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Sibirsk. Mat. Zh., 2011, Volume 52, Number 6, Pages 1234–1252 (Mi smj2270)  

This article is cited in 5 scientific papers (total in 5 papers)

An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace's equation

V. N. Belykh

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

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 $C^\infty$-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 $C^\infty$-smooth functions containing the exact solution to the problem. This asymptotics also has the form of an exponential function decaying to zero.

Keywords: Laplace equation, Neumann problem, unsaturated numerical method, exponential convergence.

Full text: PDF file (379 kB)
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English version:
Siberian Mathematical Journal, 2011, 52:6, 980–994

Bibliographic databases:

UDC: 519.644+532.582.33
Received: 08.11.2010

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

Citation in format AMSBIB
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\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}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2961751}
\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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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84855176004}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. L. Vaskevich, “Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures”, Siberian Math. J., 53:6 (2012), 996–1010  mathnet  crossref  mathscinet  isi  elib  elib
    2. 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  mathnet  crossref  mathscinet  isi
    3. V. N. Belykh, “Nonsaturable quadrature formulas on an interval (on Babenko's problem)”, Dokl. Math., 93:2 (2016), 197–201  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    4. V. N. Belykh, “Peculiarities of the numerical realization of unsaturated quadrature formulas on a finite interval”, Siberian Math. J., 58:5 (2017), 778–785  mathnet  crossref  crossref  isi  elib
    5. V. N. Belykh, “The problem of constructing unsaturated quadrature formulae on an interval”, Sb. Math., 210:1 (2019), 24–58  mathnet  crossref  crossref  adsnasa  isi  elib
  • Сибирский математический журнал Siberian Mathematical Journal
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