The collocation method for first-kind boundary integral equations on polygonal regions

Author:
Yi Yan

Journal:
Math. Comp. **54** (1990), 139-154

MSC:
Primary 65N35; Secondary 65R20

MathSciNet review:
995213

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper the collocation method for first-kind boundary integral equations, by using piecewise constant trial functions with uniform mesh, is shown to be equivalent to a projection method for second-kind 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.

**[1]**Philip M. Anselone,*Collectively compact operator approximation theory and applications to integral equations*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1971. With an appendix by Joel Davis; Prentice-Hall Series in Automatic Computation. MR**0443383****[2]**Douglas N. Arnold and Wolfgang L. Wendland,*On the asymptotic convergence of collocation methods*, Math. Comp.**41**(1983), no. 164, 349–381. MR**717691**, 10.1090/S0025-5718-1983-0717691-6**[3]**Douglas N. Arnold and Wolfgang L. Wendland,*The convergence of spline collocation for strongly elliptic equations on curves*, Numer. Math.**47**(1985), no. 3, 317–341. MR**808553**, 10.1007/BF01389582**[4]**Kendall E. Atkinson,*A survey of numerical methods for the solution of Fredholm integral equations of the second kind*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. MR**0483585****[5]**K. E. Atkinson and F. R. de Hoog,*Collocation methods for a boundary integral equation on a wedge*, in Treatment of Integral Equations by Numerical Methods (C. T. H. Baker and G. F. Miller, eds.), Academic Press, New York, 1983.**[6]**Christopher T. H. Baker,*The numerical treatment of integral equations*, Clarendon Press, Oxford, 1977. Monographs on Numerical Analysis. MR**0467215****[7]**Søren Christiansen,*On two methods for elimination of nonunique solutions of an integral equation with logarithmic kernel*, Applicable Anal.**13**(1982), no. 1, 1–18. MR**647662**, 10.1080/00036818208839372**[8]**Martin Costabel and Ernst P. Stephan,*Collocation methods for integral equations on polygons*, Innovative numerical methods in engineering (Atlanta, Ga., 1986) Comput. Mech., Southampton, 1986, pp. 43–50. MR**902858****[9]**Martin Costabel and Ernst P. Stephan,*On the convergence of collocation methods for boundary integral equations on polygons*, Math. Comp.**49**(1987), no. 180, 461–478. MR**906182**, 10.1090/S0025-5718-1987-0906182-9**[10]**F. R. de Hoog,*Product integration techniques for the numerical solution of integral equations*, Ph.D. Thesis, Australian National University, 1973.**[11]**Ivan G. Graham,*Estimates for the modulus of smoothness*, J. Approx. Theory**44**(1985), no. 2, 95–112. MR**794593**, 10.1016/0021-9045(85)90073-5**[12]**M. A. Krasnosel′skiĭ, G. M. Vaĭnikko, P. P. Zabreĭko, Ya. B. Rutitskii, and V. Ya. Stetsenko,*Approximate solution of operator equations*, Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the Russian by D. Louvish. MR**0385655****[13]**J. Saranen and W. L. Wendland,*On the asymptotic convergence of collocation methods with spline functions of even degree*, Math. Comp.**45**(1985), no. 171, 91–108. MR**790646**, 10.1090/S0025-5718-1985-0790646-3**[14]**J. Saranen,*The convergence of even degree spline collocation solution for potential problems in smooth domains of the plane*, Numer. Math.**53**(1988), no. 5, 499–512. MR**954767**, 10.1007/BF01397549**[15]**G. Schmidt,*On spline collocation methods for boundary integral equations in the plane*, Math. Methods Appl. Sci.**7**(1985), no. 1, 74–89. MR**783387**, 10.1002/mma.1670070105**[16]**Gunther Schmidt,*On 𝜀-collocation for pseudodifferential equations on a closed curve*, Math. Nachr.**126**(1986), 183–196. MR**846574**, 10.1002/mana.19861260112**[17]**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**, 10.1093/imanum/8.1.105**[18]**I. H. Sloan and A. Spence,*The Galerkin method for integral equations of the first kind with logarithmic kernel: applications*, IMA J. Numer. Anal.**8**(1988), no. 1, 123–140. MR**967847**, 10.1093/imanum/8.1.123**[19]**Y. Yan and I. H. Sloan,*Mesh grading for integral equations of the first kind with logarithmic kernel*, SIAM J. Numer. Anal.**26**(1989), no. 3, 574–587. MR**997657**, 10.1137/0726034**[20]**-,*On integral equations of the first kind with logarithmic kernel*, J. Integral Equations Appl.**1**(1988), No. 4, 1-31.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1990-0995213-6

Keywords:
Collocation method,
first-kind boundary integral equations,
polygonal regions

Article copyright:
© Copyright 1990
American Mathematical Society