Jump to ContentJump to Main Navigation
Show Summary Details
More options …

# Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2018: 0.27

Online
ISSN
1572-9176
See all formats and pricing
More options …
Volume 23, Issue 2

# Method of corrections by higher order differences for elliptic equations with variable coefficients

Givi Berikelashvili
• Corresponding author
• A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, 6 Tamarashvili Str., Tbilisi 0179; and Department of Mathematics, Georgian Technical University, 77 M. Kostava Str., Tbilisi 0175, Georgia
• Email
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Bidzina Midodashvili
• Faculty of Exact and Natural Sciences, I. Javakhishvili Tbilisi State University, 13 University Str., Tbilisi 0186; and Faculty of Education, Exact and Natural Sciences, Gori Teaching University, 5 I. Chavchavadze Str., Gori, Georgia
• Email
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
Published Online: 2016-03-16 | DOI: https://doi.org/10.1515/gmj-2016-0008

## Abstract

We consider the Dirichlet problem for an elliptic equation with variable coefficients, the solution of which is obtained by means of a finite-difference scheme of second order accuracy. We establish a two-stage finite-difference method for the posed problem and obtain an estimate of the convergence rate consistent with the smoothness of the solution. It is proved that the solution of the corrected scheme converges at rate $O\left(|h{|}^{m}\right)$ in the discrete L2-norm, when the solution of the original problem belongs to the Sobolev space with exponent $m\in \left[2,4\right]$.

MSC: 65M06; 65M12

## References

• 1

G. K. Berikelashvili, On the convergence in W22 of the difference solution of the Dirichlet problem (in Russian), Zh. Vychisl. Mat. i Mat. Fiz. 30 (1990), 3, 470–474; translation in USSR Comput. Math. and Math. Phys. 30 (1990), no. 2, 89–92. Google Scholar

• 2

G. Berikelashvili, The difference schemes of high order accuracy for elliptic equations with lower derivatives, Proc. A. Razmadze Math. Inst. 117 (1998), 1–6. Google Scholar

• 3

G. Berikelashvili, Construction and analysis of difference schemes for some elliptic problems, and consistent estimates of the rate of convergence, Mem. Differ. Equ. Math. Phys. 38 (2006), 1–131. Google Scholar

• 4

G. Berikelashvili, M. M. Gupta and M. Mirianashvili, Convergence of fourth order compact difference schemes for three-dimensional convection-diffusion equations, SIAM J. Numer. Anal. 45 (2007), 1, 443–455.

• 5

J. H. Bramble and S. R. Hilbert, Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math. 16 (1970/1971), 362–369. Google Scholar

• 6

T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), 150, 441–463. Google Scholar

• 7

L. Fox, Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations, Proc. R. Soc. Lond. Ser. A 190 (1947), 31–59. Google Scholar

• 8

B. S. Jovanović, The Finite Difference Method for Boundary-Value Problems with Weak Solutions, Posebna Izdan. 16, Matematički Institut u Beogradu, Belgrade, 1993. Google Scholar

• 9

R. D. Lazarov, V. L. Makarov and A. A. Samarskiĭ, Application of exact difference schemes for constructing and investigating difference schemes on generalized solutions (in Russian), Mat. Sb. (N.S.) 117(159) (1982), 4, 469–480. Google Scholar

• 10

R. D. Lazarov, V. L. Makarov and W. Weinelt, On the convergence of difference schemes for the approximation of solutions $u\in {W}_{2}^{m}$ ($m>0.5$) of elliptic equations with mixed derivatives, Numer. Math. 44 (1984), 2, 223–232. Google Scholar

• 11

A. A. Samarskii, R. D. Lazarov and V. L. Makarov, Difference Schemes for Differential Equations with Generalized Solutions (in Russian), Visshaja Shkola, Moscow, 1987. Google Scholar

• 12

E. A. Volkov, On a method of increasing the accuracy of the method of grids (in Russian), Dokl. Akad. Nauk SSSR (N.S.) 96 (1954), 685–688. Google Scholar

• 13

E. A. Volkov, Solving the Dirichlet problem by a method of corrections with higher order differences. I (in Russian), Differ. Uravn. 1 (1965), 7, 946–960. Google Scholar

• 14

E. A. Volkov, Solving the Dirichlet problem by a method of corrections with higher order differences. II (in Russian), Differ. Uravn. 1 (1965), 8, 1070–1084. Google Scholar

• 15

E. A. Volkov, On a two-stage difference method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped (in Russian), Zh. Vychisl. Mat. Mat. Fiz. 49 (2009), 3, 512–517; translation in Comput. Math. Math. Phys. 49 (2009), no. 3, 496–501.

## About the article

Received: 2014-05-06

Revised: 2014-07-16

Accepted: 2014-08-02

Published Online: 2016-03-16

Published in Print: 2016-06-01

Funding Source: Shota Rustaveli National Science Foundation

Award identifier / Grant number: FR/406/5-106/12

The work was supported by the Shota Rustaveli National Science Foundation under the grant FR/406/5-106/12.

Citation Information: Georgian Mathematical Journal, Volume 23, Issue 2, Pages 169–180, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X,

Export Citation

© 2016 by De Gruyter.

## Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Givi Berikelashvili and Bidzina Midodashvili
Boundary Value Problems, 2015, Volume 2015, Number 1

## Comments (0)

Please log in or register to comment.
Log in