Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Dosiyev A. A. A highly accurate difference method for solving the Dirichlet problem of the Laplace equation on a rectangular parallelepiped with boundary values in C^(k,1). Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2024, vol. 24, iss. 2, pp. 162-172. DOI: 10.18500/1816-9791-2024-24-2-162-172, EDN: WNJQOX

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.05.2024
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English
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518.517.944/947
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WNJQOX

A highly accurate difference method for solving the Dirichlet problem of the Laplace equation on a rectangular parallelepiped with boundary values in C^(k,1)

Autors: 
Dosiyev Adiguzel A., Western Caspian University
Abstract: 

A three-stage difference method for solving the Dirichlet problem of Laplace's equation on a rectangular parallelepiped is proposed and justified. In the first stage, approximate values of the sum of the pure fourth derivatives of the solution are defined on a cubic grid by the 14-point difference operator. In the second stage, approximate values of the sum of the pure sixth derivatives of the solution are defined on a cubic grid by the simplest $6$-point difference operator. In the third stage, the system of difference equations for the sought solution is constructed again by using the $6$-point difference operator with the correction by the quantities determined in the first and the second stages. It is proved that the proposed difference solution to the Dirichlet problem converges uniformly with the order $O(h^{6}(|\ln h|+1))$, when the boundary functions on the faces are from $C^{7,1}$ and on the edges their second, fourth, and sixth derivatives satisfy the compatibility conditions, which follows from the Laplace equation. A numerical experiment is illustrated to support the analysis made.

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Received: 
23.03.2023
Accepted: 
29.08.2023
Published: 
31.05.2024