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.


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.
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][12][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.At first domain decomposition for Bitsadze-Samarskii nonlocal boundary value problem was introduced in [7].In the abstract [11] the convergence of the domain decomposition 2 International Journal of Partial Differential Equations method for the second order nonlinear elliptic equation is given.In [12] the domain decomposition sequential and parallel algorithms are fixed.In [13] the sequential and parallel iterative algorithms are given.In [13] attention is devoted to the operator decomposition method and to the possibility of the variational formulation of the problem as well.In [15] the convergence of the domain decomposition parallel algorithm is fixed.Note that in the works [7,[11][12][13]15] results are mainly given without proof.
In the works [9,10,16,21,22,[31][32][33] different methods are displayed for study of nonlocal problems in the theory of ordinary differential equations and in the theory of equations with partial derivatives.
The present work is devoted to the variational formulation and domain decomposition and Schwarz-type iterative methods for Bitsadze-Samarskii nonlocal boundary value problem for Poisson's two-dimensional equation.Here we investigate the parallel algorithm as well as sequential ones.The rate of convergence is presented too.
The outline of this paper is as follows.In Section 2 for the Poisson equation in a rectangle we state Bitsadze-Samarskii nonlocal problem.In Section 3 the variational formulation of this problem is discussed.The convergence of the Schwarztype iterative sequential algorithm is studied in Section 4. The same question for parallel algorithm is considered in Section 5.In Section 6 some conclusions are given.

Formulation of Problem
In the plane , let us consider the rectangle  = {(, ) | − <  < 0, 0 <  < }, where  and  are the given positive constants.By  we denote the boundary of the rectangle  and by Γ  the intersection of the line  =  with the set  =  ∪ .

Variational Statement of Problem
We use usual  2 and Sobolev spaces   and ∘   .Let us denote by () the vector space of all real functions satisfying the following conditions: V(, ) is defined almost everywhere on , the boundary value V(0, ) is defined almost everywhere on Γ 0 , and V(, ) ∈  2 (), V(0, ) ∈  2 (0, ).
Let  = {(, ) | 0 ≤  ≤ , 0 ≤  ≤ } and T the operator which extends elements of () as follows: Operator T associates to every function V(, ) of the vector space () the following function Ṽ(, ) = TV(, ) in such a way that the function Ṽ(, ) − V(0, ) is the odd function with respect to the variable  almost everywhere for almost all  ∈ [0, ].
Let us define on vector space () the scalar product Introducing the scalar product ( 5), let us denote vector space () by () where the norm is defined as follows: The following statements take place [24].
Theorem 1.The norm defined in () by the formula is equivalent to the norm ‖ ⋅ ‖  .
Let the domain of definition of the operator  = −Δ be the vector space   () ⊂ () of the elements V(, ) defined on  for which the following conditions are fulfilled: Theorem 3. The vector space   () is dense in the space ().
Thus, the operator  acts from the dense vector space   () of the Hilbert space () to the space ().

Theorem 4.
Operator  is positively defined on the vector space   ().
To show the symmetry of the operator  we use the following two lemmas, whose proofs are not difficult.Lemma 5.For an arbitrary function V(, ) of the vector space   () the following identity is valid: Lemma 6.For two arbitrary functions V(, ) and (, ) of the vector space   () we have The scalar product given by ( 5) can be represented in the form × (2 (0, ) −  (, ))   .(10) In the case of the scalar product (5) we have the positively defined operator , but it is not symmetric.
As  is positive definite operator defined on the vector space   () which is dense in the space (), for the problem ( 1)-( 3) we can use the standard way of the variational formulation [27].
Let us introduce the new scalar product on   (): Denote by ‖ ⋅ ‖  the corresponding norm and by   (V, ) the corresponding metric.By   () we denote the space obtained after completion of   () by the metric   (V, ).
The following statement is true [24].
Theorem 7. The function V(, ) ∈ () belongs to the space   () if and only if the following relations are fulfilled: Thus, functions of the space   () satisfy boundary conditions.For every function (, ) ∈ () there exists a unique function (, ) in the space   (), which minimizes the quadratic functional For any function V(, ) ∈   () the following relation is fulfilled The function (, ) from the space   () which minimizes the functional ( 14) is called the generalized solution of the equation  = .
If the function (, ) is sufficiently smooth then (, ) is a solution in a classical sense of problem (1)-(3).

Domain Decomposition and Sequential Algorithm
In this section and next sections, for simplicity, let us consider Laplace equation with nonlocal (3) and again for simplicity homogeneous Dirichlet (2) conditions.So, we study the following problem: (, )    Γ = 0, For problem ( 15)-( 17) let us consider the following sequential iterative procedure: Here we utilize the following notations: where , and  0 2 (− 1 , ) = () is any continuous function on the segment [0, ], which satisfies the following conditions: (0) = () = 0.
The iterative procedure (18) reduces our nonlocal nonclassical problem ( 15)- (17) to the investigation of the sequence of classical Dirichlet boundary value problems on every step of the iteration.
The following statement takes place.
Proof.Note that solving problem (18), we get two sequences {  1 }, {  2 } of harmonic functions, which are defined on the domains  1 and  2 , respectively.
We have the following relations: Let us introduce the notations: If  1  ̸ = 0, then, for the harmonic function where  ∈ (0, 1) and depends only on domain  1 .
If   ̸ = 0, then, for the harmonic function 2 )/  , applying the extremum principle [30, pages 218] to the domain  2 , we have If  1  = 0 for some index , then from the extremum principle we get   = 0. So, the inequality ( 22) is also true.
If   = 0 for some index , then from the extremum principle we get again  1 +1 = 0. So, the inequality ( 23) is also true.
From the estimations ( 22) and ( 23) obtained above it follows that This means that the sequences {  } and { 1  } tend to zero, and we obtain the uniform convergence of the series in the domains  1 and  2 , respectively.
According to Weierstrass theorem [30, pages 232, 233], the functions  1 and  2 are harmonic ones, defined, respectively, in  1 and  2 , and satisfy the condition (16).As for domain  12 =  1 ∩  2 , we have the following relations: . The latter difference tends to zero uniformly.Again, according to the extremum principle, we obtain that the functions  1 and  2 coincide with each other in the domain  12 and define a regular harmonic function on , which represents the solution of the problem ( 15)- (17).Now, let us estimate the rate of convergence of the iterative process (18).Using the triangle inequality, the extremum principle and the second inequality of (24), we have Thus, If in this inequality we tend  to ∞, then we obtain in the domain  1 This completes the proof of Theorem 8.
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Domain Decomposition and Parallel Algorithm
Algorithm (18) for the solution of the problem (15) has a sequential form.Now, let us consider one more approach to the solution of the problem ( 15)- (17).In this case the search of approximate solutions on domains  1 and  2 will be carried out not by means of a sequential algorithm but in a parallel way.
The following statement takes place.
Proof.Let us prove this theorem in a similar way as Theorem 8 was proven.We should note that the boundary value problems (30) If we introduce the notation and, for the harmonic function V  1 = ( +1 1 −   1 )/ 1  , apply the lemma from [30, pages 250-254] to the domain  1 , we will have the estimation where  ∈ (0, 1) and depends on domain  1 only.
If  1  = 0, then from the extremum principle we get that   = 0 and the inequality (33) is clear.
If   ̸ = 0, then, for the harmonic function 2 )/ −1 , from the extremum principle in the domain  2 we have If  −1 = 0, then, using again the extremum principle, we have  1  = 0 and the inequality (34) holds.From the estimations (33) and (34), we obtain That means that the sequences {  } and { 1  } tend to zero.Thus, in this case, the series analogous to the series from (25) are also uniformly convergent.The corresponding harmonic functions  1 and  2 are defined in  1 and  2 , respectively, and satisfy condition (16).As for the common part of these domains, we have if  = − and  ∈ [0, ], then   2 −   1 =   2 −  +1 2 ; if  = − 1 and  ∈ [0, ], then   2 −   1 =   2 −  −1 2 and this difference tends to zero, when  → ∞.Thus, the functions  1 and  2 again coincide in the domain  12 and define a regular harmonic function on , which represents the solution of the problem ( 15)- (17).
Let us estimate the rate of convergence of the constructed sequences.