Galerkin methods for second kind integral equations with singularities

Graham, Ivan G.

doi:10.1090/S0025-5718-1982-0669644-3

Galerkin methods for second kind integral equations with singularities

Author: Ivan G. Graham
Journal: Math. Comp. 39 (1982), 519-533
MSC: Primary 65R20; Secondary 45E05, 45L10
DOI: https://doi.org/10.1090/S0025-5718-1982-0669644-3
MathSciNet review: 669644
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper discusses the numerical solution of Fredholm integral equations of the second kind which have weakly singular kernels and inhomogeneous terms. Global convergence estimates are derived for the Galerkin and iterated Galerkin methods using splines on arbitrary quasiuniform meshes as approximating subspaces. It is observed that, due to the singularities present in the solution being approximated, the resulting convergence may be slow. It is then shown that convergence will be improved greatly by the use of splines based on a mesh which has been suitably graded to accommodate these singularities. In fact, it is shown that, under suitable conditions, the Galerkin method converges optimally and the iterated Galerkin method is superconvergent. Numerical llustrations are given.

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
P. M. Anselone, Singularity subtraction in the numerical solution of integral equations, J. Austral. Math. Soc. Ser. B 22 (1980/81), no. 4, 408–418. MR 626932, DOI https://doi.org/10.1017/S0334270000002757

P. M. Anselone & W. Krabs, "Approximate solution of weakly singular integral equations," J. Integral Equations, v. 1, 1979, pp. 61-75.

Christopher T. H. Baker, The numerical treatment of integral equations, Clarendon Press, Oxford, 1977. Monographs on Numerical Analysis. MR 0467215

J. Bechlars, Glattheit und numerische Berechnung der Lösung linearer Integralgleichungen 2. Art mit schwachsingulären Kernen, Report HMI-B283, Hahn-Meitner-Institut für Kernforschung, Berlin GmbH, 1978.

Carl de Boor, A bound on the $L_{\infty }$-norm of $L_{2}$-approximation by splines in terms of a global mesh ratio, Math. Comp. 30 (1976), no. 136, 765–771. MR 425432, DOI https://doi.org/10.1090/S0025-5718-1976-0425432-1

G. A. Chandler, Global Superconvergence of Iterated Galerkin Solutions for Second Kind Integral Equations, Technical Report, Australian National University, Canberra, 1978.

G. A. Chandler, Superconvergence for second kind integral equations, 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. 103–117. MR 582986

G. A. Chandler, Superconvergence of Numerical Solutions to Second Kind Integral Equations, Ph. D. thesis, Australian National University, Canberra, 1979.

G. A. Chandler, Product Integration Methods for Weakly Singular Second Kind Integral Equations, Technical Report, Australian National University, Canberra, 1979.

Françoise Chatelin and Rachid Lebbar, The iterated projection solution for the Fredholm integral equation of second kind, J. Austral. Math. Soc. Ser. B 22 (1980/81), no. 4, 439–451. MR 626935, DOI https://doi.org/10.1017/S0334270000002782

F. Chatelin & R. Lebbar, "Superconvergence results for the iterated projection method applied to a second kind Fredholm integral equation and eigenvalue problem." (Preprint.)

L. M. Delves, L. F. Abd-Elal, and J. A. Hendry, A fast Galerkin algorithm for singular integral equations, J. Inst. Math. Appl. 23 (1979), no. 2, 139–166. MR 529362
Jim Douglas Jr., Todd Dupont, and Lars Wahlbin, Optimal $L_{\infty }$ error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comp. 29 (1975), 475–483. MR 371077, DOI https://doi.org/10.1090/S0025-5718-1975-0371077-0
Ivan G. Graham and Ian H. Sloan, On the compactness of certain integral operators, J. Math. Anal. Appl. 68 (1979), no. 2, 580–594. MR 533515, DOI https://doi.org/10.1016/0022-247X%2879%2990138-0
Ivan G. Graham, Singularity expansions for the solutions of second kind Fredholm integral equations with weakly singular convolution kernels, J. Integral Equations 4 (1982), no. 1, 1–30. MR 640534

I. G. Graham, The Numerical Solution of Fredholm Integral Equations of the Second Kind, Ph. D. thesis. University of New South Wales, Sydney, 1980.

B. Güsmann, "${L_\infty }$-bounds of ${L_2}$-projections on splines," Quantitative Approximation (R. A. De Vore and K. Scherer, Eds.), Academic Press, New York, 1980.

Alois Kufner, Oldřich John, and Svatopluk Fučík, Function spaces, Noordhoff International Publishing, Leyden; Academia, Prague, 1977. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. MR 0482102
Qun Lin and Jia Quan Liu, Extrapolation method for Fredholm integral equations with nonsmooth kernels, Numer. Math. 35 (1980), no. 4, 459–464. MR 593839, DOI https://doi.org/10.1007/BF01399011
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
G. R. Richter, On weakly singular Fredholm integral equations with displacement kernels, J. Math. Anal. Appl. 55 (1976), no. 1, 32–42. MR 407549, DOI https://doi.org/10.1016/0022-247X%2876%2990275-4

D. W. Schlitt, "Numerical solution of a singular integral equation encountered in polymer physics," J. Math. Phys., v. 9, 1968, pp. 436-439.

Claus Schneider, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Equations Operator Theory 2 (1979), no. 1, 62–68. MR 532739, DOI https://doi.org/10.1007/BF01729361
Claus Schneider, Product integration for weakly singular integral equations, Math. Comp. 36 (1981), no. 153, 207–213. MR 595053, DOI https://doi.org/10.1090/S0025-5718-1981-0595053-0
Ian H. Sloan, Error analysis for a class of degenerate-kernel methods, Numer. Math. 25 (1975/76), no. 3, 231–238. MR 443389, DOI https://doi.org/10.1007/BF01399412
Ian H. Sloan, Improvement by iteration for compact operator equations, Math. Comp. 30 (1976), no. 136, 758–764. MR 474802, DOI https://doi.org/10.1090/S0025-5718-1976-0474802-4
Ian H. Sloan, B. J. Burn, and N. Datyner, A new approach to the numerical solution of integral equations, J. Comput. Phys. 18 (1975), 92–105. MR 398137, DOI https://doi.org/10.1016/0021-9991%2875%2990104-7
Alastair Spence, Product integration for singular integrals and singular integral equations, Numerische Integration (Tagung, Math. Forschungsinst., Oberwolfach, 1978), Internat. Ser. Numer. Math., vol. 45, Birkhäuser, Basel-Boston, Mass., 1979, pp. 288–300. MR 561301
A. F. Timan, Theory of approximation of functions of a real variable, International Series of Monographs in Pure and Applied Mathematics, Vol. 34, The Macmillan Co., New York, 1963. Translated from the Russian by J. Berry; English translation edited and editorial preface by J. Cossar; A Pergamon Press Book. MR 0192238
G. Vainikko and A. Pedas, The properties of solutions of weakly singular integral equations, J. Austral. Math. Soc. Ser. B 22 (1980/81), no. 4, 419–430. MR 626933, DOI https://doi.org/10.1017/S0334270000002769
G. Vainikko and P. Uba, A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, J. Austral. Math. Soc. Ser. B 22 (1980/81), no. 4, 431–438. MR 626934, DOI https://doi.org/10.1017/S0334270000002770

W. Volk, Die numerische Behandlung Fredholm’scher Integralgleichungen zweiter Art mittels Splinefunktionen, Report HMI-B286, Hahn-Meitner-Institut für Kernforschung, Berlin GmbH, 1979.