The discrete Galerkin method for integral equations

Authors:
Kendall Atkinson and Alex Bogomolny

Journal:
Math. Comp. **48** (1987), 595-616, S11

MSC:
Primary 65R20

MathSciNet review:
878693

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A general theory is given for discretized versions of the Galerkin method for solving Fredholm integral equations of the second kind. The discretized Galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the Galerkin method. The theoretical framework that is given parallels that of the regular Galerkin method, including the error analysis of the superconvergence of the iterated Galerkin and discrete Galerkin solutions. In some cases, the iterated discrete Galerkin solution is shown to coincide with the Nyström solution with the same numerical integration method. The paper concludes with applications to finite element Galerkin methods.

**[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]**Uri Ascher,*Discrete least squares approximations for ordinary differential equations*, SIAM J. Numer. Anal.**15**(1978), no. 3, 478–496. MR**491701**, 10.1137/0715031**[3]**Kendall E. Atkinson,*The numerical solutions of the eigenvalue problem for compact integral operators*, Trans. Amer. Math. Soc.**129**(1967), 458–465. MR**0220105**, 10.1090/S0002-9947-1967-0220105-3**[4]**Kendall Atkinson,*Convergence rates for approximate eigenvalues of compact integral operators*, SIAM J. Numer. Anal.**12**(1975), 213–222. MR**0438746****[5]**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****[6]**Kendall E. Atkinson,*An introduction to numerical analysis*, John Wiley & Sons, New York-Chichester-Brisbane, 1978. MR**504339****[7]**Kendall E. Atkinson,*Piecewise polynomial collocation for integral equations on surfaces in three dimensions*, J. Integral Equations**9**(1985), no. 1, suppl., 25–48. MR**792418****[8]**Kendall E. Atkinson,*Solving integral equations on surfaces in space*, Constructive methods for the practical treatment of integral equations (Oberwolfach, 1984) Internat. Schriftenreihe Numer. Math., vol. 73, Birkhäuser, Basel, 1985, pp. 20–43. MR**882554****[9]**K. Atkinson, I. Graham, and I. Sloan,*Piecewise continuous collocation for integral equations*, SIAM J. Numer. Anal.**20**(1983), no. 1, 172–186. MR**687375**, 10.1137/0720012**[10]**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**, 10.1090/S0025-5718-1976-0425432-1**[11]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[12]**G. A. Chandler,*Superconvergence of Numerical Solutions to Second Kind Integral Equations*, Ph. D. thesis, Australian National University, Canberra, 1979.**[13]**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****[14]**Françoise Chatelin,*Spectral approximation of linear operators*, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With a foreword by P. Henrici; With solutions to exercises by Mario Ahués. MR**716134****[15]**Françoise Chatelin and Rachid Lebbar,*Superconvergence results for the iterated projection method applied to a Fredholm integral equation of the second kind and the corresponding eigenvalue problem*, J. Integral Equations**6**(1984), no. 1, 71–91. MR**727937****[16]**Philip J. Davis and Philip Rabinowitz,*Methods of numerical integration*, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR**760629****[17]**Jean Descloux,*On finite element matrices*, SIAM J. Numer. Anal.**9**(1972), 260–265. MR**0309292****[18]**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**, 10.1090/S0025-5718-1975-0371077-0**[19]**Jim Douglas Jr., Todd Dupont, and Lars Wahlbin,*The stability in 𝐿^{𝑞} of the 𝐿²-projection into finite element function spaces*, Numer. Math.**23**(1974/75), 193–197. MR**0383789****[20]**R. J. Herbold, M. H. Schultz, and R. S. Varga,*The effect of quadrature errors in the numerical solution of boundary value problems by variational techniques*, Aequationes Math.**3**(1969), 247–270. MR**0261798****[21]**G. C. Hsiao, P. Kopp, and W. L. Wendland,*A Galerkin collocation method for some integral equations of the first kind*, Computing**25**(1980), no. 2, 89–130 (English, with German summary). MR**620387**, 10.1007/BF02259638**[22]**G. C. Hsiao and W. L. Wendland,*The Aubin-Nitsche lemma for integral equations*, J. Integral Equations**3**(1981), no. 4, 299–315. MR**634453****[23]**S. Joe,*The Numerical Solution of Second Kind Fredholm Integral Equations*, Ph. D. thesis, Univ. of New South Wales, Sydney, Australia, 1985.**[24]**J. N. Lyness and D. Jespersen,*Moderate degree symmetric quadrature rules for the triangle*, J. Inst. Math. Appl.**15**(1975), 19–32. MR**0378368****[25]**John E. Osborn,*Spectral approximation for compact operators*, Math. Comput.**29**(1975), 712–725. MR**0383117**, 10.1090/S0025-5718-1975-0383117-3**[26]**Gerard R. Richter,*Superconvergence of piecewise polynomial Galerkin approximations, for Fredholm integral equations of the second kind*, Numer. Math.**31**(1978/79), no. 1, 63–70. MR**508588**, 10.1007/BF01396014**[27]**Larry L. Schumaker,*Spline functions: basic theory*, John Wiley & Sons, Inc., New York, 1981. Pure and Applied Mathematics; A Wiley-Interscience Publication. MR**606200****[28]**Ian H. Sloan,*Superconvergence and the Galerkin method for integral equations of the second kind*, Treatment of integral equations by numerical methods (Durham, 1982), Academic Press, London, 1982, pp. 197–207. MR**755355****[29]**Ian H. Sloan and Vidar Thomée,*Superconvergence of the Galerkin iterates for integral equations of the second kind*, J. Integral Equations**9**(1985), no. 1, 1–23. MR**793101****[30]**A. Spence and K. S. Thomas,*On superconvergence properties of Galerkin’s method for compact operator equations*, IMA J. Numer. Anal.**3**(1983), no. 3, 253–271. MR**723049**, 10.1093/imanum/3.3.253**[31]**A. H. Stroud,*Approximate calculation of multiple integrals*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. Prentice-Hall Series in Automatic Computation. MR**0327006****[32]**Robert Whitley,*The stability of finite rank methods with applications to integral equations*, SIAM J. Numer. Anal.**23**(1986), no. 1, 118–134. MR**821909**, 10.1137/0723008

Retrieve articles in *Mathematics of Computation*
with MSC:
65R20

Retrieve articles in all journals with MSC: 65R20

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878693-6

Article copyright:
© Copyright 1987
American Mathematical Society