Alternating direction implicit method for poisson equation with integral conditions
Abstract
In this paper, we investigate the convergence of the Peaceman-Rachford Alternating Direction Implicit method for the system of difference equations, approximating the two-dimensional elliptic equations in rectangular domain with nonlocal integral conditions. The main goal of the paper is the analysis of spectrum structure of difference eigenvalue problem with nonlocal conditions. The convergence of iterative method is proved in the case when the system of eigenvectors is complete. The main results are generalized for the system of difference equations, approximating the differential problem with truncation error O(h4).
Keyword : elliptic equation, integral boundary conditions, finite-difference method, iterative method, eigenvalue problem
How to Cite
Štikonienė, O., & Sapagovas, M. (2023). Alternating direction implicit method for poisson equation with integral conditions. Mathematical Modelling and Analysis, 28(4), 715–734. https://doi.org/10.3846/mma.2023.18029
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
A. Ashyralyev and E. Ozturk. On a difference scheme of fourth-order of accuracy for the Bitsadze-Samarskii type nonlocal boundary-value problem. Mathematical Methods in the Applied Sciences, 36(8):935–955, 2013. https://doi.org/10.1002/mma.2650
A. Ashyralyev and E. Ozturk. On a difference scheme of second order of accuracy for the Bitsadze-Samarskii type nonlocal boundary-value problem. Boundary Value Problems, 2014(14):1–19, 2014. https://doi.org/10.1186/1687-2770-2014-14
K.E. Atkinson. An Introduction to Numerical Analysis. Wiley, 1988.
G. Avalishvili, M. Avalishvili and D. Gordeziani. On a nonlocal problem with integral boundary conditions for a multidimensional elliptic equation. Applied Mathematics Letters, 24(4):566–571, 2011. https://doi.org/10.1016/j.aml.2010.11.014
G. Avalishvili, M. Avalishvili and D. Gordeziani. On integral nonlocal boundary value problems for some partial differential equations. Bulletin of the Georgian National Academy of Sciences, 5(1):31–37, 2011.
R.W. Beals. Nonlocal elliptic boundary value problems. Bulletin of the American Mathematical Society, 70:693–696, 1964. https://doi.org/10.1090/S0002-9904-1964-11166-6
G. Berikelashvili. Finite difference schemes for some mixed boundary problems. Proceedings of A.Razmadze Mathematical Institute, 127:77–87, 2001.
G. Berikelashvili and N. Khomeriki. On a numerical solution of one nonlocal boundary-value problem with mixed Dirichlet-Neumann conditions. Lithuanian Mathematical Journal, 53(4):367–380, 2013. https://doi.org/10.1007/s10986-013-9214-8
G. Berikelashvili and N. Khomeriki. On the convergence rate of a difference solution of the Poisson equation with fully nonlocal constraints. Nonlinear Analysis: Modelling and Control, 19(3):367–381, 2014. https://doi.org/10.15388/NA.2014.3.4
A.V. Bitsadze and A.A. Samarskii. Some elementary generalizations of linear elliptic boundary value problems. Soviet Math. Dokl., 10:398–400, 1969. (in Russian)
R. Čiupaila, K. Pupalaigė and M. Sapagovas. On the numerical solution for nonlinear elliptic equations with variable weight coefficients in an integral boundary conditions. Nonlinear Analysis: Modelling and Control, 26(4):738–758, 2021. https://doi.org/10.15388/namc.2021.26.23929
A.A. Dosiyev. Difference method of fourth order accuracy for the Laplace equation with multilevel nonlocal conditions. Journal of Computational and Applied Mathematics, 354:587–596, 2019. https://doi.org/10.1016/j.cam.2018.04.046
A.A. Dosiyev and R. Reis. A fourth-order accurate difference Dirichlet problem for the approximate solution of Laplace’s equation with integral boundary condition. Advances in Difference Equations, 2019(340):587–596, 2019. https://doi.org/10.1186/s13662-019-2282-2
N. Gordeziani. Solution Methods for a Class of Nonlocal Boundary Value Problems. Tbilisi, 1981. (in Russian)
V.A. Il’in and E.I. Moiseev. Two-dimensional nonlocal boundary value problems for Poisson’s operator in differential and difference variants. Mat. Mod., 2(8):139– 156, 1990. (in Russian)
S. Pečiulytė and A. Štikonas. On positive eigenfunctions of Sturm-Liouville problem with nonlocal two-point boundary condition. Mathematical Modelling and Analysis, 12(2):215–226, 2007. https://doi.org/10.3846/1392-6292.2007.12.215-226
S. Pečiulytė, O. Štikonienė and A. Štikonas. Investigation of negative critical points of the characteristic function for problems with nonlocal boundary conditions. Nonlinear Analysis: Modelling and Control, 13(4):467–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
A.A. Samarskii. The Theory of Difference Schemes. Moscow, Nauka, 1989. (in Russian)
M. Sapagovas, V. Griškonienė and O. Štikonienė. Application of M-matrices theory to numerical investigation of a nonlinear elliptic equation with an integral condition. Nonlinear Analysis: Modelling and Control, 22(4):489–504, 2017. https://doi.org/10.15388/NA.2017.4.5
M. Sapagovas, A. Štikonas and O. Štikonienė. Alternating direction method for the Poisson equation with variable weight coefficients in an integral condition. Differential Equations, 47(8):1176–1187, 2011. https://doi.org/10.1134/S0012266111080118
M. Sapagovas and O. Štikonienė. A fourth-order alternating-direction method for difference schemes with nonlocal condition. Lithuanian Mathematical Journal, 49(3):309–317, 2009. https://doi.org/10.1007/s10986-009-9057-5
M. Sapagovas, O. Štikonienė, K. Jakubėlienė and R. Čiupaila. Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions. Boundary Value Problems, 2019(94):1–16, 2019. https://doi.org/10.1186/s13661-019-1202-4
M.P. Sapagovas. The eigenvalues of some problem with a nonlocal condition. Differential Equations, 38(7):1020–1026, 2002. https://doi.org/10.1023/A:1021115915575
M.P. Sapagovas. Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions. Differential Equations, 44(7):1018–1028, 2008. https://doi.org/10.1134/S0012266108070148
M.P. Sapagovas and R.Yu. Čiegis. The numerical solution of some nonlocal problems. Litovsk. Matem. sborn., 27(2):348–356, 1987. (in Russian)
A. Štikonas and E. Şen. Asymptotic analysis of Sturm–Liouville problem with Neumann and nonlocal two-point boundary conditions. Lithuanian Mathematical Journal, 62(4):519–541, 2022. https://doi.org/10.1007/s10986-022-09577-6
O. Štikonienė, M. Sapagovas and R. Čiupaila. On iterative methods for some elliptic equations with nonlocal conditions. Nonlinear Analysis: Modelling and Control, 19(3):517–535, 2014. https://doi.org/10.15388/NA.2014.3.14
E.A. Volkov, A.A. Dosiyev and S.C. Buranay. On the solution of a nonlocal problem. Computers & Mathematics with Applications, 66(3):330–338, 2013. https://doi.org/10.1016/j.camwa.2013.05.010
Y. Wang. Solutions to nonlinear elliptic equations with a nonlocal boundary condition. Electron. J. Differential Equations, 2002(05):1–16, 2002. Available on Internet: http://ejde.math.txstate.edu/Volumes/2002/05/wang.pdf
L. Zhou and H. Yu. Error estimate of a high accuracy difference scheme for Poisson equation with two integral boundary conditions. Advances in Difference Equations, 2018(225):1–11, 2018. https://doi.org/10.1186/s13662-018-1682-z
A. Ashyralyev and E. Ozturk. On a difference scheme of second order of accuracy for the Bitsadze-Samarskii type nonlocal boundary-value problem. Boundary Value Problems, 2014(14):1–19, 2014. https://doi.org/10.1186/1687-2770-2014-14
K.E. Atkinson. An Introduction to Numerical Analysis. Wiley, 1988.
G. Avalishvili, M. Avalishvili and D. Gordeziani. On a nonlocal problem with integral boundary conditions for a multidimensional elliptic equation. Applied Mathematics Letters, 24(4):566–571, 2011. https://doi.org/10.1016/j.aml.2010.11.014
G. Avalishvili, M. Avalishvili and D. Gordeziani. On integral nonlocal boundary value problems for some partial differential equations. Bulletin of the Georgian National Academy of Sciences, 5(1):31–37, 2011.
R.W. Beals. Nonlocal elliptic boundary value problems. Bulletin of the American Mathematical Society, 70:693–696, 1964. https://doi.org/10.1090/S0002-9904-1964-11166-6
G. Berikelashvili. Finite difference schemes for some mixed boundary problems. Proceedings of A.Razmadze Mathematical Institute, 127:77–87, 2001.
G. Berikelashvili and N. Khomeriki. On a numerical solution of one nonlocal boundary-value problem with mixed Dirichlet-Neumann conditions. Lithuanian Mathematical Journal, 53(4):367–380, 2013. https://doi.org/10.1007/s10986-013-9214-8
G. Berikelashvili and N. Khomeriki. On the convergence rate of a difference solution of the Poisson equation with fully nonlocal constraints. Nonlinear Analysis: Modelling and Control, 19(3):367–381, 2014. https://doi.org/10.15388/NA.2014.3.4
A.V. Bitsadze and A.A. Samarskii. Some elementary generalizations of linear elliptic boundary value problems. Soviet Math. Dokl., 10:398–400, 1969. (in Russian)
R. Čiupaila, K. Pupalaigė and M. Sapagovas. On the numerical solution for nonlinear elliptic equations with variable weight coefficients in an integral boundary conditions. Nonlinear Analysis: Modelling and Control, 26(4):738–758, 2021. https://doi.org/10.15388/namc.2021.26.23929
A.A. Dosiyev. Difference method of fourth order accuracy for the Laplace equation with multilevel nonlocal conditions. Journal of Computational and Applied Mathematics, 354:587–596, 2019. https://doi.org/10.1016/j.cam.2018.04.046
A.A. Dosiyev and R. Reis. A fourth-order accurate difference Dirichlet problem for the approximate solution of Laplace’s equation with integral boundary condition. Advances in Difference Equations, 2019(340):587–596, 2019. https://doi.org/10.1186/s13662-019-2282-2
N. Gordeziani. Solution Methods for a Class of Nonlocal Boundary Value Problems. Tbilisi, 1981. (in Russian)
V.A. Il’in and E.I. Moiseev. Two-dimensional nonlocal boundary value problems for Poisson’s operator in differential and difference variants. Mat. Mod., 2(8):139– 156, 1990. (in Russian)
S. Pečiulytė and A. Štikonas. On positive eigenfunctions of Sturm-Liouville problem with nonlocal two-point boundary condition. Mathematical Modelling and Analysis, 12(2):215–226, 2007. https://doi.org/10.3846/1392-6292.2007.12.215-226
S. Pečiulytė, O. Štikonienė and A. Štikonas. Investigation of negative critical points of the characteristic function for problems with nonlocal boundary conditions. Nonlinear Analysis: Modelling and Control, 13(4):467–490, 2008. https://doi.org/10.15388/NA.2008.13.4.14552
A.A. Samarskii. The Theory of Difference Schemes. Moscow, Nauka, 1989. (in Russian)
M. Sapagovas, V. Griškonienė and O. Štikonienė. Application of M-matrices theory to numerical investigation of a nonlinear elliptic equation with an integral condition. Nonlinear Analysis: Modelling and Control, 22(4):489–504, 2017. https://doi.org/10.15388/NA.2017.4.5
M. Sapagovas, A. Štikonas and O. Štikonienė. Alternating direction method for the Poisson equation with variable weight coefficients in an integral condition. Differential Equations, 47(8):1176–1187, 2011. https://doi.org/10.1134/S0012266111080118
M. Sapagovas and O. Štikonienė. A fourth-order alternating-direction method for difference schemes with nonlocal condition. Lithuanian Mathematical Journal, 49(3):309–317, 2009. https://doi.org/10.1007/s10986-009-9057-5
M. Sapagovas, O. Štikonienė, K. Jakubėlienė and R. Čiupaila. Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions. Boundary Value Problems, 2019(94):1–16, 2019. https://doi.org/10.1186/s13661-019-1202-4
M.P. Sapagovas. The eigenvalues of some problem with a nonlocal condition. Differential Equations, 38(7):1020–1026, 2002. https://doi.org/10.1023/A:1021115915575
M.P. Sapagovas. Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions. Differential Equations, 44(7):1018–1028, 2008. https://doi.org/10.1134/S0012266108070148
M.P. Sapagovas and R.Yu. Čiegis. The numerical solution of some nonlocal problems. Litovsk. Matem. sborn., 27(2):348–356, 1987. (in Russian)
A. Štikonas and E. Şen. Asymptotic analysis of Sturm–Liouville problem with Neumann and nonlocal two-point boundary conditions. Lithuanian Mathematical Journal, 62(4):519–541, 2022. https://doi.org/10.1007/s10986-022-09577-6
O. Štikonienė, M. Sapagovas and R. Čiupaila. On iterative methods for some elliptic equations with nonlocal conditions. Nonlinear Analysis: Modelling and Control, 19(3):517–535, 2014. https://doi.org/10.15388/NA.2014.3.14
E.A. Volkov, A.A. Dosiyev and S.C. Buranay. On the solution of a nonlocal problem. Computers & Mathematics with Applications, 66(3):330–338, 2013. https://doi.org/10.1016/j.camwa.2013.05.010
Y. Wang. Solutions to nonlinear elliptic equations with a nonlocal boundary condition. Electron. J. Differential Equations, 2002(05):1–16, 2002. Available on Internet: http://ejde.math.txstate.edu/Volumes/2002/05/wang.pdf
L. Zhou and H. Yu. Error estimate of a high accuracy difference scheme for Poisson equation with two integral boundary conditions. Advances in Difference Equations, 2018(225):1–11, 2018. https://doi.org/10.1186/s13662-018-1682-z