On spline approximation for a class of integral equations. I: Galerkin and collocation methods with piecewise polynomials. (English) Zbl 0662.65112

The numerical solution of a class of one-dimensional noncompact integral equations by Galerkin and collocation methods, using piecewise polynomials as basis functions, is studied. Results concerning the stability of the approximation methods without any conditions on the norm of the integral operators are obtained.
Reviewer: C.L.Koul


65R20 Numerical methods for integral equations
45B05 Fredholm integral equations
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[1] A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, 1976.
[2] Atkinson, IMA J. Numer, Anal. 4 pp 19– (1984)
[3] and , ’Über die näherungsweise Lösung von linearen Funktionalgleichungen’, in and (eds) Funktionalanalysis, Approximationstheorie, Numerische Mathematik, Birkhäser Verlag, Basel, 1967.
[4] Chandler, J. Austral. Math. Soc. B26 pp 1– (1984)
[5] Chandler, SIAM J. Numer. Anal. 23 pp 1214– (1986)
[6] and , ’Product integration-collocation methods for non-compact integral operator equations’, Research Report CMA-R41-85, Australian National University, Canberra, 1985.
[7] Chandler, IMA J Numer. Anal. 7 pp 327– (1987)
[8] and , ’Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation’, Preprint No. 593, Technische Hochschule Darmstadt, 1981.
[9] Costabel, Math. Comp. 49 pp 461– (1987)
[10] Integral Equations with Fixed Singularities, Teubner, Leipzig, 1979.
[11] Elschner, Numer, Math. 43 pp 265– (1984)
[12] ’On spline approximation for singular integral equations on an interval’, Preprint P-Math-04/87, Karl-Weierstraß-Institut für Mathematik, Berlin, 1987.
[13] Elschner, Math. Nachr. 130 pp 267– (1987)
[14] and , Faltungsgleichungen und Projektionsverfahren zu ihrer Lösung. Akademie-Verlag, Berlin, 1974.
[15] and , Einführung in die Theorie der eindimensionalen singulären Integraloperatoren, Birkhäser, Basel, 1979.
[16] Graham, Math. Comp. 39 pp 519– (1982)
[17] Lewis, Comm. Part. Diff. Eq. 8 pp 477– (1983)
[18] ’Boundary integral methods for the Laplace equation’, Thesis, Australian National University, Canberra, 1985.
[19] Prößdorf, Integral Equations Oper. Theory 7 pp 536– (1984)
[20] Prößdorf, Numer. Math. 48 pp 99– (1986)
[21] Rathsfeld, Math. Nachr.
[22] and , ’Pseudodifferential and Mellin operators in spaces with conormal singularity’, Report R-Math-01/84, Karl-Weierstraß-Institut für Mathematik, Berlin, 1984.
[23] Schmidt, Numer. Math. 50 pp 337– (1987)
[24] and , Methods for the Solution of Weakly Singular Integral Equations (in Russian), State University, Tartu, 1984.
[25] ’Boundary element methods and their asymptotic convergence’. in (ed.), Theoretical Acoustics and Numerical Techniques, Springer, Wien-New York, 1983.
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