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\markboth{J. E. Santos 
}{Waves in saturated poroviscoelastic solids
}\title{\LARGE
Finite Element Approximation of Coupled Seismic and Electromagnetic Waves in Fluid-Saturated  Poroviscoelastic  Media}
%On the Solution of the coupled Biot and Maxwell Equations 
%in Fluid-Saturated  Porous Media using the Finite Element Method}
\author{
Juan E. Santos\thanks{CONICET,
Departamento de Geof\'\i sica Aplicada, Fac.\ Ciencias Astron\'omicas y
Geof\'isicas, UNLP, Paseo del Bosque S/N, La~Plata, 1900, Argentina,
and Purdue University, West Lafayette, IN 47907, USA;
E-mail: santos@fcaglp.fcaglp.unlp.edu.ar
}
%\quad\quad
%Fabio I. Zyserman\thanks{ 
%CONICET,
%Departamento de Geof\'\i sica Aplicada, Fac.\ Ciencias Astron\'omicas y
%Geof\'isicas, UNLP, Paseo del Bosque S/N, La~Plata, 1900, Argentina.
%}
}
%\date{}
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\begin{document}
\maketitle

\begin{abstract}
This work  presents a collection of global and iterative finite element procedures for 
the numerical approximation of coupled seismic and electromagnetic waves 
in 2D  bounded fluid-saturated porous media, with absorbing boundary conditions at the artificial boundaries.  
The  equations being analyzed    
are the  coupled Biot's equations of motion and Maxwell 
equations in the diffusive range. 
Both seismoelectric and electroseismic coupling are 
simultaneously included and analyzed in the model.  The case of compressional and vertically polarized shear waves  coupled with the transverse magnetic polarization  (PSVTM-mode)   
is analyzed in detail, including the  derivation of {\it a priori} error estimates on the 
{\it global} finite element procedure and results on the convergence of a domain decomposition iterative algorithm.  Later the corresponding 
results for the case of horizontally polarized shear waves coupled with the transverse electric polarization  (SHTE-mode) are stated. \\  



\end{abstract}
 
%\noindent{\bf Subject Classification:} Primary 65N30; Secondary 

\noindent{\bf Keywords:} Biot and Maxwell equations, finite element method, 
 a priori error estimates, domain decomposition. \\

\section{Introduction}
Seismic waves propagating through
near-surface layers of the Earth  may  induce 
electromagnetic disturbances 
that can be measured at the surface (seismoelectric effect)\cite{pride96,mikha97,mikha2000}. Also,  recent tests
suggest that the reciprocal process, {\it i.e.}  surface measurable  
acoustic disturbances induced by 
 electromagnetic fields (electroseismic effect),  is also possible
\cite{thompson05a,hornbostel07}.\\
In order to explain these phenomena, Thompson and  Gist \cite{thompson93}
and Pride \cite{pride_94} suggested that they are generated by an
electrokinetic coupling
mechanism which  can be
shortly explained as follows \cite{block06,haines_pride_06}.  Within a fluid saturated porous medium there
exists a nanometer-scale separation of electric charge in which a bound
charge existing on the surface of the solid matrix (normally of negative sign)
is balanced by adsorbed positive ions of the surrounding fluid, setting an immobile layer.
Further from the surface there exists a distribution of mobile counter ions,
forming the so called diffuse layer. The effective thickness of this double
layer is of about 10 nm. When an electric field is applied to
this system, the ions in the diffuse layer move, dragging the pore fluid
along with it because of the viscous traction. This is known as electro-osmosis
and is responsible for the electroseismic phenomena.
On the other hand, the reciprocal situation arises when an applied pressure gradient creates
fluid flow and hence, an ionic convection current, which in turn produces
an electric field. This is known as electrofiltration and is responsible for the
so-called seismoelectric phenomena.\\
Using a volume averaging approach, Pride \cite{pride_94} derived a set of
equations   describing  both electroseismic and seismoelectric effects in
electrolyte-saturated porous media. In these equations the coupling mechanism
acts through the (generally frequency dependent) electrokinetic coupling
coefficient $L(\o)$. When this coefficient is set to zero, Pride's
set of equations turns to the uncoupled Maxwell's and Biot's equations,
describing the latter mechanical wave propagation in a fluid
saturated porous medium \cite{biot56b,biot56c}.\\
There exist already some works implementing different numerical methods
to solve the set of equations modeling both mentioned processes.
Haartsen and Pride \cite{haartsen_pride_97}, Han and Wang \cite{han_01}, Pain et. al \cite{pain2005}, Haines and Pride
\cite{haines_pride_06}  and White \cite{bwhite_05} and White and Zhou \cite{bwhite_06} have
proposed several different approaches to numerically study these phenomena.\\

In this paper  we first define and analyze a {\it global} finite element procedure 
for the approximate solution of the coupled Maxwell in the diffusive range  and Biot's equations of motion. These equations will be solved   
in an isotropic  2D bounded fluid-saturated poroviscoelastic domain with absorbing boundary conditions for the case of compressional and vertically polarized seismic waves coupled with the transverse magnetic polarization  (PSVTM-mode).  Notice that in the  Maxwell diffusive range   being analyzed     Biot's slow wave is a diffusion-type wave  so that  
 to properly describe the slow-wave diffusion effects, at least four points per diffusion length should  be employed \cite{haines_pride_06}.  This in turn implies that   a large number of degrees of freedom are  needed to 
calculate  the  wave fields accurately, and consequently  a parallellizable    nonoverlapping 
domain decomposition procedure is defined and analyzed.  Later, the  case of horizontally polarized shear waves coupled with the transverse electric polarization   (SHTE-mode) is  formulated and the corresponding results are stated. 


The 2D finite element procedures to  discretize   the PSVTM-mode   employ   the following spaces.  In the case of rectangular elements  the vector electric field and the scalar magnetic field are computed using the rotated Raviart-Thomas-Nedelec spaces of zero order 
 \cite{rath,nedelec}, while for  triangular elements  the  2D mixed finite element space 
of Nedelec \cite{nedelec,pmonk_94} of lowest order  are used . 
 Also,   both for rectangular and triangular elements, the nonconforming  spaces  defined in  \cite{dssy-ncell} are used to 
 approximate   each component of the  displacement vector in the solid phase and  the   displacement in the   fluid phase  
is approximated using the vector part of the Raviart-Thomas-Nedelec  
mixed finite element space of zero order.  
The 2D finite element spaces for the SHTE-mode  are identical  to those of the PSVTM-mode,
except that in this mode the solid and fluid displacements  are scalar functions in 
  $H^1$ and $L^2$, respectively. Consequently, the solid displacement is  approximated 
using the nonconforming  spaces  defined in  \cite{dssy-ncell} and the fluid displacement employing piecewise constants. 

{\it A priori }  optimal error estimates for the discretization of Biot's equations of motion using the finite element spaces described above were derived 
 in \cite{santos_sinum_07}, and   the corresponding dispersion 
 analysis  was presented in \cite{zs07}. On the other hand, finite element procedures to solve Maxwell's equations
 in 2D and 3D  within the frame of magnetotelluric modeling were presented in   
\cite{dou00,santos2dmt_98,santos00,zys99,zys00}.

The organization of the paper is as follows. In Section 2  the differential system describing the propagation of coupled diffusive electromagnetic and seismic waves  are stated in the space-frequency domain,  with corresponding absorbing boundary conditions at the artificial boundaries. Section 3 gives a  variational formulation and a result on the uniqueness of the solution  the boundary value problem for each  temporal  frequency.  
In Section 4 the rectangular-based finite element spaces used for the spatial discretization are presented and their approximation properties are stated. Also,  the {\it global} finite element procedure is formulated and results on the existence and uniqueness of the approximate solution are derived.   In Section 5  we derive  {\it a priori} error estimates on the {\it global} finite element procedure.  Section 6 presents a domain-decomposition iterative procedure and  derives its   convergence.  In  Sections 7,  8  and 9 we present 
the  equations describing the SHTE mode, a  weak formulation, and the {\it global} and domain decomposition finite element methods.  {\it A  priori} error estimates for the {\it global} algorithm and convergence results for the  domain decomposition procedure 
are also stated.  Section 10 describes the changes  in the definition of the finite element spaces needed for the case of triangular elements. Finally   Section 11 presents a set of conclusions.

\section{The differential model}
Consider a poroviscoelastic  solid   saturated by a single phase, 
compressible viscous fluid and assume that the whole aggregate is isotropic.  
Let $u^s = (u_i^s)$ and 
$\tilde u^f = (\tilde u_i^f),\, i=1,\cdots,d$ denote the  
averaged displacement vectors  of the solid and phases, 
respectively, where $d$ denotes the Euclidean dimension, i.e., $d=1,2,3$.
Also let  
\[
u^f=\phi(\tilde u^f-u^s),
\]
be  the average relative fluid  displacement per unit 
volume of bulk material, where $\phi$ denotes the 
effective porosity. Also set $u = (u^s,u^f)$ and recall that 
\[
\qquad \xi =-\nabla\cdot u^f,
\]
represents the change in fluid content.

Let the Fourier transform in the time variable of a given function $f(t)$ 
be defined as usual by 
\[
\widehat f(\o) = \int_{-\infty}^{\infty} e^{-i\o t} f(t) dt.
\]
In the  diffusive range for Maxwell equations  displacement currents can be ignored,    so that 
 the electric and  magnetic fields $E = E(\o)$ and $H= H(\o)$ and the displacement vectors $u^s =u^s(\o)$  and 
$ u^f = u^f(\o)$ satisfy the 
 coupled electromagnetic-poroviscoelastic  equations \cite{pride_94},
stated in the space-frequency domain as follows, with $u=(u^s,u^f)$ 
(the $\hat \cdot$ is  omitted in all variables for notational convenience):

\begin{eqnarray} 
&&  \sigma E - \nabla\times H + L(\o)\eta (\kappa(\o))^{-1}\left[ i \o u^f - L(\o) E \right ] = J^s_e,\label{mod.1b}\\
&&\nabla\times E +  i \o \mu H = J^s_m, 
\label{mod.1a}\\
&&-\omega^2 \rho_b u^s  - \omega^2  
\rho_f u^f  -  \nabla\cdot \tau(u) = F^{(s)},
\label{mod.1c}\\
&& -\o^2 \rho_f u^s + \eta (\kappa(\o))^{-1} \left[i \o u^f - 
L(\o) E \right]+ \nabla p_f = F^{(f)},\label{mod.1da}\\
&&\tau_{lm}(u)=2 N\,\varepsilon_{lm}(u^s)+\delta_{lm}
\left(\lambda_c \,\nabla\cdot u^s - \alpha K_{av}\, \xi\right),\label{mod.1e}\\
&& p_f(u) =  - \alpha K_{av} \, \nabla\cdot u^s + K_{av} \xi.\label{mod.1f}
\end{eqnarray}
In the equations above $\tau_{lm}(u)$ is the stress tensor of the bulk material
and $p_f(u)$ the fluid pressure, while $\varepsilon_{lm}(u^s)$  
denotes the strain tensor of the solid frame. Also,   
$\mu$ is the magnetic permeability  and $\sigma$ the conductivity, while 
 $F^{(s)}$, $F^{(f)}$, $J^s_m$ and $J^s_e$  are external 
 seismic and electromagnetic sources, respectively. Furthermore,   
\begin{equation}\label{robulk} 
\rho_b = \phi \rho_f + (1-\phi)\rho_s, 
\end{equation} 
where  $\rho_s$ and  $\rho_f$ denote the mass densities of the solid 
grains composing the solid matrix and the saturant fluid and $\eta$ the fluid viscosity. 
On the other hand, $\kappa(\o)$ denotes the   dynamic permeability, given by
 \cite{johnston87}, \cite{pride_94}
\begin{eqnarray}\label{def_perm}
\kappa(\o) = \kappa_0 \left[\left(1 + i \dfrac{\o}{\o_c}\dfrac{4}{m}\right)^{\frac12} + i\dfrac{\o}{\o_c}\right]^{-1} = \kappa_r(\o) - i \kappa_i(\o),
\end{eqnarray}
where $\kappa_0$ is the effective permeability and $m$   is a dimensionless parameter given by \cite{haartsen_pride_97,pride_94} 
\begin{equation}\label{def_m}
m =  \dfrac{\phi}{\alpha_{\infty} \kappa_0} \Lambda^2.
\end{equation}
It is known \cite{pride_94} that for a  wide variety of porous media $m$ lies in the range  
\[
4\le m \le 8. 
\]
In \eqref{def_m} $\Lambda$ is a geometric  factor representing a weighted volume to surface ratio, and $\alpha_{\infty}$ is the formation tortuosity, with $\alpha_{\infty}=1$ for parallel tubes and $\alpha_{\infty}=3$ for randomly oriented tubes.  Also, 
\begin{equation}\label{omega_crit}
\o_c = \dfrac{\phi \eta}{\alpha_{\infty} \kappa_0 \rho_f}
\end{equation}
is  the critical frequency separating  low-frequency viscous flow and high-frequency inertial flow.  
Note that $\kappa_r(\o)>0, \kappa_i(\o)>0$.
The frequency dependent coupling coefficient $L(\o)$ is given by \cite{haartsen_pride_97,pride_94} \begin{eqnarray}\label{def_L_omega}
L(\o) = L_0 \left[ 1 + i \dfrac{\o}{\o_c}\dfrac{4}{m} \left(1 - 2 \dfrac{\tilde d}{\Lambda}  \right)^2 \left(1 + i^{\frac32} \tilde d \left(\frac{\o \rho_f}{\eta}\right)^{\frac12}\right)^2\right]^{\frac{-1}{2}} = L_r(\o) +i  L_i(\o).
\end{eqnarray}
The positive static coupling coefficient   
$L_0$  is defined as  \cite{haartsen_pride_97} by
\begin{eqnarray}\label{def_L0}
L_0 = - \frac{\phi}{\alpha_{\infty}} \frac{\epsilon_0 k_f \zeta}{\eta} \left( 1 - 2 \alpha_{\infty} \frac{\widetilde
d}{\Lambda}\right), 
\end{eqnarray}
with  $\zeta = 0.008 + 0.026 \text{log}_{10}(C_e)$ denoting 
 the zeta potential and $C_e$ being the
electrolyte  molarity. In \eqref{def_L0}  $\epsilon_0$ and $k_f$ are   the vacuum and   fluid permitivities and 
\begin{equation}\label{def_debye}
\widetilde d = \frac{\epsilon_0 k_f k_B T}{e^2 {\it z}^2 N_{ic}} 
\end{equation}
is the Debye length in meters. in \eqref{def_debye} $e$ is 
the electronic charge,  $k_B$ is the Boltzman constant, T is 
the absolute temperature (so that $k_B T$ is the thermal energy) {\it z} is 
the ionic valence and $N_{ic}$ the ionic concentration in ions per meters cubed. 
 

In order to introduce viscoelasticity, the coefficients in the constitutive equations 
\eqref{mod.1e} and \eqref{mod.1f} are considered to be frequency dependent. 
They can be determined as follows. First we consider the (relaxed) 
elastic limits of these coefficients, denoted by the superindex $^*$. 
In this case,  the coefficient $N^*$  is equal to the elastic shear modulus of the 
 dry matrix.  Also, 
\begin{equation}\label{deflambdac_aster}
\lambda_c^*= K_c^* - (2/d) N^*, 
\end{equation}
with $K_c^*$ being the bulk modulus of the saturated 
material. The 
 coefficients in \eqref{mod.1e}-\eqref{mod.1f}  can be obtained from the relations \cite{gassmann51}, \cite{santos92}
\begin{eqnarray}\label{mod.3}
&&\alpha = 1 - \dfrac{K_m}{K_s}, 
\qquad  K^*_{av} = \left[\dfrac{\alpha- \phi}{K_s} + \dfrac{\phi}{K_f}\right]^{-1}\\
&&K_c^* = K_m + \alpha^2 K^*_{av},\nonumber
\end{eqnarray} 
where $K_s, K_m$ and $ K_f$  denote the bulk modulus of the solid grains 
composing the solid matrix,  the dry matrix and the  saturant fluid, 
respectively.  
Next, using the correspondence principle 
stated by M. Biot \cite{biot56,biot62},
  we replace  the (real) relaxed elastic coefficients $N^*, K_c^*$ and 
$K_{av}^*$ by  complex frequency 
dependent viscoelastic modulus using the linear viscoelastic model presented  
in \cite{liu76}  as follows: 
\begin{eqnarray}\label{mod.43}
&&K_{c}(\o) = \dfrac{ K^*_{c}}{R_{K_{c}}(\o)-i T_{K_{c}}(\o)},
\qquad N(\o) = \dfrac{ N^*} {R_{N}(\o)-i T_{N}(\o)},\\
&&K_{av}(\o) = \dfrac{ K^*_{av}}{R_{K_{av}}(\o)-i T_{K_{av}}(\o)}.\nonumber 
\end{eqnarray}
The frequency dependent coefficient $\lambda_c = \lambda_c(\o)$ in \eqref{mod.1e} is defined 
 in terms of  $K_c(\o)$ and $N(\o)$ as 
\begin{equation}\label{deflambdac}
\lambda_c= K_c(\o) - (2/d) N(\o). 
\end{equation} 
  
Also, the frequency dependent functions $R_s$ and $T_s$, $s=K_c, N, K_{av}$,   associated with a continuous spectrum 
of relaxation times, characterize the viscoelastic  behavior 
and are given by 
\cite{liu76}
$$
R_s(\o)
=1-\dfrac1{\pi  Q_{m,s}} \ln\dfrac{1+\o^2 T^2_{1}}{1+\o^2 T^2_{2}},
\qquad
T_s(\o)
=\dfrac2{\pi  Q_{m,s}} \tan^{-1}\dfrac{\o(T_{1}-T_{2})}{1+\o^2 T_{1} T_{2}}.
$$
The model parameters $Q_{m,s}, s=K_c, N, K_{av}$, $T_{1}$ and 
$T_{2}$ are  taken  such that the
quality factor
\[ 
\widehat Q_s(\o)= \frac{T_s(\o)}{R_s(\o)}
\]
is 
approximately equal to the constant $Q_{m,s}$     
in the range of frequencies where the  equations are solved, which makes 
 this model convenient for geophysical applications. Values of $Q_{m,s}$ 
 range from $Q_{m,s} = 10$ for highly dissipative materials to 
 about $Q_{m,s} = 1000$ for almost elastic ones. 


We wish to consider a  2D model that includes 
 a portion of the air above the porous medium, since in some cases the electromagnetic sources may be located  in the air region.  
Thus we assume  a 3D-rectangular  domain $\O = \O_a \cup \O_p$, where $\O_a$ and $\O_p$ 
are  associated with the air and subsurface poroviscoelastic  (disjoint)  parts of $\O$, respectively. We will assume that all the coefficients  are independent of the $y$-direction (i.e., $y$ is the symmetry axis) and consider  two different types of electromagnetic $J^s_m$ and $J^s_e$  and  corresponding seismic sources  as follows.


A first choice is  $J^s_e=0$ and  
\begin{eqnarray}\label{def_Jsm}
J_m^s= - i \o \mu I(\o) S \delta(x) \delta(z) e_y.
\end{eqnarray}
Here $\delta(\cdot)$ denotes the Dirac distribution and  $e_y=(0,1,0)$. 
This source  represents an harmonic  magnetic current with  
$I(\o)$ being  a small loop of current and S  the area
of the loop.  
This source term induces electromagnetic fields of the form  
$(E_x(x,z,\o),0,E_z(x,z,\o))$ and  $(0,H_y(x,z,\o),0)$ and 
induces compressional and vertically polarized shear waves (PSV-waves). 
The PSV-waves may also be generated by the external compressional and/or  shear  seismic sources $F^{(s)}$ and $F^{(f)}$. This is     a 2D model known as PSVTM-mode.

On the other hand, a second  choice of  electromagnetic source 
 is $J^s_m= 0$.and $J^s_e$ an harmonic line source current 
$J^s_e=(J^s_{e,x},J^s_{e,y},J^s_{e,z} )$
in the $y$-direction at depth $z=0$  of amplitude
$I(\o)$, \cite{seg_book_87} , {\it i.e.}, 
\begin{eqnarray}\label{def_Jse}  
J^s_{e,y}(x,z,\o) = I(\o) \delta(x)\delta(z), \qquad J^s_{e,x}= J^s_{e,z} =0, \quad (x,y,z)
\in \O.
\end{eqnarray}
Assuming   that no seismic sources are present, {i.e.},  $F^{(s)}= F^{(f)}=0$,  
this source term induces electromagnetic fields $(0,E_y(x,z,\o),0)$ and $(H_x(x,z,\o),0,H_z(x,z,\o))$ and  horizontally polarized shear waves  (SH-waves); we get another 2D  model known as SHTE-mode.


It is convenient to formulate Maxwell's equations  \eqref{mod.1b}-\eqref{mod.1a} in terms of scattered fields. 
Let us assume that  
\begin{eqnarray}\label{def_sigma}
\sigma(x,z) = \sigma^p(z) + \sigma^s(x,z),
\end{eqnarray}
where $\sigma^p(z)$ is the background conductivity and 
$\sigma^s(x,z)$ is the conductivity anomaly.

For the SHTE-mode, let us consider the solution $(E^p, H^p)$ of the following problem:
\begin{eqnarray} 
&&\nabla\times E^p = -i \o \mu H^p, \quad\text{in}\quad  \O,
\label{max.11a}\\
&& \nabla\times H^p = \sigma^p E^p + J^s_e,\quad\text{in}\quad  \O. \label{max.11b}
\end{eqnarray}
For the PSVTM-mode, let $E^p, H^p$ be the solution of 
\begin{eqnarray} 
&&\nabla\times E^p = -i \o \mu H^p + J_m^s, \quad\text{in}\quad  \O,
\label{max.p1}\\
&& \nabla\times H^p = \sigma^p E^p\label{max.p2}, \quad\text{in}\quad  \O,.
\end{eqnarray}
In the case in which $\O$ is the whole space $R^3$ and $\sigma_p$ is constant, the solution of 
\eqref{max.11a}-\eqref{max.11b}   is given by  \cite{seg_book_87} 
\begin{eqnarray}
&&E^p_y(x,z,\o) = \frac{i \o \mu I(\o)}{2 \pi} J_0(i \gamma{\bf R})
\label{def_Eyp},\\
&&H^p_x(x,z,\o) = \frac{i \o \gamma I(\o)}{2 \pi} J_1(i \gamma {\bf R})\left(\frac{z}{{\bf
R}},\frac{-x}{{\bf R}}\right).\label{def_Hp} 
\end{eqnarray} 
Similarly,  the solution of  \eqref{max.p1}-\eqref{max.p2}  for the case  $\sigma_p$  constant
 and $\O = R^3$ is  \cite{seg_book_87}
\begin{eqnarray}
&&H^p_y(x,z,\o) = -\sigma^p \frac{i \o \mu I(\o) S}{2 \pi} J_0(i \gamma {\bf R}),\label{def_Hpy},\\
&&E^p= \frac{J_1(i \gamma {\bf R}) i \gamma}{{\bf
R}}\left(\frac{z}{{\bf R}},\frac{ x}{{\bf R}}\right).
\end{eqnarray}
In the equations above, 
\[
{\bf R} = \left(x^2 + z^2\right)^{1/2},
\] 
$J_0$ is the modified Bessel function of second kind, 
 and zero order, $J_1(\beta) = - J_0^{'}(\beta)$  and 
 $\gamma$ is the wave number 
\begin{eqnarray}
\gamma = \left(-i \mu \sigma^p \o\right)^{1/2}. 
\label{def_gamma} 
\end{eqnarray} 




In the rest of this work we will analyze in detail the PSVTM-mode and later the 
changes needed to formulate an analyze the SHTE-mode will be briefly indicated. 

Let us define the scattered electrical and magnetic  fields by
\begin{eqnarray}
E^s = E - E^p =(E^s_x,E^s_y,E^s_z), \quad H^s = H - H^p = (H^s_x,H^s_y,H^s_z). 
\label{def_scat} 
\end{eqnarray} 
Now from \eqref{mod.1a}-\eqref{mod.1b} and \eqref{max.p1}-\eqref{max.p2}, we see
that $E^s_y$ and $H^s$ satisfy the equations
\begin{eqnarray}
&&\sigma E^s - \nabla\times H^s = - \sigma^s E^p, \quad \text{in}\quad  \O_a,
\label{max.s3}\\
&& \sigma E^s - \nabla\times H^s 
+ L(\o) \eta (\kappa(\o))^{-1} \left( i \o u^f - L(\o) E^s \right) \label{max.s2}\\
&&\hskip2.5cm = -\left(\sigma^s - L^2(\o) \eta (\kappa(\o))^{-1} \right) E^p
,\quad \text{in} \quad \O_p,\nonumber\\
&&\nabla\times E^s + i \o \mu H^s =0, \quad \text{in} \quad \O.\label{max.s1}
\end{eqnarray}
Let us identify the 3D vectors $(E_x(x,z,\o),0,E_z(x,z,\o))$ and 
$(0,H_y(x,z,\o)),0)$  with the 2D vector 
$(E(x,z)=(E_x(x,z),E_z(x,z))$ and the scalar $H_y(x,z,\o)$, respectively. 
Then  recall that 
\begin{eqnarray*}
\text{curl} H_y = \left(-\frac{\partial H_y}{\partial z}, 
\frac{\partial H_y}{\partial x}\right),\quad 
\text{curl} E =  \frac{\partial E_x}{\partial z} - \frac{\partial E_z}{\partial x}.
\end{eqnarray*}
Let us identify our 3D-rectangular  domain $\O$ with   the   2D-rectangular domain $\O\cap \{y=0\}$, so that $\O$ is the union of the disjoint rectangular subdomains $\O_a$ and $\O_p$. Let $\G$ denote the boundary of $\O$ and let 
 $\G_{a,p} = \overline \O_a \cap \overline \O_p$ denote the free surface. 
Also let $\G_a = \p\O_a \setminus \G_{a,p}$, $\G_p = \p\O_p \setminus \G_{a,p}$ 
denote the artificial boundaries of $\O_a$ and $\O_p$, respectively. 
  Also, if  $\G_s$ is either an inner interface in $\O$ or a part of the
  boundaries $\G, \G_p$ or $\G_{a,p}$, set 
\begin{subeqnarray}\label{mod8}
{\mathcal G}_{\G_s}(u) &=& \bigg(\tau(u) \nu \cdot \nu, \tau(u)\nu\cdot \chi,p_f(u)\bigg)^t,\slabel{mod8a}\\
S_{\G_s}(u) &=& \left(u^s\cdot\nu,u^s\cdot\chi,
 u^f\cdot\nu\right)^t,\slabel{mod8b}
\end{subeqnarray} 
where , where $^t$ denotes the transpose,  $\nu$ is  the unit outer normal on $\G_s$  and 
$\chi$ is a unit tangent on $\G_s$ oriented counterclockwise.  

Then, for $\o>0$ consider the solution of the  following 
differential system for the PSVTM-mode:
\begin{eqnarray}
&&\sigma E^s - \text{curl} H^s_y =  -\sigma^s E^p, \quad \text{in} \quad \O_a,\label{modf.2a}\\
&&\sigma E^s - \text{curl} H^s_y 
+   L(\o)\eta (\kappa(\o))^{-1}\left[ i \o u^f 
-  L(\o) E^s \right] \label{modf.2}\\
&&\qquad =  -\left(\sigma^s - L^2(\o) \eta (\kappa(\o))^{-1}\right)  E^p, \quad \text{in} \quad \O_p,\nonumber\\
&&\text{curl} E^s + i \o \mu H^s_y =0, \quad \text{in} \quad \O, \label{modf.1}\\
&&-\omega^2 \rho_b u^s  - \omega^2  \rho_f u^f -
\nabla\cdot \tau(u) = F^{(s)},  \quad \text{in} \quad \O_p,\label{modf.3}\\
&& -\omega^2 \rho_f u^s  + \eta (\kappa(\o))^{-1} \left[i \o u^f -  L(\o) E^s\right] + \nabla p_f \label{modf.4}\\
&&\qquad = F^{(f)} +  \eta (\kappa(\o))^{-1} L(\o) E^p,    \,  \, \text{in} \,\, \O_p,\nonumber
\end{eqnarray}
with the  absorbing boundary conditions  \cite{santos2dmt_98}, \cite{santos_imajna}
\begin{eqnarray}
&&a (1-i) E^s\cdot \chi + H_y = 0, \quad \text{on} \quad  \G,\label{modf.5}\\
&& -{\mathcal G}_{\G_p}(u)  = i \o {\mathcal D}S_{\G_p}(u), 
\quad \text{on}\quad \G_p.\label{modf.6}
\end{eqnarray}
and the free surface condition
\begin{eqnarray}
&& -{\mathcal G}_{\G_p}(u)  = 0, 
\quad \text{on}\quad \G_{a,p}.\label{modf.6a}
\end{eqnarray}
Here $a$ is a positive coefficient  defined as 
\begin{eqnarray}
a = a(\o) = \left(\frac{\sigma}{2 \o \mu}\right)^{1/2}.
\label{def_a}
\end{eqnarray}
The  matrix ${\mathcal D}$ in \eqref{modf.6} is defined as:
${\mathcal D} 
= {\mathcal R}^{\frac12} {\mathcal S}^{\frac12}{\mathcal R}^{\frac12}
$, where ${\mathcal S } 
= {\mathcal R}^{-\frac12} {\mathcal  M}^{\frac12}{\mathcal R}^{-\frac12}
$
and 
\begin{eqnarray*}
{\mathcal R} =\begin{pmatrix}
\rho_b &0&\rho_f \\
0&b&0\\
\rho_f&0&\zeta 
\end{pmatrix},
\quad {\mathcal M}=\begin{pmatrix}
\lambda_c^* + 2 N^* &0 &\alpha ~ K^*_{av}\\
0 &N^*&0\\
\alpha ~ K^*_{av}&0 & K^*_{av}
\end{pmatrix},
\end{eqnarray*}
where 
\[
b=\rho_b - \frac{(\rho_f)^2}{\zeta},\quad \zeta= \frac{\rho_f \alpha_{\infty}}{\phi}.  
\]
{\emph Remark}: Note that since $\alpha_{\infty}\ge 1$, the matrix ${\mathcal R}$ 
is positive definite. Also, we will requiere that the following conditions 
 be satisfied by the coefficients defining the matrix ${\mathcal M}$:
\begin{subeqnarray}
N^*>0\slabel{cond_1},\\
\lambda^*_c + N^* - \alpha^2 K^*_{av} >0\slabel{cond_2},\\
K^*_{av} >0\slabel{cond_3}.
\end{subeqnarray}
Conditions \eqref{cond_1}, \eqref{cond_2} and \eqref{cond_3} are necessary 
and sufficient conditions for the matrix ${\mathcal  M}$ to be positive definite. 
 In particular, the condition \eqref{cond_2}  imposes that  the  inverse of the  
jacketed compressibility coefficient be strictly positive.  
The jacketed compressibility test  is an experiment 
  in which the fluid pressure is held constant 
and a sample of bulk material is subjected to an hidrostatic 
compression,   see \cite{biot62}.  
As a consequence of the positive definitess of the matrices ${\mathcal R}$
 and ${\mathcal M}$, the matrix ${\mathcal D}$ is also positive definite.

Finally, here and in what follows we choose
$d=2$ in the definition of $\lambda_c^*$ and $\lambda_c$ in
\eqref{deflambdac_aster} and \eqref{deflambdac}. The coefficient $\lambda_c$
defined in this fashion is used in the constitutive relation defining the
stress tensor components $\tau_{lm}(u)$ in \eqref{mod.1e}.   
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\section{A weak formulation}
For  $X\subset \mathbb R^d, d=1,2,3$ with boundary $\partial X$, let $(\cdot,\cdot)_X$ denote the complex $L^2(X)$ inner product for scalar, 
vector, or matrix valued functions.
Also, for $s\in\mathbb R$,
$\|\cdot\|_{s,X}$  will denote  
the usual norm  for the Sobolev space $H^s(X)$.
In addition, if $X=\O$ or $X=\G$, the subscript $X$ may be omitted such that
$(\cdot,\cdot)=(\cdot,\cdot)_\O$ or  
$\<\cdot,\cdot\>=\<\cdot,\cdot\>_\G$. Set 
\begin{eqnarray*}
&&H(\text{curl},\O)=\{\psi\in(L^2(\O))^2: \text{curl} \psi\in L^2(\O)\},\\
&&H(\text{div},\O)=\{\psi\in(L^2(\O))^2: \nabla\cdot \psi\in L^2(\O)\},\quad 
H^1(\text{div},\O)=\{\psi\in(H^1(\O))^2: \nabla\cdot \psi\in H^1(\O)\},
\end{eqnarray*}
provided with the natural norms
\begin{eqnarray*}
&&\|\psi\|_{H(\text{curl},\O})=(\|\psi\|^2_0+\| \text{curl} \psi\|^2_0)^{\frac12},\\
&& \|\psi\|_{H(\text{div},\O})=(\|\psi\|^2_0+\| \nabla\cdot \psi\|^2_0)^{\frac12},\quad 
\|\psi\|_{H^1(\text{div},\O}=(\|\psi\|^2_1+\| \nabla\cdot \psi\|^2_1)^{\frac12}.
\end{eqnarray*}
Recall the integration by parts formulas \cite{gira,dsheenb}
\begin{eqnarray}\label{by_parts1}
&&(\psi, \text{curl} \varphi) - (\text{curl} \psi,\varphi)=\la\psi\cdot\chi,\varphi\ra,\,\quad
\forall\psi\in H(\text{\text{curl}},\O),\quad \varphi\in H^1(\O),\\
&&(\nabla\cdot \psi, \varphi) 
+ (\psi,\nabla \varphi)=\la\psi\cdot\nu,\varphi\ra,\,\quad
\forall\psi\in H(\text{div},\O),\quad \varphi\in H^1(\O).\label{by_parts2}
\end{eqnarray}
To obtain a variational formulation, 
test \eqref{modf.1} against $\varphi\in L^2(\O)$  and test \eqref{modf.2} 
against $\psi\in H(\text{curl},\O)$ and use the integration by parts formula \eqref{by_parts1} and 
the boundary condition \eqref{modf.5}.
Also, test \eqref{modf.3} against $v^s\in [H^1(\O_p)]^2$ and \eqref{modf.4} 
against  $v^f\in H(\text{div},\O_p)$ 
and use the integration by parts formula \eqref{by_parts2} 
and  the boundary condition \eqref{modf.6}.  Thus, setting $v=(v^s, v^f)$, and 
\[
{\mathcal Y}=H(\text{curl}, \O)\times L^2(\O)\times [H^1(\O_p)]^2\times
H(\text{div},\O_p) 
\]
we have  the following weak formulation: Find $(E^s, H^s_y, u^s, u^f)
\in {\mathcal Y}$ such that
\begin{eqnarray}
&&\Theta\left( (E^s,H^s_y,u^s,u^f), (\psi,\varphi,v^s,v^f)\right) = 
(\sigma E^s,\psi) -(H^s_y, \text{curl} \psi) \label{eq_100}\\
&& \qquad + \left\langle a(1-i) E^s\cdot \chi, \psi\cdot\chi\right\rangle
 + \left( L(\o) \eta (\kappa(\o))^{-1} \left[i \o u^f - L(\o)  E^s\right], \psi\right)_{\O_p}\nonumber \\
&&\qquad  +(\text{curl} E^s, \varphi) + (i \o \mu H^s_y, \varphi)
- \o^2 \left(\rho_b  u^s, v^s\right)_{\O_p} 
- \o^2 \left(\rho_f  u^f, v^s\right)_{\O_p}- \o^2 \left(\rho_f  u^s, v^f\right)_{\O_p}\nonumber\\
&&\qquad  + \left(  \eta (\kappa(\o))^{-1} \left[i \o u^f - L(\o)  E^s\right], v^f\right)_{\O_p}
+  {\mathcal A}(u,v) + i \o \left\langle {\mathcal D} S_{\G}(u), S_{\G}(v)\right\rangle_{\G_p}\nonumber\\
&&\qquad = -\left(\left[\sigma^s - \chi_{\O_p}  L^2(\o) \eta (\kappa(\o))^{-1}\right]  E^p, \psi\right)\nonumber\\
&&\qquad\qquad   + (F^{(s)}, v^s)_{\O_p} + \left(\left[F^{(f)} +  L(\o)\eta (\kappa(\o))^{-1} E^p\right], v^f\right)_{\O_p},\nonumber\quad (\psi,\varphi,v^s,v^f)\in {\mathcal Y}.\nonumber
\end{eqnarray}
{\emph Remark}: Note that  since $E^s, \psi \in , H(\text{curl}, \O)$, 
the traces
 $E^s\cdot \chi$ and $\psi\cdot \chi$ only belong to $H^{-1/2}(\G)$ and
 consequently the boundary term 
 $\left\langle a(1-i) E^s\cdot \chi, \psi\cdot\chi\right\rangle$ 
 is  understood here as 
 $\left\langle \left(a(1-i) E^s\cdot \chi\right) \cdot 
 \left(\overline\psi\cdot\chi\right), 1\right\rangle$, 
 the duality pair between 
 $\left(a(1-i) E^s\cdot \chi\right) \cdot 
 \left(\overline\psi\cdot\chi\right) \in \text{Lip}(\G)^{'}$ and $1 \in
 \text{Lip}(\G)$. Here $\text{Lip}(\G)^{'}$ is the dual space of 
 Lipschitz-continuous functions on $\G$. Here we have applied in the  2D
 case  the argument given in  \cite{dsheenb} for the 
  3D case to define $\left\langle\nu\times U, V\right\rangle_{\G}$ 
 for $U, V \in H(\text{curl}, \O)$. For  additional references on traces of 
 $H(\text{curl}, \O)$  see for example \cite{buffa_01,buffa_02}.    
 
 
     
In \eqref{eq_100}, $\chi_{\O_p}$ denotes the characteristic  
function of  $\O_p$ and 
${\mathcal A}(u,v)$ is the bilinear form  defined by
\begin{eqnarray}
{\mathcal A}(u,v) = \sum_{l,m}\left(\tau_{lm}(u), \varepsilon_{lm}(v^s)\right)_{\O_p} - \left(p_f(u), \nabla\cdot v^f\right)_{\O_p}.\nonumber
\end{eqnarray}
Note that  ${\mathcal A}(u,v)$  can be written in the form 
\begin{eqnarray*}\label{bilinear2}
{\mathcal A}(u,v) =  \left({\bf M} ~ \tilde \epsilon(u),\tilde \epsilon(v)\right)_{\O_p} = \left({\bf M}_r ~ \tilde \epsilon(u),\tilde \epsilon(v)\right)_{\O_p}  + i \left({\bf M}_i ~ \tilde \epsilon(u),\tilde \epsilon(v)\right)_{\O_p}, \ u, v\in [H^1(\O_p)]^2\times
H(\text{div},\O_p),
\end{eqnarray*}
where ${\bf M} = {\bf M(\o)} = {\bf M}_r(\o) + i {\bf M}_i(\o)$ is the complex matrix defined by 
\begin{eqnarray*}
{\bf M} =\begin{pmatrix}
\lambda_c + 2 N&\lambda_c&\alpha~ K_{av}&0\\
\lambda_c&\lambda_c + 2 N& \alpha ~ K_{av} & 0\\
\alpha ~ K_{av}& \alpha ~ K_{av}& K_{av}&0\\
0&0&0&2 N
\end{pmatrix},
\end{eqnarray*}
and  $\tilde\varepsilon(u) = \left(
\e_{11}(u^{(s)}), \e_{22}(u^{(s)}), \nabla\cdot u^{(f)}, \e_{12}(u^{(s)}\right)^t$.
 
Furthermore, we assume that the real part ${\bf M}_r$ is positive definite since in the
elastic limit it is associated with the strain energy density. On the other
hand, 
the imaginary part ${\bf M}_i$ is assumed to be positive definite because of
the restriction imposed on our system  by the First and Second Laws of 
Thermodynamics \cite{rav_05}. 

To show uniqueness for the solution of \eqref{eq_100}, set $F^{(s)} = F^{(f)}=0, 
E^p =0$ and set 
\begin{eqnarray}\label{def_g}
&&g =   \eta (\kappa(\o))^{-1} \left[i \o u^f -  L(\o) E^s\right],\quad (x,z) \in \O_p, 
\end{eqnarray}
so that
\begin{eqnarray}\label{uniq.3}
i \o u^f - L(\o)\eta   E^s = \dfrac{\kappa(\o)}{\eta} g, \quad (x,z) \in \O_p. 
\end{eqnarray}
First choose $\psi=0, \varphi=H^s_y, v^s = v^f = 0$  in  \eqref{eq_100} to get
\begin{eqnarray}\label{eq_38_n}
(H^s_y, \text{curl} E^s) - (i \o \mu H^s_y, H^s_y) = 0.
\end{eqnarray}
Next, choose  $\psi=E^s, \varphi=0, v^s = v^f=0$ in  \eqref{eq_100} and use \eqref{eq_38_n} to obtain 
\begin{eqnarray}
&&(\sigma E^s, E^s) - (i \o \mu H^s_y, H^s_y) 
 + \left\langle a(1-i) E^s\cdot \chi, E^s\cdot\chi\right\rangle\label{uniq.1}
+ \left( L(\o) g, E^s\right)_{\O_p} = 0.
\end{eqnarray}
Now take $v^s = i \o u^s, v^f = i \o u^f$ in \eqref{eq_100}  
 and add the resulting equation to \eqref{uniq.1} to get 
\begin{eqnarray}
&&(\sigma E^s, E^s) - (i \o \mu H^s_y, H^s_y) 
 + \left\langle a(1-i) E^s\cdot \chi, E^s\cdot\chi\right\rangle + 
\left( L(\o) g, E^s\right)_{\O_p} \label{uniq.2}\\
&&\qquad - i\o \left[-\o^2 (\rho_b u^s, u^s)_{\O_p} - \o^2 2 \text{Re}\left[(\rho_f u^f, u^s)_{\O_p}\right] + \left({\bf M}_r ~ \tilde \epsilon(u),\tilde \epsilon(u)\right)_{\O_p} \right]\nonumber\\
&&\qquad 
+ \o \left({\bf M}_i ~ \tilde \epsilon(u),\tilde \epsilon(u)\right)_{\O_p} 
+ \o^2 \left\langle {\mathcal D} S_{\G}(u), S_{\G}(u)\right\rangle_{\G_p}
 + \left(g , i \o u^f\right)_{\O_p}=0.\nonumber
\end{eqnarray}
Set 
\begin{eqnarray}\label{def_fi0}
\Phi_p = (\sigma E^s, E^s)_{\O_p} + ( L(\o) g, E^s)_{\O_p} + (g, i \o u^f)_{\O_p},\quad
\Phi_a = (\sigma E^s, E^s)_{\O_a}, \quad \Phi= \Phi_a + \Phi_p.
\end{eqnarray}
Then, using \eqref{uniq.3} it follows that
\begin{eqnarray}\label{uniq.4}
\Phi = \Phi_a  +  (\sigma E^s, E^s)_{\O_p}  + ( 2 L_r(\o) g, E^s)_{\O_p} 
+ (\dfrac{\kappa^*(\o)}{\eta} g, g)_{\O_p}. 
\end{eqnarray}
Thus,
\begin{eqnarray}\label{uniq.5}
&&\text{Re}\left[\Phi_p\right] =  (\sigma E^s, E^s)_{\O_p}  + 2 \text{Re}\left[(L_r(\o) g, E^s)_{\O_p} \right] 
+ (\dfrac{\kappa_r(\o)}{\eta} g, g)_{\O_p}\nonumber\\
&&\ge \sigma_{\text{min}} \|E^s\|_{0,\O_p}^2 
+ \dfrac{(\kappa_r(\o))_{\text{min}}}{\eta} \|g\|_{0,\O_p}^2 
- (L_r(\o))_{\text{max}}\left[\|E^s\|^2_{0,\O_p} + \|g\|^2_{0,\O_p}\right],  
 \end{eqnarray}
where
\begin{eqnarray}
A_{\text{min}} = \text{inf}\,\, \{A(x,z), (x,z)\in \O_p\}, \,\, A=\sigma, \kappa_r,\,\,   L_r(\o)_{\text{max}} = \text{sup}\hskip.1cm  \{L_r(x,z,\o), (x,z)\in \O_p\}.\nonumber    
\end{eqnarray}
Assume that
\begin{subeqnarray}\label{eq51}
&&\sigma_{\text{min}}  - (L_r(\o))_{\text{max}} \ge C_1 >0,\\
&& \dfrac{(\kappa_r(\o))_{\text{min}}}{\eta} - (L_r(\o))_{\text{max}} \ge C_2 >0,
\end{subeqnarray}
and set
\begin{eqnarray}\label{defc4}
&&C_3 = \text{inf}\,\, \{\sigma(x,z), (x,z)\in \O_a\},\quad C_4 = \text{min}(C_1, C_2).
\end{eqnarray}
Then it follows from \eqref{uniq.5} that 
\begin{eqnarray}\label{eq54}
&&\text{Re}(\Phi_p) \ge C_4 \left[\|E^s\|^2_{0,\O_p} + \|g\|^2_{0,\O_p}\right].  
\end{eqnarray}
Thus, take real part in \eqref{uniq.2} and use \eqref{eq54}  to obtain 
\begin{eqnarray}
&&C_3 \|E^s\|^2_{0,\O_a} + C_4 \left(\|E^s\|_{0,\O_p}^2 + \|g\|^2_{0, \O_p}\right)
 +   \left\langle a E^s\cdot \chi, E^s\cdot\chi\right\rangle 
+  \o \left({\bf M}_i ~ \tilde \epsilon(u),\tilde \epsilon(u)\right)_{\O_p}\label{uniq.6} \\
&&\qquad +  \o^2 \left\langle {\mathcal D} S_{\G}(u), S_{\G}(u)\right\rangle_{\G_p}\le 0. \nonumber
\end{eqnarray}
Since each term in  the left-hand side of \eqref{uniq.6} is nonnegative, it follows that 
\begin{eqnarray}
&&E^s = 0, \quad \text{in}\quad L^2(\O),\label{uniq_E}\\
&&E^s\cdot\chi = 0, \quad \text{in}\quad H^{-1/2}(\p\O),\label{uniq_E_tau}\\
&&g =0,  \quad \text{in}\quad L^2(\O_p),\label{uniq_g}\\ 
&&\nabla\cdot u^f=0,  \quad \text{in}\quad L^2(\O_p),\label{uniq_div_uf}\\
&&\varepsilon_{11}(u^s) = \varepsilon_{12}(u^s) =\varepsilon_{22}(u^s) =0, 
\quad \text{in}\quad L^2(\O_p),\label{uniq_strain}\\
&&u^s=0,  \quad \text{in}\quad H^{1/2}(\G_p),\label{uniq_us_bry}\\
&&u^f\cdot\nu=0, \quad \text{in}\quad H^{-1/2}(\G_p).\label{uniq_uf_nu}
\end{eqnarray}
From \eqref{def_g}, \eqref{uniq_E} and \eqref{uniq_g} it follows that 
\begin{eqnarray}
&&u^f = 0, \quad \text{in}\quad L^2(\O_p).\label{uniq_uf}
\end{eqnarray}
Next, using \eqref{uniq_E} and  \eqref{uniq_E_tau} in \eqref{uniq.1} we conclude that 
\begin{eqnarray}
&&H^s_y = 0, \quad \text{in}\quad L^2(\O).\label{uniq_H}
\end{eqnarray}
Next, notice that 
\begin{eqnarray}
&&\|| f\||= \left(\sum_{l,m}\int_{\O_p} |\varepsilon_{lm}(f)|^2 d\O_p\right)^{1/2}
\end{eqnarray}
defines a norm on functions $f\in [H^1(\O_p)]^2$ that vanish on a subset of positive measure of $\p\O_p$, \cite{ciarlet_78}. Thus it follows from \eqref{uniq_strain} and 
\eqref{uniq_us_bry} that 
\begin{eqnarray}
&&u^s = 0, \quad \text{in}\quad L^2(\O_p)\label{uniq_us}.
\end{eqnarray}
Thus we have uniqueness. 
We summarize the result in the following theorem.
\begin{theorem}\label{Theorem3.1}
%\label{thrm3.1}
Under the hypothesis made above on the positive definitess of the matrices 
${\bf M}_i(\o)$, ${\mathcal D}$,  the fact 
that  the coefficient $a$ in the  boundary condition 
\eqref{modf.5} is strictly  positive and the validity 
of \eqref{eq51},  the solution of problem \eqref{modf.2a}-\eqref{modf.4} 
with the boundary conditions \eqref{modf.5}-\eqref{modf.6} is unique 
 for any $\o > 0$.
\end{theorem}

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\section{A global finite element method for the PSVTM-mode}
Let $\Tau^h(\O)$ be a nonoverlapping quasiregular 
 partition of $\O=\O_p\cup\O_a$ into rectangles $\O_j$ 
of diameter bounded by $h$ such that $\overline\Omega=\cup^J_{j=1}\overline \O_j$. 
Denote by $\xi_j$ and $\xi_{jk}$ the midpoints of $\G_j=\p \O_j\cap \G$ 
and $\G_{jk}= \G_{kj} = \p \O_j\cap\p \O_k $,
respectively. Let $\ai\cdot,\cdot\ad_{\G_{jk}}$ denote the
approximation to the (complex)  inner product
$\lma\cdot,\cdot\rma_{\G_{jk}}$ in $L^2(\G_{jk})$ computed using
the  mid-point quadrature rule; more precisely,
$$
\ai u,v\ad_{\G_{jk}} = (u\overline v)(\xi_{jk})|\G_{jk}|,
$$
where $|\G_{jk}|$ denotes  the measure of $\G_{jk}$,

Let us denote by $\nu_{jk}$ the unit outer normal on $\G_{jk}$ 
from $\O_j$ to $\O_k$ and by $\nu_j$ the unit outer normal to $\G_j$.
Let $\chi_j$ and $\chi_{jk}$ be unit tangents on $\G_j$ and 
$\G_{jk}$ so that $\{\nu_j,\chi_j\}$ and 
$\{\nu_{jk},\chi_{jk}\}$
are orthonormal systems on $\G_j$ and $\G_{jk}$, respectively.


To approximate the electromagnetic fields $E^s, H_y^s$ we will employ 
the spaces ${\mathcal V}^{h}$ and  ${\mathcal W}^{h}$, respectively,  defined 
 here as follows:
\begin{eqnarray*}
&&{\mathcal V}^{h}=\{\psi \in H(\text{curl},\O):\psi|\O_{j}\in {\mathcal V}^h_{j}
\equiv P_{0,1}(\O_j)\times P_{1,0}(\O_j)\},\\
&&{\mathcal W}^{h}=\{\varphi \in L^2(\O): \varphi|\O_{j}\in 
{\mathcal W}^h_{j} \equiv  P_0(\O_j)\}.
\end{eqnarray*}
Here, $P_{s,t}(\O_j)$   denote the polynomials of degree 
not greater than $s$ in $x$ and not greater than $t$ in $z$ on $\O_j$, while 
$P_0$ denote the constants on $\O_j$.
The functions in ${\mathcal V}^h$ have continuous tangential components across the internal boundaries $\G_{jk}$. Also, $\text{curl} {\mathcal V}^h\subset 
{\mathcal W}^{(h)}$ \cite{nedelec,pmonk_94}. 
Two--dimensional mixed finite element spaces of $H(\text{curl},\O)$ have been
presented in \cite{pmonk_94} to discretize Maxwell's equations and in \cite{rduran}
to solve an elasticity problem over a triangulation of $\O$.


\emph{Remark:} Note that the spaces ${\mathcal V}^h$ and 
${\mathcal W}^h$ are rotated Raviart--Thomas
spaces of zero order, which are usually employed to solve second order elliptic problems using mixed methods (see for example \cite{rath,dr,tho}). This
would not be the case if we should be treating the full 3-D problem.

Following \cite{pmonk_94}, the degrees of freedom for ${\mathcal V}^h$ are defined in the following way. 
Let $\O_{j}$
be a general element of the partition $\Tau^h(\O)$ and let
$\psi\in[H^1(\O_{j})]^2$. Then define the following moments on $\G_{jk}$:
\begin{eqnarray}\label{moments}
&&M_{\G_{jk}}(\psi)=\left\{\la\psi\cdot\tau,f\ra_{\G_{jk}};\quad f\in
P_{0}(\G_{jk})\right\}.
\end{eqnarray}

The argument given in \cite{nedelec}  for the three--dimensional
case can be applied here to show that \eqref{moments}  are
curl--conforming and unisolvent for elements in ${\mathcal V}^h$.

To approximate each component of  the solid displacement vector
we employ the nonconforming finite element space  ${\mathcal NC}^h$  
as in \cite{dssy-ncell}, while to approximate the fluid displacement  
vector we choose ${\mathcal M}^{h}$, the vector part of the 
Raviart-Thomas-Nedelec space 
\cite{rath,nedelec} of zero order. 
More specifically, set 
$$
\widehat R=[-1,1]^2, \qquad
\widehat {\mathcal NC}(\widehat R)=\mbox{Span}\{1,\hat x, \hat z,\widetilde  \alpha(\hat x)-
\widetilde \alpha(\hat z)\},\quad
\widetilde \alpha(\hat x) =  \hat x^2 - \frac{5}{3}  \hat x^4.
$$
with the degrees of freedom being the values at the 
midpoint of each edge of $\widehat R$. 
Next, for each $\O_j\subset\O_p$, let
$F_{\O_j} : \widehat R\rightarrow \O_j$
be an invertible affine mapping such that $F_{\O_j}(\widehat R)=\O_j$, and define
\begin{eqnarray*}
{\mathcal NC}^h_j &=&\{v:\, v = \widehat v\circ F_{\O_j}^{-1},\, 
\widehat v\in \widehat {\mathcal NC}(\widehat R)\}.\nonumber
\end{eqnarray*}
Thus,  
\begin{eqnarray*}
&&{\mathcal NC}^{h} = \{ v \, :  v_j=  v|_{\O_j}\,  \in 
{\mathcal NC}^h_j, ~ 
v_j(\xi_{jk}) =v_k(\xi_{jk})\,\,  \forall (j,k)\} ,\nonumber\\
&&{\mathcal M}^{h}=\{w \in H(\text{div},\O_p):w|\O_{j}\in {\mathcal M}^h_{j}
\equiv P_{1,0}(\O_j)\times P_{0,1}(\O_j)\}.\nonumber
\end{eqnarray*}
In order to state the approximating properties of the finite element spaces defined above it is convenient 
to introduce four projection operators as follows.

First, let 
\[
[H^1_h(\O)]^2=\{\psi:\psi|\O_{j}\in[H^1(\O_{j})]^2\}
\]
and   if $\G_{jk,p}$ denotes any inner interface $\G_{jk}$ in $\O_p$ 
let us introduce the space 
\begin{eqnarray*} 
&&\tilde \Lam^h = \left\{\tilde \lam^h : \tilde \lam^h_{{jk}} = \text{tr}_{~\G_{jk,p}}(\tilde
\lam^h|_{\O_j})\in
[P_0(\G_{jk,p} )]^2\equiv\tilde \Lam^h_{jk},
\quad\tilde\lam^h_{jk} + \tilde\lam^h_{kj} = 0 \right\},\nonumber
\end{eqnarray*}
where  $P_0(\G_{jk,p})$ denotes the constant  functions defined on
$\G_{jk,p}$.

\noindent
{\emph Remark}: Note that there are two     
copies of $[P_0(\G_{jk,p} )]^2$ assigned 
to each $\G_{jk,p}$, one from $\O_j$ to $\O_k$ and another  
from $\O_k$ to $\O_j$.

Then  we define the  projections
\begin{eqnarray}
&&\Pi_h:H(\text{curl},\O)\cap[H^1_h(\O)]^2\to {\mathcal V}^h: \quad
\la(\psi-\Pi_h\psi)\cdot\chi, 1\ra_{B}=0,\, B=\G_{jk} \, \mbox{ or } \, \G_j,\label{def_pih}\\
&&P_h : L^2(\O)\rightarrow  {\mathcal W}^{h}: 
\hskip3cm (P_h w - w, \varphi)=0, \,    
\varphi \in {\mathcal W}^{h}, \label{def_ph}\\ 
&&R_h: [H^2(\O_p)]^2\rightarrow [\NC^h]^2:
\hskip2cm (v_i^{s} - R_h v_i^{s})(\xi) = 0, \, \xi = \xi_{jk}\mbox{ or } \xi_j,\label{def_rh}\\
&&\hskip6.5cm \text{for}\quad   v^s=(v^s_1, v^s_2),\nonumber\\ 
&&Q_h: [H^1(\O_p)]^2\rightarrow {\mathcal M}^h:
\hskip2cm \la (v^{f}-Q_h v^{f})\cdot\nu, 1\ra_{B}=0;  
\, B=\G_{jk,p} \, \mbox{ or } \, \G_j,\label{def_qh}\\
&&S_h : [H^2(\O_p)]^2\times H^1(\text{div};\O_p) \rightarrow \tilde \Lam^h:\qquad
\left\langle \tau(v)\nu - S_h(v), 1\right\rangle_{B} = 0,\label{def_sh}\\
&&\hskip8cm v=(v^{s}, v^{f}),  B=\G_{jk,p} \, \mbox{ or } \, \G_j.\nonumber
\end{eqnarray}
Let us define the  broken norms
\[
\|v\|^2_{s,h,\O_p} = \sum_{\O_{j}\subset \O_p} \|v\|^2_{s,\O_j}. 
\]

Then   approximation properties of these operators can be 
stated as follows \cite{nedelec,santos_sinum_07}:
\begin{eqnarray}
&&\|\psi-\Pi_h\psi\|_0\leq C h\|\psi\|_{1},\,\psi\in [H^1(\O)]^2,\label{aprox_pih1}\\
&&\|\text{curl}(\psi-\Pi_h\psi)\|_0\leq C h\|\text{curl}\psi\|_{1},\,\psi\in [H^1(\O)]^2,\, \text{curl}\psi \in H^1(\O),\label{aprox_pih2} \\
&&\|P_h \var- \var\|_0\leq C h\|\var\|_1, \qquad \forall \var\in H^1(\O),\label{aprox_ph}\\
&&\left[\|v^s- R_h v^{s}\|_{\O_p}
+ h \|v^{s}-R_h v^{s}\|^2_{1,h,\O_p}
+ h^2 \|v^{s}
- R_h v^{s}\|^2_{2,h,\O_p} 
\nonumber\right.\\
&&\qquad
+  h^{\frac12} \bigg(\sum_{\O_{j}\subset \O_p}\|v^{s}-R_h v^{s}\|^2_{0,\p \O_j}
\bigg)^{\frac12}
+ h^{\frac{3}{2}}\bigg(\sum_{\O_{j}\subset \O_p}\|\tau(v_j)\nu_j -
S_h v_j\|^2_{0,\p \O_j}\bigg)^{1/2}\label{approx_rh} \\
&&\quad
\le C  h^2 \left(\|v^{s}\|_{2,\O_p}  + 
\|\nabla\cdot v^{f}\|_{1,\O_p}\right),\,v=\left(v^{s}, v^{f}\right)
\in [H^2(\O_p)]^2\times H^1(\text{div}, \O_p)\nonumber,\\
&&\|Q_h v^f - v^f\|_{0,\O_p}\leq C h\|\v^f\|_{1,\O_p},\,v^f\in [H^1(\O_p)]^2,\label{aprox_qh1}\\
&&\|\nabla\cdot(v^f- Q_h v^f)\|_{0,\O_p}\leq C h\|\nabla\cdot v^f\|_{1,\O_p},\,v^f \in H^1(\text{div},\O_p)\label{aprox_qh2}.
\end{eqnarray}
Note that since $\text{curl} \psi\in {\mathcal W}^h\, \forall \psi\in {\mathcal V}^h$, it follows from \eqref{def_ph} that
\begin{eqnarray}\label{orthog_1}
&&\left(P_h f - f, \text{curl} \psi \right) = 0, \quad \forall \psi \in {\mathcal V}^h.
\end{eqnarray}
Also note the orthogonality  property for functions on ${\mathcal NC}^h$:
\begin{eqnarray}\label{orth}
\< v^s_j -v^s_k, 1\>_{\G_{jk}} = 0 \mbox{ for  
all interior  interfaces } \G_{jk} ,
\quad v^s \in \NC^h.
\end{eqnarray}


Set 
\begin{eqnarray}\label{bilinear3}
&&{\mathcal A}_h(u,v)=\sum_{\O_{j}\subset \O_p}\left[\sum_{l,m}\left(\tau_{lm}(u),
\e_{lm}(v^{(s)})\right)_{\O_j} 
- \left(p_f(u),\nabla\cdot v^f)\right)_{\O_j}\right]\\
&&\qquad\qquad =\sum_{\O_{j}\subset \O_p} \left({\bf M}_r ~ \tilde \epsilon(u),\tilde \epsilon(v)\right)_{\O_j}  + i \left({\bf M}_i ~ \tilde \epsilon(u),\tilde \epsilon(v)\right)_{\O_j}\nonumber
\end{eqnarray}
and
\begin{eqnarray}
&&\Theta_h\left( (E^s,H^s_y,u^s,u^f), (\psi,\varphi,v^s,v^f)\right) = (\sigma E^s,\psi) -(H^s_y, \text{curl} \psi) \label{def_Thetah}\\
&& \qquad + \left\langle a(1-i) E^s\cdot \chi, \psi\cdot\chi\right\rangle
 + \left( L(\o) \eta (\kappa(\o))^{-1}\left[ i \o u^f - L(\o)  E^s\right], \psi\right)_{\O_p} \nonumber\\
&&\qquad  +(\text{curl} E^s, \varphi) + (i \o \mu H^s_y, \varphi)
- \o^2 \left(\rho_b  u^s, v^s\right)_{\O_p} 
- \o^2 \left(\rho_f  u^f, v^s\right)_{\O_p}- \o^2 \left(\rho_f  u^s, v^f\right)_{\O_p}\nonumber\\
&&\qquad  + \left(  \eta (\kappa(\o))^{-1} \left[i \o u^f -  L(\o)  E^s\right], v^f\right)_{\O_p}
+  {\mathcal A}_h(u,v) + i \o \left\langle {\mathcal D} S_{\G}(u), S_{\G}(v)\right\rangle_{\G_p}.\nonumber
\end{eqnarray}



Let 
\[
{\mathcal Y}^h = {\mathcal V}^h \times {\mathcal W}^h \times 
{\mathcal NC}^h \times {\mathcal M}^h.
\]   
Then the {\it global} Galerkin procedure is defined as follows: find $\left(E^{s,h}, H^{s,h}_y, u^{s,h}, u^{f,h}\right)\in {\mathcal Y}^h$ such that 
\begin{eqnarray}    
&&\Theta_h\left( (E^{s,h},H^{s,h}_y,u^{s,h},u^{f,h}), (\psi,\varphi,v^s,v^f)\right) = -\left(\left[\sigma^s - L^2(\o) \eta (\kappa(\o))^{-1}\right]  E^p, \psi\right) \label{eq_101}\\
&&\qquad + (F^{(s)}, v^s)_{\O_p} + \left(\left[F^{(f)} + L(\o)\eta (\kappa(\o))^{-1}  E^p\right], v^f\right)_{\O_p},
\quad  (\psi, \varphi, v^s, v^f)\in {\mathcal Y}^h. \nonumber
\end{eqnarray}
To show uniqueness for the solution of \eqref{eq_101}, let 
\begin{eqnarray}\label{def_gh}
&&g^h =   \eta (\kappa(\o))^{-1} \left[i \o u^{f,h} - L(\o)  E^{s,h}\right],\quad (x,z)\in \O_p, 
\end{eqnarray}
set $F^{(s)}=F^{(f)}= E^p = 0$  in \eqref{eq_101}  
 and  use the argument given to show uniqueness for the continuous problem  \eqref{eq_100} to obtain the equation
\begin{eqnarray}
&(\sigma E^{s,h}, E^{s,h}) - (i \o \mu H^{s,h}_y, H^{s,h}_y) 
 + \left\langle a(1+i) E^{s,h}\cdot \chi, E^{s,h}\cdot\chi\right\rangle\label{uniq_h.1}
% \\
%&&\qquad 
+  \left( L(\o) g^h, E^s\right)_{\O_p} = 0.
\end{eqnarray}
and the inequality
\begin{eqnarray}
&&C_3 \|E^{s,h}\|^2_{0,\O_a} + C_4 \left(\|E^{s,h}\|_{0,\O_p}^2 + \|g^h\|^2_{0, \O_p}\right)
 +   \left\langle a E^{s,h}\cdot \chi, E^{s,h}\cdot\chi\right\rangle \label{uniq_h.2}\\
&&+  \sum_{\O_{j}\subset \O_p} \o \left({\bf M}_i ~ \tilde \epsilon(u^h),\tilde \epsilon(u^h)\right)_{\O_j} +  \o^2 \left\langle {\mathcal D} S_{\G}(u), S_{\G}(u)\right\rangle_{\G_p}\le 0. \nonumber
\end{eqnarray}
Since each term in  the left-hand side of \eqref{uniq_h.2} is nonnegative, it follows that 
\begin{eqnarray}
&&E^{s,h} = 0, \quad \text{in}\quad L^2(\O),\label{uniq_Eh}\\
&&E^{s,h}\cdot\chi = 0, \quad \text{in}\quad L^2(\p\O),\label{uniq_Eh_tau}\\
&&g^h =0,  \quad \text{in}\quad L^2(\O_p),\label{uniq_gh}\\ 
&&\nabla\cdot u^{f,h}=0,  \quad \text{in}\quad L^2(\O_p),\label{uniq_div_ufh}\\
&&\varepsilon_{11}(u^{s,h}) = \varepsilon_{12}(u^{s,h}) 
=\varepsilon_{22}(u^{s,h}) =0, 
\quad \text{in}\quad L^2(\O_p),\label{uniq_strain_h}\\
&&u^{s,h}=0,  \quad \text{in}\quad L^2(\G_p),\label{uniq_ush_bry}\\
&&u^{f,h}\cdot\nu=0, \quad \text{in}\quad L^2(\G_p).\label{uniq_ufh_nu}
\end{eqnarray}
Next, using \eqref{uniq_Eh} and \eqref{uniq_gh} in \eqref{def_gh} 
we get
\begin{equation}\label{uniq_ufh}
u^{f,h} = 0, \quad \text{in}\quad  L^2(\O_p).
\end{equation}
Also, using \eqref{uniq_Eh} in \eqref{uniq_h.1} we get
\begin{equation}\label{uniq_Hh}
H^{s,h}_y=0, \quad \text{in} \quad L^2(\O).
\end{equation}
Next, as in \cite{santos_sinum_07},   using  \eqref{uniq_strain_h} and  \eqref{uniq_ush_bry} let us take 
a corner element $\O_j$ of $\O_p$  with two faces contained in $\G_p$.
Without loss of generality, after a transformation  we can assume that
$\O_j = (-1,1)^2$  with the faces $\G^{R}=\{(x,z)\in \G_p: x=1\}$ 
and $\G^{T}=\{(x,z)\in \G_p: z=1\}$ contained in $\G_p$. 
Set 
\begin{subeqnarray*}\label{uniqh3}
&&u_1^{s,h} = a_1 + b_1 x  + c_1 z + d_1 (\widetilde \alpha(x)-\widetilde \alpha(z)),\\ 
&&u_2^{f,h} = a_2 + b_2  x + c_2 z + d_2 (\widetilde \alpha(x)-\widetilde \alpha(z)).\nonumber
\end{subeqnarray*}
Thanks to \eqref{uniq_ush_bry}  we have that  $u^{s,h}$ must vanish 
on $\G^R \cup \G^T$ and   
in particular at the mid points of $\G^R$ and  $\G^T$  so  that    
\begin{subeqnarray}\label{uniqh4}
&&u^{s,h}_1(1,0) = a_1 + b_1 -\dfrac23 d_1 = 0, 
\quad u^{s,h}_1(0,1)= a_1 + c_1 +\dfrac23 d_1 = 0,\\
&&u^{s,h}_2(1,0) = a_2 + b_2 -\dfrac23 d_2 = 0,\quad 
u^{s,h}_2(0,1) = a_2 + c_2 +\dfrac23 d_2 = 0.
\end{subeqnarray} 
Next, thanks to \eqref{uniq_strain_h}  we must have 
\begin{subeqnarray}\label{uniqh5}
\e_{11}(u^{s,h}) &=& b_1 + 2 d_1 \left(x - \dfrac{10}{3} x^3\right)=0,\\ 
\e_{22}(u^{s,h}) &=& c_2 - 2 d_2 \left(z - \dfrac{10}{3}
z^3\right)=0,\\ 
\e_{12}(u^{s,h}) &=& \dfrac{1}{2}\left[c_1 + b_2  
- 2 d_1 \left(x - \dfrac{10}{3} x^3\right)  
+ 2 d_2 \left(z - \dfrac{10}{3} z^3\right)\right]=0.
\end{subeqnarray} 
From (\ref{uniqh4}) and (\ref{uniqh5}) it follows that 
$
u_1^{s,h}|_{\O_j} = u_2^{s,h}|_{\O_j} = 0.
$
Next, take an element $\O_k$ adjacent to $\O_j$ with a common 
face $\G_{jk}$ and a face $\G_k$ contained in $\G_p$.  Then  
$u_1^{s,h}$ and $u_2^{s,h}$ vanish at the mid points of $\G_k$ 
and $\G_{jk}$ and 
$\e_{11}(u^{s,h}), \e_{22}(u^{s,h})$ and $\e_{22}(u^{s,h})$ 
vanish identically
on $\O_j$, so that repeating the above argument we get  
\begin{eqnarray}\label{uniqh6}
u_1^{s,h}|_{\O_j} = u_2^{s,h}|_{\O_j} = 0. 
\end{eqnarray} 
Repeating the argument, one can show that  
(\ref{uniqh6})  holds for all elements with a face contained in $\G_p$. 
Next stripping out such boundary elements, 
take a boundary element with two faces common with the corner of the  stripped out domain
and repeat the argument to show the validity of (\ref{uniqh6}) for those elements. Then 
continue the process until the domain is exhausted. 
Thus uniqueness holds for the solution of \eqref{eq_101}. Existence follows from finite dimensionality. 
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\section{A priori error estimates for the global PSVTM-mode} 
In this section we derive the following {\it a priori}  error estimate for our {\it global} 
finite element procedure \eqref{eq_101}.   
\begin{theorem}\label{globalerror}
Let 
$(E^s, H^s_y,u^s, u^f)\in {\mathcal Y}$ and $(E^{s,h},H^{s,h}_y,u^{s,h},u^{f,h})\in {\mathcal Y}^h$ be the solutions of \eqref{eq_100} and \eqref{eq_101}, respectively. Assume that 
 $E^s \in [H^1(\O)]^2$,   $ \text{curl} ~E^s, ~ \H^s_y ~ \in H^1(\O)$   $u^s\in [H^2(\O_p)]^2$, 
$u^f \in H^1(\text{div},\O_p)$. Also assume that the matrix ${\bf M}_i$ is positive definite, that the coefficient $a$ in the boundary condition \eqref{modf.5}  is 
strictly positive and piecewise constant. Also assume the validity of 
the relations \eqref{eq51}.   
Then  the following {\it a priori}  error estimate holds: 
for $\o>0$ and $h>0$ sufficiently small,
\begin{eqnarray*}
&& \|E^s -E^{s,h}\|_0 + \|\text{curl}(E^s -E^{s,h})\|_0 +  \|H^s_y -H_y^{s,h}\|_0 
+ \|u^s -u^{s,h}\|_{1,h,\O_p} + \|u^f -u^{f,h}\|_{0,\O_p} \\
&& +\|\div (u^f-u^{f,h})\|_{0,\O_p} + \|(E^s-E^{s,h})\cdot\chi\|_{0,\G} + \|u^s -u^{s,h}\|_{0,\G_p} +  
\|(u^f-u^{f,h})\cdot \nu \|_{0,\G_p}\\
&&\qquad\qquad\le C(\o) \left[ h\left(\|E^s \|_1 +  \|\text{curl} ~E^s \|_1 + \|H_y^s \|_1\right)\right.\\
&& \qquad\qquad  + \left. h^{1/2}\left(  \|E^s \|_1 + \|u^s \|_{2,\O_p} + \|u^f \|_{1,\O_p} +   
 \|\nabla\cdot u^f \|_{1,\O_p}   \right)\right]. 
\nonumber
\end{eqnarray*}
\end{theorem}
\begin{proof}
Set  
\begin{eqnarray*}\label{error1}
%{\bf Z_h}(E^s, H^s_y, u^s, u^f) = \left (\Pi_h E^s, P_h H^s_y, {\mathcal  R}_h u^s, Q_h u^f\right),\\ 
& \delta = \left(E^s -E^{s,h}, H^s_y - H^{s,h}_y, u^s - u^{s,h}, 
u^f - u^{f,h}\right)\equiv\left(\delta^E,\delta^H_y, \delta^s, \delta^f \right),
\\
&\gamma = \left (\Pi_h E^s - E^{s,h}, P_h H^s_y -H^{s,h}_y , {\mathcal  R}_h u^s - u^{s,h}, Q_h u^f - u^{f,h}\right) \equiv\left(\gamma^E, \gamma^H_y, \gamma^s, \gamma^f \right) . 
\end{eqnarray*}
First, use integration by parts  to see that, for $\psi\in H(\text{curl},\O)$,
$\varphi\in L^2(\O)$ and  $v=(v^s,v^f)\in \left[L^2(\O_p)\right]^4$ such that 
$v^s|_{\O_j}\in [H^1(\O_j)]^2$ and  $ v^f|_{\O_j}\in H(\text{div}, \O_j)$,  
\begin{eqnarray}
&&\Theta_h\left( (E^s, H^s_y, u^s, u^f), (\psi,\varphi,v^s,v^f)\right) = 
(\sigma E^s,\psi) -(H^s_y, \text{curl} \psi) \label{eq_118}\\
&& \qquad + \left\langle a(1-i) E^s\cdot \chi, \psi\cdot\chi\right\rangle
 + \left( L(\o) \eta (\kappa(\o))^{-1} \left[i \o u^f - L(\o)\eta   E^s\right], \psi\right)_{\O_p}\nonumber\\
&&\qquad  +(\text{curl} E^s, \varphi) + (i \o \mu H^s_y, \varphi)
- \o^2 \left(\rho_b u^s, v^s\right)_{\O_p} 
- \o^2 \left(\rho_f u^f, v^s\right)_{\O_p} - \o^2 \left(\rho_f u^s, v^f\right)_{\O_p}\nonumber\\
&&\qquad + \left(\eta (\kappa(\o))^{-1} \left[i \o u^f - L(\o)  E^s\right] ,v^f\right)_{\O_p}
+  {\mathcal A}_h(u,v) + i \o \left\langle {\mathcal D} S_{\G}(u), S_{\G}(v)\right\rangle_{\G_p}\nonumber\\
&&\hskip5cm  \nonumber\\
&&\qquad =  (F^{(s)}, v^s)_{\O_p} 
+ \left(\left[ F^{(f)} + \eta (\kappa(\o))^{-1} L(\o) E^p\right], v^f \right)_{\O_p} \nonumber\\
&&\qquad + \sum_{\O_{j}\subset \O_p}\left[\la\tau(u)\nu, v^s\ra_{\p\O_j\setminus \G_p} -
 \la p_f(u), v^f\cdot\nu\ra_{\p\O_j\setminus\G_p}\right].\nonumber
\end{eqnarray}
Now subtract \eqref{eq_118} from \eqref{eq_101} to 
see that, for $\left(\psi,\varphi,v^s,v^f\right)\in {\mathcal Y}^h$  
\begin{eqnarray}
&&\Theta_h\left( (\delta^E,\delta^H_y, \delta^s, \delta^f), (\psi,\varphi,v^s,v^f)\right) =  
\sum_{\O_{j}\subset \O_p}\left[\la\tau(u)\nu, 
 v^s \ra_{\p\O_j \setminus \G_p}\label{eq_92} \right.\\
&&\left. \hskip7cm - 
 \la p_f(u), v^f\cdot\nu\ra_{\p\O_j\setminus\G_p}
\right].\nonumber
\end{eqnarray}
Next, by  hypothesis $u^s\in [H^2(\O_p)]^2, ~ u^f\in H^1(\text{div},\O_p)$ and 
consequently  $p_f(u)\in H^{1/2}(\p\O_j)$. Also, $v^f\in {\mathcal M}^h\subset
H(\text{div},\O)$ and consequently  $v^f\cdot\nu$ 
is continuous across  the interior interfaces of the elements $\O_j$. Thus, 
\begin{eqnarray}\label{eq_93}
\sum_{\O_{j}\subset \O_p} \la p_f(u), v^f\cdot \nu \ra_{\p\O_j \sm\G_p} = 0.
\end{eqnarray}
Also  use that $v^s\in \NC^h$ is  constant on $\G_{jk}$ and that  $S_h(u)$ changes sign on each side of $\G_{jk}$ (c.f. \eqref{def_sh}) to see that 
\begin{eqnarray}\label{eq_95}
\sum_{\O_{j}\subset \O_p} \la S_h(u), v^s \ra_{\p\O_j\sm\G_p} = 0.
\end{eqnarray}
Thus, \eqref{eq_92} becomes
\begin{eqnarray}
&&\Theta_h\left((\delta^E,\delta^H_y, \delta^s, \delta^f) , (\psi,\varphi,v^s,v^f)\right) =\sum_{\O_{j}\subset \O_p} 
\la\tau(u)\nu - S_h(u), v^s\ra_{\p\O_j \setminus \G_p},\label{eq_126} \\
&&\hskip7cm \left(\psi,\varphi,v^s,v^f\right)\in {\mathcal Y}^h.\nonumber 
\end{eqnarray}
Consequently from \eqref{eq_92} we get 
\begin{eqnarray}
&&\Theta_h\left((\gamma^E,\gamma^H_y, \gamma^s, \gamma^f) , (\psi,\varphi,v^s,v^f)\right)\nonumber \\ 
&&\qquad = \Theta_h\left(\left(\Pi_h E^s - E^s, P_h H^s_y - H^s_y, R_h u^s - u^s, Q_h u^f - u^f\right) , \left(\psi,\varphi,v^s,v^f\right)\right) \nonumber\\ 
&&\qquad + \sum_{\O_{j}\subset \O_p} 
\la\tau(u)\nu - S_h(u), v^s\ra_{\p\O_j \setminus \G_p}\nonumber \\
&& \qquad= ([\sigma - \chi_{\O_p} L^2(\o) \eta (\kappa(\o))^{-1}](\Pi_h E^s - E^s),\psi)
 -(P_h H^s_y - H^s_y, \text{curl} \psi)\label{eq_97}\\
&& \qquad + \left\langle a(1-i) (\Pi_h E^s - E^s)\cdot \chi, \psi\cdot\chi\right\rangle
 + \left( L(\o) \eta (\kappa(\o))^{-1} 
 i \o (Q_h u^f -u^f),\psi\right)_{\O_p}\nonumber\\
&&\qquad  +(\text{curl}(\Pi_h E^s - E^s), \varphi) 
+ (i \o \mu (P_h H^s_y - H^s_y), \varphi)
- \o^2 \left(\rho_b (R_h u^s - u^s), v^s\right)_{\O_p}\nonumber\\ 
&&\qquad - \o^2 \left(\rho_f (Q_h u^f - u^f), v^s\right)_{\O_p} 
- \o^2 \left(\rho_f (R_h u^s - u^s), v^f\right)_{\O_p}\nonumber\\
&&\qquad + \left(\eta (\kappa(\o))^{-1} i \o (Q_h u^f - u^f), v^f\right)_{\O_p} 
 - \left( L(\o)\eta (\kappa(\o))^{-1} (\Pi_h E^s - E^s) ,v^f\right)_{\O_p}\nonumber\\
&&\qquad + \sum_{\O_{j}\subset \O_p} \left[\left({\bf M}_r ~ \tilde \epsilon(R_h u^s - u^s, Q_h u^f - u^f),\tilde \epsilon(v)\right)_{\O_j}  
+ i \left({\bf M}_i ~ \tilde \epsilon(R_h u^s - u^s, Q_h u^f - u^f),\tilde \epsilon(v)\right)_{\O_j}\right] \nonumber\\
&&\qquad + i \o \left\langle {\mathcal D} S_{\G}(R_h u^s - u^s, Q_h u^f - u^f ), S_{\G}(v)\right\rangle_{\G_p}  + \sum_{\O_{j}\subset \O_p} 
\la\tau(u)\nu - S_h(u), v^s\ra_{\p\O_j \setminus \G_p}.\nonumber
 \end{eqnarray}


First we will derive an estimate for $\|\gamma^s\|_{0,\O_p}$. 
 Choose $\psi=0, \varphi=0,v^f=0$, 
$v^s=\gamma^s$ in \eqref{eq_97} to get 

\begin{eqnarray}
&&- \o^2 \left(\rho_b \gamma^s, \gamma^s\right)_{\O_p} 
- \o^2 \left(\rho_f \gamma^f, \gamma^s\right)_{\O_p}
 + \sum_{\O_{j}\subset \O_p} \left[\left({\bf M}_r ~ 
\tilde \epsilon(\gamma^s, \gamma^f),\tilde \epsilon(\gamma^s,0)\right)_{\O_j}\right.\label{eq_97_1}\\  
&&\qquad \left.+ i \left({\bf M}_i ~ \tilde \epsilon(\gamma^s, \gamma^f),\tilde \epsilon(\gamma^s,0)\right)_{\O_j}\right]
+ i \o \left\langle {\mathcal D} S_{\G}(\gamma^s, \gamma^f), S_{\G}(\gamma^s,0)\right\rangle_{\G_p}\nonumber\\ 
&&\qquad=- \o^2 \left(\rho_b (R_h u^s - u^s), \gamma^s\right)_{\O_p} 
 - \o^2 \left(\rho_f (Q_h u^f - u^f), \gamma^s\right)_{\O_p}\nonumber\\ 
&&\qquad + \sum_{\O_{j}\subset \O_p} \left({\bf M}_r ~ \tilde \epsilon(R_h u^s - u^s, Q_h u^f - u^f),\tilde \epsilon(\gamma^s,0)\right)_{\O_j}\nonumber\\
  &&\qquad + i \sum_{\O_{j}\subset \O_p}  \left({\bf M}_i ~ \tilde \epsilon(R_h u^s - u^s, Q_h u^f - u^f),\tilde \epsilon(\gamma^s,0)\right)_{\O_j}\nonumber\\
&&\qquad + i \o \left\langle {\mathcal D} S_{\G}(R_h u^s - u^s, Q_h u^f - u^f ), S_{\G}(\gamma^s,0)\right\rangle_{\G_p}  + \sum_{\O_{j}\subset \O_p} 
\la\tau(u)\nu - S_h(u), \gamma^s\ra_{\p\O_j \setminus \G_p}\nonumber\\ 
&&\qquad \equiv T_1 + T_2 + T_3 + T_4 + T_5 + T_6.\nonumber
\end{eqnarray}
Taking real part in \eqref{eq_97_1} we get the inequality 
\begin{eqnarray}
&&\|\gamma^s\|^2_{0,\O_p} \le C(\o) \left[ \|\gamma^f\|^2_{0,\O_p}  
+ \|\nabla \cdot \gamma^f\|^2_{0,\O_p} 
+ \sum_{\O_{j}\subset \O_p} \|\e_{lm}(\gamma^s)\|^2_{0,\O_j}\right.\label{eq_97_4}\\
&&\qquad\qquad \quad \left. \left|i \o \left\langle {\mathcal D} S_{\G}(\gamma^s, \gamma^f), S_{\G}(\gamma^s,0)\right\rangle_{\G_p}\right| + |T_1| + |T_2| + |T_3| + |T_4| + |T_5| + |T_6|\right]\nonumber.
\end{eqnarray}
Let us bound the boundary term and the terms $T_j, j=1,\cdots, 6$  in the right-hand side of \eqref{eq_97_4}. First,
\begin{eqnarray}\label{eq_97_5}
&&\left| i \o \left\langle {\mathcal D} S_{\G}(\gamma^s, \gamma^f), S_{\G}(\gamma^s,0)\right\rangle_{\G_p}\right| \le 
C(\o) \left[ \|\gamma^s\|^2_{0,\G_p} +\|\gamma^f\cdot \nu\|^2_{0,\G_p}\right]. 
\end{eqnarray}
Next, using the approximating properties  \eqref{approx_rh}, \eqref{aprox_qh1} and \eqref{aprox_qh2} we get 
\begin{eqnarray}\label{eq_97_6}
|T_1| + |T_2| \le  \epsilon \|\gamma^s\|^2_{0,\O_p} 
+  C(\o) h^2 \left[ h^2 \|u^s\|^2_{2,\O_p} + \|u^f\|^2_{1,\O_p}\right], 
\end{eqnarray}
and 
\begin{eqnarray}\label{eq_98}
|T_3| + |T_4| \le  
+  C(\o) \left[ h^2\left(\|u^s\|^2_{2,\O_p} + \|\nabla\cdot u^f\|^2_{1,\O_p}\right)
+\sum_{\O_{j}\subset \O_p} \|\e_{lm}(\gamma^s)\|^2_{0,\O_j} \right], 
\end{eqnarray}
Also, 
\begin{eqnarray}\label{eq_99}
|T_5|  \le  C(\o) \sum_j\left[ \|R_h u^s - u^s\|_{0,\G_j} + 
\|(Q_h u^f - u^f)\cdot\nu\|_{0,\G_j}\right] \|\gamma^s\|_{0,\G_j}\\
\le C(\o) h^{1/2}\left(\|u^s\|_{1/2,\G_j} 
+ \| u^f\cdot\nu\|_{1/2,\G_j}\right)\|\gamma^s\|_{0,\G_j}\nonumber\\
\le  C(\o) \left[ h\left( \|u^s\|^2_{1,\O_p} + \|u^f\|^2_{1,\O_p}\right) 
+ \|\gamma^s\|^2_{0,\G_p}\right].\nonumber 
\end{eqnarray}
Next, since $\tau(u)\nu - S_h(u)$ has average value zero on $\p\O_j \setminus \G_p$, if  $q^s$ is the average value of $\gamma^s$ on $\O_j$, using the trace inequality and Korn's  second inequality, \cite{duvaut-lions,nitsche81},
\begin{eqnarray}
&& \left|\la\tau(u)\nu - S_h(u), \gamma^s- q^s\ra_{\p\O_j \setminus \G_p}\right|\nonumber\\
&&\qquad \le  \|\tau(u)\nu - S_h(u)\|_{0,\p\O_j} \|\gamma^s- q^s\|_{0,\p\O_j}\label{eq_101_a}\\
&&\qquad \le C h^{1/2} \left(\|u^s\|_{2,\O_j}  + \|\nabla\cdot u^f\|_{1,\O_j} \right)
\|\gamma^s- q^s\|^{1/2}_{0,\O_j} \|\gamma^s- q^s\|^{1/2}_{1,\O_j}\nonumber\\
&&\qquad \le  C h  \left(\|u^s\|_{2,\O_j}  + \|\nabla\cdot u^f\|_{1,\O_j} \right) \|\gamma^s\|_{1,\O_j}\nonumber\\
&&\qquad \le \epsilon \|\gamma^s\|^2_{0,\O_j} + C\left[ h^2\left( \|u^s\|^2_{2,\O_j} +  \|\nabla\cdot u^f\|^2_{1,\O_j}\right) 
+ \sum_{l,m} \|\e_{lm}(\gamma^s)\|^2_{0,\O_j} 
\right].\nonumber
\end{eqnarray}
Adding over $j$ the estimate \eqref{eq_101_a} we get
\begin{eqnarray}\label{eq_102}
&&|T_6| \le \epsilon \|\gamma^s\|^2_{0,\O_p} + C\left[ h^2\left( \|u^s\|^2_{2,\O_p} +  \|\nabla\cdot u^f\|^2_{1,\O_p}\right) 
+ \sum_{\O_{j}\subset \O_p}\sum_{l,m} \|\e_{lm}(\gamma^s)\|^2_{0,\O_j} 
\right].
\end{eqnarray}
Using  the estimates \eqref{eq_97_5}, \eqref{eq_97_6}, \eqref{eq_98}, \eqref{eq_99} and \eqref{eq_102} in \eqref{eq_97_4} and choosing  $\epsilon$  sufficiently small in  \eqref{eq_102} we finally get the estimate 
\begin{eqnarray}
&&\|\gamma^s\|^2_{0,\O_p} \le  C(\o) \left[ h\left( \|u^s\|^2_{2,\O_p} + 
 \|u^f\|^2_{1,\O_p} 
+  \|\nabla\cdot u^f\|^2_{1,\O_p}\right)\right. \label{eq_104}\\ 
&&\qquad \left. + \|\gamma^f\|^2_{0,\O_p} + 
\|\nabla\cdot \gamma^f\|^2_{0,\O_p} + \sum_{\O_{j}\subset \O_p} \sum_{l,m}\|\e_{lm}(\gamma^s)\|^2_{0,\O_j} + \|\gamma^s\|^2_{0,\G_p} + \|\gamma^f\cdot \nu\|^2_{0,\G_p}\right]\nonumber.
\end{eqnarray}
Next take  $\psi = v^s = v^f = 0$, $\varphi = \gamma^H_y$ in \eqref{eq_97} and conjugate the resulting equation to get the equation
\begin{eqnarray}\label{eq_109_3}
\left(\gamma^H_y, \text{curl} \gamma^E\right) = 
\left(i \o \mu \gamma^H_y,  \gamma^H_y\right) + 
\left(\gamma^H_y, \text{curl} \left[\Pi_h E^s - E^s \right ]\right) 
+ \left(\gamma^H_y, i \o \mu \left[P_h H^s_y - H^s_y \right ]\right)  
\end{eqnarray}
Next choose  $\psi = \gamma^E$,  $v^s= v^f = 0$, $\varphi = 0$ in \eqref{eq_97} and  use \eqref{eq_109_3},   the orthogonality property \eqref{orthog_1} 
and  the fact that the coefficient $a$ in the boundary condition is piecewise constant  to see that 
\begin{eqnarray}\nonumber
\label{eq_112_114}
(P_h H^s_y - H^s_y, \text{curl} \gamma^E) = 0, \quad 
\left\langle a(1-i) (\Pi_h E^s - E^s)\cdot \chi, \gamma^E\cdot\chi\right\rangle=0.
\end{eqnarray}
Thus, setting   
\begin{eqnarray}\label{eq_109_4_1}
g^{\gamma, h} = \eta (\kappa(\o))^{-1} \left[ i \o \gamma^f - L(\o) \gamma^E\right]  
\end{eqnarray}
this choice of test functions in \eqref{eq_97}  yields 
\begin{eqnarray}
&&\left(\sigma \gamma^E, \gamma^E\right) - \left(i \o \mu \gamma^H_y,  \gamma^H_y\right) + 
 \left\langle a(1-i) \gamma^E\cdot \chi, \gamma^E\cdot\chi\right\rangle
+ \left( L g^{\gamma, h}, \gamma^E\right)_{\O_p}\label{eq_109_4}\\
&& \qquad =\left(\gamma^H_y, \text{curl} \left[\Pi_h E^s - E^s \right ]\right) 
+ \left(\gamma^H_y, i \o \mu \left[P_h H^s_y - H^s_y \right ]\right)\nonumber\\  
&&\qquad\quad   + ([\sigma - \chi_{\O_p} (L(\o))^2 \eta (\kappa(\o))^{-1}](\Pi_h E^s - E^s),\gamma^E)\nonumber\\ 
&&\qquad\quad   + \left( L(\o) \eta (\kappa(\o))^{-1} 
 i \o (Q_h u^f -u^f),\gamma^E\right)_{\O_p}\nonumber
\end{eqnarray}
Taking the imaginary part in \eqref{eq_109_4} yields the inequality
\begin{eqnarray}
&&\left(\o \mu \gamma^H_y,  \gamma^H_y\right) + 
 \left\langle a  \gamma^E\cdot \chi, \gamma^E\cdot\chi\right\rangle
\label{eq_109_5}\\
&&\qquad \le \epsilon_1 \|\gamma^H_y\|^2_0 + \epsilon_2 \|\gamma^E\|^2_0 
+ L(\o)\left( \|g^{\gamma, h}\|^2_{0,\O_p} + 
\|\gamma^E\|^2_{0,\O_p}\right)\nonumber\\
&&\qquad + C h^2 \left( \|E^s\|^2_1 + \|\text{curl} E^s\|^2_1 
+ \|H^s_y\|^2_1 + \|u^f\|^2_{1,\O_p}\right). \nonumber
\end{eqnarray} 
Thus, choosing  $\epsilon_1$ small in \eqref{eq_109_5} we obtain  
\begin{eqnarray}
&&\|\gamma^H_y\|^2_0 +  \|\gamma^E\cdot \chi\|^2_{0,\G}
\le  \epsilon_2 \|\gamma^E\|^2_0 
+ L(\o)\left( \|g^{\gamma, h}\|^2_{0,\O_p} + 
\|\gamma^E\|^2_{0,\O_p}\right)\label{eq_109_5_a}\\
&&\qquad + C h^2 \left( \|E^s\|^2_1 + \|\text{curl} E^s\|^2_1 
+ \|H^s_y\|^2_1 + \|u^f\|^2_{1,\O_p}\right).\nonumber
\end{eqnarray} 
Next, choose $\psi=0$, $\varphi=0$, $v^s = i \o \gamma^s, v^f = i \o \gamma^f$ 
in \eqref{eq_97} and add the resulting equation to \eqref{eq_109_4} to obtain
\begin{eqnarray}
&&\left(\sigma \gamma^E, \gamma^E\right) - \left(i \o \mu \gamma^H_y,  \gamma^H_y\right) + 
 \left\langle a(1-i) \gamma^E\cdot \chi, \gamma^E\cdot\chi\right\rangle
+ \left( L g^{\gamma, h}, \gamma^E\right)_{\O_p}\label{eq_110}\\
&&\qquad - i \o \left[ \o^2 \left( \rho_b \gamma^s, \gamma^s\right)_{\O_p} - 
2 \o^2 \text{Re} \left[\left( \rho_f \gamma^f, \gamma^s\right)_{\O_p}\right]\right] 
+ \left( g^{\gamma,h}, i \o \gamma^f\right)_{\O_p}\nonumber\\
&&\qquad - i \o \sum_{\O_{j}\subset \O_p} 
\left({\bf M}_r ~ \tilde \epsilon(\gamma^s, \gamma^f),\tilde \epsilon(\gamma^s, \gamma^f )\right)_{\O_j}  
+  \o  \sum_{\O_{j}\subset \O_p}\left({\bf M}_i ~ \tilde \epsilon(\gamma^s, \gamma^f ),\tilde \epsilon(\gamma^s, \gamma^f)\right)_{\O_j}\nonumber\\
&&\qquad + \o^2 \left\langle {\mathcal D} S_{\G}(\gamma^s, \gamma^f), S_{\G}(\gamma^s, \gamma^f)\right\rangle_{\G_p}\nonumber\\
&&\qquad = \left(\gamma^H_y, \text{curl} \left[\Pi_h E^s - E^s \right ]\right) 
+ \left(\gamma^H_y, i \o \mu \left[P_h H^s_y - H^s_y \right ]\right)\nonumber\\  
&&\qquad  + \left([\sigma - \chi_{\O_p} L^2(\o) \eta (\kappa(\o))^{-1}](\Pi_h E^s - E^s),\gamma^E\right) 
 + \left( L(\o) \eta (\kappa(\o))^{-1} 
 i \o (Q_h u^f -u^f),\gamma^E\right)_{\O_p}\nonumber\\
&&\qquad  -\o^2 \left(\rho_b (R_h u^s - u^s), i \o \gamma^s\right)_{\O_p} 
 - \o^2 \left(\rho_f (Q_h u^f - u^f), i \o \gamma^s\right)_{\O_p} 
- \o^2 \left(\rho_f (R_h u^s - u^s), i \o \gamma^f\right)_{\O_p}\nonumber\\
&&\qquad + \left(\eta (\kappa(\o))^{-1} i \o (Q_h u^f - u^f), i \o \gamma^f\right)_{\O_p} 
 - \left( L(\o)\eta (\kappa(\o))^{-1} (\Pi_h E^s - E^s) ,i \o \gamma^f\right)_{\O_p}\nonumber\\
&&\qquad + \sum_{\O_{j}\subset \O_p} \left[\left({\bf M}_r ~ \tilde \epsilon(R_h u^s - u^s, Q_h u^f - u^f),\tilde \epsilon(i \o \gamma^s, i \o \gamma^f)\right)_{\O_j}\right.\nonumber\\  
&&\qquad \left. + i \left({\bf M}_i ~ \tilde \epsilon(R_h u^s - u^s, Q_h u^f - u^f),\tilde \epsilon(i \o \gamma^s, i \o \gamma^f)\right)_{\O_j}\right] \nonumber\\
&&\qquad + i \o \left\langle {\mathcal D} S_{\G}(R_h u^s - u^s, Q_h u^f - u^f ), S_{\G}(i \o \gamma^s, i \o \gamma^f)\right\rangle_{\G_p}  
+ \sum_{\O_{j}\subset \O_p} 
\la\tau(u)\nu - S_h(u), i \o \gamma^s\ra_{\p\O_j \setminus \G_p}\nonumber\\
&&\qquad \equiv \sum_{s=7}^{s=19} T_s.\nonumber 
\end{eqnarray}
We will take real part in \eqref{eq_110}, but first we will bound each term $T_s$ in the right-hand side of \eqref{eq_110}.
First,  
\begin{eqnarray}\label{eq_111_113_115_116}
&&|T_7| + |T_8| + |T_9| + |T_{10}| 
\le \epsilon \left(\|\gamma^E\|^2_0 + \|\gamma^H_y\|^2_0\right)\\
&&\hskip4cm  + C  h^2 \left(\|E^s\|^2_1 + \|\text{curl} E^s\|_1^2 
+ \|H^s_y\|^2_1 +
\|u^f\|^2_{1,\O_p}\right).\nonumber 
\end{eqnarray}
Next, using \eqref{eq_104} and \eqref{eq_109_4_1}
\begin{eqnarray}\label{eq_117}
&&|T_{11}| + |T_{12}| + |T_{13}|  
\le \epsilon \|i \o \gamma^f\|^2_{0,\O_p} + h \|\gamma^s\|^2_{0,\O_p}
 + C(\o)  \left(h^3 \|u^s\|^2_{2,\O_p} + h \|u^f\|^2_{1,\O_p}\right)\\
&& \le \epsilon \left(\| \gamma^E\|^2_{0,\O_p} + \| g^{\gamma,h}\|^2_{0,\O_p}\right) 
+ C(\o) h \left[ h \|u^s\|^2_{0,\O_p} + 
 \|u^f\|^2_{1,\O_p} + \|\nabla \cdot u^f\|^2_{1,\O_p}\right.\nonumber\\
&&\qquad \left. + \|\gamma^E\|^2_{0,\O_p} + \| g^{\gamma,h}\|^2_{0,\O_p} 
+ \|\nabla\cdot \gamma^f\|^2_{0,\O_p} + \sum_{\O_{j}\subset \O_p} \sum_{l,m} 
\|\epsilon_{lm}(\gamma^s)\|^2_{0,\O_j} + \|\gamma^s\|^2_{0,\G_p} 
+\|\gamma^f\cdot\nu\|^2_{0,\G_p}\right].\nonumber  
\end{eqnarray}
Next, using again \eqref{eq_109_4_1},
\begin{eqnarray}\label{eq_118_a}
&&|T_{14}| + |T_{15}|   
\le \epsilon \|i \o \gamma^f\|^2_{0,\O_p} 
 + C(\o) h^2 \left( \|E^s\|^2_1 + \|u^f\|^2_{1,\O_p} \right)\\
&&\qquad \le \epsilon \left(\| \gamma^E\|^2_{0,\O_p} + \| g^{\gamma,h}\|^2_{0,\O_p}\right) + C(\o) h^2 \left( \|E^s\|^2_1 + \|u^f\|^2_{1,\O_p} \right).\nonumber
\end{eqnarray}
Also, 
\begin{eqnarray}\label{eq_119}
&&|T_{16}| + |T_{17}|\le \epsilon \left[  \sum_{\O_{j}\subset \O_p} \sum_{l,m} 
\|\epsilon_{lm}(\gamma^s)\|^2_{0,\O_j} + \|\nabla\cdot \gamma^f\|^2_{0,\O_p}  \right] \\
&&\qquad\qquad \qquad     + C(\o) h^2 \left( \|u^s\|^2_{2,\O_p} 
+ \|\nabla\cdot u^f\|^2_{1,\O_p}\right)\nonumber.
\end{eqnarray}
On the other hand,  proceeding as in \eqref{eq_99},
\begin{eqnarray}\label{eq_120}
|T_{18}|  \le  C(\o) \sum_j\left[ \|R_h u^s - u^s\|_{0,\G_j} + 
\|(Q_h u^f - u^f)\cdot\nu\|_{0,\G_j}\right] \left[\|\gamma^s\|_{0,\G_j} + 
\|\gamma^f\cdot\nu\|_{0,\G_j} \right]\\
\le C(\o) h^{1/2}\left[\|u^s\|_{1/2,\G_j} 
+ \| u^f\cdot\nu\|_{1/2,\G_j}\right]\left[\|\gamma^s\|_{0,\G_j} + 
\|\gamma^f\cdot\nu\|_{0,\G_j}\right]
\nonumber\\
\le  C(\o) h\left( \|u^s\|^2_{1,\O_p} + \|u^f\|^2_{1,\O_p}\right) 
+ \epsilon\left(\|\gamma^s\|^2_{0,\G_p} + \|\gamma^f\cdot\nu\|^2_{0,\G_p} \right).\nonumber 
\end{eqnarray}
Next, repeating the argument used in  \eqref{eq_101_a} involving Korn's second inequality and using again the estimate for $\|\gamma^s\|^2_{0,\O_p}$ obtained in  \eqref{eq_104},  
\begin{eqnarray}\label{eq_120_1}
&&|T_{19}|\le C \sum_{\O_{j}\subset \O_p} h \left(\|u^s\|_{2,\O_j} + 
\|\nabla\cdot u^f\|_{1,\O_j} \right)\|\gamma^s\|_{1,\O_j}\\
&&\qquad \le C h \left ( \|u^s\|^2_{2,\O_p} + 
\|\nabla\cdot u^f\|^2_{1,\O_p} + \sum_{\O_{j}\subset \O_p} \|\gamma^s\|^2_{1,\O_j}\right)\nonumber\\
&&\qquad \le C h \left ( \|u^s\|^2_{2,\O_p} + 
\|\nabla\cdot u^f\|^2_{1,\O_p} + \|\gamma^s\|^2_{0,\O_p}+ \sum_{\O_{j}\subset \O_p} 
\|\epsilon_{lm}(\gamma^s)\|^2_{0,\O_j}\right)\nonumber\\
&&\qquad \le C(\o) h \left ( \|u^s\|^2_{2,\O_p} + \|u^f\|^2_{1,\O_p} + 
\|\nabla\cdot u^f\|^2_{1,\O_p} +\|\gamma^E\|^2_{0,\O_p} \right.\nonumber\\
&&\qquad \left. + \sum_{\O_{j}\subset \O_p} 
\|\epsilon_{lm}(\gamma^s)\|^2_{0,\O_j} + \|g^{\gamma,h}\|^2_{0,\O_p} + 
\|\nabla\cdot\gamma^f\|^2_{0,\O_p} + 
\|\gamma^s\|^2_{0,\G_p}+ \|\gamma^f\cdot\nu\|^2_{0,\G_p}\right).\nonumber
\end{eqnarray} 
 Set 
\begin{eqnarray}\label{def_fihp}
&&\Phi^h_p = (\sigma \gamma^E, \gamma^E)_{\O_p} 
+ ( L(\o) g^{\gamma,h}, \gamma^E)_{\O_p} + (g^{\gamma,h}, i \o \gamma^f)_{\O_p},\\
&&\Phi_a = (\sigma \gamma^E, \gamma^E)_{\O_a}, \qquad \Phi= \Phi_a + \Phi_p.\nonumber
\end{eqnarray}
Then, as in the analysis of the uniqueness of the continuous problem,  thanks to 
the validity of \eqref{eq51},  
\begin{eqnarray}\label{eq_121_1}
&&\text{Re}(\Phi_p^h) \ge C_4 \left[\|\gamma^E\|^2_{0,\O_p} + \|g^{\gamma,h}\|^2_{0,\O_p}\right].  
\end{eqnarray}
Thus taking real part in \eqref{eq_110} and using the estimates 
\eqref{eq_111_113_115_116}, \eqref{eq_117}, \eqref{eq_118_a}, \eqref{eq_119}
\eqref{eq_120} \eqref{eq_120_1}  and \eqref{eq_121_1} we obtain 
\begin{eqnarray}
&&C_3 \|\gamma^E\|^2_{0,\O_a} + C_4 \left[ \|\gamma^E\|^2_{0,\O_p} 
+ \| g^{\gamma,h}\|^2_{0,\O_p}\right] 
+ \la a \gamma^E\cdot\chi,\gamma^E\cdot\chi\ra \label{eq_122}\\
&&\qquad  +\o  \sum_{\O_{j}\subset \O_p}\left({\bf M}_i ~ \tilde \epsilon(\gamma^s, \gamma^f ),\tilde \epsilon(\gamma^s, \gamma^f)\right)_{\O_j}
+ \o^2 \left\langle {\mathcal D} S_{\G}(\gamma^s, \gamma^f), S_{\G}(\gamma^s, \gamma^f)\right\rangle_{\G_p}\nonumber\\
&&\qquad \le \epsilon \left[  \|\gamma^E\|^2_0 + \|\gamma_y^H\|^2_0 +  
\|g^{\gamma,h}\|^2_{0,\O_p}
+ \|\nabla\cdot \gamma^f\|^2_{0,\O_p}
+ \|\gamma^s\|^2_{0,\G_p} + \|\gamma^f\cdot\nu\|^2_{0,\G_p}\right.\nonumber\\
&&\qquad \left. + \sum_{\O_{j}\subset \O_p} \sum_{l,m} 
\|\epsilon_{lm}(\gamma^s)\|^2_{0,\O_j}\right]\nonumber\\ 
&&\qquad + C(\o) \left[ h^2 \left( \|E^s\|^2_1 + \|\text{curl} E^s\|^2_1 + 
\|H^s_y\|^2_1\right) + h\left(\|u^s\|^2_{2,\O_p} + \|u^f\|^2_{1,\O_p}
+ \|\nabla\cdot u^f\|^2_{1,\O_p}\right)\nonumber\right.\\
&&\qquad \left. + h \left(\|\gamma^E\|^2_0 +   
\|g^{\gamma,h}\|^2_{0,\O_p} + \|\nabla\cdot \gamma^f\|^2_{0,\O_p}
+ \|\gamma^s\|^2_{0,\G_p} + \|\gamma^f\cdot\nu\|^2_{0,\G_p}
+ \sum_{\O_{j}\subset \O_p} \sum_{l,m} 
\|\e_{lm}(\gamma^s)\|^2_{0,\O_j}\right)\right].\nonumber
\end{eqnarray}
Next use in \eqref{eq_122} that the matrices  ${\bf M}_i$  and
${\mathcal D}$ are 
 positive definite, that the coefficient $a$ is positive, and the estimate for $\|\gamma_y^H\|_0$ derived in \eqref{eq_109_5}. Thus choosing in \eqref{eq_122}  $\epsilon$ and $h$ sufficiently small  we get 
\begin{eqnarray}
&&\|\gamma^E\|^2_0 +  
+ \| g^{\gamma,h}\|^2_{0,\O_p} 
+ \|\gamma^E\cdot\chi\|^2_{0,\G} + \|\nabla\cdot\gamma^f\|^2_{0,\O_p} + 
+ \|\gamma^s\|^2_{0,\G_p} + \|\gamma^f\cdot\nu\|^2_{0,\G_p}\label{eq_128}\\
&&\qquad + \sum_{\O_{j}\subset \O_p} \sum_{lm} \|\epsilon_{lm}(\gamma^s)\|^2_{0,\O_j}\nonumber\\ 
&&\qquad \le C(\o) \left[ h^2 \left( \|E^s\|^2_1 + \|\text{curl} E^s\|^2_1 + 
\|H^s_y\|^2_1\right) + h\left(\|u^s\|^2_{2,\O_p} + \|u^f\|^2_{1,\O_p}
+ \|\nabla\cdot u^f\|^2_{1,\O_p}\right)\right]\nonumber.
\end{eqnarray}  
Now from  \eqref{eq_109_4_1} and \eqref{eq_109_5_a} we see that $\|\gamma^f\|^2_{0,\O_p}$ and $\|\gamma^H_y\|^2_0$ are bounded by the right-hand side of \eqref{eq_128}. Thus, we conclude that for $h$ small the following inequality holds:
\begin{eqnarray}
&&\|\gamma^E\|^2_0 +  
+ \|\gamma^H_y\|^2_0 + \|\gamma^f\|^2_{0,\O_p} + \|\nabla\cdot\gamma^f\|^2_{0,\O_p} +  
+ \|\gamma^E\cdot\chi\|^2_{0,\G} + 
+ \|\gamma^s\|^2_{0,\G_p} + \|\gamma^f\cdot\nu\|^2_{0,\G_p}\nonumber\\
&&\qquad + \sum_{\O_{j}\subset \O_p} \sum_{lm} \|\epsilon_{lm}(\gamma^s)\|^2_{0,\O_j}\label{eq_133}\\ 
&&\qquad \le C(\o) \left[ h^2 \left( \|E^s\|^2_1 + \|\text{curl} E^s\|^2_1 + 
\|H^s_y\|^2_1\right) + h\left(\|u^s\|^2_{2,\O_p} + \|u^f\|^2_{1,\O_p}
+ \|\nabla\cdot u^f\|^2_{1,\O_p}\right)\right]\nonumber.
\end{eqnarray}  
Next, using  Korn's second inequality and the estimates  \eqref{eq_104} and \eqref{eq_133} 
\begin{eqnarray} 
&&C_5\|\gamma^s\|^2_{1,h}\le  \sum_{\O_{j}\subset \O_p} \sum_{lm} \|\epsilon_{lm}(\gamma^s)\|^2_{0,\O_j} + C_7 \|\gamma^s\|^2_{0,\O_p} \label{eq_136}\\
&&\qquad\qquad\quad  \le C(\o)\left[ h^2 \left( \|E^s\|^2_1 + \|\text{curl} E^s\|^2_1 + \|H^s_y\|^2_1\right) \right.\nonumber\\
&&\left. \qquad \qquad\quad  + h\left(\|u^s\|^2_{2,\O_p} + \|u^f\|^2_{1,\O_p}
+ \|\nabla\cdot u^f\|^2_{1,\O_p}\right)\right]\nonumber.
\end{eqnarray}    
Now from \eqref{eq_133} and \eqref{eq_136} we get
\begin{eqnarray}
&&\|\gamma^E\|_0 +  
+ \|\gamma^H_y\|_0 + \|\gamma^s\|_{1,h} + \|\gamma^f\|_{0,\O_p} + \|\nabla\cdot\gamma^f\|_{0,\O_p}\label{eq_138}\\
&&\qquad  +   \|\gamma^E\cdot\chi\|_{0,\G}  
+ \|\gamma^s\|_{0,\G_p} + \|\gamma^f\cdot\nu\|_{0,\G_p}\nonumber\\
&&\qquad \le C(\o) \left[ h \left( \|E^s\|_1 + \|\text{curl} E^s\|_1 + 
\|H^s_y\|_1\right) + h^{1/2}\left(\|u^s\|_{2,\O_p} + \|u^f\|_{1,\O_p}
+ \|\nabla\cdot u^f\|_{1,\O_p}\right)\right]\nonumber.
\end{eqnarray}
Next, to estimate $\|\text{curl} \gamma^E\|_0$ we proceed as follows. 
Choose $\psi=v^s=v^f=0, \varphi = \text{curl} \gamma^E$ in \eqref{eq_97} and use \eqref{eq_138}  to get the estimate
\begin{eqnarray}
&&\|\text{curl} \gamma^E\|_0 \le C\left( \|\gamma^H_y\|_0 
+ h \left(\|\text{curl} E^s\|_1 + \|H^s_y\|_1 \right) \right)\label{eq_146}\\
&&\qquad \le  C(\o) \left[ h \left( \|E^s\|_1 + \|\text{curl} E^s\|_1 + 
\|H^s_y\|_1\right) + h^{1/2}\left(\|u^s\|_{2,\O_p} + \|u^f\|_{1,\O_p}
+ \|\nabla\cdot u^f\|_{1,\O_p}\right)\right]\nonumber.
\end{eqnarray}
Finally using \eqref{eq_138}, \eqref{eq_146} the triangle inequality  and the approximating properties \eqref{def_pih}, \eqref{def_ph}, \eqref{def_rh} and \eqref{def_qh}
 we get the conclusion of the theorem. This completes the proof.
\end{proof} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%  SECTION           %%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{An iterative domain decomposition procedure for the PSVTM-mode}
We will define a Jacobi-type  algorithm for the case in 
which the domain decomposition of the computational domain 
$\O$ coincides with the finite element partition. The case of 
larger subdomains and a {\it red-black}-type iteration   can be 
formulated as in \cite{santos_sinum_07}.  Following the ideas 
in \cite{santos_sinum_07} and \cite{santos2dmt_98}, to define a  
nonoverlapping  domain decomposition
procedure  we introduce a set of Lagrange 
multipliers  $\lam^h_{jk}$ associated with $H^{s,h}_y$ on all interior
interfaces $\G_{jk}$ in $\O$ and  
$l^h_{jk}$ associated with ${\mathcal G}_{\G_{jk,p}}(u_j)$ 
at the midpoints $\xi_{jk}$ of $\G_{jk,p}$ for all interior interfaces 
$\G_{jk,p}$ in $\O_p$.
Thus, set 
\begin{eqnarray}
&&{\Lam}^h= \{\lam^h: \lam^h_{jk}= \text{tr}_{~\G_{jk}}(\lambda^h|_{\O_j})
\in P_0(\G_{jk})\equiv \Lam^h_{jk}\},\nonumber\\
&&{L}^h= \{ l^h:  l^h_{jk} = \text{tr}_{~\G_{jk,p}}(l^h|{\O_j}) 
\in [P_0(\G_{jk,p})]^3\equiv L^h_{jk}\}.\nonumber
\end{eqnarray}
Note that two copies of  $P_0(\G_{jk})$ and  $[P_0(\G_{jk,p})]^3$ 
are assigned to  $\G_{jk}$ 
 and $\G_{jk,p}$, respectively,    
 one from $\O_j$ to $\O_k$ and another from $\O_k$ to $\O_j$. 
 
Let $(\cdot, \cdot)_{\O_{j,p}}$ denote the $L^2$-inner product 
$(\cdot, \cdot)_{\O_{j}}$ for any $\O_j\subset \O_p$.  
The Jacobi-type {\it domain decomposition}  procedure is defined as follows:
\newline
 find 
$\left( E^{s,h,n}_j,  H^{s,h,n}_{y,j},  \lambda^{h,n}_{jk}, 
 u^{s,h,n}_j,  u^{f,h,n}_j,  l^{h,n}_{jk}\right)\in {\mathcal V}^h_j\times {\mathcal W}^h_j\times {\Lam}^h_{jk}\times {\mathcal NC}^h_j\times{\mathcal M}^h_{j}\times L^h_{jk}$ such that 
\begin{eqnarray}
&&(\sigma  E^{s,h,n}_j,\psi)_{\O_j} 
- (H^{s,h,n}_{y,j}, \text{curl} \psi)_{\O_j} 
-\sum_k\left\langle  \lambda_{jk}^{h,n},  \psi\cdot\chi_{jk}\right\rangle_{\G_{jk}}
 + \left\langle a(1-i) E^{s,h,n}_j\cdot \chi_j, 
\psi\cdot\chi_j\right\rangle_{\G_j}
%\label{eq_102_103_105}
\nonumber\\
&& \qquad + \left( L(\o) \eta (\kappa(\o))^{-1} \left[i \o  u_j^{f,h,n} 
- L(\o) E_j^{s,h,n}\right], \psi\right)_{\O_{j,p}} 
+(\text{curl}  E^{s,h,n}_j, \varphi)_{\O_j} \label{eq_181_182_184}\\
&&\qquad + (i \o \mu H^{s,h,n}_{y,j}, \varphi)_{\O_j}
-  \o^2 \left(\rho_b  u_j^{s,h,n}, v^s\right)_{\O_{j,p}} 
- \o^2 \left(\rho_f  u_j^{f,h,n}, v^s\right)_{\O_{j,p}}
- \o^2 \left(\rho_f  u_j^{s,h,n}, v^f\right)_{\O_{j,p}}\nonumber\\
&&\qquad  + \left(  \eta (\kappa(\o))^{-1} \left[i \o  u_j^{f,h,n} 
-  L(\o)  E_j^{s,h,n}\right], v^f\right)_{\O_{j,p}}
 +\left({\bf M}_r ~ \tilde \epsilon(u^{h,n}_j),\tilde \epsilon(v)\right)_{\O_{j,p}}  
\nonumber\\
&&\qquad + i \left({\bf M}_i ~ \tilde \epsilon(u^{h,n}_j),\tilde \epsilon(v)\right)_{\O_{j,p}}
- \sum_{k}\left\langle \left\langle  l^{h,n}_{jk} , S_{\G_{jk}}(v)\right\rangle\right\rangle
_{\G_{jk,p}}
 + i \o \left\langle {\mathcal D} S_{\G}(u_j^{h,n}),
 S_{\G}(v)\right\rangle_{\G_j\cap \G_p}\nonumber\\
&&\qquad = -\left(\left[\sigma^s - \chi_{\O_p} L^2(\o) \eta (\kappa(\o))^{-1}\right]  E^p, \psi\right)_{\O_j}  + (F^{(s)}, v^s)_{\O_{j,p}}\nonumber\\
&&\qquad \quad + \left(\left[F^{(f)} +  L(\o)\eta (\kappa(\o))^{-1} E^p\right], v^f\right)_{\O_{j,p}}, \quad 
 (\psi, \varphi, v^s, v^f)\in  
 {\mathcal V}^h_j\times {\mathcal W}^h_j\times {\mathcal NC}^h_j\times{\mathcal M}^h_{j},\nonumber\\
&&\lambda^{h,n}_{jk} = \lambda^{h,n-1}_{kj} 
- \beta \left(E^{s,h,n}_j\cdot\chi_{jk}+ 
E^{s,h,n-1}_k\cdot\chi_{kj} \right), \quad \text{on} \quad \G_{jk},
\label{eq_183}\\
&&l^{h,n}_{jk} = l^{h,n-1}_{kj} - i \o \beta \left(S_{\G_{jk,p}}(u^{h,n}_j) +
S_{\G_{kj,p}}(u^{h,n-1}_k)\right)(\xi_{jk}), \quad \text{on} \quad \G_{jk,p}
\label{eq_185_dd}.
\end{eqnarray}
In \eqref{eq_183}  
$\beta$ is a  positive  scalar iteration parameter and  $u^{h,n}_j = (u_j^{s,h,n}, u_j^{f,h,n})$.  

To demonstrate the convergence of the 
iterative procedure \eqref{eq_181_182_184}-\eqref{eq_183}-\eqref{eq_185_dd}, 
we need to introduce some additional spaces.
Let 
\begin{eqnarray*}
&&{\mathcal V}^{h}_{-1}=\{\psi \in [L^2(\O)]^2:\psi|\O_{j}\in {\mathcal V}^h_{j}\},\qquad 
{\mathcal NC}^h_{-1} = \{v\in [L^2(\O)]^2: v|{\O_j}\in {\mathcal NC}^h_j\}\\  
&&{\tilde \Lam}^h_{-1}= \{\tilde \lam^h:  
\tilde \lam^h|{\G_{jk}}= \tilde\lam_{jk} \in P_0(\G_{jk}), \tilde \lam_{jk}= \tilde\lam_{kj}\}, 
\\ 
&&{\tilde L}^h_{-1}= \{\tilde l^h:  
\tilde l^h|{\G_{jk,p}}= \tilde l_{jk} \in [P_0(\G_{jk,p})]^3, 
\tilde l^h_{jk} = \tilde l^h_{kj}\}.
\end{eqnarray*}
We will show that the iterations    
approximate a fixed point of  an operator    $T_{F}$ from
${\mathcal V}^h_{-1}\times 
{\mathcal W}^h\times\tilde \Lam^h_{-1}\times{\mathcal NC}^h_{-1}
\times{\mathcal M}^h\times\tilde  L^h_{-1}$ into itself, 
with $F= \left(E^p, F^{(s)}, F^{(f)}\right)$, defined as follows. 
Let
$T_F: {\mathcal V}^h_{-1}\times {\mathcal W}^h\times\tilde \Lambda^h_{-1}\times
{\mathcal NC}^h_{-1}\times{\mathcal M}^h\times\tilde  L^h_{-1} \to {\mathcal V}^h_{-1}\times {\mathcal W}^h\times\tilde \Lam^h_{-1}\times{\mathcal NC}^h_{-1}\times{\mathcal M}^h\times\tilde  L^h_{-1}$  be the
affine map such that for any $(e,h_y,\delta,u^s, u^f,\zeta)\in {\mathcal V}^h_{-1}\times {\mathcal W}^h\times\tilde\Lam^h_{-1}\times{\mathcal NC}^h_{-1}\times{\mathcal M}^h\times\tilde L^h_{-1}$,
$(E, H_y,\lambda, U^s, U^f, l)= T_F(e,h_y,\delta,u^s, u^f,\zeta)$ is the solution of
\begin{eqnarray}
&&(\sigma E,\psi) -\Sum_j (H_{y,j}, \text{curl} \psi)_{\O_j} 
-  \sum_{j,k}\left\langle \lambda_{jk}, \psi\cdot\chi_{jk}\right\rangle_{\G_{jk}}
 + \left\langle a(1-i) (E_j\cdot \chi, \psi\cdot\chi\right\rangle_{\G}\label{eq_236_237_239}\\
&& \qquad 
+ \left( L(\o) \eta (\kappa(\o))^{-1} \left[i \o U^{f} -  L(\o)  E\right], \psi\right)_{\O_p} 
 + \Sum_j (\text{curl} E_j, \varphi)_{\O_j}\nonumber\\
&&\qquad  + (i \o \mu  H_{y}, \varphi)
-  \o^2 \left(\rho_b   U^s, v^s\right)_{\O_p} 
- \o^2 \left(\rho_f  U^{f}, v^s\right)_{\O_p}
- \o^2 \left(\rho_f  U^{s}, v^f\right)_{\O_p}\nonumber\\
&&\qquad  + \left( \eta (\kappa(\o))^{-1} \left[i \o 
 U^{f} -  L(\o) E\right], v^f\right)_{\O_p} 
+ {\mathcal A}_h(U,v)
\nonumber\\
&&\qquad 
 - \sum_{j,k}\left\langle\left\langle l_{jk}, S_{\G_{jk,p}}(v)
 \right\rangle\right\rangle_{\G_{jk,p}}
+ i \o \left\langle {\mathcal D} S_{\G_p}(U), S_{\G}(v)\right\rangle_{\G_p}\nonumber\\
&&\qquad = -\left(\left[\sigma^s - \chi_{\O_p}  L^2(\o) \eta (\kappa(\o))^{-1}\right]  E^p, \psi\right) \nonumber\\
&&\qquad\qquad  + (F^{(s)}, v^s)_{\O_p} 
+ \left(\left[F^{(f)} +  L(\o)\eta (\kappa(\o))^{-1} E^p\right], v^f\right)_{\O_p},\nonumber\\
&&\hskip6cm \quad (\psi, \varphi, v^s, v^f)\in {\mathcal V}^h_{-1}\times {\mathcal W}^h_j\times {\mathcal NC}^h_{-1}\times{\mathcal M}^h_{j}.\nonumber\\
&& \lambda_{jk} = \delta_{kj} - \beta\left(E_j\cdot\chi_{jk}+ 
e_k\cdot\chi_{kj} \right), \quad \text{on} \quad \G_{jk}, \quad \forall \, j,k
\label{eq_238}\\
&&l_{jk} =  \zeta_{kj} - i \o \beta\left(S_{\G_{jk,p}}(U_j) +
S_{\G_{kj,p}}(u_k)\right)(\xi_{jk}), \quad \text{on} \quad \G_{jk,p}, \quad \forall\, j,k
\label{eq_240}.
\end{eqnarray}
First we observe that if $E\in {\mathcal V}^h_{-1}, U^s \in {\mathcal NC}^h_{-1}$, then  $E\in {\mathcal V}^h, U^s\in {\mathcal NC}^h$  
if and only if, for $U^f \in {\mathcal M}^h$, 
\begin{eqnarray} 
&&\sum_{jk}\left\langle E\cdot\chi_{jk}, \tilde \theta^h\right\rangle_{\G_{jk}} = 0, \quad \tilde \theta^h\in {\tilde\Lambda}^h_{-1},\label{eq_78}\\
&&\sum_{j,k}\left\langle \left\langle\tilde l^h, S_{\G_{jk}}(U^s, U^f)\right\rangle\right\rangle_{\G_{jk,p}} = 0, \quad \tilde l^h\in {\tilde L}^h_{-1}.\label{eq_79}
\end{eqnarray}
Consequently,  thanks to \eqref{eq_238} and \eqref{eq_240},  $\left(E, H_y, \lambda, U^s, U^f, l\right)$  is a fixed point of $T_{F}$ if and only if $\left(E, H_y, \lambda, U^s, U^f, l\right)$  is a solution of the equations \eqref{eq_236_237_239}, \eqref{eq_78} and \eqref{eq_79}. Thus in particular  $\left(E, H_y,  U^s, U^f\right)
\in {\mathcal Y}^h$ solves  the {\it global} Galerkin procedure
\eqref{eq_101}.

\noindent 
{\emph Remark:} Equations  \eqref{eq_236_237_239}, \eqref{eq_78} and \eqref{eq_79}. 
define the {\it Global hybridized} procedure associated with the 
Galerkin procedure  \eqref{eq_101}.

On the other hand, since 
\begin{equation}\label{eq_252_a}
\left(E, H_y, \lambda, U^s, U^f, l\right) = T_F(e, h_y, \delta, u^s, u^f, \zeta)=T_0(e, h_y, \delta, u^s, u^f, \zeta) + T_F(0, 0, 0, 0, 0), 
\end{equation}
we see that $\left(E, H_y, \lambda, U^s, U^f, l\right)$ is a fixed point of $T_F$ if and only if
\begin{equation}\label{eq6.8}
(I-T_0)(\left(E, H_y, \lambda, U^s, U^f, l\right))=T_F(0,0,0,0,0).
\end{equation}
The next theorem gives a result on the convergence of  the iterative procedure \eqref{eq_181_182_184}-\eqref{eq_183}-\eqref{eq_185_dd} to the solution 
of the {\it global} Galerkin procedure
\eqref{eq_101}.

\begin{theorem}\label{conv_dd} Let $\rho(T_0)$ be the spectral radius of $T_0$. 
Then, under the assumption of the hypothesis of \thmref{Theorem3.1}, 
\[
\rho(T_0)<1,
\]
and  consequently  the iterative procedure \eqref{eq_181_182_184}-\eqref{eq_183}-\eqref{eq_185_dd} is
convergent.
\end{theorem}
\begin{proof} Let $\Upsilon$ be an eigenvalue of $T_0$ and let $\left(E, H_y, \lambda, U^s, U^f, l\right) $ be the
associated eigenvector, so that 
\[
T_0\left(E, H_y, \lambda, U^s, U^f, l\right)=\Upsilon \left(E, H_y, \lambda, U^s, U^f, l\right).
\]
Let us define the boundary  energy $R$  as 
\begin{eqnarray}\label{eq_130}
&&R\left( E, H_y, \lambda, U^s, U^f, l\right)=\Sum_{j,k} \|\lambda_{jk}+  \b E_j\cdot\chi_{jk}\|^2_{0,\G_{jk}} \\
&&\hskip4cm + \o \Sum_{j,k}  \|l_{jk}
+ i \o \b S_{\G_{jk,p}}(U_j)(\xi_{jk})\|^2_{0,\G_{jk,p}}\nonumber.
\end{eqnarray}
Then it follows from \eqref{eq_130} that
\begin{equation}\label{eq_256}
R\left(T_0\left(E, H_y, \lambda, U^s, U^f, l\right)\right) 
= |\Upsilon|^2 R\left(E, H_y, \lambda, U^s, U^f, l\right),
\end{equation}
On the other hand, since    for any pair of complex numbers $p$ and $q$
 and $\b>0$, we have
\[
|p \pm i \b q|^2=|p|^2 + \b^2 |q|^2 \pm 2 \b \Im(p \overline q), \quad 
|p \pm \b q|^2=|p|^2 + \b^2 |q|^2 \pm 2 \b \Re(p \overline q),
\]
from  \eqref{eq_236_237_239} for $F=\left(E^p, F^{(s)}, F^{(f)}\right)
=\left(0,0,0\right)$ and  \eqref{eq_238}-\eqref{eq_240}
we get 
\begin{eqnarray}\label{eq_614}
&&R\left( T_0\left(E, H_y, \lambda, U^s, U^f, l\right)\right)\\
&&\qquad =\Sum_{j,k} \|\lambda_{kj}-  \b E_k\cdot\chi_{kj}\|^2_{0,\G_{jk}} 
+ \o \Sum_{j,k}  \|l_{kj}- i \o \b
S_{\G_{jk,p}}(U_k)(\xi_{jk})\|^2_{0,\G_{jk,p}} \nonumber\\
&&\qquad = R\left( E, H_y, \lambda, U^s, U^f, l\right) 
-  4 \b \Re \left[\Sum_{jk } \la \lambda_{kj}, E_k \cdot \chi_{kj}\ra_{\G_{jk}}\right]\nonumber\\
&&\qquad \quad - 4 \omega \b \Im \left[\Sum_{jk} 
\la \la l_{kj},  S_{\G_{jk,p}}(U_k)(\xi_{kj})\ra\ra_{\G_{jk,p}}\right].\nonumber
\end{eqnarray}
Let us compute the last two terms in the right-hand side of \eqref{eq_614}.  
Choose $\psi=E_j, \varphi=H_{y,j}, v=(v^s, v^f)=i \o U_j = (i \o U_j^{s}, i\o U^{f}_j)$ 
in \eqref{eq_236_237_239} for $F=0$ 
 and use the argument leading to \eqref{uniq.2} in the proof of 
 uniqueness of the solution of the variational problem  \eqref{eq_100} 
 to get the equation
\begin{eqnarray}
&&(\sigma  E_j,  E_j)_{\O_j} 
- (i \o \mu H_{y,j}, H_{y,j})_{\O_j} - \sum_k\left\langle  \lambda_{jk}, E_j\cdot\chi_{jk}\right\rangle_{\G_{jk}} 
 + \left\langle a(1-i)  E_j\cdot \chi_{jk}, 
 E_j\cdot\chi_{jk}\right\rangle_{\G_j} \nonumber\\
&&\qquad + 
\left( L(\o) g_{0,p,j},  E_j\right)_{\O_{j,p}} 
- i\o \left[-\o^2 (\rho_b U_j^{s}, U_j^{s})_{\O_{j,p}} - \o^2 2 \text{Re}\left[\left(\rho_f U_j^{f}, U_j^{s}\right)_{\O_{j,p}}\right] \right.\nonumber\\
&&\qquad \left.+ i \o \left({\bf M}_r ~ \tilde \epsilon(U_j),\tilde \epsilon(U_j)\right)_{\O_{j,p}}\right]
+ \o \left({\bf M}_i ~ \tilde \epsilon(U_j),\tilde \epsilon(U_j)\right)_{\O_{j,p}} 
+ i \o  \sum_{k}\left\langle\la  l_{jk}, S_{\G_{jk,p}}(U_j)(\xi_{jk})\right\rangle\ra_{\G_{jk,p}} \nonumber\\
&&\qquad + \o^2 \left\langle {\mathcal D} S_{\G_j}(U_j), S_{\G_j}(U_j)
\right\rangle_{\G_j\cap\G_p}+ \left( g_{0,p,j}, i \o U^{f}_j\right)_{\O_{j,p}}=0.\label{eq_615}
\end{eqnarray}
where
\begin{eqnarray}\label{def_g0j}
&&g_{0,p,j} =   \eta (\kappa(\o))^{-1} i \o U_j^{f}
 -  L(\o)\eta (\kappa(\o))^{-1} E_j, \quad  \O_j\subset \O_p, \quad g_{0,p} = \sum_j g_{0,p,j}.
\end{eqnarray}
Taking real part in \eqref{eq_615} and adding over $j$ the resulting equation  it follows that
\begin{eqnarray}\label{eq_621a}
&&\text{Re}\left[\Sum_{j,k}\la \lambda_{jk},  E_j \cdot \chi_{jk}\ra_{\G_{jk}}\right] 
+ \o \text{Im}\left[ \Sum_{j,k} \la\la l_{jk}, 
S_{\G_{jk,p}}(U_j)\ra\ra_{\G_{jk,p}}\right] \\
&& \qquad= (\sigma E, E)_{\O_a} + \text{Re}\left[\Phi_{0,p}\right] +
\left\langle a  E\cdot \chi, E\cdot\chi\right\rangle_{\G}\nonumber\\
&&\qquad+  \o \Sum_j \left({\bf M}_i ~ \tilde \epsilon(U_j),
\tilde \epsilon(U_j)\right)_{\O_{j,p}} 
+  \o^2 \left\langle {\mathcal D} S_{\G_p}(U), S_{\G_p}(U)
\right\rangle_{\G_p},\nonumber
\end{eqnarray}
with
\begin{eqnarray}\label{def_phi0p}
&&\Phi_{0,p} = (\sigma  E,  E)_{\O_p} +  ( L(\o) g_{0,p},  E)_{\O_p} 
+ (g_{0,p},  i \o U^{f})_{\O_p}.
\end{eqnarray}
Using  \eqref{eq_621a} in  \eqref{eq_614}, 
\begin{eqnarray}\label{eq_622}
&&R( T_0\left(E, H_y, \lambda, U^s, U^f, l)\right)\\
&&\qquad =R\left( E, H_y, \lambda, U^s, U^f, l\right) 
-  4 \beta \left[ (\sigma E, E)_{\O_a} +\text{Re}\left[\Phi_{0,p}\right] +
\left\langle a  E\cdot \chi, E\cdot\chi\right\rangle_{\G}\nonumber\right.\\
&&\qquad+  \left.\Sum_j \o \left({\bf M}_i ~ \tilde \epsilon(U_j),
\tilde \epsilon(U_j)\right)_{\O_{j,p}} 
+  \o^2 \left\langle {\mathcal D} S_{\G_p}(U), S_{\G_p}(U)
\right\rangle_{\G_p}\right],\nonumber
\end{eqnarray}
From  \eqref{eq_622} and  \eqref{eq_256} we obtain
\begin{eqnarray}\label{eq_261}
&&|\Upsilon|^2=1-\dfrac{4\beta}{R\left(E, H_y, \lambda, U^s, U^f, l\right)}\sum_j\left[ (\sigma E, e)_{\O_a} + \text{Re}\left[\Phi_{0,p}\right]  +
\left\langle a E\cdot \chi, E\cdot\chi\right\rangle_{\G}\right.\\
&&\qquad \left. \qquad  +  \o \Sum_j \left({\bf M}_i ~ \tilde \epsilon(U_j),
\tilde \epsilon(U_j)\right)_{\O_{j,p}} 
+  \o^2 \left\langle {\mathcal D} S_{\G_p}(U), S_{\G_p}(U)
\right\rangle_{\G_p}\right].\nonumber 
\end{eqnarray}
Thus, $|\Upsilon|\leq 1$ and $|\Upsilon|=1$ if and only if
\begin{eqnarray}\label{eq_262}
&&(\sigma E, E)_{\O_a} +  \text{Re}\left[\Phi_{0,p}\right]  +
\left\langle a E\cdot \chi, E\cdot\chi\right\rangle_{\G}
  +  \o \Sum_j \left({\bf M}_i ~ \tilde \epsilon(U_j),
\tilde \epsilon(U_j)\right)_{\O_{j,p}}\\ 
&& \qquad\qquad +  \o^2 \left\langle {\mathcal D} S_{\G_p}(U), S_{\G_p}(U)
\right\rangle_{\G_p}=0.\nonumber 
\end{eqnarray}
Now we show that $|\Upsilon |=1$ implies that 
\begin{equation}\label{zero}
\left(E, H_y, \lambda, U^s, U^f, l\right) = (0, 0, 0, 0, 0, 0).
\end{equation}  
Since  (see  \eqref{eq54})
\begin{eqnarray}\label{ineq_Phi0p} 
&&\text{Re}\left[\Phi_{0,p}\right]\ge C_4 
\left[\|E\|_{0,\O_p}^2 + \|g_{0,p}\|^2_{0,\O_p}\right],
\end{eqnarray}
applying  in \eqref{eq_262} the argument given in the proof of the uniqueness of the  {\it global}  Galerkin  procedure \eqref{eq_101}  it follows that   
\begin{eqnarray}
&&E = 0 \quad \text{in}\quad [L^2(\O)]^2,\label{uniq_E0}\\
&&E\cdot\chi = 0  \quad \text{in}\quad L^2(\G_j),\label{uniq_E0_chi}\\
&&g_{0,p} = 0 \quad \text{in}\quad L^2(\O_p), \label{uniq_g0p}\\ 
&&\nabla\cdot U^f = 0    \quad \text{in}\quad L^2(\O_p),\label{uniq_div_uf0}\\
&&\varepsilon_{11}(U^s_j)= 0,\,   \varepsilon_{12}(U^s_j) = 0, \,  
\varepsilon_{22}(U^s_j) = 0, \quad    
\quad \text{in}\quad L^2(\O_j), \quad \O_j\subset \O_p,\label{uniq_strain_j0}\\
&&U^s = 0 \quad  \text{in}\quad L^2(\G_p),\label{uniq_us0_bry}\\
&&U^f\cdot\nu= 0  \quad \text{in}\quad L^2(\G_p).\label{uniq_uf0_nu}
\end{eqnarray}
Combining \eqref{uniq_E0}, \eqref{uniq_g0p} and \eqref{def_g0j} 
we get
\begin{equation}\label{uniq_uf0}
U^f = 0  \quad \text{in} \quad [L^2(\O_p)]^2.
\end{equation}
Next, using  \eqref{uniq_strain_j0}and  \eqref{uniq_us0_bry}, the argument used in the proof of the uniqueness of the discrete problem  \eqref{eq_101} 
can be used here to show that 
\begin{equation}\label{uniq_us0}
U^s = 0 \quad \text{in} \quad [L^2(\O_p)]^2.
\end{equation}
Next, by quasiregularity, 
\begin{eqnarray}
\|\text{curl}  E\|_0 \le C h^{-1} \| E\|_0 =0, \label{quasi1}
\end{eqnarray}
so that  
the choice $\psi=0, v^s=0, v^f=0, \varphi=H_{y,j}$ in \eqref{eq_236_237_239}  (for $F=0$)   implies  that
\begin{eqnarray}
H_{y}= 0 \quad  \text{in} \quad L^2(\O). \label{eq_271}
\end{eqnarray}
Consequently using  \eqref{uniq_E0}, \eqref{uniq_E0_chi}, \eqref{uniq_uf0}, \eqref{uniq_us0}, 
\eqref{quasi1}  and \eqref{eq_271}  in \eqref{eq_236_237_239} for $F=0$  we get
\begin{eqnarray}\label{eq_274_275}
&& \sum_{k}\left\langle \Upsilon \lambda_{jk}, \psi\cdot\chi_{jk}\right\rangle_{\G_{jk}}
 + \sum_k\left\langle\left\langle \Upsilon l_{jk}, 
 S_{\G_{jk,p}}(v)\right\rangle\right\rangle_{\G_{jk,p}}=0,\\
&&\hskip6cm  \psi\in {\mathcal V}^h_j, v^s\in {\mathcal NC}^h_j, v^f\in {\mathcal M}^h_j.\nonumber
\end{eqnarray}
To show that \eqref{eq_274_275} implies that the Lagrange multipliers vanish identically, for any 
element $\O_j$ take $v^s = v^f = 0$  and $\psi$ in \eqref{eq_274_275} such that 
$\psi\cdot\chi_{jk}= \lambda_{jk}$ on one  interior side of $\p\O_j$ and $\psi\cdot\chi_{jk}=0$ on the other three sides of  $\p\O_j$ to see that 
\begin{equation}\label{eq_276}
\lambda_{jk} = 0, \quad\text{for all} \quad j,k.
\end{equation}
Similarly, choose $\psi=0$ and $v=(v^s, v^f)\in 
{\mathcal NC}^h\times {\mathcal M}^h$ such that 
$
S_{\G_{jk,p}}(v)(\xi_{jk}) = l_{jk}
$
on one  interior side of $\p\O_j$ and 
$
 S_{\G_{jk,p}}(v)(\xi_{jk}) =0 
$
at the  mid points of the other three sides of  $\p\O_j$ to see that
\begin{equation}\label{eq_277}
l_{jk} = 0, \quad\text{for all} \quad j,k.
\end{equation}
Thus \eqref{zero} holds. This completes the proof.
 
\end{proof}
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\section{Formulation of the  SHTE-mode}
As explained in Section 2, in this case   only horizontally polarized SH waves
 are present and 
particle motions are only   in the
y-direction. Thus,  
\begin{eqnarray}\label{def_us_uf} 
u^s = (0,u^s_y(x,z),0), \qquad  \ u^f = (0,u^f_y(x,z),0),
\end{eqnarray}
and consequently
\begin{eqnarray}\label{def_div} 
\nabla\cdot u^s = \nabla\cdot u^f = 0.
\end{eqnarray}
Hence,  it follows from  \eqref{mod.1f} and \eqref{def_div} that 
\begin{eqnarray} 
p_f = 0.
\end{eqnarray}
Thus, in the SHTE-mode we have the following differential system:
\begin{eqnarray}
&&\sigma E^s - \text{curl} H^s + \chi_{\O_p} L(\o) \eta \kappa(\o))^{-1}\left[ i \o u^f_y -  L(\o) E^s_y\right]\nonumber \\
&&\qquad\qquad = 
-(\sigma^s - L^2(\o) \eta \kappa(\o))^{-1} E^p_y,\quad \text{in}\quad \O,\label{mod.te1}\\
&&\text{curl} E^s_y + i \o \mu H^s = 0, \quad \text{in}\quad \O\label{mod.te2}\\
&&-\omega^2 \rho_b u^s_y - \omega^2  \rho_f u^f_y  -
\nabla\cdot \left( N \nabla u^s_y\right)= 0,\quad \text{in}\quad \O_p,  \label{mod.te3}\\
&& -\omega^2 \rho_f u^s_y   
+  \eta (\kappa(\o))^{-1}  \left[i \o  u^f_y -   L(\o)  E^s_y\right]\nonumber\\
&&\qquad  = \eta (\kappa(\o))^{-1} L(\o)  E^p_y \quad \text{in}\quad \O_p\label{mod.te4},
\end{eqnarray}
with the absorbing boundary conditions
\begin{eqnarray}
&& a (1-i) E^s_y - H^s \cdot \chi = 0, \quad \text{on} \quad \G,\label{mod.te5}\\
&&-N^*\nabla u^s_y \cdot \nu = i \o  ((N^* b))^{1/2}  u^s_y, \quad \text{on} \quad
\G_p.\label{mod.te6}
\end{eqnarray}  
Let us consider again a rectangular  domain $\O=\O_{a}\cup\O_p$ in the $(x,z)$ plane where the boundary value problem  \eqref{mod.te1}-\eqref{mod.te6} need to be solved.
Test \eqref{mod.te2} against 
$\psi\in H(\text{curl},\O)$ and use \eqref{mod.te5}; test 
\eqref{mod.te1} against $\varphi\in L^2(\O)$. 
Also, test \eqref{mod.te3} against $v\in H^1(\O_p)$ 
and use integration by parts and  \eqref{mod.te6}. Finally, test 
\eqref{mod.te4} against $w\in L^2(\O_p)$. 
This yields the following weak formulation: Find $(H^s, E^s_y, u^s_y, u^f_y)
\in H(\text{curl}, \O)\times L^2(\O)\times H^1(\O_p)\times
L^2(\O_p)$ such that
\begin{eqnarray}
&& (i \o \mu H^s, \psi) + (E^s_y, curl \psi) 
+ \left\langle \left(\frac{1+i}{2 a}\right) H^s\cdot \chi, \psi\cdot\chi\right\rangle\label{weak.1}\\
&&\qquad +(\sigma E^s_y,\varphi) -(curl H^s,\varphi) 
+ \left( L(\o)\eta (\kappa(\o))^{-1} \left[i \o u^f_y - 
 L(\o) E^s_y\right], \varphi\right)_{\O_p}\nonumber\\ 
&&\qquad -\omega^2 (\rho_b u^s_y, v^s)_{\O_p}  - \omega^2  (\rho_f u^f_y, v^s)_{\O_p} +
\left( N \nabla u^s_y, \nabla v^s\right)_{\O_p} 
+ \left\langle i \o (b N^*)^{1/2} u^s_y, v^s\right\rangle_{\G_p} 
\nonumber\\
&& \qquad -\omega^2 (\rho_f u^s_y, v^f)_{\O_p} 
+\left(\eta (\kappa(\o))^{-1} \left[i \o u^f_y -  L(\o) E^s_y\right], v^f\right)_{\O_p}\nonumber\\
&&\qquad = - (\left[\sigma^s - \chi_{\O_p} L^2(\o)\eta (\kappa(\o))^{-1} E^p_y\right],\varphi) 
+ (\eta (\kappa(\o))^{-1} L(\o) E^p_y , v^f)_{\O_p},\nonumber\\
&&\hskip6cm   (\psi,\varphi,v^s,v^f)\in  H(\text{curl}, \O)\times L^2(\O)\times H^1(\O_p)\times L^2(\O_p)\nonumber.
\end{eqnarray}
{\emph Remark:} the boundary term $\left\langle 
\left(\frac{1+i}{2 a}\right) H^s\cdot \chi, 
\psi\cdot\chi\right\rangle$ in \eqref{weak.1} must be interpreted as
explained in the remark below \eqref{eq_100}. 
 
An  argument similar to that given in the proof of \thmref{Theorem3.1} can be used 
to show the validity of the following theorem.
\begin{theorem}\label{thm;uniqueness}
Assume the validity of \eqref{eq51}. Then for any $\o>0$, uniqueness holds for the  solution of  \eqref{mod.te1}-\eqref{mod.te4}  
with the boundary 
conditions \eqref{mod.te5} and \eqref{mod.te6}.
\end{theorem}
\section{The Finite Element Method for the SHTE-mode}
We will employ the spaces  ${\mathcal V}^h $ and ${\mathcal W}^h$  to approximate the magnetic vector field $H^s$ and the scalar field $E^s_y$. Also,  to approximate the solid displacement  $u^s_y$ in $\O_p$ 
we employ the nonconforming finite element space ${\mathcal NC}^h$, while to approximate the fluid displacement  
$u^s_y$  we choose the space of  piecewise constants over the 
restriction of the  partition $\Tau^h(\O)$ to $\O_p$.
Thus,  for the fluid displacement, the space ${\mathcal {\widehat M}_j}^h $ is defined as  
\begin{eqnarray*}
&&{\mathcal {\widehat  M}_j^h = \{w:\, w|_{\O_j}\in P_0(\O_j)\},\qquad
{\mathcal {\widehat M}}^h = \{w \in L^2(\O):\, w_j=  w|_{\O_j} \in {\mathcal {\widehat M}_j}^h} \}
\end{eqnarray*}
Let 
\[
{\mathcal {\widehat Y}}^h = {\mathcal V}^h \times {\mathcal W}^h\times {\mathcal NC}^h \times
{\widehat{\mathcal M}}^h. 
\]
Then the  global finite element procedure for the approximate solution of 
\eqref{weak.1} is defined as follows: 
Find $(H^{s,h}, E^{s,h}_y, u^{s,h}_y, u^{f,h}_y)
\in   {\mathcal {\widehat Y}}^h $ such that
\begin{eqnarray}
&& (i \o \mu H^{s,h}, \psi) + (E^{s,h}_y, curl \psi) 
+ \left\langle \left(\frac{1+i}{2 a}\right) H^{s,h}\cdot \chi, \psi\cdot\chi\right\rangle
\label{fe.1}\\
&&\qquad +(\sigma E^{s,h}_y,\varphi) -(curl H^{s,h},\varphi)   + 
\left( L(\o) \eta (\kappa(\o))^{-1}\left[ i \o u^{f,h}_y - 
 L(\o) E^{s,h}_y\right], \varphi\right)_{\O_p} \nonumber\\
&&\qquad -\omega^2 (\rho_b u^{s,h}_y, v^s)_{\O_p}  - \omega^2  (\rho_f u^{f,h}_y, v^s)_{\O_p} +
\sum_{j}\left( N \nabla u^{s,h}_y, \nabla v^s\right)_{\O_{j,p}} + 
\left\langle i \o (\zeta N^*)^{1/2} u^{s,h}_y, v^s\right\rangle_{\G_p} 
\nonumber\\
&&\qquad  -\omega^2 (\rho_f u^{s,h}_y, v^f)_{\O_p}  
+ \left(\eta (\kappa(\o))^{-1} \left[i \o u^{f,h}_y -  L(\o) E^{s,h}_y\right] , v^f\right)_{\O_p}\nonumber\\  
&&\qquad = - (\left[\sigma^s - \chi_{\O_p} L^2(\o) \eta (\kappa(\o))^{-1}\right] E^p_y,\varphi)  
+ (\eta (\kappa(\o))^{-1} L(\o) E^p_y, v^f)_{\O_p},\quad 
 (\psi,\varphi,v^s,v^f)\in {\widehat{\mathcal Y}}^h.\nonumber
\end{eqnarray}
Uniqueness for the solution of \eqref{fe.1} follows with an argument similar to that presented in the analysis of the global Galerkin procedure \eqref{eq_101} for the PSVTM-mode.
Also, the ideas used in the demonstration of the validity of \thmref{globalerror} 
can be used with minor changes to show 
the validity of the   following theorem. 
\begin{theorem}\label{te_globalerror}
Let 
$(H^s, E_y^s, u_y^s, u_y^f)\in  H(curl, \O)\times L^2(\O)\times 
H^1(\O_p)\times L^2(\O_p)$ 
and  
\newline
$(H^{s,h}_y, E_y^{s,h}, u_y^{s,h}, u_y^{f,h})\in {\mathcal V}^h \times {\mathcal W}^h\times {\mathcal NC}^h \times
{\mathcal {\widehat M}}^h$ be the solutions of \eqref{weak.1} 
and \eqref{fe.1}, respectively. Assume that 
 $H^s \in [H^1(\O)]^2$,   $ \text{curl} ~H^s, ~ E^s_y ~ \in H^1(\O)$   $u_y^s\in H^2(\O_p)$, 
$u_y^f \in H^1(\O_p)$. Also assume  that the coefficient $a$ in the boundary condition \eqref{mod.te5}  is 
strictly positive and piecewise constant and the validity of 
the relations \eqref{eq51}.   
Then  the following  {\it a priori}  error estimate holds: 
for $\o > 0$ and sufficiently small $h>0$,

\begin{eqnarray*}
&& \|H^s -H^{s,h}\|_0 + \|\text{curl}(H^s -H^{s,h})\|_0 +  \|E^s_y - E_y^{s,h}\|_0 
+ \|u^s -u^{s,h}\|_{1,h,\O_p} + \|u^f -u^{f,h}\|_{0,\O_p} \\
&& + \|(H^s - H^{s,h})\cdot\chi\|_{0,\G} + \|u^s -u^{s,h}\|_{0,\G_p} \\
&&\qquad\qquad\le C(\o) \left[ h\left(\|H^s \|_1 +  \|\text{curl} ~H^s \|_1 + \|E_y^s \|_1\right)\right.\\
&& \qquad\qquad  + \left. h^{1/2}\left(\|u_y^s \|_{2,\O_p} 
+ h^{1/2} \|u_y^f \|_{1,\O_p}  +  \|H^s \|_1 \right)\right]. 
\nonumber
\end{eqnarray*}
\end{theorem}
\section{An iterative domain decomposition procedure for the  SHTE-mode}
As in the PSVTM-mode the  iteration will be defined for the case in which the domain decomposition of the computational domain $\O$ coincides with the finite element partition.  We introduce the   Lagrange 
multipliers  $\lam^h_{jk}\in\Lam^h_{jk} $ associated with $E^{s,h}_y$ 
on $\G_{jk}$. Thus the definition of the set ${\Lam}^h$ remains unchanged.  
Also, 
 we introduce the multipliers $l^h_{jk}$ associated with 
$- N \nabla u^{s,h}_{y,j}\cdot \nu_{jk}$ 
at the midpoints $\xi_{jk}$ of $\G_{jk,p}$. The  multipliers $l_{jk}$ 
belong to the space  
\[
{\widehat L}^h= \{ l^h:  l^h|{\G_{jk,p}}= l_{jk} \in P_0(\G_{jk,p})
=\widehat L^h_{jk}\}.
\]

The  Jacobi-type {\it domain decomposition}  
procedure is defined as follows:
\newline
 find 
$\left( E_{y,j}^{s,h,n},  H^{s,h,n}_{j},  \lambda^{h,n}_{jk}, 
 u^{s,h,n}_{y,j},  u^{f,h,n}_{y,j},  l^{h,n}_{jk}\right)\in {\mathcal V}^h_j\times 
 {\mathcal W}^h_j\times {\Lam}^h_{jk}\times {\mathcal NC}^h_j
\times{\mathcal {\widehat M}_j}^h \times {\widehat L}^h_{jk}$ such that 
\begin{eqnarray}
&&(i \o \mu H^{s,h,n}_j, \psi)_{\O_j} + (E^{s,h,n}_{y,j}, \text{curl} \psi)_{\O_j}
+\Sum_k\left\langle  \lambda_{jk}^{h,n},  \psi\cdot\chi_{jk}\right\rangle_{\G_{jk}}
 + \left\langle \left(\frac{1+i}{2 a} \right)H^{s,h,n}_j\cdot \chi_j, 
\psi\cdot\chi_j\right\rangle_{\G_j}\nonumber\\ 
&&\qquad + (\sigma  E_{y,j}^{s,h,n},\varphi)_{\O_j}  - (\text{curl} H^{s,h,n}_j, \varphi)_{\O_j}  + \left( L(\o) \eta (\kappa(\o))^{-1} \left[ i \o  u_{y,j}^{f,h,n} -  E_{y,j}^{s,h,n}\right], \varphi\right)_{\O_{j,p}}\nonumber\\
&&\qquad   -\o^2 \left(\rho_b  u_{y,j}^{s,h,n}, v^s\right)_{\O_{j,p}} 
- \o^2 \left(\rho_f  u_{y,j}^{f,h,n}, v^s\right)_{\O_{j,p}}
- \o^2 \left(\rho_f  u_{y,j}^{s,h,n}, v^f\right)_{\O_{j,p}}\nonumber\\
 &&\qquad +\left(N \nabla u^{s,h,n}_{y,j}),\nabla v^s\right)_{\O_{j,p}} 
 + i \o \left\langle (N^* b)^{1/2} u_{y,j}^{s,h,n}, v^s\right\rangle_{\G_j\cap\G_p} + \Sum_{k}\left\langle \left\langle  l^{h,n}_{jk}, v^s\right\rangle\right\rangle_{\G_{jk,p}}
\label{eq_dd_te1}\\
&&\qquad + \left(\eta (\kappa(\o))^{-1}\left[ i \o  u_{y,j}^{f,h,n} -  L(\o) E_{y,j}^{s,h,n}\right], v^f\right)_{\O_{j,p}}\nonumber\\
&&\qquad = -\left(\left[\sigma^s - \chi_{\O_p} L^2(\o) \eta (\kappa(\o))^{-1}\right]  E_y^p, \varphi\right)_{\O_j} \nonumber\\
&&\qquad \quad + \left( L(\o)\eta (\kappa(\o))^{-1} E_y^p, v^f\right)_{\O_{j,p}}, \quad 
 (\psi, \varphi, v^s, v^f)\in  {\mathcal V}^h_j\times {\mathcal W}^h_j\times {\mathcal NC}^h_j\times{\mathcal {\widehat M}_j}^h,\nonumber\\
&&\lambda^{h,n}_{jk} = \lambda^{h,n-1}_{kj} 
+\beta \left(H^{s,h,n}_j\cdot\chi_{jk}+ 
H^{s,h,n-1}_k\cdot\chi_{kj} \right), \quad \text{on} \quad \G_{jk},
\label{eq_dd_te2}\\
&&l^{h,n}_{jk} = l^{h,n-1}_{kj} 
+ i \o \beta \left(u^{s,h,n}_{y,j} +u^{s,h,n-1}_{y,k})\right)(\xi_{jk}), 
\quad \text{on} \quad \G_{jk,p}
\label{eq_dd_te3}.
\end{eqnarray}
In \eqref{eq_dd_te2},\eqref{eq_dd_te3} 
$\beta$ is a scalar positive  iteration parameter.

The convergence of  the solution of the iterative procedure \eqref{eq_dd_te1}-\eqref{eq_dd_te3}  to the solution of the {\it global} Galerkin procedure for the SHTE-mode \eqref{weak.1} can be shown with the ideas given in the proof of   \thmref{conv_dd}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\section{The triangular nonconforming element case}

Let $\overline \O =  \cup^J_{j=1}\overline \O_j$, with the  $\O_j's$ being
a quasiregular partition ${\mathcal T}^h$ of $\O$ into triangles; here, $\O$ can be a convex
polygon.  First we analyze the PSVTM-mode.
 
We will  employ the edge-constant  elements of Nedelec \cite{pmonk_94}  for the vector electric  field and the constant elements for the  scalar magnetic field. 
Thus we redefine the  spaces  ${\mathcal V}^h$ and ${\mathcal W}^h$ as follows.
\[
{\mathcal V}^h_j = \left\{\psi = \left( a + b z, c - b z\right), \quad a,b,c \quad   \text{constants on }  \quad    \O_j\right\},
\]
\begin{equation}\label{def_vh_tri}
{\mathcal V}^h = \left\{\psi\in H(\text{curl},\O): \psi|_{\O_j} \in  {\mathcal V}^h_j  \right\},
\end{equation}
\begin{equation}\label{def_wh_tri}
{\mathcal W}^h = \left\{\varphi \in L^2(\O): \varphi|_{\O_j} \in {\mathcal W}^h_j\equiv   P_0(\O_j)  \right\}.
\end{equation}
The degrees of freedom associated with  ${\mathcal V}$ are the tangential components of the  electric field  at the mid-points of each edge of  the triangles $\O_j$ , while for ${\mathcal W}^h$    are the values  of the magnetic field at the centroid of the triangles $\O_j$.

Also, as in the nonconforming  space  based on
rectangular elements to approximate each component of the solid displacement  vector , we wish again
to impose the continuity  at the midpoints $\xi_{jk}$ of the interior
interfaces $\G_{jk}$. Thus, in the triangular case we change
 the definition of the set ${\mathcal NC}^h$   to 
\begin{equation}\label{def_nch_tri}
{\mathcal NC}^h  = \left\{v | v_j = v|_{\O_j} \in {\mathcal NC}^h_j\equiv  [P_1(\O_j)], \, \O_j\subset \O_p, \,  v_j(\xi_{jk}) = v_k(\xi_{jk}),\, \forall \{j,k\}\right\}.
\end{equation}
Finally the  space ${\mathcal M}^h$ to approximate the fluid displacement vector 
is taken to be the lowest order  Raviart-Thomas-Nedelec space over  our triangulation. Thus, 
we redefine the  space  ${\mathcal M}^h$ as follows.
\begin{equation}\label{def_mhj_tri}
{\mathcal M}^h_j = \left\{v = \left( a + c  z,  b + c  z\right), \quad a,b,c \quad   \text{constants on }  \quad    \O_j\right\},
\end{equation}
\begin{equation}\label{def_mh_tri}
{\mathcal M}^h = \left\{v \in H(\text{div},\O):  v|_{\O_j} \in  {\mathcal M}^h_j  \right\}.
\end{equation}
As in the rectangular case, let 
\[
{\mathcal Y}^h = {\mathcal V}^h \times {\mathcal W}^h \times 
{\mathcal NC}^h\times {\mathcal M}^h.
\]   
Then the definition of the {\it global} Galerkin  and domain decomposition procedures \eqref{eq_101} and  \eqref{eq_181_182_184}-\eqref{eq_183}-\eqref{eq_185_dd} remain unchanged as well as the validity of the  {\it a priori}  error estimates derived in  \thmref{globalerror} and the convergence results for the domain decomposition iteration in \thmref{conv_dd}.
  
For the  SHTE-mode, we will employ  the spaces  ${\mathcal V}^h$ and  ${\mathcal W}^h$  defined \eqref{def_vh_tri} and \eqref{def_wh_tri}  to approximate the vector magnetic field $H^s$ and the 
scalar field $E^s_y$. Also,  the solid displacement $u^s_y$ will be approximated employing the space ${\mathcal NC}^h$ in \eqref{def_nch_tri}.  The fluid displacement will be computed using the space of  piecewise constants over the partition $\O_j$, i.e.,  
\begin{equation}\label{def_mh_tri_te}
{\widehat{\mathcal M}}^h = \left\{v \in L^2(\O_p): v|_{\O_j} \in {\mathcal M}^h_j\equiv   P_0(\O_j)  \right\}.
\end{equation}
Let 
\[
{\widehat{\mathcal Y}}^h = {\mathcal V}^h \times {\mathcal W}^h \times 
{\mathcal NC}^h \times {\widehat{\mathcal M}}^h.
\]
With this definitions, the formulation  of the  {\it global} finite element procedure \eqref{fe.1} and the iterative procedure 
\eqref{eq_dd_te1}-\eqref{eq_dd_te2}-\eqref{eq_dd_te3} are unchanged. Also the corresponding {\it a priori} error estimates  stated in \thmref{te_globalerror} and  the results on the    convergence of the domain decomposition algorithm stay valid. 

\section{Conclusions}
We presented and analyzed {\it global} and domain decomposed procedures for solving the system of partial differential equations describing the oscillatory  motion of coupled  electromagnetic and seismic waves in 2D isotropic fluid-saturated poroviscoelastic  media.
 The analysis included the derivation of {\it a priori} error 
 estimates for the {\it global} finite element procedures and   
 results  on the convergence of the domain decomposition iteration.  In 
 particular our iterative procedure is a new  approach to solve 
approximately this differential system  
containing a Helmholtz-type  part  associated with Biot's equations and a diffusive-type  
part related  to Maxwell's equations in the diffusive range. 
The model analyzed  includes the  particular cases of electroseismic, in which only electromagnetic sources are present, and seismoelectric, in which only seismic sources are present, both 
of recent interest in  exploration geophysics.


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