text:
\section{Introduction}

In this work, we consider numerical solutions of the time-harmonic fluid-solid interaction (FSI) problem, the scattering of a time-harmonic acoustic incident wave by an elastic body immersed in a fluid. This problem plays an important role in many fields of application, including exploration seismology, oceanography, and non-destructive testing, and to name a few. We consider the linear, isotropic and bounded elastic body  whose displacement field is modeled by the time-harmonic Navier equation. The medium outside the elastic body is filled with a homogeneous compressible inviscid fluid and the acoustic scattered field is described by the Helmholtz equation together with the Sommerfeld's radiation condition at infinity.

It has been studied extensively in the past several decades to develop efficient and accurate  numerical methods for the solution of time-harmonic FSI problems. One of the popular methods is the boundary integral equation (BIE) method (\cite{LM95,YHX}) which leads us to solving  a system of boundary integral equations defined on the transmission boundary.  In addition to the well-known advantages associated with BIE methods, there also exist some restrictions,  such as eigenvalue issues and hyper-singularity features associated with boundary integral operators, which need to be extraordinarily considered for this method. The second method is the coupling methods, for instance, using the boundary element method (BEM) to solve the acoustic problem while the finite element methods (FEM) is employed to approximate the interior elastic wave leads to a coupled FEM-BEM method (\cite{BM91,DSM121,DSM122,HKR00,MMS04,YRX}).   Advantages of this method include the feasibility  of dealing with elastic mediums with variable coefficients, comparatively less requirement on the shape of artificial boundary and the fact of naturally and exactly  taking into account the Sommerfeld's radiation condition, and to name a few.  The third  way is to introduce an artificial boundary or an absorbing layer so that one can reduce the original unbounded problem to a nonlocal boundary value problem  which could be solved using field equation solvers such as the FEM. In this case, proper absorbing or non-reflected boundary conditions are required to be defined on the artificial boundaries or layers. There are mainly two strategies for the definition of such artificial boundary conditions applied to the exterior time-harmonic  acoustic scattering. One is called the perfectly matched layer (\cite{B94,BP13,CL06}), and the other is  the Dirichlet-to-Neumann (DtN) mapping based on Fourier series expansions (\cite{F83,F84,KG89}).  Coupling of the DtN mapping and the FEM leads to the so-called DtN-FEM (\cite{G99}) which is the method we use in this work to solve the FSI problem.

There exsiting some issues for the DtN-FEM solving the scattering problem. The first one is the limitation on the shape of the artificial boundary, and it has been  considered in earlier works (\cite{NN04,NN06}) where the authors apply   perturbation methods to the DtN mapping so that it can be defined on a perturbed curve of the artificial boundary with regular shape such as a circle or an ellipse.  The second issue is the potential loss of uniqueness for the solution of modified nonlocal boundary value problem due to the truncation of infinite series of DtN mapping. In \cite{GK95}, the authors defined a modified form of the DtN mapping to circumvent this difficulty. This modified form of DtN  mapping is equivalent to the original DtN mapping, and however is not equivalent to the one after truncation (\cite{HW13}).  From the numerical point of view, authors of \cite{HH92} suggested to make a choice of $N\ge kR$, where $N$ is the truncation order of infinite series, $k$ is the wave number and $R$ is the radius of circular artificial boundary.  The third problem is related to the error estimate due to   truncation of the DtN mapping, and the errors are indeed understood (\cite{NN06}) to have an order of exponential decay. For problems of Laplace and  linear elasto-statics, authors of  \cite{HW85,HW92} have derived an priori error estimates indicating such effects  based on the fact that the corresponding sesquilinear form is positive definite. However, it is nontrivial to apply the technique  in \cite{HW85,HW92} to the acoustic and elastic  scattering problems since the corresponding sesquilinear form  is usually indefinite. In this work, we take the FSI problem as the model problem to investigate the last two issues mentioned above for the DtN-FEM.   Without doing  modification, we will give a novel and strait-forward procedure to prove the unique solvability (Section 4.2) of the truncated variational equation for FSI problem instead of a proof of contradiction in (\cite{HNPX11})  where the exterior acoustic scattering problem was considered.  In order to investigate the errors due to the truncation of the infinite series DtN mapping,  we derive a new and more apparent truncation error estimates declaring exponential attenuation between the exact DtN mapping and its truncated one (Theorem 4.2). Based on these two results, we are allowed to derive a priori error estimates (Theorem 5.3 and  5.4) involving the effects of both finite element discretization and infinite series truncation, and in particular, explicitly indicating the order of exponential decay due to the truncation of DtN mapping.

The remainder of the paper is organized as follows. We first describe the classical FSI problem in Section 2, and then reduce the transmission problem to an equivalent nonlocal boundary value problem in Section 3. Essential mathematical analysis for the corresponding variational equation of the nonlocal boundary value problem and its modification due to  truncation of the DtN mapping are discussed in Section 3 and 4, respectively. In Section 5, we establish a priori error estimates including both effects of numerical discretization and for the finite element solution of the modified variational equation. Several numerical experiments are presented to confirm our theoretical results in Section 6 and finally we make a conclusion in Section 7.

\section{Statement of the problem}
\begin{figure}
\centering
{\includegraphics[scale=1.5]{FSFigure.jpg}
\caption{The geometry settings for the fluid-solid interaction problem (\ref{Navier1})-(\ref{RadiationCond}).}\label{fig1}
\end{figure}

Let $\Omega\subset {\mathbb R}^2$ denote a bounded and simply connected domain with sufficiently smooth interface  $\Gamma= \partial\Omega$,  and $\Omega^c= {\mathbb R}^2\setminus\overline{\Omega}\subset {\mathbb R}^2$ be the unbounded exterior region. The domain $\Omega$ is occupied by a linear and isotropic elastic solid determined through the Lam\'e constants $\lambda$ and $\mu$ ($\mu$>0, $\lambda+\mu>0$) and its mass density $\rho>0$, and $\Omega^c$ is filled with compressible, inviscid fluids. We denote by $c_0>0$ the speed of sound in the fluid, $\rho_f>0$ the density of the fluid, $\omega$ the frequency and $k=\omega/c_0$ the wave number. The fluid-solid interaction problem to be considered in this work is to determine the elastic displacement ${\bf u}$ in the solid and the acoustic scattered field $p$ in the fluid provided an incident field $p^{inc}$. It can be described mathematically as follows. Given $p^{inc}$, find ${\bf u}=(u_x,u_y)\in \left(C^2(\Omega)\cap C^1(\overline\Omega)\right)^2$ and $p\in C^2(\Omega^c)\cap C^1(\overline{\Omega^c})$ satisfying
\begin{eqnarray}
\label{Navier1}
\Delta^{*}{\bf u} +   \rho \omega^2{\bf u} &=& {\bf 0}\quad\quad\quad\quad\quad\quad \mbox{in}\quad \Omega,\\
\label{Helmholtz1}
\Delta p + k^2p &=& 0\quad\quad\quad\quad\quad\quad\, \mbox{in}\quad \Omega^c,\\
\label{TransCond1.1}
\omega^2\rho_f{\bf u}\cdot {\bf n} &=& \frac{\partial}{\partial {\bf n}}(p+p^{inc}) \quad \mbox{on}\quad \Gamma,\\
\label{TransCond2.1}
{\bf T}{\bf u} &=& -{\bf n}(p + p^{inc}) \quad \mbox{on}\quad \Gamma,
\end{eqnarray}
and the Sommerfeld radiation condition
\begin{eqnarray}
\label{RadiationCond}
\lim_{r \to \infty} r^{\frac{1}{2}}\left(\frac{\partial p}{\partial r}-ikp\right) =  0,\quad r=|x|,
\end{eqnarray}
uniformly with respect to all $\hat{x}=x/|x|\in\mathbb{S}:=\{x\in{\mathbb R}^2:|x|=1\}$. Here, $\partial/\partial{\bf n}$ is the normal derivative on $\Gamma$ (here and in the sequel, $\bf{n}$ is always the outward unit normal to the boundary), $i=\sqrt{-1}$ is the imaginary unit. $\Delta^{*}$ is the operator defined by
\begin{eqnarray*}
\Delta^* = \mu\Delta + (\lambda + \mu)\nabla\nabla\cdot,
\end{eqnarray*}
and the standard stress operator ${\bf T}$ is defined by
\begin{eqnarray*}
{\bf T}{\bf u} = 2\mu\frac{\partial{\bf u}}{\partial {\bf n}} + \lambda{\bf n}\nabla\cdot {\bf u} + \mu{\bf n}\times\mbox{curl}\,{\bf u}.
\end{eqnarray*}

It is known (\cite{JDS83}) that the problem (\ref{Navier1})-(\ref{RadiationCond}) is not always uniquely solvable due to the occurrence of traction free oscillations for certain geometries and some frequencies $\omega$. These $\omega$ are also known as the Jones frequencies which are inherent to the original model. We conclude from \cite{LM95} the following uniqueness result.

\begin{theorem}
If the surface $\Gamma$ and the material parameters $(\mu,\lambda,\rho)$ are such that there are no traction free solutions, the boundary value problem (\ref{Navier1})-(\ref{RadiationCond}) has at most one solution. Here, we call a nontrivial ${\bf u}_0$ a traction free solution if it solves
\begin{eqnarray*}
\label{Eq:TracEq1}
\Delta^{*}{\bf u}_0 + \rho\omega^2{\bf u}_0 &=& {\bf 0} \quad\mbox{in}\quad \Omega,\\
\label{Eq:TracEq2}
{\bf T}{\bf u}_0 &=&{\bf 0}\quad \mbox{on}\quad\Gamma,\\
\label{Eq:TracEq3}
{\bf u}_0\cdot{\bf n} &=& 0\quad\mbox{on}\quad\Gamma.
\end{eqnarray*}
\end{theorem}

To simplify the presentation throughout the dissertation, we shall denote by $c>0$, $\alpha>0$ and $\beta\ge0$ generic constants whose precise values are not required and may change line by line.

\section{Nonlocal boundary value problem}
\subsection{DtN mapping}
We introduce an artificial circular boundary $\Gamma_R:=\{x\in{\mathbb R}^2:|x|=R\}$ such that $\overline{\Omega}\subset B_R:=\{x\in{\mathbb R}^2:|x|<R\}$. The artificial boundary decomposes the exterior domain $\Omega^c$ into two subdomains. One is the annular region $\Omega_R=B_R\backslash\overline{\Omega}$ between $\Gamma$ and $\Gamma_R$, and the other is the unbounded exterior region $\Omega_R^c=\mathbb{R}^{2}\backslash \overline{B_R}$. On $\Gamma_R$, we impose the exact transparent boundary condition
\begin{eqnarray}
\label{DtNBoundaryCond}
\frac{\partial p}{\partial {\bf n}} = Sp \quad \mbox{on}\quad  \Gamma_R,
\end{eqnarray}
where $S$ is known as the DtN mapping from $H^s(\Gamma_R)$ to $H^{s-1}(\Gamma_R)$ defined by
\begin{eqnarray}
\label{DtNmap}
S\varphi:={\sum\limits_{n=0}^{\infty}} \ '\frac{ {k H_{n}^{(1)}}^{\prime}{(kR)}}{\pi H_{n}^{(1)}(kR)}\int_ {0}^{2 \pi}\varphi(R,\phi)\cos(n(\theta-\phi))d\phi \quad\mbox{for all}\quad \varphi \in H^{s}(\Gamma_R),1/2\leq s\in {\mathbb R},
\end{eqnarray}
or equivalently,
\begin{eqnarray}
\label{DtNmap1}
S\varphi:=\sum\limits_{n\in{\mathbb Z}}\frac{ {k H_{n}^{(1)}}^{\prime}{(kR)}}{2\pi H_{n}^{(1)}(kR)}\int_ {0}^{2 \pi}\varphi(R,\phi)e^{in(\theta-\phi)}d\phi \quad\mbox{for all}\quad \varphi \in H^{s}(\Gamma_R),1/2\leq s\in {\mathbb R}.
\end{eqnarray}
Here and in the sequel, the prime behind the summation means that the first term in the summation is multiplied by $1/2$. The transparent boundary condition (\ref{DtNBoundaryCond}) defines a nonlocal condition for $p$ on $\Gamma_R$ since the Dirichlet data $p$ over the entire boundary $\Gamma_R$ are required to compute the Neumann date ${\partial p}/{\partial {\bf n}}$ at a single point $x\in \Gamma_R$. We have the following mapping property for $S$ (see Theorem 3.1 in \cite{HNPX11}).
\begin{theorem}
\label{Theorem3.1}
For any constant $s\ge 1/2$, the DtN mapping $S$ defined by (\ref{DtNmap}) or (\ref{DtNmap1}) is a bounded linear operator from $H^{s}(\Gamma_{R})$ to $H^{s-1}(\Gamma_R)$, that is, there exists a constant $c>0$ independent of $\varphi\in H^{s}(\Gamma_R)$ such that
\begin{eqnarray}
\label{DtNcontinuity}
\|S\varphi\|_{H^{s-1}(\Gamma_R)} \leq c\|\varphi\|_{H^s(\Gamma_R)}.
\end{eqnarray}
\end{theorem}

\subsection{Reduced problem}
Now, the transmission problem (\ref{Navier1})-(\ref{RadiationCond}) can be equivalently replaced by the following nonlocal boundary value problem: Given $p^{inc}$, find ${\bf u}\in \left(C^2(\Omega)\cap C^1(\overline\Omega)\right)^2$ and $p\in C^2(\Omega_R)\cap C^1(\overline{\Omega_R})$ such that
\begin{eqnarray}
\label{Navier2}
\Delta^{*}{\bf u} +   \rho \omega^2{\bf u} &=& {\bf 0}\quad\quad\quad\quad\quad\quad \mbox{in}\quad \Omega,\\
\label{Helmholtz2}
\Delta p + k^2p &=& 0\quad\quad\quad\quad\quad\quad\, \mbox{in}\quad \Omega_R,\\
\label{TransCond1.2}
\omega^2\rho_f{\bf u}\cdot {\bf n} &=& \frac{\partial}{\partial {\bf n}}(p+p^{inc}) \quad \mbox{on}\quad \Gamma,\\
\label{TransCond2.2}
{\bf T}{\bf u} &=& -{\bf n}(p + p^{inc}) \quad \mbox{on}\quad \Gamma,\\
\label{DtNCond}
\frac{\partial p}{\partial {\bf n}} &=& Sp \quad\quad\quad\quad\quad\;\; \mbox{on}\quad  \Gamma_R.
\end{eqnarray}
The   uniqueness result for the nonlocal boundary value problem (\ref{Navier2})-(\ref{DtNCond}) is established in the next Theorem.
\begin{theorem}
\label{Theorem3.2}
If the surface $\Gamma$ and the material parameter $(\mu,\lambda,\rho)$ are such that there are no traction free solutions, the nonlocal boundary value problem (\ref{Navier2})-(\ref{DtNCond}) has at most one solution.
\end{theorem}
\begin{proof}
It is sufficient to prove that the corresponding homogeneous boundary value problem of (\ref{Navier2})-(\ref{DtNCond}) has only the trivial solution. Suppose that $({\bf u}_0,p_0)$ is the solution of the corresponding homogeneous value problem of (\ref{Navier2})-(\ref{DtNCond}). Now let $p_1$ be the solution of the following exterior Dirichlet problem for the Helmholtz equation
\begin{eqnarray}
\label{ExDiriP1}
\Delta p_1 +k^{2} p_1 &=& 0  \quad\;\; \mbox{in}\quad \Omega_R^c,\nonumber \\
\label{ExDiriP2}
p_1&=&p_0 \quad \mbox{on}\quad \Gamma_R,\\
\label{ExDiriP3}
\lim\limits _{r\rightarrow \infty}r^{\frac{1}{2}}\left(\frac{\partial p_1}{\partial r}-ikp_1\right)&=& 0,\quad r=|x|.\nonumber
\end{eqnarray}
Then $p_1$ has the explicit representation form
\begin{eqnarray}
\label{Reprep1}
p_1(r,\theta)=\sum\limits_{n\in\mathbb{Z}}a_{n}H_{n}^{(1)}(kr)e^{in\theta},\quad\forall \,r\geq R,\quad \theta\in[0,2\pi).
\end{eqnarray}
Calculating the normal derivative of (\ref{Reprep1}) and taking the limit as $r\rightarrow R,$ we obtain on $\Gamma_R$,
\begin{eqnarray*}
\frac{\partial p_1}{\partial {\bf n}}&=&\sum\limits_{n\in\mathbb{Z}}\frac{k{ H_{n}^{(1)}}^{\prime}{(kR)}}{2\pi H_{n}^{(1)}(kR)}\int_{0}^{2\pi}{p_1(R,\theta)e^{in(\theta-\phi)}d\phi} \nonumber \\
&=& Sp_1 \nonumber \\
&=& Sp_0
\end{eqnarray*}
because of the boundary condition (\ref{ExDiriP2}). In addition, the nonlocal boundary condition (\ref{DtNCond}) gives
\begin{eqnarray*}
\frac{\partial p_0}{\partial {\bf n}}= Sp_0 \quad  \mbox{on }\Gamma_R.
\end{eqnarray*}
Therefore,
\begin{eqnarray}
\label{Dtiffp01}
\frac{\partial p_1}{\partial {\bf n}}-\frac{\partial p_0}{\partial {\bf n}}= Sp_0-Sp_0=0 \quad  \mbox{on}\quad\Gamma_R.
\end{eqnarray}
If we define the function $({\bf u},p)\in \left(C^{2}(\Omega)\cap C^{1}(\overline{\Omega})\right)^2\times (C^{2}(\Omega_R\cup \Omega_R^c)\cap  C^{1}(\overline{\Omega_R}))$ as
\begin{eqnarray*}
\boldsymbol({\bf u},p)=
\begin{cases}
({\bf u}_0,p_0)\quad &\mbox{in}\quad \Omega \times \Omega_R ,\cr ({\bf u}_0,p_1)\quad &\mbox{in}\quad \Omega \times\Omega_R^c,\nonumber
\end{cases}
\end{eqnarray*}
then by (\ref{ExDiriP2}) and (\ref{Dtiffp01}), we can see that both $p$ and ${\partial p}/{\partial {\bf n}}$ are continuous across the interface $\Gamma_R$. Hence, $({\bf u},p)$ is the solution of the corresponding homogeneous boundary value problem of (\ref{Navier1})-(\ref{RadiationCond}). Then the assumption leads to
\begin{eqnarray*}
({\bf u}_0,p_0)=({\bf 0},0).
\end{eqnarray*}
This completes the proof.
\end{proof}

\subsection{Weak formulation}
Now we study the weak formulation of (\ref{Navier2})-(\ref{DtNCond}) which reads: Given $p^{inc}$, find ${\bf U}=({\bf u},p)\in \mathcal{H}^1= \left(H^{1}(\Omega)\right)^2\times H^{1}(\Omega_R)$ such that
\begin{eqnarray}
\label{Weak1}
A({\bf U},{\bf V})=B({\bf U},{\bf V})+b(p,q)=\ell({\bf V}),\quad\forall\ \bf{V}=({\bf v},q)\in\mathcal{H}^1,
\end{eqnarray}
where
\begin{eqnarray*}
\label{B}
B({\bf U},{\bf V})&=& a_{1}({\bf u},{\bf v})+a_{2}(p,q)+a_{3}({\bf u},q)+a_{4}(p,{\bf v}),
\end{eqnarray*}
and
\begin{eqnarray*}
\label{a1}
a_{1}({\bf u},{\bf v}) &=&\int_{\Omega}{ \left[\lambda(\nabla\cdot{\bf u})(\nabla\cdot\overline{\bf v})+\frac{\mu}{2}\left(\nabla{\bf u}+(\nabla{\bf u})^T\right):\left(\nabla\overline{\bf v}+(\nabla\overline{\bf v})^T\right)- \rho\omega^{2}{\bf u} \cdot \overline {\bf v}\right] dx},\\
\label{a2}
a_{2}(p,q)&=&\int_{\Omega_R}{ \left(\nabla p\cdot\nabla\overline{q}-k^2p\overline{q}\right) dx},\\
\label{a3}
a_{3}({\bf u},q) &=& \rho_f\omega^2\int_{\Gamma}{{\bf u}\cdot {\bf n} \overline q ds},\\
\label{a4}
a_{4}(p,{\bf v}) &=& \int_{\Gamma}{{\bf n} p\cdot \overline {\bf v} ds},\\
\label{b}
b(p,q) &=& -\int_{\Gamma_R}{(Sp)\overline q ds}
\end{eqnarray*}
are sesquilinear forms defined on $\left(H^{1}(\Omega)\right)^2\times \left(H^{1}(\Omega)\right)^2$, $H^{1}(\Omega_R)\times H^{1}(\Omega_R)$, $\left(H^{1}(\Omega)\right)^2\times H^{1}(\Omega_R)$, $H^{1}(\Omega_R)\times \left(H^{1}(\Omega)\right)^2$ and $H^{1}(\Omega_R)\times H^{1}(\Omega_R)$, respectively, and $\ell$ defined by
\begin{eqnarray*}
\label{lv}
\ell({\bf V})=\int_{\Gamma}{\frac{\partial p^{inc}}{\partial {\bf n}}\overline qds}-\int_{\Gamma}{{\bf n}p^{inc}\cdot\overline {\bf v}ds},
\end{eqnarray*}
is a linear functional on $\mathcal{H}^1$ dependent on $\left(p^{inc},{\partial p^{inc}}/{\partial {\bf n}}\right)\in H^{1/2}(\Gamma)\times H^{-1/2}(\Gamma).$

\begin{remark}
The double dot notation appeared in the above formulation is understood in the following way. If tensors ${\bf A}$ and ${\bf B}$ have rectangular Cartesian components $a_{ij}$ and $b_{ij}$, $i,j=1,2$, respectively, then the double contraction of ${\bf A}=(a_{ij})$ and ${\bf B}=(b_{ij})$ is
\begin{eqnarray*}
{\bf A}:{\bf B}=\sum\limits_{i = 1}^2 {\sum\limits_{j = 1}^2 {{a_{ij}}{b_{ij}}}}.
\end{eqnarray*}
\end{remark}

\begin{theorem}
\label{Theorem3.4}
The sesquilinear form $A(\cdot,\cdot)$ in (\ref{Weak1}) satisfies
\begin{eqnarray*}
\label{Weak1boundness}
A({\bf U},{\bf V})|\leq c\|{\bf U}\|_{\mathcal{H}^1}\|{\bf V}\|_{\mathcal{H}^1},\quad\forall\ {\bf U},{\bf V}\in{\mathcal{H}^1}.
\end{eqnarray*}
where $c$ is the continuity constant independent of ${\bf U}$ and ${\bf V}$.
\end{theorem}
\begin{proof}
We begin with the sesquilinear form $a_{1}(\cdot,\cdot)$. The Cauchy-Schwarz inequality yields
\begin{eqnarray}
\label{a1continuity}
|a_{1}({\bf u},{\bf v})| &\leq& (\lambda+8\mu)\|{\bf u}\|_{\left(H^1(\Omega)\right)^2}\|{\bf v}\|_{\left(H^1(\Omega)\right)^2}+\rho\omega^{2}\|{\bf u}\|_{\left(H^0(\Omega)\right)^2}\|{\bf v}\|_{\left(H^0(\Omega)\right)^2} \nonumber\\
&\leq& c\|{\bf u}\|_{\left(H^1(\Omega)\right)^2}\|{\bf v}\|_{\left(H^1(\Omega)\right)^2},
\end{eqnarray}
where $c>0$ is a constant independent of ${\bf u}$ and ${\bf v}$. Similarly, we have
\begin{eqnarray}
\label{a2continuity}
|a_{2}(p,q)| \leq c\|p\|_{H^1(\Omega_R)}\|q\|_{H^1(\Omega_R)},
\end{eqnarray}
where $c>0$ is a constant independent of $p$ and $q$. Also, according to the Cauchy-Schwarz inequality and the trace theorem, we know
\begin{eqnarray}
\label{a3continuity}
|a_{3}({\bf u},q)| &\leq& c\|{\bf u}\|_{\left(H^{1/2}(\Gamma)\right)^2}\|q\|_{H^{1/2}(\Gamma)} \nonumber\\
&\leq& c\|{\bf u}\|_{\left(H^1(\Omega)\right)^2}\|q\|_{H^1(\Omega_R)}
\end{eqnarray}
and
\begin{eqnarray}
\label{a4continuity}
|a_{4}(p,{\bf v})| \leq c\|p\|_{H^1(\Omega_R)}\|{\bf v}\|_{\left(H^1(\Omega)\right)^2},
\end{eqnarray}
where $c>0$ in (\ref{a3continuity})-(\ref{a4continuity}) are all constants independent of ${\bf U}$ and ${\bf V}$. Next, we consider the sesquilinear form $b(\cdot,\cdot)$. The Cauchy-Schwarz inequality, the boundedness of the DtN mapping $S$ and the trace theorem give
\begin{eqnarray}
\label{bcontinuity}
|b(p,q)| &\leq& \|Sp\|_{H^{-1/2}(\Gamma_R)}\|q\|_{H^{1/2}(\Gamma_R)}\nonumber\\
&\leq& c\|p\|_{H^{1/2}(\Gamma_R)}\|q\|_{H^{1/2}(\Gamma_R)}\nonumber\\
&\leq& c\|p\|_{H^{1}(\Omega_R)}\|q\|_{H^{1}(\Omega_R)},
\end{eqnarray}
where $c>0$ is a constant independent of $p$ and $q$. Therefore, by combining (\ref{a1continuity})-(\ref{bcontinuity}) and applying the arithmetic-geometric mean inequality, we complete the proof.
\end{proof}

\begin{theorem}
\label{Theorem3.5}
The sesquilinear form $A(\cdot,\cdot)$ in (\ref{Weak1}) satisfies a Garding's inequality taking the form
\begin{eqnarray*}
\label{Weak1garding}
\mbox{Re}\,\{A({\bf V},{\bf V})\}\geq \alpha\|{\bf V}\|_{\mathcal{H}^1}^2-\beta\left(\|{\bf v}\|_{(H^{1/2+\varepsilon}(\Omega))^2}^2+\|q\|_{H^{1/2+\varepsilon}(\Omega_R)}^2\right)  ,\quad\forall\,{\bf V}=({\bf v},q)\in{\mathcal{H}^1}.
\end{eqnarray*}
where $\alpha>0,\beta\geq0$ and $0<\varepsilon<1/2$ are constants independent of ${\bf V}$.
\end{theorem}
\begin{proof}
We first consider the sesquilinear form $a_{1}(\cdot,\cdot)$. Suppose ${\bf v}=(v_1,v_2)$ and from the Korn's inequality (\cite{H95}) we know that there exists a constant $\alpha>0$ such that
\begin{eqnarray}
\label{Korn'sInequality}
\sum\limits_{i,j = 1}^2 {\left\| {{\epsilon _{ij}}({\bf v})} \right\|_{{H^0}(\Omega )}^2}  + \left\| {\bf v} \right\|_{{{\left( {{H^0}(\Omega)} \right)}^2}}^2 \geq \alpha \left\| {\bf v} \right\|_{{{\left( {{H^1}(\Omega)} \right)}^2}}^2,
\end{eqnarray}
where
\begin{eqnarray*}
{{\epsilon _{ij}}({\bf v})}=\frac{1}{2}\left( \frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i} \right),\quad \forall\; 1\leq i,j\leq2.
\end{eqnarray*}
If $\lambda\ge0$, (\ref{Korn'sInequality}) yields
\begin{eqnarray}
\label{a1vv1}
a_{1}({\bf v},{\bf v})+\rho\omega^{2}\left\| {\bf v} \right\|_{{{\left( {{H^0}(\Omega)} \right)}^2}}^2 &=& \lambda\int_{\Omega}{\left|\nabla\cdot{\bf v}\right|^2dx} +\frac{\mu}{2}\int_{\Omega}{\left(\nabla{\bf v}+(\nabla{\bf v})^T\right):\left(\nabla\overline{\bf v}+(\nabla\overline{\bf v})^T\right)dx} \nonumber\\
&\geq& \frac{\mu}{2}\int_{\Omega}{\left(4{\left| {\frac{{\partial {v_1}}}{{\partial {x_1}}}} \right|^2} + 2{\left| {\frac{{\partial {v_1}}}{{\partial {x_2}}} + \frac{{\partial {v_2}}}{{\partial {x_1}}}} \right|^2}+4{\left| {\frac{{\partial {v_2}}}{{\partial {x_2}}}} \right|^2}\right)dx} \nonumber\\
&\geq& 2\mu\left(\sum\limits_{i,j = 1}^2 {\left\| {{\epsilon _{ij}}({\bf v})} \right\|_{{H^0}(\Omega)}^2}  + \left\| {\bf v} \right\|_{{{\left( {{H^0}(\Omega)} \right)}^2}}^2 \right)-2\mu\left\| {\bf v} \right\|_{{{\left( {{H^0}(\Omega)} \right)}^2}}^2 \nonumber\\
&\geq& 2\mu\alpha \left\| {\bf v} \right\|_{{{\left( {{H^1}(\Omega)} \right)}^2}}^2-2\mu \left\| {\bf v} \right\|_{{{\left( {{H^0}(\Omega)} \right)}^2}}^2 .
\end{eqnarray}
If $\lambda<0$, we have
\begin{eqnarray}
\label{a1vv2}
a_{1}({\bf v},{\bf v})+\rho\omega^{2}\left\| {\bf v} \right\|_{{{\left( {{H^0}(\Omega)} \right)}^2}}^2 &\geq& 2(\lambda+\mu)\int_{\Omega}{\left({\left| {\frac{{\partial {v_1}}}{{\partial {x_1}}}} \right|^2}+{\left| {\frac{{\partial {v_2}}}{{\partial {x_2}}}} \right|^2}\right)dx}+\mu\int_{\Omega}{{\left| {\frac{{\partial {v_1}}}{{\partial {x_2}}} + \frac{{\partial {v_2}}}{{\partial {x_1}}}} \right|^2}dx} \nonumber\\
&\geq& 2(\lambda+\mu)\left(\sum\limits_{i,j = 1}^2 {\left\| {{\epsilon _{ij}}({\bf v})} \right\|_{{H^0}(\Omega)}^2}  + \left\| {\bf v} \right\|_{{{\left( {{H^0}(\Omega)} \right)}^2}}^2 \right)-2(\lambda+\mu)\left\| {\bf v} \right\|_{{{\left( {{H^0}(\Omega)} \right)}^2}}^2 \nonumber\\
&\geq& 2(\lambda+\mu)\alpha \left\| {\bf v} \right\|_{{{\left( {{H^1}(\Omega)} \right)}^2}}^2-2(\lambda+\mu)\left\| {\bf v} \right\|_{{{\left( {{H^0}(\Omega)} \right)}^2}}^2.
\end{eqnarray}
Then a combination of (\ref{a1vv1}) and (\ref{a1vv2}) leads to
\begin{eqnarray}
\label{a1vv}
a_{1}({\bf v},{\bf v})+\rho\omega^{2}\left\| {\bf v} \right\|_{{{\left( {{H^0}(\Omega)} \right)}^2}}^2 \geq 2(\lambda+\mu)\alpha \left\| {\bf v} \right\|_{{{\left( {{H^1}(\Omega)} \right)}^2}}^2-2\mu\left\| {\bf v} \right\|_{{{\left( {{H^0}(\Omega)} \right)}^2}}^2.
\end{eqnarray}
Therefore, the Sobolev embedding theorem gives
\begin{eqnarray}
\label{a1garding}
\mbox{Re}\,\{a_{1}({\bf v},{\bf v})\} \geq \alpha\left\| {\bf v} \right\|_{{{\left( {{H^1}(\Omega )} \right)}^2}}^2-\beta\left\| {\bf v} \right\|_{{{\left( {{H^{1/2+\varepsilon}}(\Omega )} \right)}^2}}^2,
\end{eqnarray}
where $\alpha>0,\beta>0$ and $0<\varepsilon<1/2$ are constants. Similarly, for the sesquilinear form $a_{2}(\cdot,\cdot)$, since
\begin{eqnarray*}
\label{a2qq}
a_{2}(q,q) &=& \left\|q \right\|_{H^1(\Omega_R)}^2-(k^2+1)\left\|q \right\|_{H^0(\Omega_R)}^2,
\end{eqnarray*}
we conclude that
\begin{eqnarray}
\label{a2garding}
\mbox{Re}\,\{a_{2}(q,q)\} \geq \|q \|_{{H^1}(\Omega_R)}^2-\beta\left\|q \right\|_{H^{1/2+\varepsilon}(\Omega_R)}^2,
\end{eqnarray}
where $\beta>0$ and $0<\varepsilon<1/2$ are constants. Next, we consider the sesquilinear form $a_{3}(\cdot,\cdot)$. The H\"{o}lder inequality and the arithmetic-geometric mean inequality give
\begin{eqnarray*}
|a_{3}({\bf v},q)| &\leq& \rho_f\omega^2 \left\| {\bf v} \right\|_{{{\left( {{H^0}(\Gamma)} \right)}^2}} \|q \|_{{H^0}(\Gamma)} \\
&\leq& \beta \left(\left\| {\bf v} \right\|_{{{\left( {{H^0}(\Gamma)} \right)}^2}}^2 +\|q \|_{{H^0}(\Gamma)}^2\right).
\end{eqnarray*}
Then by the trace theorem and the Sobolev embedding theorem we have
\begin{eqnarray}
\label{a3garding}
\mbox{Re}\,\{a_{3}({\bf v},q)\} &\geq& -\beta \left(\left\| {\bf v} \right\|_{{{\left( {{H^0}(\Gamma)} \right)}^2}}^2 +\|q \|_{{H^0}(\Gamma)}^2\right) \nonumber\\
&\geq& -\beta \left(\left\| {\bf v} \right\|_{{{\left( {{H^{1/2}}(\Omega)} \right)}^2}}^2+ \|q \|_{{H^{1/2}}(\Omega_R )}^2 \right)\nonumber\\
&\geq& -\beta \left(\left\| {\bf v} \right\|_{{{\left( {{H^{1/2+\varepsilon}}(\Omega)} \right)}^2}}^2 +\left\|q \right\|_{H^{1/2+\varepsilon}(\Omega_R)}^2\right),
\end{eqnarray}
where $\beta>0$ and $0<\varepsilon<1/2$ are constants. Similarly, for the sesquilinear form $a_{4}(\cdot,\cdot)$, it follows that
\begin{eqnarray}
\label{a4garding}
\mbox{Re}\,\{a_{4}(q,{\bf v})\}\geq -\beta \left(\left\| {\bf v} \right\|_{{{\left( {{H^{1/2+\varepsilon}}(\Omega)} \right)}^2}}^2 +\left\|q \right\|_{H^{1/2+\varepsilon}(\Omega_R )}^2\right).
\end{eqnarray}
Finally, for the the sesquilinear form $b(\cdot,\cdot)$, we define
\begin{eqnarray*}
b_{1}(p, q)=\frac{1}{\pi}{\sum\limits_{n=1}^{\infty}}\ n \int_{0}^{2 \pi}\int_ {0}^{2 \pi}p(R,\phi)\overline{q(R,\theta)}\cos(n(\theta-\phi))d\theta d\phi
\end{eqnarray*}
and
\begin{eqnarray*}
b_{2}(p, q)=\frac{kR}{\pi}{\sum\limits_{n=0}^{\infty}} \ '\frac{{H_{n}^{(n-1)}}{(kR)}}{H_{n}^{(1)}(kR)}\int_{0}^{2 \pi}\int_ {0}^{2 \pi}p(R,\phi)\overline{q(R,\theta)}\cos(n(\theta-\phi))d\theta d\phi
\end{eqnarray*}
which reads
\begin{eqnarray}
\label{b=b1-b2}
b(p, q)=b_{1}(p, q)-b_{2}(p, q).
\end{eqnarray}
It follows from Theorem 4.2 in \cite{HNPX11} that
\begin{eqnarray}
\label{b1}
b_{1}(q, q) \ge0,
\end{eqnarray}
and
\begin{eqnarray}
\label{b2}
\left|\mbox{Re}\,\{b_{2}(q, q)\}\right| \le |b_{2}(q, q)| \le c \left\|q \right\|_{H^{0}(\Gamma_R)}^2.
\end{eqnarray}
A combination of (\ref{b=b1-b2})-(\ref{b2}) implies that
\begin{eqnarray}
\label{bgarding}
\mbox{Re}\,\{b(q,q)\}\geq -\beta \left\|q \right\|_{H^{1/2+\varepsilon}(\Omega_R)}^2,
\end{eqnarray}
because of the Sobolev embedding theorem where $\beta>0$ and $0<\varepsilon<1/2$ are constants. Therefore, a combination of (\ref{a1garding})-(\ref{a4garding}) and (\ref{bgarding}) yields (\ref{Weak1garding}). This completes the proof.
\end{proof}

Now, the existence result follows immediately from the Fredholm Alternative theorem: Uniqueness implies the existence. As a consequence of Theorem \ref{Theorem3.4} and \ref{Theorem3.5}, we have the following theorem.

\begin{theorem}
\label{Theorem3.6}
Let the surface $\Gamma$ and the material parameter $(\mu,\lambda,\rho)$ be such that there are no traction free solutions, then the variational equation (\ref{Weak1}) admits a unique solution.
\end{theorem}

\section{Modified nonlocal boundary value problem}
\subsection{Truncated DtN mapping}
In practical computing, one needs to truncate the infinite series of the exact DtN mapping at a finite order to obtain an approximate DtN mapping written as
\begin{eqnarray}
\label{TruncDtNmap}
S^{N}\varphi:={\sum\limits_{n=0}^{N}}\ '\frac{ {k H_{n}^{(1)}}'{(kR)}}{\pi H_{n}^{(1)}(kR)}\int_ {0}^{2 \pi}\varphi(R,\phi)\cos(n(\theta-\phi))d\phi,
\end{eqnarray}
or
\begin{eqnarray}
\label{TruncDtNmap1}
S^N\varphi:=\sum\limits_{n=-N}^N\frac{ {k H_{n}^{(1)}}^{\prime}{(kR)}}{2\pi H_{n}^{(1)}(kR)}\int_ {0}^{2 \pi}\varphi(R,\phi)e^{in(\theta-\phi)}d\phi,
\end{eqnarray}
for all $\varphi \in H^{s}(\Gamma_R), \ s\geq1/2$. Obviously, $S^N$ is also a linear bounded operator. Here, the non-negative integer $N$ is called the truncation order of the DtN mapping. Consequently, we arrive at a truncated nonlocal boundary value problem consisting of (\ref{Navier2})-(\ref{TransCond2.2}) and
\begin{eqnarray}
\label{TruncDtNCond}
\frac{\partial p}{\partial {\bf n}} = S^{N}p \quad  \text {on } \Gamma_R.
\end{eqnarray}
We include the point estimate for the difference of $S$ and $S^N$ proposed in \cite{HNPX11} in the next theorem.
\begin{theorem}
\label{Theorem4.1}
Suppose that the DtN mappings $S$ and $S^{N}$ are defined as in (\ref{DtNmap}) and (\ref{TruncDtNmap}), respectively. Then, for given $\varphi \in H^{s}(\Gamma_R),s \in \mathbb{R}$, there holds, for all $t\geq 0$,
\begin{eqnarray}
\label{EstDiffSSN}
\left\|(S-S^{N})\varphi\right\|_{H^{s-1}(\Gamma_R)}\leq c\frac{\epsilon(N,\varphi)}{N^{t}}\|\varphi\|_{H^{s+t}(\Gamma_R)},
\end{eqnarray}
where $c>0$ is a constant independent of $\varphi$ and $N$, $\epsilon (N,\varphi )$ is a function of the truncation order $N$ and $\varphi$ for given values of $s$ and $t$, generated by the addition of leading terms to the summation with positive terms for the construction of the norm on the space $H^{s+t}(\Gamma_R)$, satisfying $\epsilon(N,\varphi)\le 1$ and $\epsilon(N,\varphi)\rightarrow 0$ as $N\rightarrow \infty$ for all $\varphi\in H^{s}(\Gamma_R)$.
\end{theorem}

From the estimation (\ref{EstDiffSSN}) we can observe that the truncation error between $S$ and $S^N$ tends to zero as $N\rightarrow\infty$. However, when $t=0$, the attenuation property depends on $\epsilon(N,\varphi)$ and we know few about this function. We now give a new and more apparent truncation error estimate for some special $\varphi\in H^{s}(\Gamma_R)$ which will be used in the subsequent discussions.
\begin{theorem}
\label{Theorem4.2}
Suppose that the DtN mappings $S$ and $S^{N}$ are defined as in (\ref{DtNmap1}) and (\ref{TruncDtNmap1}), respectively. Let $p$ be a solution of Helmholtz equation outside $\Omega$ satisfying either (\ref{DtNBoundaryCond}) or (\ref{TruncDtNCond}). Then there exists a $N_0>0$ such that for all $N>N_0$,
\begin{eqnarray}
\label{EstDiffSSN1}
\left\|(S-S^N)p\right\|_{H^{s-1}(\Gamma_R)}\leq cq^N\|p\|_{H^{s+t+1/2}(\Omega_R)},\quad\forall\; t\ge 0,\;s\ge 1/2,
\end{eqnarray}
where $0<q<1$ is a constant independent of $N$.
\end{theorem}
\begin{proof}
{\bf Case 1:} Suppose that $p$ satisfies (\ref{DtNBoundaryCond}).

We know that $p$ admits an expansion taking the form
\begin{eqnarray*}
p(x)=\sum_{n\in{\mathbb Z}}p_n\frac{H_n^{(1)}(k|x|)}{H_n^{(1)}(k|R|)}e^{in\theta}\quad\mbox{for all} \quad |x|>\Gamma^+.
\end{eqnarray*}
Here, $\Gamma^+:=\max\{|x|, x\in\Gamma\}$. It easily follows that
\begin{eqnarray*}
\left\|(S-S^N)p\right\|_{H^{s-1}(\Gamma_R)} &=& \left\{\sum_{|n|>N} (1+n^2)^s|p_n|^2
\frac{k^2}{1+n^2}\left|\frac{{H_n^{(1)}}^{\prime}(kR)}{H_n^{(1)}(kR)}\right|^2\right\}^{1/2}\\
&\leq& c\left\{\sum_{|n|>N}(1+n^2)^s|p_n|^2\right\}^{1/2}\\
&\leq& c\sum_{|n|>N}(1+n^2)^{s/2}|p_n|.
\end{eqnarray*}
For some small $\varepsilon>0$, we have
\begin{eqnarray*}
(1+n^2)^{s/2}|p_n|\left|\frac{H_n^{(1)}(k(\Gamma^++2\varepsilon))}{H_n^{(1)}(kR)}\right|
&=& (1+n^2)^{(s+t)/2}|p_n|\left|\frac{H_n^{(1)}(k(\Gamma^++\varepsilon))}{H_n^{(1)}(kR)}\right|\\
&\times& (1+n^2)^{-t/2}\left|\frac{H_n^{(1)}(k(\Gamma^++2\varepsilon))}{H_n^{(1)}(k(\Gamma^++\varepsilon))}\right|\\
&\le& \left\{\sum_{n\in{\mathbb Z}}(1+n^2)^{s+t}|p_n|^2\left|\frac{H_n^{(1)}(k(\Gamma^++\varepsilon))}{H_n^{(1)}(kR)}\right|^2\right\}^{1/2}\\
&\times& \left\{\sum_{n\in{\mathbb Z}}(1+n^2)^{-t}\left|\frac{H_n^{(1)}(k(\Gamma^++2\varepsilon))}{H_n^{(1)}(k(\Gamma^++\varepsilon))}\right|^2\right\}^{1/2}\\
\end{eqnarray*}
Note that for sufficiently large $|n|$ and $z>0$,
\begin{eqnarray*}
H_{|n|}^{(1)}(z)\sim -i\sqrt{\frac{2}{|n|\pi}}\left(\frac{e}{2|n|}\right)^{-|n|}z^{-|n|}.
\end{eqnarray*}
Then we deduce that there exists a $N_0>0$ such that for $|n|>N_0$,
\begin{eqnarray*}
(1+n^2)^{-t}\left|\frac{H_n^{(1)}(k(\Gamma^++2\varepsilon))}{H_n^{(1)}(k(\Gamma^++\varepsilon))}\right|^2 \le c(1+n^2)^{-t}\left(\frac{\Gamma^++\varepsilon}{\Gamma^++2\varepsilon}\right)^n,
\end{eqnarray*}
and
\begin{eqnarray*}
\left|\frac{H_n^{(1)}(kR)}{H_n^{(1)}(k(\Gamma^++2\varepsilon))}\right| \le cq^{|n|},
\end{eqnarray*}
where $q:=(\Gamma^++2\varepsilon)/R<1$. These further imply that
\begin{eqnarray*}
\sum_{n\in{\mathbb Z}}(1+n^2)^{-t}\left|\frac{H_n^{(1)}(k(\Gamma^++2\varepsilon))}{H_n^{(1)}(k(\Gamma^++\varepsilon))}\right|^2 <\infty.
\end{eqnarray*}
Then for $|n|>N_0$, we deduce from the trace theorem that
\begin{eqnarray*}
(1+n^2)^{s/2}|p_n|\le cq^{|n|}\|p\|_{H^{s+t}(\Gamma_{\Gamma^++\varepsilon})}\le cq^{|n|}\|p\|_{H^{s+t+1/2}(\Omega_R)}.
\end{eqnarray*}
We conclude that
\begin{eqnarray*}
\left\|(S-S^N)p\right\|_{H^{s-1}(\Gamma_{R})}&\le& c\sum_{|n|>N}q^{|n|}\|p\|_{H^{s+t+1/2}(\Omega_R)}\\
&\le& cq^N\|p\|_{H^{s+t+1/2}(\Omega_R)}.
\end{eqnarray*}

{\bf Case 2:} Suppose that $p$ satisfies (\ref{TruncDtNCond}).

We know that $p$ admits an expansion taking the form
\begin{eqnarray*}
p(x)=\sum_{n\in{\mathbb Z}}\left(p_n^{(1)}\frac{H_n^{(1)}(k|x|)}{H_n^{(1)}(k|R|)}+p_n^{(2)}\frac{H_n^{(2)}(k|x|)}{H_n^{(2)}(k|R|)}\right)e^{in\theta}\quad\mbox{for all} \quad |x|>\Gamma^+.
\end{eqnarray*}
According to the condition (\ref{TruncDtNCond}), we conclude that
\begin{eqnarray*}
p_n^{(1)}\frac{{H_n^{(1)}}^{\prime}(kR)}{H_n^{(1)}(kR)} +p_n^{(2)}\frac{{H_n^{(2)}}^{\prime}(kR)}{H_n^{(2)}(kR)}=
\begin{cases}
(p_n^{(1)}+p_n^{(2)})\frac{{H_n^{(1)}}^{\prime}(kR)}{H_n^{(1)}(kR)},\quad &|n|\le N ,\cr 0,\quad &|n|>N,\nonumber
\end{cases}
\end{eqnarray*}
which implies that
\begin{eqnarray*}
p_n^{(2)}=
\begin{cases}
0,\quad &|n|\le N ,\cr -p_n^{(1)}\frac{{H_n^{(1)}}^{\prime}(kR)}{H_n^{(1)}(kR)} \frac{H_n^{(2)}(kR)}{{H_n^{(2)}}^{\prime}(kR)},\quad &|n|>N,\nonumber
\end{cases}
\end{eqnarray*}
It follows that
\begin{eqnarray*}
\left\|(S-S^N)p\right\|_{H^{s-1}(\Gamma_R)} &=& \left\{\sum_{|n|>N} (1+n^2)^s|p_n^{(1)}+p_n^{(2)}|^2
\frac{k^2}{1+n^2}\left|\frac{{H_n^{(1)}}^{\prime}(kR)}{H_n^{(1)}(kR)}\right|^2\right\}^{1/2}\\
&\leq& c\left\{\sum_{|n|>N}(1+n^2)^s|p_n^{(1)}+p_n^{(2)}|^2\right\}^{1/2}\\
&\leq& c\sum_{|n|>N}(1+n^2)^{s/2}|p_n^{(1)}| \left|1-\frac{{H_n^{(1)}}^{\prime}(kR)}{H_n^{(1)}(kR)} \frac{H_n^{(2)}(kR)}{{H_n^{(2)}}^{\prime}(kR)}\right|.
\end{eqnarray*}
The proof of the remaining assertions is analogous to that of Case 1.
\end{proof}

\subsection{Modified  weak formulation}
We now consider the modified variational equation of (\ref{Weak1}) due to the truncation of the DtN mapping: looking for ${\bf U}_N=({\bf u}_N,p_N) \in{\mathcal{H}^1}$ such that
\begin{eqnarray}
\label{Weak2}
A^N({\bf U}_N,{\bf V})=B({\bf U}_N,{\bf V})+b^N(p_N,q)=\ell({\bf V}),\quad \forall\; {\bf V}=({\bf v},q)\in{\mathcal{H}^1},
\end{eqnarray}
where
\begin{eqnarray*}
b^N(p_N,q)=-\int_{\Gamma_R}{(S^Np_N)\overline q ds}.
\end{eqnarray*}

\begin{theorem}
\label{Theorem4.3}
The sesquilinear form $A^N(\cdot,\cdot)$ in (\ref{Weak2}) satisfies
\begin{align}
\label{Weak2continuity}
& 1.\quad\left|A^N({\bf U},{\bf V})\right|\leq c\|{\bf U}\|_{\mathcal{H}^1}\|{\bf V}\|_{\mathcal{H}^1},\quad\forall\;{\bf U},{\bf V}\in{\mathcal{H}^1},\\
\label{Weak2garding}
& 2.\quad\mbox{Re}\,\{A^N({\bf V},{\bf V})\}\geq \alpha\|{\bf V}\|_{\mathcal{H}^1}^2-\beta\left(\|{\bf v}\|_{(H^{1/2+\varepsilon}(\Omega))^2}^2+\|q\|_{H^{1/2+\varepsilon}(\Omega_{R})}^2\right)  ,\quad\forall\;{\bf V}=({\bf v},q)\in{\mathcal{H}^1},
\end{align}
where $c>0, \alpha>0, \beta\geq0$ and $0<\varepsilon<1/2$ are constants independent of ${\bf U}$ and ${\bf V}$.
\end{theorem}
\begin{proof}
It is easy to deduce (\ref{Weak2continuity}) due to the fact that $S^N$ is also bounded. We now show the proof of (\ref{Weak2garding}). Following (\ref{b=b1-b2}), we define
\begin{eqnarray*}
b_1^N(p, q)=\frac{1}{\pi}{\sum\limits_{n=1}^{N}}\ n \int_{0}^{2 \pi}\int_ {0}^{2 \pi}p(R,\phi)\overline{q(R,\theta)}\cos(n(\theta-\phi))d\theta d\phi
\end{eqnarray*}
and
\begin{eqnarray*}
b_2^N(p, q)=\frac{kR}{\pi}{\sum\limits_{n=0}^{N}} \ '\frac{{H_{n-1}^{(1)}}{(kR)}}{H_{n}^{(1)}(kR)}\int_{0}^{2 \pi}\int_ {0}^{2 \pi}p(R,\phi)\overline{q(R,\theta)}\cos(n(\theta-\phi))d\theta d\phi
\end{eqnarray*}
which yields
\begin{eqnarray*}
b^N(p, q)=b_1^N(p, q)-b_2^N(p, q).
\end{eqnarray*}
It follows from Theorem 4.4 in \cite{HNPX11} that
\begin{eqnarray*}
\label{b1N}
b_1^N(q, q)\ge0,
\end{eqnarray*}
and
\begin{eqnarray*}
\label{b2N}
\left|\mbox{Re}\,\{b_2^N(q, q)\}\right| \le |b_2^N(q, q)| \le c \left\|q \right\|_{H^{0}(\Gamma_R )}^2
\end{eqnarray*}
and these further imply that
\begin{eqnarray}
\label{bNgarding}
\mbox{Re}\,\{b^N(q,q)\}\geq -\beta \left\|q \right\|_{H^{1/2+\varepsilon}(\Omega_R )}^2,
\end{eqnarray}
where $\beta>0$ and $0<\varepsilon<1/2$ are constants. Consequently, by the combination of (\ref{a1garding})-(\ref{a4garding}) and (\ref{bNgarding}), we completes the proof of (\ref{Weak2garding}).
\end{proof}

Now, we  present some   preliminary  results before  showing the uniqueness (Theorem \ref{Theorem4.11}) of the modified variational equation (\ref{Weak2}) which is the key ingredient of this work.
\begin{definition}
\label{Definition4.1}
Let $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ be inner products on $(H^1(\Omega))^2\times (H^1(\Omega))^2$ and $H^1(\Omega_R)\times H^1(\Omega_R)$ defined by, for $\forall\; {\bf u},{\bf v}\in (H^1(\Omega))^2$ and $\forall\; p,q\in H^1(\Omega_R)$,
\begin{eqnarray*}
\label{innerproduct1}
\langle{\bf u},{\bf v}\rangle_1 &=& \int_{\Omega}{ \left[\lambda(\nabla\cdot{\bf u})(\nabla\cdot\overline{\bf v})+\frac{\mu}{2}\left(\nabla{\bf u}+(\nabla{\bf u})^T\right):\left(\nabla\overline{\bf v}+(\nabla\overline{\bf v})^T\right)+2\mu{\bf u} \cdot \overline {\bf v}\right] dx},\\
\label{innerproduct2}
\langle p,q\rangle_2 &=& \int_{\Omega_R}{\left(\nabla p\cdot \nabla \overline{q}+p \overline{q}\right) dx}+b_1(p,q),
\end{eqnarray*}
which further induce the norms $|||\cdot|||_1$ on $(H^1(\Omega))^2$ and $|||\cdot|||_2$ on $H^1(\Omega_R)$, respectively, i.e., $\forall\;{\bf u}\in (H^1(\Omega))^2, p\in H^1(\Omega_R)$,
\begin{eqnarray*}
|||{\bf u}|||_1^2=\langle{\bf u},{\bf u}\rangle_1, \quad|||p|||_2^2=\langle p,p\rangle_2.
\end{eqnarray*}
\end{definition}

From the above definition together with (\ref{a1vv}) and (\ref{b1}), we conclude that there exist constants $\alpha_0>0,\beta_0>0,\gamma_0>0$ such that
\begin{eqnarray}
\label{equivilent1}
\alpha_0\|{\bf u}\|_{\left(H^1(\Omega)\right)^2}^2 \le |||{\bf u}|||_1^2 \le \beta_0\|{\bf u}\|_{\left(H^1(\Omega)\right)^2}^2,\\
\label{equivilent2}
\|p\|_{H^1(\Omega_R)}^2\le |||p|||_2^2 \le \gamma_0\|p\|_{H^1(\Omega_R)}^2.
\end{eqnarray}
Now, let $\{\widetilde{\Phi}_i,\widetilde{\lambda}_i\}$ and $\{\widehat{\varphi}_i,\widehat{\lambda}_i\}$ be eigenpairs satisfying
\begin{eqnarray}
\label{eigen1}
\langle\widetilde{\Phi}_i,\Theta\rangle_1  &=& \widetilde{\lambda}_i
\left(\widetilde{\Phi}_i,\Theta\right)_{\left(L^2(\Omega)\right)^2}  , \quad\forall\; \Theta\in \left(H^1(\Omega)\right)^2,\\
\label{eigen2}
\langle\widehat{\varphi_i},\theta\rangle_2 &=& \widehat{\lambda}_i\left(\widehat{\varphi}_i,\theta\right)_{L^2(\Omega_R)}, \quad\forall\; \theta\in H^1(\Omega_R),
\end{eqnarray}
where $\left(\cdot,\cdot\right)_{H}$ is the classical $L^2$ inner product on $H$. Without loss of generalities, we assume that $0<\widetilde{\lambda}_1\le\widetilde{\lambda}_2\le \ldots$, $0<\widehat{\lambda}_1\le\widehat{\lambda}_2\le \ldots$, and $(\widetilde\Phi_i,\widetilde\Phi_j)_{(L^2(\Omega))^2} =(\widehat\varphi_i,\widehat\varphi_j)_{L^2(\Omega_R)} =\delta_{ij}$.

\begin{lemma}
\label{Lemma4.5}
Let ${\bf u}=\sum_{n=1}^{\infty}{\widetilde c_n\widetilde{\Phi}_n}$, $p=\sum_{n=1}^{\infty}{\widehat{c}_n\widehat{\varphi}_n}$ and ${\bf U}=({\bf u},p)$.\\
(1). If ${\bf U}\in \mathcal{H}^0$,
\begin{eqnarray}
\label{L4.1-1}
\|{\bf U}\|_{\mathcal{H}^0}^2=\sum_{n=1}^{\infty}{(|\widetilde{c}_n|^2+|\widehat{c}_n|^2)}.
\end{eqnarray}
(2). If ${\bf U}\in \mathcal{H}^1$,
\begin{eqnarray}
\label{L4.1-2}
|||{\bf u}|||_1^2=\sum_{n=1}^{\infty}{\widetilde{\lambda}_i|\widetilde{c}_n|^2},\quad|||p|||_2^2=\sum_{n=1}^{\infty}{\widehat{\lambda}_i|\widehat{c}_n|^2}.
\end{eqnarray}
(3). Let $H_{M_1}=\underset{\widetilde\lambda_i\le M_1}{span}\{\widetilde\Phi_i\},H_{M_2}=\underset{\widehat\lambda_i\le M_2}{span}\{\widehat\varphi_i\}$ and we define
\begin{eqnarray*}
H_{M_1}^{\perp}&=&\{{\bf v}\in (H^1(\Omega))^2:\langle{\bf v},\Theta\rangle_1=0, \quad\forall\; \Theta\in H_{M_1}\},\\
H_{M_2}^{\perp}&=&\{q\in H^1(\Omega_R): \langle q,\theta\rangle_1=0, \quad\forall\; \theta\in H_{M_2}\}.
\end{eqnarray*}
Then we have
\begin{eqnarray}
\label{L4.1-3}
|||{\bf u}|||_1^2\le M_1\|{\bf u}\|_{(H^0(\Omega))^2}^2,\quad|||p|||_2^2\le M_2\|p\|_{H^0(\Omega_R)}^2,\quad\forall\; {\bf U}\in H_{M_1}\times H_{M_2},
\end{eqnarray}
and
\begin{eqnarray}
\label{L4.1-4}
\|{\bf u}\|_{(H^0(\Omega))^2}^2\le\frac{1}{M_1}|||{\bf u}|||_1^2,\quad\|p\|_{H^0(\Omega_R)}^2\le \frac{1}{M_2}|||p|||_2^2,\quad\forall\; {\bf U}\in H_{M_1}^{\perp}\times H_{M_2}^{\perp}.
\end{eqnarray}
\end{lemma}

\begin{lemma}
\label{Lemma4.6}
Suppose ${\bf U}=({\bf u},p)\in \mathcal{H}^1$ satisfying
\begin{eqnarray*}
\label{L4.2-1}
\langle{\bf u},\Theta\rangle_1&=&(\widetilde{\bf f}_1,\Theta),\quad\forall\; \Theta\in (H^1(\Omega))^2,\\
\label{L4.2-2}
\langle p,\theta\rangle_2&=&(\widehat f_2,\theta),\quad\forall\; \theta\in H^1(\Omega_R)
\end{eqnarray*}
with $(\widetilde{\bf f}_1,\widehat f_2)\in \mathcal{H}^0$. Then there exists a positive constant $c$ such that
\begin{eqnarray}
\label{L4.2-3}
\|{\bf u}\|_{(H^2(\Omega))^2}\le c\|\widetilde{\bf f}_1\|_{(H^0(\Omega))^2},\quad
\|p\|_{H^2(\Omega_R)}\le c\|\widehat f_2\|_{H^0(\Omega_R)}.
\end{eqnarray}
\end{lemma}
\begin{proof}
The first assertion for ${\bf u}$ follows from the interior regularity estimates (\cite{E98}). We now prove the regularity results for $p$. It follows that $b_1(\cdot,\cdot)$ is the corresponding sesquilinear form of the DtN mapping for the solution of homogeneous Laplace equation outside $\Omega$. Now we define
\begin{eqnarray*}
\widetilde{p}(x)=
\begin{cases}
p(x),\quad &|x|\le R ,\cr \sum_{n\ge 0}\left(\frac{|x|}{R}\right)^n(a_n\cos n\theta+b_n\sin n\theta),\quad &|x|>R,\nonumber
\end{cases}
\end{eqnarray*}
where $p(R,\theta)=\sum_{n\ge 0}(a_n\cos n\theta+b_n\sin n\theta)$. Note that for $\varphi\in C_0^\infty({\mathbb R}^2)$,
\begin{eqnarray*}
\int_{{\mathbb R}^2}\nabla\widetilde{p}\cdot\nabla\overline{\varphi}dx +\int_{\Omega_R}\widetilde{p}\,\overline{\varphi}dx
&=& \langle \widetilde{p},\varphi\rangle_2-\int_{|x|\ge R}\Delta\widetilde{p}\,\overline{\varphi}dx\\
&=& \langle p,\varphi\rangle_2\\
&=& (\widehat f_2,\varphi).
\end{eqnarray*}
Then we conclude from interior regularity estimates that $\widetilde{p}=p\in H^2(\Omega_R)$ and
\begin{eqnarray*}
\|p\|_{H^2(\Omega_R)} &\le& c\{\|\widetilde{p}\|_{H^1(\Omega_{R+1})}+\|\widehat{f}_2\|_{H^0(\Omega_R)}\}\\
&\le& c\|\widehat{f}_2\|_{H^0(\Omega_R)},
\end{eqnarray*}
in which the last inequality follows from the fact that $\widetilde{p}$ satisfies a stable variational problem on $H^1(\Omega_{R+1})$.
\end{proof}

\begin{corollary}
\label{Corollary4.7}
If ${\bf U}=({\bf u},p)\in H_{M_1}\times H_{M_2}$,
\begin{eqnarray}
\label{C4.1-1}
\|{\bf u}\|_{(H^2(\Omega))^2}\le c M_1^{1/2}|||{\bf u}|||_1,\quad \|p\|_{H^2(\Omega_R)}\le cM_2^{1/2}|||p|||_2.
\end{eqnarray}
\end{corollary}
\begin{proof}
For ${\bf U}=({\bf u},p)\in H_{M_1}\times H_{M_2}$, we can see that
\begin{eqnarray*}
{\bf u}=\sum_{\widetilde\lambda_n\le M_1}{\widetilde{c}_n\widetilde{\Phi}_n}, \quad p=\sum_{\widehat\lambda_n\le M_2}{\widehat{c}_n\widehat{\varphi}_n}.
\end{eqnarray*}
Then (\ref{eigen1}) and (\ref{eigen2}) imply that
\begin{eqnarray*}
\langle{\bf u},\Theta\rangle_1&=&\left(\sum_{\widetilde\lambda_n\le M_1}{\widetilde\lambda_n\widetilde{c}_n\widetilde{\Phi}_n},\Theta\right),\quad\forall\; \Theta\in (H^1(\Omega))^2,\\
\langle p,\theta\rangle_2&=&\left(\sum_{\widehat\lambda_n\le M_2}{\widehat\lambda_n\widehat{c}_n\widehat{\varphi}_n},\theta\right),\quad\forall\; \theta\in H^1(\Omega_R).
\end{eqnarray*}
Therefore, (\ref{C4.1-1}) follows from Lemma 4.6 and (\ref{L4.1-2}).
\end{proof}

\begin{corollary}
\label{Corollary4.8}
If ${\bf U}=({\bf u},p)\in H_{M_1}^{\perp}\times H_{M_2}^{\perp}$, there exist positive constants $M_0,C_1$ such that for $M_1,M_2\ge M_0$,
\begin{eqnarray}
\label{C4.2-1}
\|{\bf U}\|_{\mathcal{H}^1}^2\le C_1 \mbox{Re}\{A^N({\bf U},{\bf U})\}.
\end{eqnarray}
\end{corollary}
\begin{proof}
From (\ref{equivilent1}) and (\ref{equivilent2}), we know
\begin{eqnarray*}
\label{C4.2-2}
\alpha_0\|{\bf U}\|_{\mathcal{H}^1}^2 \le |||{\bf u}|||_1^2+\alpha_0|||p|||_2^2 \le \max\{\alpha_0,1\}(|||{\bf u}|||_1^2 +|||p|||_2^2).
\end{eqnarray*}
According to Definition 4.1 and $b_1^N(p, p)\ge0$, we have
\begin{eqnarray}
\label{C4.2-3}
\alpha_0\|{\bf U}\|_{\mathcal{H}^1}^2 &\le&
\max\{\alpha_0,1\}\mbox{Re}\{A^N({\bf U},{\bf U})\} +\max\{\alpha_0,1\}(2\mu+\rho\omega^2)\|{\bf u}\|_{(H^0(\Omega))^2}^2+\max\{\alpha_0,1\}(k^2+1)\|p\|_{H^0(\Omega_R)}^2 \nonumber\\
&+& \max\{\alpha_0,1\}\mbox{Re}\{b_2^N(p,p)-a_3({\bf u},p)-a_4(p,{\bf u})\}.
\end{eqnarray}
The Sobolev embedding theorem and the the interpolation inequality for $0<\theta=1/2+\varepsilon< 1$, $0<\varepsilon<1/2$ gives
\begin{eqnarray}
\label{C4.2-4}
|b_2^N(p,p)-a_3({\bf u},p)-a_4(p,{\bf u})|&\le& c(\|{\bf u}\|_{(H^0(\Gamma ))^2}^2+\|p\|_{H^0(\Gamma_R)}^2)\nonumber\\
&\le& c(\|{\bf u}\|_{(H^{1/2}(\Omega))^2}^2+\|p\|_{H^{1/2}(\Omega_R)}^2)\nonumber\\
&\le& c(\|{\bf u}\|_{(H^{1/2+\varepsilon}(\Omega))^2}^2+\|p\|_{H^{1/2+\varepsilon}(\Omega_R )}^2)\nonumber\nonumber\\
&\le& c(\|{\bf u}\|_{(H^0(\Omega))^2}^{1-2\varepsilon}\|{\bf u}\|_{(H^1(\Omega ))^2}^{1+2\varepsilon} +\|p\|_{H^0(\Omega_R )}^{1-2\varepsilon}\|p\|_{H^1(\Omega_R )}^{1+2\varepsilon}).
\end{eqnarray}
A combination of (\ref{equivilent1}), (\ref{equivilent2}) and (\ref{L4.1-4}) leads to
\begin{eqnarray}
\label{C4.2-5}
\|{\bf u}\|_{(H^0(\Omega ))^2}^2\le\frac{\beta_0}{M_1}\|{\bf u}\|_{(H^1(\Omega ))^2}^2,\|p\|_{H^0(\Omega_R )}^2\le\frac{\gamma_0}{M_2}\|p\|_{H^1(\Omega_R )}^2.
\end{eqnarray}
Thus, we conclude from (\ref{C4.2-3})-(\ref{C4.2-5}) that
\begin{eqnarray}
\label{C4.2-6}
\|{\bf U}\|_{\mathcal{H}^1}^2 &\le&
C_\alpha \mbox{Re}\{A^N({\bf U},{\bf U})\}+C_\alpha(2\mu+\rho\omega^2)\|{\bf u}\|_{(H^0(\Omega))^2}^2+C_\alpha(k^2+1)\|p\|_{H^0(\Omega_R)}^2 \nonumber\\
&+& C_\alpha|b_2^N(p,p)-a_3({\bf u},p)-a_4(p,{\bf u})|\nonumber\\
&\le& C_\alpha \mbox{Re}\{A^N({\bf U},{\bf U})\}\nonumber\\
&+& C_\alpha\frac{\beta_0^2(2\mu+\rho\omega^2)}{M_1^2} \|{\bf u}\|_{(H^1(\Omega ))^2}^2+ C_\alpha\frac{\gamma_0^2(1+k^2)}{M_2^2}\|p\|_{H^1(\Omega_R )}^2 \nonumber\\
&+& C_\alpha \frac{c\beta_0^{1/2-\varepsilon}}{M_1^{1/2-\varepsilon}}\|{\bf u}\|_{(H^1(\Omega))^2}^2+C_\alpha \frac{c\gamma_0^{1/2-\varepsilon}}{M_2^{1/2-\varepsilon}}\|p\|_{H^1(\Omega_R)}^2,
\end{eqnarray}
where $C_\alpha=\max\{1,1/\alpha_0\}$. Let $M_0>0$ be such that
\begin{eqnarray}
\label{C4.2-7}
1-C_\alpha\frac{\beta_0^2(2\mu+\rho\omega^2)}{M_0^2}-C_\alpha \frac{c\beta_0^{1/2-\varepsilon}}{M_0^{1/2-\varepsilon}} >\frac{1}{2},\\
\label{C4.2-8}
1-C_\alpha\frac{\gamma_0^2(1+k^2)}{M_0^2}-C_\alpha \frac{c\gamma_0^{1/2-\varepsilon}}{M_0^{1/2-\varepsilon}} >\frac{1}{2}.
\end{eqnarray}
Then for $M_1,M_2\ge M_0$, there exists a constant $C_1=2C_\alpha$ such that (\ref{C4.2-1}) holds.
\end{proof}

\begin{corollary}
\label{Corollary4.9}
If ${\bf U}=({\bf u},p)\in H_{M_1}\times H_{M_2}$ and ${\bf V}=({\bf v},q)\in H_{M_1}^{\perp}\times H_{M_2}^{\perp}$, we have
\begin{eqnarray}
\label{C4.3-1}
|A({\bf U},{\bf V})|\le c\gamma_0^{1/4-\varepsilon/2} M_2^{\varepsilon/2-1/4} \|p\|_{H^1(\Omega_R)}\|q\|_{H^1(\Omega_R)}.
\end{eqnarray}
where $c>0$ and $0<\varepsilon<1/2$ are constants independent of ${\bf U}$ and ${\bf V}$.
\end{corollary}
\begin{proof}
It easily follows that
\begin{eqnarray*}
|A({\bf U},{\bf V})| &=& |\langle{\bf u},{\bf v}\rangle_1-(2\mu+\rho\omega^2)({\bf u},{\bf v})_{(H^0(\Omega))^2}+\langle p,q\rangle_2-(k^2+1)(p,q)_{H^0(\Omega_R)}-b_2(p,q)|\\
&=& |b_2(p,q)|\\
&\le& c\|p\|_{H^{1/2+\varepsilon}(\Omega_R)}\|q\|_{H^{1/2+\varepsilon}(\Omega_R)},
\end{eqnarray*}
where $c>0$ is a constant. Then the interpolation inequality and (\ref{C4.2-5}) leads to (\ref{C4.3-1}).
\end{proof}

\begin{corollary}
\label{Corollary4.10}
If ${\bf U}=({\bf u},p)\in H_{M_1}\times H_{M_2}$, there exist ${\bf V}=({\bf v},q)\in H_{M_1}\times H_{M_2}$ and a positive constant $c$ such that
\begin{eqnarray}
\label{C4.4-1}
\|{\bf U}\|_{\mathcal{H}^1}\le c\frac{\mbox{Re}\{A({\bf U},{\bf V})\}}{\|{\bf V}\|_{\mathcal{H}^1}}.
\end{eqnarray}
\end{corollary}
\begin{proof}
Firstly, we observe that for ${\bf W}=({\bf w},\varpi)\in \mathcal{H}^1$, we can rewrite it as
\begin{eqnarray}
\label{C4.4-2}
{\bf W}={\bf W}_M+{\bf W}_M^{\perp}=({\bf w}_{M_1}+{\bf w}_{M_1}^{\perp},\varpi_{M_2}+\varpi_{M_2}^{\perp}),
\end{eqnarray}
where ${\bf w}_{M_1}\in H_{M_1}$, ${\bf w}_{M_1}^{\perp}\in H_{M_1}^{\perp}$, $\varpi_{M_2}\in H_{M_2}$ and $\varpi_{M_2}^{\perp}\in H_{M_2}^{\perp}$. Then from (\ref{equivilent1}) and (\ref{equivilent2}) we know that
\begin{eqnarray}
\label{C4.4-3}
\|{\bf W}_M\|_{\mathcal{H}^1}^2 &=& \|{\bf w}_{M_1}\|_{(H^1(\Omega ))^2}^2+\|\varpi_{M_2}\|_{H^1(\Omega_R)}^2\nonumber\\
&\le& \frac{1}{\alpha_0}|||{\bf w}_{M_1}|||_1 +|||\varpi_{M_2}|||_2\nonumber\\
&\le& \frac{1}{\alpha_0}|||{\bf w}|||_1 +|||\varpi|||_2\nonumber\\
&\le& \frac{\beta_0}{\alpha_0}\|{\bf w}\|_{(H^1(\Omega))^2}^2+\gamma_0\|\varpi\|_{H^1(\Omega_R)}^2 \nonumber\\
&\le& \max\left\{\frac{\beta_0}{\alpha_0},\gamma_0\right\}\|{\bf W}\|_{\mathcal{H}^1}^2.
\end{eqnarray}
Similarly, we have
\begin{eqnarray}
\label{C4.4-4}
\|{\bf W}_M^{\perp}\|_{\mathcal{H}^1}^2 \le \max\left\{\frac{\beta_0}{\alpha_0},\gamma_0\right\}\|{\bf W}\|_{\mathcal{H}^1}^2.
\end{eqnarray}
From the stability of (\ref{Weak1}), we know that there exists a ${\bf W}=({\bf w},\varpi)\in \mathcal{H}^1$ such that
\begin{eqnarray*}
\|{\bf U}\|_{\mathcal{H}^1}\le c\frac{\mbox{Re}\{A({\bf U},{\bf W})\}}{\|{\bf W}\|_{\mathcal{H}^1}},
\end{eqnarray*}
where $c>0$ is a constant. Let ${\bf V}={\bf W}_M$. Therefore, Corollary 4.3, (\ref{C4.4-3}) and (\ref{C4.4-4}) yield
\begin{eqnarray*}
\label{C4.4-5}
\|{\bf U}\|_{\mathcal{H}^1} &\le& c\frac{\mbox{Re}\{A({\bf U},{\bf W})\}}{\|{\bf W}\|_{\mathcal{H}^1}} \nonumber\\
&=& c\frac{\mbox{Re}\{A({\bf U},{\bf V})+A({\bf U},{\bf W}-{\bf V})\}}{\|{\bf W}\|_{\mathcal{H}^1}} \nonumber\\
&\le& c \max\left\{\frac{\beta_0}{\alpha_0},\gamma_0\right\}\frac{\mbox{Re}\{A({\bf U},{\bf V})\}}{\|{\bf V}\|_{\mathcal{H}^1}}+ c\gamma_0^{1/4-\varepsilon/2} M_2^{\varepsilon/2-1/4}\max\left\{\frac{\beta_0}{\alpha_0},\gamma_0\right\} \|{\bf U}\|_{\mathcal{H}^1}.
\end{eqnarray*}
Let $M_0>0$ be such that
\begin{eqnarray}
\label{C4.4-6}
1-c\gamma_0^{1/4-\varepsilon/2} M_0^{\varepsilon/2-1/4}\max\left\{\frac{\beta_0}{\alpha_0},\gamma_0\right\}>\frac{1}{2}.
\end{eqnarray}
Thus, (\ref{C4.4-1}) holds for $M_2\ge M_0$.
\end{proof}

\begin{theorem}
\label{Theorem4.11}
Let the surface $\Gamma$ and the material parameter $(\mu,\lambda,\rho)$ be such that there are no traction free solutions, then there exists a constant $N_{0}=N_0\geq 0$ such that the modified variational equation (\ref{Weak2}) has at most one solution for $N\geq N_0$.
\end{theorem}
\begin{proof}
Here we prove that for all ${\bf U}=({\bf u},p)\in \mathcal{H}^1$, there exists a constant $N_{0}=N_0\geq 0$ such that
\begin{eqnarray}
\label{T4.3-1}
\|{\bf U}\|_{\mathcal{H}^1} \le c \sup_{({\bf 0},0)\neq {\bf V}\in \mathcal{H}_{h}}\frac{|A^N({\bf U},{\bf V})|}{\|{\bf V}\|_{\mathcal{H}^1}},
\end{eqnarray}
where $c>0$ is a constant. Then the Theorem follows immediately. We use the same form as (\ref{C4.4-2}) to represent ${\bf U}$ as
\begin{eqnarray*}
\label{T4.3-2}
{\bf U}={\bf U}_M+{\bf U}_M^{\perp}=({\bf u}_{M_1}+{\bf u}_{M_1}^{\perp},p_{M_2}+p_{M_2}^{\perp}),
\end{eqnarray*}
where ${\bf u}_{M_1}\in H_{M_1}$, ${\bf u}_{M_1}^{\perp}\in H_{M_1}^{\perp}$, $p_{M_2}\in H_{M_2}$ and $p_{M_2}^{\perp}\in H_{M_2}^{\perp}$.

If ${\bf U}={\bf 0}$, then (\ref{T4.3-1}) holds due to Corollary 4.8.

If ${\bf U}\ne{\bf 0}$, we can obtain from Corollary 4.10 that there exist ${\bf V}_M=({\bf v}_{M_1},q_{M_2})\in H_{M_1}\times H_{M_2}$ and a positive constant $c$ such that
\begin{eqnarray}
\label{T4.3-3}
\|{\bf U}_M\|_{\mathcal{H}^1}\le c\frac{\mbox{Re}\{A({\bf U}_M,{\bf V}_M)\}}{\|{\bf V}_M\|_{\mathcal{H}^1}}
\end{eqnarray}
with $\|{\bf U}_M\|_{\mathcal{H}^1}=\|{\bf V}_M\|_{\mathcal{H}^1}$ after scaling. Now we define ${\bf V}={\bf V}_M+{\bf U}_M^{\perp}$. Then (\ref{T4.3-3}) together with Corollary 4.8 give
\begin{eqnarray}
\label{T4.3-4}
\|{\bf U}_M\|_{\mathcal{H}^1}^2+\|{\bf U}_M^{\perp}\|_{\mathcal{H}^1}^2 &\le& c(\mbox{Re}\{A({\bf U}_M,{\bf V}_M)\}+\mbox{Re}\{A^N({\bf U}_M^{\perp},{\bf U}_M^{\perp})\}) \nonumber\\
&=& c|A^N({\bf U},{\bf V})|+c|A^N({\bf U}_M,{\bf U}_M^{\perp}|+c|A^N({\bf U}_M^{\perp},{\bf V}_M)| \nonumber\\
&+& c|b(p_{M_2},q_{M_2})-b^N(p_{M_2},q_{M_2})|.
\end{eqnarray}

According to Theorem 4.1, we obtain
\begin{eqnarray}
\label{T4.3-5}
|b(p_{M_2},q_{M_2})-b^N(p_{M_2},q_{M_2})| &\le& \|(S-S^N)p_{M_2}\|_{H^{-3/2}(\Gamma_R)}\|q_{M_2}\|_{H^{3/2}(\Gamma_R)}  \nonumber\\
&\le& c\frac{\epsilon(N,p_M)}{N^2}\|p_{M_2}\|_{H^{3/2}(\Gamma_R)} \|q_{M_2}\|_{H^{3/2}(\Gamma_R)}\nonumber\\
&\le& cN^{-2}\|p_{M_2}\|_{H^2(\Omega_R)}\|q_{M_2}\|_{H^2(\Omega_R)} \nonumber\\
&\le& c_1N^{-2}M_2\gamma_0\|{\bf U}_M\|_{\mathcal{H}^1}^2.
\end{eqnarray}
Additionally, we have that
\begin{eqnarray}
\label{T4.3-6}
|A^N({\bf U}_M,{\bf U}_M^{\perp})|&\le& |A({\bf U}_M,{\bf U}_M^{\perp})|+
|b(p_{M_2},p_{M_2}^{\perp})-b^N(p_{M_2},q_{M_2}^{\perp})|.
\end{eqnarray}
Then Theorem 4.1 gives
\begin{eqnarray}
\label{T4.3-7}
|b(p_{M_2},p_{M_2}^{\perp})-b^N(p_{M_2},q_{M_2}^{\perp})| &\le& \|(S-S^N)p_{M_2}\|_{H^{-1/2}(\Gamma_R)}\|q_{M_2}^{\perp}\|_{H^{1/2}(\Gamma_R)}  \nonumber\\
&\le& c\frac{\epsilon(N,p_M)}N\|p_{M_2}\|_{H^{3/2}(\Gamma_R)} \|q_{M_2}^{\perp}\|_{H^{1/2}(\Gamma_R)}\nonumber\\
&\le& cN^{-1}\|p_{M_2}\|_{H^2(\Omega_R)}\|q_{M_2}^{\perp}\|_{H^1(\Omega_R)} \nonumber\\
&\le& c_2N^{-1}M_2^{1/2}\gamma_0^{1/2}\|{\bf U}_M\|_{\mathcal{H}^1} \|{\bf U}_M^{\perp}\|_{\mathcal{H}^1}.
\end{eqnarray}
Therefore, Corollary, (\ref{T4.3-6}), (\ref{T4.3-7}) and the arithmetic-geometric mean inequality lead to
\begin{eqnarray}
\label{T4.3-8}
|A^N({\bf U}_M,{\bf U}_M^{\perp})|&\le& (c_3\gamma_0^{1/4-\varepsilon/2} M_2^{\varepsilon/2-1/4}+c_4N^{-1}M_2^{1/2}\gamma_0^{1/2})(\|{\bf U}_M\|_{\mathcal{H}^1}^2+\|{\bf U}_M^{\perp}\|_{\mathcal{H}^1}^2).
\end{eqnarray}
Similarly, we have
\begin{eqnarray}
\label{T4.3-9}
|A^N({\bf U}_M^{\perp},{\bf V}_M)|&\le& (c_5\gamma_0^{1/4-\varepsilon/2} M_2^{\varepsilon/2-1/4}+c_6N^{-1}M_2^{1/2}\gamma_0^{1/2})(\|{\bf U}_M\|_{\mathcal{H}^1}^2+\|{\bf U}_M^{\perp}\|_{\mathcal{H}^1}^2).
\end{eqnarray}
Here, we need that $M_1,M_2\ge M_0$ where $M_0>0$ satisfies (\ref{C4.2-7}), (\ref{C4.2-8}) and (\ref{C4.4-6}). In additionally, we suppose that $M_0>0$ and $N_0\ge0$ are large enough such that
\begin{eqnarray}
\label{T4.3-10}
1-c_1N_0^{-2}M_0\gamma_0-(c_3+c_5)\gamma_0^{1/4-\varepsilon/2} M_0^{\varepsilon/2-1/4}-(c_5+c_6)N_0^{-1}M_0^{1/2}\gamma_0^{1/2}>\frac{1}{2},
\end{eqnarray}
which further implies that there is a positive constant $c$ such that
\begin{eqnarray}
\label{T4.3-11}
\|{\bf U}_M\|_{\mathcal{H}^1}^2+\|{\bf U}_M^{\perp}\|_{\mathcal{H}^1}^2 \le c|A^N({\bf U},{\bf V})|.
\end{eqnarray}
Now, since
\begin{eqnarray*}
\alpha_0\|{\bf u}\|_{(H^1{\Omega})^2}^2+\|p\|_{H^1(\Omega_R)}^2 &\le& |||{\bf u}|||_1^2+|||p|||_2^2 \nonumber\\
&=& |||{\bf u}_{M_1}|||_1^2+|||{\bf u}_{M_1}^{\perp}|||_1^2+|||p_{M_2}|||_2^2+|||p_{M_2}^{\perp}|||_2^2 \nonumber\\
&\le& \beta_0(\|{\bf u}_{M_1}\|_{(H^1{\Omega})^2}^2+\|{\bf u}_{M_1}^{\perp}\|_{(H^1{\Omega})^2}^2)+ \gamma_0(\|p_{M_2}\|_{H^1(\Omega_R)}^2+\|p_{M_2}^{\perp}\|_{H^1(\Omega_R)}^2) \nonumber\\
&\le& \max\{\beta_0,\gamma_0\}(\|{\bf U}_M\|_{\mathcal{H}^1}^2+\|{\bf U}_M^{\perp}\|_{\mathcal{H}^1}^2),
\end{eqnarray*}
we have
\begin{eqnarray}
\label{T4.3-12}
\|{\bf U}\|_{\mathcal{H}^1}^2 \le \frac{\max\{\beta_0,\gamma_0\}}{\min\{1,\alpha_0\}}(\|{\bf U}_M\|_{\mathcal{H}^1}^2+\|{\bf U}_M^{\perp}\|_{\mathcal{H}^1}^2).
\end{eqnarray}
Thus,
\begin{eqnarray}
\label{T4.3-13}
\|{\bf U}\|_{\mathcal{H}^1}^2 \le c|A^N({\bf U},{\bf V})|
\end{eqnarray}
for some constant $c>0$.
Finally, we obtain from (\ref{T4.3-12}) similarly that
\begin{eqnarray}
\label{T4.3-14}
\|{\bf V}\|_{\mathcal{H}^1} &\le& \sqrt{\frac{\max\{\beta_0,\gamma_0\}}{\min\{1,\alpha_0\}}\left(\|{\bf V}_M\|_{\mathcal{H}^1}^2+\|{\bf U}_M^{\perp}\|_{\mathcal{H}^1}^2\right)}\nonumber\\
&=& \sqrt{\frac{\max\{\beta_0,\gamma_0\}}{\min\{1,\alpha_0\}}\left(\|{\bf U}_M\|_{\mathcal{H}^1}^2+\|{\bf U}_M^{\perp}\|_{\mathcal{H}^1}^2\right)}\nonumber\\
&\le& \sqrt{\frac{\max\{\beta_0,\gamma_0\}}{\min\{1,\alpha_0\}} \left(\frac{1}{\alpha_0}|||{\bf u}|||_1^2+|||p|||_2^2\right)}\nonumber\\
&\le& \sqrt{\frac{\max\{\beta_0,\gamma_0\}}{\min\{1,\alpha_0\}} \max\left\{\frac{\beta_0}{\alpha_0},\gamma_0\right\}}\|{\bf U}\|_{\mathcal{H}^1}
\end{eqnarray}
Therefore, (\ref{T4.3-13}) and (\ref{T4.3-14}) give
\begin{eqnarray*}
\|{\bf U}\|_{\mathcal{H}^1} &\le& c\frac{|A^N({\bf U},{\bf V})|}{\|{\bf V}\|_{\mathcal{H}^1}} \nonumber\\
&\le& c \sup_{({\bf 0},0)\neq {\bf V}\in \mathcal{H}_{h}}\frac{|A^N({\bf U},{\bf V})|}{\|{\bf V}\|_{\mathcal{H}^1}},
\end{eqnarray*}
where $c>0$ is a constant. This completes the proof.
\end{proof}

According to Theorem 4.3, Theorem 4.11 and the Fredholm alternative theorem, we are let to the following result.
\begin{theorem}
\label{Theorem4.12}
Let the surface $\Gamma$ and the material parameter $(\mu,\lambda,\rho)$ be such that there are no traction free solutions, then there exists a constant $N_0\ge0$ such that the modified variational equation (\ref{Weak2}) admits a unique solution $({\bf u}_N,p_N)\in\mathcal{H}^1$ for $N\geq N_0$.
\end{theorem}

\section{Finite element analysis}
Our main goal in this section is to establishing a priori error estimates (\cite{HW85,HW92,HNPX11,GYXZ}) for the finite element solution of (\ref{Weak2}) in terms of the finite element mesh size $h$ and the truncation order $N$ in  appropriate Sobolev spaces. It has been known that numerical errors induced from the truncation of DtN mapping are exponentially decaying (\cite{NN06}), and thanks to the error estimate \eqref{EstDiffSSN1}, we are able to derive a new upper bound of numerical errors indicating such effect explicitly.

\subsection{Galerkin formulation}
Let $\mathcal{H}_{h}=({\bf S}_{h},S'_{h})\subset \mathcal{H}^1$ be the standard finite element space. Now we consider the Galerkin formulation of (\ref{Weak2}): Given $p^{inc}$, find ${\bf U}_h=({\bf u}_h,p_h)\in \mathcal{H}_h$, ${\bf u}_h=(u_x^h,u_y^h)$ such that
\begin{equation}
\label{Weak3}
A^N({\bf U}_h,{\bf V}_h)=\ell({\bf V}_h),\quad \forall\; {\bf V}_h=({\bf v}_h,q_h)\in\mathcal{H}_h,\;{\bf v}_h=(v_x^h,v_y^h).
\end{equation}
It can be shown (\cite{HW04}) that the discrete sesquilinear form $A^N(\cdot,\cdot)$ satisfies the BBL-condition as implication of the following:

{\it G{\aa}rding's inequality + Uniqueness + Approximation property of $\mathcal{H}_{h}\Rightarrow$ BBL-condition.}

\begin{theorem}
\label{Theorem5.1}
Let the surface $\Gamma$ and the material parameter $(\mu,\lambda,\rho)$ be such that there are no traction free solutions and suppose that the finite element space $\mathcal{H}_h\subset\mathcal{H}^1$ satisfies the standard approximation property, then there exist constants $N_{0}\geq 0$ and $h_{0}>0$ such that $A^N(\cdot,\cdot)$ for $0<h\leq h_{0},N\ge N_0$ satisfies the BBL condition in the form
\begin{equation}
\label{BBLcondition}
\sup_{({\bf 0},0)\neq {\bf W}_{h}\in \mathcal{H}_{h}} \frac{|{A}^{N}({\bf V}_{h},{\bf W}_{h})|}{\|{\bf W}_{h})\|_ {\mathcal{H}^1}}\geq \gamma\|{\bf V}_{h}\|_{\mathcal{H}^1},\quad \forall\;{\bf V}_{h}\in\mathcal{H}_{h}.
\end{equation}
Here $\gamma>0$ is the inf-sup constant independent of $h$.
\end{theorem}

From the BBL condition (\ref{BBLcondition}), we are allowed to derive a priori error estimates for the finite element solution ${\bf U}_{h}\in\mathcal{H}_h$.

\subsection{Asymptotic error estimates}
In this subsection, we mainly derive a reasonable priori error estimates on appropriate Sobolev spaces including error effects of both the numerical discretization and the truncation of infinite series. We first establish an upper bound of numerical errors analogous to the well-known C\'ea's lemma in the positive definite case.

\begin{theorem}
\label{Theorem5.2}
There exist constants  $h_{0}>0$ and $N_0\ge0$ such that for any  $h\in(0,h_0]$ and $N\geq N_0$
\begin{eqnarray}
\label{uppererror}
\|{\bf U}-{\bf U}_{h}\|_{\mathcal{H}^1}\leq c\left\{\inf_{{\bf V}_{h}\in \mathcal{H}_{h}}\|{\bf U}-{\bf V}_{h}\|_ {\mathcal{H}^1}+\sup_{0\neq w_2\in {S}'_{h}} \frac{|(b(p,w_2)-b^{N}(p,w_2)|}{\|w_2\|_{H^{1}(\Omega_R)}}\right\}
\end{eqnarray}
where $c>0$ is a constant independent of $h$ and $N$.
\end{theorem}
\begin{proof}
From the BBL condition (\ref{BBLcondition}), we know that
\begin{eqnarray}
\label{5-2-1}
\gamma\|{\bf U}_h-{\bf V}_h\|_ {\mathcal{H}^1}\leq\sup_{({\bf 0},0)\neq {\bf W}_h\in \mathcal{H}_h} \frac{|A^N({\bf U}_h-{\bf V}_h,{\bf W}_h)|}{\|{\bf W}_h\|_ {\mathcal{H}^1}},\quad \forall\; {\bf V}_h\in\mathcal{H}_h.
\end{eqnarray}
by replacing ${\bf V}_h$ with ${\bf U}_h-{\bf V}_h\in \mathcal{H}_h$. According to (\ref{Weak1}) and (\ref{Weak3}), we have
\begin{eqnarray}
\label{5-2-2}
B({\bf U},{\bf W}_h)+b^{N}(p,w_2)=\ell({\bf W}_{h})+b^{N}(p,w_2)-b(p,w_2),\quad \forall\; {\bf W}_{h}=({\bf w}_{1},w_{2})\in\mathcal{H}_{h},
\end{eqnarray}
and
\begin{eqnarray}
\label{5-2-3}
B({\bf U}_h,{\bf W}_h)+b^{N}(p_h,w_2)=\ell({\bf W}_{h}),\quad \forall\ \textbf{W}_{h}=({\bf w}_{1},w_{2})\in\mathcal{H}_{h},
\end{eqnarray}
respectively. Therefore, substracting (\ref{5-2-2}) from (\ref{5-2-3}) leads to
\begin{eqnarray}
\label{5-2-4}
{A}^{N}(\textbf{U}_{h}-\textbf{U},\textbf{W}_{h})=b(p,w_2)-b^{N}(p,w_2).
\end{eqnarray}
Thus, by (\ref{5-2-4}), the inequality (\ref{5-2-1}) implies that
\begin{eqnarray}
\label{5-2-5}
\gamma\|\textbf{U}_{h}-\textbf{V}_{h}\|_{\mathcal{H}^1}& \leq&\sup_{({\bf 0},0)\neq \textbf{W}_{h}\in \mathcal{H}_{h}} \frac{|\textit{A}^{N}(\textbf{U}_{h}-\textbf{V}_{h},\textbf{W}_{h})|}{\|\textbf{W}_{h}\|_ {\mathcal{H}^1}}\nonumber\\
&=&\sup_{({\bf 0},0)\neq \textbf{W}_{h}\in \mathcal{H}_{h}}\frac{|\textit{A}^{N}(\textbf{U}-\textbf{V}_{h},\textbf{W}_{h}) +\textit{A}^{N}(\textbf{U}_{h}-\textbf{U},\textbf{W}_{h})|}{\|\textbf{W}_{h}\|_ {\mathcal{H}^1}}\nonumber\\
& =& \sup_{({\bf 0},0)\neq \textbf{W}_{h}\in \mathcal{H}_{h}} \frac{|\textit{A}^{N}(\textbf{U}-\textbf{V}_{h},\textbf{W}_{h})+b(p,w_2)-b^{N}(p,w_2)|}{\|\textbf{W}_{h}\|_ {\mathcal{H}^1}}\nonumber\\
& \leq&  \sup_{({\bf 0},0)\neq {\bf W}_{h}\in \mathcal{H}_{h}} \frac{|\textit{A}^{N}(\textbf{U}-\textbf{V}_{h},\textbf{W}_{h})|}{\|\textbf{W}_{h}\|_{\mathcal{H}^1}}+ \sup_{({\bf 0},0)\neq {\bf W}_{h}\in \mathcal{H}_{h}} \frac{|b(p,w_2)-b^{N}(p,w_2)|}{\|{\bf W}_{h}\|_ {\mathcal{H}^1}}\nonumber\\
& \leq& c\|\textbf{U}-\textbf{V}_{h}\|_ {\mathcal{H}^1}+\sup_{0\neq w_2\in {S}'_{h}} \frac{|b(p,w_2)-b^{N}(p,w_2)|}{\|w_2\|_{H^{1}(\Omega_R)}},
\end{eqnarray}
where $c$ is a positive constant. Consequently, the triangular inequality and the formulation (\ref{5-2-5}) yield, $\forall\ \textbf{V}_{h}\in \mathcal{H}_{h},$
\begin{eqnarray*}
\|\textbf{U}-\textbf{U}_{h}\|_{\mathcal{H}^1} &\leq& \|\textbf{U}-\textbf{V}_{h}\|_ {\mathcal{H}^1}+\|\textbf{U}_{h}-\textbf{V}_{h}\|_{\mathcal{H}^1}\nonumber\\
&\leq& (1+\frac{c}{\gamma})\|\textbf{U}-\textbf{V}_{h}\|_ {\mathcal{H}^1}+\frac{1}{\gamma}\sup_{0\neq w_2\in {S}'_{h}} \frac{|b(p,w_2)-b^{N}(p,w_2)|}{\|w_2\|_{H^{1}(\Omega_R)}}
\end{eqnarray*}
and this further leads to
\begin{eqnarray*}
\|\textbf{U}-\textbf{U}_{h}\|_ {\mathcal{H}^1}\leq c\left\{\inf_{\textbf{V}_{h}\in \mathcal{H}_{h}}\|\textbf{U}-\textbf{V}_{h}\|_{\mathcal{H}^1}+\sup_{0\neq w_2\in {S}'_{h}} \frac{|b(p,w_2)-b^{N}(p,w_2)|}{\|w_2\|_{H^{1}(\Omega_R)}}\right\}
\end{eqnarray*}
where $c>0$ is a constant independent of $h$ and $N$. This completes the proof.
\end{proof}

From the estimate (\ref{uppererror}), we can see that the numerical errors is constructed by two single terms. We study the first term correlated with the finite element meshsize $h$ by using the approximation theory, and the second term dependent on the truncation order $N$ by employing the point estimate (\ref{EstDiffSSN1}). Let $\mathcal{H}^t=\left(H^{t}(\Omega)\right)^2\times H^{t}(\Omega_R)$. In the following, starting with the estimate (\ref{uppererror}), we derive a priori error estimates measured in $\mathcal{H}^1$-norm and $\mathcal{H}^0$-norm respectively to conclude this section.

\begin{theorem}
\label{Theorem5.3}
Suppose that ${\bf U}\in \mathcal{H}^{t}$ for $2\leq t \in \mathbb{R}$. Then there exist constants  $h_0>0$ and $N_0\geq 0$ such that for any  $h\in(0,h_0]$ and $N\geq N_0$
\begin{eqnarray}
\label{energyest}
\|{\bf U}-{\bf U}_{h}\|_ {\mathcal{H}^1}\leq c \left\{ h^{t-1}\|{\bf U}\|_{\mathcal{H}^{t}}+q^N\|p\|_{{H}^{t}(\Omega_{R})}\right\},
\end{eqnarray}
where $c>0$ and $0<q<1$ are constants independent of $h$ and $N$.
\end{theorem}
\begin{proof}
We have known from Theorem \ref{Theorem5.2} that
\begin{eqnarray}
\label{5-3-1}
\|\textbf{U}-\textbf{U}_{h}\|_{\mathcal{H}^1}\leq c\left\{\inf_{\textbf{V}_{h}\in \mathcal{H}_{h}}\|\textbf{U}-\textbf{V}_{h}\|_{\mathcal{H}^1}+\sup_{0\neq w_2\in {S}'_{h}} \frac{|b(p,w_2)-b^{N}(p,w_2)|}{\|w_2\|_{H^{1}(\Omega_R)}}\right\},
\end{eqnarray}
where $c>0$ is a positive constant. The approximation property of the finite element space $\mathcal{H}_{h}$ gives
\begin{eqnarray}
\label{5-3-2}
\inf_ {\textbf{V}_{h}\in \mathcal{H}_{h}}\|\textbf{U}-\textbf{V}_{h}\|_ {\mathcal{H}^1}\leq c h^{t-1}\|\textbf{U}\|_{\mathcal{H}^{t}},
\end{eqnarray}
where $c>0$ is a positive constant independent of $h$. We now consider the second term in (\ref{5-3-1}). The trace theorem implies that there exists a bounded linear operator $\gamma:{H}^1(\Omega_{R})\rightarrow {H}^{1/2}(\Gamma_R)$ such that
\begin{eqnarray}
\label{5-3-3}
\frac{|b(p,w_2)-b^{N}(p,w_2)|}{\|w_2\|_{H^{1}(\Omega_R)}}=\frac{|\langle (S-S^{N})\gamma p,\gamma w_2\rangle_{\Gamma_R}|}{\|w_2\|_{H^{1}(\Omega_R)}}=\frac{|\langle \gamma^{*}(S-S^{N})\gamma p, w_2\rangle_{\Omega_R}|}{\|w_2\|_{H^{1}(\Omega_R)}},
\end{eqnarray}
where $\langle \cdot,\cdot\rangle_{\Gamma_R}$ is the standard $L^{2}$ duality pairing between $H^{-1/2}(\Gamma_R)$ and $H^{1/2}(\Gamma_R),$ and $\gamma^{*}:H^{-1/2}(\Gamma_R)\rightarrow ({H}^{1}(\Omega_R))'$ is the
adjoint operator of $\gamma$. Therefore, we have
\begin{eqnarray}
\label{5-3-4}
\sup_{0\neq w_2\in {S}'_{h}} \frac{|b(p,w_2)-b^{N}(p,w_2)|}{\|w_2\|_{H^{1}(\Omega_R)}}
&=& \sup_{0\neq w_2\in {S}'_{h}} \frac{|\langle \gamma^{*}(S-S^{N})\gamma p, w_2\rangle_{\Omega_R}|}{\|w_2\|_{H^{1}(\Omega_{R})}}\nonumber\\
&=& \|\gamma^{*}(S-S^{N})\gamma p\|_{ ({H}^{1}(\Omega_R))'}\nonumber\\
&\leq& c\|(S-S^{N})\gamma p\|_{H^{-1/2}(\Gamma_R)}\nonumber\\
&\leq& cq^N\|\gamma p\|_{{H}^{t-1/2}(\Gamma_R)}\nonumber\\
&\leq& cq^N\|p\|_{{H}^{t}(\Omega_R)}
\end{eqnarray}
because of Theorem \ref{Theorem4.2} and the boundedness of operators $\gamma$ and $\gamma^*$. The proof is hence established by following a
combination of (\ref{5-3-2}) and (\ref{5-3-4}).
\end{proof}

We now extend the error estimate in the energy space to the one measured in the $\mathcal{H}^0=\left(L_{2}(\Omega)\right)^2\times L_{2}(\Omega_R)$ space.

\begin{theorem}
\label{Theorem5.4}
Suppose that ${\bf U}\in \mathcal{H}^{t}$ for $2\leq t \in \mathbb{R}$. Then there exist constants  $h_{0}>0$ and $N_{0}\geq 0$ such that for any  $h\in(0,h_{0}]$ and $N\geq N_{0}$
\begin{eqnarray}
\label{L2est}
\|{\bf U}-{\bf U}_{h}\|_ {\mathcal H^0}\leq c \left\{ h^{t}\|{\bf U}\|_{\mathcal{H}^{t}}+q^N\|p\|_{{H}^{t}(\Omega_{R})}\right\}
\end{eqnarray}
where $c>0$ and $0<q<1$ are constants independent of $h$ and $N$.
\end{theorem}
\begin{proof}
Let $\textbf{E}=({\bf e}_{1},e_{2})={\bf U}-{\bf U}_h$ be the finite element error. Then by (\ref{5-2-4}) in Theorem 5.2, we have
\begin{eqnarray}
\label{5-4-1}
B({\bf E},{\bf V}_h)+b^{N}(e_{2},v_2)+b(p,v_2)-b^N(p,v_2)=0,\quad\forall\ {\bf V}_h=({\bf v}_1,v_2)\in\mathcal{H}_{h}.
\end{eqnarray}
Now, we consider the following boundary value problem: Find $\textbf{W}=({\bf w}_1,w_2)\in
(C^{2}(\Omega)\cap C^{1}(\bar{\Omega}))^2\times(C^{2}(\Omega_{R})\bigcap C^{1}(\overline{\Omega_{R}}))$ satisfying
\begin{eqnarray}
\label{duality1}
\Delta^{*}{\bf w}_1 +   \rho \omega^2{\bf w}_1 &=& {\bf e}_1\quad \text{in}\quad \Omega,\\
\label{duality2}
\Delta w_2 + k^2w_2 &=& e_2\quad \text{in}\quad {\Omega_R},\\
\label{duality3}
\rho_f\omega^2{\bf w}_1\cdot {\bf n} &=& \frac{\partial w_2}{\partial n} \quad \text{on}\quad \Gamma,\\
\label{duality4}
{\bf T}{\bf w}_1 &=& -w_2{\bf n} \quad \text{on}\quad \Gamma,\\
\label{duality5}
\frac{\partial w_2}{\partial n} &=& Sw_2 \quad \text {on}\quad  \Gamma_{R}.
\end{eqnarray}
Let $\textbf{W}=({\bf w}_1,w_2)$ be a weak solution of nonlocal boundary value problem (\ref{duality1})-(\ref{duality5}), hence $\textbf{W}$ satisfies
\begin{eqnarray}
\label{5-4-2}
{A}^{N}(\textbf{V},\textbf{W})+b(v_2,w_2)-b^{N}(v_2,w_2)=({\bf w}_1,{\bf e}_1)_{\left(L^2(\Omega)\right)^2}+(w_2,e_2)_{L^2(\Omega_R)},\quad \forall\; \textbf{V}=({\bf v}_1,v_2)\in{\mathcal{H}^1}.
\end{eqnarray}
Replacing $\textbf{V}$ by $\textbf{E}$ in (\ref{5-4-2}) gives
\begin{eqnarray}
\label{5-4-3}
{A}^{N}(\textbf{E},\textbf{W})+b(e_2,w_2)-b^{N}(e_2,w_2)=({\bf e}_1,{\bf e}_1)_{\left(L^2(\Omega)\right)^2}+(e_2,e_2)_{L^2(\Omega_R)}=\|\textbf{E}\|_{\mathcal H^0}^2.
\end{eqnarray}
Then subtracting (\ref{5-4-3}) from (\ref{5-4-1}) leads to, for $\forall\ \textbf{V}_{h}\in\mathcal{H}_{h},$
\begin{eqnarray}
\label{5-4-4}
\|{\bf E}\|_{\mathcal H^0}^2=A^N({\bf E},{\bf W}-{\bf V}_h)+b(e_{2},w_2)-b^{N}(e_{2},w_2)+b^{N}(p,v_2)-b(p,v_2).
\end{eqnarray}
Theorem 4.4, the approximation property of $\mathcal{H}_{h}$ and the regularity theory imply that
\begin{eqnarray}
\label{5-4-5}
|\textit{A}^{N}(\textbf{E},\textbf{W}-\textbf{V}_{h})|&\leq& c\|\textbf{E}\|_{\mathcal{H}^1}\|\textbf{W}-\textbf{V}_{h}\|_{\mathcal{H}^1}\nonumber\\
&\leq& ch\|\textbf{E}\|_{\mathcal{H}^1}\|\textbf{W}\|_{\mathcal{H}^{2}}\nonumber\\
&\leq& ch\|\textbf{E}\|_{\mathcal{H}^1}\|\textbf{E}\|_{\mathcal{H}^{0}},
\end{eqnarray}
where $c>0$ is a constant. Following the same argument in Theorem 5.3 and choosing $t = 2$, we arrive at, by the regularity theory,
\begin{eqnarray}
\label{5-4-6}
|b(e_{2},w_2)-b^{N}(e_{2},w_2)|&\leq& \|\gamma^{*}(S-S^{N})\gamma e_2\|_{ ({H}^{2}(\Omega_{R}))'}\|w_2\|_{H^{2}(\Omega_{R})} \nonumber\\
&\leq& c\|(S-S^{N})\gamma e_2\|_{H^{-3/2}(\Gamma_{R})}\|\textbf{W}\|_{\mathcal{H}^{2}}\nonumber\\
&\leq& cq^N\|\gamma e_2\|_{{H}^{1/2}(\Gamma_{R})}\|\textbf{E}\|_{\mathcal{H}^{0}}\nonumber\\
&\leq& cq^N\|e_2\|_{{H}^{1}(\Omega_{R})}\|\textbf{E}\|_{\mathcal{H}^{0}}
\end{eqnarray}
Similarly, we also have
\begin{eqnarray*}
|b^{N}(p,w_2-v_2)-b(p,w_2-v_2)|&\leq& \|\gamma^{*}(S-S^{N})\gamma p\|_{ ({H}^{1}(\Omega_{R}))'}\|w_2-v_2\|_{H^{1}(\Omega_{R})} \nonumber\\
&\leq& c\|(S-S^{N})\gamma p\|_{H^{-1/2}(\Gamma_{R})}\|\textbf{W}-\textbf{V}_{h}\|_{\mathcal{H}^1}\nonumber\\
&\leq& chq^N\|\gamma p\|_{{H}^{t-1/2}(\Gamma_{R})} \|\textbf{W}\|_{\mathcal{H}^{2}}\nonumber\\
&\leq& chq^N\|p\|_{{H}^{t}(\Omega_{R})}\|\textbf{E}\|_{\mathcal{H}^{0}}.
\end{eqnarray*}
and
\begin{eqnarray*}
|b(p,w_2)-b^{N}(p,w_2)|&\leq& \|\gamma^{*}(S-S^{N})\gamma p\|_{ ({H}^{2}(\Omega_{R}))'}\|w_2\|_{H^{2}(\Omega_{R})} \nonumber\\
&\leq& c\|(S-S^{N})\gamma p\|_{H^{-3/2}(\Gamma_{R})}\|\textbf{W}\|_{\mathcal{H}^{2}}\nonumber\\
&\leq& cq^N\|\gamma p\|_{{H}^{t-1/2}(\Gamma_{R})} \|\textbf{E}\|_{\mathcal{H}^{0}}\nonumber\\
&\leq& cq^N\|p\|_{{H}^{t}(\Omega_{R})}\|\textbf{E}\|_{\mathcal{H}^{0}},
\end{eqnarray*}
Thus, by the triangular inequality we know
\begin{eqnarray}
\label{5-4-7}
|b^{N}(p,v_2)-b(p,v_2)|&\leq& |b^{N}(p,w_2-v_2)-b(p,w_2-v_2)|+|b^{N}(p,w_2)-b(p,w_2)|\nonumber\\
&\leq& c_1hq^N\|p\|_{H^{t}(\Omega_{R})}\|\textbf{E}\|_{\mathcal{H}^{0}} +c_2q^N\|p\|_{H^{t}(\Omega_{R})}\|\textbf{E}\|_{\mathcal{H}^{0}}.
\end{eqnarray}
Therefore, by the combination of the inequalities (\ref{5-4-5})-(\ref{5-4-7}) and (\ref{energyest}) we have
\begin{eqnarray*}
\|{\bf E}\|_{\mathcal H^0} &=& A^N({\bf E},{\bf W}-{\bf V}_h)+b(e_{2},w_2)-b^{N}(e_{2},w_2)+b^{N}(p,v_2)-b(p,v_2) \nonumber\\
&\le& c h\left\{ h^{t-1}\|\textbf{U}\|_{\mathcal{H}^{t}}+q^N\|p\|_{{H}^{t}(\Omega_{R})}\right\}
+cq^N\left\{ h^{t-1}\|\textbf{U}\|_{\mathcal{H}^{t}} +q^N\|p\|_{{H}^{t}(\Omega_{R})}\right\}\nonumber\\
&+& chq^N\|p\|_{{H}^{t}(\Omega_{R})}
+cq^N\|p\|_{{H}^{t}(\Omega_{R})}.
\end{eqnarray*}
Finally, according to the fact that $h\in(0,h_{0}]$ and $N\geq N_{0}$, we arrive at (\ref{L2est}).
\end{proof}

\section{Numerical experiments}

\begin{figure}
\centering
{\includegraphics[scale=0.3]{NDomain.jpg}
\caption{The computational domain of the model problem.}\label{fig2}
\end{figure}

In this section, we present several numerical tests to validate our theoretical results, and,  unless otherwise stated, we always take the parameters $\omega=1$, $\mu=1$, $\lambda=1$, $\rho=1$, $\rho_f=1$ and $R_0=1$.  We first introduce a model problem whose analytical solution is available  so that we are able to evaluate the accuracy of the numerical solution. We consider the scattering of a plane incident wave $p^{inc}=e^{ikx\cdot d}$ with direction $d=(1,0)$ by a disc-shaped elastic body of radius $R_0$ (see Figure \ref{fig2}). For the exact solution of this model problem, please see the Appendix. In numerical computations, the computational domain $\Omega$ and $\Omega_R$ are discretized by uniform triangle elements and we employ piecewise linear basis functions $\{\varphi_i\}_{i=1}^{i=N_1}$ and $\{\psi_i\}_{i=1}^{i=N_2}$ in $\Omega$ and $\Omega_R$, respectively,  to construct the finite element space $\mathcal{H}_{h}$. Here, $N_1$ and $N_2$ are the total number of elements in $\Omega$ and $\Omega_R$, respectively. To find the finite element solution of (\ref{Weak3}), one needs to compute the integrals
\begin{eqnarray}
\label{numbn1}
\int_{\Gamma_R}{(S^N\psi_j)\overline{\psi_i} ds},
\end{eqnarray}
for those $\psi_i$ and $\psi_j$  not vanishing on $\Gamma_R$. For the sake of simplicity, (\ref{numbn1}) can be approximated by
\begin{eqnarray}
\label{numbn2}
\int_{\Gamma_R}{(S^N\psi_j)\overline{\psi_i} ds}\approx \int_{\Gamma_R}{(S^N\zeta_j)\overline{\zeta_i} ds},
\end{eqnarray}
where $\{\zeta_i\}$ are the corresponding piecewise linear basis functions in terms of  $\theta$ on $\Gamma_R$. Thus, the computation of integrals (\ref{numbn1}) amounts to evaluating a series as
\begin{eqnarray*}
\int_{\Gamma_{R}}{(S^N\psi_j)\overline{\psi_i} ds} \approx {\sum\limits_{n=0}^N} \ '\frac{ {kR H_{n}^{(1)}}^{\prime}{(kR)}}{\pi H_{n}^{(1)}(kR)}\int_ {0}^{2 \pi}\int_ {0}^{2 \pi}\zeta_j(R,\phi)\overline{\zeta_i(R,\theta)}\cos(n(\theta-\phi)) d\theta d\phi,
\end{eqnarray*}
or
\begin{eqnarray*}
\int_{\Gamma_{R}}{(S^N\psi_j)\overline{\psi_i} ds} \approx {\sum\limits_{n=-N}^N}\ \frac{ {kR H_{n}^{(1)}}^{\prime}{(kR)}}{\pi H_{n}^{(1)}(kR)}\int_ {0}^{2\pi}\zeta_j(R,\phi)e^{-in\phi}d\phi \int_ {0}^{2 \pi}\overline{\zeta_i(R,\theta)}e^{in\theta} d\theta.
\end{eqnarray*}

In the numerical experiments, we firstly check the accuracy of the DtN-FEM. We choose $R=2$, $N=20$ and consider three different wave numbers $k=1$, 2 and 4. Figure 3 shows the exact and numerical solutions of the elastic displacement ${\bf u}=(u_x,u_y)$ in the solid and the acoustic scattered field $p$ in the fluid for the problem (\ref{Navier1})-(\ref{RadiationCond}) with $k=1$ and meshsize $h=0.1076$. We can observe that the numerical solutions are in a perfect agreement with the exact series solutions from the qualitative point of view. According to Theorems 5.3 and 5.4, as the truncation order $N$ of the DtN mapping is large enough that the domain discretization error is dominant, we should be able to observe that
\begin{eqnarray*}
\|{\bf U}-{\bf U}_{h}\|_ {\mathcal H^0}=O(h^2),\quad\|{\bf U}-{\bf U}_{h}\|_ {{\mathcal H}^1}=O(h)
\end{eqnarray*}
for the finite element space $\mathcal H_h$. In Figure 4, we show the log-log plot of errors ${\bf U}-{\bf U}_{h}$ measured in $\mathcal H^0$ and $\mathcal H^1$-norms with respect to $1/h$ which verifies the convergence order of $O(h^2)$ and $O(h)$. Finally, we are concerned with the effects of truncation order $N$ on the total numerical errors. We choose $k=1$ and compute the numerical errors measured in $\mathcal H^0$-norm for three different meshsizes $h=$0.4304, 0.2151 and 0.1076, respectively. The log-log plots of errors are presented in Figure 5 showing that the errors due to the truncation of the DtN mapping decay extremely fast, arriving at the low bound correlating to each finite element mesh size.

\begin{figure}[ht]
\centering
\begin{tabular}{ccc}
&\includegraphics[scale=0.2]{FigureMesh.jpg}&  \\
& Mesh I & \\
\includegraphics[scale=0.2]{Exactux.jpg} &
\includegraphics[scale=0.2]{Exactuy.jpg} &
\includegraphics[scale=0.2]{Exactp.jpg} \\
(a) $|u_x|$ & (b) $|u_y|$ & (c) $|p|$ \\
\includegraphics[scale=0.2]{Numericux.jpg} &
\includegraphics[scale=0.2]{Numericuy.jpg} &
\includegraphics[scale=0.2]{Numericp.jpg} \\
(d) $|u_x|$ & (e) $|u_y|$ & (f) $|p|$ \\
\end{tabular}
\caption{Exact solutions (a,b,c) and numerical solutions of   DtN-FEM (d,e,f) using Mesh I.}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tabular}{cc}
\includegraphics[scale=0.4]{L2ErrorU.jpg} &
\includegraphics[scale=0.4]{H1ErrorU.jpg} \\
\end{tabular}
\caption{Log-log plots for numerical errors (vertical) of ${\bf U}$   vs. $1/h$ (horizontal). Left: ${\mathcal H}^0$-norm; right: ${\mathcal H}^1$-norm.}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[scale=0.4]{L2ErrorN.jpg}
\caption{Log-log plot for numerical errors $\|{\bf U}-{\bf U}_{h}\|_ {\mathcal H^0}$ (vertical) vs. $N$ (horizontal).}
\end{figure}

\section{Conclusion}

In this paper, we consider the DtN-FEM solving the two dimensional FSI problem in which the DtN mapping is defined by a Fourier series on a circle. Uniqueness and existence of solutions for the corresponding variational problem are established. We also consider a truncated variational problem due to the truncation of DtN mapping and its solvability results are given by using a novel proof technique. A priori error estimates involving the effects of both finite element meshsize and the truncation order of series for the proposed DtN-FEM are derived. The investigation strategy in this article will be extended to other scattering problems in the future.

\section*{Appendix}
Here, we derive the exact solution for the considered model problem in Section 6. We can represent the solution of (\ref{Navier1})-(\ref{RadiationCond}) as
\begin{eqnarray*}
p(r,\theta) &=& \sum\limits_{n = 0}^\infty  {{A_n}H_n^{(1)}(kr)\cos (n\theta )},\\
{\bf u} &=& \nabla \varphi  -  \nabla\times \psi,
\end{eqnarray*}
with
\begin{eqnarray*}
\varphi(r,\theta) &=& \sum\limits_{n = 0}^\infty  {{B_n}J_n(k_pr)\cos (n\theta )},\\
\psi(r,\theta) &=& \sum\limits_{n = 0}^\infty  {{C_n}J_n(k_sr)\sin (n\theta )},
\end{eqnarray*}
where the coefficients $A_n$, $B_n$ and $C_n$ are to be determined. According to the transmission condition (\ref{TransCond1.1})-(\ref{TransCond2.1}), we are able to obtain a linear system of equations as
\begin{eqnarray*}
\left[ {E_n^{ij}} \right]{X_n} = \left[ {e_n^j} \right],i,j=1,2,3,
\end{eqnarray*}
where $X_n=(A_n,C_n,D_n)^T$. For the system matrix $E_n$ and vector $e_n$, the elements (identified by the super-script) are computed as follows:

\begin{eqnarray*}
E_n^{11} &=& -H_{n-1}^{(1)}(kR_0)+\frac{n}{kR_0}H_{n}^{(1)}(kR_0),\\
E_n^{12} &=& \frac{\rho_f\omega^2k_p}{k}\left[J_{n-1}(k_pR_0)-\frac{n}{k_pR_0}J_{n}(k_pR_0)\right],\\
E_n^{13} &=& \frac{\rho_f\omega^2n}{kR_0}J_{n}(k_sR_0),\\
E_n^{21} &=& 0,\\
E_n^{22} &=& \frac{2\mu nk_p}{R_0}J_{n-1}(k_pR_0)-\frac{2\mu (n^2+n)}{R_0^2}J_{n}(k_pR_0),\\
E_n^{23} &=& \frac{2\mu (n^2+n)-\mu k_s^2R_0^2}{R_0^2}J_{n}(k_sR_0)-\frac{2\mu k_s}{R_0}J_{n-1}(k_sR_0),\\
E_n^{31} &=& H_{n}^{(1)}(kR_0),\\
E_n^{32} &=& \frac{2\mu (n^2+n)-\mu k_s^2R_0^2}{R_0^2}J_{n}(k_pR_0)-\frac{2\mu k_p}{R_0}J_{n-1}(k_pR_0),\\
E_n^{33} &=& \frac{2\mu nk_s}{R_0}J_{n-1}(k_sR_0)-\frac{2\mu (n^2+n)}{R_0^2}J_{n}(k_sR_0),\\
e_n^{1} &=& \epsilon_ni^n\left[J_{n-1}(kR_0)-\frac{n}{kR_0}J_{n}(kR_0) \right],\\
e_n^{2} &=& 0,\\
e_n^{1} &=& -\epsilon_ni^nJ_{n}(kR_0).
\end{eqnarray*}

\section*{Acknowledgments}
The work of T. Yin is partially supported by the NSFC Grant (11371385). The work of L. Xu is partially supported by the NSFC Grant (11371385), the Start-up fund of Youth 1000 plan of China and that of Youth 100 plan of Chongqing University.

redpajama_set_name: RedPajamaArXiv


In Through the Bible Day by Day, F. B. Meyer shares his insights in small sections that correspond to a daily reading from the scriptures. Each of his comments are written with a focus on application in the reader's daily life. He also includes an introduction, outline, and review questions of each Bible book (see the e-Sword Book Comments). This 7 volume set contains more than 1520 pages for the printed edition.
The commentary is devotional and/or "how to apply the Bible to your life" in nature. The comments are concise but nearly every verse in the Bible is covered, with the exception of 1 Chronicles and parts of Numbers.
There is need everywhere for the cultivation of the habit of reading some portion of God's Word each day. Such reading should be consecutive, for only thus can continuous interest be maintained. It may also be greatly helped by an interpreter, who will explain what is obscure as well as suggest applications of the message to the daily life.
To meet such need this series of seven volumes provides an arrangement of the books of the Bible in daily portions, with concise devotional comments. It includes all portions of the Bible most suitable for daily reading, either individually or in family groups.
As a commentary on the whole Bible, these volumes will be found valuable because they omit points of merely scholarly interest and fix attention upon the central message of each passage and its application to daily needs. There are frequent references to other parts of Scripture, especially from Old Testament truths to their New Testament fulfillment and interpretation.
Mr. Meyer's life-long experience in interpreting the Bible to the common people makes him pre-eminently fitted for this service-a crowning one in his world-wide ministry by voice and pen.
Introductions and outlines are provided for each Bible book. Review questions have been added at appropriate points, to enable readers to sum up and better preserve in memory what they have learned. The illustrations, selected from paintings of Bible scenes by modern artists, aid in the interpretation of the passages which they portray.
It is hoped that these volumes will prove of much value to Sunday-school teachers and adult scholars; will promote profitable Bible reading in connection with both family and private devotions; and will everywhere deepen a love for and an intelligent acquaintance with the one Book which can provide a fresh and helpful message for the needs of each new day.
The Bible is full of God's wisdom. It is the Word of God. We must read it not only with our head, but with our heart. An analysis of its contents is good and desirable; but when all this is done, we need to allow time for the inner wisdom, power and grace which are stored in the Book, to soak into our inner consciousness. Therefore, for some reasons, it is better to have a shorter portion for daily study, to give us time to get all the juice and marrow out of what we read.
This is the method followed in this system. Busy people, students, commercial men, Sunday-school workers and older scholars, soldiers and sailors, people on the hills and on the prairies, living on the edge of the great tides of human activity, will find these passages, selected for daily reading, ample to furnish a daily meal of truth, and yet deep and wide enough for profound inquiry and interest.
It has always seemed to me that the true way to know the Bible is to read it through. It is only so that we can follow the unveiling of the divine purpose, from the Dawn of Genesis to the Perfect Day of the divine writings of John. It is only so that we can obtain a connected view of the wide scope of Revelation. To read in different, unconnected passages, now of Poetry, now of History, now of Doctrine, prevents the mind from getting proper focus. We are apt by that method to concentrate on a few familiar passages, instead of listening to all that God has spoken to men's hearts. There is, probably, nothing more vital to an accurate knowledge of the Bible than the good old practice of reading it thoughtfully and steadily through with such brief notes as this volume provides.
Frederick Brotherton Meyer (1847–1929). Baptist pastor, Bible conference speaker and writer. Born in London, Meyer was convinced from his childhood that he would be a preacher.
While at Priory Street Church in 1873, Meyer befriended Dwight L. Moody and Ira D. Sankey, who had arrived in England to minister but had found that the two men who had invited them had died. Invited to York, Moody found Meyer very helpful in getting his campaign started. This friendship resulted in Moody inviting Meyer to America in 1891, the first of twelve trips. He spoke at the East Northfield Summer Conference for two weeks and proved to be so popular that he was asked to give post-conference addresses. In one meeting at Northfield, J. Wilbur Chapman was touched by Meyer's preaching and testified later to a life-changing commitment made there.
Prior to his coming to the U.S., his written works had received much attention. His biographies of Bible characters gained great popularity, as did his expositional and devotional works.
Perhaps F.B. Meyer could be regarded as one of the greatest Bible teachers of his time. Some have placed him in the league of C.H. Spurgeon as the "prince of Bible expositors." Whether he or not he could be regarded in this light, having read some of his works in the STEP reader, I know we are in for a real treat of exceptional Biblical wisdom in this Commentary.
I have said this before with regards to the 49 volume set of the Expositor's Bible that works such as this never become redundant. And so it is such with the works by F.B. Meyer.
I love great Christian literary works by such men of God which has stood the test of time. The wealth of knowledge they have, and the uncompromising truth they hold to gives attests to the credibility of these men of God.
You know, Josh, I think I know where you are from in your walk in Christ when you put up works like these. I praise God for you, and thank him for men of God like you whose desire is to bring the best out of us through these wonderful works.
To read in different, unconnected passages, now of Poetry, now of History, now of Doctrine, prevents the mind from getting proper focus. We are apt by that method to concentrate on a few familiar passages, instead of listening to all that God has spoken to men's hearts.
This resource is now included in e-Sword official module repository! You can download it here, or from within e-Sword using the Download command from the main menu.
Thanks Brother. It is very useful. and help me to rightly dividing the Word of God.


text:
Poland 2016: 70 songs submitted for Krajowe Eliminacje
Giannis Argyriou / 9th February 2016 at 13:43 / National Selection / Poland
Polish Television closed yesterday the submission period for Krajowe Eliminacje – their National Final, which is scheduled for the 5th of March in Warsaw. According to JastrzabPost, almost 70 songs have been submitted. Amongst them are well known Polish artists.
Yesterday it was the last day of submissions for the Polish national final. Aside from Edyta Górniak who confirmed her intentions to take part at the upcoming Krajowe Eliminacje, a lot of other famous Polish singers submitted songs to TVP in order to win the Polish ticket for the 2016 Eurovision Song Contest in Stockholm.
Let's take a look at some of them:
Margaret submitted her "energetic" pop song the last day of the submission period just like Edyta. She is well-known in local music industry thanks hits such as Thank You Very Much, Wasted, Start a Fire and Heartbeat". Margaret has been mentioned as a candidate to represent Poland for years. Her song this time is called Cool Me Down and it's mentioned that Arash (Azerbaijan 2009) co-wrote her song for the Polish National Final this year.
Michal Szpak
Michał Szpak is a young Polish singer who found fame on the first series of the Polish X Factor in June 2011. He ranked second behind Alexandra Burke. Michal confirmed that the song he submitted to Krajowe Eliminacje is Color Of Your Life.
Kasia Nova
Kasia started her career as a model. She competed for the title of Miss Poland in 2005. In 2008 she released her first single. She also tried to represent Poland at the 2008 Eurovision Song Contest without success as she only reached the 11th place in the Polish national final. She will fight for Stockholm with Love is Music submitted by American Label from Los Angeles.
Szymon Wydra
Szymon Wydra is a Polish rock vocalist, poet and finalist on the first edition of Polish Idol. On December 13, 2002, his band released their first album. He strongly criticized illegal downloading of his work from the Internet. With his band, he has released five albums.
Monika became famous after her participation in the talentshow The Voice of Poland in 2014. Her name was rumoured to represent Poland last year.
Krajowe Eliminacje as the Polish national final is called will be held on the 5th of March in Warsaw. The Polish representative in the 61st Eurovision Song Contest will be selected by televoting 100%, between 5 to 10 entries. Poland returns to a national selection format for the first time since 2011.
Poland in the Eurovision Song Contest
Poland debuted in the Eurovision Song Contest in 1994, the same year they achieved their best placement in the contest when their superstar Edyta Górniak with her song To Nie Ja gained a second place. Since then Poland have only achieved a top 10 place in the final once, in 2003, when Ich Troje performed Keine Grenzen in Riga.
This will be their 19th entry in the contest. This year Monika Kuszyńska represented the country in Vienna. With 10 points she ended 23rd out of 27 in the Grand Final.
Below you can watch Monika Kuszyńska performing her song In The Name Of Love at the Eurovision Song Contest in Vienna:
Who will represent the Czech Republic in Stockholm?
Your guide to understanding Sanremo 2016
Written by Giannis Argyriou from Greece
View profile and more posts from Giannis
Edited by Charlotte Jensen
9th February 2016 in National Selection. Tags: Edyta Górniak, Eurovision Song Contest, Kasia Nova, Krajowe Eliminacje, Margaret, Michal Szpak, Monika Urlik, national final, Szymon Wydra, TVP Countries: Poland
Source: EuroVisionary, Wikipedia, JastrzabPost
Polish Television closed yesterday the submission period for Krajowe Eliminacje – their National Final, which is scheduled for the 5th of March in Warsaw. According to JastrzabPost, almost 70 songs have been submitted. Amongst them are well known Polish artists. Yesterday it was the last day of submissions for the Polish national final. Aside from Edyta Górniak… Read more
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text:
Austin-Morris
Austin-Rover
Asia-Motors
Zumikon ( Switzerland ), 16 February 2015
For the 21st time, the Swiss company Rinspeed concept at the Geneva Motor Show ( March 5 to 15 2015), as her thoughts on automotive future look like this. This year will Frank M. Rinderknecht, the strong man and genius behind it Rinspeed, provide a self-navigating small electric car based on the BMW i3 stir. The Rinspeed Budii to become the "best buddy ", that is the best friend of his driver. His autopilot wants to ensure adapt quickly to the habits and preferences of its occupant, reducing the maintenance of the car to a minium. There is an infotainment system with many large screens and the latest craze when it comes to networking technologies. Thus saith the Budii with other networked vehicles or automatically pay the parking ticket. Also possible: wireless mobile phone charging and smart access solutions to open the vehicle to start, or to personalize.
Budii learn
In addition Budii should also be capable of learning. Thus, the concept car information from its environment as well as their own experiences or those of other vehicles wants to consider along the route. Rinderknecht says: "In the future, the car will do the same as we do. It is a day to learn and thereby meet the complex requirements of the modern individual traffic getting better "
Steering wheel with a difference
Rinspeed calls the Budii a " trans-urban concept car ". This is intended to express that the car on the road or off-road has its place. While Budii so annoying takes daily trips independently in attack, the driver should also engage in conscious fun rides at the weekend quite self into the steering wheel. But what does this actually driver? For the latest Rinspeed study that is not so clearly defined in the act. The reason: Budiis steer-by-wire steering wheel is carried by a robotic arm that is able to swiftly switch sides. Compared to last year's study Xchange, the system was revised. The seven-axis unit is not only steering column, but will also serve as a side table or courteous butler if necessary. Driver and front passenger can pick up the tax. And if none of them feel like the valance is easily maneuvered in the middle and Budii moves back by itself.
Air suspension and Terrain scanner
Rinspeed Budii is driven by the electric motor of the BMW i3. He sends 170 hp and 250 newton meters to the rear axle. In 7.2 seconds to go from zero to 100 km / h top speed of 150 km / h and 200 kilometers are in there too. The study is available on Borbet 19-inch wheels and has an air suspension with an adjustment range of 100 millimeters. An up to 70 centimeters extendable and sensors - stocked telescope called Track View provides an accurate 3D perspective of the underground. This will ?? for example in the area ?? Height and suspension adjusted. Also, the driver can detect and avoid obstacles so early, and even autonomous.
Induction, fans and electric scooters
If it goes to Rinspeed must " be sexy and emotionally charged, to be purchased frequent e-Mobile. " But one has plunged the Budii in a light - anthracite finish, the " sparkles like a sparkler. Similarly opulent it comes to the interior of the futuristic crossover. There is a privacy-enhancing fan system or multi-function panels, which include not only the lights but also with other road users communicate. For the cognac-colored lining the interior noble tuner Mansory responsible. work is carried out during the autonomous driving to be inserted plexiglass tables and the mobile phone is of course inductively charged. Per smartphone or-watch also works the operation of the heater or the Batterievorwärmung. and if are no longer to realize the last meters of the journey to the Budii, you just take the two mini electric scooter (including helmet), driving on lateral drawers out of the car . (sw)
Lifted in Geneva
Red - black Renner
ground giant
strong increase
Space - athletes
Get rid of the angle
Passat with a beard
So is the fastest Lotus ever
The longest S-Class in the world
Hardcore small series
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Zitlally's family is undocumented, and her father has just been arrested for speeding and deported back to Mexico. As her family waits for him to return—they've paid a coyote to guide him back across the border—they receive news that he and the coyote's other charges have been kidnapped and are being held for ransom. Meanwhile, Zitlally and a new friend find a dog in the forest near their trailer park. They name it Star for the star-shaped patch over its eye. As time goes on, Zitlally starts to realize that Star is her father's "spirit animal," and that as long as Star is safe, her father will be also. But what will happen to Zitlally's dad when Star disappears?


text:
Tore Devos (Kortrijk, 11 mei 1996) is een Belgisch handbalspeler.

Levensloop
In 2018 speelde Devos bij Apolloon Kortrijk.

Belgisch handballer

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