\documentclass[%a4paper 11pt %draft ,twoside ,reqno]{article} \usepackage{subeqnarray,amsmath,amssymb,amsthm,amsfonts,graphicx,epsfig,comment%,psfig }\pagestyle{myheadings} %\usepackage{datetime} \date{ %\today%: \currenttime } % descomentar esta linea para doble espacio \def\baselinestretch{2} \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{} \headheight 20mm \oddsidemargin 2.5mm \evensidemargin 2.5mm \topmargin -15mm \textheight 220mm \textwidth 160mm % Some definitions useful in producing this sort of documentation: \chardef\bslash=`\\ % p. 424, TeXbook \newcommand{\ntt}{\normalfont\ttfamily} \newcommand{\cn}[1]{{\protect\ntt\bslash#1}} \newcommand{\pkg}[1]{{\protect\ntt#1}} \newcommand{\fn}[1]{{\protect\ntt#1}} \newcommand{\env}[1]{{\protect\ntt#1}} \hfuzz1pc % Don't bother to report overfull boxes if overage is < 1pc % Theorem environments %\newenvironment{proof}{\noindent{\bf Proof:\ }}{\hspace*{\fill} \QED} \newtheorem{ax}{Axiom} \newtheorem{axiom}{Axiom} \newtheorem{Thm}{Theorem}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{thm}[Thm]{Theorem} \newtheorem{Lem}[Thm]{Lemma} \newtheorem{lemma}[Thm]{Lemma} \newtheorem{Prop}[Thm]{Proposition} \newtheorem{proposition}[Thm]{Proposition} %\theoremstyle{Remark} \newtheorem{Rmk}[Thm]{Remark} \newtheorem{remark}[Thm]{Remark} \newtheorem{Cor}[Thm]{Corollary} \newtheorem{corollary}[Thm]{Corollary} \newtheorem{Def}[Thm]{Definition} \newcommand{\thmrefs}[2]{Theorems~\ref{#1},~\ref{#2}} \newcommand{\thmref}[1]{Theorem~\ref{#1}} \newcommand{\secref}[1]{\S\ref{#1}} \newcommand{\lemref}[1]{Lemma~\ref{#1}} \newcommand{\rmkref}[1]{Remark~\ref{#1}} \newcommand{\figref}[1]{Figure~\ref{#1}} \numberwithin{equation}{section} \newcounter{saveeqn} \newcommand{\subeqn}{\setcounter{saveeqn}{\value{eqnarray}}% \stepcounter{saveeqn}\setcounter{eqnarray}{0}% \renewcommand{\theequation}{\mbox{\arabic{section}.\arabic{saveeqn}\alph {eqnarray}}}} %\alph, \roman \newcommand{\reseteqn}{\setcounter{eqnarray}{\value{saveeqn}}% \renewcommand{\theequation}{\arabic{section}.\arabic{eqnarray}}} \theoremstyle{definition} \newtheorem{defn}{Definition}[section] \theoremstyle{remark} \newtheorem{rem}{Remark}[section] \newtheorem*{notation}{Notation} \newcommand{\be}{\begin{eqnarray*}} \newcommand{\ee}{\end{eqnarray*}} \newcommand{\bes}{\begin{eqnarray*}} \newcommand{\ees}{\end{eqnarray*}} \newcommand{\bs}{\begin{subeqnarray*}} \newcommand{\es}{\end{subeqnarray*}} \newcommand{\bss}{\begin{eqnarray*}} \newcommand{\ess}{\end{eqnarray*}} \newcommand{\interval}[1]{\mathinner{#1}} \newcommand{\eval}[2][\right]{\relax \ifx#1\right\relax \left.\fi#2#1\rvert} \newcommand{\envert}[1]{\left\lvert#1\right\rvert} \let\abs=\envert \newcommand{\enVert}[1]{\left\lVert#1\right\rVert} \let\norm=\enVert %\renewcommand{\baselinestretch}{1.3} \renewcommand\tilde{\widetilde} \renewcommand\a{\alpha} \def\b{\beta} \newcommand\bA{\mathbf{A}} \def\bmu{\boldsymbol\mu} %\newcommand\bysame{\rule[-1pt]{.75in}{.4pt}} \renewcommand\d{\delta} \newcommand\D{\Delta} \def\dfrac{\displaystyle\frac} \def\Sum{\displaystyle\sum} \def\ai{\left\langle\langle} \def\ad{\right\rangle\rangle} \def\b{\beta} \def\k{\kappa} \def\l{\ell} \def\D{\Delta} \def\d{\delta} \def\e{\varepsilon} \def\dint{\displaystyle\int} \def\g{\gamma} \def\G{\Gamma} \def\h{{\mathbf h}} \def\hf{ f} \def\hR{\widehat R} \def\hu{ u} \def\hz{\widehat z} \def\hZ{\widehat Z} \def\Im{\operatorname{Im}} \def\v{{\mathbf v}} \def\H{{\mathbf H}} \def\n{{\mathbf n}} \def\R{{\mathbb R}} \def\b{\beta} \def\lla{\langle\langle} \def\Lam{\Lambda} \def\lam{\lambda} \def\some{{x,y,z}} \def\Lip{\operatorname{Lip}} \def\tr{\operatorname{tr}} \def\curl{{\nabla\times}} \def\div{{\nabla\cdot}} \def\opcurl{\operatorname{curl}} \def\opdiv{\operatorname{div}} \def\nab{{\nabla}} \def\grad{{\nabla}} \def\g{\gamma} \def\q{\quad} \def\2q{\quad\quad} \def\3q{\quad\quad\quad} \def\s{\sigma} \def\sG{\mathcal{G}} \def\sL{\mathcal{L}} \def\sB{\mathcal{B}} \def\sA{\mathcal{A}} \def\cV{\mathcal{V}} \def\sm{\setminus} \def\NC{\mathcal{NC}} \def\sG{\mathcal{G}} \def\sm{\setminus} \def\t{\theta} \def\T{\Theta} \def\Tau{\mathcal T} \def\talpha{\widetilde\alpha} \def\tA{\tilde A} \def\tB{\tilde B} \def\tC{\tilde C} \def\V{\Vert} \def\z{\zeta} \def\xo{(x,\omega)} \def\<{\left\langle} \def\>{\right\rangle} \def\div{\nabla\cdot} \def\grad{\nabla} \def\lma{\left\langle} \def\la{\left\langle} \def\lla{\langle\langle} \def\rra{\rangle\rangle} \def\rma{\right\rangle} \def\ra{\right\rangle} \def\qq{\quad\quad} \def\dt{{\varDelta t}} \def\dx{{\varDelta x}} \def\dy{{\varDelta y}} \def\dz{{\varDelta z}} \def\BbbR{\mathbb R} \def\O{\Omega} \def\o{\omega} \def\Oh{{\mathcal O}} \def\mD{{\mathcal D}} \def\mS{{\mathcal S}} \def\mT{{\mathcal T}} \def\mC{{\mathcal C}} \def\mN{{\mathcal N}} \def\p{\partial} \def\for{\,\forall} \def\a{\alpha} \def\nb{\nonumber} \def\var{\varphi} \def\w{\omega} \def\G{\Gamma} \def\R{{\mathbb R}} \def\:{{\,:\,}} \def\Set#1{{\left\{#1\right\}}} \def\norm#1{{\left\|#1\right\|}} \def\qnorm#1{{\left|\left|\left|#1\right|\right|\right|}} \def\abs#1{{\left|#1\right|}} \def\th#1{\theta_{jk}^{#1}} \renewcommand{\hat}{\widehat} \newcommand{\pp}[2]{\frac{\partial {#1}}{\partial {\nu_{#2}}}} \newcommand{\pu}[2]{P_{\tau}u_{#1}^{#2}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \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}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% SECTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% SECTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% SECTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% SECTION %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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. 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