Abstract
Basing on the fundamental ideas of Babenko, we construct a fundamentally new, unsaturated, numerical method for solving the axially symmetric exterior Neumann problem for Laplace’s equation. The distinctive feature of this method is the absence of the principal error term enabling us to automatically adjust to every class of smoothness of solutions natural in the problem.
This result is fundamental since in the case of C ∞-smooth solutions the method, up to a slowly increasing factor, realizes an absolutely unimprovable exponential error estimate. The reason is the asymptotics of the Aleksandroff widths of the compact set of C ∞-smooth functions containing the exact solution to the problem. This asymptotics also has the form of an exponential function decaying to zero.
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References
Belykh V. N., “To the problem of the evolutionary “blow-up” of an axially symmetric gas bubble in an ideal incompressible fluid (the main constructive hypothesis),” in: Proc. Intern. Conf. dedicated to M. A. Lavrentyev on the occasion on his birthday centenary, Kiev, 2000, pp. 6–8.
Babenko K. I., “Some remarks on discretization of elliptic problems,” Dokl. Akad. Nauk SSSR, 221, No. 1, 11–14 (1975).
Babenko K. I., Fundamentals of Numerical Analysis [in Russian], RCD, Moscow and Izhevsk (2002).
Algazin S. D. and Kiĭko I. A., Flutter of Plates and Shells [in Russian], Nauka, Moscow (2006).
Belykh V. N., “On solving the problem of an ideal incompressible fluid flow around large-aspect-ratio axisymmetric bodies,” J. Appl. Mech. Tech. Phys., 47, No. 5, 661–670 (2006).
Belykh V. N., “Exterior axisymmetric Neumann problem for the Laplace equation: Numerical algorithms without saturation,” Dokl. Math., 76, No. 3, 882–885 (2007).
Belykh V. N., “On the best approximation properties of C ∞-smooth functions on an interval of the real axis (to the phenomenon of unsaturated numerical methods),” Siberian Math. J., 46, No. 3, 373–387 (2005).
Gyunter N. M., Theory of the Potential and Its Application to the Basic Problems of Mathematical Physics [in Russian], Gostekhteoretizdat, Moscow (1953).
Agmon S., Douglis A., and Nirenberg L., Estimates near the Boundary for Solutions of Elliptic Partial Differential Equations [Russian translation], Izdat. Inostr. Lit., Moscow (1962).
Belykh V. N., “On calculation on computers the complete elliptic integrals K(x) and E(x),” in: Boundary Value Problems for Partial Differential Equations [in Russian], IM SO RAN, Novosibirsk, 1988, pp. 3–15.
Godunov S. K., Antonov A. G., Kirilyuk O. P., and Kostin V. I., Guaranteed Accuracy in Solving Systems of Linear Equations in Euclidean Spaces [in Russian], Nauka, Novosibirsk (1992).
Nikol’skiĭ S. M., Approximation of Functions in Several Variables and Embedding Theorems [in Russian], Nauka, Moscow (1969).
Mikhlin S. G., Multidimensional Singular Integrals and Integral Equations [in Russian], Fizmatgiz, Moscow (1962).
Babenko K. I., “Saturation of numerical analysis,” Dokl. Akad. Nauk SSSR, 241, No. 3, 505–508 (1978).
Theoretical Foundations and Construction of Numerical Algorithms of Problems of Mathematical Physics [in Russian], (Eds.: N. N. Anuchina, K. I. Babenko, S. K. Godunov et al.), Nauka, Moscow (1977).
Bers L., John F., and Schechter M., Partial Differential Equations [Russian translation], Mir, Moscow (1966).
Belykh V. N., “Asymptotics of Kolmogorov’s ɛ-entropy for some classes of infinitely differentiable periodic functions (Babenko’s problem),” Dokl. Math., 81, No. 2, 293–297 (2010).
Bernstein S. N., Collected Works. Vol. 2 [in Russian], Izdat. Akad. Nauk SSSR, Moscow (1954).
Belykh V. N., “Algorithms without saturation in the problem of numerical integration,” Soviet Math. Dokl., 39, No. 1, 95–98 (1989).
Nikol’skiĭ S. M., “On the best approximation by polynomials of functions which satisfy the Lipschitz condition,” Izv. Akad. Nauk SSSR Ser. Mat., 10, No. 4, 295–318 (1946).
Dzyadyk V. K., Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).
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Original Russian Text Copyright © 2011 Belykh V. N.
The author was supported by the Russian Foundation for Basic Research (Grants 08-01-00207-a and 11-01-00147-a) and the Interdisciplinary Project of the Presidium of the Siberian Division of the Russian Academy of Sciences (No. 40).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 52, No. 6, pp. 1234–1252, November–December, 2011.
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Belykh, V.N. An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace’s equation. Sib Math J 52, 980–994 (2011). https://doi.org/10.1134/S0037446611060036
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DOI: https://doi.org/10.1134/S0037446611060036