On a difference scheme of second order of accuracy for the Bitsadze-Samarskii type nonlocal boundary-value problem
© Ashyralyev and Ozturk; licensee Springer. 2014
Received: 11 October 2013
Accepted: 16 December 2013
Published: 13 January 2014
In this study, the Bitsadze-Samarskii type nonlocal boundary-value problem with integral condition for an elliptic differential equation in a Hilbert space H with self-adjoint positive definite operator A is considered. The second order of the accuracy difference scheme for the approximate solutions of this nonlocal boundary-value problem is presented. The well-posedness of this difference scheme in Hölder spaces with a weight is proved. The theoretical statements for the solution of this difference scheme are supported by the results of numerical example.
is a twice continuously differentiable on the segment .
The element belongs to for all , and the function is continuous on the segment .
satisfies the equation and nonlocal boundary conditions (1).
The paper is organized as follows. In Section 2 the second order of the accuracy difference scheme for the approximate solution (1) is presented. The stability, the almost coercive stability, and the coercive stability estimates for the solution of the difference scheme for an approximate solution of the nonlocal boundary-value problem with integral condition for elliptic equations are obtained. Section 3 contains the applications of Section 2. The final section is devoted to the numerical result. Theoretical statements for the solution of the second order of the accuracy difference scheme is supported by a numerical experiment.
2 The second order of the accuracy difference scheme
In this paper, positive constants, which can differ in time (hence they are not a subject of precision considerations) will be indicated with M. On the other hand is used to focus on the fact that the constant depends only on .
where M does not depend on τ.
The estimate (6) follows from the spectral representation of A and the Cauchy inequality.
Theorem 2 is proved. □
Theorem 3 is proved. □
From the last estimate and the estimate (15) follows the estimate (12). Theorem 4 is proved. □
Combining estimates for , we get the estimate (19). Theorem 5 is proved. □
for the numerical solution of (21).
and on the symmetry properties of the difference operator defined by the formula (22) in .
The proof of Theorem 7 is based on Theorem 5 and the symmetry properties of the difference operator defined by formula (22).
for the numerical solution of (26).
The proof of Theorem 8 is based on Theorem 3 and the symmetry properties of the difference operator defined by (27) in .
The proof of Theorem 9 is based on Theorem 4, on the estimate (25), on the symmetry properties of the difference operator defined by (27) in , and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in .
The proof of Theorem 11 is based on Theorem 5, on the symmetry properties of the difference operator defined by the formula (27), and on Theorem 10 on the coercivity inequality for the solution of the elliptic difference equation in .
4 Numerical results
The errors for first- and second-order difference scheme
N = M = 20
N = M = 40
N = M = 80
First-order difference scheme
Second-order difference scheme
6.245e − 004
1.562e − 004
3.906e − 005
In this paper, the second order of the accuracy difference scheme for the approximate solution of the Bitsadze-Samarskii type nonlocal boundary-value problem with the integral condition for elliptic equations is presented. Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference scheme for elliptic equations are proved. The theoretical statements for the solution of this difference scheme are supported by the result of a numerical example. As can be seen from Table 1, the second order of the accuracy difference scheme is more accurate than the first order of the accuracy difference scheme.
The authors would like to thank Prof. Dr. PE Sobolevskii for his helpful suggestions on the improvement of this paper.
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