Geodesic
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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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},
     year = {2005},
     volume = {45},
     number = {9},
     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
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_9_a6/
LA  - ru
ID  - ZVMMF_2005_45_9_a6
ER  - 
%0 Journal Article
%A E. A. Volkov
%T On the convergence in $C^1_h$ of the difference solution to the Laplace equation in a rectangular parallelepiped
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2005
%P 1587-1593
%V 45
%N 9
%U http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_9_a6/
%G ru
%F ZVMMF_2005_45_9_a6
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/

[1] Wasow W., “On the truncation error in the solution of Laplace's equation by finite differences”, J. Res. Nat. Bur. Standards, 48:4 (1952), 345–348 | MR

[2] Volkov E. A., “On the solution of the Dirichlet problem for the Laplace equation on a rectangular parallelepiped by the grid method”, Russ. J. Numer. Anal. Math. Modelling, 16:6 (2001), 519–527 | MR | Zbl

[3] Lebedev V. I., “Otsenka pogreshnosti metoda setok dlya dvumernoi zadachi Neimana”, Dokl. AN SSSR, 132:5 (1960), 1016–1018 | Zbl

[4] Volkov E. A., “Approximation of the derivatives of the solution of the Dirichlet problem for the Laplace equation on a rectangular parallelepiped by the grid method”, Russ. J. Numer. Anal. Math. Modelling, 19:3 (2004) | DOI | MR

[5] Volkov E. A., “O differentsialnykh svoistvakh reshenii uravnenii Laplasa i Puassona na parallelepipede i effektivnykh otsenkakh pogreshnosti metoda setok”, Tr. MI AN SSSR, 105, 1969, 46–65 | Zbl

[6] Volkov E. A., “K voprosu o reshenii metodom setok vnutrennei zadachi Dirikhle dlya uravneniya Laplasa”, Vychislitelnaya matematika, 1, Izd-vo AN SSSR, 1957, 34–61

[7] Samarskii A. A., Andreev V. B., Raznostnye metody dlya ellipticheskikh uravnenii, Nauka, M., 1976 | MR | Zbl

[8] Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M., Chislennye metody, Nauka, M., 1987 | MR | Zbl

[9] Volkov E. A., “On the grid method for solving the Dirichlet problem for the Laplace equation in a cylinder with lateral surface of class $C_{1, 1}$”, Russ. J. Numer. Anal. Math. Modelling, 15:6 (2000), 539–551 | DOI | MR

[10] Volkov E. A., “Effektivnyi metod kubicheskikh setok resheniya uravneniya Laplasa na parallelepipede pri razryvnykh granichnykh usloviyakh”, Tr. MI AN SSSR, 156 (1980), 30–46 | Zbl