%0 Journal Article
%T Variational Statement and Domain Decomposition Algorithms for Bitsadze-Samarskii Nonlocal Boundary Value Problem for Poisson＊s Two-Dimensional Equation
%A Temur Jangveladze
%A Zurab Kiguradze
%A George Lobjanidze
%J International Journal of Partial Differential Equations
%D 2014
%R 10.1155/2014/680760
%X 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每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每15, 19每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每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每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.
%U http://www.hindawi.com/journals/ijpde/2014/680760/