Abstract
We discuss an a posteriori error estimate for collocation methods applied to boundary value problems in ordinary differential equations with a singularity of the first kind. As an extension of previous results we show the asymptotical correctness of our error estimate for the most general class of singular problems where the coefficient matrix is allowed to have eigenvalues with positive real parts. This requires a new representation of the global error for the numerical solution obtained by piecewise polynomial collocation when applied to our problem class.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auzinger, W., Kneisl, G., Koch, O., Weinmüller, E.: A collocation code for boundary value problems in ordinary differential equations. Numer. Algorithms 33, 27–39 (2003)
Auzinger, W., Koch, O., Weinmüller, E.: Analysis of a new error estimate for collocation methods applied to singular boundary value problems. To appear in SIAM J. Numer. Anal. Also available at http://www.math.tuwien.ac.at/~inst115/preprints.htm/.
Auzinger, W., Koch, O., Weinmüller, E.: Efficient mesh selection for collocation methods applied to singular BVPs. To appear in J. Comput. Appl. Math. Also available at http://www.math.tuwien.ac.at/~inst115/preprints.htm/.
Auzinger, W., Koch, O., Weinmüller, E.: Efficient collocation schemes for singular boundary value problems. Numer. Algorithms 31, 5–25 (2002)
Badralexe, E., Freeman, A.J.: Eigenvalue equation for a general periodic potential and its multipole expansion solution. Phys. Rev. B 37(3), 1067–1084 (1988)
de Boor, C., Swartz, B.: Collocation at Gaussian points. SIAM J. Numer. Anal. 10, 582–606 (1973)
Carr, T.W., Erneux, T.: Understanding the bifurcation to traveling waves in a class-b laser using a degenerate Ginzburg-Landau equation. Phys. Rev. A 50, 4219–4227 (1994)
Coddington, E., Levison, N.: Theory of ordinary differential equations. McGraw-Hill, New York, 1955
Fernandez, F.M., Ogilvie, J.F.: Approximate solutions to the Thomas-Fermi equation. Phys. Rev. A 42, 149–154 (1990)
Gräff, M., Scheidl, R., Troger, H., Weinmüller, E.: An investigation of the complete post-buckling behavior of axisymmetric spherical shells. ZAMP 36, 803–821 (1985)
Hildebrand, F.B.: Introduction to numerical analysis. McGraw-Hill, New York, 2nd edition, 1974
de Hoog, F.R., Weiss, R.: Difference methods for boundary value problems with a singularity of the first kind. SIAM J. Numer. Anal. 13, 775–813 (1976)
de Hoog, F.R., Weiss, R.: Collocation methods for singular boundary value problems. SIAM J. Numer. Anal. 15, 198–217 (1978)
de Hoog, F.R., Weiss, R.: The application of Runge-Kutta schemes to singular initial value problems. Math. Comp. 44, 93–103 (1985)
Jinqiao, D., Ly, H.V., Titi, E.S.: The effect of nonlocal interactions on the dynamics of the Ginzburg-Landau equation. Z. Ang. Math. Phys. 47, 432–455 (1996)
Keller, H., Wolfe, A.: On the nonunique equilibrium states and buckling mechanism of spherical shells. J. Soc. Indust. Applied Math. 13, 674–705 (1965)
Koch, O., Kofler, P., Weinmüller, E.: Initial value problems for systems of ordinary first and second order differential equations with a singularity of the first kind. Analysis 21, 373–389 (2001)
Koch, O., Weinmüller, E.: Iterated Defect Correction for the solution of singular initial value problems. SIAM J. Numer. Anal. 38(6), 1784–1799 (2001)
Koch, O., Weinmüller, E.: Analytical and numerical treatment of a singular initial value problem in avalanche modeling. Appl. Math. Comput. 148(2), 561–570 (2003)
Stetter, H.J.: Analysis of discretization methods for ordinary differential equations. Springer-Verlag, Berlin-Heidelberg-New York, 1973
Stetter, H.J.: The defect correction principle and discretization methods. Numer. Math. 29, 425–443 (1978)
Tholfsen, P., Meissner, H.: Cylindrically symmetric solutions of the Ginzburg-Landau equations. Phys. Rev. 169, 413–416 (1968)
Weinmüller, E.: On the boundary value problems for systems of ordinary second order differential equations with a singularity of the first kind. SIAM J. Math. Anal. 15, 287–307 (1984)
Weinmüller, E.: Collocation for singular boundary value problems of second order. SIAM J. Numer. Anal. 23, 1062–1095 (1986)
Chin-Yu Yeh, A.-B. Chen, Nicholson, D.M., Butler, W.H.: Full-potential Korringa-Kohn-Rostoker band theory applied to the Mathieu potential. Phys. Rev. B 42(17), 10976–10982 (1990)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Koch, O. Asymptotically correct error estimation for collocation methods applied to singular boundary value problems. Numer. Math. 101, 143–164 (2005). https://doi.org/10.1007/s00211-005-0617-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-005-0617-2