The collocation method for firstkind boundary integral equations on polygonal regions
Author:
Yi Yan
Journal:
Math. Comp. 54 (1990), 139154
MSC:
Primary 65N35; Secondary 65R20
MathSciNet review:
995213
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Abstract: In this paper the collocation method for firstkind boundary integral equations, by using piecewise constant trial functions with uniform mesh, is shown to be equivalent to a projection method for secondkind Fredholm equations. In a certain sense this projection is an interpolation projection. By introducing this technique of analysis, we particularly consider the case of polygonal boundaries. We give asymptotic error estimates in norm on the boundaries, and some superconvergence results for the single layer potential.
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 D. N. Arnold and W. L. Wendland, On the asymptotic convergence of collocation methods, Math. Comp. 41 (1983), 349381. MR 717691 (85h:65254)
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 , The convergence of spline collocation for strongly elliptic equations on curves, Numer. Math. 47 (1985), 317341. MR 808553 (87f:65142)
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 K. E. Atkinson, A survey of numerical methods for the solution of Fredholm integral equations of the second kind, SIAM, Philadelphia, Pa., 1976. MR 0483585 (58:3577)
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 S. Christiansen, On two methods for elimination of nonunique solutions of an integral equation with logarithmic kernel, Applicable Anal. 13 (1982), 118. MR 647662 (83e:65204)
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 M. Costabel and E. P. Stephan, Collocation methods for integral equations on polygons, in Innovative Numerical Methods in Engineering (R. P. Shaw et al., eds.), Proc. 4th Internat. Sympos., Georgia Tech., Atlanta, GA, SpringerVerlag, Berlin and New York, 1986, pp. 4350. MR 902858 (88h:65242)
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 , On the convergence of collocation methods for boundary integral equations on polygons, Math. Comp. 49 (1987), 461478. MR 906182 (88j:65292)
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 F. R. de Hoog, Product integration techniques for the numerical solution of integral equations, Ph.D. Thesis, Australian National University, 1973.
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 I. G. Graham, Estimates for the modulus of smoothness, J. Approx. Theory 44 (1985), 95112. MR 794593 (86k:41007)
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 M. A. Krasnoselskii et al., Approximate solution of operator equations, Noordhoff, Groningen, 1972. MR 0385655 (52:6515)
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 J. Saranen and W. L. Wendland, On the asymptotic convergence of collocation methods with spline functions of even degree, Math. Comp. 171 (1985), 91108. MR 790646 (86m:65159)
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 J. Saranen, The convergence of even degree spline collocation solution for potential problems in smooth domains of the plane, Numer. Math. 53 (1988), 499512. MR 954767 (89h:65199)
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 , On collocation for pseudodifferential equations on a closed curve, Math. Nachr. 126 (1986), 183196. MR 846574 (87m:65202)
 [17]
 I. Sloan and A. Spence, The Galerkin method for integral equations of the first kind with logarithmic kernel: Theory, IMA J. Numer. Anal. 8 (1988), 105122. MR 967846 (90d:65230a)
 [18]
 , The Galerkin method for integral equations of the first kind with logarithmic kernel: Applications, IMA J. Numer. Anal. 8 (1988), 123140. MR 967847 (90d:65230b)
 [19]
 Y. Yan and I. Sloan, Mesh grading for integral equations of the first kind with logarithmic kernel, SIAM J. Numer. Anal. 26 (1989), 574587. MR 997657 (90f:65241)
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 , On integral equations of the first kind with logarithmic kernel, J. Integral Equations Appl. 1 (1988), No. 4, 131.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199009952136
PII:
S 00255718(1990)09952136
Keywords:
Collocation method,
firstkind boundary integral equations,
polygonal regions
Article copyright:
© Copyright 1990
American Mathematical Society
