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 Zh. Vychisl. Mat. Mat. Fiz., 2007, Volume 47, Number 4, Pages 665–670 (Mi zvmmf305)

On a combined grid method for solving the Dirichlet problem for the Laplace equation in a rectangular parallelepiped

E. A. Volkov

Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Vavilova 42, Moscow, 119991, Russia

Abstract: A combined grid method for solving the Dirichlet problem for the Laplace equation in a rectangular parallelepiped is proposed. At the grid points that are at the distance equal to the grid size from the boundary, the 6-point averaging operator is used. At the other grid points, the 26-point averaging operator is used. It is assumed that the boundary values have the third derivatives satisfying the Lipschitz condition on the faces; on the edges, they are continuous and their second derivatives satisfy the compatibility condition implied by the Laplace equation. The uniform convergence of the grid solution with the fourth order with respect to the grid size is proved.

Key words: Numerical solution of the Laplace equation, convergence of grid solutions, rectangular parallelepiped domain.

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English version:
Computational Mathematics and Mathematical Physics, 2007, 47:4, 638–643

Bibliographic databases:

UDC: 519.632.4

Citation: E. A. Volkov, “On a combined grid method for solving the Dirichlet problem for the Laplace equation in a rectangular parallelepiped”, Zh. Vychisl. Mat. Mat. Fiz., 47:4 (2007), 665–670; Comput. Math. Math. Phys., 47:4 (2007), 638–643

Citation in format AMSBIB
\Bibitem{Vol07} \by E.~A.~Volkov \paper On a~combined grid method for solving the Dirichlet problem for the Laplace equation in a~rectangular parallelepiped \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2007 \vol 47 \issue 4 \pages 665--670 \mathnet{http://mi.mathnet.ru/zvmmf305} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2376630} \zmath{https://zbmath.org/?q=an:05200950} \transl \jour Comput. Math. Math. Phys. \yr 2007 \vol 47 \issue 4 \pages 638--643 \crossref{https://doi.org/10.1134/S0965542507040094} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34248136895} 

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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. E. A. Volkov, “A two-stage difference method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped”, Comput. Math. Math. Phys., 49:3 (2009), 496–501
2. E. A. Volkov, “A modified combined grid method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped”, Comput. Math. Math. Phys., 50:2 (2010), 274–284
3. E. A. Volkov, “Application of a 14-point averaging operator in the grid method”, Comput. Math. Math. Phys., 50:12 (2010), 2023–2032
4. Dosiyev A.A., “New properties of 9-point finite difference solution of the Laplace equation”, Mediterr. J. Math., 8:3 (2011), 451–462
5. Comput. Math. Math. Phys., 52:6 (2012), 879–886
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