The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs
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- by Kendall E. Atkinson and Ian H. Sloan PDF
- Math. Comp. 56 (1991), 119-139 Request permission
Abstract:
Consider solving the Dirichlet problem \[ \begin {array}{*{20}{c}} {\Delta u(P) = 0,} \hfill & {P \in {\mathbb {R}^2}\backslash S,} \hfill \\ {u(P) = h(P),} \hfill & {P \in S,} \hfill \\ {\sup |u(P)| < \infty ,} \hfill & {} \hfill \\ {P \in {\mathbb {R}^2}} \hfill & {} \hfill \\ \end {array} \] 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 \[ \int _S {g(Q)\log |P - Q|dS(Q) = h(P),\quad P \in S.} \] 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
- Philip M. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. With an appendix by Joel Davis. MR 0443383 K. Atkinson, A survey of numerical methods for Fredholm integral equations of the second kind, SIAM, Philadelphia, Pa., 1976.
- Kendall E. Atkinson, A discrete Galerkin method for first kind integral equations with a logarithmic kernel, J. Integral Equations Appl. 1 (1988), no. 3, 343–363. MR 1003700, DOI 10.1216/JIE-1988-1-3-343
- Kendall Atkinson and Alex Bogomolny, The discrete Galerkin method for integral equations, Math. Comp. 48 (1987), no. 178, 595–616, S11–S15. MR 878693, DOI 10.1090/S0025-5718-1987-0878693-6 G. Chandler, Private communication, 1988.
- M. Costabel, V. J. Ervin, and E. P. Stephan, On the convergence of collocation methods for Symm’s integral equation on open curves, Math. Comp. 51 (1988), no. 183, 167–179. MR 942148, DOI 10.1090/S0025-5718-1988-0942148-1 G. Gladwell and S. Coen, A Chebyshev approximation method for microstrip problems, IEEE Trans. Microwave Theory Tech. 23 (1975), 865-870. 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.
- M. A. Jaswon and G. T. Symm, Integral equation methods in potential theory and elastostatics, Computational Mathematics and Applications, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1977. MR 0499236 W. McLean, Boundary integral methods for the Laplace equation, Ph.D. Thesis, Australian National University, Canberra, 1985.
- I. H. Sloan and A. Spence, The Galerkin method for integral equations of the first kind with logarithmic kernel: theory, IMA J. Numer. Anal. 8 (1988), no. 1, 105–122. MR 967846, DOI 10.1093/imanum/8.1.105
- Y. Yan and I. H. Sloan, On integral equations of the first kind with logarithmic kernels, J. Integral Equations Appl. 1 (1988), no. 4, 549–579. MR 1008406, DOI 10.1216/JIE-1988-1-4-549
Additional Information
- © Copyright 1991 American Mathematical Society
- 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