Product integrationcollocation methods for noncompact integral operator equations
Authors:
G. A. Chandler and I. G. Graham
Journal:
Math. Comp. 50 (1988), 125138
MSC:
Primary 65R20
MathSciNet review:
917821
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Abstract: We discuss the numerical solution of a class of secondkind 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 twodimensional 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.
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 [1]
 K. E. Atkinson, A Survey of Numerical Methods for the Solution of Integral Equations of the Second Kind, SIAM, Philadelphia, Pa., 1976.
 [2]
 K. E. Atkinson & F. R. de Hoog, "The numerical solution of Laplace's equation on a wedge," IMA J. Numer. Anal., v. 4, 1984, p. 19. MR 740782 (86a:65124)
 [3]
 C. de Boor, "Good approximation by splines with variable knots," in Spline Functions and Approximation Theory (A. Meir and A. Sharma, eds.), Birkhäuser, Basel, 1972, pp. 5772. MR 0403169 (53:6982)
 [4]
 C. de Boor, A Practical Guide to Splines, SpringerVerlag, New York, 1978. MR 507062 (80a:65027)
 [5]
 C. A. Brebbia, J. C. F. Telles & L. C. Wrobel, Boundary Element Techniques, SpringerVerlag, Berlin and New York, 1984.
 [6]
 H. G. Burchard, "The degree of convergence of piecewise polynomial approximation on optimal meshes," Trans. Amer. Math. Soc., v. 234, 1977, pp. 531559. MR 0481758 (58:1857)
 [7]
 G. A. Chandler, Superconvergence of Numerical Solutions to Second Kind Integral Equations, Thesis, Australian National University, 1979.
 [8]
 G. A. Chandler, "Galerkin's method for boundary integral equations on polygonal domains," J. Austral. Math. Soc. Ser. B, v. 26, 1984, pp. 113. MR 750551 (86a:65116)
 [9]
 G. A. Chandler, "Superconvergent approximations to the solution of a boundary integral equation," SIAM. J. Numer. Anal., v. 23, 1986, pp. 12141229. MR 865952 (88d:45014)
 [10]
 G. A. Chandler & I. G. Graham, Product IntegrationCollocation Methods for NonCompact Integral Operator Equations, Research Report CMAR4185, Centre for Mathematical Analysis, Australian National University, Canberra, 1985.
 [11]
 M. Costabel & E. Stephan, "Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation," in Mathematical Models and Methods in Mechanics, Banach Centre Publications, vol. 15, PWN, Warsaw, 1985, pp. 175251. MR 874845 (88f:35037)
 [12]
 M. Costabel, "Boundary integral operators on curved polygons," Ann. Mat. Pura Appl. (4), v. 33, 1983, pp. 305326. MR 725031 (85m:44001)
 [13]
 M. L. Dow & D. Elliott, "The numerical solution of singular integral equations over ," SIAM. J. Numer. Anal., v. 16, 1979, pp. 115134. MR 518688 (80b:65151)
 [14]
 R. De Vore & K. Scherer, "Variable knot, variable degree spline approximation to ," in Quantitative Approximation (R. De Vore and K. Scherer, eds.), Academic Press, New York, 1980, pp. 121131.
 [15]
 I. G. Graham & G. A. Chandler, "High order methods for linear functionals of solutions of second kind integral equations," SIAM J. Numer. Anal. (To appear.) MR 960869 (90b:65246)
 [16]
 M. A. Jaswon & G. I. Symm, Integral Equation Methods in Potential Theory and Electrostatics, Academic Press, New York, 1977.
 [17]
 V. A. Kondrat'ev, "Boundary problems for elliptic equations in domains with conical or angular points," Trans. Moscow Math. Soc., v. 16, 1967, pp. 227313. MR 0226187 (37:1777)
 [18]
 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. 163288.
 [19]
 U. Lamp, T. Schleicher, E. Stephan & W. L. Wendland, "Galerkin collocation for an improved boundary element method on a plane mixed boundary value problem," Computing, v. 33, 1984, pp. 269296. MR 773929 (86k:65112)
 [20]
 W. McLean, Boundary Integral Methods for the Laplace Equation, Thesis, Australian National University, Canberra, 1985.
 [21]
 J. R. Rice, "On the degree of convergence of nonlinear spline approximation," in Approximation with Special Emphasis on Spline Functions (I. J. Schoenberg, ed.), Academic Press, New York, 1969. MR 0267324 (42:2226)
 [22]
 C. Schneider, "Product integration for weakly singular integral equations," Math. Comp., v. 36, 1981, pp. 207213. MR 595053 (82c:65090)
 [23]
 I. H. Sloan, "A review of numerical methods for Fredholm equations of the second kind," in The Application and Numerical Solution of Integral Equations (R. S. Anderssen et al., eds.), Sijthoff and Noordhoff, Groningen, 1980, pp. 5174. MR 582984 (81h:65133)
 [24]
 I. H. Sloan & A. Spence, "Projection methods for integral equations on the half line," IMA J. Numer. Anal., v. 6, 1986, pp. 153172. MR 967661 (89h:65225)
 [25]
 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. 2536.
 [26]
 I. N. Sneddon & M. Lowengrub, Crack Problems in the Classical Theory of Elasticity, Wiley, New York, 1969. MR 0258339 (41:2986)
 [27]
 M. P. Stallybrass, "A pressurised crack in the form of a cross," Quart. J. Mech. Appl. Math., v. 23, 1970, pp. 3548. MR 0261843 (41:6454)
 [28]
 M. P. Stallybrass, "A crack perpendicular to an elastic half plane," Internat. J. Engrg. Sci., v. 8, 1970, pp. 351362. MR 0261842 (41:6453)
 [29]
 F. Stenger, "Numerical methods based on Whittaker cardinal, or sinc functions," SIAM Rev., v. 23, 1981, pp. 165224. MR 618638 (83g:65027)
 [30]
 W. L. Wendland, "On some mathematical aspects of boundary element methods for elliptic problems," in MAFELAP V (J. R. Whiteman, ed.), Academic Press, New York, 1985. MR 811035 (87c:65154)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809178211
PII:
S 00255718(1988)09178211
Keywords:
Secondkind integral equations,
product integration,
boundary integral equations,
collocation
Article copyright:
© Copyright 1988
American Mathematical Society
