Well-Posed and Ill-Posed Boundary Value Problems for PDE 2013View this Special Issue
Research Article | Open Access
Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination
A finite difference method for the approximate solution of the inverse problem for the multidimensional elliptic equation with overdetermination is applied. Stability and coercive stability estimates of the fi rst and second orders of accuracy difference schemes for this problem are established. The algorithm for approximate solution is tested in a two-dimensional inverse problem.
It is well known that inverse problems arise in various branches of science (see [1, 2]). The theory and applications of well-posedness of inverse problems for partial differential equations have been studied extensively by many researchers (see, e.g., [3–17] and the references therein). One of the effective approaches for solving inverse problem is reduction to nonlocal boundary value problem (see, e.g., [6, 8, 11]). Well-posedness of the nonlocal boundary value problems of elliptic type equations was investigated in [18–25] (see also the references therein).
In , 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:
In , 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.
Let be the open cube in the -dimensional Euclidean space with boundary and . In , we consider the inverse problem of finding functions and for the multidimensional elliptic equation
Here, and are known numbers, , , , , and are given smooth functions, and also .
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.
2. Well-Posedness of Inverse Problem with Overdetermination
It is known that the differential expression  defines a self-adjoint positive definite operator acting on with the domain .
Theorem 1. Assume that is defined by formula (3), . Then, for the solutions of inverse boundary value problem (2), the stability estimates are satisfied:
where is independent of , and .
Here, is the space obtained by completion of the space of all smooth -valued functions on with the norm
3. Difference Schemes and Their Well-Posedness
Now we present the following lemmas, which will be used later.
Lemma 3. The following estimates are satisfied (see ):
Lemma 5. For and for the operator the operator has an inverse and the estimate is valid, where is independent of .
Proof. We have that
By using estimates of Lemma 3, we have that where 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.
Further, we discretize problem (2) in two steps. In the first step, we define the grid spaces
Introduce the Hilbert space and of grid functions , defined on , equipped with the norms respectively.
To the differential operator generated by problem (2) we assign the difference operator defined by formula (3), acting in the space of grid functions , satisfying the condition for all . Here, is an approximation of .
By using , for obtaining functions, we arrive at problem For finding a solution of problem (19) we apply the substitution where is the solution of nonlocal boundary value problem; a system of ordinary differential equations and unknown function 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 . In the second stage, we put and find . Then, using (22), we obtain . Finally, in the third stage, we use formula (20) for obtaining the solution of problem (19).
In the second step, we approximate (19) in variable . Let be the uniform grid space with step size , where is a fixed positive integer. Applying the approximate formulas for , 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 respectively.
Theorem 7. Let and be sufficiently small positive numbers. Then, for the solutions of difference schemes (24) and (25) the following almost coercive stability estimate holds, where is independent of , , and .
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 .
Theorem 8 (see ). For the solution of the elliptic difference problem the following coercivity inequality holds: where does not depend on and .
4. 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 It is clear that and are the exact solutions of (34).
We can obtain by formula , where is the solution of the nonlocal boundary value problem and is the solution of the boundary value problem
Introduce small parameters and such that . For approximate solution of nonlocal boundary value problem (35) we consider the set of a family of grid points
Applying (21), we obtain difference schemes of the first order of accuracy in and the second order of accuracy in for the approximate solutions of the nonlocal boundary value problem (35), and for the approximate solutions of the boundary value problem (36).
By using (22) and second order of accuracy in approximation of , we get the following values of in grid points:
We can rewrite difference scheme (38) in the matrix form Here, is the identity matrix, , , are square matrices, and is a column matrix which are defined by
For solving (41) we use the modified Gauss elimination method (see ). Namely, we seek solution of (41) by the formula where , are square matrices and are column matrices. For , we get formulas where is the identity matrix and is the zero column vector.
Futher, we rewrite difference scheme (39) in the matrix form
Now we present second order of accuracy in and difference schemes for problems (35) and (36). Applying (27) and formulas for sufficiently smooth function we get difference scheme for nonlocal problem (35), and difference scheme for boundary value problem (36).
We seek solution of (54) by the formula where are square matrices and are column matrices. For the solution of difference equation (41) we need to use the following formulas for : where and are the zero column vector. For and we have where
Now, we give the results of the numerical realization of finite difference method for (34) by using MATLAB programs. The numerical solutions are recorded for and different values of . Grid functions , represent the numerical solutions of difference schemes for auxiliary nonlocal problem (35) and inverse problem (34) at , respectively. Grid function calculated by (40) represents numerical solution at for unknown function . The errors are computed by the norms