Abstract
In the present study, the inverse problem for a multidimensional elliptic equation with mixed boundary conditions and overdetermination is considered. The first and second orders of accuracy in t and the second order of accuracy in space variables for the approximate solution of this inverse problem are constructed. Stability, almost coercive stability, and coercive stability estimates for the solution of these difference schemes are established. For the two-dimensional inverse problems with mixed boundary value conditions, numerical results are presented in test examples.
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1 Introduction
Inverse problems for partial differential equations are frequently encountered in various branches of science (see [1–3] and the bibliography therein).
In recent years, theory and methods of solutions of inverse problems of determining the parameter of partial differential equations have been extensively studied by several researchers (see [1–25] and the references therein).
We will denote the set of natural numbers from 1 to n by \(\mathbb{N}_{n}=\{1,2,\ldots ,n\}\), and closure of set Ω by \(\overline{\Omega}=\Omega \cup S\).
Let \(n\in\mathbb{N}\), \(\delta ,T,L>0\), \(\lambda \in (0,T) \) be given numbers and \(\Omega =(0,L)\times \cdots \times (0,L)\) be the open cube in the n-dimensional Euclidean space with boundary \(S=\partial \Omega =S_{1}\cup S_{2}\), where
and let \(a_{r},\varphi ,\xi ,\psi :\overline{\Omega }\rightarrow \mathbb{R}\), \(f:(0,T)\times \Omega \rightarrow \mathbb{R}\) be given smooth functions such that \(a_{r}(x)\geq a>0\) for \(x\in \overline{\Omega }\), \(r\in\mathbb{N}_{n}\). Consider the multidimensional elliptic equation
Here \(u(t,x)\) and \(p(x)\) are unknown functions.
In \([0,T]\times \Omega \), consider the inverse problem of finding functions \(u(t,x)\) and \(p(x)\) for the multidimensional elliptic equation (1.1) with overdetermined boundary conditions
and one of the following usual boundary conditions on \([ 0,T ] \times S\):
For the problem of determining the parameter of an elliptic equation in a Banach space, the exact estimates of solution in Hölder norms are established in [21]. Well-posedness of inverse problem (1.1), (1.2), (1.3) and its approximations were investigated in [18]. In [21], the third and fourth order stable difference schemes for approximate solution of this problem were presented. Stability and coercive stability estimates for the solution of the first and second order difference schemes for inverse problem (1.1), (1.2), (1.4) were established in [19]. In [23], the authors studied inverse problem (1.1), (1.2), (1.5).
Our aim in this work is to construct the first and second orders of accuracy difference schemes for an approximate solution of inverse problem (1.1), (1.2), (1.6) and to establish stability, coercive stability, almost coercive stability estimates for the solution of these difference schemes.
This paper is planned as follows. In Section 2, we give theorems on well-posedness of inverse problems with mixed boundary conditions and overdetermination. Section 3 is devoted to the construction of the first and second order difference schemes for approximate solution of problem (1.1), (1.2), (1.6). In this section, we establish stability, almost coercive, and coercive stability inequalities for the solution of difference schemes. In Section 4, we present numerical results for a two-dimensional elliptic equation. The conclusion is given in the final Section 5.
2 Well-posedness of inverse problems with mixed boundary conditions and overdetermination
Let \(\mathcal{C}_{0T}^{\alpha ,\alpha }(L_{2}(\overline{\Omega }))\) be the space obtained by completion of the space of all smooth \(L_{2}(\overline{ \Omega })\)-valued functions g on \([0,T]\) with the norm
Denote the differential expression generated by (1.1) [26]
where \(\rho \in W_{2}^{2} ( \overline{\Omega } ) \).
The differential expression (1.1) defines a self-adjoint positive definite operator A acting on \(L_{2} ( \overline{\Omega } ) \) with the domain
Therefore, by using abstract Theorems 2.1 and 2.2 of paper [19] for \(H=L_{2} ( \overline{\Omega } ) \), we get the following theorems on well-posedness of inverse problem (1.1), (1.2), (1.5).
Theorem 2.1
Suppose that A is defined by formula (2.1), \(\varphi ,\xi ,\psi \in D(A)\). Then, for the solutions \((u,p)\) of inverse problem (1.1), (1.2), (1.5), the stability inequalities
are satisfied, where \(M_{1}\) is independent of α, \(\varphi (x)\), \(\xi (x)\), \(\psi (x)\), and \(f(t,x)\).
Theorem 2.2
Assume that A is defined by formula (2.1), \(\varphi ,\psi ,\xi \in D(A)\). Then, for the solution of inverse problem (1.1), (1.2), (1.5), the coercive stability estimate
holds, where \(M_{2}\) does not depend on α, \(\varphi (x)\), \(\xi (x)\), \(\psi (x)\), and \(f(t,x)\).
Now, consider inverse problem (1.1), (1.2), (1.6).
The differential expression (1.1) defines a self-adjoint positive definite operator A acting on \(L_{2} ( \overline{\Omega } ) \) with the domain
So, in a similar manner, applying abstract Theorems 2.1 and 2.2 of paper [19] for \(H=L_{2} ( \overline{\Omega } ) \), we can get the theorems on well-posedness of inverse problem (1.1), (1.2), (1.6).
Theorem 2.3
Suppose that A is defined by formula (2.1), \(\varphi ,\xi ,\psi \in D(A)\). Then, for the solution \((u,p)\) of inverse problem (1.1), (1.2), (1.6), the stability inequalities (2.2), (2.3), (2.4) hold.
Theorem 2.4
Assume that A is defined by formula (2.1), \(\varphi ,\psi ,\xi \in D(A)\). Then, for the solution \((u,p)\) of inverse problem (1.1), (1.2), (1.6), coercive stability estimate (2.5) is valid.
3 Difference schemes and their well-posedness
We discretize inverse problem (1.1), (1.2), (1.6) in two steps. In the first step, define the set of grid points in space variables,
Introduce the Hilbert spaces \(L_{2h}=L_{2}(\widetilde{\Omega }_{h})\) and \(W_{2h}^{2}=W_{2}^{2}(\widetilde{\Omega }_{h})\) of grid functions \(g^{h}(x)=\{g(h_{1}m_{1},\ldots ,h_{n}m_{n})\mid m_{i}=0,\ldots ,M_{i}, i=1,\ldots ,n\}\) defined on \(\widetilde{\Omega }_{h}\) equipped with the norms
respectively.
To the differential operator A (2.1) assign the difference operator \(A_{h}^{x}\), defined by the formula
acting in the space of grid functions \(u^{h}(x)\) satisfying the conditions \(u^{h}(x)=0\) for all \(x\in S_{h}^{1}\) and \(D^{h}u^{h}(x)=0 \) for all \(x\in S_{h}^{2}\). Here, \(D^{h}u^{h}(x)\) is an approximation of \(\frac{ \partial u}{\partial \vec {n}}\). Note that [26, 27] \(A_{h}^{x}\) is a self-adjoint positive definite operator in \(L_{2} ( \widetilde{\Omega }_{h} ) \).
In this step, by using \(A_{h}^{x}\), for obtaining \(u^{h}(t,x)\) functions, we arrive at the problem
In the second step, applying the approximate formula
for \(u^{h}(\lambda ,x)=\xi ^{h}(x)\), we replace inverse problem (1.1), (1.2), (1.6) with the first order of accuracy difference scheme
Here, \(l= [ \frac{\lambda }{\tau } ] \), \([\cdot ]\) is a notation for the greatest integer function.
In this step, by using the approximate formula
for \(u^{h}(\lambda ,x)=\xi ^{h}(x)\), we construct the second order of accuracy difference scheme for inverse problem (1.1), (1.2), (1.6)
Applying
we can reduce difference problems (3.3) and (3.4) to the auxiliary nonlocal difference problems
and
respectively.
The difference schemes for nonlocal boundary value problems for the multidimensional elliptic equation were studied in [28, 29].
Denote
Let H be the Hilbert space \(L_{2}(\overline{\Omega })\).
Lemma 3.1
The following estimates are satisfied [27]:
where \(M_{3}\) is independent of τ.
Lemma 3.2
([19])
For \(1\leq l\leq N-1\) and for the operator \(S=R^{2N}+R^{l}-R^{2N-l}+R^{N-l}-R^{N+l}\), the operator \(I-S\) has an inverse \(G=(I-S)^{-1}\) and the estimate
is satisfied, where \(M_{4}\) does not depend on τ.
Lemma 3.3
([19])
For \(1\leq l\leq N-1\) and for the operator
the operator \(I-S_{1}\) has an inverse
and the estimate
is valid, where \(M_{5}\) does not depend on τ.
In a similar manner as [19], we can obtain the following representation formula
with
for the solution of difference scheme (3.3) and
for difference scheme (3.4).
Now, we consider the linear spaces of mesh functions \(\theta ^{\tau }= \{ \theta _{k} \} _{1}^{N-1}\) with values in the Hilbert space H. We denote by \(C([0,T]_{\tau },H)\) a normed space with the norm
and by \(\mathcal{C}_{0T}^{\alpha ,\alpha }([0,T]_{\tau },H)\) a normed space with the norm
Let τ and \(|h|=\sqrt{h_{1}^{2}+\cdots +h_{n}^{2}}\) be sufficiently small positive numbers.
Theorem 3.1
The solutions \(( \{ u_{k}^{h} \} _{1}^{N-1},p^{h} ) \) of difference schemes (3.3) and (3.4) obey the following stability estimates:
where \(M_{6}\) is independent of τ, α, h, \(\varphi ^{h}\), \(\psi ^{h}\), \(\xi ^{h}\), and \(\{ f_{k}^{h} \} _{1}^{N-1}\).
Theorem 3.2
The solutions of difference schemes (3.3) and (3.4) obey the following almost coercive stability estimate:
where \(M_{7}\) does not depend on τ, α, h, \(\varphi ^{h}\), \(\psi ^{h}\), \(\xi ^{h}\), and \(\{ f_{k}^{h} \} _{1}^{N-1}\).
Theorem 3.3
The solutions of difference schemes (3.3) and (3.4) obey the following coercive stability estimate:
where \(M_{8}\) is independent of τ, α, h, \(\varphi ^{h}\), \(\psi ^{h}\), \(\xi ^{h}\), and \(\{ f_{k}^{h} \} _{1}^{N-1}\).
The proofs of Theorems 3.1-3.3 are based on the symmetry property of operator \(A_{h}^{x}\) in \(L_{2h}\), representation formulas (3.10), (3.11), (3.12), Lemmas 3.1-3.3 and the following theorem on the coercivity estimate for the solution of the elliptic difference problem in \(L_{2h}\).
Theorem 3.4
([30])
For the solution of the elliptic difference problem
the following coercivity inequality holds:
where M does not depend on h and \(\omega ^{h}\).
4 Numerical example
We consider the inverse problem
for the elliptic equation. It is easy to see that \(u(t,x)= ( \exp ( -t ) +t+1 ) (\cos (x)-1)\) and \(p(x)=3\cos ( x ) +\cos ( 2x ) -1\) are the exact solutions of (4.1).
For this example, we have the following auxiliary nonlocal boundary value problem for an unknown function \(v(t,x)\):
Consider the set \([ 0,T ] _{\tau }\times [ 0,\pi ] _{h}\) of grid points depending on the small parameters τ and h,
We get difference schemes of the first order of accuracy in t and the second order of accuracy in x,
for the approximate solutions of the nonlocal boundary value problem (4.2).
Now, it is easy to rewrite difference scheme (4.3) in the following matrix form:
Here, \(\theta _{n}\) is an \((N+1)\times 1\) column matrix, \(I_{N+1}\) is the \((N+1)\times (N+1)\) identity matrix, \(A_{n}\), \(B_{n}\), \(C_{n}\) are \((N+1)\times (N+1)\) square matrices
For numerical solving of this system equation, we use the modified Gauss elimination method [31]. Solution of (4.4) is presented by the formula
where \(\alpha _{n}\) (\(n=1,\ldots ,M-1\)) are \((N+1)\times (N+1)\) square matrices and \(\beta _{n}\) (\(n=1,\ldots ,M-1\)) are \((N+1)\times 1\) column matrices. For \(\alpha _{n+1}\), \(\beta _{n+1}\), we have the following formulas:
where
By using the second order of accuracy in x approximation of A, we get the following values of p in grid points:
For the approximate solution of problem (4.1), we use the formula
Now, we consider again nonlocal boundary value problem (4.2).
Applying (3.7), we get the second order of accuracy in t and xdifference scheme
for the approximate solution of nonlocal boundary value problem (4.2).
Therefore, we can rewrite difference scheme (4.9) in the matrix form
where \(A_{n}\), \(C_{n}\) are defined by (4.5), (4.6), (4.7) and \(B_{n}\) is an \((N+1)\times (N+1)\) square matrix
For solution of (4.10), we use the formula
where \(\alpha _{n}\), \(\beta _{n}\) (\(n=0,\ldots ,M-2\)) are \((N+1)\times (N+1)\) square matrices and \(\gamma _{n}\) (\(n=0,\ldots ,M-2\)) are \((N+1)\times 1\) column matrices. For the coefficients \(\alpha _{n}\), \(\beta _{n}\), \(\gamma _{n}\), we get formulas
Here
and \(\gamma _{0}\), \(\gamma _{1}\), \(\gamma _{M-2}\), \(\gamma _{M-3} \) are the \((N+1)\times 1\) zero column vector. For \(v_{M}\) and \(v_{M-1}\), we have the following formulas:
where
Now, by using MATLAB programs, we present numerical results for this example (4.1). The numerical solutions are recorded for different values of N and M.
The numerical solutions are recorded for different values of \(N=M\) for \(T=2\). The grid function \(u_{n}^{k}\) represents the numerical solution of difference schemes for inverse problem at \((t_{k},x_{n})\), respectively. Grid function \(p_{n}\) represents the numerical solutions at \(x_{n}\) for unknown function p. For their comparison, the errors are computed by
Tables 1 and 2 give the error analysis between the exact solution and solutions derived by difference schemes. Tables 1 and 2 are constructed for \(N=M=20, 40, 80, \mbox{and } 160\). Hence, the second order of accuracy difference scheme is more accurate comparing with the first order of accuracy difference scheme.
Table 1 is the error analysis between the exact solution p and solutions derived by difference schemes in the second stage of algorithm.
Table 2 presents the error analysis between the exact solution u and solutions derived by first order and second order accuracy of difference schemes.
5 Conclusion
In this paper, the inverse problem for a multidimensional elliptic equation with Dirichlet-Neumann conditions and overdetermination is considered. The first and second orders of accuracy difference schemes for approximate solution of this problem are presented. Theorems on the stability, almost coercive stability, and coercive stability inequalities for the solutions of difference schemes for inverse problem are proved. Numerical example in a two-dimensional case is given. As it can be seen from Tables 1 and 2, the second order of accuracy difference scheme is more accurate comparing with the first order of accuracy difference scheme.
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Ashyralyyev, C., Dedeturk, M. Approximation of the inverse elliptic problem with mixed boundary value conditions and overdetermination. Bound Value Probl 2015, 51 (2015). https://doi.org/10.1186/s13661-015-0312-x
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DOI: https://doi.org/10.1186/s13661-015-0312-x