Abstract and Applied Analysis

Volume 2012 (2012), Article ID 454831, 22 pages

doi:10.1155/2012/454831

Research Article

FDM for Elliptic Equations with Bitsadze-Samarskii-Dirichlet Conditions

Allaberen Ashyralyev1,2 and Fatma Songul Ozesenli Tetikoglu¹

¹Department of Mathematics, Fatih University, 34500 Istanbul, Turkey

²Department of Mathematics, ITTU, 74400 Ashgabat, Turkmenistan

Received 8 April 2012; Accepted 6 May 2012

Academic Editor: Ravshan Ashurov

Copyright © 2012 Allaberen Ashyralyev and Fatma Songul Ozesenli Tetikoglu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A numerical method is proposed for solving nonlocal boundary value problem for the multidimensional elliptic partial differential equation with the Bitsadze-Samarskii-Dirichlet condition. The first and second-orders of accuracy stable difference schemes for the approximate solution of this nonlocal boundary value problem are presented. The stability estimates, coercivity, and almost coercivity inequalities for solution of these schemes are established. The theoretical statements for the solutions of these nonlocal elliptic problems are supported by results of numerical examples.

1. Introduction

Many problems in fluid mechanics, dynamics, elasticity, and other areas of engineering, physics, and biological systems lead to partial differential equations of elliptic type. The role played by coercive inequalities in the study of local boundary-value problems for elliptic and parabolic differential equations is well known (see, e.g., [1, 2]).

In the present paper, we consider the Bitsadze-Samarskii type nonlocal boundary value problem

for elliptic differential equations in a Hilbert space with self-adjoint positive definite operator . It is known (see, e.g., [3–11]) that various nonlocal boundary value problems for elliptic equations can be reduced to the boundary value problem (1.1). The simply nonlocal boundary value problem was presented and investigated for the first time by Bitsadze and Samarskii [12]. Further, methods of solutions of Bitsadze-Samarskii nonlocal boundary value problems for elliptic differential equations have been studied extensively by many researchers (see [13–21] and the references given therein).

A function is called a solution of problem (1.1) if the following conditions are satisfied.

(i) is twice continuously differentiable on the segment . Derivatives at the endpoints of the segmentare understood asthe appropriate unilaterial derivatives.(ii)The element belongs to for all , and the function is continuous on .(iii) satisfies the equation and nonlocal boundary condition in (1.1).

Let be the open unit cube in with boundary . In present paper, we are interested in studying the stable difference schemes for the numerical solution of the following nonlocal boundary value problem for the multidimensional elliptic equation

Here , and are given smooth functions, is a large positive constant and .

In the present paper, the first and second-orders of accuracy difference schemes are presented for the approximate solution of problem (1.2). The stability and coercive stability estimates for the solution of these difference schemes are obtained. A numerical method is proposed for solving nonlocal boundary value problem for the multidimensional elliptic partial differential equation with the Bitsadze-Samarskii-Dirichlet condition. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of two-dimensional elliptic partial differential equations.

2. Difference Schemes: Well-Posedness

The discretization of problem (1.2) is carried out in two steps. In the first step let us define the grid sets as follows:

We introduce the Hilbert space of the grid functions defined on , equipped with the norm

and the Hilbert space defined on , equipped with the norm

Finally, we introduce the Banach spaces and of grid abstract function defined on with values , equipped with the following norms:

To the differential operator generated by the problem (1.2) we assign the difference operator by the formula

acting in the space of grid functions , satisfying the condition for all . It is known that is a self-adjoint positive definite operator in (). With the help of , we arrive at the nonlocal boundary value problem for an infinite system of the following ordinary differential equations:

In the second step, we replaced problem (2.6) by the first-order of accuracy difference scheme as follows:

and the second order of accuracy difference scheme as follows:

Now, we will study the well-posedness of (2.7) and (2.8). We have the following theorem on stability of (2.7) and (2.8).

Theorem 2.1. Let and be sufficiently small positive numbers. Then the solutions of difference schemes (2.7) and (2.8) satisfy the following stability estimate:

where does not depend on and .

Proof. The proof of (2.9) is based on the following formula:

where

for (2.7), and

for (2.8), and the symmetry properties of the difference operator defined by the formula (2.5).

Difference schemes (2.7) and (2.8) are ill-posed in . We have the following theorem on almost coercive stability.

Theorem 2.2. Let and be sufficiently small positive numbers. Then the solutions of difference schemes (2.7) and (2.8) satisfy the following almost coercive stability estimate:

where does not depend on , and .

Proof. The proof of (2.14) is based on the formulas (2.11), (2.12), (2.13), the symmetry properties of the difference operator defined by the formula (2.5) and on the following theorem on well-posedness of the elliptic difference problem.

Theorem 2.3. For the solutions of the elliptic difference problem

the following coercivity inequality holds (see [21]):

where M does not depend on and .

Theorem 2.4. Let . Then the difference problems (2.7) and (2.8) are well-posed in Hölder spaces and the following coercivity inequality holds:

Here does not depend on , and .

Proof. The proof of (2.17) is based on the formula

for (2.7) and

for (2.8), the symmetry properties of the difference operator defined by the formula (2.5) and on Theorem 2.3 on well-posedness of the elliptic difference problem.

3. Numerical Analysis

We have not been able to obtain a sharp estimate for the constants figuring in the stability inequality. Therefore, we will give the following results of numerical experiments of the Bitsadze-Samarskii-Dirichlet problem:

for the two-dimensional elliptic equation.

The exact solution of this problem is

For the approximate solution of problem (3.1), we consider the set of a family of grid points depending on small parameters and as follows:

Applying (2.7), we present the following first-order of accuracy difference scheme for the approximate solution of problem (3.1):

We have system of linear equations in (3.4) and we will write them in the following matrix form:

Here,

where and

We seek a solution of the matrix equation in following form:

where are square matrices and are column matrices defined by (see, [17]) as follows:

where

Now, applying (2.8) for even number, we can present the following second-order of accuracy difference scheme:

for the approximate solution of problem (3.1).

So, again we have system of linear equations in (3.11) and we will write them in the following matrix form:

where

where and

Here,

So, we have the second-order difference equation with respect to with matrix coefficients. To solve this difference equation, we use the same algorithm (3.8) and (3.9).

Now, we will give the results of the numerical experiments.

The errors in numerical solutions are computed by

for different values of and , where represents the exact solution and represents the numerical solution at . The results are shown in Table  1 for , , and .

Table 1: Comparison of the errors of difference schemes.

Difference schemes

First-order difference scheme (2.7)
0.05384197882300
0.02631633782987
0.01740639813932

Second-order difference scheme (2.8)
0.00631619894867
0.00164455475890
7.4144589892985e-004

Thus, second-order of accuracy difference scheme is more accurate compared with the first-order of accuracy difference scheme.

4. Conclusion

The first and second-orders of accuracy difference schemes for approximate solutions of the Bitsadze-Samarskii-Dirichlet type nonlocal boundary value problem for the multidimensional elliptic partial differential equation are presented. Theorems on the stability, almost coercive stability, and coercive stability estimates for the solution of these difference schemes are established. Numerical experiments are given.

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