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 Zh. Vychisl. Mat. Mat. Fiz., 2005, Volume 45, Number 9, Pages 1587–1593 (Mi zvmmf596)

On the convergence in $C^1_h$ of the difference solution to the Laplace equation in a rectangular parallelepiped

E. A. Volkov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The Dirichlet problem for the Laplace equation in a rectangular parallelepiped is considered. It is assumed that the boundary values have the third derivatives on the faces that satisfy the Hölder condition, the boundary values are continuous on the edges, and their second derivatives satisfy the compatibility condition that is implied by the Laplace equation. The uniform convergence of the grid solution of the Dirichlet problem and of its difference derivative on the cubic grid at the rate $O(h^2)$, where $h$ is the grid size, is proved. A piecewise polylinear continuation of the grid solution and of its difference derivative uniformly approximate the solution of the Dirichlet problem and its second derivative on the close parallelepiped with the second order of accuracy with respect to $h$.

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

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English version:
Computational Mathematics and Mathematical Physics, 2005, 45:9, 1531–1537

Bibliographic databases:
UDC: 519.632.4

Citation: E. A. Volkov, “On the convergence in $C^1_h$ of the difference solution to the Laplace equation in a rectangular parallelepiped”, Zh. Vychisl. Mat. Mat. Fiz., 45:9 (2005), 1587–1593; Comput. Math. Math. Phys., 45:9 (2005), 1531–1537

Citation in format AMSBIB
\Bibitem{Vol05} \by E.~A.~Volkov \paper On the convergence in $C^1_h$ of the difference solution to the Laplace equation in a rectangular parallelepiped \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2005 \vol 45 \issue 9 \pages 1587--1593 \mathnet{http://mi.mathnet.ru/zvmmf596} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2216070} \zmath{https://zbmath.org/?q=an:1117.65364} \transl \jour Comput. Math. Math. Phys. \yr 2005 \vol 45 \issue 9 \pages 1531--1537 

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This publication is cited in the following articles:
1. Dosiyev A.A. Sadeghi H.M.-M., “On a highly accurate approximation of the first and pure second derivatives of the Laplace equation in a rectangular parallelpiped”, Adv. Differ. Equ., 2016, 145
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