Boundary integral equation methods for solving Laplace’s equation with nonlinear boundary conditions: the smooth boundary case

Atkinson, Kendall E.; Chandler, Graeme

doi:10.1090/S0025-5718-1990-1035924-X

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6842(online) ISSN 0025-5718(print)

Boundary integral equation methods for solving Laplace's equation with nonlinear boundary conditions: the smooth boundary case

Authors: Kendall E. Atkinson and Graeme Chandler
Journal: Math. Comp. 55 (1990), 451-472
MSC: Primary 65N38; Secondary 65R20
MathSciNet review: 1035924
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A nonlinear boundary value problem for Laplace's equation is solved numerically by using a reformulation as a nonlinear boundary integral equation. Two numerical methods are proposed and analyzed for discretizing the integral equation, both using product integration to approximate the singular integrals in the equation. The first method uses the product Simpson's rule, and the second is based on trigonometric interpolation. Iterative methods (including two-grid methods) for solving the resulting nonlinear systems are also discussed extensively. Numerical examples are included.

[1] E. L. Allgower, K. Böhmer, F. A. Potra, and W. C. Rheinboldt, A mesh-independence principle for operator equations and their discretizations, SIAM J. Numer. Anal. 23 (1986), no. 1, 160–169. MR 821912 (87h:65107), http://dx.doi.org/10.1137/0723011
[2] Kendall E. Atkinson, The numerical evaluation of fixed points for completely continuous operators, SIAM J. Numer. Anal. 10 (1973), 799–807. MR 0346602 (49 #11327)
[3] Kendall Atkinson, Iterative variants of the Nyström method for the numerical solution of integral equations, Numer. Math. 22 (1973/74), 17–31. MR 0337038 (49 #1811)
[4] -, A survey of numerical methods for the solution of Fredholm integral equations of the second kind, SIAM, Philadelphia, Pa., 1976.
[5] Frank de Hoog and Richard Weiss, Asymptotic expansions for product integration, Math. Comp. 27 (1973), 295–306. MR 0329207 (48 #7549), http://dx.doi.org/10.1090/S0025-5718-1973-0329207-0
[6] M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR 0159197 (28 #2414)
[7] R. Kress, Boundary integral equations in time-harmonic acoustic scattering, Tech. Rep. #66, Institut für Numerische und Angewandte Mathematik, Universität Göttingen, 1989.
[8] K. Ruotsalainen and W. Wendland, On the boundary element method for some nonlinear boundary value problems, Numer. Math. 53 (1988), no. 3, 299–314. MR 948589 (89h:65189), http://dx.doi.org/10.1007/BF01404466
[9] 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 (90f:45008), http://dx.doi.org/10.1216/JIE-1988-1-4-549
[10] A. Zygmund, Trigonometric series, Vols. I and II, Cambridge Univ. Press, Cambridge, 1959.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|