of Mathematical Sciences, Academia Sinica, Chengdu, China, 1994, see also Systems Sci. Math. Sci., 10:3 (1997), pp. 253–260.Google Scholar

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© Swets & Zeitlinger 1997