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

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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.

**[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]**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**, https://doi.org/10.1017/S0334270000002757**[3]**P. M. Anselone & W. Krabs, "Approximate solution of weakly singular integral equations,"*J. Integral Equations*, v. 1, 1979, pp. 61-75.**[4]**Christopher T. H. Baker,*The numerical treatment of integral equations*, Clarendon Press, Oxford, 1977. Monographs on Numerical Analysis. MR**0467215****[5]**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.**[6]**Carl de Boor,*A bound on the 𝐿_{∞}-norm of 𝐿₂-approximation by splines in terms of a global mesh ratio*, Math. Comp.**30**(1976), no. 136, 765–771. MR**0425432**, https://doi.org/10.1090/S0025-5718-1976-0425432-1**[7]**G. A. Chandler,*Global Superconvergence of Iterated Galerkin Solutions for Second Kind Integral Equations*, Technical Report, Australian National University, Canberra, 1978.**[8]**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****[9]**G. A. Chandler,*Superconvergence of Numerical Solutions to Second Kind Integral Equations*, Ph. D. thesis, Australian National University, Canberra, 1979.**[10]**G. A. Chandler,*Product Integration Methods for Weakly Singular Second Kind Integral Equations*, Technical Report, Australian National University, Canberra, 1979.**[11]**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**, https://doi.org/10.1017/S0334270000002782**[12]**F. Chatelin & R. Lebbar, "Superconvergence results for the iterated projection method applied to a second kind Fredholm integral equation and eigenvalue problem." (Preprint.)**[13]**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****[14]**Jim Douglas Jr., Todd Dupont, and Lars Wahlbin,*Optimal 𝐿_{∞} error estimates for Galerkin approximations to solutions of two-point boundary value problems*, Math. Comp.**29**(1975), 475–483. MR**0371077**, https://doi.org/10.1090/S0025-5718-1975-0371077-0**[15]**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**, https://doi.org/10.1016/0022-247X(79)90138-0**[16]**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****[17]**I. G. Graham,*The Numerical Solution of Fredholm Integral Equations of the Second Kind*, Ph. D. thesis. University of New South Wales, Sydney, 1980.**[18]**B. Güsmann, " -bounds of -projections on splines,"*Quantitative Approximation*(R. A. De Vore and K. Scherer, Eds.), Academic Press, New York, 1980.**[19]**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****[20]**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**, https://doi.org/10.1007/BF01399011**[21]**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****[22]**G. R. Richter,*On weakly singular Fredholm integral equations with displacement kernels*, J. Math. Anal. Appl.**55**(1976), no. 1, 32–42. MR**0407549**, https://doi.org/10.1016/0022-247X(76)90275-4**[23]**D. W. Schlitt, "Numerical solution of a singular integral equation encountered in polymer physics,"*J. Math. Phys.*, v. 9, 1968, pp. 436-439.**[24]**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**, https://doi.org/10.1007/BF01729361**[25]**Claus Schneider,*Product integration for weakly singular integral equations*, Math. Comp.**36**(1981), no. 153, 207–213. MR**595053**, https://doi.org/10.1090/S0025-5718-1981-0595053-0**[26]**Ian H. Sloan,*Error analysis for a class of degenerate-kernel methods*, Numer. Math.**25**(1975/76), no. 3, 231–238. MR**0443389**, https://doi.org/10.1007/BF01399412**[27]**Ian H. Sloan,*Improvement by iteration for compact operator equations*, Math. Comp.**30**(1976), no. 136, 758–764. MR**0474802**, https://doi.org/10.1090/S0025-5718-1976-0474802-4**[28]**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**0398137**, https://doi.org/10.1016/0021-9991(75)90104-7**[29]**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****[30]**A. F. Timan,*Theory of approximation of functions of a real variable*, Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, Vol. 34, A Pergamon Press Book. The Macmillan Co., New York, 1963. MR**0192238****[31]**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**, https://doi.org/10.1017/S0334270000002769**[32]**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**, https://doi.org/10.1017/S0334270000002770**[33]**W. Volk,*Die numerische Behandlung Fredholm'scher Integralgleichungen zweiter Art mittels Splinefunktionen*, Report HMI-B286, Hahn-Meitner-Institut für Kernforschung, Berlin GmbH, 1979.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669644-3

Keywords:
Second kind Fredholm integral equation,
weak singularities,
Galerkin method,
iterated Galerkin method,
spline approximation,
graded mesh

Article copyright:
© Copyright 1982
American Mathematical Society