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Collocation with Chebyshev polynomials for Symm's integral equation on an interval

Published online by Cambridge University Press:  17 February 2009

I. H. Sloan  and
E. P. Stephan
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Abstract

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A collocation method for Symm's integral equation on an interval (a first-kind integral equation with logarithmic kernel), in which the basis functions are Chebyshev polynomials multiplied by an appropriate singular function and the collocation points are Chebyshev points, is analysed. The novel feature lies in the analysis, which introduces Sobolev norms that respect the singularity structure of the exact solution at the ends of the interval. The rate of convergence is shown to be faster than any negative power of n, the degree of the polynomial space, if the driving term is smooth.

Type
Research Article
Information
The ANZIAM Journal , Volume 34 , Issue 2 , October 1992 , pp. 199 - 211
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Arnold, D. N. and Wendland, W. L., “The convergence of spline collocation for strongly elliptic equations on curves”, Numer. Math. 47 (1985) 317–341.CrossRefGoogle Scholar
[2]Atkinson, K. E. and Sloan, I. H., “The numerical solution of first kind logarithmic-kernel integral equations on smooth open arcs,” Math. Comp. 56 (1991) 119–139.CrossRefGoogle Scholar
[3]Bourlard, M., Nicaise, S. and Paquet, L., “An adapted boundary element method for the Dirichlet problem in polygonal domains”, Preprint, Math. Department, State University, Mons, Belgium (1988).Google Scholar
[4]Costabel, M., Ervin, V. J. and Stephan, E. P., “On the convergence of collocation methods for Symm's integral equation on open curves”, Math. Comp. 51 (1988) 167–179.Google Scholar
[5]Guo, B., von, T. Petersdorff and Stephan, E. P., “An hp version for BEM for plane mixed boundary value problems.” in Proc. Conference BEM-11, 1989, Cambridge USA (ed.C. A. Brebbia) (1989).Google Scholar
[6]Jorgens, K., Lineare Integral Operatoren, (Teubner-Verlag, Stuttgart, 1970).CrossRefGoogle Scholar
[7]Petersdorff, T. von, “Elasticity problems in polyhedra-Singularities and approximations with boundary elements”, Dissertation, TH Darmstadt (1989).Google Scholar
[8]Reade, J. B., “Asymptotic behavior of eigenvalues of certain integral equations”, Proc. Edin. Math. Soc. 12 (1979) 137–144.CrossRefGoogle Scholar
[9]Saranen, J. and Wendland, W. L., “On the asymptotic convergence of collocation methods with spline functions of even degree”, Math. Comp. 45 (1985) 91–108.CrossRefGoogle Scholar
[10]Sloan, I. H. and Spence, A., “The Galerkin method for integral equations of the first kind with logarithmic kernel: Theory”. IMA Journal of Numerical Analysis 8 (1988) 105–122.CrossRefGoogle Scholar
[11]Stephan, E. P., “The hp version of the Galerkin boundary element method for the integral equations on polygons and open arcs”. In Proc. Conference BEM-10, Southhampton, U.K. (ed. C. A. Brebbia) (1988).Google Scholar
[12]Stephan, E. P. and Suri, M., “On the convergence of the p-version of some boundary element Galerkin methods”, Math. Comp. 52 (1989) 31–48.Google Scholar
[13]Stephan, E. P. and Wendland, W. L., “An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems”, Applicable Analysis 18 (1984) 183–219.CrossRefGoogle Scholar
[14]Symm, G. T., “Integral equation methods in potential theory. II”, Proc. Roy. Soc. (London) A275 (1963) 33–46.Google Scholar
[15]Yan, Y. and Sloan, I. H., “Mesh grading for integral equations of the first kind with logarithmic kernel”. SIAM J. Numer. Anal., 26 (1989) 574–587.CrossRefGoogle Scholar
[16]Yan, Y. and Sloan, I. H., “On integral equations of the first kind with logarithmic kernels”, J. Integral Equations and Applications, 1 (1988) 517–548.CrossRefGoogle Scholar