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The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs


Authors: Kendall E. Atkinson and Ian H. Sloan
Journal: Math. Comp. 56 (1991), 119-139
MSC: Primary 65R20; Secondary 31A10, 35C15
DOI: https://doi.org/10.1090/S0025-5718-1991-1052084-0
MathSciNet review: 1052084
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Abstract: Consider solving the Dirichlet problem

\begin{displaymath}\begin{array}{*{20}{c}} {\Delta u(P) = 0,} \hfill & {P \in {\... ...ll \\ {P \in {\mathbb{R}^2}} \hfill & {} \hfill \\ \end{array} \end{displaymath}

with S a smooth open curve in the plane. We use single-layer potentials to construct a solution $ u(P)$. This leads to the solution of equations of the form

$\displaystyle \int_S {g(Q)\log \vert P - Q\vert dS(Q) = h(P),\quad P \in S.} $

This equation is reformulated using a special change of variable, leading to a new first-kind equation with a smooth solution function. This new equation is split into a principal part, which is explicitly invertible, and a compact perturbation. Then a discrete Galerkin method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. A complete convergence analysis is given; numerical examples conclude the paper.

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

  • [1] P. Anselone, Collectively compact operator approximation theory, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR 0443383 (56:1753)
  • [2] K. Atkinson, A survey of numerical methods for Fredholm integral equations of the second kind, SIAM, Philadelphia, Pa., 1976.
  • [3] -, A discrete Galerkin method for first kind integral equations with a logarithmic kernel, J. Integral Equations Appl. 1 (1988), 343-363. MR 1003700 (90m:65222)
  • [4] K. Atkinson and A. Bogomolny, The discrete Galerkin method for integral equations, Math. Comp. 48 (1987), 595-616. MR 878693 (88k:65125)
  • [5] G. Chandler, Private communication, 1988.
  • [6] M. Costabel, V. Ervin, and E. Stephan, On the convergence of collocation methods for Symm's integral equation on open curves, Math. Comp. 51 (1988), 167-179. MR 942148 (89h:65218)
  • [7] G. Gladwell and S. Coen, A Chebyshev approximation method for microstrip problems, IEEE Trans. Microwave Theory Tech. 23 (1975), 865-870.
  • [8] G. Hsiao, E. Stephan, and W. Wendland, On the Dirichlet problem in elasticity for a domain exterior to an arc, Math. Institut A, Universität Stuttgart, Tech. Rep. 15, 1989.
  • [9] M. Jaswon and G. Symm, Integral equation methods in potential theory and elastostatics, Academic Press, London, 1977. MR 0499236 (58:17147)
  • [10] W. McLean, Boundary integral methods for the Laplace equation, Ph.D. Thesis, Australian National University, Canberra, 1985.
  • [11] I. H. Sloan and A. Spence, The Galerkin method for integral equations of the first kind with logarithmic kernel: Theory, IMA J. Numerical Anal. 8 (1988), 105-122. MR 967846 (90d:65230a)
  • [12] Y. Yan and I. H. Sloan, On integral equations of the first kind with logarithmic kernels, J. Integral Equations Appl. 1 (1988), 549-579. MR 1008406 (90f:45008)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1991-1052084-0
Article copyright: © Copyright 1991 American Mathematical Society

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