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A general mesh independence principle for Newton's method applied to second order boundary value problems

Ein allgemeines Gitterunabhängigkeitsprinzip für das Newton-Verfahren bei Randwertproblemen zweiter Ordnung

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Abstract

Recent work has established that for certain classes of nonlinear boundary value problems, the number of Newton iterations applied to the related standard discrete problem for a given tolerance is independent of the mesh size when the mesh is sufficiently fine. This paper develops an extension of the mesh independence principle by relaxing the assumption on the differential equation, its boundary conditions, and the related difference approximation.

Zusammenfassung

Wie kürzlich gezeigt wurde, ist bei gewissen Klassen von nichtlinearen Randwertproblemen die Anzahl der Newton-Iterationen bei der Lösung der zugehörigen Standard-Diskretisierung für eine gegebene Toleranz unabhängig von der Gitterweite, wenn das Gitter hinreichend fein ist. In der Arbeit wird dieses Gitterunabhängigkeitsprinzip durch Abschwächung der Voraussetzungen über die Differentialgleichung, die Randbedingungen und die zugehörige Differenzenapproximation verallgemeinert.

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References

  1. Allgower, E. L., McCormick, S. F.: Newton's method with mesh refinements for numerical solution of nonlinear two-point boundary value problems. Numer. Math.29, 237–260 (1978).

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  3. Daniel, J. W., Martin, A. J.: Implementing deferred corrections for Numerov's difference method for second-order two-point boundary value problems. Report CNA-107, November, 1975, Center for Numerical Analysis, University of Texas, at Austin.

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  4. McCormick, S. F.: A revised mesh refinement strategy for Newton's method applied to non-linear two-point boundary value problems, in: Numerical treatment of differential equations in applications (Ansorge, R., Tornig, W., Hrsg.)., (Lecture Notes in Mathematics, Vol. 679). Berlin-Heidelberg-New York: Springer 1978.

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Authors and Affiliations

  1. Department of Mathematics, Colorado State University, 80523, Fort Collins, CO, U.S.A.

    E. L. Allgower, F. St. McCormick & D. V. Pryor

Authors
  1. E. L. Allgower
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  2. F. St. McCormick
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  3. D. V. Pryor
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Additional information

This work was supported by the National Science Foundation under grant number MCS76-09215.

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Allgower, E.L., St. McCormick, F. & Pryor, D.V. A general mesh independence principle for Newton's method applied to second order boundary value problems. Computing 23, 233–246 (1979). https://doi.org/10.1007/BF02252130

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  • DOI: https://doi.org/10.1007/BF02252130

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