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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Product integration-collocation methods for noncompact integral operator equations
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by G. A. Chandler and I. G. Graham PDF
Math. Comp. 50 (1988), 125-138 Request permission

Abstract:

We discuss the numerical solution of a class of second-kind integral equations in which the integral operator is not compact. Such equations arise, for example, when boundary integral methods are applied to potential problems in a two-dimensional domain with corners in the boundary. We are able to prove the optimal orders of convergence for the usual collocation and product integration methods on graded meshes, provided some simple modifications are made to the underlying basis functions. These are sufficient to ensure stability, but do not damage the rate of convergence. Numerical experiments show that such modifications are necessary in certain circumstances.
References
    K. E. Atkinson, A Survey of Numerical Methods for the Solution of Integral Equations of the Second Kind, SIAM, Philadelphia, Pa., 1976.
  • K. Atkinson and F. de Hoog, The numerical solution of Laplace’s equation on a wedge, IMA J. Numer. Anal. 4 (1984), no. 1, 19–41. MR 740782, DOI 10.1093/imanum/4.1.19
  • Carl de Boor, Good approximation by splines with variable knots, Spline functions and approximation theory (Proc. Sympos., Univ. Alberta, Edmonton, Alta., 1972) Internat. Ser. Numer. Math., Vol. 21, Birkhäuser, Basel, 1973, pp. 57–72. MR 0403169
  • Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062
    • C. A. Brebbia, J. C. F. Telles & L. C. Wrobel, Boundary Element Techniques, Springer-Verlag, Berlin and New York, 1984.
  • H. G. Burchard, On the degree of convergence of piecewise polynomial approximation on optimal meshes, Trans. Amer. Math. Soc. 234 (1977), no. 2, 531–559. MR 481758, DOI 10.1090/S0002-9947-1977-0481758-4
    • G. A. Chandler, Superconvergence of Numerical Solutions to Second Kind Integral Equations, Thesis, Australian National University, 1979.
  • G. A. Chandler, Galerkin’s method for boundary integral equations on polygonal domains, J. Austral. Math. Soc. Ser. B 26 (1984), no. 1, 1–13. MR 750551, DOI 10.1017/S033427000000429X
  • G. A. Chandler, Superconvergent approximations to the solution of a boundary integral equation on polygonal domains, SIAM J. Numer. Anal. 23 (1986), no. 6, 1214–1229. MR 865952, DOI 10.1137/0723082
    • G. A. Chandler & I. G. Graham, Product Integration-Collocation Methods for Non-Compact Integral Operator Equations, Research Report CMA-R41-85, Centre for Mathematical Analysis, Australian National University, Canberra, 1985.
  • Martin Costabel and Ernst Stephan, Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, Mathematical models and methods in mechanics, Banach Center Publ., vol. 15, PWN, Warsaw, 1985, pp. 175–251. MR 874845
  • Martin Costabel, Boundary integral operators on curved polygons, Ann. Mat. Pura Appl. (4) 133 (1983), 305–326. MR 725031, DOI 10.1007/BF01766023
  • M. L. Dow and David Elliott, The numerical solution of singular integral equations over $(-1,\,1)$, SIAM J. Numer. Anal. 16 (1979), no. 1, 115–134. MR 518688, DOI 10.1137/0716009
    • R. De Vore & K. Scherer, "Variable knot, variable degree spline approximation to ${x^\beta }$," in Quantitative Approximation (R. De Vore and K. Scherer, eds.), Academic Press, New York, 1980, pp. 121-131.
  • I. G. Graham and G. A. Chandler, High-order methods for linear functionals of solutions of second kind integral equations, SIAM J. Numer. Anal. 25 (1988), no. 5, 1118–1137. MR 960869, DOI 10.1137/0725064
    • M. A. Jaswon & G. I. Symm, Integral Equation Methods in Potential Theory and Electrostatics, Academic Press, New York, 1977.
  • V. A. Kondrat′ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209–292 (Russian). MR 0226187
    • M. G. Krein, Integral Equations on a Half Line with Kernel Depending Upon the Difference of the Arguments, Amer. Math. Soc. Transl., vol. 22, Amer. Math. Soc., Providence, R.I., 1963, pp. 163-288.
  • U. Lamp, T. Schleicher, E. Stephan, and W. L. Wendland, Galerkin collocation for an improved boundary element method for a plane mixed boundary value problem, Computing 33 (1984), no. 3-4, 269–296 (English, with German summary). MR 773929, DOI 10.1007/BF02242273
    • W. McLean, Boundary Integral Methods for the Laplace Equation, Thesis, Australian National University, Canberra, 1985.
  • John R. Rice, On the degree of convergence of nonlinear spline approximation, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ. of Wisconsin, Madison, Wis., 1969) Academic Press, New York, 1969, pp. 349–365. MR 0267324
  • Claus Schneider, Product integration for weakly singular integral equations, Math. Comp. 36 (1981), no. 153, 207–213. MR 595053, DOI 10.1090/S0025-5718-1981-0595053-0
  • Ian H. Sloan, A review of numerical methods for Fredholm equations of the second kind, Application and numerical solution of integral equations (Proc. Sem., Australian Nat. Univ., Canberra, 1978) Monographs Textbooks Mech. Solids Fluids: Mech. Anal., vol. 6, Nijhoff, The Hague, 1980, pp. 51–74. MR 582984
  • I. H. Sloan and A. Spence, Projection methods for integral equations on the half-line, IMA J. Numer. Anal. 6 (1986), no. 2, 153–172. MR 967661, DOI 10.1093/imanum/6.2.153
    • I. N. Sneddon & S. C. Das, "The stress intensity factor at the tip of an edge crack in an elastic half plane," Internat. J. Engrg. Sci., v. 9, 1971, pp. 25-36.
  • I. N. Sneddon and M. Lowengrub, Crack problems in the classical theory of elasticity, John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0258339
  • M. P. Stallybrass, A pressurized crack in the form of a cross, Quart. J. Mech. Appl. Math. 23 (1970), 35–48. MR 261843, DOI 10.1093/qjmam/23.1.35
  • M. P. Stallybrass, A crack perpendicular to an elastic half-plane, Internat. J. Engrg. Sci. 8 (1970), 351–362 (English, with French, German, Italian and Russian summaries). MR 0261842, DOI 10.1016/0020-7225(70)90073-X
  • Frank Stenger, Numerical methods based on Whittaker cardinal, or sinc functions, SIAM Rev. 23 (1981), no. 2, 165–224. MR 618638, DOI 10.1137/1023037
  • W. L. Wendland, On some mathematical aspects of boundary element methods for elliptic problems, The mathematics of finite elements and applications, V (Uxbridge, 1984) Academic Press, London, 1985, pp. 193–227. MR 811035
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Math. Comp. 50 (1988), 125-138
  • MSC: Primary 65R20
  • DOI: https://doi.org/10.1090/S0025-5718-1988-0917821-1
  • MathSciNet review: 917821