The discrete Galerkin method for integral equations
Authors:
Kendall Atkinson and Alex Bogomolny
Journal:
Math. Comp. 48 (1987), 595616, S11
MSC:
Primary 65R20
MathSciNet review:
878693
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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, PrenticeHall, Inc., Englewood
Cliffs, N. J., 1971. With an appendix by Joel Davis; PrenticeHall Series
in Automatic Computation. MR 0443383
(56 #1753)
 [2]
Uri
Ascher, Discrete least squares approximations for ordinary
differential equations, SIAM J. Numer. Anal. 15
(1978), no. 3, 478–496. MR 491701
(81e:65043), http://dx.doi.org/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
(36 #3172), http://dx.doi.org/10.1090/S00029947196702201053
 [4]
Kendall
Atkinson, Convergence rates for approximate eigenvalues of compact
integral operators, SIAM J. Numer. Anal. 12 (1975),
213–222. MR 0438746
(55 #11653)
 [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
(58 #3577)
 [6]
Kendall
E. Atkinson, An introduction to numerical analysis, John Wiley
& Sons, New YorkChichesterBrisbane, 1978. MR 504339
(80a:65001)
 [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
(87g:65161)
 [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
(85a:65175), http://dx.doi.org/10.1137/0720012
 [10]
Carl
de Boor, A bound on the
𝐿_{∞}norm of 𝐿₂approximation by splines in
terms of a global mesh ratio, Math. Comp.
30 (1976), no. 136, 765–771. MR 0425432
(54 #13387), http://dx.doi.org/10.1090/S00255718197604254321
 [11]
Philippe
G. Ciarlet, The finite element method for elliptic problems,
NorthHolland Publishing Co., AmsterdamNew YorkOxford, 1978. Studies in
Mathematics and its Applications, Vol. 4. MR 0520174
(58 #25001)
 [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
(81h:45027)
 [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
(86d:65071)
 [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
(85i:65167)
 [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 (86d:65004)
 [17]
Jean
Descloux, On finite element matrices, SIAM J. Numer. Anal.
9 (1972), 260–265. MR 0309292
(46 #8402)
 [18]
Jim
Douglas Jr., Todd
Dupont, and Lars
Wahlbin, Optimal 𝐿_{∞} error
estimates for Galerkin approximations to solutions of twopoint boundary
value problems, Math. Comp. 29 (1975), 475–483. MR 0371077
(51 #7298), http://dx.doi.org/10.1090/S00255718197503710770
 [19]
Jim
Douglas Jr., Todd
Dupont, and Lars
Wahlbin, The stability in 𝐿^{𝑞} of the
𝐿²projection into finite element function spaces, Numer.
Math. 23 (1974/75), 193–197. MR 0383789
(52 #4669)
 [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
(41 #6410)
 [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
(83e:65210), http://dx.doi.org/10.1007/BF02259638
 [22]
G.
C. Hsiao and W.
L. Wendland, The AubinNitsche lemma for integral equations,
J. Integral Equations 3 (1981), no. 4, 299–315.
MR 634453
(83j:45019)
 [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 (51 #14536)
 [25]
John
E. Osborn, Spectral approximation for compact
operators, Math. Comput. 29 (1975), 712–725. MR 0383117
(52 #3998), http://dx.doi.org/10.1090/S00255718197503831173
 [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
(80a:65273), http://dx.doi.org/10.1007/BF01396014
 [27]
Larry
L. Schumaker, Spline functions: basic theory, John Wiley &
Sons, Inc., New York, 1981. Pure and Applied Mathematics; A
WileyInterscience Publication. MR 606200
(82j:41001)
 [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
(86j:65184)
 [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
(85c:65074), http://dx.doi.org/10.1093/imanum/3.3.253
 [31]
A.
H. Stroud, Approximate calculation of multiple integrals,
PrenticeHall, Inc., Englewood Cliffs, N.J., 1971. PrenticeHall Series in
Automatic Computation. MR 0327006
(48 #5348)
 [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
(87j:65067), http://dx.doi.org/10.1137/0723008
 [1]
 P. M. Anselone, Collectively Compact Operator Approximation Theory, PrenticeHall, Englewood Cliffs, N. J., 1971. MR 0443383 (56:1753)
 [2]
 U. Ascher, "Discrete least squares approximations for ordinary differential equations," SIAM J. Numer. Anal., v. 15, 1978, pp. 478496. MR 491701 (81e:65043)
 [3]
 K. Atkinson, "The numerical solution of the eigenvalue problem for compact integral operators," Trans. Amer. Math. Soc., v. 129, 1967, pp. 458465. MR 0220105 (36:3172)
 [4]
 K. Atkinson, "Convergence rates for approximate eigenvalues of compact integral operators," SIAM J. Numer. Anal., v. 12, 1975, pp. 213222. MR 0438746 (55:11653)
 [5]
 K. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, Pa., 1976. MR 0483585 (58:3577)
 [6]
 K. Atkinson, An Introduction to Numerical Analysis, Wiley, New York, 1978. MR 504339 (80a:65001)
 [7]
 K. Atkinson, "Piecewise polynomial collocation for integral equations on surfaces in three dimensions," J. Integral Equations, v. 9, 1985, pp. 2548. MR 792418 (87g:65161)
 [8]
 K. Atkinson, "Solving integral equations on surfaces in space," in Constructive Methods for the Practical Treatment of Integral Equations (G. Hämmerlin and K.H. Hoffmann, eds.), Birkhäuser Verlag, Basel, 1985, pp. 2043. MR 882554
 [9]
 K. Atkinson, I. Graham & I. Sloan, "Piecewise continuous collocation for integral equations," SIAM J. Numer. Anal., v. 20, 1983, pp. 172186. MR 687375 (85a:65175)
 [10]
 C. de Boor, "A bound on the norm of approximation by splines in terms of a global mesh ratio," Math. Comp., v. 30, 1976, pp. 765771. MR 0425432 (54:13387)
 [11]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [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," in The Application and Numerical Solution of Integral Equations (R. S. Anderssen, F. de Hoog, and M. Lukas, eds.), Sijthoff & Noordhoff, Groningen, 1980, pp. 103118. MR 582986 (81h:45027)
 [14]
 F. Chatelin, Spectral Approximation of Linear Operators, Academic Press, New York, 1984. MR 716134 (86d:65071)
 [15]
 F. Chatelin & R. 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, v. 6, 1984, pp. 7191. MR 727937 (85i:65167)
 [16]
 P. J. Davis & P. Rabinowitz, Methods of Numerical Integration, 2nd ed., Academic Press, New York, 1984. MR 760629 (86d:65004)
 [17]
 J. Descloux, "On finite element matrices," SIAM. J. Numer. Anal., v. 9, 1972, pp. 260265. MR 0309292 (46:8402)
 [18]
 J. Douglas Jr., Todd Dupont & Lars Wahlbin, "Optimal error estimates for Galerkin approximations to solutions of twopoint boundary value problems," Math. Comp., v. 29, 1975, pp. 475483. MR 0371077 (51:7298)
 [19]
 J. Douglas Jr., Todd Dupont & Lars Wahlbin, "The stability in of the projection into finite element function spaces," Numer. Math., v. 23, 1975, pp. 193197. MR 0383789 (52:4669)
 [20]
 R. J. Herbold, M. H. Schultz & R. S. Varga, "The effect of quadrature errors in the numerical solution of boundary value problems by variational techniques," Aequationes Math., v. 3, 1969, pp. 247270. MR 0261798 (41:6410)
 [21]
 G. C. Hsiao, P. Kopp & W. L. Wendland, "A Galerkin collocation method for some integral equations of the first kind," Computing, v. 25, 1980, pp. 89130. MR 620387 (83e:65210)
 [22]
 G. C. Hsiao & W. L. Wendland, "The AubinNitsche lemma for integral equations," J. Integral Equations, v. 3, 1981, pp. 299315. MR 634453 (83j:45019)
 [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 & D. C. Jespersen, "Moderate degree symmetric quadrature rules for the triangle," J. Inst. Math. Appl., v. 15, 1975, pp. 1932. MR 0378368 (51:14536)
 [25]
 J. E. Osborn, "Spectral approximation for compact operators," Math. Comp., v. 29, 1975, pp. 712725. MR 0383117 (52:3998)
 [26]
 G. R. Richter, "Superconvergence of piecewise polynomial Galerkin approximations for Fredholm integral equations of the second kind," Numer. Math., v. 31, 1978, pp. 6370. MR 508588 (80a:65273)
 [27]
 L. L. Schumaker, Spline Functions: Basic Theory, Wiley, New York, 1981. MR 606200 (82j:41001)
 [28]
 I. H. Sloan, "Superconvergence and the Galerkin method for integral equations," in Treatment of Integral Equations by Numerical Methods (C. T. H. Baker and G. F. Miller, eds.), Academic Press, London, 1982, pp. 197208. MR 755355
 [29]
 I. H. Sloan & Vidar Thomée, "Superconvergence of the Galerkin iterates for integral equations of the second kind," J. Integral Equations, v. 9, 1985, pp. 123. MR 793101 (86j:65184)
 [30]
 A. Spence & K. S. Thomas, "On superconvergence properties of Galerkin's method for compact operator equations," IMA J. Numer. Anal., v. 3, 1983, pp. 253271. MR 723049 (85c:65074)
 [31]
 A. H. Stroud, Approximate Calculation of Multiple Integrals, PrenticeHall, Englewood Cliffs, N.J., 1971. MR 0327006 (48:5348)
 [32]
 R. Whitley, "The stability of finite rank methods with applications to integral equations," SIAM J. Numer. Anal., v. 23, 1986, pp. 118134. MR 821909 (87j:65067)
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DOI:
http://dx.doi.org/10.1090/S00255718198708786936
PII:
S 00255718(1987)08786936
Article copyright:
© Copyright 1987
American Mathematical Society
