Abstract
The overdetermination problem for elliptic differential equation with Dirichlet boundary condition is considered. The third and fourth orders of accuracy stable difference schemes for the solution of this inverse problem are presented. Stability, almost coercive stability, and coercive inequalities for the solutions of difference problems are established. As a result of the application of established abstract theorems, we get wellposedness of high order difference schemes of the inverse problem for a multidimensional elliptic equation. The theoretical statements are supported by a numerical example.
MSC: 35N25, 39A14, 39A30, 65J22.
Keywords:
difference scheme; inverse elliptic problem; high order accuracy; wellposedness; stability; almost coercive stability; coercive stability1 Introduction
Many problems in various branches of science lead to inverse problems for partial differential equations [13]. Inverse problems for partial differential equations have been investigated extensively by many researchers (see [318] and the references therein).
Consider the inverse problem of finding a function u and an element p for the elliptic equation
in an arbitrary Hilbert space H with a selfadjoint positive definite operator A. Here, λ is a known number, φ, ξ, and ψ are given elements of H.
Existence and uniqueness theorems for problem (1.1) in a Banach space are presented in [5]. The first and second accuracy stable difference schemes for this problem have been constructed in [15]. High order of accuracy stable difference schemes for nonlocal boundary value elliptic problems are presented in [1921].
Our aim in this work is the construction of the third and fourth order stable accuracy difference schemes for the inverse problem (1.1).
In the present paper, the third and fourth orders of accuracy difference schemes for the approximate solution of problem (1.1) are presented. Stability, almost coercive stability, and coercive stability inequalities for the solution of these difference schemes are established.
In the application, we consider the inverse problem for the multidimensional elliptic equation with Dirichlet condition
Here, is the open cube in the ndimensional Euclidean space with boundary S, , (), , , (), (, ) are given smooth functions, (), and , are given numbers.
The first and second orders of accuracy stable difference schemes for equation (1.2) are presented in [15]. We construct the third and fourth orders of accuracy stable difference schemes for problem (1.2).
The remainder of this paper is organized as follows. In Section 2, we present the third and fourth order difference schemes for problem (1.1) and obtain stability estimates for them. In Section 3, we construct the third and fourth order difference schemes for problem (1.2) and establish their wellposedness. In Section 4, the numerical results are given. Section 5 is our conclusion.
2 High order of accuracy difference schemes for (1.1) and stability inequalities
We use, respectively, the third and fourth order accuracy approximate formulas
for . Here, , is a notation for the greatest integer function. Applying formulas (2.1) and (2.2) to , we get, respectively,
the third order of accuracy difference problem and
the fourth order of accuracy difference problem for inverse problem (1.1).
For solving of problems (2.3) and (2.4), we use the algorithm [14], which includes three stages. For finding a solution of difference problems (2.3) and (2.4) we apply the substitution
In the first stage, applying approximation (2.5), we get a nonlocal boundary value difference problem for obtaining . In the second stage, we put and find . Then, using the formula
we define an element p. In the third stage, by using approximation (2.5), we can obtain the solution of difference problems (2.3) and (2.4). In the framework of the above mentioned algorithm for , we get the following auxiliary nonlocal boundary value difference scheme:
for the third order of accuracy difference problem (2.3) and
for the fourth order of accuracy difference problem (2.4).
For a selfadjoint positive definite operator A, it follows that [22] is a selfadjoint positive definite operator, where , , I is the identity operator. Moreover, the bounded operator D is defined on the whole space H.
Now we give some lemmas that will be needed below.
Lemma 2.1The following estimates hold[23]:
Lemma 2.2The following estimate holds[23]:
where
has an inverse such that
and the estimate
is valid.
Proof We have
where
Applying estimates of Lemma 2.1, we have
By using the triangle inequality, formula (2.10), estimates (2.9), (2.12), and Lemma 2.2 of paper [15], we obtain
for any small positive parameter τ. From that follows estimate (2.9). Lemma 2.3 is proved. □
has an inverse
and the estimate
is satisfied.
Proof We can get
where G is defined by formula (2.11) and
Applying estimates of Lemma 2.1, we have
Using the triangle inequality, formula (2.14), estimates (2.13), (2.15), and Lemma 2.3 of paper [15], we get
for any small positive parameter τ. From that follows estimate (2.13). Lemma 2.4 is proved. □
Let and be the spaces of all Hvalued grid functions in the corresponding norms,
Theorem 2.1Assume thatAis a selfadjoint positive definite operator, and (). Then, the solutionof difference problem (2.3) obeys the following stability estimates:
Proof We will obtain the representation formula for the solution of problem (2.7). Applying the formula [23], we get
By using formula (2.19) and nonlocal boundary conditions
we get the system of equations
Solving system (2.20), we obtain
Therefore, difference problem (2.7) has a unique solution which is defined by formulas (2.19), (2.21), and (2.22). Applying formulas (2.19), (2.21), (2.22), and the method of the monograph [23], we get
The proofs of estimates (2.17), (2.18) are based on formula (2.5) and estimate (2.23). Using formula (2.5) and estimates (2.23), (2.17), we obtain inequality (2.16). Theorem 2.1 is proved. □
Theorem 2.2Suppose thatAis a selfadjoint positive definite operator, and (). Then, the solutionof difference problem (2.4) obeys the stability estimates (2.16), (2.17), and (2.18).
Proof By using the representation formula (2.19) for the solution of (2.8), formula (2.19), and the nonlocal boundary conditions
we obtain the system of equations
Solving system (2.24), we have
So, the difference problem (2.8) has a unique solution , which is defined by formulas (2.19), (2.25), and (2.26). By using formulas (2.19), (2.25), (2.26), and the method of the monograph [23], we can get the stability estimate (2.23) for the solution of difference problem (2.8). The proofs of estimates (2.17), (2.18) are based on (2.5) and (2.23). Applying formula (2.5) and estimates (2.23), (2.17), we get estimate (2.16). Theorem 2.2 is proved. □
Theorem 2.3Assume thatAis a selfadjoint positive definite operator, and. Then, the solutionsof difference problems (2.3) and (2.4) obey the following almost coercive inequality:
Theorem 2.4Assume thatAis a selfadjoint positive definite operator, and (). Then, the solutionsof difference problems (2.3) and (2.4) obey the following coercive inequality:
The proofs of Theorems 2.3 and 2.4 are based on formulas (2.5), (2.19), (2.21), (2.22), (2.25), (2.26), Lemmas 2.1 and 2.2.
3 High order of accuracy difference schemes for the problem (1.2) and their wellposedness
Now, we consider problem (1.2). The differential expression [22,23]
defines a selfadjoint strongly positive definite operator acting on with the domain
The discretization of problem (1.2) is carried out in two steps. In the first step, we define the grid spaces
To the differential operator generated by problem (1.2) we assign the difference operator defined by the formula
acting in the space of grid functions , satisfying the condition for all .
To formulate our results, let and be spaces of the grid functions defined on , equipped with the norms
Applying formula (2.5) to , we arrive for functions, at auxiliary nonlocal boundary value problem for a system of ordinary differential equations
In the second step, auxiliary nonlocal problem (3.2) is replaced by the third order of accuracy difference scheme
and by the fourth order of accuracy difference scheme
Let τ and be sufficiently small positive numbers.
Theorem 3.1The solutions of difference schemes (3.4) and (3.5) obey the following stability estimates:
Theorem 3.2The solutions of difference schemes (3.4) and (3.5) obey the following almost coercive stability estimate:
Theorem 3.3The solutions of difference schemes (3.4) and (3.5) obey the following coercive stability estimate:
The proofs of Theorems 3.13.3 are based on the abstract Theorems 2.12.4, symmetry properties of the operator in and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in .
Theorem 3.4[24]
For the solution of the elliptic difference problem
the following coercivity inequality holds:
4 Numerical results
In this section, by using the third and fourth order of the accuracy approximation, we obtain an approximate solution of the inverse problem
for the elliptic equation. Note that and are the exact solutions of equation (4.1).
For the approximate solution of the nonlocal boundary value problem (3.2), consider the set of grid points
which depends on the small parameters and .
Applying approximations (3.4) and (3.5), we get, respectively, the third order of the accuracy difference scheme
and the fourth order of the accuracy difference scheme
for the approximate solutions of the auxiliary nonlocal boundary value problem (3.2). Applying approximation (3.3) and the second order of the accuracy in x in the approximation of A, we get the following values of the p function in the grid points:
In this step, applying to the boundary value problem for the function for the third and fourth order approximation in the variable t, we get, respectively, the third order of the accuracy difference scheme
and the fourth order of the accuracy difference scheme
We can rewrite the difference scheme (4.2) in the following matrix form:
Here, I is the identity matrix, is column matrix, A, B, C, D, E are square matrices. Moreover,
For the solution of the linear matrix equation (4.7), we use the modified Gauss elimination method [25]. Namely, we seek a solution of equation (4.7) by the formula
Here, and () are square matrices, () are column matrices which are defined by
with , and are the zero matrix, , . and are defined by formulas
We rewrite the difference scheme (4.5) in matrix form,
Here, is an column matrix, A, B, D, E are defined by formulas (4.8) and (4.9). We use square matrices, and C is the following matrix:
We can write the difference scheme (4.3) in matrix form (4.12), where A, B, D, E are defined by formulas (4.8) and (4.9), is defined by equation (4.10), C is defined by
We have the difference scheme (4.6) in the matrix form of equation (4.12), where A, B, D, E are defined by formulas (4.8) and (4.9), is defined by formula (4.10), C is defined by equation (4.13), is defined by
Now we give the results of the numerical analysis using MATLAB programs. The numerical solutions are recorded for different values of N, M; and represents the numerical solutions of these difference schemes at the grid points of , and represents the numerical solutions at . For comparison with the exact solutions, the errors are computed by
Tables 13 are constructed for , , , . Hence, the third order and fourth order of the accuracy difference schemes are more accurate than the second order of the accuracy difference schemes (ADS). Table 1 gives the error between the exact solution and solutions derived by difference schemes for the nonlocal problem. Table 2 includes the error between the exact p solution and approximate p derived by the difference schemes. Table 3 gives the error between the exact u solution and solutions derived by the difference schemes.
5 Conclusion
In this paper, the overdetermination problem for an elliptic differential equation with Dirichlet boundary condition is considered. The third and fourth orders of accuracy difference schemes for approximate solutions of this problem are presented. Theorems on the stability, almost coercive stability, and coercive stability estimates for the solutions of difference schemes for the elliptic equation are proved. As a result of the application of established abstract theorems, we get wellposedness of high order difference schemes of the inverse problem for a multidimensional elliptic equation. Numerical experiments are given. As can be seen from Tables 13, the third and fourth orders of the accuracy difference schemes are more accurate than the second order of the accuracy difference scheme.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author would like to thank Prof. Allaberen Ashyralyev (Fatih University, Turkey) for his helpful suggestions in improving the quality of this paper.
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