Variational Statement and Domain Decomposition Algorithms for Bitsadze-Samarskii Nonlocal Boundary Value Problem for Poisson’s Two-Dimensional Equation
The Bitsadze-Samarskii nonlocal boundary value problem is considered. Variational formulation is done. The domain decomposition and Schwarz-type iterative methods are used. The parallel algorithm as well as sequential ones is investigated. 1. Introduction In applied sciences different problems with nonlocal boundary conditions arise very often. In some nonlocal problems, unlike classical boundary value problems, instead of boundary conditions, the dependence between the value of an unknown function on the boundary and some of its values inside of the domain is given. Modern investigation of nonlocal elliptic boundary value problems originates from Bitsadze and Samarskii work [1], in which by means of the method of integral equations the theorems are proved on the existence and uniqueness of a solution for the second order multidimensional elliptic equations in rectangular domains. Some classes of problems for which the proposed method works are given. Many works are devoted to the investigation of nonlocal problems for elliptic equations (see, e.g., [2–18] and references therein). It is known how a great role takes place in the variational formulation of classical and nonlocal boundary value problems in modern mathematics (see, e.g., [13–15, 19–27]). It is also well known that in order to find the approximate solutions, it is important to construct useful economical algorithms. For constructing such algorithms, the method of domain decomposition has a great importance (see, e.g., [23, 28, 29]). In the work [6] the iterative method of proving the existence of a solution of Bitsadze-Samarskii problem for Laplace equation was proposed. This iterative method is based on the idea of Schwarz alternating method [30, pages 249–254]. It should be noted that the usage of Schwarz alternating method not only gives us the existence of a solution, but also allows finding effective algorithms for numerical resolution of such problems. By this approach the nonlocal problem reduces to classical Dirichlet problems on whole domain that yields the possibility to apply the already developed effective methods for numerical resolution of these problems. In [7, 11–13, 15] using Schwarz alternating method and domain decomposition algorithms Bitsadze-Samarskii nonlocal problem is studied for Laplace equation. The domain decomposition algorithms are more economical than the method which was proposed in [6]. In the work [6] the reduction of nonlocal problem to the sequence of Dirichlet problems is studied. For investigating author used Schwarz lemma but not domain decomposition.
References
[1]
A. V. Bitsadze and A. A. Samarskii, “On some simplified generalization of the linear elliptic problems,” Doklady Akademii Nauk SSSR, vol. 185, pp. 739–740, 1969 (Russian).
[2]
A. Ashyralyev, “On well-posedness of the nonlocal boundary value problems for elliptic equations,” Numerical Functional Analysis and Optimization, vol. 24, no. 1-2, pp. 1–15, 2003.
[3]
A. Ashyralyev and E. Ozturk, “The numerical solution of the Bitsadze-Samarskii nonlocal boundary value problems with the Dirichlet-Neumann condition,” Abstract and Applied Analysis, vol. 2012, Article ID 730804, 13 pages, 2012.
[4]
A. Ashyralyev, O. Tetikoglu, and F. Songul, “FDM for elliptic equations with Bitsadze-Samarskii-Dirichlet conditions,” Abstract and Applied Analysis, vol. 2012, Article ID 454831, 22 pages, 2012.
[5]
G. Berikelashvili, D. Gordeziani, and S. Kharibegashvili, “Finite difference scheme for one mixed problem with integral condition,” in Proceedings of the 2nd WSEAS International Conference on Finite Differences, Finite Elements, Finite Volumes, Boundary Elements, pp. 118–120, 2009.
[6]
D. G. Gordeziani, “On one method of solving the Bitsadze-Samarskii boundary value problem,” in Proceedings of the Industrial and Applied Mathematics Seminar, pp. 39–41, 1970, 1970 (Russian).
[7]
T. A. Jangveladze, “On one iterative method of solution of Bitsadze-Samarskii boundary value problem,” Manuscript, Faculty of Applied Mathematics and Cybernetics of Tbilisi State University, Tbilisi, Georgia, 1975 (Georgian).
[8]
D. G. Gordeziani, On the Methods of Solution for One Class of Non-Local Boundary Value Problems, University Press, Tbilisi, Georgia, 1981 (Russian).
[9]
A. K. Gushin and V. P. Mikhailov, “On the continuity of the solutions of a class of non-local problems for an elliptic equation,” Sbornik: Mathematics, vol. 186, no. 2, pp. 197–219, 1995 (Russian).
[10]
V. A. Il’in and E. I. Moiseev, “Two-dimensional nonlocal boundary value problem for Poisson’s operator in the differential and difference forms,” Mathematical Modelling, vol. 2, pp. 139–156, 1990 (Russian).
[11]
T. A. Jangveladze, “Domain decomposition method for Bitsadze-Samarskii boundary value problem for second order nonlinear elliptic equation,” in Proceedings of the Postgraduate Students' Scientific Conference of Ivane Javakhishvili Tbilisi State University, p. 7, Tbilisi State University, Tbilisi, Georgia, 1977.
[12]
T. A. Jangveladze and Z. V. Kiguradze, “Domain decomposition for Bitsadze-Samarskii boundary value problem,” Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, vol. 16, pp. 16–19, 2001.
[13]
T. Jangveladze, Z. Kiguradze, and G. Lobjanidze, “On decomposition methods and variational formulation for Bitsadze-Samarskii nonlocal boundary value problem for two-dimensional second order elliptic equations,” in Proccedings of the 15th WSEAS International Conference on Applied Mathematics (MATH '10), pp. 116–121, Athens, Greece, December 2010.
[14]
J. Mo and C. Ouyang, “A class of nonlocal boundary value problems of nonlinear elliptic systems in unbounded domains,” Acta Mathematica Scientia, vol. 21, no. 1, pp. 93–97, 2001.
[15]
Z. V. Kiguradze, “Domain decomposition and parallel algorithm for Bitsadze-Samarskii boundary value problem,” Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, vol. 10, pp. 49–51, 1995.
[16]
B. P. Paneyakh, “On some nonlocal boundary value problems for linear differential operators,” Matematicheskie Zametki, vol. 35, no. 3, pp. 425–433, 1984 (Russian).
[17]
M. P. Sapagovas and P. U. Chegis, “On some boundary value problems with nonlocal conditions,” Differential Equations, vol. 23, pp. 1268–1274, 1987 (Russian).
[18]
A. L. Skubachevskii, “On a spectrum of some nonlocal elliptic boundary value problems,” Sbornik: Mathematics, vol. 117, pp. 548–558, 1982 (Russian).
[19]
T. A. Jangveladze and G. B. Lobjanidze, “On a variational statement of a nonlocal boundary value problem for a fourth-order ordinary differential equation,” Differentsial'nye Uravneniya, vol. 45, no. 3, pp. 325–333, 2009 (Russian), English translation in Differential Equations vol. 45, no. 3, pp. 335–343, 2009.
[20]
T. A. Jangveladze and G. B. Lobjanidze, “On a nonlocal boundary value problem for a fourth-order ordinary differential equation,” Differentsial'nye Uravneniya, vol. 47, no. 2, pp. 181–188, 2011 (Russian), English translation in Differential Equations vol. 47, no. 2, pp. 179–186, 2011.
[21]
G. L. Karakostas and P. Ch. Tsamatos, “Sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem,” Applied Mathematics Letters, vol. 15, no. 4, pp. 401–407, 2002.
[22]
I. Kiguradze and T. Kiguradze, “Conditions for the well-posedness of nonlocal problems for higher order linear differential equations with singularities,” Georgian Mathematical Journal, vol. 18, no. 4, pp. 735–760, 2011.
[23]
P. L. Lions, “On the Schwarz alternating method I,” in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Periaux, Eds., pp. 2–42, SIAM, Philadelphia, Pa, USA, 1988.
[24]
G. Lobjanidze, “On the variational formulation of Bitsadze-Samarskii problem for the equation ,” Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, vol. 18, pp. 39–42, 2003.
[25]
G. Lobjanidze, “On variational formulation of some nonlocal boundary value problems by symmetric continuation operation of a function,” Journal of Applied Mathematics and Mechanics, vol. 12, pp. 15–22, 2006.
[26]
G. Lobjanidze, “On variational formulation of some nonlocal boundary value problem,” Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, vol. 25, pp. 80–85, 2011.
[27]
K. Rektorys, Variational Methods in Mathematics, Science and Engineering, Springer, Prague, Czech Republic, 1980.
[28]
S. Nepomnyaschikh, “Domain decomposition methods,” Radon Series on Computational and Applied Mathematics, vol. 1, pp. 89–160, 2007.
[29]
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, Oxford, UK, 1999.
[30]
R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 2, Leningrad, Moscow, Russia, 1951 (Russian).
[31]
Ch. P. Gupta and S. I. Trofimchuk, “A sharper condition for the solvability of a three-point second order boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 205, no. 2, pp. 586–597, 1997.
[32]
A. Lomtatidze and L. Malaguti, “On a nonlocal boundary value problem for second order nonlinear singular differential equations,” Georgian Mathematical Journal, vol. 7, no. 1, pp. 133–154, 2000.
[33]
R. Ma and N. Castaneda, “Existence of solutions of nonlinear m-point boundary-value problems,” Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 556–567, 2001.
