On the convergence in $C^1_h$ of the difference solution to the Laplace equation in a rectangular parallelepiped
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 9, pp. 1587-1593
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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 Hцlder 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 $h$.
@article{ZVMMF_2005_45_9_a6,
author = {E. A. Volkov},
title = {On the convergence in $C^1_h$ of the difference solution to the {Laplace} equation in a rectangular parallelepiped},
journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
pages = {1587--1593},
publisher = {mathdoc},
volume = {45},
number = {9},
year = {2005},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_9_a6/}
}
TY - JOUR AU - E. A. Volkov TI - On the convergence in $C^1_h$ of the difference solution to the Laplace equation in a rectangular parallelepiped JO - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki PY - 2005 SP - 1587 EP - 1593 VL - 45 IS - 9 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_9_a6/ LA - ru ID - ZVMMF_2005_45_9_a6 ER -
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E. A. Volkov. On the convergence in $C^1_h$ of the difference solution to the Laplace equation in a rectangular parallelepiped. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 9, pp. 1587-1593. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_9_a6/