1 Introduction

The theory and methods of the solutions of inverse problems of determining the parameter of partial differential equations have been extensively studied by several researchers (see [1–19] and the references therein).

The well-posedness of source identification problems for elliptic type differential and difference equations was studied in [12–19]. These papers were devoted to various inverse elliptic problems and their approximations. For abstract elliptic equations in general \(L_{p}\) spaces and some exposition on practical applications of inverse problems we refer to [20, 21] and the references therein.

In recent years, Bitsadze-Samarskii type NBVPs have been investigated extensively (see [22–27]).

Let A be a self-adjoint positive definite operator (SAPD) in an arbitrary Hilbert space H. Then \(B=A^{\frac{1}{2}}\) is also an SAPD operator. One assumes that the smooth function \(f ( \cdot ) \), elements \(\varphi ,\psi,\zeta\in D(B)\), numbers \(\lambda_{0}\), \(\lambda_{i}\), \(i=1,\ldots,q\), and nonnegative real coefficients \(k_{i}\), \(i=1,\ldots,q\), are given. In [8], the Bitsadze-Samarskii type inverse elliptic problem of finding an element \(p\in H\) and a function \(u ( \cdot ) \in C^{2} ( [ 0,T ] , H ) \cap C ( [ 0,T ] , D(A) ) \) such that

$$ \textstyle\begin{cases} -u_{tt}(t)+Au(t)=f(t)+p,\quad 0< t< T, \\ u_{t}(0)=\varphi, \quad\quad u_{t}(T)=\sum_{i=1}^{q}k_{i}u_{t}(\lambda _{i})+\psi, \quad\quad u(\lambda_{0})=\zeta, \end{cases} $$
(1)

was studied. Here \(\lambda_{1},\ldots,\lambda_{q} \) such that \(0<\lambda _{1}<\lambda_{2}<\cdots<\lambda_{q}<T\).

Under the conditions

$$ \sum_{i=1}^{q}k_{i}\leq1 $$
(2)

the theorem on the solvability and uniqueness of the solution of problem (1) was proved.

Denote by \(C(H)\), \(C^{\alpha}(H)\), and \(C_{0T}^{\alpha,\alpha}(H)\) the corresponding Banach spaces of smooth H-valued functions u on \([0,T]\) with the following norms:

$$\begin{aligned}& \Vert u \Vert _{C(H)}=\max_{0\leq t\leq T} \bigl\Vert u(t) \bigr\Vert _{H} , \\& \Vert u \Vert _{C^{\alpha}(H)}= \Vert u \Vert _{C(H)}+\sup _{0\leq t< t+s\leq T} s^{-\alpha} \bigl\Vert u(t+s)-u(t) \bigr\Vert _{H}, \\& \Vert u \Vert _{C_{0T}^{\alpha,\alpha}(H)}= \Vert u \Vert _{C(H)}+\sup _{0\leq t< t+s\leq T} \biggl[ \frac{ ( t+s ) (T-t)}{s} \biggr] ^{\alpha} \bigl\Vert u(t+s)-u(t) \bigr\Vert _{H}. \end{aligned}$$

Lemma 1

The following inequalities are fulfilled ([28]):

$$ \begin{gathered} \bigl\Vert Be^{-tB} \bigr\Vert _{H\rightarrow H}\leq t^{-\alpha}, \quad 0\leq\alpha\leq1, t>0, \quad\quad \bigl\Vert \bigl( I-e^{-2TB} \bigr) ^{-1} \bigr\Vert _{H\rightarrow H}\leq M( \delta), \\ \bigl\Vert B^{\beta} \bigl( e^{-tB}-e^{- ( t+\tau ) B} \bigr) \bigr\Vert _{H\rightarrow H}\leq M(\delta)\frac{\tau^{\alpha}}{ ( t+\tau ) ^{\alpha+\beta}}, \quad 0\leq\alpha,\beta \leq1, \\ 0< t\leq t+\tau\leq1, \quad \bigl\Vert \bigl( I\pm e^{-\lambda B} \bigr) ^{-1} \bigr\Vert _{H\rightarrow H}\leq M(\lambda), \quad 0< \lambda \leq T.\end{gathered} $$
(3)

In beginning of this paper, we discuss stability estimates for the solution of the overdetermined problem (1). Next, we apply the finite difference method and establish stability estimates for the first and second order of accuracy difference schemes. Later, we consider Bitsadze-Samarskii type overdetermined elliptic multidimensional differential and difference problems with multipoint NBVC and we obtain stability estimates for their solutions. Finally, we present numerical results with an explanation on the realization for two dimensional and three dimensional elliptic overdetermined multipoint NBVP in test examples.

2 Overdetermined differential problem

Let us take \(Q= ( I-e^{-2TB} ) ^{-1}\). It is well known that ([9]) the function

$$ G(t,s)=-\frac{1}{2B}Q \bigl[ e^{- ( t+s ) B}+e^{- ( 2T-t-s ) B}+e^{- \vert t-s \vert B}+e^{- ( 2T- \vert t-s \vert ) B} \bigr] $$
(4)

is the Green function for the following elliptic problem with second kind boundary conditions:

$$ \textstyle\begin{cases} -v_{tt}(t)+Av(t)=f(t),\quad 0< t< T, \\ v_{t}(0)=\varphi,\quad\quad v_{t}(T)=\xi. \end{cases} $$
(5)

Under the assumptions that \(\varphi,\xi\in D(B)\), \(f ( \cdot ) \in C^{1} ( [ 0,T ] ,H ) \), the solution of problem (5) is defined by the equation ([9])

$$ \begin{aligned}[b] v(t)&=-Q \bigl[ \bigl(e^{-tB}+e^{- ( 2T-t ) B} \bigr)B^{-1}\varphi - \bigl(e^{- ( T-t ) B}+e^{- ( T+t ) B} \bigr)B^{-1}\xi \bigr] \\ &\quad{} + \int_{0}^{T}G(t,s)f(s)\,ds.\end{aligned} $$
(6)

Moreover, the derivative is given by the formula

$$\begin{aligned}& v_{t}(t)= Q \bigl[ \bigl(e^{-tB}-e^{- ( 2T-t ) B} \bigr)\varphi+ \bigl(e^{- ( T-t ) B}-e^{- ( T+t ) B} \bigr)\xi \bigr] \\& \hphantom{v_{t}(t)=}{} +F(B,f,t), \\& F(B,f,t)=\frac{1}{2}Q \int_{0}^{T} \bigl( e^{- ( t+s ) B}-e^{- ( 2T-t-s ) B} \bigr) f(s)\,ds \\& \hphantom{F(B,f,t)=}{} +\frac{1}{2}Q \int_{0}^{t} \bigl( e^{- ( t-s ) B}-e^{- ( 2T-t+s ) B} \bigr) f(s)\,ds \\& \hphantom{F(B,f,t)=}{} -\frac{1}{2}Q \int_{t}^{T} \bigl( e^{- ( s-t ) B}-e^{- ( 2T+t-s ) B} \bigr) f(s)\,ds. \end{aligned}$$
(7)

Let us take \(\alpha\in(0,1)\). To formulate our results, we use the notation

$$P_{H}^{\gamma} ( \varphi,\psi,\zeta ) = \bigl\Vert A^{\gamma }\varphi \bigr\Vert _{H}+ \bigl\Vert A^{\gamma}\psi \bigr\Vert _{H}+ \bigl\Vert A^{\gamma} \zeta \bigr\Vert _{H}, \quad \gamma=-1,-\frac {1}{2},0,\frac{1}{2},1. $$

Theorem 1

Suppose that A is an SAPD operator, condition (2) for coefficients is valid, \(\varphi,\psi,\zeta\in D(A)\cap D(A^{-\frac{1}{2}})\), and \(f(t)\in C_{0T}^{\alpha,\alpha}(H)\). Then for the solution \((p,u(t))\) of problem (1) the following stability estimates hold:

$$\begin{aligned}& \Vert u \Vert _{C(H)}\leq M ( \delta ) \bigl[ P_{H}^{0} ( \varphi,\psi,\zeta ) +P_{H}^{-\frac{1}{2}} ( \varphi,\psi,\zeta ) + \Vert f \Vert _{C(H)} \bigr] , \end{aligned}$$
(8)
$$\begin{aligned}& \bigl\Vert A^{-1}p \bigr\Vert _{H}\leq M ( \delta ) \bigl[ P_{H}^{0} ( \varphi,\psi,\zeta ) + \Vert f \Vert _{C(H)} \bigr] , \end{aligned}$$
(9)
$$\begin{aligned}& \Vert p \Vert _{H}\leq M ( \delta ) \biggl[ P_{H}^{1} ( \varphi,\psi,\zeta ) +\frac{1}{\alpha(1-\alpha )} \Vert f \Vert _{C_{0T}^{\alpha,\alpha}(H)} \biggr] , \end{aligned}$$
(10)

where \(M ( \delta ) \) does not depend on α, φ, ψ, ζ, and \(f(t)\).

Proof

Applying the substitution

$$ u(t)=v(t)+A^{-1} ( p ) , $$
(11)

we get the auxiliary nonlocal boundary value problem:

$$ \textstyle\begin{cases} -v_{tt}(t)+Av(t)=f(t), \quad 0< t< T, \\ v_{t}(0)=\varphi, \quad\quad v_{t}(T)=\sum_{i=1}^{q}k_{i}v_{t}(\lambda _{i})+\psi,\end{cases} $$
(12)

to find a function \(v ( \cdot ) \).

Thus, we consider the algorithm to solve (1) which includes three stages. In the first stage, we solve the nonlocal boundary value problem (12). In the second stage, we put \(t=\lambda_{0}\), and find \(v(\lambda_{0})\). Then we obtain an element p by the formula

$$ p=A\zeta-Av(\lambda_{0}). $$
(13)

In the third stage, we use equation (11) to find \(u(t)\).

In [22–27], the well-posedness of the Bitsadze-Samarskii type nonlocal BVP (12) in the case of \(q=1\) and its applications were investigated. The stability and coercive stability estimates for the solution of problem (12) were established.

Taking into account equation (7) in the nonlocal condition of problem (12), we get the equation

$$ \begin{aligned}[b] & \Biggl[ I-\sum_{i=1}^{q}k_{i}Q \bigl(e^{- ( T-\lambda_{i} ) B}-e^{- ( T+\lambda_{i} ) B} \bigr) \Biggr] v_{t}(T) \\ &\quad =\sum_{i=1}^{q}k_{i} \bigl[ Q \bigl(e^{-\lambda_{i}B}-e^{- ( 2T-\lambda _{i} ) B} \bigr)\varphi+F(B,f, \lambda_{i}) \bigr] +\psi \end{aligned} $$
(14)

to find \(v_{t}(T)\). In ([23]), the authors proved that the operator

$$G=I-\sum_{i=1}^{q}k_{i}Q \bigl(e^{- ( T-\lambda_{i} ) B}-e^{- ( T+\lambda_{i} ) B} \bigr) $$

has an inverse and its norm is bounded by

$$ \bigl\Vert G^{-1} \bigr\Vert _{H\rightarrow H} \leq M ( \delta ) . $$
(15)

So, we can obtain a solution of problem (5) by

$$\begin{aligned} v(t)&=-Q \bigl(e^{-tB}+e^{- ( 2T-t ) B} \bigr)B^{-1} \varphi-Q \bigl(e^{- ( T-t ) B}+e^{- ( T+t ) B} \bigr)G^{-1} \\ &\quad{} \times \Biggl\{ \sum_{i=1}^{q}k_{i} \bigl[ Q \bigl(e^{-\lambda _{i}B}-e^{- ( 2T-\lambda_{i} ) B} \bigr)B^{-1}\varphi +B^{-1}F(B,f,\lambda_{i}) \bigr] \\ &\quad{} +B^{-1}\psi \Biggr\} + \int_{0}^{T}G(t,s)f(s)\,ds. \end{aligned}$$
(16)

Applying estimates (2), (3), (15), and the Cauchy-Schwartz and the triangle inequalities, we can obtain

$$ \begin{gathered} \bigl\Vert Q \bigl(e^{-tB}+e^{- ( 2T-t ) B} \bigr)B^{-1}\varphi \bigr\Vert _{C(H)}\leq M ( \delta ) \bigl\Vert B^{-1}\varphi \bigr\Vert _{H}, \\ \Biggl\Vert Q \bigl(e^{- ( T-t ) B}+e^{- ( T+t ) B} \bigr)G^{-1} \Biggl\{ \sum_{i=1}^{q}k_{i}B^{-1} \varphi+B^{-1}\psi \Biggr\} \Biggr\Vert _{C(H)} \leq M ( \delta ) \bigl\Vert B^{-1}\varphi \bigr\Vert _{H}, \\ \Biggl\Vert Q \bigl(e^{- ( T-t ) B}+e^{- ( T+t ) B} \bigr)G^{-1} \sum_{i=1}^{q}k_{i}B^{-1}F(B,f, \lambda_{i}) \Biggr\Vert _{C(H)}\leq M ( \delta ) \Vert f \Vert _{C(H)}, \\ \biggl\Vert \int_{0}^{T}G(t,s)f(s)\,ds \biggr\Vert _{C(H)}\leq M ( \delta ) \Vert f \Vert _{C(H)}.\end{gathered} $$
(17)

From (17) it follows that

$$ \Vert v \Vert _{C(H)}\leq M ( \delta ) \bigl[ P_{H}^{-\frac{1}{2}} ( \varphi,\psi,\zeta ) + \Vert f \Vert _{C(H)} \bigr] . $$
(18)

By using (13),(16) and (3), the estimates (9) and (10) can be established.

From (11), (13), (16), (18), and the triangle inequalities the estimate (8) follows. □

Theorem 2

Under the assumptions that A is an SAPD operator, condition (2) for the coefficients is valid, \(\varphi,\psi ,\zeta\in D(A)\cap D(A^{\frac{1}{2}}) \) and \(f(t)\in C_{0T}^{\alpha,\alpha }(H)\) (\(0<\alpha<1 \)), and the solution \((p,u(t))\) of problem (1) obeys the following coercive inequality:

$$\begin{aligned}& \bigl\Vert u^{\prime\prime} \bigr\Vert _{C_{0T}^{\alpha,\alpha}(H)} + \Vert Au \Vert _{C_{01}^{\alpha,\alpha}(H)}+ \Vert p \Vert _{H} \end{aligned}$$
(19)
$$\begin{aligned}& \quad \leq M ( \delta ) \biggl[ P_{H}^{1} ( \varphi,\psi,\zeta ) +P_{H}^{\frac{1}{2}} ( \varphi,\psi,\zeta ) +\frac {1}{\alpha(1-\alpha)} \Vert f \Vert _{C_{0T}^{\alpha,\alpha }(H)} \biggr] , \end{aligned}$$
(20)

where \(M ( \delta ) \) is independent of α, φ, ψ, ζ, and \(f(t)\).

Proof

Applying (11) and (16), we have

$$\begin{aligned} Au(t)&=-Q \bigl(e^{-tB}+e^{- ( 2T-t ) B}-e^{-\lambda_{0}B }-e^{- ( 2T-\lambda_{0} ) B} \bigr)B\varphi \\ &\quad{} -Q \bigl(e^{- ( T-t ) B}+e^{- ( T+t ) B}-e^{- ( T-\lambda _{0} ) B}-e^{- ( T+\lambda_{0} ) B} \bigr)G^{-1} \\ &\quad{} \times \Biggl\{ \sum_{i=1}^{q}k_{i} \bigl[ Q \bigl(e^{-\lambda _{i}B}-e^{- ( 2T-\lambda_{i} ) B} \bigr)B\varphi+BF(B,f,\lambda _{i}) \bigr] +B\psi \Biggr\} -\frac{1}{2}QB \\ &\quad{} \times \int_{0}^{T} \bigl[ e^{- ( t+s ) B}+e^{- ( 2T-t-s ) B}+e^{- \vert t-s \vert B}+e^{- ( 2T- \vert t-s \vert ) B} \bigr] f(s)\,ds \\ &\quad{} +\frac{1}{2}QB \int_{0}^{T} \bigl[ e^{- ( \lambda_{0}+s ) B}+e^{- ( 2T-\lambda_{0}-s ) B}+e^{- \vert \lambda _{0}-s \vert B} \\ &\quad{} +e^{- ( 2T- \vert \lambda_{0}-s \vert ) B} \bigr] f(s)\,ds. \end{aligned}$$
(21)

The proof of estimate (20) is based on equation (21) and the estimates (3). □

3 Difference schemes for problem (1)

Since A is an SAPD operator, the operator \(C=\frac{1}{2}(\tau A+\sqrt{ 4A+\tau^{2}A^{2}})\) will be an SAPD operator, too ([29]). Denote \(R=(I+\tau C)^{-1}\), \(P=(I-R^{2N})^{-1}\), \(D=(I+\tau C)(2I+\tau C)^{-1}C^{-1}\).The bounded operator R is defined on the whole space H.

Lemma 2

[28]

The following estimates hold:

$$\begin{aligned}& \bigl\Vert R^{k} \bigr\Vert _{H\rightarrow H}\leq M ( \delta ) \bigl(1+\delta^{\frac{1}{2}}\tau \bigr)^{-k}, \quad\quad \bigl\Vert CR^{k} \bigr\Vert _{H\rightarrow H}\leq\frac{M ( \delta ) }{k\tau},\quad k\geq1, \\& \Vert P \Vert _{H\rightarrow H}\leq M ( \delta ) , \quad \delta >0. \end{aligned}$$

Lemma 3

Under the assumptions that (2) is valid, the operator

$$ \Delta_{1}= \bigl( I-R^{2N} \bigr) \Biggl[ I-R^{2N-2}-\sum _{i=1}^{q}k_{i} \bigl( R^{N-l_{i}-1}-R^{N+l_{i}-1} \bigr) \Biggr] $$
(22)

has inverse \(G_{1}^{-1}\) and the following estimate holds:

$$ \bigl\Vert \Delta_{1}^{-1} \bigr\Vert _{H\rightarrow H} \leq M ( \delta ,\lambda_{1},\ldots,\lambda_{q} ) . $$
(23)

Lemma 4

Under the assumptions that (2) is valid, the operator

$$\begin{aligned} \Delta_{2}&= \bigl[ R-3I+R^{2N-2}(I-3R) \bigr] \Biggl[ 3I-R-R^{2N-2} ( I-3R ) \\ &\quad{} -\sum_{i=1}^{q}k_{i} \bigl[ R^{N-l_{i}-1} \bigl( 3I-R+\mu _{i} ( I-R ) \bigr) -R^{2N+l_{i}-1} \bigl( I-3R+\mu_{i} ( I-R ) \bigr) \bigr] \Biggr] \\ &\quad{} - ( I+R ) R^{N-2} \bigl( -R^{2}+4R-I \bigr) \Biggr[ ( I+R ) R^{N-2} \bigl( R^{2}-4R+I \bigr) \\ &\quad{} -\sum_{i=1}^{q}k_{i} \bigl[ R^{l_{i}-1} \bigl( I-3R+\mu_{i} ( I-R ) \bigr) -R^{2N-l_{i}-1} \bigl( 3I-R+\mu_{i} ( I-R ) \bigr) \bigr] \Biggr] \end{aligned}$$

has inverse \(\Delta_{2}^{-1}\) and the following estimate holds:

$$ \bigl\Vert \Delta_{2}^{-1} \bigr\Vert _{H\rightarrow H} \leq M ( \delta ,\lambda_{1},\ldots,\lambda_{q} ) . $$
(24)

It is well known that [28]

$$ \begin{aligned}[b] v_{m}&=P \bigl[ \bigl( R^{m}-R^{2N-m} \bigr) v_{0}+ \bigl( R^{N-m}-R^{N+m} \bigr) v_{N} \bigr] \\ &\quad{} -P \bigl( R^{N-m}-R^{N+m} \bigr) D\sum _{j=1}^{N-1} \bigl( R^{N-j}-R^{N+j} \bigr) f_{j}\tau \\ &\quad{} +D\sum_{j=1}^{N-1} \bigl( R^{ \vert m-j \vert }-R^{m+j} \bigr) f_{j}\tau\quad (1\leq m\leq N-1)\end{aligned} $$
(25)

is a solution of the direct problem

$$ \textstyle\begin{cases} -\tau^{-2}(v_{m+1}-2v_{m}+v_{m-1})+Av_{m}=f_{m}, \quad 1\leq m\leq N-1, \\ v_{0}\text{ and }v_{N}\text{ are given}. \end{cases} $$
(26)

Let \([\cdot]\) be the notation of the greatest integer function, and

$$l_{i}= \biggl[\frac{\lambda_{i}}{\tau} \biggr],\quad\quad \mu_{i}= \frac{\lambda_{i}}{\tau}-l_{i},\quad 0\leq i\leq q. $$

We consider the first and second order of accuracy difference schemes (ADSs) for the nonlocal problem (1). First, by using the approximate formulas

$$ u_{t}(\lambda_{i})=\frac{u_{l_{i}+1}-u_{l_{i}}}{\tau}+o(\tau ),\quad\quad u_{l_{o}}=\zeta+o(\tau), $$
(27)

problem (1) is reduced to the first order of the ADS

$$ \textstyle\begin{cases} -\tau^{-2}(u_{k+1}-2u_{k}+u_{k-1})+Au_{k}=f_{k}+p, \quad 1\leq k\leq N-1, \\ u_{1}-u_{0}=\tau\varphi, \\ u_{N}-u_{N-1}=\sum_{i=1}^{q}k_{i} ( u_{l_{i}+1}-u_{l_{i}} ) +\tau\psi,\quad\quad u_{l_{o}}=\zeta.\end{cases} $$
(28)

Second, applying the approximate formulas

$$ \begin{gathered} u_{t}(\lambda_{i})= \frac{3u_{l_{i}+1}-4u_{l_{i}}+u_{l_{i}-1}}{2\tau }+\mu _{i}\frac{u_{l_{i}+1}-2u_{l_{i}}+u_{l_{i}-1}}{\tau}+o \bigl( \tau^{2} \bigr), \\ u(\lambda_{0})=u_{l_{o}}+\mu_{i}(u_{l_{o}+1}-u_{l_{o}})+o \bigl(\tau^{2} \bigr),\end{gathered} $$
(29)

problem (1) can be reduced to the second order of the ADS,

$$ \textstyle\begin{cases} -\tau^{-2}(u_{k+1}-2u_{k}+u_{k-1})+Au_{k}=f_{k}+p, \quad 1\leq k\leq N-1, \\ \frac{-3u_{0}+4u_{1}-u_{2}}{2\tau}=\varphi, \\ \frac{3u_{N}-4u_{N-1}+u_{N-2}}{2\tau}=\psi +\sum_{i=1}^{q}k_{i} [ \frac{3u_{l_{i}+1}-4u_{l_{i}}+u_{l_{i}-1}}{2\tau}+2\mu_{i}\frac{u_{l_{i}+1}-2u_{l_{i}}+u_{l_{i}-1}}{\tau} ] , \\ u_{l_{o}}+\mu_{i}(u_{l_{o}+1}-u_{l_{o}})=\zeta.\end{cases} $$
(30)

Introduce the set of grid points \(\{ t_{k}=k\tau, 1\leq k\leq N-1, N\tau=T \} \) and the spaces \(C_{\tau}(H)\), \(C_{\tau}^{\alpha }(H)\), and \(C_{\tau}^{\alpha,\alpha}(H)\) (\(\alpha\in (0,1)\)) of H-valued grid functions \(f_{\tau}= \{ f_{k} \} _{k=1}^{N-1}\) with the corresponding norms,

$$\begin{aligned}& \Vert f_{\tau} \Vert _{C_{\tau}(H)}=\max_{1\leq k\leq N-1} \Vert f_{k} \Vert _{H}, \\& \Vert f_{\tau} \Vert _{C_{\tau}^{\alpha}(H)}= \Vert f_{\tau } \Vert _{C_{\tau}(H)}+\sup_{1\leq k< k+n\leq N-1}\frac{ \Vert f_{k+n}-f_{k} \Vert _{H}}{ ( n\tau ) ^{\alpha}}, \\& \Vert f_{\tau} \Vert _{C_{\tau}^{\alpha,\alpha}(H)}= \Vert f_{\tau} \Vert _{C_{\tau}(H)}+\sup_{1\leq k< k+n\leq N-1}\frac{(k\tau+n\tau)^{\alpha}(1-k\tau)^{\alpha} \Vert f_{k+n}-f_{k} \Vert _{H}}{ ( n\tau ) ^{\alpha}}. \end{aligned}$$

Theorem 3

Under the assumptions that \(\varphi,\psi,\zeta\in D(A)\), \(f_{\tau}\in C_{\tau}^{\alpha,\alpha}(H) \) the solution \(( p, \{ u_{k} \} _{k=1}^{N-1} ) \) of the difference problems (28) and (30) in \(C_{\tau}(H)\times H \) obeys the following stability estimates:

$$\begin{aligned}& \bigl\Vert \{ u_{k} \} _{k=1}^{N-1} \bigr\Vert _{C_{\tau }(H)}\leq M ( \delta,\lambda_{1},\ldots, \lambda_{q} ) \bigl[ P_{H}^{0} ( \varphi,\psi, \zeta ) + \Vert f_{\tau } \Vert _{C_{\tau}(H)} \bigr] , \end{aligned}$$
(31)
$$\begin{aligned}& \bigl\Vert A^{-1}p \bigr\Vert _{H}\leq M ( \delta, \lambda _{1},\ldots ,\lambda_{q} ) \bigl[ P_{H}^{0} ( \varphi,\psi,\zeta ) + \Vert f_{\tau} \Vert _{C_{\tau}(H)} \bigr] , \end{aligned}$$
(32)
$$\begin{aligned}& \Vert p \Vert _{H}\leq M ( \delta,\lambda_{1},\ldots ,\lambda_{q} ) \biggl[ P_{H}^{1} ( \varphi,\psi, \zeta ) +\frac{1}{\alpha(1-\alpha)} \Vert f_{\tau} \Vert _{C_{\tau }^{\alpha,\alpha}(H)} \biggr] , \end{aligned}$$
(33)

where \(M ( \delta,\lambda_{1},\ldots,\lambda_{q} ) \) does not depend on parameters τ, α, elements φ, ψ, ξ, and the grid function \(f_{\tau}\).

Proof

Applying equation (11) at the point \(t=t_{k}\) to the difference problem (28) , we get the auxiliary nonlocal difference problem

$$ \textstyle\begin{cases} -\tau^{-2}(v_{k+1}-2v_{k}+v_{k-1})+Av_{k}=f_{k}, \quad 1\leq k\leq N-1, \\ v_{1}-v_{0}=\tau\varphi, \\ v_{N}-v_{N-1}=\sum_{i=1}^{q}k_{i} ( v_{l_{i}+1}-v_{l_{i}} ) +\tau\psi. \end{cases} $$
(34)

By using (25), and the local and nonlocal conditions of problem (34), one can get the system of equations

$$ \begin{gathered} \bigl( I+R^{2N-1} \bigr) v_{0}- \bigl( R^{N-1}+R^{N} \bigr) v_{N} = ( R-I ) ^{-1} \bigl( I-R^{2N} \bigr) F_{1}, \\ \Biggl( -R^{N-1}-R^{N}-\sum_{i=1}^{q}k_{i} \bigl( R^{l_{i}}+R^{2N-l_{i}-1} \bigr) \Biggr) v_{0} \\ \quad\quad{} + \Biggl( \bigl( I+R^{2N-1} \bigr) +\sum _{i=1}^{q}k_{i} \bigl( R^{N-l_{i}-1}+R^{N+l_{i}} \bigr) \Biggr) v_{N} \\ \quad =- ( R-I ) ^{-1} \bigl( I-R^{2N} \bigr) F_{2},\end{gathered} $$
(35)

to find \(v_{0} \) and \(v_{N}\), where

$$ \begin{gathered} F_{1}=\tau\varphi+P \bigl( R^{N-1}-R^{N+1} \bigr) D\sum_{j=1}^{N-1} \bigl( R^{N-j}-R^{N+j} \bigr) f_{j}\tau -D\sum_{j=1}^{N-1} \bigl( R^{ \vert 1-j \vert }-R^{1+j} \bigr) f_{j}\tau, \\ F_{2}=\tau\psi+P \Biggl[ R-R^{2N-1} \\ \hphantom{F_{2}=}{} -\sum_{i=1}^{q}k_{i} \bigl( R^{N-l_{i}-1}-R^{N+l_{i}+1}-R^{N-l_{i}}+R^{N+l_{i}} \bigr) \Biggr] \\ \hphantom{F_{2}=}{}\times D\sum_{j=1}^{N-1} \bigl( R^{N-j}-R^{N+j} \bigr) f_{j}\tau -D\sum _{j=1}^{N-1} \Biggl[ R^{ \vert N-1-j \vert }-R^{N-1+j} \\ \hphantom{F_{2}=}{} -\sum_{i=1}^{q}k_{i} \bigl( R^{ \vert l_{i}+1-j \vert }-R^{l_{i}+1+j}-R^{ \vert l_{i}-j \vert }+R^{l_{i}+j} \bigr) \Biggr] f_{j}\tau.\end{gathered} $$
(36)

It easy to check that the determinant of system (35) is defined by equation (22). Since the operator \(\Delta_{1}\) has a bounded inverse, we obtain the solution of (35),

$$ \begin{gathered} v_{0}=\Delta_{1}^{-1} \Biggl\{ - \bigl( I+R^{2N-1} \bigr) +\sum_{i=1}^{q}k_{i} \bigl( R^{N-l_{i}-1}+R^{N+l_{i}} \bigr) ( R-I ) ^{-1} \bigl( I-R^{2N} \bigr) F_{1} \\ \hphantom{v_{0}=}{} + \bigl( R^{N-1}+R^{N} \bigr) ( R-I ) ^{-1} \bigl( I-R^{2N} \bigr) F_{2} \Biggr\} , \\ v_{N}=\Delta_{1}^{-1} \Biggl\{ \Biggl( R^{N-1}+R^{N}+\sum_{i=1}^{q}k_{i} \bigl( R^{l_{i}}+R^{2N-l_{i}-1} \bigr) \Biggr) \\ \hphantom{v_{N}=}{} \times ( R-I ) ^{-1} \bigl( I-R^{2N} \bigr) F_{1}- \bigl( I+R^{2N-1} \bigr) ( R-I ) ^{-1} \bigl( I-R^{2N} \bigr) F_{2} \Biggr\} ,\end{gathered} $$
(37)

Hence, problem (34) has a unique solution \(\{ v_{k} \} _{k=0}^{N}\) and it is defined by equations (25), (37).

Applying equation (11) to the difference problems (30), we have the auxiliary nonlocal difference problem

$$ \textstyle\begin{cases} -\tau^{-2}(v_{k+1}-2v_{k}+v_{k-1})+Av_{k}=f_{k}, \quad 1\leq k\leq N-1, \\ \frac{-3v_{0}+4v_{1}-v_{2}}{2\tau}=\varphi, \\ \frac {3v_{N}-4v_{N-1}+v_{N-2}}{2\tau}=\psi +\sum_{i=1}^{q}k_{i} [ \frac{3v_{l_{i}+1}-4v_{l_{i}}+v_{l_{i}-1}}{2\tau}+2\mu_{i}\frac{v_{l_{i}+1}-2v_{l_{i}}+v_{l_{i}-1}}{\tau} ] .\end{cases} $$
(38)

Later, by using the local and nonlocal conditions of problem (38), we have the following system of equations:

$$ \begin{gathered} P ( I-R ) \bigl\{ \bigl[ R-3I+R^{2N-2}(I-3R) \bigr] v_{0} \bigr\} \\ \quad{} + ( I+R ) R^{N-2} \bigl( -R^{2}+4R-I \bigr) v_{N} \} =F_{3}, \\ P ( I-R ) \Biggl\{ \Biggl[ ( I+R ) R^{N-2} \bigl( R^{2}-4R+I \bigr) \\ \quad{} -\sum_{i=1}^{q}k_{i} \bigl[ R^{l_{i}-1} \bigl( I-3R+\mu _{i} ( I-R ) \bigr) -R^{2N-l_{i}-1} \bigl( 3I-R+\mu_{i} ( I-R ) \bigr) \bigr] \Biggr] v_{0} \\ \quad{} + \Biggl[ 3I-R-R^{2N-2} ( I-3R ) -\sum _{i=1}^{q}k_{i} \bigl[ R^{N-l_{i}-1} \bigl( 3I-R+\mu_{i} (I-R ) \bigr) \\ \quad{} -R^{2N+l_{i}-1} \bigl( I-3R+ \mu_{i} ( I-R ) \bigr) \bigr] \Biggr] v_{N} \Biggr\} =F_{4},\end{gathered} $$
(39)

where

$$\begin{aligned}& \begin{aligned}[b] F_{3}&=2\tau\varphi+P ( I-R ) ( I+R ) R^{N-2} \bigl( -R^{2}+4R-I \bigr) \\ &\quad{} -D\sum_{j=1}^{N-1} \bigl[ 4 \bigl( R^{ \vert N-1-j \vert }-R^{N-1+j} \bigr) + \bigl( R^{ \vert N-2-j \vert }-R^{N-2+j} \bigr) \bigr] f_{j}\tau,\end{aligned} \end{aligned}$$
(40)
$$\begin{aligned}& \begin{aligned}[b] F_{4}&=2\tau\psi+P ( I-R ) \Biggl[ 3I-R-R^{N-2} ( I-3R ) \\ &\quad{} -\sum_{i=1}^{q}k_{i} \bigl[ R^{N-l_{i}-1} \bigl( 3I-R+\mu _{i} ( I-R ) \bigr) -R^{2N+l_{i}-1} \bigl( I-3R+\mu_{i} ( I-R ) \bigr) \bigr] \Biggr] \\ &\quad{} -D\sum_{j=1}^{N-1} \Biggl[ -4 \bigl( R^{ \vert 1-j \vert }-R^{1+j} \bigr) + \bigl( R^{ \vert 2-j \vert }-R^{2+j} \bigr) \\ &\quad{} +\sum_{i=1}^{q}k_{i} \biggl[ \biggl( 3+2\frac{\mu_{i}}{\tau} \biggr) \bigl( R^{ \vert l_{i}+1-j \vert }-R^{l_{i}+1+j} \bigr) + \biggl( -4+4\frac{\mu_{i}}{\tau} \biggr) \bigl( R^{ \vert l_{i}-j \vert }-R^{l_{i}+j} \bigr) \\ &\quad{} + \biggl( 1+2\frac{\mu_{i}}{\tau} \biggr) \bigl( R^{ \vert l_{i}-j-1 \vert }-R^{l_{i}+j-1} \bigr) \biggr] \Biggr] f_{j} \tau,\end{aligned} \\& \begin{aligned}[b] \Delta_{2}&= \bigl[ R-3I+R^{2N-2}(I-3R) \bigr] \Biggl[ 3I-R-R^{2N-2} ( I-3R ) \\ &\quad{} -\sum_{i=1}^{q}k_{i} \bigl[ R^{N-l_{i}-1} \bigl( 3I-R+\mu _{i} ( I-R ) \bigr) -R^{2N+l_{i}-1} \bigl( I-3R+\mu_{i} ( I-R ) \bigr) \bigr] \Biggr] \\ &\quad{} - ( I+R ) R^{N-2} \bigl( -R^{2}+4R-I \bigr) \Biggr[ ( I+R ) R^{N-2} \bigl( R^{2}-4R+I \bigr) \\ &\quad{} -\sum_{i=1}^{q}k_{i} \bigl[ R^{l_{i}-1} \bigl( I-3R+\mu_{i} ( I-R ) \bigr) -R^{2N-l_{i}-1} \bigl( 3I-R+\mu_{i} ( I-R ) \bigr) \bigr] \Biggr]. \end{aligned} \end{aligned}$$
(41)

Since the operator \(\Delta_{2}\) has bounded inverse, we obtain the solution of (39) in the following form:

$$ \begin{gathered} v_{0}=\Delta_{2}^{-1} ( I-R )^{-1} \bigl( I-R^{2N} \bigr) \\ \hphantom{v_{0}=}{} \times \Biggl\{ \bigl[ 3I-R-R^{2N-2} ( I-3R ) -\sum _{i=1}^{q}k_{i} \bigl[ R^{N-l_{i}-1} \bigl( 3I-R+\mu_{i} ( I-R ) \bigr) \\ \hphantom{v_{0}=}{} -R^{2N+l_{i}-1} \bigl( I-3R+ \mu_{i} ( I-R ) \bigr) \bigr] \bigr] F_{3}- ( I+R ) R^{N-2} \bigl( -R^{2}+4R-I \bigr) F_{4} \Biggr\} , \\ v_{N}=\Delta_{2}^{-1} ( I-R )^{-1} \bigl( I-R^{2N} \bigr) \Biggl\{ - \Biggl[ ( I+R ) R^{N-2} \bigl( R^{2}-4R+I \bigr) \\ \hphantom{v_{N}=}{} -\sum_{i=1}^{q}k_{i} \bigl[ R^{l_{i}-1} \bigl( I-3R+\mu _{i} ( I-R ) \bigr) -R^{2N-l_{i}-1} \bigl( 3I-R+\mu_{i} ( I-R ) \bigr) \bigr] \Biggr] F_{3} \\ \hphantom{v_{N}=}{} + \bigl[ R-3I+R^{2N-2}(I-3R) \bigr] F_{4} \Biggr\} .\end{gathered} $$
(42)

Thus, problem (34) has a unique solution \(\{ v_{k} \} _{k=0}^{N}\) and it is defined by equations (25), (42). Applying equations (25), (37), (42) and the method of [28], we get

$$\begin{aligned}& \bigl\Vert \{ v_{k} \} _{k=1}^{N-1} \bigr\Vert _{C_{\tau }(H)}\leq M ( \delta,\lambda_{1},\ldots, \lambda_{q} ) \bigl[ P_{H}^{0} ( \varphi,\psi, \zeta ) + \Vert f_{\tau } \Vert _{C_{\tau}(H)} \bigr] , \end{aligned}$$
(43)
$$\begin{aligned}& \begin{aligned}[b] &\bigl\Vert \bigl\{ \tau^{-2}(v_{k+1}-2v_{k}+v_{k-1}) \bigr\} _{k=1}^{N-1} \bigr\Vert _{C_{\tau}^{\alpha,\alpha}(H)}+ \bigl\Vert \{ Av_{k} \} _{k=1}^{N-1} \bigr\Vert _{C_{\tau}^{\alpha ,\alpha }(H)} \\ &\quad \leq M ( \delta,\lambda_{1},\ldots,\lambda_{q} ) \biggl[ P_{H}^{1} ( \varphi,\psi,\zeta ) +\frac{1}{\alpha(1-\alpha )} \Vert f_{\tau} \Vert _{C_{\tau}^{\alpha,\alpha }(H)} \biggr], \end{aligned} \end{aligned}$$
(44)

for the solutions of both difference problems (34), (38). The proofs of the estimates (32), (33) for the solutions of the difference problems (34), (38) are based on equation (11) and estimates (43), (44). By using equation (11) and estimates (43), (32), we can get estimate (31). □

Denote by F an SAPD operator in a Hilbert space H, introduce \(E_{\alpha }=E_{\alpha} ( D ( F ) ,H ) \), the Banach space of such functions \(u\in H\) for which the norm

$$\Vert u \Vert _{E_{\alpha}}=\sup_{z>0}z^{1-\alpha} \bigl\Vert Fe^{-zF}u \bigr\Vert _{H} + \Vert u \Vert _{H} $$

is finite.

Theorem 4

Under the assumptions \(\varphi,\psi,\zeta\in D(C)\), \(f_{\tau}\in C_{\tau}^{\alpha,\alpha}(H)\) (\(\alpha\in(0,1)\)) the solution \(( p, \{ u_{k} \} _{k=1}^{N-1} ) \) of the difference problems (28) and (30) obeys the coercive stability estimate

$$ \begin{aligned}[b] & \biggl\Vert \biggl\{ \frac{u_{k+1}-2u_{k}+u_{k-1}}{\tau^{2}} \biggr\} _{k=1}^{N-1} \biggr\Vert _{C_{\tau}(H)}+ \bigl\Vert \{ Au_{k} \} _{k=1}^{N-1} \bigr\Vert _{C_{\tau}(H)}+ \Vert p \Vert _{H} \\ &\quad \leq M ( \delta,\lambda_{1},\ldots,\lambda_{q} ) \biggl[ \Vert C\varphi \Vert _{E_{\alpha}}+ \Vert C\psi \Vert _{E_{\alpha}}+ \Vert C\zeta \Vert _{E_{\alpha}}+\frac {1}{\alpha ( 1-\alpha ) } \Vert f_{\tau} \Vert _{C_{\tau}(H)} \biggr] ,\end{aligned} $$
(45)

where \(M ( \delta,\lambda_{1},\ldots,\lambda_{q} ) \) is independent from parameters τ, α, elements φ, ψ, ξ, and the grid function \(f_{\tau}\).

4 Multidimensional elliptic problem

Now, we give the application of the abstract Theorems 1 and 3.

Let \(\Omega=(0,\ell)\times\cdots\times(0,\ell)\) be the open cube in the n-dimensional Euclidean space with boundary S, \(\overline {\Omega}=\Omega\cup S \) and numbers \(\lambda_{0}\), \(\lambda_{i}\), \(i=1,\ldots,q\) (\(0<\lambda_{1}<\lambda_{2}<\cdots<\lambda_{q}<T\)) and nonnegative real coefficients \(k_{i}\), \(i=1,\ldots,q\), be given, condition (2) for coefficients be valid, \(x=(x_{1},\ldots,x_{n})\). Consider the inverse problem of finding functions \(p(x)\) and \(u(t,x)\) for the multidimensional elliptic equation in \([0,T]\times\Omega\),

$$ \textstyle\begin{cases} -u_{tt}(t,x)-\sum_{r=1}^{n}(a_{r}(x)u_{x_{r}})_{x_{r}}+\sigma u(t,x)=f(t,x)+p(x), \\ \quad 0< t< T, x\in\Omega, \\ u_{t}(0,x)=\varphi(x), \quad\quad u_{t}(T,x)=\sum_{i=1}^{q}k_{i}u_{t}(\lambda _{i},x)+\psi(x), \\ u(\lambda_{0},x)=\zeta(x),\quad x\in\overline{\Omega}, \\ u(t,x)=0, \quad x\in S,0\leq t\leq T.\end{cases} $$
(46)

Here, \(\sigma>0\) is a known number, \(a_{r}(x)\) (\(x\in\Omega\)), \(\varphi (x)\), \(\psi(x)\), \(\zeta(x)\) (\(x\in\overline{\Omega}\)), and \(f(t,x)\) (\(t\in (0,1)\), \(x\in\Omega\)) are given smooth functions, \(a_{r}(x)\geq a>0\) (\(x\in \Omega\)).

It is well known that the differential expression [28]

$$ A^{x}u(x)=-\sum_{r=1}^{n} \bigl(a_{r}(x)u_{x_{r}} \bigr)_{x_{r}}+\sigma u(x) $$
(47)

defines a self-adjoint positive definite operator \(A^{x}\) acting on \(L_{2}(\overline{\Omega})\) with the domain \(D(A^{x})= \{ u(x)\in W_{2}^{2}(\overline{\Omega}), u=0\text{ on }S \} \).

Let H be the Hilbert space \(L_{2}(\overline{\Omega})\). Denote by \(C_{0T}^{\alpha,\alpha}(L_{2}(\overline{\Omega}))\) the space obtained by completion of the space of all smooth \(L_{2}(\overline{\Omega})\)-valued functions ρ on \([0,1]\) with the norm

$$\Vert \rho \Vert _{C_{0T}^{\alpha,\alpha}(L_{2}(\overline{\Omega}))} = \Vert \rho \Vert _{C(L_{2}(\overline{\Omega}))}+ \sup_{0\leq t< t+\tau\leq1} \frac{(t+\tau)^{\alpha}(1-t)^{\alpha } \Vert \rho(t+\tau)-\rho(t) \Vert _{L_{2}(\overline{\Omega})}}{\tau ^{\alpha}}. $$

Applying the abstract Theorems 1 and 3, we get the following estimates for the solution of problem (46).

Theorem 5

Assume that \(A^{x}\) is defined by equation (47), condition (2) for coefficients is valid, \(f\in \mathcal{C}_{0T}^{\alpha,\alpha}(L_{2}(\overline{\Omega})) \) and \(\varphi ,\zeta,\psi\in D(A^{x})\cap D ( ( A^{x} ) ^{-\frac{1}{2} } ) \). Then, for the solutions \((p,u)\) of the inverse boundary value problem (46), the stability estimates

$$\begin{aligned}& \Vert u \Vert _{C(L_{2}(\overline{\Omega}))}\leq M(\delta ) \bigl[ P_{L_{2}(\overline{\Omega})}^{0} ( \varphi,\psi,\zeta ) +P_{L_{2}(\overline{\Omega})}^{-\frac{1}{2}} ( \varphi,\psi,\zeta ) + \Vert f \Vert _{C(L_{2}(\overline{\Omega}))} \bigr], \\& \bigl\Vert \bigl(A^{x} \bigr)^{-1}p \bigr\Vert _{L_{2}(\overline{\Omega})}\leq M(\delta) \bigl[ P_{L_{2}(\overline{\Omega})}^{0} ( \varphi, \psi ,\zeta ) + \Vert f \Vert _{C(L_{2}(\overline{\Omega}))} \bigr] , \\& \Vert p \Vert _{L_{2}(\overline{\Omega})}\leq M(\delta ) \biggl[ P_{L_{2}(\overline{\Omega})}^{1} ( \varphi,\psi,\zeta ) +\frac{1}{\alpha(1-\alpha)} \Vert f \Vert _{C_{0T}^{\alpha,\alpha }(L_{2}(\overline{\Omega}))} \biggr], \end{aligned}$$

are satisfied, where \(M(\delta)\) is independent of α, \(\varphi (x)\), \(\zeta(x)\), \(\psi(x)\), and \(f(t,x)\).

Now, we will discretize problem (46) into two steps. In the first step, we define the grid spaces

$$\begin{aligned}& \widetilde{\Omega}_{h} = \bigl\{ x=x_{m}=(h_{1}m_{1}, \ldots ,h_{n}m_{n});m=(m_{1}, \ldots,m_{n}), \\& \hphantom{\widetilde{\Omega}_{h} =} m_{r} =0,\ldots,M_{r}, h_{r}M_{r}= \ell,r=1,\ldots,n \bigr\} , \\& \Omega_{h}=\widetilde{\Omega}_{h}\cap \Omega,\quad\quad S_{h}=\widetilde{\Omega} _{h}\cap S. \end{aligned}$$

To the differential operator \(A^{x}\) generated by problem (46) we assign the difference operator \(A_{h}^{x}\) defined by the formula

$$A_{h}^{x} u^{h}(x)=-\sum _{r=1}^{n} \bigl( a_{r}(x)u_{\overline{x}_{r}}^{h} \bigr) _{x_{r},j_{r}} $$

acting in the space of grid functions \(u^{h}(x)\), satisfying the condition \(u^{h}(x)=0\) for all \(x\in S_{h}\). It is well known that \(A_{h}^{x}\) is a self-adjoint positive definite operator.

By using \(A_{h}^{x}\), we arrive at the following BVP:

$$ \textstyle\begin{cases} -\frac{d^{2}u^{h}(t,x)}{dt^{2}}+A_{h}^{x}u^{h}(t,x)=f^{h}(t,x)+p^{h}(x),\quad 0< t< T, x\in\Omega_{h}, \\ \frac{du^{h}}{dt}(0,x)=\varphi^{h}(x), \\ \frac{du^{h}}{dt}(T,x)-\sum_{i=1}^{q}k_{i}\frac{du^{h}}{dt}(\lambda _{i},x)=\psi^{h}(x),\quad x\in\widetilde{\Omega}_{h}, \end{cases} $$
(48)

for a system of ordinary differential equations.

In the second step of discretization, problem (48) is replaced by the following difference schemes:

$$\begin{aligned}& \textstyle\begin{cases} -\frac{u_{k+1}^{h}(x)-2u_{k}^{h}(x)+v_{k-1}^{h}(x)}{\tau^{2}}+A_{h}^{x}u_{k}^{h}(x)=f_{k}^{h}(x)+p^{h}(x), \\ f_{k}^{h}(x)=f^{h}(t_{k},x),\quad t_{k}=k\tau, 1\leq k\leq N-1, x\in\Omega _{h}, \\ u_{1}^{h}(x)-u_{0}^{h}(x) =\tau\varphi^{h}(x), \\ u_{N}^{h}(x)-u_{N-1}^{h}(x)-\sum_{i=1}^{q}k_{i} ( u_{l_{i}+1}^{h}(x)-u_{l_{i}}^{h}(x) ) =\tau\psi^{h}(x), \\ u_{l_{o}}^{h}(x)=\zeta^{h}(x),\quad x\in\widetilde{\Omega}_{h},\end{cases}\displaystyle \end{aligned}$$
(49)
$$\begin{aligned}& \textstyle\begin{cases} -\frac{u_{k+1}^{h}(x)-2u_{k}^{h}(x)+v_{k-1}^{h}(x)}{\tau^{2}}+A_{h}^{x}u_{k}^{h}(x)=f_{k}^{h}(x)+p^{h}(x), \\ f_{k}^{h}(x)=f^{h}(t_{k},x),\quad t_{k}=k\tau, 1\leq k\leq N, x\in\Omega_{h}, \\ -3u_{0}^{h}(x)+4u_{1}^{h}(x)-2u_{2}^{h}(x)=2\tau\varphi^{h}(x), \\ 3u_{N}^{h}(x)-4u_{N-1}^{h}(x)+u_{N-2}^{h}(x) \\ \quad \quad{} -\sum_{i=1}^{q}k_{i} [ ( 3+2\mu_{i} ) u_{l_{i}+1}^{h}(x)- ( 4+4\mu_{i} ) u_{l_{i}}^{h}(x)+ ( 1+2\mu_{i} ) u_{l_{i}-1}^{h}(x) ] \\ \quad =2\tau\psi^{h}(x),u_{l_{o}}^{h}(x)+\mu _{i}(u_{l_{o}+1}^{h}(x)-u_{l_{o}}^{h}(x))=\zeta(x),\quad x\in\widetilde {\Omega}_{h}.\end{cases}\displaystyle \end{aligned}$$
(50)

For the calculation of \(p^{h}(x)\) we have

$$ p^{h}(x)=A_{h}^{x}\zeta^{h}(x)-A_{h}^{x}v^{h}( \lambda_{0},x),\quad x\in \widetilde{\Omega}_{h}. $$
(51)

To formulate our results, let \(L_{2h}=L_{2}(\widetilde{\Omega}_{h})\) and \(W_{2h}^{2}=W_{2}^{2}(\widetilde{\Omega}_{h})\) be spaces of the grid functions \(\rho^{h}(x)=\{\rho(h_{1}m_{1},\ldots,h_{n}m_{n})\}\) defined on \(\widetilde{\Omega}_{h}\), equipped with the norms

$$\begin{aligned}& \bigl\Vert \rho^{h} \bigr\Vert _{L_{2h}}= \biggl(\sum _{x\in\widetilde{\Omega }_{h}} \bigl\vert \rho^{h}(x) \bigr\vert ^{2}h_{1}\cdots h_{n} \biggr)^{1/2}, \\& \bigl\Vert \rho^{h} \bigr\Vert _{W_{2h}^{2}} = \bigl\Vert \rho ^{h} \bigr\Vert _{L_{2h}}+ \Biggl(\sum _{x\in\widetilde{\Omega}_{h}}\sum_{r=1}^{n} \bigl\vert \bigl(\rho^{h} \bigr)_{x_{r}} \bigr\vert ^{2}h_{1}\cdots h_{n} \Biggr)^{1/2} \\& \hphantom{\bigl\Vert \rho^{h} \bigr\Vert _{W_{2h}^{2}} =}{} + \Biggl(\sum_{x\in\widetilde{\Omega}_{h}}\sum _{r=1}^{n} \bigl\vert \bigl(\rho ^{h}(x) \bigr)_{x_{r}\overline{x_{r}}, m_{r}} \bigr\vert ^{2}h_{1} \cdots h_{n} \Biggr)^{1/2}. \end{aligned}$$

Let τ and \(\vert h \vert =\sqrt{h_{1}^{2}+\cdots +h_{n}^{2}}\) be sufficiently small positive numbers.

Theorem 6

For the solutions of the difference schemes (49) and (50) the following stability estimates hold:

$$\begin{aligned}& \begin{aligned} \bigl\Vert \bigl\{ u_{k}^{h} \bigr\} _{1}^{N-1} \bigr\Vert _{\mathcal{C} _{\tau}(L_{2h})} &\leq M ( \delta, \lambda_{1},\ldots,\lambda _{q} ) \bigl[ \bigl\Vert \varphi^{h} \bigr\Vert _{L_{2h}}+ \bigl\Vert \psi^{h} \bigr\Vert _{L_{2h}} \\ &\quad {} + \bigl\Vert \zeta^{h} \bigr\Vert _{L_{2h}}+ \bigl\Vert \bigl\{ f_{k}^{h} \bigr\} _{1}^{N-1} \bigr\Vert _{C_{\tau}(L_{2h})} \bigr] , \end{aligned} \\& \begin{aligned} \bigl\Vert p^{h} \bigr\Vert _{L_{2h}} &\leq M ( \delta, \lambda _{1},\ldots,\lambda_{q} ) \biggl[ \bigl\Vert \varphi^{h} \bigr\Vert _{W_{2h}^{2}}+ \bigl\Vert \psi^{h} \bigr\Vert _{W_{2h}^{2}}+ \bigl\Vert \zeta ^{h} \bigr\Vert _{W_{2h}^{2}} \\ &\quad{} +\frac{1}{\alpha(1-\alpha)} \bigl\Vert \bigl\{ f_{k}^{h} \bigr\} _{1}^{N-1} \bigr\Vert _{\mathbf{C}_{\tau}(L_{2h})} \biggr], \end{aligned} \end{aligned}$$

where \(M ( \delta,\lambda_{1},\ldots,\lambda_{q} ) \) is independent of τ, α, h, \(\varphi^{h}(x)\), \(\psi^{h}(x)\), \(\zeta^{h}(x)\) and \(\{ f_{k}^{h}(x) \} _{1}^{N-1}\).

The proof of Theorem 6 is based on the symmetry property of the operator \(A_{h}^{x}\) in \(L_{2h}\) and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in \(L_{2h}\).

Theorem 7

[30]

For the solution of the elliptic difference problem

$$ \textstyle\begin{cases} A_{h}^{x} u^{h}(x)=\omega^{h}(x), \quad x\in\widetilde{\Omega}_{h}, \\ u^{h}(x)=0, \quad x\in S_{h}, \end{cases} $$
(52)

the following coercivity inequality holds:

$$\sum_{r=1}^{n} \bigl\Vert \bigl(u_{k}^{h} \bigr)_{\overline{x}_{r}\overline{x}_{r},j_{r}} \bigr\Vert _{L_{2h}}\leq M \bigl\Vert \omega^{h} \bigr\Vert _{L_{2h}}, $$

where M is independent of h and ω.

5 Numerical results

In this section, we present numerical results with explanation on the realization for two dimensional and three dimensional examples of the Bitsadze-Samarskii type overdetermined elliptic multipoint NBVP. The MATLAB program is used to get numerical results.

5.1 Two dimensional example

It is easy to check that pair functions \(p(x)= [ (1+x)\pi^{2}+1 ] \sin(\pi x)-\pi\cos(\pi x)\) and \(u(t,x)= ( \exp ( -t ) +1 ) \sin(\pi x)\) is an exact solution of the following two dimensional overdetermined elliptic three-point NBVP:

$$ \textstyle\begin{cases} -\frac{\partial^{2}u}{\partial t^{2}}(t,x)-\frac{\partial}{\partial x} ( (1+x)\frac{\partial u}{\partial x}(t,x) ) +u(t,x)=f(t,x)+p(x), \\ \quad 0< x< 1,0< t< 1, \\ u_{t}(1,x)-\frac{1}{4}u_{t}(\frac{1}{4},x)-\frac{1}{4}u_{t}(\frac {1}{2},x)-\frac{1}{4}u_{t}(\frac{3}{4},x)=\psi(x), \\ u_{t}(0,x)=\varphi(x), \quad\quad u(\frac{2}{5},x)=\zeta(x),\quad 0\leq x\leq1, \\ u(t,0)=u(t,1)=0, \quad 0\leq t\leq1,\end{cases} $$
(53)

where

$$\begin{aligned}& f(t,x)=(1+x)\pi^{2}e^{-t}\sin(\pi x)-\pi e^{-t} \cos(\pi x), \\& \psi(x)= \biggl[ -e^{-1}+\frac{1}{4} \bigl( e^{-\frac{1}{4}}+e^{-\frac {1}{2}}+e^{-\frac{3}{4}} \bigr) \biggr] \sin(\pi x), \\& \zeta(x)= \bigl( e^{-\frac{2}{5}}+1 \bigr) \sin(\pi x). \end{aligned}$$

Denote by \([ 0,1 ] _{\tau}\times [ 0,1 ] _{h}\) the set of grid points depending on the small parameters τ and h

$$[ 0,1]_{\tau}\times[ 0,1]_{h}= \bigl\{ (t_{k},x_{n}):t_{k}=k \tau , k=\overline{0,N}, x_{n}=nh, n=\overline{0,M} \bigr\} , $$

where \(N \tau=1\), \(Mh=1\). Let us

$$\begin{aligned}& \lambda_{0}=\frac{2}{5},\quad\quad \lambda_{1}= \frac{1}{4},\quad\quad \lambda_{2}=\frac {1}{2},\quad\quad \lambda_{3}=\frac{3}{4}; \\& l_{i}= \biggl[ \frac{\lambda_{i}}{\tau} \biggr] ,\quad\quad \mu_{i}= \frac{\lambda _{i}}{\tau}-l_{i} ,\quad i=0,1,2,3; \\& \varphi_{n}=\varphi ( x_{n} ) ,\quad\quad \psi_{n}=\psi ( x_{n} ) ,\quad\quad \zeta_{n}=\zeta ( x_{n} ) , \quad n= \overline{0,M}; \\& f_{n}^{k}=f(t_{k},x_{n}),\quad k= \overline{0,N}, n=\overline{0,M}. \end{aligned}$$

To approximately solve (53), we use an algorithm with three stages. In the first stage, we find a numerical solution of the auxiliary NBVP. We can write

$$ \textstyle\begin{cases} \frac{v_{n}^{k+1}-2v_{n}^{k}+v_{n}^{k-1}}{\tau^{2}}+(1+x_{n})\frac{v_{n+1}^{k}-2v_{n}^{k}+v_{n-1}^{k}}{h^{2}}+\frac {v_{n+1}^{k}-v_{n-1}^{k}}{2h}=f(t_{k},x_{n}), \\ \quad k=\overline{1,N-1}, n=\overline{1,M-1}, \\ v_{0}^{k}=v_{M}^{k}=0, \quad k=0,\ldots,N, \quad\quad v_{n}^{1}-v_{n}^{0}=\tau\varphi _{n}, \\ v_{n}^{N}-v_{n}^{N-1}-\frac{1}{4} ( v_{n}^{l_{1}+1}-v_{n}^{l_{1}} ) -\frac{1}{4} ( v_{n}^{l_{2}+1}-v_{n}^{l_{2}} ) -\frac {1}{4} ( v_{n}^{l_{3}+1}-v_{n}^{l_{3}} ) =\psi_{n}, \quad n=\overline{0,M},\end{cases} $$
(54)

the first order of accuracy in t and the second order of accuracy in x and

$$ \textstyle\begin{cases} \frac{v_{n}^{k+1}-2v_{n}^{k}+v_{n}^{k-1}}{\tau^{2}}+(1+x_{n})\frac{v_{n+1}^{k}-2v_{n}^{k}+v_{n-1}^{k}}{h^{2}}+\frac {v_{n+1}^{k}-v_{n-1}^{k}}{2h}=f(t_{k},x_{n}), \\ \quad k=\overline{1,N-1}, n=\overline{1,M-1}, \\ v_{0}^{k}=v_{M}^{k}=0, \quad k=\overline{0,N},\quad\quad -3v_{n}^{0}+4v_{n}^{1}-v_{n}^{2}=2\tau\varphi_{n}, \\ 3v_{n}^{N}-4v_{n}^{N-1}+v_{n}^{N-2}-\frac{1}{4} [ ( 3+2\mu _{1} ) v_{n}^{l_{1}+1}- ( 4+4\mu_{1} ) v_{n}^{l_{1}}+ ( 1+2\mu_{1} ) v_{n}^{l_{1}} ] \\ \quad{} -\frac{1}{4} [ ( 3+2\mu_{2} ) v_{n}^{l_{2}+1}- ( 4+4\mu _{2} ) v_{n}^{l_{2}}+ ( 1+2\mu_{2} ) v_{n}^{l_{2}} ] \\ \quad{} -\frac{1}{4} [ ( 3+2\mu_{3} ) v_{n}^{l_{1}+1}- ( 4+4\mu _{3} ) v_{n}^{l_{3}}+ ( 1+2\mu_{3} ) v_{n}^{l_{3}} ] =2\tau\psi_{n}, \quad n=\overline{0,M},\end{cases} $$
(55)

the second order of accuracy in t and the second order of accuracy in x difference schemes for corresponding NBVP.

In the second stage, we find \(p_{n}\). It is carried out by

$$\begin{aligned} p_{n}&=-(1+x_{n})\frac{(\zeta_{n+1}-v_{n+1}^{l_{0}})-2(\zeta _{n}-v_{n}^{l_{0}})+(\zeta_{n-1}-v_{n-1}^{l_{0}})}{h^{2}} \\ &\quad{} -\frac{(\zeta_{n+1}-v_{n+1}^{l_{0}})-(\zeta _{n-1}-v_{n-1}^{l_{0}})}{2h}+v_{n}^{l_{0}},\quad n= \overline{1,M-1}, \end{aligned}$$

for the first order approximation, and

$$\begin{aligned} p_{n}&=-\frac{1+x_{n}}{h^{2}} \bigl\{ \bigl[ \zeta_{n+1}- \bigl( \mu _{0}v_{n+1}^{l_{0}+1}- ( \mu_{0}-1 ) v_{n+1}^{l_{0}} \bigr) \bigr] \\ &\quad{} -2 \bigl[ \zeta_{n}- \bigl( \mu_{0}v_{n}^{l_{0}+1}- ( \mu _{0}-1 ) v_{n}^{l_{0}} \bigr) \bigr] \\ &\quad{} + \bigl[ \zeta_{n-1}- \bigl( \mu_{0}v_{n-1}^{l_{0}+1}- ( \mu _{0}-1 ) v_{n-1}^{l_{0}} \bigr) \bigr] \bigr\} \\ &\quad{}-\frac{1}{2h} \bigl\{ \bigl[ \zeta_{n+1}- \bigl( \mu _{0}v_{n+1}^{l_{0}+1}- ( \mu_{0}-1 ) v_{n+1}^{l_{0}} \bigr) \bigr] \\ &\quad{} - \bigl[ \zeta_{n-1}- \bigl( \mu_{0} v_{n-1}^{l_{0}+1}- ( \mu _{0}-1 ) v_{n-1}^{l_{0}} \bigr) \bigr] \bigr\} \\ &\quad{}+\zeta_{n}- \bigl( \mu_{0}v_{n}^{l_{0}+1}- ( \mu_{0}-1 ) v_{n}^{l_{0}} \bigr) ,\quad n= \overline{1,M-1}, \end{aligned}$$

for the second order approximation.

Both difference problems (54) and (55) can be rewritten in the matrix form

$$ \begin{gathered} A_{n}v_{n+1}+B_{n}v_{n}+C_{n}v_{n-1}=Ig_{n}, \quad n=\overline{1,M-1}, \\ v_{0}=\overrightarrow{0},\quad\quad v_{M}= \overrightarrow{0}.\end{gathered} $$
(56)

Here, \(g_{n}\) is \((N+1)\times1\) a column matrix, \(A_{n}\), \(B_{n}\), \(C_{n}\) are \((N+1)\times(N+1)\) square matrices, and I is the \((N+1)\times(N+1)\) identity matrix, \(v_{s}\) is the \((N+1)\times1\) matrix \(v_{s}= [ v_{s}^{0} \ \cdots\ v_{s}^{N} ] ^{t}\), \(s=n-1,n,n+1\). Denote

$$a_{n}=\frac{1+x_{n}}{h^{2}}+\frac{1}{2h},\quad\quad c_{n}= \frac {1+x_{n}}{h^{2}}-\frac{1}{2h},\quad\quad b_{n}=-\frac{2}{\tau^{2}}- \frac{2(1+x_{n})}{h^{2}},\quad \quad d=\frac {1}{\tau ^{2}}. $$

Then

$$\begin{aligned}& A_{n}=\operatorname{diag}(0,a_{n},a_{n},\ldots ,a_{n},0),\quad\quad C_{n}=\operatorname{diag}(0,c_{n},c_{n}, \ldots ,c_{n},0), \\& g_{n}^{k}=-f(t_{k},x_{n}), \quad k= \overline{1,N-1}, n=\overline{1,M-1},\\& B_{n} [ i,i ] =b_{n},\quad\quad B_{n} [ i-1,i ] =d,\quad\quad B_{n} [ i,i-1 ] =d,\quad i=\overline{2,N}, \\& B_{n} [ 1,1 ] =-1,\quad\quad B_{n} [ 1,2 ] =1,\quad\quad B_{n} [ N+1,N+1 ] =1, \\& B_{n} [ N+1,N ] =-1, \\& B_{n} [ N+1,l_{1} ] =-\frac{1}{4},\quad\quad B_{n} [ N+1,l_{2} ] =-\frac{1}{4},\quad\quad B_{n} [ N+1,l_{3} ] =-\frac{1}{4} \\& B_{n} [ N+1,l_{1}+1 ] =\frac{1}{4},\quad\quad B_{n} [ N+1,l_{2}+1 ] =\frac{1}{4}, \\& B_{n} [ N+1,l_{3}+1 ] =\frac{1}{4}, \\& B_{n} [ i,j ] =0, \quad \text{for other }i\text{ and }j, \\& g_{n}^{0}=\tau\varphi_{n},\quad\quad g_{n}^{N}=\tau\psi_{n},\quad n=\overline{1,M-1}, \end{aligned}$$

for the first order approximation, and

$$\begin{aligned}& B_{n} [ i,i ] =b_{n},\quad\quad B_{n} [ i-1,i ] =d,\quad\quad B_{n} [ i,i-1 ] =d,\quad i=\overline{2,N}, \\& B_{n} [ 1,1 ] =-3,\quad\quad B_{n} [ 1,2 ] =4,\quad\quad B_{n} [ 1,3 ] =-1,\quad\quad B_{n} [ N+1,N+1 ] =3, \\& B_{n} [ N+1,N ] =-4,\quad\quad B_{n} [ N+1,N-1 ] =-1, \\& B_{n} [ N+1,l_{1}+1 ] =-\frac{1}{4} ( 3+2 \mu_{1} ) ,\quad\quad B_{n} [ N+1,l_{1}+1 ] = \frac{1}{4} ( 3+2\mu_{1} ) , \\& B_{n} [ N+1,l_{1}-1 ] =-\frac{1}{4} ( 1+2 \mu_{1} ) ,\quad\quad B_{n} [ N+1,l_{2}+1 ] =- \frac{1}{4} ( 3+2\mu_{2} ) , \\& B_{n} [ N+1,l_{2}+1 ] =\frac{1}{4} ( 3+2 \mu_{2} ) ,\quad\quad B_{n} [ N+1,l_{2}-1 ] =- \frac{1}{4} ( 1+2\mu_{2} ) , \\& B_{n} [ N+1,l_{3}+1 ] =-\frac{1}{4} ( 3+2 \mu_{3} ) ,\quad\quad B_{n} [ N+1,l_{3}+1 ] = \frac{1}{4} ( 3+2\mu_{3} ) , \\& B_{n} [ N+1,l_{3}-1 ] =-\frac{1}{4} ( 1+2 \mu_{3} ) , \\& B_{n} [ i,j ] =0, \quad \text{for other }i\text{ and }j, \\& g_{n}^{0}=2\tau\varphi_{n},\quad\quad g_{n}^{N}=2\tau\psi_{n}, \quad n=\overline{1,M-1}, \end{aligned}$$

for the second order approximation.

Finally, in the third stage, we calculate \(\{ u_{n}^{k} \} \) by \(u_{n}^{k}=v_{n}^{k}+\zeta_{n}-v_{n}^{l_{0}}\), and \(u_{n}^{k}=v_{n}^{k}+ \zeta_{n}- ( \mu_{0}v_{n}^{l_{0}+1}- ( \mu_{0}-1 ) v_{n}^{l_{0}} ) \), for the first and second order approximations, respectively.

To solve (56), we use a modification of the Gauss elimination method ([31]). Let \(v_{M}=\overrightarrow{0}\), \(\alpha_{n}\) (\(n=1,\ldots,M-1\)) be \((N+1)\times(N+1)\) square matrices and \(\beta_{n}\) (\(n=1,\ldots,M-1\)) be \((N+1)\times1\) column vectors, \(\alpha_{1}\) be the zero matrix and \(\beta_{1}\) be zero column vector. Searching the solution of (56) by

$$v_{n}=\alpha_{n+1}v_{n+1}+\beta_{n+1},\quad n=M-1, \ldots,1, $$

we get formulas for \(\alpha_{n+1}\), \(\beta_{n+1}\):

$$\begin{aligned}& \alpha_{n+1} = -(B_{n}+C_{n} \alpha_{n})^{-1}A_{n}, \\& \beta_{n+1} = -(B_{n}+C_{n} \alpha_{n})^{-1}(I\theta_{n}-C_{n} \beta _{n}),\quad n=1,\ldots,M-1. \end{aligned}$$

Results of numerical calculations are presented in Tables 1-3 for both the first and second order approximations in the cases \(N=M=20, 40, 80\) and 160. Table 1 gives the error between the exact solution of NBVP and the solutions derived by difference schemes. Table 2 contains the error between exact and approximate u. Tables 3 includes the error for p. It can be seen from Tables 1-3 that the second order of the ADS is more accurate compared to the first order of the ADS.

Table 1 Error of v in example ( 53 )
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Table 2 Error of u in example ( 53 )
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Table 3 Error of p in example ( 53 )
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5.2 Three dimensional example

Now, consider the three dimensional overdetermined elliptic two point NBVP

$$ \textstyle\begin{cases} -\frac{\partial^{2}u}{\partial t^{2}}(t,x,y)-\frac{\partial ^{2}u}{\partial x^{2}}(t,x,y)-\frac{\partial^{2}u}{\partial y^{2}}(t,x,y)+u(t,x,y)=f(t,x,y)+p(x,y), \\ \quad 0< x< 1,0< y< 1,0< t< 1, \\ u_{t}(0,x,y)=\varphi(x,y), \\ u_{t}(1,x,y)-\frac{1}{3}u_{t}(\frac{1}{3},x,y)-\frac{1}{3}u_{t}(\frac {2}{3},x,y)=\psi(x,y), \\ u(\frac{3}{5},x,y)=\zeta(x,y), \\ \quad 0\leq x\leq1,0\leq y\leq1,u(t,0)=u(t,1)=0, 0\leq t\leq1,\end{cases} $$
(57)

where

$$\begin{aligned}& f(t,x,y)=2\pi^{2}e^{-t}\sin(\pi x)\sin(\pi y), \\& \varphi(x,y)=-\sin(\pi x)\sin(\pi y), \\& \psi(x,y)= \biggl[ -e^{-1}+\frac{1}{3} \bigl( e^{-\frac{1}{3}}+e^{-\frac {2}{3}} \bigr) \biggr] \sin(\pi x)\sin(\pi y), \\& \zeta(x,y)= \bigl( e^{-\frac{3}{5}}+1 \bigr) \sin(\pi x)\sin(\pi y). \end{aligned}$$

It is easy to see that the pair functions \(p(x,y)= ( 2\pi ^{2}+1 ) \sin(\pi x)\sin(\pi y)\) and \(u(t,x,y)= ( e^{-t}+1 ) \sin(\pi x)\sin(\pi y)\) are an exact solution of (57).

Denote by \([ 0,1 ] _{\tau}\times [ 0,1 ] _{h} \times [ 0,1 ] _{h}\) the set of grid points depending on the small parameters τ and h,

$$\begin{aligned}& [ 0,1]_{\tau}\times[ 0,1]_{h}\times[ 0,1]_{h}= \bigl\{ (t_{k},x_{n},y_{m}):t_{k}=k \tau, k=\overline{0,N}, \\& \hphantom{[ 0,1]_{\tau}\times[ 0,1]_{h}\times[ 0,1]_{h}=} x_{n}=nh, n=\overline{0,M},y_{m}=mh, m=\overline{0,M}, N \tau=1,Mh=1 \bigr\} . \end{aligned}$$

Let us take

$$\begin{aligned}& \lambda_{0}=\frac{3}{5},\quad\quad \lambda_{1}= \frac{1}{4},\quad \quad \lambda_{2}=\frac{3}{4}; \\& l_{i}= \biggl[ \frac{\lambda_{i}}{\tau} \biggr] ,\quad\quad \mu_{i}= \frac{\lambda _{i}}{\tau}-l_{i} ,\quad i=0,1,2; \\& \varphi_{m,n}=\varphi ( x_{n},y_{m} ) ,\quad\quad \psi_{m,n}=\psi ( x_{n},y_{m} ) ,\quad\quad \zeta_{m,n}=\xi ( x_{n},y_{m} ) , \\& \quad n=\overline{0,M}, m=\overline{0,M}; \\& f_{m,n}^{k}=f(t_{k},x_{n},y_{m}),\quad k=\overline{0,N}, n=\overline{0,M,} m= \overline{0,M}. \end{aligned}$$

In the first stage, the difference schemes for the approximate solution of NBVP can be written in the following forms:

$$ \textstyle\begin{cases} -\frac{v_{m,n}^{k+1}-2v_{m,n}^{k}+v_{m,n}^{k-1}}{\tau^{2}}-\frac{v_{m,n+1}^{k}-2v_{m,n}^{k}+v_{m,n-1}^{k}}{h^{2}} \\ \quad{} -\frac{v_{m+1,n}^{k}-2v_{m,n}^{k}+v_{m-1,n}^{k}}{\tau^{2}}+v_{m,n}^{k}=f_{m,n}^{k}, \\ \quad k=\overline{1,N-1}, n=\overline{1,M-1}, m=\overline{1,M-1}, \\ v_{0,n}^{k}=v_{M,n}^{k}=v_{m,n}^{k}=v_{m,M}^{k}=0, \\ \quad k=0,\ldots,N, n=\overline{1,M-1}, m=\overline{1,M-1}, \\ v_{m,n}^{1}-v_{m,n}^{0}=\tau\varphi_{m,n}, \\ v_{m,n}^{N}-v_{m,n}^{N-1}-\frac{1}{3} ( v_{m,n}^{l_{1}+1}-v_{m,n}^{l_{1}} ) -\frac{1}{3} ( v_{m,n}^{l_{2}+1}-v_{n}^{l_{2}} ) =\psi_{m,n}, \\ \quad n=\overline{1,M-1},n=\overline{1,M-1},\end{cases} $$
(58)

and

$$ \textstyle\begin{cases} -\frac{v_{m,n}^{k+1}-2v_{m,n}^{k}+v_{m,n}^{k-1}}{\tau^{2}}-\frac{v_{m,n+1}^{k}-2v_{m,n}^{k}+v_{m,n-1}^{k}}{h^{2}} \\ \quad{} -\frac{v_{m+1,n}^{k}-2v_{m,n}^{k}+v_{m-1,n}^{k}}{\tau^{2}}+v_{m,n}^{k}=f_{m,n}^{k}, \\ \quad k=\overline{1,N-1}, n=\overline{1,M-1}, m=\overline{1,M-1}, \\ v_{0,n}^{k}=v_{M,n}^{k}=v_{m,n}^{k}=v_{m,M}^{k}=0, \\ \quad k=0,\ldots,N, n=\overline{1,M-1}, m=\overline{1,M-1}, \\ -3v_{m,n}^{0}+4v_{m,n}^{1}-v_{m,n}^{2} \\ \quad =2\tau\varphi_{m,n}, 3v_{m,n}^{N}-4v_{m,n}^{N-1}+v_{m,n}^{N-2} \\ \quad \quad{} -\frac{1}{3} [ ( 3+2\mu_{1} ) v_{m,n}^{l_{1}+1}- ( 4+4\mu _{1} ) v_{m,n}^{l_{1}}+ ( 1+2\mu_{1} ) v_{m,n}^{l_{1}} ] \\ \quad \quad{} -\frac{1}{3} [ ( 3+2\mu_{2} ) v_{m,n}^{l_{2}+1}- ( 4+4\mu _{2} ) v_{m,n}^{l_{2}}+ ( 1+2\mu_{2} ) v_{m,n}^{l_{2}} ] =2\tau\psi_{m,n}, \\ \quad n=\overline{1,M-1},n=\overline{1,M-1},\end{cases} $$
(59)

respectively.

In the second stage, the calculation of \(p_{n}\) (\(n=\overline {1,M-1}\), \(m=\overline{1,M-1}\)) is carried out by

$$\begin{aligned} p_{m,n}&=-\frac{(\zeta_{m,n+1}-v_{m,n+1}^{l_{0}})-2(\zeta _{m,n}-v_{m,n}^{l_{0}})+(\zeta_{m,n-1}-v_{m,n-1}^{l_{0}})}{h^{2}} \\ &\quad{} -\frac{(\zeta_{m+1,n}-v_{m+1,n}^{l_{0}})-2(\zeta _{m,n}-v_{m,n}^{l_{0}})+(\zeta_{m-1,n}-v_{m-1,n}^{l_{0}})}{h^{2}}+v_{m,n}^{l_{0}}, \end{aligned}$$

for the first order approximation, and

$$\begin{aligned} p_{m,n}&=-\frac{1}{h^{2}} \bigl\{ \bigl[ \zeta_{m,n+1}- \bigl( \mu _{0}v_{m,n+1}^{l_{0}+1}- ( \mu_{0}-1 ) v_{m,n+1}^{l_{0}} \bigr) \bigr] \\ &\quad{} -2 \bigl[ \zeta_{m,n}- \bigl( \mu_{0}v_{m,n}^{l_{0}+1}- ( \mu _{0}-1 ) v_{m,n}^{l_{0}} \bigr) \bigr] \\ &\quad{} + \bigl[ \zeta_{m,n-1}- \bigl( \mu_{0}v_{m,n-1}^{l_{0}+1}- ( \mu _{0}-1 ) v_{m,n-1}^{l_{0}} \bigr) \bigr] \bigr\} \\ &\quad{}-\frac{1}{h^{2}} \bigl\{ \bigl[ \zeta_{m+1,n}- \bigl( \mu _{0}v_{m+1,n}^{l_{0}+1}- ( \mu_{0}-1 ) v_{m+1,n}^{l_{0}} \bigr) \bigr] \\ &\quad{} -2 \bigl[ \zeta_{m,n}- \bigl( \mu_{0}v_{m,n}^{l_{0}+1}- ( \mu _{0}-1 ) v_{m,n}^{l_{0}} \bigr) \bigr] \\ &\quad{} + \bigl[ \zeta_{m-1,n}- \bigl( \mu_{0}v_{m-1,n}^{l_{0}+1}- ( \mu _{0}-1 ) v_{m-1,n}^{l_{0}} \bigr) \bigr] \bigr\} \end{aligned}$$

for the second order approximation.

In the third stage, we calculate \(\{ u_{n}^{k} \} \) by

$$u_{m,n}^{k}=v_{m,n}^{k}+ \zeta_{n}-v_{m,n}^{l_{0}}, \quad \text{and}\quad u_{m,n}^{k}=v_{m,n}^{k}+ \zeta_{m,n}- \bigl( \mu _{0}v_{m,n}^{l_{0}+1}- ( \mu_{0}-1 ) v_{m,n}^{l_{0}} \bigr) $$

for the first and second order approximation, respectively.

The difference problems (58) and (59) can be rewritten in the matrix form (56). In this case, \(g_{n}\) is \((N+1)(M+1)\times1\) a column matrix, A, B, C, I are \((N+1)(M+1)\times (N+1)(M+1)\) square matrices, and I is the identity matrix, \(v_{s}\) is the \((N+1)(M+1)\times1\) column matrix such that

$$\begin{aligned}& v_{s}= \left[ \textstyle\begin{array}{cccccccccc} v_{0,s}^{0} & \cdots& v_{0,s}^{N} & v_{1,s}^{0} & \cdots& v_{1,s}^{N} & \cdots& v_{M,s}^{0} & \cdots& v_{M,s}^{N}\end{array}\displaystyle \right] ^{t}, \\ & \quad s=n-1,n,n+1. \end{aligned}$$

Denote

$$a=\frac{1}{h^{2}},\quad\quad b=1+\frac{2}{\tau^{2}}+\frac{4}{h^{2}},\quad\quad d= \frac {1}{\tau ^{2}}. $$

Then

$$\begin{aligned}& A=C= \left[ \textstyle\begin{array}{ccccc} O & O & \cdots& O & O \\ O & E & \cdots& O & O \\ \cdots& \cdots& \ddots& \cdots& \cdots\\ O & O & \cdots& E & \\ O & O & \cdots& O & O\end{array}\displaystyle \right] , \quad \quad B= \left[ \textstyle\begin{array}{ccccc} Q & O & \cdots& O & O \\ O & D & \cdots& O & O \\ \cdots& \cdots& \ddots& \cdots& \cdots\\ O & O & \cdots& D & \\ O & O & \cdots& O & Q\end{array}\displaystyle \right] ,\\& E=\operatorname{diag}(0,a,a,\ldots,a,0),\quad\quad Q=I_{ ( N+1 ) \times ( N+1 ) },\quad\quad O=O_{ ( N+1 ) \times ( N+1 ) }, \\& g_{m,n}^{k}=-f(t_{k},x_{n},y_{m}),\quad k=\overline{1,N-1}, n=\overline {1,M-1},m=\overline{1,M-1},\\& D [ i,i ] =b,\quad\quad D [ i-1,i ] =d,\quad\quad D [ i,i-1 ] =d,\quad i= \overline{2,N}, \\& D [ 1,1 ] =-1,\quad\quad D [ 1,2 ] =1,\quad\quad D [ N+1,N+1 ] =1, \\& D [ N+1,N ] =-1, \\& D [ N+1,l_{1} ] =-\frac{1}{4},\quad\quad D [ N+1,l_{2} ] =- \frac {1}{4},\quad\quad D [ N+1,l_{3} ] =-\frac{1}{4} \\& D [ N+1,l_{1}+1 ] =\frac{1}{4},\quad\quad D [ N+1,l_{2}+1 ] = \frac{1}{4}, \\& D [ N+1,l_{3}+1 ] =\frac{1}{4}, \\& D [ i,j ] =0, \quad \text{for other }i\text{ and }j, \\& g_{m,n}^{0}=\tau\varphi_{m,n},\quad\quad g_{m,n}^{N}=\tau\psi_{m,n},\quad n= \overline{1,M-1} ,m=\overline{1,M-1}, \end{aligned}$$

for the first order approximation, and

$$\begin{aligned}& D [ i,i ] =b,\quad\quad D [ i-1,i ] =d,\quad\quad D [ i,i-1 ] =d,\quad i= \overline{2,N}, \\& D [ 1,1 ] =-3,\quad\quad D [ 1,2 ] =4,\quad\quad D [ 1,3 ] =-1,\quad\quad D [ N+1,N+1 ] =3, \\& D [ N+1,N ] =-4,\quad\quad D [ N+1,N-1 ] =-1, \\& D [ N+1,l_{1}+1 ] =-\frac{1}{3} ( 3+2\mu_{1} ) ,\quad\quad D [ N+1,l_{1}+1 ] =\frac{1}{3} ( 3+2\mu_{1} ) , \\& D [ N+1,l_{1}-1 ] =-\frac{1}{3} ( 1+2\mu_{1} ) ,\quad\quad D [ N+1,l_{2}+1 ] =-\frac{1}{3} ( 3+2\mu_{2} ) , \\& D [ N+1,l_{2}+1 ] =\frac{1}{3} ( 3+2\mu_{2} ) ,\quad\quad D [ N+1,l_{2}-1 ] =-\frac{1}{3} ( 1+2\mu_{2} ) , \\& D [ i,j ] =0, \quad \text{for other }i\text{ and }j; \\& g_{m,n}^{0}=2\tau\varphi_{m,n},\quad\quad g_{m,n}^{N}=2\tau\psi_{m,n}, \quad n=\overline{1,M-1} ,m=\overline{1,M-1}, \end{aligned}$$

for the second order approximation.

In Tables 4-6, we give the results of the numerical calculations for both first and second order approximations in the cases \(N=M=10, 20, 40\). Table 1 presents the error between exact and approximately solutions of NBVP . Table 2 includes the error between exact and approximate u. Table 3 gives the error for p. It can be seen from Tables 1-3 that the second order of the ADS is more accurate compared with the first order of the ADS.

Table 4 Error of v in example ( 57 )
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Table 5 Error of u in example ( 57 )
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Table 6 Error of p in example ( 57 )
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6 Conclusion

In the present paper, we discuss stability estimates for the solution of a Bitsadze-Samarskii type elliptic overdetermined multipoint NBVP. We apply the finite difference method to construct the first and second order of the ADSs for this problem and establish stability estimates for its solutions. The abstract results established are applied to get stability estimates for the solution of Bitsadze-Samarskii type overdetermined elliptic multidimensional differential and difference problems with multipoint NBVCs. Stability estimates for the solution of difference schemes are obtained. Finally, we present numerical results with explanation on the realization for two dimensional and three dimensional elliptic overdetermined multipoint NBVP in test examples.

Moreover, applying the results of [28, 32, 33] the high order ADSs for the numerical solution of a Bitsadze-Samarskii type elliptic overdetermined multipoint NBVP can be presented.