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.
Citation:
V. N. Belykh, “An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace's equation”, Sibirsk. Mat. Zh., 52:6 (2011), 1234–1252; Siberian Math. J., 52:6 (2011), 980–994
\Bibitem{Bel11}
\by V.~N.~Belykh
\paper An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace's equation
\jour Sibirsk. Mat. Zh.
\yr 2011
\vol 52
\issue 6
\pages 1234--1252
\mathnet{http://mi.mathnet.ru/smj2270}
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\transl
\jour Siberian Math. J.
\yr 2011
\vol 52
\issue 6
\pages 980--994
\crossref{https://doi.org/10.1134/S0037446611060036}
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Linking options:
https://www.mathnet.ru/eng/smj2270
https://www.mathnet.ru/eng/smj/v52/i6/p1234
This publication is cited in the following 7 articles:
V. N. Belykh, “Superconvergent algorithms for the numerical solution of the Laplace equation in smooth axisymmetric domains”, Comput. Math. Math. Phys., 60:4 (2020), 545–557
V. N. Belykh, Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov's Legacy, 2020, 13
V. N. Belykh, “The problem of constructing unsaturated quadrature formulae on an interval”, Sb. Math., 210:1 (2019), 24–58
V. N. Belykh, “Peculiarities of the numerical realization of unsaturated quadrature formulas on a finite interval”, Siberian Math. J., 58:5 (2017), 778–785
V. N. Belykh, “Nonsaturable quadrature formulas on an interval (on Babenko's problem)”, Dokl. Math., 93:2 (2016), 197–201
V. N. Belykh, “Particular features of implementation of an unsaturated numerical method for the exterior axisymmetric Neumann problem”, Siberian Math. J., 54:6 (2013), 984–993
V. L. Vaskevich, “Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures”, Siberian Math. J., 53:6 (2012), 996–1010
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