Variational Statement and Domain Decomposition Algorithms for Bitsadze-Samarskii Nonlocal Boundary Value Problem for Poisson’s Two-Dimensional Equation

Abstract

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. 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 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, 11–13, 15] results are mainly given without proof.

In the works [9, 10, 16, 21, 22, 31–33] 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 Schwarz-type 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.

2. Formulation of Problem

In the plane 𝑂𝑥𝑦, let us consider the rectangle 𝐺={(𝑥,𝑦)∣−𝑎<𝑥<0,0<𝑦<𝑏}, where 𝑎 and 𝑏 are the given positive constants. By 𝜕𝐺 we denote the boundary of the rectangle 𝐺 and by Γ_𝑡 the intersection of the line 𝑥=𝑡 with the set 𝐺=𝐺∪𝜕𝐺.

Consider the nonlocal Bitsadze-Samarskii boundary value problem [1] −Δ𝑢𝑥,𝑦=𝑓𝑥,𝑦,𝑥,𝑦∈𝐺,𝑢(𝑥,𝑦)_Γ=0,𝑢(𝑥,𝑦)_Γ−𝜉=𝑢(𝑥,𝑦)_Γ0, (1)(2)(3) where Δ=𝜕²/𝜕𝑥²+𝜕²/𝜕𝑦² is the Laplace operator,  Γ=𝜕𝐺⧵Γ₀, 𝜉∈(0,𝑎); 𝑓(𝑥,𝑦) is a given function, 𝑓(𝑥,𝑦)∈𝐶(𝐺); and 𝑢(𝑥,𝑦)∈𝐶(𝐺)∩𝐶⁽²⁾(𝐺) is an unknown function.

Uniqueness of the solution of problem (1)–(3) follows from the extremum principle. It is known [1] that if 𝑓 is a continuous function on 𝐺, then there exists unique regular solution of problem (1)–(3) 𝑢(𝑥,𝑦)∈𝐶²(𝐺)∩𝐶(𝐺).

3. Variational Statement of Problem

We use usual 𝐿₂ and Sobolev spaces 𝐻^𝑘 and 𝐻^∘𝑘. Let us denote by 𝐷(𝐺) the vector space of all real functions satisfying the following conditions: 𝑣(𝑥,𝑦) is defined almost everywhere on 𝐺, the boundary value 𝑣(0,𝑦) is defined almost everywhere on Γ₀, and 𝑣(𝑥,𝑦)∈𝐿₂(𝐺), 𝑣(0,𝑦)∈𝐿₂(0,𝑏).

Functions 𝑣₁(𝑥,𝑦) and 𝑣₂(𝑥,𝑦) are assumed as the same element of 𝐷(𝐺) if 𝑣₁(𝑥,𝑦)=𝑣₂(𝑥,𝑦) almost everywhere on 𝐺 and 𝑣₁(0,𝑦)=𝑣₂(0,𝑦) almost everywhere on Γ₀.

Let 𝑄={(𝑥,𝑦)∣0≤𝑥≤𝜉,0≤𝑦≤𝑏} and 𝒯 the operator which extends elements of 𝐷(𝐺) as follows: =𝑣∈𝒯𝑣𝑥,𝑦𝑥,𝑦,if𝑥,𝑦∈𝐺,−𝑣−𝑥,𝑦+2𝑣0,𝑦,if𝑥,𝑦𝑄. (4)

Operator 𝒯 associates to every function 𝑣(𝑥,𝑦) of the vector space 𝐷(𝐺) the following function 𝑣̃(𝑥,𝑦)=𝒯𝑣(𝑥,𝑦) in such a way that the function 𝑣̃(𝑥,𝑦)−𝑣(0,𝑦) is the odd function with respect to the variable 𝑥 almost everywhere for almost all 𝑦∈[0,𝑏].

Let us define on vector space 𝐷(𝐺) the scalar product [𝑣,𝑤]=^𝑏₀^𝜉_−𝜉^𝑥_−𝑎𝑣̃𝑠,𝑦𝑤𝑠,𝑦𝑑𝑠𝑑𝑥𝑑𝑦. (5)

Introducing the scalar product (5), let us denote vector space 𝐷(𝐺) by 𝐻(𝐺) where the norm is defined as follows: ‖𝑣‖²_𝐻=^𝑏₀^𝜉_−𝜉^𝑥_−𝑎𝑣̃²𝑠,𝑦𝑑𝑠𝑑𝑥𝑑𝑦. (6)

The following statements take place [24].

Theorem 1. The norm defined in 𝐻(𝐺) by the formula ‖𝑣‖²=𝑣(𝑥,𝑦)²_𝐿2(𝐺)+𝑣(0,𝑦)²_{𝐿2(0,𝑏)} (7) is equivalent to the norm ‖⋅‖_𝐻.

Theorem 2. Space 𝐻(𝐺) is complete with the metric 𝜌(𝑣,𝑤)=‖𝑣−𝑤‖_𝐻.

Let the domain of definition of the operator 𝐴=−Δ be the vector space 𝐷_𝐴(𝐺)⊂𝐻(𝐺) of the elements 𝑣(𝑥,𝑦) defined on 𝐺 for which the following conditions are fulfilled:(1)𝑣(𝑥,𝑦)∈𝐶^(∞)(𝐺),  𝜕^𝑘𝑣(0,𝑦)/𝜕𝑥^𝑘=0,  𝑦∈[0,𝑏], 𝑘=1,2,…;(2)𝑣(𝑥,𝑦)|_Γ=0,  𝑣(𝑥,𝑦)|_Γ−𝜉=𝑣(𝑥,𝑦)|_Γ0.

Theorem 3. The vector space 𝐷_𝐴(𝐺) is dense in the space 𝐻(𝐺).

Thus, the operator 𝐴 acts from the dense vector space 𝐷_𝐴(𝐺) of the Hilbert space 𝐻(𝐺) to the space 𝐻(𝐺).

Theorem 4. Operator 𝐴 is positively defined on the vector space 𝐷_𝐴(𝐺).

To show the symmetry of the operator 𝐴 we use the following two lemmas, whose proofs are not difficult.

Lemma 5. For an arbitrary function 𝑣(𝑥,𝑦) of the vector space 𝐷_𝐴(𝐺) the following identity is valid: 𝒯𝐴𝑣=𝐴𝒯𝑣. (8)

Lemma 6. For two arbitrary functions 𝑣(𝑥,𝑦) and 𝑤(𝑥,𝑦) of the vector space 𝐷_𝐴(𝐺) we have ^𝜉_−𝜉𝜕𝑣̃𝑥,𝑦𝑤[].𝜕𝑥𝑥,𝑦𝑑𝑥=0,𝑦∈0,𝑏 (9)

The scalar product given by (5) can be represented in the form [𝑣,𝑤]=^𝑏₀⁰_−𝜉^𝑥_−𝑎𝑣𝑠,𝑦𝑤𝑠,𝑦𝑑𝑠𝑑𝑥𝑑𝑦+𝜉^𝑏₀⁰_−𝑎𝑣𝑤+𝑠,𝑦𝑠,𝑦𝑑𝑥𝑑𝑦_{𝑏0𝜉00−𝑥}×2𝑣0,𝑦−𝑣𝑠,𝑦2𝑤0,𝑦−𝑤𝑠,𝑦𝑑𝑠𝑑𝑥𝑑𝑦. (10)

In the case of the scalar product (5) we have the positively defined operator 𝐴, but it is not symmetric.

As 𝐴 is positive definite operator defined on the vector space 𝐷_𝐴(𝐺) which is dense in the space 𝐻(𝐺), for the problem (1)–(3) we can use the standard way of the variational formulation [27].

Let us introduce the new scalar product on 𝐷_𝐴(𝐺): [𝑣,𝑤]_𝐴=[]=𝐴𝑣,𝑤_{𝑏0𝜉−𝜉𝑥−𝑎}𝜕𝑣̃𝑠,𝑦𝜕𝑤𝜕𝑠𝑠,𝑦+𝜕𝑣̃𝜕𝑠𝑠,𝑦𝜕𝑤𝜕𝑦𝑠,𝑦𝜕𝑦𝑑𝑠𝑑𝑥𝑑𝑦. (11)

Denote by ‖⋅‖_𝐴 the corresponding norm and by 𝜌_𝐴(𝑣,𝑤) the corresponding metric. By 𝐻_𝐴(𝐺) we denote the space obtained after completion of 𝐷_𝐴(𝐺) by the metric 𝜌_𝐴(𝑣,𝑤).

The following statement is true [24].

Theorem 7. The function 𝑣(𝑥,𝑦)∈𝐻(𝐺) belongs to the space 𝐻_𝐴(𝐺) if and only if the following relations are fulfilled: 𝑣𝑥,𝑦∈𝐻¹(𝐺),𝑣0,𝑦∈𝐻^∘1(),𝑣0,𝑏𝑥,𝑦_Γ=0,𝑣𝑥,𝑦_Γ−𝜉=𝑣𝑥,𝑦_Γ0=𝑣0,𝑦. (12)

Thus, functions of the space 𝐻_𝐴(𝐺) satisfy boundary conditions. For every function 𝑓(𝑥,𝑦)∈𝐻(𝐺) there exists a unique function 𝑢(𝑥,𝑦) in the space 𝐻_𝐴(𝐺), which minimizes the quadratic functional 𝐹(𝑣)=‖𝑣‖²_𝐴−2𝑓,𝑣. (13)

For any function 𝑣(𝑥,𝑦)∈𝐻_𝐴(𝐺) the following relation is fulfilled [𝑢,𝑣]_𝐴=𝑓,𝑣. (14)

The function 𝑢(𝑥,𝑦) from the space 𝐻_𝐴(𝐺) which minimizes the functional (14) is called the generalized solution of the equation 𝐴𝑢=𝑓.

If the function 𝑢(𝑥,𝑦) is sufficiently smooth then 𝑢(𝑥,𝑦) is a solution in a classical sense of problem (1)–(3).

4. 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: Δ𝑢𝑥,𝑦=0,𝑥,𝑦∈𝐺,𝑢𝑥,𝑦_Γ=0,𝑢𝑥,𝑦_Γ−𝜉=𝑢𝑥,𝑦_Γ0. (15)(16)(17)

For problem (15)–(17) let us consider the following sequential iterative procedure: Δ𝑢^𝑛₁𝑥,𝑦=0,𝑥,𝑦∈𝐺₁,𝑢^𝑛₁𝑥,𝑦_Γ1𝑢=0,_𝑛1𝑥,𝑦_Γ1−𝜉=𝑢₂^𝑛−1𝑥,𝑦_Γ1−𝜉,𝑛=1,2,…;Δ𝑢^𝑛₂𝑥,𝑦=0,𝑥,𝑦∈𝐺₂,𝑢^𝑛₂𝑥,𝑦_Γ2𝑢=0,_𝑛2𝑥,𝑦_Γ−𝜉=𝑢^𝑛₂𝑥,𝑦_Γ0=𝑢^𝑛₁𝑥,𝑦_Γ−𝜉,𝑛=1,2,…. (18)

Here we utilize the following notations: 𝐺₁=−𝑎<𝑥<−𝜉₁,𝐺,0<𝑦<𝑏₂=−𝜉<𝑥<0,0<𝑦<𝑏, (19) where −𝜉₁ is a fixed point of the interval (−𝜉,0), Γ¹=𝜕𝐺₁⧵Γ_−𝜉1, Γ²=𝜕𝐺₂⧵(Γ_−𝜉∪Γ₀), and 𝑢⁰₂(−𝜉₁,𝑦)=𝜑(𝑦) is any continuous function on the segment [0,𝑏], which satisfies the following conditions: 𝜑(0)=𝜑(𝑏)=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.

Theorem 8. The sequential iterative process (18) converges to a solution of problem (15)–(17) uniformly in the domain 𝐺, and the following estimations are true: 𝑢𝑥,𝑦−𝑢^𝑛₁𝑥,𝑦≤𝐶𝑞^𝑛−1,𝑥,𝑦∈𝐺₁,𝑢𝑥,𝑦−𝑢^𝑛₂𝑥,𝑦≤𝐶𝑞^𝑛−1,𝑥,𝑦∈𝐺₂, (20) where 𝑞∈(0,1) is a constant independent of functions 𝑢(𝑥,𝑦), 𝑢^𝑛₁(𝑥,𝑦), 𝑢^𝑛₂(𝑥,𝑦), and constant 𝐶 depends on 𝜑(𝑦).

Proof. Note that solving problem (18), we get two sequences {𝑢^𝑛₁},{𝑢^𝑛₂} of harmonic functions, which are defined on the domains 𝐺₁ and 𝐺₂, respectively.
We have the following relations: if 𝑥=−𝜉  and 𝑦∈[0,𝑏], then 𝑢^𝑛₂=𝑢^𝑛₁, that is 𝑢₁^𝑛+1−𝑢^𝑛₂=𝑢₂^𝑛+1−𝑢^𝑛₂; if 𝑥=−𝜉₁ and 𝑦∈[0,𝑏], then 𝑢^𝑛₂=𝑢₁^𝑛+1, that is 𝑢₁^𝑛+1−𝑢^𝑛₁=𝑢^𝑛₂−𝑢₂^𝑛−1.
Let us introduce the notations: 𝑀_𝑛=max_{𝑦∈[0,𝑏]}𝑢₁^𝑛+1−𝜉,𝑦−𝑢^𝑛₁−𝜉,𝑦=max_{𝑦∈[0;𝑏]}𝑢₂^𝑛+1−𝜉,𝑦−𝑢^𝑛₂,𝑀−𝜉,𝑦_1𝑛=max_{𝑦∈[0;𝑏]}𝑢₁^𝑛+1−𝜉₁,𝑦−𝑢^𝑛₁−𝜉₁,𝑦=max_{𝑦∈[0;𝑏]}𝑢^𝑛₂−𝜉₁,𝑦−𝑢₂^𝑛−1−𝜉₁,𝑦. (21)
If 𝑀¹_𝑛≠0, then, for the harmonic function 𝑣^𝑛₁=(𝑢₁^𝑛+1−𝑢^𝑛₁)/𝑀¹_𝑛, applying the lemma from [30, pages 250–254] to the domain 𝐺₁, we have 𝑀_𝑛≤𝑞𝑀¹_𝑛, (22) where 𝑞∈(0,1) and depends only on domain 𝐺₁.
If 𝑀_𝑛≠0, then, for the harmonic function 𝑣^𝑛₂=(𝑢₂^𝑛+1−𝑢^𝑛₂)/𝑀_𝑛, applying the extremum principle [30, pages 218] to the domain 𝐺₂, we have 𝑀¹_𝑛+1≤𝑀_𝑛. (23)
If 𝑀¹_𝑛=0 for some index 𝑛, then from the extremum principle we get 𝑀_𝑛=0. So, the inequality (22) is also true.
If 𝑀_𝑛=0 for some index 𝑛, then from the extremum principle we get again 𝑀¹_𝑛+1=0. So, the inequality (23) is also true.
From the estimations (22) and (23) obtained above it follows that 𝑀_𝑛+1≤𝑞𝑀_𝑛,𝑀¹_𝑛+1≤𝑞𝑀¹_𝑛. (24)
This means that the sequences {𝑀_𝑛} and {𝑀¹_𝑛} tend to zero, and we obtain the uniform convergence of the series 𝑢¹₁+^∞_𝑘=1𝑢₁^𝑘+1−𝑢^𝑘₁=lim_𝑛→∞𝑢^𝑛₁=𝑢₁,𝑢¹₂+^∞_𝑘=1𝑢₂^𝑘+1−𝑢^𝑘₂=lim_𝑛→∞𝑢^𝑛₂=𝑢₂, (25) in the domains 𝐺₁ and 𝐺₂, respectively.
According to Weierstrass theorem [30, pages 232, 233], the functions 𝑢₁ and 𝑢₂ are harmonic ones, defined, respectively, in 𝐺₁ and 𝐺₂, and satisfy the condition (16). As for domain 𝐺₁₂=𝐺₁∩𝐺₂, we have the following relations: if 𝑥=−𝜉 and 𝑦∈[0,𝑏], then 𝑢^𝑛₂−𝑢^𝑛₁=0; if 𝑥=−𝜉₁ and 𝑦∈[0,𝑏], then 𝑢^𝑛₂−𝑢^𝑛₁=𝑢^𝑛₂−𝑢₂^𝑛−1.
The latter difference tends to zero uniformly.
Again, according to the extremum principle, we obtain that the functions 𝑢₁ and 𝑢₂ coincide with each other in the domain 𝐺₁₂ and define a regular harmonic function on 𝐺, 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 𝑢₁^𝑛+𝑝−𝑢^𝑛₁=𝑢₁^𝑛+𝑝−𝑢₁^{𝑛+𝑝−1}+𝑢₁^{𝑛+𝑝−1}−𝑢₁^{𝑛+𝑝−2}+⋯+𝑢₁^𝑛+2−𝑢₁^𝑛+1+𝑢₁^𝑛+1−𝑢^𝑛₁≤𝑀¹_{𝑛+𝑝−1}+𝑀¹_{𝑛+𝑝−2}+⋯+𝑀¹_𝑛+1+𝑀¹_𝑛≤𝑞^{𝑛+𝑝−2}𝑀¹₁+𝑞^{𝑛+𝑝−3}𝑀¹₁+⋯+𝑞^𝑛𝑀¹₁+𝑞^𝑛−1𝑀¹₁=𝑞^𝑛−1𝑀¹₁𝑞^𝑝−1+𝑞^𝑝−2+⋯+𝑞+1=𝑞^𝑛−1𝑀¹₁1−𝑞^𝑝,𝑀1−𝑞₁₁=max_{[]𝑦∈0;𝑏}𝑢²₁−𝜉₁,𝑦−𝑢¹₁−𝜉₁,𝑦≤max_{[]𝑦∈0;𝑏}𝑢²₁−𝜉₁,𝑦+𝑢¹₁−𝜉₁,𝑦=max_{[]𝑦∈0;𝑏}𝑢¹₂−𝜉₁,𝑦+max_{[]𝑦∈0;𝑏}𝑢⁰₂−𝜉₁,𝑦≤2max_{[]𝑦∈0;𝑏}𝜑𝑦=𝑀. (26) Thus, 𝑢₁^𝑛+𝑝−𝑢^𝑛₁≤𝑞^𝑛−1𝑀1−𝑞^𝑝.1−𝑞 (27)
If in this inequality we tend 𝑝 to ∞, then we obtain in the domain 𝐺₁𝑢₁−𝑢^𝑛₁≤𝑞^𝑛−1𝑀.1−𝑞 (28)
Analogous estimation is true for 𝑢₂ in the domain 𝐺₂.
Consequently, we get 𝑢₁𝑥,𝑦−𝑢^𝑛₁𝑥,𝑦≤𝐶𝑞^𝑛−1,𝑥,𝑦∈𝐺₁,𝑢₂𝑥,𝑦−𝑢^𝑛₂𝑥,𝑦≤𝐶𝑞^𝑛−1,𝑥,𝑦∈𝐺₂. (29)
This completes the proof of Theorem 8.

5. 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 𝐺₁ and 𝐺₂ will be carried out not by means of a sequential algorithm but in a parallel way.

Consider the following overlapping parallel iterative process: Δ𝑢^𝑛₁𝑥,𝑦=0,𝑥,𝑦∈𝐺₁,𝑢^𝑛₁𝑥,𝑦_Γ1𝑢=0,_𝑛1𝑥,𝑦_Γ1−𝜉=𝑢₂^𝑛−1𝑥,𝑦_Γ1−𝜉,𝑛=1,2,…;Δ𝑢^𝑛₂𝑥,𝑦=0,𝑥,𝑦∈𝐺₂,𝑢^𝑛₂𝑥,𝑦_Γ2𝑢=0,_𝑛2𝑥,𝑦_Γ−𝜉=𝑢^𝑛₂𝑥,𝑦_Γ0=𝑢₁^𝑛−1𝑥,𝑦_Γ−𝜉,𝑛=1,2,…, (30) where 𝑢⁰₂(−𝜉₁,𝑦)=𝜑₁(𝑦) and 𝑢⁰₁(𝜉,𝑦)=𝜑₂(𝑦) are any continuous functions on the segment [0,𝑏], which satisfy the following conditions: 𝜑₁(0)=𝜑₁(𝑏)=0,  𝜑₂(0)=𝜑₂(𝑏)=0.

The following statement takes place.

Theorem 9. The parallel iterative process (30) converges to a solution of the problem (15)–(17) uniformly in the domain 𝐺, and the following estimations are true: 𝑢𝑥,𝑦−𝑢^𝑛₁𝑥,𝑦≤𝐶𝑞^(𝑛/2)−1,𝑥,𝑦∈𝐺₁,𝑢𝑥,𝑦−𝑢^𝑛₂𝑥,𝑦≤𝐶𝑞^(𝑛/2)−1,𝑥,𝑦∈𝐺₂, (31) where 𝑞∈(0,1) is a constant independent of functions 𝑢(𝑥,𝑦), 𝑢^𝑛₁(𝑥,𝑦), and 𝑢^𝑛₂(𝑥,𝑦), and constant 𝐶 depends on 𝜑₁(𝑦), 𝜑₂(𝑦).

Proof. Let us prove this theorem in a similar way as Theorem 8 was proven. We should note that the boundary value problems (30) are simultaneously solved in domains 𝐺₁ and 𝐺₂, respectively. Sequences of harmonic functions {𝑢^𝑛₁}, {𝑢^𝑛₂} are defined on 𝐺₁ and 𝐺₂, respectively.
The following relations are satisfied: if 𝑥=𝜉 and 𝑦∈[0;𝑏], then 𝑢^𝑛₂=𝑢₁^𝑛−1; that is, 𝑢^𝑛₁−𝑢₁^𝑛−1=𝑢₂^𝑛+1−𝑢^𝑛₂; if 𝑥=−𝜉₁ and 𝑦∈[0;𝑏], then 𝑢₂^𝑛−1=𝑢^𝑛₁; that is, 𝑢₁^𝑛+1−𝑢^𝑛₁=𝑢^𝑛₂−𝑢₂^𝑛−1.
If we introduce the notation 𝑀_𝑛−1=max_{𝑦∈[0;𝑏]}𝑢^𝑛₁−𝜉,𝑦−𝑢₁^𝑛−1−𝜉,𝑦=max_{𝑦∈[0;𝑏]}𝑢₂^𝑛+1−𝜉,𝑦−𝑢^𝑛₂,𝑀−𝜉,𝑦_1𝑛=max_{𝑦∈[0;𝑏]}𝑢₁^𝑛+1−𝜉₁,𝑦−𝑢^𝑛₁−𝜉₁,𝑦=max_{[]𝑦∈0;𝑏}𝑢^𝑛₂−𝜉₁,𝑦−𝑢₂^𝑛−1−𝜉₁,𝑦, (32) and, for the harmonic function 𝑣^𝑛₁=(𝑢₁^𝑛+1−𝑢^𝑛₁)/𝑀¹_𝑛, apply the lemma from [30, pages 250–254] to the domain 𝐺₁, we will have the estimation 𝑀_𝑛≤𝑞𝑀¹_𝑛, (33) where 𝑞∈(0,1) and depends on domain 𝐺₁ only.
If 𝑀¹_𝑛=0, then from the extremum principle we get that 𝑀_𝑛=0 and the inequality (33) is clear.
If 𝑀_𝑛≠0, then, for the harmonic function 𝑣₂^𝑛−1=(𝑢₂^𝑛+1−𝑢^𝑛₂)/𝑀_𝑛−1, from the extremum principle in the domain 𝐺₂ we have 𝑀¹_𝑛+1≤𝑀_𝑛−1. (34)
If 𝑀_𝑛−1=0, then, using again the extremum principle, we have 𝑀¹_𝑛=0 and the inequality (34) holds.
From the estimations (33) and (34), we obtain 𝑀_𝑛+1≤𝑞𝑀_𝑛−1,𝑀¹_𝑛+2≤𝑞𝑀¹_𝑛. (35) That means that the sequences {𝑀_𝑛} and {𝑀¹_𝑛} tend to zero.
Thus, in this case, the series analogous to the series from (25) are also uniformly convergent. The corresponding harmonic functions 𝑢₁ and 𝑢₂ are defined in 𝐺₁ and 𝐺₂, respectively, and satisfy condition (16). As for the common part of these domains, we have if 𝑥=−𝜉 and 𝑦∈[0,𝑏], then 𝑢^𝑛₂−𝑢^𝑛₁=𝑢^𝑛₂−𝑢₂^𝑛+1; if 𝑥=−𝜉₁ and 𝑦∈[0,𝑏], then 𝑢^𝑛₂−𝑢^𝑛₁=𝑢^𝑛₂−𝑢₂^𝑛−1and this difference tends to zero, when 𝑛→∞. Thus, the functions 𝑢₁ and 𝑢₂ again coincide in the domain 𝐺₁₂ and define a regular harmonic function on 𝐺, which represents the solution of the problem (15)–(17).
Let us estimate the rate of convergence of the constructed sequences.
We should remark that from the second inequality of (35) we have the following: if 𝑛=2𝑘, then 𝑀¹_𝑛≤𝑞^(𝑛−2)/2𝑀¹₂; (36) if 𝑛=2𝑘−1, then 𝑀¹_𝑛≤𝑞^(𝑛−1)/2𝑀¹₁. (37)
Using the triangle inequality, we get 𝑢₁^𝑛+𝑝−𝑢^𝑛₁=𝑢₁^𝑛+𝑝−𝑢₁^{𝑛+𝑝−1}+𝑢₁^{𝑛+𝑝−1}−𝑢₁^{𝑛+𝑝−2}+⋯+𝑢₁^𝑛+2−𝑢₁^𝑛+1+𝑢₁^𝑛+1−𝑢^𝑛₁≤𝑀¹_{𝑛+𝑝−1}+𝑀¹_{𝑛+𝑝−2}+⋯+𝑀¹_𝑛+1+𝑀¹_𝑛. (38)
Taking into account inequalities (36) and (37) for 𝑛=2𝑘 and 𝑝=2𝑚, we obtain from (36) 𝑢₁^𝑛+𝑝−𝑢^𝑛₁≤𝑞^{(𝑛+𝑝−1−1)/2}𝑀¹₁+𝑞^{(𝑛+𝑝−2−2)/2}𝑀¹₂+𝑞^{(𝑛+𝑝−3−1)/2}𝑀¹₁+𝑞^{(𝑛+𝑝−4−2)/2}𝑀¹₂+⋯+𝑞^{(𝑛+1−1)/2}𝑀¹₁+𝑞^(𝑛−2)/2𝑀¹₂=𝑀¹₁+𝑀¹₂𝑞^𝑛/2𝑞^(𝑝−4)/2+𝑞^(𝑝−6)/2+⋯+𝑞+1+𝑞^{(𝑛+𝑝−2)/2}𝑀¹₁+𝑞^(𝑛−2)/2𝑀¹₂=𝑞^(𝑛−2)/2𝑞𝑀¹₁+𝑀¹₂1−𝑞^(𝑝−2)/21−𝑞+𝑞^𝑝/2𝑀¹₁+𝑀¹₂. (39)
If 𝑛=2𝑘−1 and 𝑝=2𝑚−1, then from (38) we have 𝑢₁^𝑛+𝑝−𝑢^𝑛₁≤𝑞^{(𝑛+𝑝−1−1)/2}𝑀¹₁+𝑞^{(𝑛+𝑝−2−2)/2}𝑀¹₂+𝑞^{(𝑛+𝑝−3−1)/2}𝑀¹₁+𝑞^{(𝑛+𝑝−4−2)/2}𝑀¹₂+⋯+𝑞^{(𝑛+1−2)/2}𝑀¹₂+𝑞^(𝑛−1)/2𝑀¹₁=𝑞^{(𝑛+𝑝−1)/2}𝑀¹₁+𝑀¹₁+𝑀¹₂𝑞^(𝑛−2)/2×𝑞^(𝑝−2)/2+𝑞^(𝑝−4)/2+⋯+𝑞^3/2+𝑞^1/2=𝑞^(𝑛−2)/2𝑞^𝑝/2𝑀¹₁+𝑀¹₁+𝑀¹₂𝑞^1/2−𝑞^𝑝/2.1−𝑞 (40)
If 𝑛=2𝑘 and 𝑝=2𝑚−1, then 𝑢₁^𝑛+𝑝−𝑢^𝑛₁≤𝑞^{(𝑛+𝑝−1−2)/2}𝑀¹₂+𝑞^{(𝑛+𝑝−2−1)/2}𝑀¹₁+⋯+𝑞^{(𝑛+3−1)/2}𝑀¹₁+𝑞^{(𝑛+2−2)/2}𝑀¹₂+𝑞^{(𝑛+1−1)/2}𝑀¹₁+𝑞^(𝑛−2)/2𝑀¹₂=𝑞^(𝑛−2)/2𝑀¹₁+𝑀¹₂𝑞^(𝑝−1)/2+𝑞^(𝑝−3)/2+⋯+𝑞+𝑀¹₂=𝑞^(𝑛−2)/2𝑀¹₁+𝑀¹₂𝑞−𝑞^(𝑝+1)/21−𝑞+𝑀¹₂, (41) and at last, supposing 𝑛=2𝑘−1 and 𝑝=2𝑚, we have 𝑢₁^𝑛+𝑝−𝑢^𝑛₁≤𝑞^{(𝑛+𝑝−1−2)/2}𝑀¹₂+𝑞^{(𝑛+𝑝−2−1)/2}𝑀¹₁+⋯+𝑞^{(𝑛+3−2)/2}𝑀¹₂+𝑞^{(𝑛+2−1)/2}𝑀¹₁+𝑞^{(𝑛+1−2)/2}𝑀¹₂+𝑞^(𝑛−1)/2𝑀¹₁=𝑞^(𝑛−2)/2𝑀¹₁+𝑀¹₂𝑞^(𝑛−2)/2×𝑞^(𝑝−1)/2+𝑞^(𝑝−3)/2+⋯+𝑞^3/2+𝑞^1/2=𝑞^(𝑛−2)/2𝑀¹₁+𝑀¹₂𝑞^1/2−𝑞^(𝑝+1)/2.1−𝑞 (42)
If in the obtained inequalities we tend to the limit, when 𝑝→∞, we get the following estimations.
If  𝑛=2𝑘 and 𝑝=2𝑚, then 𝑢−𝑢^𝑛₁≤𝑞^(𝑛−2)/2𝑀¹₁+𝑀¹₂𝑞1−𝑞+𝑀¹₂; (43) if  𝑛=2𝑘−1 and 𝑝=2𝑚−1, then 𝑢−𝑢^𝑛₁≤𝑞^(𝑛−1)/2𝑀¹₁+𝑀¹₂;1−𝑞 (44) if  𝑛=2𝑘 and 𝑝=2𝑚−1, then 𝑢−𝑢^𝑛₁≤𝑞^(𝑛−2)/2𝑀¹₁+𝑀¹₂𝑞1−𝑞+𝑀¹₂; (45) and if  𝑛=2𝑘−1 and 𝑝=2𝑚, then 𝑢−𝑢^𝑛₁≤𝑞^(𝑛−1)/2𝑀¹₁+𝑀¹₂.1−𝑞 (46)
Let us estimate 𝑀¹₁ and 𝑀¹₂. We have 𝑀¹₁=max_{𝑦∈[0;𝑏]}𝑢²₁−𝜉₁,𝑦−𝑢¹₁−𝜉₁,𝑦≤max_{[]𝑦∈0;𝑏}𝑢²₁−𝜉₁,𝑦+𝑢¹₁−𝜉₁,𝑦=max_{[]𝑦∈0;𝑏}𝑢¹₂−𝜉₁,𝑦+𝑢⁰₂−𝜉₁,𝑦≤2max_{[]𝑦∈0;𝑏}𝜑₁𝑦,𝜑₂𝑦,𝑀¹₂=max_{𝑦∈[0;𝑏]}𝑢³₁−𝜉₁,𝑦−𝑢²₁−𝜉₁,𝑦≤max_{𝑦∈[0;𝑏]}𝑢³₁−𝜉₁,𝑦+𝑢²₁−𝜉₁,𝑦=max_{𝑦∈[0;𝑏]}𝑢²₁−𝜉₁,𝑦+𝑢¹₁−𝜉₁,𝑦≤2max_{[]𝑦∈0;𝑏}𝜑₁𝑦,𝜑₂𝑦. (47)
So, for any 𝑛, we obtain the following relation in the domain 𝐺₁: 𝑢₁𝑥,𝑦−𝑢^𝑛₁𝑥,𝑦≤𝐶𝑞^(𝑛−2)/2. (48)
Analogous estimation is true for 𝑢₂ in the domain 𝐺₂𝑢₂𝑥,𝑦−𝑢^𝑛₂𝑥,𝑦≤𝐶𝑞^(𝑛−2)/2. (49)
Theorems similar to Theorems 8 and 9 are valid for the sequential as well as parallel algorithms for multigrid domain decomposition case too.

6. Conclusion

Because of importance of nonlocal problems many scientific articles are devoted for their investigation. We gave one problem to illustrate the variational formulation and domain decomposition of the problem. Nonlocal problems for the second and the fourth order ordinary and partial differential equations are also studied by authors. Nonlocal boundary value problems with some kind of nonlocal integral conditions are studied as well (see, e.g., [15, 24–26] and references therein).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author thanks Fulbright Visiting Scholar Program (Grant no. AY 2012-2013, USA) and the Shota Rustaveli National Science Foundation (Grant no. DI/16/4-120/11, Georgia) for the financial support and the Naval Postgraduate School in Monterey, CA, USA, for hosting him during the nine months of his tenure in 2012-2013. The second author thanks the Shota Rustaveli National Scientific Foundation (Grant no. YS/40/5-106/12, Georgia) for the financial support and the Naval Postgraduate School in Monterey, CA, USA, for hosting him during the four months of his tenure in 2013.