On the convergence of collocation methods for Symm’s integral equation on open curves

Costabel, M.; Ervin, V. J.; Stephan, E. P.

doi:10.1090/S0025-5718-1988-0942148-1

On the convergence of collocation methods for Symm’s integral equation on open curves
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by M. Costabel, V. J. Ervin and E. P. Stephan PDF

Math. Comp. 51 (1988), 167-179 Request permission

Abstract:

Recently, Costabel and Stephan in [8] presented convergence proofs for collocation with piecewise linear trial functions for Symm’s integral equation on plane closed curves with corners. In this article we prove the convergence of the above collocation method in the case of open curves. We derive asymptotic error estimates in Sobolev norms and analyze the effect of graded meshes. Numerical experiments based on the implementation of [6] show experimental orders of convergence which confirm our theoretical results on the asymptotic rates of convergence.

References

Philip M. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. With an appendix by Joel Davis. MR 0443383
Douglas N. Arnold and Wolfgang L. Wendland, On the asymptotic convergence of collocation methods, Math. Comp. 41 (1983), no. 164, 349–381. MR 717691, DOI 10.1090/S0025-5718-1983-0717691-6
Douglas N. Arnold and Wolfgang L. Wendland, The convergence of spline collocation for strongly elliptic equations on curves, Numer. Math. 47 (1985), no. 3, 317–341. MR 808553, DOI 10.1007/BF01389582
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
Christopher T. H. Baker, The numerical treatment of integral equations, Monographs on Numerical Analysis, Clarendon Press, Oxford, 1977. MR 0467215

J. Comput. Mech.

Martin Costabel and Ernst Stephan, Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, Mathematical models and methods in mechanics, Banach Center Publ., vol. 15, PWN, Warsaw, 1985, pp. 175–251. MR 874845
Martin Costabel and Ernst P. Stephan, On the convergence of collocation methods for boundary integral equations on polygons, Math. Comp. 49 (1987), no. 180, 461–478. MR 906182, DOI 10.1090/S0025-5718-1987-0906182-9
Martin Costabel and Ernst P. Stephan, Collocation methods for integral equations on polygons, Innovative numerical methods in engineering (Atlanta, Ga., 1986) Comput. Mech., Southampton, 1986, pp. 43–50. MR 902858
Johannes Elschner, On spline collocation for singular integral equations on an interval, Seminar analysis, Akad. Wiss. DDR, Berlin, 1986, pp. 31–54. MR 890525
I. C. Gohberg and I. A. Fel′dman, Convolution equations and projection methods for their solution, Translations of Mathematical Monographs, Vol. 41, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by F. M. Goldware. MR 0355675
Roland Hagen and Bernd Silbermann, A finite element collocation method for bisingular integral equations, Applicable Anal. 19 (1985), no. 2-3, 117–135. MR 800163, DOI 10.1080/00036818508839538
M. A. Krasnosel′skiĭ, G. M. Vaĭnikko, P. P. Zabreĭko, Ya. B. Rutitskii, and V. Ya. Stetsenko, Approximate solution of operator equations, Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the Russian by D. Louvish. MR 0385655, DOI 10.1007/978-94-010-2715-1

Non-Homogeneous Boundary Value Problems and Applications I

Siegfried Prössdorf, Ein Lokalisierungsprinzip in der Theorie der Spline-Approximationen und einige Anwendungen, Math. Nachr. 119 (1984), 239–255 (German). MR 774194, DOI 10.1002/mana.19841190123
Siegfried Prössdorf and Bernd Silbermann, Gestörte Projektionsverfahren und einige ihrer Anwendungen, Theory of nonlinear operators (Proc. Fifth Internat. Summer School, Central Inst. Math. Mech. Acad. Sci. GDR, Berlin, 1977) Abh. Akad. Wiss. DDR, Abt. Math. Naturwiss. Tech., 1978, vol. 6, Akademie-Verlag, Berlin, 1978, pp. 229–237 (German, with English summary). MR 540463
Siegfried Prössdorf and Andreas Rathsfeld, A spline collocation method for singular integral equations with piecewise continuous coefficients, Integral Equations Operator Theory 7 (1984), no. 4, 536–560. MR 757987, DOI 10.1007/BF01238865
J. Saranen and W. L. Wendland, On the asymptotic convergence of collocation methods with spline functions of even degree, Math. Comp. 45 (1985), no. 171, 91–108. MR 790646, DOI 10.1090/S0025-5718-1985-0790646-3
G. Schmidt, On spline collocation for singular integral equations, Math. Nachr. 111 (1983), 177–196. MR 725777, DOI 10.1002/mana.19831110108
W. L. Wendland, Boundary element methods and their asymptotic convergence, Theoretical acoustics and numerical techniques, CISM Courses and Lect., vol. 277, Springer, Vienna, 1983, pp. 135–216. MR 762829