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Galerkin's method for boundary integral equations on polygonal domains

Published online by Cambridge University Press:  17 February 2009

G. A. Chandler
G. A. Chandler
Affiliation:
Department of Mathematics, University of Queensland, St. Lucia, Queensland 4067.
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Abstract

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A harmonic function in the interior of a polygon is the double layer potential of a distribution satisfying a second kind integral equation. This may be solved numerically by Galerkin's method using piecewise polynomials as basis functions. But the corners produce singularities in the distribution and the kernel of the integral equation; and these reduce the order of convergence. This is offset by grading the mesh, and the orders of convergence and superconvergence are restored to those for a smooth boundary.

Type
Research Article
Information
The ANZIAM Journal , Volume 26 , Issue 1 , July 1984 , pp. 1 - 13
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Anselone, P. M., Collectively compact operator approximation theory (Prentice-Hall, Englewood Cliffs, N. J., 1971).Google Scholar
[2]Atkinson, K. E., A survey of numerical methods for the solution of Fredholm integral equations of the second kind (SIAM, 1976).Google Scholar
[3]Atkinson, K. E. and de Hoog, F. R., “Collocation methods for a boundary integral equation on a wedge”, in Treatment of integral equations by numerical methods (eds. Baker, C.T.H. and Miller, G. F.), (Academic Press, 1983).Google Scholar
[4]Atkinson, K. E. and de Hoog, F. R., “The numerical solution of Laplace's equation on a wedge”, IMA J. Numer. Anal., (to appear).Google Scholar
[5]Babuska, I. and Rheinboldt, W. C., “Error estimates for adaptive finite element computations”, SIAM J. Numer. Anal. 15 (1978), 736–754.CrossRefGoogle Scholar
[6]Baker, C. T. H., The numerical treatment of integral equations (Oxford University Press, 1977).Google Scholar
[7]Brebbia, C. A., Progress in boundary element methods, Vol. 1 (Pentech Press, 1981).CrossRefGoogle Scholar
[8]Chandler, G. A., “Superconvergence for second kind integral equations” in The application and numerical solution of integral equations (eds. Anderssen, R. S.et al.), (Sijthoff and Noordhoff, 1980), 103–117.CrossRefGoogle Scholar
[9]Chandler, G. A., “Superconvergence of numerical solutions to second kind integral equations”, Ph.D. Thesis, Australian National University, 1979.CrossRefGoogle Scholar
[10]Chandler, G. A., “Superconvergent approximations to the solution of a boundary integral equation on polygonal domains”, (in preparation).Google Scholar
[11]Christiansen, S. and Hansen, E. B., “Numerical solution of boundary problems through integral equations”, Z. Angew. Math. Mech. 58 (1978), T14–T25.Google Scholar
[12]Cryer, C. W., “The solution of the Dirichlet problem for Laplace's equation when the boundary data is discontinuous and the domain has a boundary which is of bounded rotation by means of the Lebesgue-Stieltjes integral equation for the double layer potential”, Technical Report 99, Computer Sciences Department, University of Wisconsin (1970).Google Scholar
[13]Djaoua, M., “A method of calculation of lifting flows around two dimensional corner shaped bodies”, Math. Comp. 36 (1981), 405–521.Google Scholar
[14]Graham, I. G., “Galerkin methods for second kind integral equations with singularities”, Math. Comp. 39 (1982), 519–533.CrossRefGoogle Scholar
[15]Grisvard, P., “Behaviour of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain”, in Numerical solution of partial differential equations—III, Synspade, 1975 (ed. Hubbard, B.), (Academic Press, 1976), 207–247.Google Scholar
[16]Jaswon, M. A. and Symm, G. I., Integral equation methods in potential theory and electrostatics (Academic Press, 1977).Google Scholar
[17]Kellogg, O. D., Foundations of potential theory (Springer, J., 1929; Dover, 1954).CrossRefGoogle Scholar
[18]Kondrat'ev, V. A., “Boundary problems for elliptic equations in domains with conical or angular points”, Trans. Moscow Math. Soc. 16 (1967), 227–313.Google Scholar
[19]Král, J., “Integral operators in potential theory”, in Lecture Notes in Mathematics 823, (Springer-Verlag, 1980).Google Scholar
[20]Qun, Lin, “Some problems about the approximate solution for operator equations”, Acta Math. Sinica 22 (1979), 219–230.Google Scholar
[21]Rice, J. R., “On the degree of convergence of nonlinear spline approximation” in Approximation with special emphasis on spline functions (ed. Schoenberg, I. J.), (Academic Press, New York 1969).Google Scholar
[22]Schneider, C., “Product integration for weakly singular integral equations”, Z. Angew. Math. Mech. 61 (1981), T317–319.Google Scholar
[23]Sloan, I. H., “A review of numerical methods for Fredholm equations of the second kind” in The application and numerical solution of integral equations (eds. Anderssen, R. S.et al.), (Sijthoff and Noordhoff, 1980), 51–74.CrossRefGoogle Scholar
[24]White, A. B., “On selection of equidistributing meshes for two point boundary value problems, SIAM. J. Numer. Anal. 16 (1979), 472–502.CrossRefGoogle Scholar