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 Sibirsk. Mat. Zh., 2013, Volume 54, Number 6, Pages 1237–1249 (Mi smj2490)

Particular features of implementation of an unsaturated numerical method for the exterior axisymmetric Neumann problem

V. N. Belykh

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

Abstract: Using Babenko's profound ideas, we construct a fundamentally new unsaturated numerical method for solving the spectral problem for the operator of the exterior axisymmetric Neumann problem for Laplace's equation. We estimate the deviation of the first eigenvalue of the discretized problem from the eigenvalue of the Neumann operator. More exactly, the unsaturated discretization of the spectral Neumann problem yields an algebraic problem with a good matrix, i.e., a matrix inheriting the spectral properties of the Neumann operator. Thus, its spectral portrait lacks “parasitic” eigenvalues provided that the discretization error is sufficiently small. The error estimate for the first eigenvalue involves efficiently computable parameters, which in the case of $C^\infty$-smooth data provides a foundation for a guaranteed success.

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

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English version:
Siberian Mathematical Journal, 2013, 54:6, 984–993

Bibliographic databases:

UDC: 519.632.6+532.582.33

Citation: V. N. Belykh, “Particular features of implementation of an unsaturated numerical method for the exterior axisymmetric Neumann problem”, Sibirsk. Mat. Zh., 54:6 (2013), 1237–1249; Siberian Math. J., 54:6 (2013), 984–993

Citation in format AMSBIB
\Bibitem{Bel13} \by V.~N.~Belykh \paper Particular features of implementation of an unsaturated numerical method for the exterior axisymmetric Neumann problem \jour Sibirsk. Mat. Zh. \yr 2013 \vol 54 \issue 6 \pages 1237--1249 \mathnet{http://mi.mathnet.ru/smj2490} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3184089} \transl \jour Siberian Math. J. \yr 2013 \vol 54 \issue 6 \pages 984--993 \crossref{https://doi.org/10.1134/S0037446613060037} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000329110700003} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84891291781} 

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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. B. V. Semisalov, “Razrabotka i analiz bystrogo psevdospektralnogo metoda resheniya nelineinykh zadach Dirikhle”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 11:2 (2018), 123–138
2. V. N. Belykh, “The problem of constructing unsaturated quadrature formulae on an interval”, Sb. Math., 210:1 (2019), 24–58
3. Belykh V.N., “Numerical Implementation of Nonstationary Axisymmetric Problems of An Ideal Incompressible Fluid With a Free Surface”, J. Appl. Mech. Tech. Phys., 60:2 (2019), 382–391
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