Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination

and Applied Analysis 3 the operator I − S 1 has an inverse G 1 = (I − R 2N + ( λ

In [4], Orlovsky proved existence and uniqueness theorems for the inverse problem of finding a function  and an element  for the elliptic equation in an arbitrary Hilbert space  with the self-adjoint positive definite operator A: −  () +  () =  () + , 0 <  < ,  (0) = ,  () = ,  () = , 0 <  < . (1) In [11], the authors established stability estimates for this problem and studied inverse problem for multidimensional elliptic equation with overdetermination in which the Dirichlet condition is required on the boundary.
In present work, we study inverse problem for multidimensional elliptic equation with Dirichlet-Neumann boundary conditions.
The aim of this paper is to investigate inverse problem (2) for multidimensional elliptic equation with Dirichlet-Neumann boundary conditions.We obtain well-posedness of problem (2).For the approximate solution of problem (2), we construct first and second order of accuracy in  and difference schemes with second order of accuracy in space variables.Stability and coercive stability estimates for these difference schemes are established by applying operator approach.The modified Gauss elimination method is applied for solving these difference schemes in a two-dimensional case.
The remainder of this paper is organized as follows.In Section 2, we obtain stability and coercive stability estimates for problem (2).In Section 3, we construct the difference schemes for (2) and establish their well-posedness.In Section 4, the numerical results in a two-dimensional case are presented.Section 5 is conclusion.

Difference Schemes and Their Well-Posedness
Suppose that   is defined by formula (3).Then (see [26]), ) is a self-adjoint positive definite operator and  = ( + ) −1 which is defined on the whole space  =  2 (Ω) is a bounded operator.Here,  is the identity operator.Now we present the following lemmas, which will be used later.
Lemma 3. The following estimates are satisfied (see [27]):            →  ≤ (1 + Lemma 5.For 1 ≤  ≤  − 1 and for the operator the operator  −  1 has an inverse and the estimate is valid, where  is independent of .
Proof.We have that where By using estimates of Lemma 3, we have that where  1 is independent of .Using the triangle inequality, formula (13), and estimates ( 8) and ( 15), we obtain for sufficiently small positive .From that it follows estimate (11).Lemma 5 is proved.
To the differential operator   generated by problem (2) we assign the difference operator   ℎ defined by formula (3), acting in the space of grid functions  ℎ (), satisfying the condition  ℎ  ℎ () = 0 for all  ∈  ℎ .Here,  ℎ  ℎ () is an approximation of / ⃗ .By using   ℎ , for obtaining  ℎ (, ) functions, we arrive at problem For finding a solution  ℎ (, ) of problem (19) we apply the substitution where V ℎ (, ) is the solution of nonlocal boundary value problem; a system of ordinary differential equations and unknown function  ℎ () is defined by formula Thus, we consider the algorithm for solving problem (19) which includes three stages.In the first stage, we get the nonlocal boundary value problem (21) and obtain V ℎ (, ).In the second stage, we put  = 0 and find V ℎ (0, ).Then, using (22), we obtain  ℎ ().Finally, in the third stage, we use formula (20) for obtaining the solution  ℎ (, ) of problem (19).
In the second step, we approximate (19) in variable .Let [0, ]  = {  = ,  = 1, . . ., ,  = } be the uniform grid space with step size  > 0, where  is a fixed positive integer.Applying the approximate formulas for  ℎ (, ) =  ℎ (), problem ( 19) is replaced by first order of accuracy difference scheme and second order of accuracy difference scheme For approximate solution of nonlocal problem (21), we have first order of accuracy difference scheme and second order of accuracy difference scheme holds, where  is independent of , , ℎ,  ℎ (), ℎ (),  ℎ (), .
Proofs of Theorems 6 and 7 are based on the symmetry property of operator   , on Lemmas 3-5, the formulas for difference scheme (24), for difference scheme (25), and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in  2ℎ .
Theorem 8 (see [28]).For the solution of the elliptic difference problem the following coercivity inequality holds: where  does not depend on ℎ and  ℎ .

Numerical Results
We have not been able to obtain a sharp estimate for the constants figuring in the stability estimates.Therefore, we will give the following results of numerical experiments of the inverse problem for the two-dimensional elliptic equation with Dirichlet-Neumann boundary conditions (34) It is clear that (, ) = (exp(−) +  + 1) cos() and () = sin() + ( + 2) cos() are the exact solutions of (34).
By difference scheme (52), we write in matrix form where   ,   are defined by ( 42), ( 43), (44), and   is defined by We seek solution of (54) by the formula where   ,   ( = 0, . . .,  − 2) are ( + 1) × ( + 1) square matrices and   ( = 0, . . .,  − 2) are ( + 1) × 1 column matrices.For the solution of difference equation (41) we need to use the following formulas for   ,   : where and  0 ,  1 ,  −2 ,  −3 are the ( + 1) × 1 zero column vector.For V  and V −1 we have where We can rewrite difference scheme (53) in the matrix form where   ,   ,   are defined by ( 42), ( 49     Tables 1-3 present the error between the exact solution and numerical solutions derived by corresponding difference schemes.The results are recorded for  =  = 20, 40, 80 and 160, respectively.The tables show that the second order of accuracy difference scheme is more accurate than the first order of accuracy difference scheme for both auxiliary nonlocal and inverse problems.Table 1 contains error between the exact and approximate solutions V of auxiliary nonlocal boundary value problem (35).Table 2 includes error between the exact and approximate solutions  of inverse problem (34).Table 3 represents error between the exact solution  of inverse problem (34) and approximate solution which is derived by the first and second orders accuracy of difference schemes.

Conclusion
In this paper, inverse problem for multidimensional elliptic equation with Dirichlet-Neumann conditions is considered.The stability and coercive stability estimates for solution of this problem are established.First and second order of accuracy difference schemes are presented for approximate solutions of inverse problem.Theorems on the stability and coercive stability inequalities for difference schemes are proved.The theoretical statements for the solution of these difference schemes are supported by the results of numerical example in a two-dimensional case.As it can be seen from Tables 1-3, second order of accuracy difference scheme is more accurate compared with the first order of accuracy difference scheme.Moreover, applying the result of the monograph [29] the high order of accuracy difference schemes for the numerical solution of the boundary value problem (2) can be presented.

Table 1 :
Error analysis for nonlocal problem.

Table 2 :
Error analysis for p.

Table 3 :
Error analysis for u.