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An extrapolation method for a class of boundary integral equations

Authors: Yuesheng Xu and Yunhe Zhao
Journal: Math. Comp. 65 (1996), 587-610
MSC (1991): Primary 65R20, 65B05, 45L10
MathSciNet review: 1333328
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Abstract: Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green's formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for the approximate solutions obtained by the Nyström scheme are developed to analyze the extrapolation method. Some computational aspects of the methods are considered, and two numerical examples are given to illustrate the acceleration of convergence.

References [Enhancements On Off] (What's this?)

  • 1. P. M. Anselone, Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice-Hall, Englewood Cliffs, N.J., 1971. MR 56:1753
  • 2. K. E. Atkinson, The numerical solution of Fredholm integral equations of the second kind with singular kernels, Numer. Math. 19 (1972), 248-259. MR 46:6632
  • 3. K. E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, 1976. MR 58:3577
  • 4. K. E. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solutions of Integral Equations, (Edited by Michael A. Golberg), Plenum, 1990. MR 91j:65169
  • 5. K. E. Atkinson and G. Chandler, Boundary integral equation methods for solving Laplace's equation with nonlinear boundary conditions: the smooth boundary case, Math. Comp. 55 (1990), 451-472. MR 91d:65181
  • 6. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd Edition, Academic Press, San Diego, 1984. MR 86d:65004
  • 7. W. F. Ford and A. Sidi, An algorithm for a generalization of the Richardson extrapolation process, SIAM J. Numer. Anal. 24 (1987), 1212-1232. MR 89a:65006
  • 8. L. Navot, An extension of the Euler-Maclaurin summation formula to functions with a branch singularity, J. Math. and Phys. 40 (1961), 271-276. MR 25:4290
  • 9. L. Navot, A further extension of the Euler-Maclaurin summation formula, J. Math. and Phys. 41 (1962), 155-163.
  • 10. K. Ruotsalainen and W. Wendland, On the boundary element method for some nonlinear boundary value problems, Numer. Math. 53 (1988), 299-314. MR 89h:65189
  • 11. C. Schneider, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Int. Eqs. Op. Thy. 2 (1979), 62-68. MR 80f:45002
  • 12. A. Sidi, Comparison of some numerical quadrature formulas for weakly singular periodic Fredholm integral equations, Computing 43 (1989), 159-170. MR 91c:65086
  • 13. A. Sidi and M. Israeli, Quadrature methods for periodic singular and weakly singular
    Fredholm integral equations, J. Scientific Computing 3 (1988), 201-231. MR 90e:65194
  • 14. I. Sloan, Private communication, 1993.

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Additional Information

Yuesheng Xu
Affiliation: Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105

Yunhe Zhao
Affiliation: Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105

Keywords: Boundary value problem, boundary integral equations, Euler-Maclaurin formula, extrapolation scheme, Nystr\"om method, periodic logarithmic Fredholm integral equations, asymptotic expansion
Received by editor(s): February 21, 1994
Received by editor(s) in revised form: October 4, 1994
Additional Notes: This work is partially supported by NASA under grant NAG 3-1312
Article copyright: © Copyright 1996 American Mathematical Society

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