The Numerical Solution of the Bitsadze-Samarskii Nonlocal Boundary Value Problems with the Dirichlet-Neumann Condition

and Applied Analysis 3 acting in the space of grid functions u x , satisfying the conditions u x 0 for all x ∈ Sh andDhu x 0 for all x ∈ Sh. Here,Dhu x is an approximation to ∂u/∂ n. It is known that A h is a self-adjoint positive definite operator in L2 Ω̃h . With the help of Axh, we arrive at the nonlocal boundary value problem − 2uh t, x dt2 Axhu h t, x f t, x , 0 < t < 1, x ∈ Ωh, u 0, x φ x , u 1, x J ∑ j 1 αju hλj , x ) ψ x , x ∈ Ω̃h,


Introduction
Methods of solution of the Bitsadze-Samarskii nonlocal boundary value problems for elliptic differential equations have been studied extensively by many researchers see 1-22 and the references given therein .

Difference Schemes: The Stability and Coercive Stability Estimates
The discretization of problem 1 is carried out in two steps.In the first step, let us define the grid sets

2.1
We introduce the Hilbert space L 2h L 2 Ω h and W 2 2h W 2 2 Ω h of the grid functions ϕ h x {ϕ h 1 m 1 , . . ., h m m m } defined on Ω h , equipped with the norms

2.2
To the differential operator A generated by problem 1 , we assign the difference operator A x h by the formula for an infinite system of ordinary differential equations.In the second step, we replace problem 2.4 by the first and second orders of accuracy difference schemes

2.6
To formulate our result on well-posedness, we will give definition of C α 01 0, 1 τ , H and C 0, 1 τ , H . Let F 0, 1 τ , H be the linear space of mesh functions ϕ τ {ϕ k } N−1 1 with values in the Hilbert space H.We denote C 0, 1 τ , H normed space with the norm and C α 01 0, 1 τ , H normed space with the norm Theorem 2.1.Let τ and |h| be sufficiently small positive numbers.Then, the solutions of difference schemes 2.5 and 2.6 satisfy the following stability and almost coercive stability estimates .

2.9
Here, M 1 and M 2 do not depend on τ, h, ψ h x , ϕ h x , and Theorem 2.2.Let τ and |h| be sufficiently small positive numbers.Then, the solution of difference schemes 2.5 and 2.6 satisfies the following coercive stability estimate: Proofs of Theorems 2.1 and 2.2 are based on the symmetry properties of operator A x h defined by formula 2.3 and on the following formulas: for difference scheme 2.5 , and 2.12 for difference scheme 2.6 .Here,

2.13
and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L 2h .

2.14
the following coercivity inequality holds: where M 4 does not depend on h and ω h x .
Abstract and Applied Analysis 7 Note that we have not been able to obtain sharp estimate for the constants figuring in the stability estimates.Hence, in the following section, we study difference schemes 2.5 and 2.6 by numerical experiments.

Numerical Results
For the numerical result, we consider the nonlocal boundary value problem for the elliptic equation.The exact solution of 3.1 is For the approximate solution of the nonlocal boundary Bitsadze-Samarskii problem 3.1 , we consider the set 0, 1 τ × 0, 1 h of a family of grid points depending on the small parameters τ and h

3.3
Firstly, applying difference scheme 2.5 , we present the first order of accuracy difference scheme for the approximate solution of problem 3.1 is

3.4
Then, we have an N 1 × M 1 system of linear equations and we will write them in the matrix form where and C A, D is an N 1 × N 1 identity matrix and Abstract and Applied Analysis 9 where s n − 1, n, n 1, .

3.8
Here, 3.9 So, we have a second-order difference equation with respect to n matrix coefficients.To solve this difference equation, we have applied a procedure of modified Gauss elimination method for difference equation with respect to n matrix coefficients.Hence, we seek a solution of the matrix equation in the following form:

3.10
where α j j 1, . . ., M are N 1 × N 1 square matrix and β j j 1, . . ., M are N 1 ×1 column matrix and α 1 is the N 1 × N 1 zero matrix and β j is the N 1 × 1 zero matrix.Secondly, applying difference scheme 2.6 , we present the following second order of accuracy difference scheme for the approximate solutions of problem 3.1 :

3.11
So, we have again an N 1 × M 1 system of linear equations and we will write in the matrix form

3.12
where Abstract and Applied Analysis 11

3.14
Thus, we have a second-order difference equation with respect to n matrix coefficients.
To solve this difference equation, we have applied the same procedure of modified Gauss elimination method 3.10 for difference equation with respect to n matrix coefficients with Now, we will give the results of the numerical analysis.The errors computed by of the numerical solutions for different values of M and N, where u t k , x n represents the exact solution and u k n represents the numerical solution at t k , x n .Table 1 gives the error analysis between the exact solution and solutions derived by difference schemes for N M 20, 40, and 60, respectively.

Conclusion
In this work, the first and second orders of accuracy difference schemes for the approximate solution of the Bitsadze-Samarskii nonlocal boundary value problem for elliptic equations are presented.Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference schemes for elliptic equations are It is known that A x h is a self-adjoint positive definite operator in L 2 Ω h .With the help of A x h , we arrive at the nonlocal boundary value problem r ,j r ηu h x , 2.3Abstract and Applied Analysis 3 acting in the space of grid functions u h x , satisfying the conditions u h x 0 for all x ∈ S 1 h and D h u h x 0 for all x ∈ S 2 h .Here, D h u h x is an approximation to ∂u/∂ n.

Table 1 :
Error analysis.The theoretical statements for the solution of these difference schemes are supported by the results of numerical examples.The second order of accuracy difference scheme is more accurate comparing with the first order of accuracy difference scheme.As a future work, high orders of accuracy difference schemes for the approximate solutions of this problem could be established.Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference schemes for elliptic equations could be proved.