DOI 10.15507/2079-6900.23.202101.58–71

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 65N06

On a method for approximate solution of a mixed boundary value problem for an elliptic equation

M. E. Fairuzov, F. V. Lubyshev

Bashkir State University (Ufa, Russian Federation)

Abstract. A mixed boundary value problem for an elliptic equation of divergent type with variable coefficients is considered. It is assumed that the integration region is a rectangle, and the boundary of the integration region is the union of two disjoint pieces. The Dirichlet boundary condition is set on the first piece, and the Neumann boundary condition is set on the other one. The given problem is a problem with a discontinuous boundary condition. Such problems with mixed conditions at the boundary are most often encountered in practice in process modeling, and the methods for solving them are of considerable interest. This work is related to the paper [1] and complements it. It is focused on the approbation of the results established in [1] on the convergence of approximations of the original mixed boundary value problem with the main boundary condition of the third boundary value problem already with the natural boundary condition. On the basis of the results obtained in this paper and in [1], computational experiments on the approximate solution of model mixed boundary value problems are carried out.

Key Words: elliptic equation, mixed boundary value problem, Sobolev spaces, embedding theorem, approximation, convergence of approximations, difference scheme, iterative method, grid method

For citation: M. E. Fairuzov, F. V. Lubyshev. On a method for approximate solution of a mixed boundary value problem for an elliptic equation. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 23:1(2021), 58–71. DOI: https://doi.org/10.15507/2079-6900.23.202101.58–71

Submitted: 26.12.2020; Revised: 20.02.2021; Accepted: 25.02.2021

Information about the authors:

Mahmut E. Fairuzov, Associate Professor, Department of Information Technology and Computer Mathematics, «Bashkir State University» (32 Zaki Validi St., Ufa 450076, Russia), Ph. D. (Phys.-Math.), ORCID: http://orcid.org/0000-0002-9118-660X, fairuzovme@mail.ru

Fedor V. Lubyshev, Professor, Department of Information Technology and Computer Mathematics, «Bashkir State University» (32 Zaki Validi St., Ufa 450076, Russia), Dr. Sci. (Phys.-Math.), ORCID: http://orcid.org/0000-0002-3279-4293, maxam721@mail.ru

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.