\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 197, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/197\hfil
Inverse problem for multidimensional elliptic equation]
{Approximate solution for an inverse problem \\ of multidimensional
 elliptic equation with multipoint nonlocal and Neumann boundary conditions}

\author[C. Ashyralyyev, G. Akyuz, M. Dedeturk \hfil EJDE-2017/197\hfilneg]
{Charyyar Ashyralyyev, Gulzipa Akyuz, Mutlu Dedeturk}

\address{Charyyar Ashyralyyev \newline
Department of Mathematical Engineering,
Gumushane University,
Gumushane, Turkey}
\email{charyyar@gumushane.edu.tr}

\address{Gulzipa Akyuz \newline
Department of Mathematical Engineering,
Gumushane University,
Gumushane, Turkey}
\email{gulzipaakyuz@gmail.com}

\address{Mutlu Dedeturk \newline
Department of Mathematical Engineering,
 Gumushane University,
Gumushane, Turkey}
\email{mutludedeturk@gumushane.edu.tr}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted June 7, 2017. Published August 9, 2017.}
\subjclass[2010]{35N25, 39A30}
\keywords{Difference scheme; inverse elliptic problem; stability;
\hfill\break\indent overdetermination; nonlocal problem}

\begin{abstract}
 In this work, we consider an inverse elliptic problem with Bitsadze-Samarskii
 type multipoint nonlocal and Neumann boundary conditions. We construct the
 first and second order of accuracy difference schemes (ADSs)
 for problem considered. We stablish stability and coercive stability estimates
 for solutions of these difference schemes. Also, we give numerical results
 for overdetermined elliptic problem with multipoint Bitsadze-Samarskii type
 nonlocal and Neumann boundary conditions in two and three dimensional
 test examples. Numerical results are carried out by MATLAB program
 and brief explanation on the realization of algorithm is given.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks


\section{Introduction}

Theory and methods of solving inverse problems for differential and
difference equations have been comprehensively studied by several
researchers 
(see \cite{1,2,5,6,7,11,12,13,13,15,16,17,18,19,21,22,23,24,25,26,27,28,29,30,31,
32,38} and the references therein). In
papers \cite{6,11,12,13,14,15,16,30,32} well-posedness of various overdetermined elliptic
type differential and difference problems are studied. Dirichlet type overdetermined
problems for elliptic partial differential equation (PDE) were
investigated in \cite{6,15,16}. Neumann type overdetermined elliptic problems
were studied in papers \cite{11,12,14}.

In recent years, different types of elliptic nonlocal boundary value problems and
generalizations of such type problems to various differential and difference
equations have been extensively investigated (see \cite{3,8,9,13,32,34} and
the bibliography therein).

In this article, we study approximation of Bitsadze-Samarskii type overdetermined
elliptic differential problem with Neumann boundary conditions.

Given an integer $q\geq 2$, we assume that the nonnegative numbers
$k_{1},\ldots ,k_{q}$, $\lambda _0, \lambda _{1},\ldots ,\lambda _{q}$
satisfy the conditions
\begin{equation}
\sum_{i=1}^{q}k_{i}=1,\quad k_{i}\geq 0,\; i=1,\ldots ,q,
\quad 0<\lambda _{1}<\ldots <\lambda _{q}<1,\quad
0<\lambda _0<1.  \label{klam}
\end{equation}

Let $\Omega =(0,\ell )^{n}\ \subset R_{n}$ be the open cube with boundary $S$,
 $\overline{\Omega }=\Omega \cup S$. In $[0,T]\times \Omega $, we consider
the inverse problem of finding function $u(t,x)$ and function $p(x)$ in $
\Omega $ for the following multidimensional elliptic PDE with multipoint
nonlocal and Neumann boundary conditions
\begin{equation}
\begin{gathered}
-v_{tt}(t,x)-\sum_{r=1}^{n}(a_{r}(x)v_{x_{r}})_{x_{r}}+\sigma
v(t,x)=g(t,x)+p(x), \\
x=(x_{1},\dots ,x_{n})\in \Omega ,\quad 0<t<T; \\
v(T,x)-\sum_{i=1}^{q}k_{i}v(\lambda _{i},x)=\eta (x),\quad
v(0,x)=\phi(x),\quad
v(\lambda _0,x)=\zeta (x),\quad x\in \overline{\Omega },  \\
\frac{\partial v(t,x)}{\partial \overrightarrow{n}}=0,\quad
x\in S,\quad 0\leq t\leq T.
\end{gathered}
\label{mdep}
\end{equation}
Here, $\overrightarrow{n}$ is the normal vector to $S$; $a_{r}$,
 $\varphi,\psi ,\xi $, and $g$ are given smooth functions,  $a_{r}(x)\geq
a>0$ for all $x\in \Omega $.

Well-posedness of problem \eqref{mdep} was established in \cite{13}.
In this article, we apply a finite difference method to approximate the
solution of problem \eqref{mdep}. Namely, we construct the first and
second order of ADSs
with respect to $t$ and second order of ADS with respect to $x$ for the
approximate solution of problem. Stability and coercive stability estimates
for solutions of both difference schemes are established. Later, we give two
and three dimensional numerical examples with brief explanation on the
realization for inverse elliptic problem with multipoint Bitsadze-Samarskii
type nonlocal and Neumann boundary conditions.

The differential operator \cite{10}
\begin{equation}
A^{x}v(x)=-\sum_{r=1}^{n}(a_{r}(x)v_{x_{r}})_{x_{r}}+\sigma v(x)
\label{operatorAx}
\end{equation}
is a self-adjoint positive definite (SAPD) operator $A=A^{x}$ acting on
Hilbert space $H=L_{2}(\overline{\Omega })$ with the domain\ $D(A^{x})$=$
\{ v(x)\in W_{2}^2(\overline{\Omega }),\text{ }\frac{\partial v}{
\partial \overrightarrow{n}}=0\text{\ on }S\} $.

Therefore, primal problem \eqref{mdep} corresponds to the following
Bitsadze-Samarskii type inverse elliptic problem of finding an element
$p\in H$ and a function
$v\in C([0,T],D(A)) \cap C^2([0,T],H) $:
\begin{equation}
\begin{gathered}
-v_{tt}(t)+Av(t)=g(t)+p,\quad t\in (0,T), \\
v(0)=\phi , \quad v(\lambda_0)=\zeta,\quad
 v(T)=\sum_{i=1}^{q}\alpha _{i}v(\lambda _{i})+\eta.
\end{gathered} \label{invd}
\end{equation}

Let $\ [0,T]_{\tau }=\{ t_{k}=k\tau ,\;k=\overline{0,N},\text{ }N\tau
=T\} $ be the set of grid points. Introduce the notation
\begin{gather*}
C=\frac{1}{2}(\tau A+\sqrt{4A+\tau ^2A^2}),\quad
R=(I+\tau C)^{-1}, \\
P=(I-R^{2N})^{-1}, \quad
D=(I+\tau C)(2I+\tau C)^{-1}C^{-1},
\end{gather*}
where $I$ is the identity operator. It is known that $A>\delta I$
$(\delta >0)$,
$C$ is SAPD operator and the bounded operator $R$ is defined on the whole
space $H$ \cite{10,30}.

\begin{lemma}[\cite{10}] \label{lem1.1}
 The following estimates hold:
\begin{gather*}
\| R^k\| _{H\to H}\leq M(\delta)
(1+\delta ^{\frac{1}{2}}\tau )^{-k}, \| CR^k\|
_{H\to H}\leq \frac{M(\delta) }{k\tau }, \\
k\geq 1,\| P\| _{H\to H}\leq M(\delta),\quad \delta >0.
\end{gather*}
\end{lemma}

The remainder of this article  is organized as follows: In Section 2, we
present two difference schemes for approximate solution of inverse elliptic
problem \eqref{mdep} with Bitsadze-Samarskii type multipoint nonlocal and
Neumann boundary conditions. In Section 3, we obtain the stability and
coercive stability estimates for the solution of both presented difference
schemes. Numerical results for two dimensional and three dimensional
elliptic equations are presented in Section 4. Finally, the conclusion is
given in Section 5.

\section{Difference problems}

The approximation of problem \eqref{mdep} is carried out in two steps. In the
first step, we define the grid spaces
\begin{gather*}
\begin{aligned}
\widetilde{\Omega }_{h}
=\Big\{&x : x=x_{m}=(h_{1}m_{1},\dots ,h_{n}m_{n}),\;
 m=(m_{1},\dots ,m_{n}),\\
& 0\leq m_{r}\leq M_{r},  \;  h_{r}M_{r}=\ell ,\; r=1,\dots ,n\Big\} ,
\end{aligned} \\
\Omega _{h}=\widetilde{\Omega }_{h}\cap \Omega ,\quad
S_{h}=\widetilde{\Omega }_{h}\cap S,\quad
h=(h_{1},\dots ,h_{n}),
\end{gather*} %\label{overd}
and assign the difference operator $A_{h}^{x}$  to operator $A^{x}$
\eqref{operatorAx} by the formula
\begin{equation*}
A_{h}^{x}v^{h}(x)=-\sum_{r=1}^{n}(a_{r}(x)v_{x_{r}}^{h})
_{x_{r},m_{r}}+\sigma v^{h}(x),
\end{equation*}
acting in the space of grid functions $v^{h}(x)$, satisfying the condition
$D^{h}v^{h}(x)=0$ for all $x\in S_{h}$. Here and in future $D^{h}$ is the
approximation of operator $\frac{\partial }{\partial \overrightarrow{n}}$.
It is known that $A_{h}^{x}$  is a SAPD operator (see \cite{36,37}).

By using $A_{h}^{x}$, the overdetermined problem \eqref{mdep} is reduced to the
boundary value problem for the system of ordinary differential
equations
\begin{equation}
\begin{gathered}
-\frac{d^2v^{h}(t,x)}{dt^2}+A_{h}^{x}v^{h}(t,x)=g^{h}(t,x)+p^{h}(x),\quad
 t\in (0,T),\; x\in \Omega _{h}, \\
v^{h}(0,x)=\phi (x),\quad  v^{h}(\lambda_0,x)=\zeta ^{h}(x), \\
v^{h}(T,x)-\sum_{i=1}^{q}k_{i}v^{h}(\lambda _{i},x)
=\eta ^{h}(x),x\in \widetilde{\Omega }_{h}.
\end{gathered}
\label{1stepappr}
\end{equation}

Denote
\[
l_{i}=[\frac{\lambda _{i}}{\tau }],\quad \mu _{i}=\frac{\lambda _{i}}{\tau }
-l_{i},\quad i=0,1,\ldots ,q,
\]
where $[\cdot ]$ is standard notation for greatest integer function.

Let $v_{k}^{h}(x)=v^{h}(t_{k},x) $, $g_{k}^{h}(x)=g^{h}(
t_{k},x) ,k=\overline{0,N}$.

In the second step, we apply the following approximation formulas
\begin{gather*}
v^{h}(\lambda _{i},x)=v_{l_{i}}^{h}(x)+o(\tau ), \\
v^{h}(\lambda _{i},x)=v_{l_{i}}^{h}(x)+\mu
_{i}(v_{l_{i}+1}^{h}(x)-v_{l_{i}}^{h}(x))+o(\tau ^2)
\end{gather*}
for $v^{h}(\lambda _{i},x)$, $i=0,1,\ldots ,q$. Then problem \eqref{1stepappr}
 is replaced by
\begin{equation}
\begin{gathered}
-\tau ^{-2}\left[ v_{k+1}^{h}(x)-2v_{k}^{h}(x)+v_{k-1}^{h}(x)\right]
+A_{h}^{x}v_{k}^{h}(x)=g_{k}^{h}(x)+p^{h}(x),\text{ }\;\; \\
1\leq k\leq N-1,\quad x\in \Omega _{h}, \\
v_{N}^{h}(x)=\sum_{i=1}^{q}k_{i}v_{l_{i}}^{h}(x)+\eta ^{h}(x), \\
v_{l_0}^{h}(x)=\zeta ^{h}(x),\quad
v_0^{h}(x)=\phi ^{h}(x),\quad
x\in \widetilde{\Omega }_{h},
\end{gathered}  \label{dsm1}
\end{equation}
and
\begin{equation}
\begin{gathered}
-\tau ^{-2}\left[ v_{k+1}^{h}(x)-2v_{k}^{h}(x)+v_{k-1}^{h}(x)\right]
+A_{h}^{x}v_{k}^{h}(x)=g_{k}^{h}(x)+p^{h}(x),\text{ }\;\; \\
1\leq k\leq N-1,\quad x\in \Omega _{h}, \\
v_{N}^{h}(x)=\sum_{i=1}^{q}k_{i}(
v_{l_{i}}^{h}(x)+\mu _{i}(v_{l_{i}+1}^{h}(x)-v_{l_{i}}^{h}(x))
) +\eta ^{h}(x), \\
v_{l_0}^{h}(x)+\mu _0(v_{l_0+1}^{h}(x)-v_{l_0}^{h}(x))
=\zeta ^{h}(x), \quad
v_0^{h}(x)=\phi ^{h}(x), \quad x\in \widetilde{\Omega }_{h},
\end{gathered}  \label{dsm2}
\end{equation}
respectively.

By substituting
\begin{equation}
v_{k}^{h}(x)=u_{k}^{h}(x)+(A_{h}^{x}) ^{-1}p^{h}(x),\quad
x\in \widetilde{\Omega }_{h},\quad 1\leq k\leq N-1,  \label{vkh}
\end{equation}
difference scheme \eqref{dsm1} is reduced to the auxiliary difference
scheme
\begin{equation}
\begin{gathered}
-\tau ^{-2}\left[ u_{k+1}^{h}(x)-2u_{k}^{h}(x)+u_{k-1}^{h}(x)\right]
+A_{h}^{x}u_{k}^{h}(x)=g_{k}^{h}(x), \\
 1\leq k\leq N-1,\quad x\in \Omega _{h}, \\
u_0^{h}(x)-u_{l_0}^{h}(x)=\phi ^{h}(x)-\zeta
^{h}(x),\\
u_{N}^{h}(x)=\sum_{i=1}^{q}k_{i}u_{l_{i}}^{h}(x)+\eta
^{h}(x), \quad x\in \widetilde{\Omega }_{h}.
\end{gathered}   \label{dsu1}
\end{equation}

The solution of system \eqref{dsu1} is defined by the formula
\begin{equation}
\begin{aligned}
u_{k}^{h}(x)
&=P\left[ (R^k-R^{2N-k}) u_0^{h}(x)+(
R^{N-k}-R^{N+k}) \right] u_{N}^{h}(x) \\
&\quad -P(R^{N-k}-R^{N+k}) D\sum_{j=1}^{N-1}(
R^{N-j}-R^{N+j}) g_{j}^{h}(x)\tau   \\
&\quad +D\sum_{j=1}^{N-1}(R^{| k-j|
}-R^{k+j}) g_{j}^{h}(x)\tau, \quad k=\overline{1,N-1},
\end{aligned} \label{uk}
\end{equation}
where
\begin{equation}
\begin{gathered}
\begin{aligned}
u_0^{h}(x)
&=F_{1}^{-1}\Big[ \Big(
I-R^{2N}-\sum_{i=1}^{q}k_{i}(R^{N-l_{i}}-R^{N+l_{i}})\Big)
G_{1}^{h}(x) \\
&\quad +(R^{N-s}-R^{N+s}) G_{2}^{h}(x)\Big] ,u_{N}^{h}(x) \\
&= \Delta _{1}^{-1}\Big[ (I-R^{2N}-R^{s}+R^{2N-s})
G_{2}^{h}(x)
 +\sum_{i=1}^{q}k_{i}(
R^{l_{i}}-R^{2N-l_{i}}) G_{1}^{h}(x)\Big] ,
\end{aligned} \\
F_{1}=(I-R^{2N}) (I-R^{l_0})\Big(
I-\sum_{i=1}^{q}k_{i}R^{N-l_{i}}\Big) \Big(
I-\sum_{i=1}^{q}k_{i}R^{N-(l_0-l_{i})}\Big) ,
\\
\begin{aligned}
G_{1}^{h}(x)
&=P^{-1}(\phi ^{h}(x)-\zeta ^{h}(x)) +(R^{N-s}-R^{N+s}) \\
&\quad \times D\sum_{j=1}^{N-1}(R^{N-j-1}-R^{N+j-1})g_{j}^{h}(x)\tau \\
&\quad -P^{-1}D\sum_{j=1}^{N-1}(R^{| s-j|
-1}-R^{s+j-1}) g_{j}^{h}(x)\tau ,
\end{aligned}\\
\begin{aligned}
G_{2}^{h}(x)&=k\Big\{ (R^{N-l_{i}}-R^{N+l_{i}})
D\sum_{j=1}^{N-1}(R^{N-j-1}-R^{N+j-1}) g_{j}^{h}(x)\tau \\
&\quad -P^{-1}D\sum_{j=1}^{N-1}(R^{|
l_{i}-j| -1}-R^{l_{i}+j-1}) g_{j}^{h}(x)\tau \Big\}
+P^{-1}\eta ^{h}(x).
\end{aligned}
\end{gathered}\label{uouN1}
\end{equation}

Using \ \eqref{vkh}, difference scheme \eqref{dsm2} can be
reduced to the auxiliary difference scheme
\begin{equation}
\begin{gathered}
-\tau ^{-2}\left[ u_{k+1}^{h}(x)-2u_{k}^{h}(x)+u_{k-1}^{h}(x)\right]
+A_{h}^{x}u_{k}^{h}(x)=g_{k}^{h}(x),  \\
1\leq k\leq N-1,\quad  x\in \Omega _{h}, \\
u_0^{h}(x)+(\mu _0-1) u_{l_0}^{h}(x)-\mu
_0u_{l_0+1}^{h}(x)=\phi ^{h}(x)-\zeta ^{h}(x), \\
u_{N}^{h}(x)+\sum_{i=1}^{q}k_{i}\left[ (\mu _{i}-1)
u_{l_{i}}^{h}(x)-\mu _{i}u_{l_{i}+1}^{h}(x)\right] =\eta ^{h}(x), \quad
x\in \widetilde{\Omega }_{h}.
\end{gathered} \label{dsu2}
\end{equation}
The solution of system \eqref{dsu2} is defined by formula \eqref{uk}, where
\begin{align*}
u_0^{h}(x)
&=F_{2}^{-1}\Big\{ \Big[ I-R^{2N}+\sum_{i=1}^{q}k_{i}
(\mu _{i}-1) (R^{N-l_{i}}-R_{i}^{N+l})   \\
&\quad   -\sum_{i=1}^{q}k_{i}\mu _{i}(
R^{N-l_{i}-1}-R^{N+l_{i}+1}) \Big] G_{3}^{h}(x)  \\
&\quad -[ (\mu _0-1) (
R^{N-l_0}-R^{N+l_0}) -\mu _0(
R^{N-l_0-1}-R^{N+l_0+1}) ] G_{4}^{h}(x)\Big\} ,
\end{align*}
\begin{align*}
u_{N}^{h}(x)
&=F_{2}^{-1}\Big\{ \Big[ I-R^{2N}+(\mu _0-1)
(R^{l_0}-R^{2N-l_0})    \\
&\quad  -\mu _0(R^{l_0+1}-R^{2N-l_0-1}) \Big]
G_{4}^{h}(x)-\Big[ \sum_{i=1}^{q}k_{i}(\mu _{i}-1)
(R^{l_{i}}-R^{2N-l_{i}})    \\
&\quad   -\sum_{i=1}^{q}k_{i}\mu _{i}(
R^{l_{i}+1}-R^{2N-l_{i}-1}) \Big] G_{3}^{h}(x)\Big\} ,
\end{align*}
\begin{equation}  \label{uouN2}
\begin{aligned}
F_{2}&=[ I-R^{2N}+(\mu _0-1) (
R^{l_0}-R^{2N-l_0}) -\mu _0(
R^{l_0+1}-R^{2N-l_0-1})] \\
&\quad \times \Big[ I-R^{2N}+\sum_{i=1}^{q}k_{i}(\mu _{i}-1)
(R^{N-l_{i}}-R_{i}^{N+l})  \\
&\quad -\sum_{i=1}^{q}k_{i}\mu _{i}(
R^{N-l_{i}-1}-R^{N+l_{i}+1}) \Big] \\
&\quad -[ (\mu _0-1) (R^{N-l_0}-R^{N+l_0}) -\mu
_0(R^{N-l_0-1}-R^{N+l_0+1}) ] \\
&\quad \times \Big[ \sum_{i=1}^{q}k_{i}(\mu _{i}-1) (
R^{l_{i}}-R^{2N-l_{i}}) -\sum_{i=1}^{q}k_{i}\mu _{i}(
R^{l_{i}+1}-R^{2N-l_{i}-1}) \Big] .
\end{aligned}
\end{equation}
\begin{align*}
G_{3}^{h}(x)
&=P^{-1}(\phi ^{h}(x)-\zeta ^{h}(x)) \\
&\quad +\left[ (\mu _0-1) (R^{N-l_0}-R^{N+l_0}) -\mu
_0(R^{N-l_0-1}-R^{N+l_0+1}) \right] \\
&\quad \times D\sum_{j=1}^{N-1}(R^{N-j}-R^{N+j}) g_{j}\tau
-P^{-1}D \\
&\quad \times \sum_{j=1}^{N-1}\left[ (\mu _0-1) (
R^{| l_0-j| }-R^{l_0+j}) -\mu _0(
R^{| l_0+1-j| }-R^{l_0+j+1}) \right]
g_{j}^{h}(x)\tau ,
\end{align*}
\begin{align*}
G_{4}^{h}(x)
&=\sum_{i=1}^{q}k_{i}\left[ (\mu _{i}-1)
(R^{N-l_{i}}-R^{N+l_{i}}) -\mu _{i}(
R^{N-l_{i}-1}-R^{N+l_{i}+1}) \right] \\
&\quad \times D\sum_{j=1}^{N-1}(R^{N-j}-R^{N+j})
g_{j}^{h}(x)\tau +P^{-1}\eta ^{h}(x)-P^{-1}D \\
&\quad \times \sum_{j=1}^{N-1}\sum_{i=1}^{q}k_{i}\Big[ (\mu
_{i}-1) (R^{| l_{i}-j| }-R^{l_{i}+j}) \\
&\quad -\mu _0(R^{| l_{i}+1-j| }-R^{l_{i}+j+1})
\Big] g_{j}^{h}(x)\tau .
\end{align*}

So, to find an approximate solution of \eqref{mdep}, we consider the algorithm
which contains three stages. We find $\{ u_{k}^{h}(x)\} _0^{N}$
 as solution of \eqref{dsu1} or \eqref{dsu2} in the first stage. Putting $
k=l_0\ $and $k=l_0+1$, we get $u_{l_0}^{h}(x)\ $and $
u_{l_0+1}^{h}(x)$, respectively. In the second stage, we obtain $p^{h}(x)$
by
\begin{equation}
p^{h}(x)=A_{h}^{x}\zeta ^{h}(x)-A_{h}^{x}u_{l_0}^{h}(x),\quad x\in
\widetilde{\Omega }_{h},  \label{p1}
\end{equation}
for \eqref{dsm1}, and
\begin{equation}
p^{h}(x)=A_{h}^{x}\zeta ^{h}(x)-A_{h}^{x}\left[ (1-\mu _0)
u_{l_0}^{h}(x)+\mu _0u_{l_0+1}^{h}(x)\right] ,\quad x\in \widetilde{
\Omega }_{h},  \label{p2}
\end{equation}
for \eqref{dsm2}.

In the third stage, we use formulas
\begin{equation}
v_{k}^{h}(x)=u_{k}^{h}(x)+\zeta ^{h}(x)-u_{l_0}^{h}(x),\quad x\in
\widetilde{\Omega }_{h},\; 1\leq k\leq N-1,  \label{vkh1}
\end{equation}
and
\begin{equation}
v_{k}^{h}(x)=u_{k}^{h}(x)+\zeta ^{h}(x)-\left[ (1-\mu _0)
u_{l_0}^{h}(x)+\mu _0u_{l_0+1}^{h}(x)\right] , \label{vkh2}
\end{equation}
for $x\in \widetilde{\Omega }_{h}$, $1\leq k\leq N-1$,
to obtain the solution $\{ v_{k}^{h}(x)\} _0^{N}$
of corresponding difference problems \eqref{dsm1} and \eqref{dsm2}.

\section{Stability and coercive stability estimates}

 Let $L_{2h}=L_{2}(\widetilde{\Omega }_{h})$ and
$W_{2h}^2=W_{2}^2(\widetilde{\Omega }_{h})$ be Banach spaces of the grid
functions $f^{h}(x)=\{f(h_{1}m_{1},\dots ,h_{n}m_{n})\}$ defined on
$\widetilde{\Omega }_{h}$, equipped with the following  norms
\begin{gather*}
\| f^{h}\| _{L_{2h}}=\Big(\sum_{x\in \widetilde{\Omega }
_{h}}|f^{h}(x)|^2h_{1}\dots h_{n}\Big)^{1/2}, \\
\begin{aligned}
\| f^{h}\| _{W_{2h}^2}
&=\| f^{h}\|_{L_{2h}} + \Big[
\sum_{x\in \widetilde{\Omega }_{h}}\sum_{r=1}^{n}
| (f^{h})_{x_{r}}| ^2h_{1}\dots h_{n}\Big]^{1/2}  \\
&\quad + \Big[\sum_{x\in \widetilde{\Omega }_{h}}\sum_{r=1}^{n}|
(f^{h}(x))_{x_{r}\overline{x_{r}}, m_{r}}| ^2h_{1}\dots
h_{n})\Big] ^{1/2},
\end{aligned}
\end{gather*}
respectively.
Denote by $C_{\tau }(H) $ and $C_{\tau }^{\alpha ,\alpha
}(H) $, the corresponding Banach spaces of $H$-valued
mesh functions  $\varphi _{\tau }^{h}=\{ \varphi _{k}^{h}\}
_{1}^{N}\ $  on $[0,T]_{\tau }$ with the following norms
\begin{gather*}
\| \varphi _{\tau }^{h}\| _{C_{\tau }(H)
}=\max_{1\leq t\leq N-1}\quad
\| \varphi _{k}^{h}\| _{H}, \\
\| \varphi _{\tau }^{h}\| _{C_{\tau }^{\alpha ,\alpha}(H)}
=\| \varphi _{\tau }^{h}\| _{C_{\tau }(H) }
 +\sup_{1\leq k\leq k+s\leq N-1}\frac{((N-s) \tau) ^{\alpha }((k+s) \tau)
^{\alpha }}{(s\tau) ^{\alpha }}\Vert \varphi _{k+s}^{h}-\varphi
_{k}^{h}\Vert _{H}.
\end{gather*}

Let $\tau $ and $|h|=\sqrt{h_{1}^2+\dots +h_{n}^2}$ be sufficiently
small positive numbers.

\begin{theorem} \label{thm3.1}
Under conditions \eqref{klam}, for the
solution of difference problems \eqref{dsm1} and \eqref{dsm2} the next
stability inequalities hold:
\begin{gather*}
\begin{aligned}
\| \{ v_{k}^{h}\} _{1}^{N-1}\| _{\mathcal{C}_{\tau }(L_{2h})}
&\leq M(\delta ,\lambda _{1},\ldots ,\lambda
_{q}) \Big[ \| \phi ^{h}\| _{L_{2h}}+\|\zeta ^{h}\| _{L_{2h}} \\
&\quad  +\| \eta ^{h}\| _{L_{2h}}+\| \{
g_{k}^{h}\} _{1}^{N-1}\| _{C_{\tau }(L_{2h})}\Big] ,
\end{aligned} \\
\begin{aligned}
\| p^{h}\| _{L_{2h}}
&\leq M(\delta ,\lambda_{1},\ldots ,\lambda _{q}) \Big[ \| \phi ^{h}\|
_{W_{2h}^2}+\| \zeta ^{h}\| _{W_{2h}^2} \\
&\quad  +\| \eta ^{h}\| _{W_{2h}^2}+\frac{1}{\alpha
(1-\alpha )}\| \{ g_{k}^{h}\} _{1}^{N-1}\|
_{C_{\tau }^{\alpha ,\alpha }(L_{2h})}\Big] ,
\end{aligned}
\end{gather*}
where $M(\delta ,\lambda _{1},\ldots ,\lambda _{q}) $ does not
depend on $\tau ,\alpha ,h,\phi ^{h}(x),\zeta ^{h}(x)$, $\eta ^{h}(x)$ and
$\{ g_{k}^{h}(x)\} _{1}^{N-1}$.
\end{theorem}

\begin{theorem}\label{thm3.2}
Under conditions \eqref{klam}, for the
solution of difference problems \eqref{dsm1} and \eqref{dsm2} the coercive
stability inequality holds:
\begin{align*}
&\| \{ \frac{v_{k+1}^{h}-2v_{k}^{h}+v_{k-1}^{h}}{\tau ^2}
)\} _{1}^{N-1}\| _{C_{\tau }^{\alpha ,\alpha }(L_{2h})}+\| \{
v_{k}^{h}\} _{1}^{N-1}\| _{C_{\tau }^{\alpha ,\alpha }(W_{2h}^2)} \\
&\leq M(\delta ,\lambda _{1},\ldots ,\lambda _{q}) [\|
\phi ^{h}\| _{W_{2h}^2}+\| \zeta ^{h}\|
_{W_{2h}^2}+\| \eta ^{h}\| _{W_{2h}^2}+\frac{1}{\alpha
(1-\alpha )}\| \{ g_{k}^{h}\} _{1}^{N}\|
_{C_{\tau }^{\alpha ,\alpha }(L_{2h})}],
\end{align*}
where $M(\delta ,\lambda _{1},\ldots ,\lambda _{q}) $ does not
depend on $\tau ,\alpha ,h,\phi ^{h}(x),\eta ^{h}(x)$, $\zeta ^{h}(x)$, or
 $\{ g_{k}^{h}(x)\} _{1}^{N-1}$.
\end{theorem}

The proofs of Theorems \ref{thm3.1} and \ref{thm3.2} are based on the symmetry property of
operator $A_{h}^{x}$ in $L_{2h}$,  the formulas \eqref{uk}, \eqref{uouN1},
\eqref{uouN2}, \eqref{p1}, \eqref{p2}, \eqref{vkh1},\eqref{vkh2}
for solution of corresponding
difference schemes  and the following theorem on well-posedness of
the elliptic difference problem.

\begin{theorem} \cite{35} \label{thm3.3}
 For the solution of the elliptic difference problem
\begin{gather*}
A_{h}^{x}u^{h}(x)=\omega ^{h}(x),\quad  x\in \widetilde{\Omega
}_{h}, \\
D^{h}u^{h}(x)=0,\quad  x\in S_{h},
\end{gather*}
the following coercivity inequality holds:
\[
\sum_{q=1}^{n}\| (u^{h})_{\overline{x}_{q}x_{q},j_{q}}\| _{L_{2h}}
\leq M||\omega ^{h}||_{L_{2h}},
\]
here $M$ does not depend on $h$ and $\omega ^{h}$.
\end{theorem}

\section{Numerical Examples}

Now, we give two and three dimensional numerical examples with brief explanation
on the realization  for Bitsadze-Samarskii type inverse elliptic multipoint NBVP.
These numerical results are carried out by
using MATLAB program.

\subsection{Two dimensional example}

Consider the following two dimensional Bitsadze-Samarskii type
overdetermined problem with three point nonlocal boundary conditions,
\begin{equation}
\begin{gathered}
-\frac{\partial ^2v(t,x) }{\partial t^2}-\frac{\partial }{
\partial x}((3+\sin (\pi x)) \frac{\partial v(
t,x) }{\partial x}) +v(t,x)=g(t,x) +p(x), \\
t,x\in (0,1), \quad v(0,x) =\phi (x) ,\quad
v(0.1,x) =\zeta (x) ,  \\
v(1,x) -\frac{1}{10}\quad v(0.3,x) -\frac{1}{5}
v(0.7,x) -\frac{7}{10}v(0.8,x) =\eta (x) , \\
x\in[ 0,1] ,\quad v(t,0)=0,\quad v(t,1)=0,\quad  t\in [ 0,1] ,
\end{gathered}   \label{n2}
\end{equation}
where
\begin{gather*}
g(t,x)=\left[ (1+4\pi ^2) \cos (\pi t)+(3\pi
^2+1) t\right] \sin (\pi x) -\pi ^2(\cos (\pi
t)+t) \cos (2\pi x) , \\
\phi (x) =2\sin (\pi x) ,\quad
\zeta (x)=(\cos (\frac{\pi }{10})+\frac{\pi }{10}+1) \sin (\pi x) ,
\\
\eta (x) =-\Big(\frac{1}{10}\cos (\frac{3\pi }{10})+\frac{1}{5}
\cos (\frac{7\pi }{10})+\frac{7}{10}\cos (\frac{4\pi }{5})+\frac{73}{100}\Big) 
\sin (\pi x) ,\quad x\in [ 0,1] .
\end{gather*}
It is easy to show that exact solution of problem \eqref{n2} is the pair of
functions
 $v(t,x) =(\cos (\pi t)+t+1) \sin ( \pi x) $ and
$p(x)=(3\pi ^2+1)\sin (\pi x) -\pi ^2\cos(2\pi x)$.

Denote by $[0,1] _{\tau }\times [0,1] _{h}$ set of
grid points
\[
[ 0,1]_{\tau }\times [ 0,1]_{h}=\{(t_{k},x_{n}):t_{k}=k\tau ,\quad
 k=\overline{0,N};\; x_{n}=nh,\; n=\overline{0,M}\},
\]
where $\tau $ and $h$ such that $N\tau =1$, $Mh=1$. Moreover,
\begin{gather*}
\lambda _0=\frac{1}{10},\quad \lambda _{1}=\frac{1}{10},
\quad \lambda _{2}=\frac{1}{5},\quad \lambda _{3}=\frac{7}{10},\quad
l_{i}=[ \frac{\lambda _{i}}{\tau }],\quad
\mu _{i}=\frac{\lambda _{i}}{\tau }-l_{i},\\
i=0,1,2,3; \quad
\phi _{n}=\phi (x_{n}) ,\quad
\zeta _{n}=\zeta (x_{n}),\quad
\eta _{n}=\eta (x_{n}) ,  \\
p_{n}=p(x_{n}) ,\quad
n=\overline{0,M},\quad
g_{n}^k=g(t_{k},x_{n}),\quad k=0, \ldots , N,\; n=0, \ldots  M.
\end{gather*}
The algorithm for solving \eqref{n2} contains three corresponding stages.
In the first stage, we find numerical solutions
 $\{ u_{n}^k : n=\overline{1,M-1},k=\overline{1,N-1}\} $ of
corresponding the first and second order of ADSs  for auxiliary problem
\begin{equation}
\begin{gathered}
\begin{aligned}
&\frac{u_{n}^{k+1}-2u_{n}^k+u_{n}^{k-1}}{\tau ^2}+(3+\sin (\pi
x_{n}) )\frac{u_{n+1}^k-2u_{n}^k+u_{n-1}^k}{h^2} \\
&+\frac{u_{n+1}^k-u_{n-1}^k}{2h}
=-g_{n}^k, \quad n=\overline{1,M-1},k=
\overline{1,N-1};
\end{aligned} \\
u_0^k=u_{1}^k,\quad u_{M}^k=u_{M-1}^k,\quad k=\overline{0,N}; \\
u_{n}^0-u_{n}^{l_0}=\phi _{n}-\zeta _{n},\quad
u_{n}^{N}-\frac{1}{10} u_{n}^{l_{1}}-\frac{1}{5}u_{n}^{l_{2}}
-\frac{7}{10}u_{n}^{l_{3}}=\eta _{n},\quad
n=\overline{0,M}
\end{gathered}  \label{app1v}
\end{equation}
and
\begin{equation}
\begin{gathered}
\begin{aligned}
&\frac{u_{n}^{k+1}-2u_{n}^k+u_{n}^{k-1}}{\tau ^2}+(3+\sin (\pi
x_{n}) )\frac{u_{n+1}^k-2u_{n}^k+u_{n-1}^k}{h^2} \\
&+\frac{u_{n+1}^k-u_{n-1}^k}{2h}
=-g_{n}^k,\quad n=\overline{1,M-1},\;
k= \overline{1,N-1};
\end{aligned} \\
3u_0^k-4u_{1}^k+u_{2}^k=0,\quad
3u_{M}^k-4u_{M-1}^k+u_{M-2}^k=0,\quad
k=\overline{0,N}; \\
u_{n}^0+(\mu _0-1) u_{n}^{l_0}-\mu
_0u_{n}^{l_0+1}=\phi _{n}-\zeta _{n},  \\
\begin{aligned}
&u_{n}^{N}+\frac{1}{10}\left[ (\mu _{1}-1) u_{n}^{l_{1}}-\mu
_{1}u_{n}^{l_{1}+1}\right] +\frac{1}{5}\left[ (\mu _{2}-1)
u_{n}^{l_{2}}-\mu _{2}u_{n}^{l_{2}+1}\right] \\
&+\frac{7}{10}\left[ (\mu _{3}-1) u_{n}^{l_{3}}-\mu
_{3}u_{n}^{l_{3}+1}\right]\\
& =\eta _{n},\quad n=\overline{0,M}.
\end{aligned}
\end{gathered}  \label{app2v}
\end{equation}
Difference schemes \eqref{app1v} and \eqref{app2v} can be
presented in the matrix form
\begin{equation}
\begin{gathered}
A^{(n) }u_{n+1}+B^{(n) }u_{n}+C^{(n)
}u_{n-1}=I g_{n},\quad  n=1, \ldots , M-1, \\
u_0-u_{1}=\overrightarrow{0},\quad u_{M}-u_{M-1}=\overrightarrow{0},
\end{gathered}  \label{app1vmatr}
\end{equation}
and
\begin{equation}
\begin{gathered}
A^{(n) }u_{n+1}+B^{(n) }u_{n}+C^{(n)
}u_{n-1}=I g_{n},\quad  n=1, \ldots ,M-1, \\
3u_0-4u_{1}+u_{1}=\overrightarrow{0},\quad 3u_{M}-4u_{M-1}+u_{M-1}=
\overrightarrow{0},
\end{gathered}  \label{app2vmatr}
\end{equation}
respectively. Here, $A^{(n)}, B^{(n) }, C^{(n) }$, and $I$ are
$(N+1)\times (N+1)$ matrices. Moreover, $I$ is
identity matrix,
$g_{s}=[ g_{s}^0\; \ldots \;  g_{s}^{N}] ^{t}$ and
$u_{s}=[ u_{s}^0 \; \ldots \; u_{s}^{N} ] ^{t}$ ,
$(s=n-1, n, n+1)$ are $(N+1)\times 1$ column matrices. Let
\begin{gather*}
a^{(n) }=(3+\sin (\pi x_{n})) h^{-2}+h^{-1}/2,c^{(n) }
=(3+\sin (\pi x_{n})) h^{-2}-h^{-1}/2,\  \\
z^{(n) }=-2\tau ^{-2}-2(3+\sin (\pi x_{n}))h^{-2},r=\tau ^{-2}.
\end{gather*}
Then, we have
\begin{gather*}
A^{(n) }=\operatorname{diag}\{0, a^{(n) }, a^{(n)},\ldots , a^{(n) }, 0\},
\\C^{(n) }=\operatorname{diag}\{0, c^{(n) }, c^{(n) },\ldots , c^{(n)
}, 0\} ,  \\
g_{n}^0=\phi _{n}-\zeta _{n},\quad
g_{n}^{N}=\eta _{n},\quad  n=\overline{1,M-1}
\end{gather*}
for both schemes \eqref{app1v} and \eqref{app2v}.
The elements $b_{i,j}^{(n) }$ of matrix $B^{(n) }$ are defined by
\begin{gather*}
b_{i,i}^{(n) }=z^{(n) },\quad
b_{i-1,i}^{(n)}=b_{i,i-1}^{(n) }=r,\quad i=\overline{2,N};b_{1,1}^{(n)
}=1,b_{1,l_0}^{(n) }=-1, \quad b_{N+1,N+1}^{(n) }=1, \\
b_{N+1,l_{1}}^{(n) }=-\frac{1}{5},\quad
b_{N+1,l_{2}}^{(n)}=-\frac{3}{10},\quad
b_{N+1,l_{3}}^{(n) }=-\frac{1}{2},\quad
b_{N+1,l_{3}+1}^{(n) }=\frac{1}{4}, \\
b_{i,j}^{(n) }=0\quad\text{in other cases }
\end{gather*}
for problem \eqref{app1v}, and
\begin{gather*}
b_{i,i}^{(n) }=z^{(n) }, \quad
b_{i-1,i}^{(n)}=b_{i,i-1}^{(n) }=r, \quad i=\overline{2,N};\quad
b_{1,1}^{(n) }=1,\quad b_{1,l_0}^{(n) }=\mu _0-1, \\
b_{1,l_0+1}^{(n) }=-\mu _0,\quad
b_{N+1,N+1}^{(n)}=1,b_{N+1,l_{1}+1}^{(n) }=-\frac{\mu _{1}}{5},\quad
b_{N+1,l_{1}}^{(n) }=\frac{\mu _{1}-1}{5}, \\
b_{N+1,l_{2}+1}^{(n) }=-\frac{3\mu _{2}}{10},\quad
b_{N+1,l_{2}}^{(n) }=\frac{3(\mu _{2}-1) }{10},b_{N+1,l_{3}+1}^{
(n) }=-\frac{\mu _{3}}{2},\quad
b_{N+1,l_{3}}^{(n) }=\frac{\mu _{3}-1}{2}, \\
b_{i,j}^{(n) }=0\quad \text{in other cases }
\end{gather*}
for problem \eqref{app2v}.

In the second stage, we find $\{ p_{n}\}$ by \eqref{p1} and
\eqref{p2}, respectively.

 In the third stage, $\{ v_{n}^k\} $
are calculated by $v_{n}^k=u_{n}^k+\zeta _{n}-v_{n}^{l_0}$, and
$v_{n}^k=v_{n}^k+\zeta _{n}-(\mu _0u_{n}^{l_0+1}-(\mu
_0-1) u_{n}^{l_0}) $, for the first and second order of
approximations, respectively.

By using MATLAB program and modified Gauss method (\cite{33}), numerical
calculations are carried out for $N=M=20, 40, 80, 160$.
In the Tables \ref{table1}--\ref{table3},
we give error of  numerical solution for inverse problem \eqref{n2} and
auxiliary NBVP. Table 1 contains error between exact solution of NBVP and
solutions derived by difference schemes \eqref{app1v} and \eqref{app2v} .
Table \ref{table2} and Table \ref{table3} contain error between exact and approximately solution
of overdetermined problem \eqref{n2} for $p$ and $u$, respectively. Tables
\ref{table1}--\ref{table3} show that the second order of ADS is more accurate
comparing with the first order of ADS.

\begin{table}[htb]
\caption{Error for NBVP} \label{table1}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|c|c|c|c|c|}
\hline
order of ADS & $N=M=20$ & $N=M=40$ & $N=M=80$ & $N=M=160$ \\ \hline
first & 0.65402 & 0.31258 & 0.1528 &
7.55$\times 10^{-2}$ \\
second  & 0.10305 & 1.37$\times 10^{-2}$ & 1.98$\times 10^{-3}$
 & 3.50$\times 10^{-4}$ \\ \hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[htb]
\caption{Error of $p$ for problem \eqref{n2}} \label{table2}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|c|c|c|c|c|} \hline
order of ADS & $N=M=20$ & $N=M=40$ & $N=M=80$ & $N=M=160$ \\ \hline
first & 0.70016 & 0.35855 & 0.18181 &
9.15$\times 10^{-2}$ \\
second  & 0.13998 & 2.32$\times 10^{-2}$ &
4.78$\times 10^{-3}$ & 1.13$\times 10^{-3}$ \\ \hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[htb]
\caption{Error of $v$ for problem \eqref{n2}} \label{table3}
\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
order of ADS& $N=M=20$ & $N=M=40$ &$ N=M=8$0 & $N=M=160$ \\ \hline
first  & 5.31$\times 10^{-2}$ & 2.40$\times 10^{-2}$
& 1.16$\times 10^{-2}$ & 5.69$\times 10^{-3}$ \\
second  & 5.45$\times 10^{-3}$ & 6.45$
\times 10^{-4}$ & 8.51$\times 10^{-5}$ &
1.49$\times 10^{-5}$ \\ \hline
\end{tabular}
\end{center}
\end{table}

\subsection{Three dimensional example}

Consider the three dimensional overdetermined elliptic two point NBVP
\begin{equation}
\begin{gathered}
\begin{aligned}
&-\frac{\partial ^2v}{\partial t^2}(t,x,y)-\frac{\partial ^2v}{\partial
x^2}(t,x,y)-\frac{\partial ^2v}{\partial y^2}(t,x,y)+v(t,x,y) \\
&=g(t,x,y)+p(x,y),\quad  0<x<1,\quad 0<y<1,\quad 0<t<1,
\end{aligned} \\
v(0,x,y)=\phi (x,y),\quad  v(0.26,x,y)=\zeta (x,y) , \\
v(1,x,y)-\frac{1}{2}v(0.38,x,y)-\frac{1}{2}v(0.88,x,y)=\eta (x,y)  \\
0\leq x\leq 1, \quad 0\leq y\leq 1, \\
v_{x}(t,0,y)=v_{x}(t,1,y)=0,\quad  0\leq y\leq 1, 0<t<1, \\
v_{y}(t,x,0)=v_{y}(t,x,1)=0,\quad  0\leq x\leq 1, 0<t<1,
\end{gathered}   \label{aaaa3}
\end{equation}
where
\begin{gather*}
g(t,x,y)=2\pi ^2e^{-t}\cos (\pi x)\cos (\pi y), \quad
 \phi (x,y) =2\cos (\pi x)\cos (\pi y), \\
\zeta (x,y)=(e^{-0.26}+1) \cos (\pi x)\cos (\pi y),\\
\eta (x,y)=(e^{-1}-\frac{1}{2}e^{-0.38}-\frac{1}{2}
e^{-0.88}) \cos (\pi x)\cos (\pi y).
\end{gather*}
The pair of functions 
\[
p(x,y)=(2\pi ^2+1)\cos (\pi x)\cos (\pi y),\quad
v(t,x,y)=(e^{-t}+1) \cos (\pi x)\cos (\pi y)
\]
is an exact solution of \eqref{aaaa3}.

We use the notation $[0,1] _{\tau }\times [0,1] _{h}^2$  for set of grid
 points depending on the small parameters $\tau $ and $h$
\begin{gather*}
[0,1]_{\tau }\times [0,1]_{h}^2=\{(t_{k},x_{n},y_{m}):t_{k}=k\tau ,\quad
 k=0, \ldots, N, \\
x_{n}=nh,\quad  y_{m}=mh,\quad  n,m=0, \ldots, M,\; N\tau =1,Mh=1\}.
\end{gather*}
Also suppose that
\begin{gather*}
\lambda _0=0.26,\quad \lambda _{1}=0.38,\lambda _{2}=0.88,\quad
l_{i}=[ \frac{\lambda _{i}}{\tau }] ,\quad
\mu _{i}=-l_{i}+\frac{\lambda _{i}}{\tau },\quad i=0,1,2; \\
\varphi _{m,n}=\varphi (x_{n},y_{m}) ,\quad
\psi _{m,n}=\psi (x_{n},y_{m}) ,\quad \zeta _{m,n}=\xi (x_{n},y_{m}) ,\quad
n,m=\overline{0,M}; \\
g_{m,n}^k=g(t_{k},x_{n}, y_{m}),\quad  k=\overline{0,N},\; n,m=\overline{0,M}.
\end{gather*}

In the first stage, we can write the first and order of ADSs for approximately
solution of corresponding NBVP in the following forms:
\begin{equation}
\begin{gathered}
\begin{aligned}
&-\frac{u_{m,n}^{k+1}-2u_{m,n}^k+u_{m,n}^{k-1}}{\tau ^2}-\frac{
u_{m,n+1}^k-2u_{m,n}^k+u_{m,n-1}^k}{h^2} \\
&-\frac{u_{m+1,n}^k-2u_{m,n}^k+u_{m-1,n}^k}{h^2}
+u_{m,n}^k\\
&=g_{m,n}^k,\quad k=\overline{1,N-1}, \; m,n=\overline{1,M-1},
\end{aligned} \\
u_{0,n}^k-u_{1,n}^k=0,\quad
u_{M,n}^k-u_{M-1,n}^k=0,\quad k=\overline{1,N-1},\; n=\overline{1,M-1},  \\
u_{m,0}^k-u_{m,1}^k=0, \quad u_{m,M}^k-u_{m,M-1}^k=0,
\quad k=\overline{1,N-1},\; m=\overline{1,M-1},  \\
u_{m,n}^{1}-u_{m,n}^0=\tau \varphi _{m,n},\quad
u_{m,n}^{N}-u_{m,n}^{N-1}-\frac{1}{2}(u_{m,n}^{l_{1}+1}-u_{m,n}^{l_{1}})  \\
-\frac{1}{2}(u_{m,n}^{l_{2}+1}-u_{n}^{l_{2}}) =\psi _{m,n},\quad
 m,n=\overline{1,M-1},
\end{gathered}   \label{app1v3}
\end{equation}
and
\begin{equation}
\begin{gathered}
\begin{aligned}
&-\frac{u_{m,n}^{k+1}-2u_{m,n}^k+u_{m,n}^{k-1}}{\tau ^2}-\frac{
u_{m,n+1}^k-2u_{m,n}^k+u_{m,n-1}^k}{h^2} \\
&-\frac{u_{m+1,n}^k-2u_{m,n}^k+u_{m-1,n}^k}{h^2}+u_{m,n}^k
=g_{m,n}^k, \quad k=\overline{1,N-1},\; m,n=\overline{1,M-1},
\end{aligned} \\
3u_{0,n}^k-4u_{1,n}^k+u_{2,n}^k=0, \quad
3u_{M,n}^k-4u_{M-1,n}^k+u_{M-2,n}^k=0,
\\
k=\overline{1,N-1},n=\overline{1,M-1},  \\
3u_{m,0}^k-4u_{m,1}^k+u_{m,2}^k=0, \quad
3u_{m,M}^k-4u_{m,M-1}^k+u_{m,M-2}^k=0, \\
k=\overline{1,N-1},\; m=\overline{1,M-1},  \\
-3u_{m,n}^0+4u_{m,n}^{1}-u_{m,n}^2
 =2\tau \varphi _{m,n},\\
\begin{aligned}
&3u_{m,n}^{N}-4u_{m,n}^{N-1}+u_{m,n}^{N-2}
 -\frac{1}{2}\Big[ (3+2\mu _{1}) u_{m,n}^{l_{1}+1}-(4+4\mu
_{1}) u_{m,n}^{l_{1}}\\
&+(1+2\mu _{1}) u_{m,n}^{l_{1}}\Big]
 -\frac{1}{2}\left[ (3+2\mu _{2}) u_{m,n}^{l_{2}+1}-(4+4\mu
_{2}) u_{m,n}^{l_{2}}+(1+2\mu _{2}) u_{m,n}^{l_{2}}\right] \\
&=2\tau \psi _{m,n},\quad m,n=\overline{1,M-1},
\end{aligned}
 \end{gathered}  \label{app2v3}
\end{equation}
respectively.

In the second stage, $p_{m,n}$ is calculated by formulas by
\eqref{p1} and \eqref{p2}, respectively.

In the last stage, calculation of $\{ v_{n}^k\} $ is carried out by
\[
v_{m,n}^k=u_{m,n}^k+\zeta _{n}-u_{m,n}^{l_0},
v_{m,n}^k=u_{m,n}^k+\zeta _{m,n}-(\mu _0u_{m,n}^{l_0+1}-(
\mu _0-1) u_{m,n}^{l_0}) \text{ }
\]
in the cases corresponding to first and second order approximations.

Problems \eqref{app1v3} and \eqref{app2v3} can be presented in
the matrix form
\begin{equation}
\begin{gathered}
Au_{n+1}+Bu_{n}+Cu_{n-1}=I g_{n},\quad n=\overline{1,M-1}, \\
u_0-u_{1}=\overrightarrow{0},\quad u_{M}-u_{M-1}=\overrightarrow{0},
\end{gathered}  \label{app1v3matr}
\end{equation}
and
\begin{equation}
\begin{gathered}
Au_{n+1}+Bu_{n}+Cu_{n-1}=I g_{n},\quad n=\overline{1,M-1}, \\
3u_0-4u_{1}+u_{1}=\overrightarrow{0},\quad 3u_{M}-4u_{M-1}+u_{M-1}=
\overrightarrow{0},
\end{gathered}   \label{app2v3matr}
\end{equation}
respectively.

Note that $A, B, C, I$ are  square matrices with $(N+1)^2(M+1)^2$ elements,
and $I$ is the identity matrix, $g_{s}$ and $u_{s}$ $(s=n-1,n,n+1)$ are the
column matrices with $(N+1)(M+1)$ elements such that
\begin{gather*}
u_{s} =\begin{bmatrix}
u_{0,s}^0 & \dots & u_{0,s}^{N} & u_{1,s}^0 & \dots & u_{1,s}^{N} &
\dots & u_{M,s}^0 & \dots & v_{M,s}^{N}
\end{bmatrix} ^{t}, \\
g_{s} = \begin{bmatrix}
g_{0,s}^0 & \dots & g_{0,s}^{N} & g_{1,s}^0 & \dots & g_{1,s}^{N} &
\dots & g_{M,s}^0 & \dots & g_{M,s}^{N}
\end{bmatrix}^{t}.
\end{gather*}
Denote
\begin{gather*}
a = \frac{1}{h^2},b=1+\frac{2}{\tau ^2}+\frac{4}{h^2},r=\frac{1}{\tau
^2}, \\
E =\operatorname{diag}(0, a, a,\ldots , a, 0),\quad
  O=O_{(N+1) \times (N+1)}.
\end{gather*}
Then
\begin{gather*}
A=C=\begin{bmatrix}
O & O & \dots & O & O \\
O & E & \dots & O & O \\
\dots & \dots & \ddots & \dots & \dots \\
O & O & \dots & E &  \\
O & O & \dots & O & O
\end{bmatrix}, \\
B=\begin{bmatrix}
Q & W & Z & \dots & O & O & O \\
O & D & O & \dots & O & O & O \\
O & O & D & \dots & O & O & O \\
\dots & \dots & \dots & \ddots & \dots & \dots & \dots \\
O & O & O & \dots & O & O & O \\
O & O & O & \dots & O & D & O \\
O & O & O & \dots & Z & W & Q
\end{bmatrix},\\
Q=I_{(N+1) \times (N+1) }, \quad
W=-I_{(N+1) \times (N+1) },\quad Z=O, \\
d_{i,i}=b,\quad d_{i-1,i}=r,\quad d_{i,i-1}=r,\quad i=\overline{2,N};\quad
 d_{1,1}=-1, \quad d_{1,2}=1,\\
d_{N+1,N+1}=1,\quad d_{N+1,N}=-1,\quad
d_{N+1,l_{1}}=-\frac{1}{2}, \quad d_{N+1,l_{2}}=-\frac{1}{2}, \\
d_{N+1,l_{1}+1}=\frac{1}{2},\quad d_{N+1,l_{2}+1}=\frac{1}{2}, \\
d_{i,j}=0,\quad \text{for other cases,} \\
g_{m,n}^0=\tau \varphi _{m,n},\quad  g_{m,n}^{N}=\tau \psi _{m,n},\quad
 n,m=1,\ldots ,M-1
\end{gather*}
for first order of ADS, and
\begin{gather*}
Q=3I_{(N+1) \times (N+1) },W=-4I_{(N+1)\times (N+1) }, \quad
Z=I_{(N+1) \times (N+1) },\\
d_{i,i}=b,\quad d_{i-1,i}=r,\quad d_{i,i-1}=r,\quad i=\overline{2,N};\quad
d_{1,1}=-3, \\
d_{1,2}=4,\quad d_{1,3}=-1,\quad d_{N+1,N+1}=3,\quad d_{N+1,N}=-4,\quad
d_{N+1,N-1}=-1, \\
d_{N+1,l_{1}+1}=-\frac{1}{2}(3+2\mu _{1}) ,\quad
d_{N+1,l_{1}}=2+2\mu _{1}, \\
d_{N+1,l_{1}-1}=-\frac{1}{2}(1+2\mu _{1}) ,\quad
d_{N+1,l_{2}+1}=- \frac{1}{2}(3+2\mu _{2}) , \\
d_{N+1,l_{2}}=2+2\mu _{2},\quad
d_{N+1,l_{2}-1}=-\frac{1}{2}(1+2\mu_{2}) , \\
d_{i,j}=0,\quad \text{for other $i$ and }j; \\
g_{m,n}^0=2\tau \varphi _{m,n}, \quad
g_{m,n}^{N}=2\tau \psi _{m,n},\quad  n,m=\overline{1,M-1}
\end{gather*}
for second order of ADS.

Numerical calculations are carried out by using MATLAB program and modified
Gauss method \cite{33} for $N=M=10,20,40$. In Tables
\ref{table4}--\ref{table6}, the numerical results for both order of ADSs
are given. Table \ref{table4} contains error between exact and approximately
solutions of NBVP. Table \ref{table5} presents error for $u$.
Tables \ref{table6} includes error for $p$. These tables show that the
second order of ADS is more
accurate comparing to the first order of ADS.

\begin{table}[htb]
\caption{Error analysis for NBVP} \label{table4}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|c|c|c|c|}
\hline
Difference scheme & $N=M=10$ & $N=M=20$ & $N=M=40$ \\ \hline
First order of ADS &  0.0822 & 0.0392 & 0.0169 \\
Second order of ADS & 0.0226 & 2.02$\times 10^{-3}$ & 1.33$\times 10^{-4}$
\\ \hline
\end{tabular}
\end{center}
\end{table}


\begin{table}[htb]
\caption{Error analysis for $p$ in example  \eqref{aaaa3}} \label{table5}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|c|c|c|c|c|}
\hline
Difference scheme & $N=M=10$ & $N=M=20$ & $N=M=40$ \\ \hline
First order of ADS & 0.8207 & 0.1693 & 0.1029 \\
Second order of ADS & 0.3266 & 0.0592 & 0.0106 \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[htb]
\caption{Error analysis for v in example \eqref{aaaa3}} \label{table6}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|c|c|c|c|}
\hline
Difference scheme & N=10,M=10 & N=20,M=20 & N=40,M=40 \\ \hline
First order of ADS & 0.0291 & 0.0135 & 4.06$\times 10^{-3}$ \\ \hline
Second order of ADS & 0.0053 & 4.68$\times 10^{-4}$ & 3.03$\times 10^{-5}$ \\ \hline
\end{tabular}
\end{center}
\end{table}

\subsection*{Conclusion}

In this research work, inverse elliptic problem with Bitsadze-Samarskii type
multipoint nonlocal and Neumann boundary conditions are discussed. First
and second order of accuracy difference schemes for this problem are
presented. Stability and coercive stability estimates for solutions of
corresponding difference schemes are established. Then, numerical results
for inverse elliptic problem with multipoint Bitsadze-Samarskii type
nonlocal and Neumann boundary conditions in two and three
dimensional test examples are illustrated. Numerical results are carried out
by MATLAB program and short explanation on the realization of algorithm is
given.

Moreover, applying the results of papers \cite{4,12,20} the high order of ADSs
for the numerical solution to the
Bitsadze-Samarskii type overdetermined elliptic problem with Neumann conditions
can be presented.

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