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 Tr. Mat. Inst. Steklova, 2010, Volume 269, Pages 63–70 (Mi tm2892)

On a grid-method solution of the Laplace equation in an infinite rectangular cylinder under periodic boundary conditions

E. A. Volkov

Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

Abstract: We study the Dirichlet problem for the Laplace equation in an infinite rectangular cylinder. Under the assumption that the boundary values are continuous and bounded, we prove the existence and uniqueness of a solution to the Dirichlet problem in the class of bounded functions that are continuous on the closed infinite cylinder. Under an additional assumption that the boundary values are twice continuously differentiable on the faces of the infinite cylinder and are periodic in the direction of its edges, we establish that a periodic solution of the Dirichlet problem has continuous and bounded pure second-order derivatives on the closed infinite cylinder except its edges. We apply the grid method in order to find an approximate periodic solution of this Dirichlet problem. Under the same conditions providing a low smoothness of the exact solution, the convergence rate of the grid solution of the Dirichlet problem in the uniform metric is shown to be on the order of $O(h^2\ln h^{-1})$, where $h$ is the step of a cubic grid.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2010, 269, 57–64

Bibliographic databases:

UDC: 519.632.4

Citation: E. A. Volkov, “On a grid-method solution of the Laplace equation in an infinite rectangular cylinder under periodic boundary conditions”, Function theory and differential equations, Collected papers. Dedicated to Academician Sergei Mikhailovich Nikol'skii on the occasion of his 105th birthday, Tr. Mat. Inst. Steklova, 269, MAIK Nauka/Interperiodica, Moscow, 2010, 63–70; Proc. Steklov Inst. Math., 269 (2010), 57–64

Citation in format AMSBIB
\Bibitem{Vol10} \by E.~A.~Volkov \paper On a~grid-method solution of the Laplace equation in an infinite rectangular cylinder under periodic boundary conditions \inbook Function theory and differential equations \bookinfo Collected papers. Dedicated to Academician Sergei Mikhailovich Nikol'skii on the occasion of his 105th birthday \serial Tr. Mat. Inst. Steklova \yr 2010 \vol 269 \pages 63--70 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm2892} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2729973} \zmath{https://zbmath.org/?q=an:1205.65296} \elib{http://elibrary.ru/item.asp?id=15109751} \transl \jour Proc. Steklov Inst. Math. \yr 2010 \vol 269 \pages 57--64 \crossref{https://doi.org/10.1134/S0081543810020057} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000281705900005} \elib{http://elibrary.ru/item.asp?id=15337096} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77956644894} 

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This publication is cited in the following articles:
1. E. A. Volkov, “About a local grid method of a solution of Laplace’s equation in the infinite rectangular cylinder”, Comput. Math. Math. Phys., 52:1 (2012), 90–97
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