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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.

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 [

Many works are devoted to the investigation of nonlocal problems for elliptic equations (see, e.g., [

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., [

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., [

In the work [

In the work [

In the works [

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

In the plane

Consider the nonlocal Bitsadze-Samarskii boundary value problem [

Uniqueness of the solution of problem (

We use usual

Functions

Let

Operator

Let us define on vector space

Introducing the scalar product (

The following statements take place [

The norm defined in

Space

Let the domain of definition of the operator

The vector space

Thus, the operator

Operator

To show the symmetry of the operator

For an arbitrary function

For two arbitrary functions

The scalar product given by (

In the case of the scalar product (

As

Let us introduce the new scalar product on

Denote by

The following statement is true [

The function

Thus, functions of the space

For any function

The function

If the function

In this section and next sections, for simplicity, let us consider Laplace equation with nonlocal (

For problem (

Here we utilize the following notations:

The iterative procedure (

The following statement takes place.

The sequential iterative process (

Note that solving problem (

We have the following relations:

if

if

Let us introduce the notations:

If

If

If

If

From the estimations (

This means that the sequences

According to Weierstrass theorem [

if

if

The latter difference tends to zero uniformly.

Again, according to the extremum principle, we obtain that the functions

Now, let us estimate the rate of convergence of the iterative process (

If in this inequality we tend

Analogous estimation is true for

Consequently, we get

This completes the proof of Theorem

Algorithm (

Consider the following overlapping parallel iterative process:

The following statement takes place.

The parallel iterative process (

Let us prove this theorem in a similar way as Theorem

The following relations are satisfied:

if

if

If we introduce the notation

If

If

If

From the estimations (

Thus, in this case, the series analogous to the series from (

if

if

Let us estimate the rate of convergence of the constructed sequences.

We should remark that from the second inequality of (

Using the triangle inequality, we get

Taking into account inequalities (

If

If

If in the obtained inequalities we tend to the limit, when

If

Let us estimate

So, for any

Analogous estimation is true for

Theorems similar to Theorems

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., [

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

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.