#  A new variational formulation with high order impedance boundary condition
for the scattering problem in electromagnetism

Soumaya Oueslati<EMAIL_ADDRESS>Department of Mathematics, CNRS(UMR
8088), University CY Cergy Paris,2 avenue Adolphe Chauvin, 95302 Cergy-
Pontoise, France Christian Daveau<EMAIL_ADDRESS>Department of
Mathematics, CNRS(UMR 8088), University CY Cergy Paris,2 avenue Adolphe
Chauvin, 95302 Cergy-Pontoise, France Abil Aubakirov<EMAIL_ADDRESS>Department of Mathematics, CNRS(UMR 8088), University CY Cergy Paris,2 avenue
Adolphe Chauvin, 95302 Cergy-Pontoise, France

###### Abstract

In this paper, we propose some variational formulations with the use of high
order impedance boundary condition (HOIBC) to solve the scattering problem. We
study the existence and uniqueness of the solution. Then, a discretization of
these formulations is done. We give validations of the HOIBC obtained with a
MoM code that show the improvement in accuracy over the standard impedance
boundary condition (SIBC) computations.

Keywords: boundary element method, scattering problem, Lagrange multipliers,
high order impedance boundary condition.

##  1 Introduction

The paper deals with computational electromagnetic to model thin coatings on
perfect conducting objects. The method of integral equations is often used to
calculate the far field radiated by a homogeneous object [1]. It allows to set
the equations on the surface of the scattering obstacle which lowers the
dimension of the problem. In counterpart, the equations are non local and then
lead after discretization to dense linear systems. This method reduces
drastically the number of unknowns compared to other finite elements methods
where thin coatings is meshed.
We sometimes use with integral methods the perfectly conducting assumption
neglecting the losses. However, it is important to model the losses and finite
conductivity of the metal by an impedance boundary condition. This impedance
boundary condition defines a relationship between the tangential electric and
magnetic fields on a surface. Usually, the impedance of the coating is assumed
to be the same for all incidence angles and polarizations. It is called
standard impedance boundary condition or Leontovicht condition. We can see the
papers [2, 30] about it, but the list is not exhaustive.
This assumption is valid for thin coatings with high refractive index or
significant losses. However, it is known to be poorly efficient for coatings
with smaller refractive index or lower losses [19]. In this framework, some
approximations of the HOIBC are proposed in [11, 22, 23] which involve at most
a first derivative of the field. Other strategies have also been established
in [26, 3, 4, 5, 7] that based on a polynomial approximation of the impedance
operator is done in the spectral domain. Then, the tangential traces of
electric and magnetic fields are related with the incident angle at each point
of the surface since the approximation of the impedance is defined as a ratio
of polynomials of the sine of the incidence angle for each polarization.
Numerical results for two-dimensional cylinders with lossy and lossless
dielectric coatings and for three-dimensional bodies of revolution is given in
these papers. Later, in [8] sufficient uniqueness conditions (SUC) have been
established for the solutions of Maxwell’s equations associated withe the
particular HOIBCs proposed in [5].

This paper deals with the implementation of high order impedance boundary
condition implemented in an integral equation. An outline of the article is as
follows: Section 2 presents the physical model and the framework. In section
4, we give an approximation of the impedance operator with Hodge operator and
we give approximation of the impedance boundary condition. To my knowledge,
this is the first time that the impedance condition is written in this form.
In section 5, we determine SUC for the solutions of Maxwell’s equations
associated with the impedance conditions. In the following, we derive a
formulation and in section 8.1 we prove the existence and uniqueness of a
solution for this variational formulation. Sections 8.2 and
LABEL:Lagrange_multipliers is devoted to the description of the
discretization. Numerical results are presented in section 8.3 and in section
9 concluding remarks are done.

##  2 Mathematical model of physical problem

In this section, we recall the model problem on which we work. We consider the
scattering problem of electromagnetic waves by a perfect conducting body with
a complex coating. Let $\Omega$ be a bounded domain with a Lipschitz-
continuous boundary $\Gamma$. Let $n$ be the unit normal vector to $\Gamma$
directed to the exterior of $\Omega$. We define $\Omega_{+}$ as the space of
radiating fields $(\textbf{E},\textbf{H})$ solutions of Maxwell equations. The
total electromagnetic fields $(\textbf{E},\textbf{H})$ in $\Omega_{+}$
therefore have the expression:

$\left\\{\begin{array}[]{llll}\mathbf{E}=\mathbf{E}^{inc}+\mathbf{E}^{sc}\\\
\mathbf{H}=\mathbf{H}^{inc}+\mathbf{H}^{sc}\\\ \end{array}\right.$

wehre $(\textbf{E}^{inc},\textbf{H}^{inc})$ is incident fields and the fields
$(\textbf{E}^{sc},\textbf{H}^{sc})$ scattered by the object $\Omega$.

Figure 1: Equivalent impedance

While the coating is replaced by the following impedance boundary condition on
surface $\Gamma$, (see Figure 1):

$\textbf{E}_{tg}-Z(n\times\textbf{H})=0\,\,\,\,\,on\,\,\,\,\Gamma,$ (1)

where $Z$ is the impedance operator, subscript tg denotes tangent component on
the surface defined as:

$\textbf{E}_{tg}=n\times(\textbf{E}\times n).$

An approximation of this impedance operator is given in the Section 4, which
depends on layer thickness, on dielectric characteristics of medium of this
layer. Also it depends on incident angle of incident electromagnetic plane
wave.

Waves propagate with constant wave number $k$ in the exterior unbounded domain
$\Omega$. Clearly that the electromagnetic fields $(\textbf{E},\textbf{H})$ is
solution the following problem:

######  Problem 2.1.

$\left\\{\begin{array}[]{llll}\nabla\times(\nabla\times\textbf{E})-k^{2}\textbf{E}=0\,\,\,\,\,\text{in}\,\,\,\Omega_{+},\\\
\nabla\times(\nabla\times\textbf{H})-k^{2}\textbf{H}=0\,\,\,\,\,\text{in}\,\,\,\Omega_{+},\\\
\textbf{E}_{tg}-Z(n\times\textbf{H})=0\,\,\,\,\,on\,\,\,\,\Gamma.\\\
\lim_{r\rightarrow\infty}r(\mathbf{E}\times\mathbf{n}_{r}+\mathbf{H})=0.\end{array}\right.$
(2)

where $r=|\mathbf{x}|$ and $\mathbf{n}_{r}=\dfrac{\mathbf{x}}{|\mathbf{x}|}$.

Now, we shall show that problem 2.1 is well posed, using the result:

######  Lemma 2.1.

(Rellich): Let $\Omega$ be the open complement of a closed domain and $u\in
L^{2}(\mbox{\bf R}^{3}-\Omega)$ is a solution of the Helmholtz equation,
satisfying

$\lim_{r\rightarrow\infty}\int_{|x|=r}|u(x)|^{2}dx=0.$ (3)

Then $u=0$, in $\mbox{\bf R}^{3}-\Omega$.

######  Theorem 2.1.

The problem 2.1 admits a unique solution, if following relations are verified:

$\begin{cases}\Im(\mu)\leq 0,\\\ \Im(\epsilon)\leq 0,\\\
\Re(k_{0}\int_{\Gamma}\mathbf{E}^{*}\cdot(\mathbf{n}\times\mathbf{H})ds)\geq
0.\end{cases}$

###### Proof.

This theorem is an adaptation of uniqueness that is determined in [8]. It
gives us sufficient condition on characteristics of medium to get unique
solution of the problem 2.1. That is how we get sufficient uniqueness
condition (SUC) in general form

$\Re(k_{0}\int_{\Gamma}(\mathbf{n}\times\mathbf{H})\cdot\textbf{E}^{*})\geq 0$
(4)

∎

##  3 Framework

We present preliminary results concerning some basic functional spaces and
operators which will be used throughout this work.

###  3.1 Notation and spaces

######  Definition 3.1.

Let be $L_{D}$ and $L_{R}$ operators defined for all vector function
$\mathbf{A}$ sufficiently smooth, such that $\mathbf{A}\cdot\mathbf{n}=0$ by:

$L_{D}(\mathbf{A})=\nabla_{\Gamma}(\mathrm{div}_{\Gamma}\mathbf{A}),\ \ \ \
L_{R}(\mathbf{A})=\mathbf{\mathrm{rot}}_{\Gamma}(\mathrm{rot}_{\Gamma}\mathbf{A}).$
(5)

Now, we consider some the following Sobolev spaces on the surface $\Gamma$:

  1. 1.

$H^{-1/2}(\mathrm{div},\Gamma)$ defined by

$H^{-1/2}(\mathrm{div},\ \Gamma)=\\{\mathbf{g}\in(H^{-1/2}(\Gamma))^{3},\
\mathbf{g}\cdot\mathbf{n}=0,\
\mathrm{div}_{\Gamma}(\mathbf{g})\in(H^{-1/2}(\Gamma))^{3}\\}.$

It is an Sobolev spaces equipped with the norm:

$\|\mathbf{g}\|_{-1/2,\mathrm{div}_{\Gamma}}=\left(\|\mathbf{g}\|^{2}_{-1/2,H(\Gamma)}+\|\mathrm{div}_{\Gamma}(\mathbf{g})\|^{2}_{-1/2,H(\Gamma)}\right)^{1/2}$

  2. 2.

And also the classical space $H^{-1/2}(\mathrm{rot},\Gamma)$ defined on the
boundary of $\Omega$

$H^{-1/2}(\mathrm{rot},\ \Gamma)=\\{\mathbf{g}\in H^{-1/2}(\Gamma)^{3},\
\mathbf{g}\cdot\mathbf{n}=0,\ \mathbf{rot}_{\Gamma}(\mathbf{g})\in
H^{-1/2}(\Gamma)^{3}\\}$

equipped with the norm denoted by

$\|\mathbf{g}\|_{-1/2,\mathrm{rot}_{\Gamma}}=\left(\|\mathbf{g}\|^{2}_{-1/2,H(\Gamma)}+\|\mathbf{rot}_{\Gamma}(\mathbf{g})\|^{2}_{-1/2,H(\Gamma)}\right)^{1/2}$

######  Proposition 3.1.

We have duality relations:

$\left(H^{-1/2}(\mathrm{rot},\ \Gamma)\right)^{\prime}=H^{-1/2}(\mathrm{div},\
\Gamma)\ \ and\ \ \left(H^{-1/2}(\mathrm{div},\
\Gamma)\right)^{\prime}=H^{-1/2}(\mathrm{rot},\ \Gamma)$

and $\forall\mathbf{f}\in H^{-1/2}(\mathrm{div},\ \Gamma);\ \
\forall\mathbf{g}\in H^{-1/2}(\mathrm{rot},\ \Gamma)$ we have

$\|\mathbf{f}\|_{(-1/2,\mathrm{rot}_{\Gamma})^{\prime}}=\sup_{\mathbf{g}\neq
0\in
H^{-1/2}(\mathrm{rot},\Gamma)}{\frac{<\mathbf{f},\mathbf{g}>}{\|\mathbf{g}\|_{-1/2,\mathrm{rot}_{\Gamma}}}}=\|\mathbf{f}\|_{-1/2,\mathrm{div}_{\Gamma}}$
$\|\mathbf{g}\|_{(-1/2,\mathrm{div}_{\Gamma})^{\prime}}=\sup_{\mathbf{f}\neq
0\in
H^{-1/2}(\mathrm{div},\Gamma)}{\frac{<\mathbf{g},\mathbf{f}>}{\|\mathbf{f}\|_{-1/2,\mathrm{div}_{\Gamma}}}}=\|\mathbf{g}\|_{-1/2,\mathrm{rot}_{\Gamma}}$

######  Theorem 3.1.

We define a linear application $\gamma_{0}$ by:

$\forall\mathbf{u}\in H^{1}(\Omega),\ \
\gamma_{0}\mathbf{u}=\mathbf{u}_{|_{\Gamma}}$

Then $\gamma_{0}$ is continuous from $H^{1}(\Omega)$ to $H^{1/2}(\Gamma)$
equipped with the norm and also we have:

$\forall\mathbf{u}\in
H^{1}(\Omega),\,\,\|\gamma_{0}\mathbf{u}\|_{1/2,H(\Gamma)}\leq
C(\Gamma)\|\mathbf{u}\|_{H^{1}(\Omega)}$

$\forall\,\,\mathbf{f}\in H^{1/2}(\Gamma),\ \ \exists\mathbf{u}\in
H^{1}(\Omega)$ such that

$\mathbf{f}=\gamma_{0}\mathbf{u}\ \ and\ \ \|\mathbf{u}\|_{H^{1}(\Omega)}\leq
C(\Gamma)\|\mathbf{f}\|_{1/2,H(\Gamma)}$

###### Proof.

: See [31] pp.110-113. ∎

######  Theorem 3.2.

We define linear application $\gamma_{n}$ as:

$\forall\mathbf{u}\in[H^{1}(\Omega)]^{3},\ \
\gamma_{n}\mathbf{u}=-\mathbf{n}\times\mathbf{u}_{|_{\Gamma}}$

Then, $\gamma_{n}$ extends in unique form to a continuous linear application
from $H(\mathrm{rot},\Omega)$ to $H^{-1/2}(\mathrm{div},\Gamma)$ and also we
have:

$\forall\mathbf{u}\in H(\mathrm{rot},\Omega),\ \
\|\gamma_{n}\mathbf{u}\|_{-1/2,\mathrm{div}_{\Gamma}}\leq
C(\Gamma)\|\mathbf{u}\|_{H(\mathrm{rot},\Omega)}$

$\forall\mathbf{f}\in H^{-1/2}(\mathrm{div},\Gamma),\ \exists\mathbf{u}\in
H(\mathrm{rot},\Omega)$ such that

$\mathbf{f}=\gamma_{n}\mathbf{u}\ \ and\ \
\|\mathbf{u}\|_{H(\mathrm{rot},\Omega)}\leq
C(\Gamma)\|\mathbf{f}\|_{-1/2,\mathrm{div}_{\Gamma}}$

###### Proof.

: See [31] pp.124-125 ∎

######  Theorem 3.3.

We define linear application $\Pi_{\Gamma}$ as:

$\forall\mathbf{u}\in[H^{1}(\Omega)]^{3},\ \
\Pi_{\Gamma}\mathbf{u}=-\mathbf{n}\times(\mathbf{n}\times\mathbf{u}_{|_{\Gamma}})$

Then, $\Pi_{\Gamma}$ extends in unique form to a continuous linear application
from $H(\mathrm{rot},\Omega)$ to $H^{-1/2}(\mathrm{rot},\Gamma)$ and also we
have:

$\forall\mathbf{u}\in H(\mathrm{rot},\Omega),\ \
\|\Pi_{\Gamma}\mathbf{u}\|_{-1/2,H(\mathrm{rot},\Gamma)}\leq
C(\Gamma)\|\mathbf{u}\|_{H(\mathrm{rot},\Omega)}$

$\forall\mathbf{g}\in H^{-1/2}(\mathrm{rot},\Gamma),\ \exists\mathbf{u}\in
H(\mathrm{rot},\Omega)$ such that

$\mathbf{g}=\Pi_{\Gamma}\mathbf{u}\ \ and\ \
\|\mathbf{u}\|_{H(\mathrm{rot},\Omega)}\leq
C(\Gamma)\|\mathbf{g}\|_{-1/2,\mathrm{rot}_{\Gamma}}$

###### Proof.

: See [31] pp.125-126 ∎

###  3.2 Properties of integral operators

Some solutions of the problems of diffraction of electromagnetic waves can be
expressed with the help of potential and in particular the potential of single
and dual layer defined on the surface of the obstacle. We present here the
properties of the harmonic potential.

######  Definition 3.2.

We introduce the potentials $(B-S)$ and $Q$, that are defined by

$(B-S)\mathbf{J}:=\int_{\Gamma}\left(G(x,y)\mathbf{J}(y)+\frac{1}{k^{2}}\nabla_{x}G(x,y)\mathrm{div}_{\Gamma}\mathbf{J}\right)d\Gamma(y)$
(6) $Q\mathbf{M}:=\int_{\Gamma}\nabla_{y}G(x,y)\times\mathbf{M}d\Gamma(y)$ (7)

and $G(x,y)$ is the Green kernel giving the outgoing solutions to the scalar
Helmholtz equation.

We have some results [29],

######  Theorem 3.4.

The operator $Q$ is continuous from $H^{-1/2}(\mathrm{div},\Gamma)$ to
$H^{-1/2}(\mathrm{rot},\Gamma)$ and we have that:

$|(\mathbf{n}\times Q+\frac{I}{2})\mathbf{M}|_{-1/2,\mathrm{div}_{\Gamma}}\leq
C|\mathbf{M}|_{-1/2,\mathrm{div}_{\Gamma}}\ \ \forall\mathbf{M}\in
H^{-1/2}(\mathrm{div},\Gamma)$ (8)

######  Theorem 3.5.

The operator $(B-S)$ is an isomorphisme from $H^{-1/2}(\mathrm{div},\Gamma)$
to $H^{-1/2}(\mathrm{rot},\Gamma)$ and it verifies the inequality:

$\|(B-S)\bm{\phi}\|_{-1/2,\mathrm{rot}_{\Gamma}}\leq
C\|\bm{\phi}\|_{-1/2,\mathrm{div}_{\Gamma}}$ (9)

and the coercivity relation $\forall\bm{\phi}\in
H^{-1/2}(\mathrm{div},\Gamma)$:

$\Re(<\bm{\phi},(B-S)\bm{\phi}>)\geq
C\|\bm{\phi}\|^{2}_{-1/2,\mathrm{div}_{\Gamma}}$ (10)

##  4 Obtaining of the higher order impedance boundary

Generally, we take the constant impedance operator, known as standard or
Leontovitch impedance boundary conditions [26, 28, 29]. Here, we are
interested in approximation of the impedance operator which depends on
incident angle. We consider a plane isotropic medium with a local orthogonal
basis (x,y,z) on a tangent plane where a normal vector n is in z-direction and
(x,y) is a tangent plane [26, 4]. The exact impedance tensor is defined
obtained for the dielectric plane layer [26] by:

$Z_{xy}(k_{x},k_{y})=Z_{yx}(k_{x},k_{y})=i\sqrt{\frac{\mu}{\epsilon}}\dfrac{k_{x}k_{y}}{kk_{z}}\tan[k_{z}d],$
(11)
$Z_{xx}(k_{x},k_{y})=-i\sqrt{\frac{\mu}{\epsilon}}\dfrac{k^{2}_{x}k^{2}_{z}+k^{2}_{y}k^{2}}{kk_{z}(k^{2}_{x}+k^{2}_{y})}\tan[k_{z}d],$
(12)

and

$Z_{yy}(k_{x},k_{y})=-i\sqrt{\frac{\mu}{\epsilon}}\dfrac{k^{2}_{y}k^{2}_{z}+k^{2}_{x}k^{2}}{kk_{z}(k^{2}_{y}+k^{2}_{x})}\tan[k_{z}d],$
(13)

$Z$ is a impedance tensor of wave numbers ($k_{x}$, $k_{y}$), wave frequency
and coating at each point of the surface. In [26] a method based on a spectral
domain approach is presented to derive high order impedance boundary
condition. The spectral domain approach is applicable to complex coatings and
surface treatments as well as simple dielectric. So, the impedance boundary
condition is approximated as a ratio of second order polynomials for a coating
invariant under rotation. Then they transform the polynomial approximation of
the spectral domain boundary condition into the spatial domain using
elementary properties of the Fourier transform. Then they get

$(1+b_{1}\partial^{2}_{x}+b_{2}\partial^{2}_{y})E_{x}+(b_{1}-b_{2})\partial^{2}_{xy}E_{y}=(a_{1}-a_{2})\partial^{2}_{xy}H_{x}-(a_{0}+a_{1}\partial^{2}_{x}+a_{2}\partial^{2}_{y})H_{y},$
(14)

and

$(b_{1}-b_{2})\partial^{2}_{xy}E_{x}+(1+b_{2}\partial^{2}_{x}+b_{1}\partial^{2}_{y})E_{y}=(a_{0}+a_{2}\partial^{2}_{x}+a_{1}\partial^{2}_{y})H_{x}+(a_{2}-a_{1})\partial^{2}_{xy}H_{y}.$
(15)

where $E_{tg}=(E_{x},E_{y})$ and $H_{tg}=(H_{x},H_{y})$.

###  4.1  The impedance boundary condition form in 3D case

We use formula (5) to rewrite the equations (14) and (15) with operators
$L_{D}$ and $L_{R}$ as following equations:

$\mathbf{x}\cdot(\mathbf{E}_{tg}+b_{1}L_{D}\mathbf{E}_{tg}-b_{2}L_{R}\mathbf{E}_{tg})=\mathbf{x}\cdot(a_{0}(\mathbf{n}\times\mathbf{H})+a_{1}L_{D}(\mathbf{n}\times\mathbf{H})-a_{2}L_{R}(\mathbf{n}\times\mathbf{H})),$

and

$\mathbf{y}\cdot(\mathbf{E}_{tg}+b_{1}L_{D}\mathbf{E}_{tg}-b_{2}L_{R}\mathbf{E}_{tg})=\mathbf{y}\cdot(a_{0}(\mathbf{n}\times\mathbf{H})+a_{1}L_{D}(\mathbf{n}\times\mathbf{H})-a_{2}L_{R}(\mathbf{n}\times\mathbf{H})).$

Then, we obtain the following approximation of the high order impedance
boundary condition:

$Z_{3D}:\ \
(I+b_{1}L_{D}-b_{2}L_{R})\textbf{E}_{tg}=(a_{0}I+a_{1}L_{D}-a_{2}L_{R})(\mathbf{n}\times\textbf{H}).$
(16)

###  4.2  The impedance boundary condition form in 2D case

Here we need to consider two different situations. In the first one, we
consider case when the electric field is perpendicular to the incident plane,
as shown on (Figure 2 a). Incident, scattered and transverse electric fields
are directed toward the viewer. The direction of magnetic field was chosen
such that energy current has positive direction, i.e. direction of wave
propagation. We call this case, transverse-electric (TE) polarization. In the
second case electric fields are parallel to incident plane, as shown in
(Figure 2 b). In this case we call it transverse-magnetic (TM) polarization.

Figure 2: Reflection and refraction with (a) TE and (b) TM polarizations.

We assume that the incident fields propagate perpendicular to the cylinder
axis, so that $\partial/\partial y=0$. These relations (14) and (15) are
written in matrix form:

$\left(\begin{array}[]{cc}1+b_{1}\partial^{2}_{x}&0\\\
0&1+b_{2}\partial^{2}_{x}\end{array}\right)\left(\begin{array}[]{c}E_{x}\\\
E_{y}\end{array}\right)=\left(\begin{array}[]{cc}a_{0}+a_{1}\partial^{2}_{x}&0\\\
0&a_{0}+a_{2}\partial^{2}_{x}\end{array}\right)\left(\begin{array}[]{c}-H_{y}\\\
H_{x}\end{array}\right)$ (17)

We get first order approximation of impedance in two dimensional cases for
each polarization

$Z_{2Dj}:\ \
(1+b_{j}\partial^{2}_{x})\mathbf{E}_{tg}=(a_{0}+a_{j}\partial^{2}_{x})\mathbf{n}\times\mathbf{H}$
(18)

where $j=1,2$ correspond to TE and TM polarizations, respectively. In the same
manner, the equation (18) can be extended to second order polynomials in
$\partial^{2}_{x}$. Also, we obtain the following approximation of the high
order impedance boundary in two dimensional:

$\mathbf{E}_{tg}+b_{j}\partial^{2}_{x}\mathbf{E}_{tg}+b^{\prime}_{j}\partial^{4}_{x}\mathbf{E}_{tg}=a_{0}(\mathbf{n}\times\mathbf{H})+a_{j}\partial^{2}_{x}(\mathbf{n}\times\mathbf{H})+a^{\prime}_{j}\partial^{4}_{x}(\mathbf{n}\times\mathbf{H})$
(19)

or it can be reduced to constant:

$\mathbf{E}_{tg}=a_{0}(\mathbf{n}\times\mathbf{H}).$ (20)

We will call the equation (18) as first order IBC (IBC1). And we will call the
equation (19) second order IBC (IBC2), the equation (20) zeroth order IBC
(IBC0), which is also known as Leontovich IBC.

##  5 Study of Maxwell’s equations with HOIBC

We want to establish SUC for the solutions of Maxwell’s equations associated
with these IBCs, using of the uniqueness theorem mentioned earlier. We begin
this section by recalling the Maxwell’s equations

######  Problem 5.1.

$\left\\{\begin{array}[]{llll}\nabla\times(\nabla\times\textbf{E})-k^{2}\textbf{E}=0\,\,\,\,\,\text{in}\,\,\,\Omega_{+},\\\
\nabla\times(\nabla\times\textbf{H})-k^{2}\textbf{H}=0\,\,\,\,\,\text{in}\,\,\,\Omega_{+},\\\
\lim_{r\rightarrow\infty}r(\mathbf{E}\times\mathbf{n}_{r}+\mathbf{H})=0.\end{array}\right.$
(21)

On the first time, we are going to study the problem 5.1 associated with a
more complicated boundary condition, that is called first order IBC1:

$\mathbf{E}_{tg}+b_{j}L(\mathbf{E}_{tg})=a_{0}(\mathbf{n}\times\mathbf{H})+a_{j}L(\mathbf{n}\times\mathbf{H})\
\ on\ \Gamma,$ (22)

where $L$ is a complex differential operator, also known as Hodge operator. We
assume that

$\begin{cases}(\mathbf{n}\times\mathbf{H})\cdot\tau=(\mathbf{n}\times\mathbf{H})\cdot\nu=0\
\ \text{on}\ \partial{\Gamma}\ \ \text{if}\ \ \Gamma\ \ \text{is}\ \
\text{open},\\\ \mathbf{E}_{tg}\cdot\tau=\mathbf{E}_{tg}\cdot\nu=0\ \
\text{on}\ \partial{\Gamma}\ \ \text{if}\ \ \Gamma\ \ \text{is}\ \
\text{open}.\\\ \end{cases}$ (23)

We have the following result

######  Theorem 5.1.

The Maxwell problem (5.1) with the boundary conditions (22) and (23) has a
unique solution if the following relations are verified

$\begin{cases}\Im(\mu)\leq 0,\\\ \Im(\epsilon)\leq 0,\\\
a_{j}-b_{j}^{*}a_{0}\neq 0,\\\ \Re(a_{j}-b_{j}^{*}a_{0})=0,\\\
\Im(a_{0}^{*}a_{j})\Im(a_{j}-b_{j}^{*}a_{0})\geq 0,\\\
\Im(b_{j})\Im(a_{j}-b_{j}^{*}a_{0})\geq 0.\end{cases}$ (24)

###### Proof.

See [8]. ∎

On the other hand, we study yet another more complicated HOIBC, which is
called second-order IBC that increases power of Hodge operator. For the SUC of
the problem (5.1) with IBC2, we assume that

$\mathbf{E}_{tg}+b_{j}L(\mathbf{E}_{tg})+b^{\prime}_{j}L^{2}(\mathbf{E}_{tg})=a_{0}(\mathbf{n}\times\mathbf{H})+a_{j}L(\mathbf{n}\times\mathbf{H})+a^{\prime}_{j}L^{2}(\mathbf{n}\times\mathbf{H})\
\ on\ \Gamma,$ (25)

where $L^{2}(\cdot)=L\circ L(\cdot)$. And all coefficients
$a_{j},a^{\prime}_{j},b_{j}$ and $b^{\prime}_{j}$ are defined locally and
depend on incident angle. With a next conditions on a bound

$\begin{cases}(\mathbf{n}\times\mathbf{H})\cdot\tau=(\mathbf{n}\times\mathbf{H})\cdot\nu=0\
\ on\ \partial{\Gamma}\ \ if\ \ \Gamma\ \ is\ \ open,\\\
\mathbf{E}_{tg}\cdot\tau=\mathbf{E}_{tg}\cdot\nu=0\ \ on\ \partial{\Gamma}\ \
if\ \ \Gamma\ \ is\ \ open\\\
X_{1}\cdot\nu=X_{2}\cdot\nu=Y_{1}\cdot\nu=Y_{2}\cdot\nu=0\ \ on\
\partial{\Gamma}\ \ if\ \ \Gamma\ \ is\ \ open\\\
X_{1}\cdot\tau=X_{2}\cdot\tau=Y_{1}\cdot\tau=Y_{2}\cdot\tau=0\ \ on\
\partial{\Gamma}\ \ if\ \ \Gamma\ \ is\ \ open.\end{cases}$ (26)

where

$\begin{cases}X_{1}=\nabla_{\Gamma}\mathrm{div}_{\Gamma}\mathbf{E}_{tg}=L_{D}\mathbf{E}_{tg}\\\
X_{2}=\nabla_{\Gamma}\mathrm{div}_{\Gamma}(\mathbf{n}\times\mathbf{H})=L_{D}(\mathbf{n}\times\mathbf{H})\\\
Y_{1}=\mathbf{rot}(\mathbf{n}(\mathrm{rot}_{\Gamma}\mathbf{E}_{tg}))=L_{R}\mathbf{E}_{tg}\\\
Y_{2}=\mathbf{rot}(\mathbf{n}(\mathrm{rot}_{\Gamma}(\mathbf{n}\times\mathbf{H})))=L_{R}(\mathbf{n}\times\mathbf{H})\end{cases}$
(27)

So for the IBC2 we have following uniqueness theorem

######  Theorem 5.2.

The Maxwell problem (5.1) with a boundary condition (25), (26) and (27) admits
a unique solution if next relations are verified

$\begin{cases}\Im(\mu)\leq 0,\\\ \Im(\epsilon)\leq 0,\\\ \Delta\neq 0,\\\
\Re[\Delta^{*}(a_{j}b^{\prime*}_{j}-a^{\prime}_{j}b_{j}^{*})]=0,\\\
\Re[\Delta^{*}(a^{\prime}_{j}-a_{0}b^{\prime*}_{j})]=0,\\\
\alpha\Im{b^{\prime}_{j}}+\beta\Im(b_{j}b^{\prime*}_{j})\leq 0,\\\
\alpha\Im(a^{\prime}_{j}a_{0}^{*})-\beta\Im(a^{\prime}_{j}a_{j}^{*})\leq 0,\\\
-\alpha\Im{b_{j}}+\beta\Im{b^{\prime}_{j}}\leq 0,\\\
\alpha\Im(a_{j}a_{0}^{*})-\beta\Im(a^{\prime}_{j}a_{0}^{*})\geq 0\end{cases}$
(28)

with

$\Delta=(a_{0}b_{j}^{*}-a_{j})(a_{j}b^{\prime*}_{j}-a^{\prime}_{j}b_{j}^{*})-(a^{\prime}_{j}-a_{0}b^{\prime*}_{j})^{2}$
$\alpha=\Im[\Delta^{*}(a_{j}b^{\prime*}_{j}-a^{\prime}_{j}b_{j}^{*})]$
$\beta=\Im[\Delta^{*}(a_{0}b^{\prime*}_{j}-a^{\prime}_{j})]$

###### Proof.

see [8]. ∎

All these three theorems are based on the theorem 2.1 and its condition

$\Re(k_{0}\int_{\Gamma}\mathbf{E}^{*}\cdot(\mathbf{n}\times\mathbf{H})ds)\geq
0.$

So, to verify this condition, coefficients of HOIBC should verify conditions
mentioned in these theorems.

##  6 Calculation of the coefficients

###  6.1 The coefficients of the IBC1

Now, we propose some different methods to calculate these coefficients
$(a_{0},a_{j},b_{j})$. The impedance $Z_{2Dj}$ (18) is the following rational
function of $k^{2}_{x}$

$Z_{2Dj}=\frac{a_{0}-a_{j}k^{2}_{x}}{1-b_{j}k^{2}_{x}},\ \ j=1,2$ (29)

The coefficients indicated by $j=1,2$ correspond to polarizations TE and TM
respectively. From (11) and (12) we can express exact impedance for TE and TM
polarization:

$Z^{ex}_{TE}=\sqrt{\frac{\mu}{\epsilon}}\frac{k_{z}}{k}\tan{(k_{z}d)}=z_{0}\sqrt{\mu_{r}\epsilon_{r}-\left(\frac{k_{x}}{k_{0}}\right)^{2}}\tan{\left(\sqrt{\mu_{r}\epsilon_{r}-\left(\frac{k_{x}}{k_{0}}\right)^{2}}k_{0}d\right)}/\epsilon_{r}$
(30)
$Z^{ex}_{TM}=\sqrt{\frac{\mu}{\epsilon}}\frac{k}{k_{z}}\tan{(k_{z}d)}=\frac{z_{0}\mu_{r}\tan{\left(\sqrt{\mu_{r}\epsilon_{r}-\left(\frac{k_{x}}{k_{0}}\right)^{2}}k_{0}d\right)}}{\sqrt{\mu_{r}\epsilon_{r}-\left(\frac{k_{x}}{k_{0}}\right)^{2}}}$
(31)

  * •

First method
The accuracy of the boundary condition may be improved by including the
coefficient $a_{1}$, thus approximation derive first order polynomial form
[26]. Wave formulas of electromagnetic fields give the quantity of wave number
$x$-component, at calculation coefficients of approximate polynomials. Where
from we denote $\xi=-(\frac{k_{x}}{k_{0}})^{2}=-\sin^{2}(\theta)$ and input it
in the exact impedance values (30), (31), we also obtain

$Z^{ex}_{TE}(\xi)=z_{0}\sqrt{\mu_{r}\epsilon_{r}+\xi}\tan{\left(\sqrt{\mu_{r}\epsilon_{r}+\xi}k_{0}d\right)}/\epsilon_{r}$
$Z^{ex}_{TM}(\xi)=\frac{z_{0}\mu_{r}\tan{\left(\sqrt{\mu_{r}\epsilon_{r}+\xi}k_{0}d\right)}}{\sqrt{\mu_{r}\epsilon_{r}+\xi}}.$

Here we approximate impedance as ratio of polynomials of $\xi$.

If $\theta=0$, we get that $a_{0}=Z(0)=Z^{ex}(0)$, for a normally incident
wave, which is known as the Leontovich boundary condition and we get

$a_{0}=\sqrt{\frac{\mu_{0}\mu_{r}}{\epsilon_{0}\epsilon_{r}}}\tan{\left(\omega\sqrt{\mu_{0}\mu_{r}\epsilon_{0}\epsilon_{r}}d\right)}$

We calculate other coefficients $a_{j}$ and $b_{j}$, using two arbitrary
angles $\theta_{1}$ and $\theta_{2}$ by:

$\left(\begin{array}[]{cl}a_{j}\\\
b_{j}\end{array}\right)=\left[\begin{array}[]{clcr}-k^{2}_{x}(\theta_{1})&k^{2}_{x}Z_{j}^{ex}(\theta_{1})\\\
-k^{2}_{x}(\theta_{2})&k^{2}_{x}Z_{j}^{ex}(\theta_{2})\end{array}\right]^{-1}\left(\begin{array}[]{cl}Z_{j}^{ex}(\theta_{1})-a_{0}\\\
Z_{j}^{ex}(\theta_{2})-a_{0}\end{array}\right)$

The indices correspond to TE and TM polarizations, as in (29). Therefore,
$a_{j}$ is defined for arbitrary $\theta_{1}\in]0,\pi/2[$

$a_{j}=\frac{Z^{ex}_{j}(\theta_{1})-a_{0}}{\xi(\theta_{1})}\ \ \text{and}\ \
b_{j}=0.$

We can approximate impedance $Z^{ex}$ as the first order Taylor polynomial for
$\xi$ near to zero, which yields

$a_{1}=z_{0}\frac{k_{0}d}{2\epsilon_{r}}+z_{0}\frac{\tan(\sqrt{\mu_{r}\epsilon_{r}}k_{0}d)}{2\sqrt{\mu_{r}\epsilon_{r}}\epsilon_{r}}+z_{0}\frac{k_{0}d\tan^{2}(\sqrt{\mu_{r}\epsilon_{r}}k_{0}d)}{2\epsilon_{r}}\
\ \ (Taylor)$ $b_{1}=0.$

Now we suppose that $b_{1}$ is different to zero and we approximate the
impedance as ratio of polynomials. Stupfel mentioned Padé approximation, which
considers second order Taylor approximation as follows

$Z^{ex}_{1}(\xi)=c_{0}+c_{1}\xi+c_{2}\xi^{2}+O(\xi^{2})\approx\frac{a_{0}+a_{1}\xi}{1+b_{1}\xi},$
(32)

where coefficients $c_{0},c_{1}$ and $c_{2}$ are Taylor coefficients:

$c_{0}=z_{0}\sqrt{\frac{\mu_{r}}{\epsilon_{r}}}\tan{\left(\sqrt{\mu_{r}\epsilon_{r}}k_{0}d\right)}$
$c_{1}=z_{0}\frac{k_{0}d}{2\epsilon_{r}}+z_{0}\frac{\tan(\sqrt{\mu_{r}\epsilon_{r}}k_{0}d)}{2\sqrt{\mu_{r}\epsilon_{r}}\epsilon_{r}}+z_{0}\frac{k_{0}d\tan^{2}(\sqrt{\mu_{r}\epsilon_{r}}k_{0}d)}{2\epsilon_{r}}$
$c_{2}=\frac{z_{0}k_{0}d}{8\epsilon^{2}_{r}\mu_{r}}+\left(\frac{z_{0}k^{2}_{0}d^{2}}{4\epsilon_{r}\sqrt{\epsilon_{r}\mu_{r}}}-\frac{z_{0}}{8\epsilon_{r}(\epsilon_{r}\mu_{r})^{3/2}}\right)\tan(\sqrt{\epsilon_{r}\mu_{r}}k_{0}d)$
$+\frac{z_{0}k_{0}d}{8\epsilon^{2}_{r}\mu_{r}}\tan^{2}(\sqrt{\epsilon_{r}\mu_{r}}k_{0}d)+\frac{z_{0}k^{2}_{0}d^{2}}{2\epsilon_{r}\sqrt{\epsilon_{r}\mu_{r}}}\tan^{3}(\sqrt{\epsilon_{r}\mu_{r}}k_{0}d).$

By multiplying this Taylor polynomial by denominator polynomial and equating
the product to the numerator polynomial, we derive equations, where
coefficients of Pade approximation may be determined through coefficients of
Taylor approximation. So, from (32) we have

$a_{0}+a_{1}\xi\approx
c_{0}+(c_{1}+c_{0}b_{1})\xi+(c_{2}+c_{1}b_{1})\xi^{2}+O(\xi^{2}),$

Finally, we get the coefficients of Padé approximation

$b_{1}=-\frac{c_{2}}{c_{1}},\,\,\,\,\,\,a_{1}=c_{1}-c_{0}\frac{c_{2}}{c_{1}}.$

  * •

Second method
We present another method to calculate the coefficients. For different
$\theta_{1}$, $\theta_{2}$ from $]0,\pi/2[$, we get different values of
$\xi_{i}=-k^{2}_{0}\sin^{2}(\theta_{i})$ and two linear equations [26]

$Z_{1}^{ex}(\xi_{1})-b_{1}\xi_{1}Z_{1}^{ex}(\xi_{1})-a_{0}+a_{1}\xi_{1}=0$
$Z_{1}^{ex}(\xi_{2})-b_{1}\xi_{2}Z_{1}^{ex}(\xi_{2})-a_{0}+a_{1}\xi_{2}=0.$

That can be solved in matrix form as follows

$\left(\begin{array}[]{cl}a_{j}\\\
b_{j}\end{array}\right)=\left[\begin{array}[]{clcr}\xi_{1}&-\xi_{1}Z_{j}^{ex}(\xi_{1})\\\
\xi_{2}&-\xi_{2}Z_{j}^{ex}(\xi_{2})\end{array}\right]^{-1}\left(\begin{array}[]{cl}Z_{j}^{ex}(\xi_{1})-a_{0}\\\
Z_{j}^{ex}(\xi_{2})-a_{0}\end{array}\right)$

and we get the coefficients of so-called Collocation approximation [6]

$a_{1}=\frac{-\xi_{2}Z_{j}^{ex}(\xi_{2})(Z_{j}^{ex}(\xi_{2})-a_{0})+\xi_{1}Z_{j}^{ex}(\xi_{1})(Z_{j}^{ex}(\xi_{2})-a_{0})}{\xi_{1}\xi_{2}(Z_{j}^{ex}(\xi_{1})-Z_{j}^{ex}(\xi_{2}))}$
$b_{1}=\frac{-\xi_{2}(Z_{j}^{ex}(\xi_{1})-a_{0})+\xi_{1}(Z_{j}^{ex}(\xi_{2})-a_{0})}{\xi_{1}\xi_{2}(Z_{j}^{ex}(\xi_{1})-Z_{j}^{ex}(\xi_{2}))}.$

Note that in two dimensional case we suppose that $\partial_{y}=0$. The three
dimensional impedance $Z_{3D}$ should correspond to $Z_{2Dj}$ in two
dimensional case. Coefficients $a_{1},b_{1}$ and $a_{2},b_{2}$ should
correspond to those in two dimension, TE and TM coefficients, respectively.

###  6.2 The coefficients of the IBC2

We can apply all methods that were proposed earlier for IBC2, too. We get
Taylor approximation, if we admit that $b_{j}$ and $b^{\prime}_{j}$ are zero.
If they are not zero, we get ratio of polynomials approximation. Padé
approximation uses fourth order Taylor expansion. Whence we get five equation
for five unknown coefficients. And in the last method, the coefficients are
calculated by solving system of linear equations for different
$\theta_{k}\in]0,\pi/2[,\ k=1,2,3,4$

$\left(\begin{array}[]{cccc}a_{j}\\\ a^{\prime}_{j}\\\ b_{j}\\\
b^{\prime}_{j}\end{array}\right)=\left[\begin{array}[]{cccc}\xi_{1}&\xi^{2}_{1}&-\xi_{1}Z_{j}^{ex}(\xi_{1})&-\xi^{2}_{1}Z_{j}^{ex}(\xi_{1})^{2}\\\
...&...&...&...\\\
\xi_{4}&\xi^{2}_{4}&-\xi_{4}Z_{j}^{ex}(\xi_{4})&-\xi^{2}_{4}Z_{j}^{ex}(\xi_{4})^{2}\\\
\end{array}\right]^{-1}\left(\begin{array}[]{cl}Z_{j}^{ex}(\xi_{1})-a_{0}\\\
...\\\ ...\\\ Z_{j}^{ex}(\xi_{4})-a_{0}\end{array}\right)$

Here, we suppose that in the last two methods we get invertible matrices.

###  6.3 Numerical results

Let us consider a mono-layer dielectric coating with characteristics
$\varepsilon_{r}=4.0$, $\mu_{r}=1.0$ and $d=0.005\lambda_{0}$. Figure 3 shows
values exact IBC, Leontovich IBC, first order and second order impedance
boundary conditions, in TE polarization and TM polarization. Where the angle
of incidence of the plane wave $\phi$ has angle range $]0,\pi[$.

(a) TE polarisation

(b) TM polarisation

Figure 3: Comparison of the exact impedance, Leontovich impedance, first-order
and second-order IBC.

We can easily see that the difference between IBC0 and exact IBC increases.
While the difference between exact IBC and IBC1 is very small, as the
difference between exact IBC and IBC2. But we can see the error of IBC1 and
IBC2 approximations on the Figure 4. As the angle of incidence increases the
error of first-order IBC approximation reaches $0.39\Omega$.

(a) TE polarisation

(b) TM polarisation

Figure 4: Errors of first-order (IBC1) and second-order (IBC2) IBC.

In the next sections, we want to establish some variational formulations,
applying a boundary integral method for the problem in 2D case with different
IBCs. Then, we introduce current densities J and M on the boundary $\Gamma$ as
follows:

$\mathbf{M}=\mathbf{E}\times\mathbf{n}\ \ ,\
\mathbf{J}=\mathbf{n}\times\mathbf{H}.$

Thanks to the Stratton-Chu formula, we know that $J$ and $M$ are solutions of
EFIE and MFIE, as[25]:

${}<Z_{0}(B-S)\mathbf{J},\bm{\Psi}_{J}>+<(P+Q)\mathbf{M},\bm{\Psi}_{J}>=<E^{inc},\bm{\Psi}_{J}>,$
(33)
${}-<(P+Q)\mathbf{J},\bm{\Psi}_{M}>+<\frac{1}{Z_{0}}(B-S)\mathbf{M},\bm{\Psi}_{M}>=<H^{inc},\bm{\Psi}_{M}>$
(34)

##  7 Variational formulation for the 2D problem

Two dimensional case system is invariant in one direction, In the two
dimensional case, so object surface $\Gamma$ becomes a curved contour, that we
will call $C$. We have the curvilinear abscissa $l$ along $C$ and normal to
the contour unit vector $\mathbf{n}$. We set the local frame ($\bm{\tau},\
\bm{\nu},\ \mathbf{n}$), where $\bm{\tau}$ is a unit vector tangent to the
contour $C$ in $l$ direction, and $\bm{\nu}$ can be defined as
$\bm{\nu}=\mathbf{n}\times\bm{\tau}$. We suppose that our two dimensional
system does not depend on $\nu$ parameter, however variable $\nu$ component is
depend on $l$.

So, we write the first order IBC (18) as follows

$(1+b_{j}d_{l}^{2})(\mathbf{n}\times\mathbf{M})=(a_{0}+a_{j}d_{l}^{2})\mathbf{J},$
(35)

and we can write the second order IBC (19) as follows

$(1+b_{j}d_{l}^{2}+b^{\prime}_{j}d_{l}^{4})(\mathbf{n}\times\mathbf{M})=(a_{0}+a_{j}d_{l}^{2}+a^{\prime}_{j}d_{l}^{4})\mathbf{J},$
(36)

where $j=1,2$ correspond to TE and TM polarizations, respectively. The
invariance in one direction for two dimensional model allow us to simplify the
Hodge operator as the second partial derivative on the contour. Where the
electromagnetic current densities $\mathbf{n}\times\mathbf{M}$ and
$\mathbf{J}$ have $\bm{\tau}$ direction for TE polarization, and $\bm{\nu}$
direction for TM polarization.

  * •

TE:
$\partial_{x}^{2}\mathbf{J}=\bm{\tau}\partial_{x}^{2}J_{\tau}=\bm{\tau}d_{l}^{2}J_{\tau}$
and
$\partial_{x}^{2}(\mathbf{n}\times\mathbf{M})=-\bm{\tau}\partial_{x}^{2}M_{\nu}=-\bm{\tau}d_{l}^{2}M_{\nu}$;

  * •

TM:
$\partial_{x}^{2}\mathbf{J}=\bm{\nu}\partial_{x}^{2}J_{\nu}=\bm{\nu}d_{l}^{2}J_{\nu}$
and
$\partial_{x}^{2}(\mathbf{n}\times\mathbf{M})=\bm{\nu}\partial_{x}^{2}M_{\tau}=\bm{\nu}d_{l}^{2}M_{\tau}$.

###  7.1 Variational formulation with first order IBC

We weakly write the equation (35), we multiply it by a test function and
integrate along the contour $C$. In EFIE test function is noted as $\Psi_{j}$,
so we use it to insert our boundary condition in this equation.

$\int_{C}(1+b_{j}d_{l}^{2})(\mathbf{n}\times\mathbf{M})\cdot\bm{\Psi}_{J}dl=\int_{C}(a_{0}+a_{j}d_{l}^{2})\mathbf{J}\cdot\bm{\Psi}_{J}dl.$

That gives us

$\int_{C}(\mathbf{n}\times\mathbf{M})\cdot\bm{\Psi}_{J}dl=\int_{C}(a_{0}+a_{j}d_{l}^{2})\mathbf{J}\cdot\bm{\Psi}_{J}dl-\int_{C}b_{j}d_{l}^{2}(\mathbf{n}\times\mathbf{M})\cdot\bm{\Psi}_{J}dl.$

We put it in the operator $P$ and obtain:

$<P\mathbf{M},\bm{\Psi}_{J}>=\frac{1}{2}\int_{C}(\mathbf{n}\times\mathbf{M})\cdot\bm{\Psi}_{J}dl=\frac{a_{0}}{2}\int_{C}\mathbf{J}\cdot\bm{\Psi}_{J}dl$
$+\frac{a_{j}}{2}\int_{C}d_{l}^{2}\mathbf{J}\cdot\bm{\Psi}_{J}dl-\frac{b_{j}}{2}\int_{C}d_{l}^{2}(\mathbf{n}\times\mathbf{M})\cdot\bm{\Psi}_{J}dl$
(37)

We use $(\mathbf{n}\times\bm{\Psi}_{M})$ as a test function, to insert IBC1 in
MFIE

$\int_{C}(1+b_{j}d^{2}_{l})(\mathbf{n}\times\mathbf{M})\cdot(\mathbf{n}\times\bm{\Psi}_{M})dl=\int_{C}(a_{0}+a_{j}d^{2}_{l})\mathbf{J}\cdot(\mathbf{n}\times\bm{\Psi}_{M})dl.$

We take the first part of right side

$\int_{C}\mathbf{J}\cdot(\mathbf{n}\times\bm{\Psi}_{M})dl=\frac{1}{a_{0}}\int_{C}(1+b_{j}d^{2}_{l})(\mathbf{n}\times\mathbf{M})\cdot(\mathbf{n}\times\bm{\Psi}_{M})dl-\frac{1}{a_{0}}\int_{C}a_{j}d^{2}_{l}\mathbf{J}\cdot(\mathbf{n}\times\bm{\Psi}_{M})dl.$

And using the formula of vector analysis

$\bm{\Psi}_{M}\cdot(\mathbf{n}\times\mathbf{J})=-\mathbf{J}\cdot(\mathbf{n}\times\bm{\Psi}_{M}),$

we put it in P operator with weakly form of IBC1

$<P\mathbf{J},\bm{\Psi}_{M}>=\frac{1}{2}\int_{C}(\mathbf{n}\times\mathbf{J})\cdot\bm{\Psi}_{M}dl=-\frac{1}{2}\int_{C}\mathbf{J}\cdot(\mathbf{n}\times\bm{\Psi}_{M})dl=$
$=-\frac{1}{2a_{0}}\int_{C}(\mathbf{n}\times\mathbf{M})\cdot(\mathbf{n}\times\bm{\Psi}_{M})dl-\frac{b_{j}}{2a_{0}}\int_{C}d_{l}^{2}(\mathbf{n}\times\mathbf{M})\cdot(\mathbf{n}\times\bm{\Psi}_{M})dl$
$+\frac{a_{j}}{2a_{0}}\int_{C}d_{l}^{2}\mathbf{J}\cdot(\mathbf{n}\times\bm{\Psi}_{M})dl$
(38)

First, we observe TE polarization, where $P$ operator becomes:

$\int_{C}PM_{\nu}\ \Psi_{J\tau}dl=\frac{a_{0}}{2}\int_{C}J_{\tau}\
\Psi_{J\tau}dl+\frac{a_{1}}{2}\int_{C}d_{l}^{2}J_{\tau}\
\Psi_{J\tau}dl+\frac{b_{1}}{2}\int_{C}d_{l}^{2}M_{\nu}\ \Psi_{J\tau}dl$

and

$\int_{C}PJ_{\tau}\ \Psi_{M\nu}dl=-\frac{1}{2a_{0}}\int_{C}M_{\nu}\
\Psi_{M\nu}dl-\frac{b_{1}}{2a_{0}}\int_{C}d_{l}^{2}M_{\nu}\
\Psi_{M\nu}dl-\frac{a_{1}}{2a_{0}}\int_{C}d_{l}^{2}J_{\tau}\ \Psi_{M\nu}dl$

for EFIE and MFIE, respectively.

We put them in the variational equations (33) and (34) and get:

$iZ_{0}\iint_{C}kG(l,l^{\prime})\ J_{\tau}(l^{\prime})\ \Psi_{J\tau}\
[\bm{\tau}(l^{\prime})\cdot\bm{\tau}(l)]-\frac{1}{k}G(l,l^{\prime})\
d^{\prime}_{l}J_{\tau}(l^{\prime})\ d_{l}\Psi_{J\tau}(l)\ dl^{\prime}dl$
$+\iint_{C}\Psi_{J\tau}(l)\ M_{\nu}(l^{\prime})\
[\bm{\tau}(l)\times\bm{\nu}(l^{\prime})]\cdot\nabla_{\Gamma}G(l,l^{\prime})\
dl^{\prime}dl$ $+\frac{a_{0}}{2}\int_{C}J_{\tau}\
\Psi_{J\tau}dl+\frac{a_{1}}{2}\int_{C}d_{l}^{2}J_{\tau}\ \Psi_{J\tau}dl$
$+\frac{b_{1}}{2}\int_{C}d_{l}^{2}M_{\nu}\
\Psi_{J\tau}dl=\int_{C}E_{\tau}^{inc}\ \Psi_{J\tau}dl$ (39)

and

$-\iint_{C}\Psi_{M\nu}(l)\ J_{\tau}(l^{\prime})\
[\bm{\nu(l)}\times\bm{\tau(l^{\prime})}]\cdot\nabla_{\Gamma}G(l,l^{\prime})dl^{\prime}dl$
$+\frac{i}{Z_{0}}\iint_{C}kG(l,l^{\prime})\ M_{\nu}(l^{\prime})\
\Psi_{M\nu}(l)[\bm{\nu(l^{\prime})}\cdot\bm{\nu(l)}]dl^{\prime}dl$
$+\frac{1}{2a_{0}}\int_{C}M_{\nu}\
\Psi_{M\nu}dl+\frac{b_{j}}{2a_{0}}\int_{C}d_{l}^{2}M_{\nu}\ \Psi_{M\nu}dl$
$+\frac{a_{j}}{2a_{0}}\int_{C}d_{l}^{2}J_{\tau}\
\Psi_{M\nu}dl=\int_{C}H_{\nu}^{inc}\ \Psi_{M\nu}dl$ (40)

for EFIE and MFIE, respectively.

In the equations (39) and (40) we have scalar products
$[\bm{\tau}(l^{\prime})\cdot\bm{\tau}(l)]=1$ and
$[\bm{\nu}(l)\cdot\bm{\nu}(l^{\prime})]=1$, and vector products
$[\bm{\tau}(l)\times\bm{\nu}(l^{\prime})]=\mathbf{n}(l)$ and
$[\bm{\nu}(l)\times\bm{\tau}(l^{\prime})]=-\mathbf{n}(l^{\prime})$. The
operator $S$ contains surface divergence operator that becomes differential
operator

$\mathrm{div}_{\Gamma}\mathbf{J}=\mathrm{div}_{\Gamma}(\bm{\tau}J_{\tau})=d_{l}J_{\tau};\,\,\,\,\,\,\,\mathrm{div}_{\Gamma}\mathbf{M}=\mathrm{div}_{\Gamma}(\bm{\nu}M_{\nu})=d_{\nu}M_{\nu}\equiv
0,$

because the model is invariance in $\nu$ parameter.

By doing integration by parts, we have

$\frac{b_{1}}{2}\int_{C}d_{l}^{2}M_{\nu}(l)\Psi_{J\tau}(l)dl=-\frac{b_{1}}{2}\int_{C}d_{l}M_{\nu}(l)\
d_{l}\Psi_{J\tau}(l)dl.$ (41)

Finally we combine two equations (39)-(40) to present next variational
problem:

######  Problem 7.1.

Find $U=(J_{\tau},M_{\nu})\in[H^{1}(C)]^{2}$ such that:

$A(U,\Psi)=\int_{C}E_{\tau}^{inc}\Psi_{J\tau}dl+\int_{C}H_{\nu}^{inc}\Psi_{M\nu}dl$

for all $\Psi=(\Psi_{J\tau},\Psi_{M\nu})\in[H^{1}(C)]^{2}$, where the bilinear
form $A$ is defined as:

$A(U,\Psi)=iZ_{0}\iint_{C}kG(l,l^{\prime})J_{\tau}(l^{\prime})\Psi_{J\tau}\
[\tau(l)\cdot\tau(l^{\prime})]-\frac{1}{k}G(l,l^{\prime})d^{\prime}_{l}J_{\tau}(l^{\prime})d_{l}\Psi_{J\tau}(l)dl^{\prime}dl$
$+\iint_{C}\Psi_{J\tau}(l)M_{\nu}(l^{\prime})\
\mathbf{n}(l)\cdot\nabla_{\Gamma}G(l,l^{\prime})dl^{\prime}dl+\iint_{C}\Psi_{M\nu}J_{\tau}\
\mathbf{n}(l^{\prime})\cdot\nabla_{\Gamma}G(l,l^{\prime})dl^{\prime}dl$
$+\frac{i}{Z_{0}}\iint_{C}kG(l,l^{\prime})M_{\nu}(l^{\prime})\Psi_{M\nu}(l)dl^{\prime}dl+\frac{a_{0}}{2}\int_{C}J_{\tau}\Psi_{J\tau}dl+\frac{1}{2a_{0}}\int_{C}M_{\nu}\Psi_{M\nu}dl$
$-\frac{a_{1}}{2}\int_{C}d_{l}J\
d_{l}\Psi_{J\tau}dl-\frac{b_{1}}{2}\int_{C}d_{l}M\
d_{l}\Psi_{J\tau}dl-\frac{b_{1}}{2a_{0}}\int_{C}d_{l}M\
d_{l}\Psi_{M\nu}dl-\frac{a_{1}}{2a_{0}}\int_{C}d_{l}J\ d_{l}\Psi_{M\nu}dl$
(42)

We present similar variational problem for TM polarization:

######  Problem 7.2.

Find $U=(J_{\nu},M_{\tau})\in[H^{1}(C)]^{2}$ such that:

$A(U,\Psi)=\int_{C}E_{\nu}^{inc}\Psi_{J\nu}dl+\int_{C}H_{\tau}^{inc}\Psi_{M\tau}dl$

for all $\Psi=(\Psi_{J\nu},\Psi_{M\tau})\in[H^{1}(C)]^{2}$, where the bilinear
form $A$ is defined as:

$A(U,\Psi)=iZ_{0}\iint_{C}kG(l,l^{\prime})J_{\nu}(l^{\prime})\Psi_{J\nu}dl^{\prime}dl-\iint_{C}\Psi_{J\nu}(l)M_{\tau}(l^{\prime})\
\mathbf{n}(l^{\prime})\cdot\nabla_{\Gamma}G(l,l^{\prime})dl^{\prime}dl$
$-\iint_{C}\Psi_{M\tau}J_{\nu}\
\mathbf{n}(l)\cdot\nabla_{\Gamma}G(l,l^{\prime})dl^{\prime}dl+\frac{i}{Z_{0}}\iint_{C}kG(l,l^{\prime})M_{\tau}(l^{\prime})\Psi_{M\tau}(l)[\tau(l)\cdot\tau(l^{\prime})]$
$-\frac{1}{k}G(l,l^{\prime})d^{\prime}_{l}M_{\tau}(l^{\prime})d_{l}\Psi_{M\tau}(l)dl^{\prime}dl+\frac{a_{0}}{2}\int_{C}J_{\nu}\Psi_{J\nu}dl-\frac{1}{2a_{0}}\int_{C}M_{\tau}\Psi_{M\tau}dl$
$-\frac{a_{2}}{2}\int_{C}d_{l}J\
d_{l}\Psi_{J\nu}dl+\frac{b_{2}}{2}\int_{C}d_{l}M\
d_{l}\Psi_{J\nu}dl+\frac{b_{2}}{2a_{0}}\int_{C}d_{l}M\
d_{l}\Psi_{M\tau}dl-\frac{a_{2}}{2a_{0}}\int_{C}d_{l}J\ d_{l}\Psi_{M\tau}dl$
(43)

###  7.2 Variational formulation with second order IBC

The equation (36) passes the same way as IBC1 to become weak. The weak
formulations replace operator $P$ in EFIE and MFIE equations. Finally, we
assemble them to define the bilinear form:

$A(U,\Psi)=iZ_{0}\iint_{C}kG(l,l^{\prime})J_{\tau}(l^{\prime})\Psi_{J\tau}\
[\bm{\tau}(l)\cdot\bm{\tau}(l^{\prime})]-\frac{1}{k}G(l,l^{\prime})d^{\prime}_{l}J_{\tau}(l^{\prime})d_{l}\Psi_{J\tau}(l)dl^{\prime}dl$
$+\iint_{C}\Psi_{J\tau}(l)M_{\nu}(l^{\prime})\
\mathbf{n}(l)\cdot\nabla_{\Gamma}G(l,l^{\prime})dl^{\prime}dl+\iint_{C}\Psi_{M\nu}J_{\tau}\
\mathbf{n}(l^{\prime})\cdot\nabla_{\Gamma}G(l,l^{\prime})dl^{\prime}dl$
$+\frac{i}{Z_{0}}\iint_{C}kG(l,l^{\prime})M_{\nu}(l^{\prime})\Psi_{M\nu}(l)dl^{\prime}dl+\frac{a_{0}}{2}\int_{C}J_{\tau}\Psi_{J\tau}dl+\frac{1}{2a_{0}}\int_{C}M_{\nu}\Psi_{M\nu}dl$
$+\frac{a_{1}}{2}\int_{C}d^{2}_{l}J_{\tau}\Psi_{J\tau}dl+\frac{b_{1}}{2}\int_{C}d^{2}_{l}M_{\nu}\Psi_{J\tau}dl+\frac{b_{1}}{2a_{0}}\int_{C}d^{2}_{l}M_{\nu}\Psi_{M\nu}dl+\frac{a_{1}}{2a_{0}}\int_{C}d^{2}_{l}J_{\tau}\Psi_{M\nu}dl$
$+\frac{a^{\prime}_{1}}{2}\int_{C}d^{4}_{l}M_{\nu}\
\Psi_{J\tau}dl+\frac{b^{\prime}_{1}}{2}\int_{C}d^{4}_{l}M_{\nu}\
\Psi_{J\tau}+\frac{b^{\prime}_{1}}{2a_{0}}\int_{C}d^{4}_{l}M_{\nu}\
\Psi_{M\nu}dl+\frac{a^{\prime}_{1}}{2a_{0}}\int_{C}d^{4}_{l}J_{\tau}\
\Psi_{M\nu}dl$

for TE polarization. And with integration by parts, we get

$A(U,\Psi)=iZ_{0}\iint_{C}kG(l,l^{\prime})J_{\tau}(l^{\prime})\Psi_{J\tau}\
[\bm{\tau}(l)\cdot\bm{\tau}(l^{\prime})]-\frac{1}{k}G(l,l^{\prime})d^{\prime}_{l}J_{\tau}(l^{\prime})d_{l}\Psi_{J\tau}(l)dl^{\prime}dl$
$+\iint_{C}\Psi_{J\tau}(l)M_{\nu}(l^{\prime})\
\mathbf{n}(l)\cdot\nabla_{\Gamma}G(l,l^{\prime})dl^{\prime}dl+\iint_{C}\Psi_{M\nu}J_{\tau}\
\mathbf{n}(l^{\prime})\cdot\nabla_{\Gamma}G(l,l^{\prime})dl^{\prime}dl$
$+\frac{i}{Z_{0}}\iint_{C}kG(l,l^{\prime})M_{\nu}(l^{\prime})\Psi_{M\nu}(l)dl^{\prime}dl+\frac{a_{0}}{2}\int_{C}J_{\tau}\Psi_{J\tau}dl+\frac{1}{2a_{0}}\int_{C}M_{\nu}\Psi_{M\nu}dl$
$-\frac{a_{1}}{2}\int_{C}d_{l}J_{\tau}\
d_{l}\Psi_{J\tau}dl-\frac{b_{1}}{2}\int_{C}d_{l}M_{\nu}\
d_{l}\Psi_{J\tau}dl-\frac{b_{1}}{2a_{0}}\int_{C}d_{l}M_{\nu}\
d_{l}\Psi_{M\nu}dl-\frac{a_{1}}{2a_{0}}\int_{C}d_{l}J_{\tau}\
d_{l}\Psi_{M\nu}dl$ $+\frac{a^{\prime}_{1}}{2}\int_{C}d^{2}_{l}M_{\nu}\
d^{2}_{l}\Psi_{J\tau}dl+\frac{b^{\prime}_{1}}{2}\int_{C}d^{2}_{l}M_{\nu}\
d^{2}_{l}\Psi_{J\tau}+\frac{b^{\prime}_{1}}{2a_{0}}\int_{C}d^{2}_{l}M_{\nu}\
d^{2}_{l}\Psi_{M\nu}dl+\frac{a^{\prime}_{1}}{2a_{0}}\int_{C}d^{2}_{l}J_{\tau}\
d^{2}_{l}\Psi_{M\nu}dl$

We thus obtain a variational formulation for the problem in $\mathbb{R}^{2}$
with second order IBC written as follows:

######  Problem 7.3.

Find $U=(J_{\tau},M_{\nu})\in[H^{1}(C)]^{2}$ such that

$A(U,\Psi)=\int_{C}E_{\tau}^{inc}\Psi_{J\tau}dl+\int_{C}H_{\nu}^{inc}\Psi_{M\nu}dl$

for all $\Psi=(\Psi_{J\tau},\Psi_{M\nu})\in[H^{1}(C)]^{2}$.

In the following, we study the well-posedness of the formulation (7.1)

###  7.3 Existence and uniqueness theorem

We are going to show that our variational problem in TE has a unique solution
using theorem from [14]. For TM problem 7.2, it will be analogous. It is
necessary to determine the continuity and the coercivity of the bilinear form
$A(U,\Psi)$.

For the sake of simplicity we consider the operator $A(U,\Psi)$ as a sum of
three bilinear operator

$A_{1}(U,\Psi)=\iint_{C}Z_{0}(B-S)J_{\tau}\Psi_{J\tau}dl^{\prime}dl+\iint_{C}\frac{1}{Z_{0}}(B-S)M_{\nu}\Psi_{M\nu}dl^{\prime}dl+\iint_{C}QM_{\nu}\Psi_{J\tau}dl^{\prime}dl$
$+\iint_{C}QJ_{\tau}\Psi_{M\nu}dl^{\prime}dl+\frac{a_{0}}{2}\int_{C}J_{\tau}\Psi_{J\tau}dl+\frac{1}{2a_{0}}\int_{C}M_{\nu}\Psi_{M\nu}dl$
$A_{2}(U,\Psi)=-\frac{a_{1}}{2}\int_{C}d_{l}J\
d_{l}\Psi_{J\tau}dl-\frac{b_{1}}{2a_{0}}\int_{C}d_{l}M\ d_{l}\Psi_{M\nu}dl$

and

$A_{3}(U,\Psi)=-\frac{b_{1}}{2}\int_{C}d_{l}M\
d_{l}\Psi_{J\tau}dl-\frac{a_{1}}{2a_{0}}\int_{C}d_{l}J\ d_{l}\Psi_{M\nu}dl$

where

$A=A_{1}+A_{2}+A_{3}$

We have the following result about continuous linear form $A(U,\Psi)$.

######  Lemma 7.1.

The bilinear form $A(U,\Psi)$ (42) is continuous on $[H^{1}(C)]^{2}$.

###### Proof.

: We have to show that there exists $\beta>0$ such that

$|A(U,\Psi)|\leq\beta\|U\|_{H^{1}(C)}\|\Psi\|_{H^{1}(C)}$ (44)

for all $U,\Psi\in H^{1}(C)$.

From the works in the past [29], we have that for operator $A_{1}(U,\Psi)$
there exists constant $\beta_{1}>0$ such that

$|A_{1}(U,\Psi)|\leq\beta_{1}\|U\|_{H^{1}(C)}\|\Psi\|_{H^{1}(C)}$

It remains to prove continuity for last integrals in bilinear form. Using
Cauchy-Schwarz inequality we get:

$|A_{2}(U,\Psi)+A_{3}(U,\Psi)|\leq$ $\left|\frac{a_{1}}{2}\int_{C}d_{l}J\
d_{l}\Psi_{J\tau}dl\right|+\left|\frac{b_{1}}{2}\int_{C}d_{l}M\
d_{l}\Psi_{J\tau}dl\right|+\left|\frac{b_{1}}{2a_{0}}\int_{C}d_{l}M\
d_{l}\Psi_{M\nu}dl\right|+\left|\frac{a_{1}}{2a_{0}}\int_{C}d_{l}J\
d_{l}\Psi_{M\nu}dl\right|\leq$
$\left|\frac{a_{1}}{2}\right|\|d_{l}J\|_{L^{2}}\|\Psi_{J\tau}\|_{L^{2}}+\left|\frac{b_{1}}{2}\right|\|d_{l}M\|_{L^{2}}\|\Psi_{J\tau}\|_{L^{2}}+\left|\frac{b_{1}}{2a_{0}}\right|\|d_{l}M\|_{L^{2}}\|\Psi_{M\nu}\|_{L^{2}}+\left|\frac{a_{1}}{2a_{0}}\right|\|d_{l}J\|_{L^{2}}\|\Psi_{M\nu}\|_{L^{2}}$
$\leq\beta_{2}\|U\|_{H^{1}(C)}\|\Psi\|_{H^{1}(C)}\ \ ,\ where\ \beta_{2}\geq
0.$

Finally, we take $\beta=\beta_{1}+\beta_{2}\geq 0$ for which (44) is true. ∎

It remains to show the existence and uniqueness for the variational problem
7.1. Therefore, we give the following result about coercivity of the bilinear
form $A(U,\Psi)$

######  Lemma 7.2.

The bilinear form $A(U,\Psi)$ is coercive on $H^{1}(C)$; i.e., there exists
$\gamma>0$ and $\gamma^{\prime}$ such that

$\Re[A(U,U^{*})]\geq\gamma\|U\|^{2}_{H^{1}(C)}-\gamma^{\prime}\|U\|^{2}_{L^{2}(C)},\
\ \ \forall U\in[H^{1}(C)]^{4}.$

###### Proof.

First of all, we take $\Psi=U^{*}$ and get

$A(U,U^{*})=iZ_{0}\iint_{C}kGJ_{\tau}J^{*}_{\tau}[\bm{\tau}(l^{\prime})\cdot\bm{\tau}(l)]-\frac{1}{k}d^{\prime}_{l}J_{\tau}(l^{\prime})d_{l}J^{*}_{\tau}(l)dl^{\prime}dl$
$+\iint_{C}J^{*}_{\tau}M_{\nu}\mathbf{n}(l)\cdot\nabla_{\Gamma}Gdl^{\prime}dl+\iint_{C}M^{*}_{\nu}J_{\tau}\mathbf{n}(l^{\prime})\cdot\nabla_{\Gamma}Gdl^{\prime}dl$
$+\frac{i}{Z_{0}}\iint_{C}kGM_{\nu}M^{*}_{\nu}[\bm{\nu}(l^{\prime})\cdot\bm{\nu}(l)]dl^{\prime}dl+\frac{a_{0}}{2}\int_{C}J_{\tau}J^{*}_{\tau}dl+\frac{1}{2a_{0}}\int_{C}M_{\nu}M^{*}_{\nu}dl$
$-\frac{a_{1}}{2}\int_{C}d_{l}J_{\tau}\
d_{l}J^{*}_{\tau}dl-\frac{b_{1}}{2}\int_{C}d_{l}M_{\nu}\
d_{l}J^{*}_{\tau}dl-\frac{b_{1}}{2a_{0}}\int_{C}d_{l}M_{\nu}\
d_{l}M^{*}_{\nu}dl-\frac{a_{1}}{2a_{0}}\int_{C}d_{l}J_{\tau}\
d_{l}M^{*}_{\nu}dl$ (45)

As in continuity, from works in the past [29], we have that there exists
$\gamma_{1}>0$ for the operator $A_{1}(U,\Psi)$, such that $\forall U\in V$

$\Re[A_{1}(U,U^{*})]\geq\frac{\Re(a_{0})}{2}\|J_{\tau}\|^{2}_{L^{2}(C)}+\frac{\Re(a_{0})}{2|a_{0}|^{2}}\|M_{\nu}\|^{2}_{L^{2}(C)}+\gamma_{1}\left(\|J_{\tau}\|^{2}_{H^{1}(C)}+\|M_{\nu}\|^{2}_{H^{1}(C)}\right)$

Next we take operator $A_{2}$

$A_{2}=-\frac{a_{1}}{2}\int_{C}d_{l}J_{\tau}\
d_{l}J^{*}_{\tau}dl-\frac{b_{1}}{2a_{0}}\int_{C}d_{l}M_{\nu}\
d_{l}M^{*}_{\nu}dl$

where real part

$\Re(A_{2})=-\frac{\Re(a_{1})}{2}\|d_{l}J_{\tau}\|^{2}_{L^{2}(C)}-\Re(\frac{b_{1}}{2a_{0}})\|d_{l}M_{\nu}\|^{2}_{L^{2}(C)}$

And it remains

$A_{3}=-\frac{b_{1}}{2}\int_{C}d_{l}M_{\nu}\
d_{l}J^{*}_{\tau}dl-\frac{a_{1}}{2a_{0}}\int_{C}d_{l}J_{\tau}\
d_{l}M^{*}_{\nu}dl$

where real part

$\Re(A_{3})=\Re\left(-\frac{b_{1}}{2}\int_{C}d_{l}M_{\nu}\
d_{l}J^{*}_{\tau}dl-\frac{a_{1}}{2a_{0}}\int_{C}d_{l}J_{l}\
d_{l}M^{*}_{\nu}dl\right)=$
$=-\Re\left[\left(\frac{b_{1}}{2}+\frac{a^{*}_{1}a_{0}}{2|a_{0}|^{2}}\right)\int_{C}d_{l}M_{\nu}\
d_{l}J^{*}_{\tau}dl\right]=$
$=-\Re\left[\int_{C}\frac{1}{|a_{0}|^{1/2}}\left(\frac{b_{1}}{2}+\frac{a^{*}_{1}a_{0}}{2|a_{0}|^{2}}\right)^{1/2}d_{l}M_{\nu}\cdot|a_{0}|^{1/2}\left(\frac{b_{1}}{2}+\frac{a^{*}_{1}a_{0}}{2|a_{0}|^{2}}\right)^{1/2}d_{l}J^{*}_{l}dl\right]$

We note $q=b_{1}|a_{0}|+a^{*}_{1}a_{0}/|a_{0}|$, so

$\Re(A_{3})\geq-\frac{|q|}{4}\|d_{l}J_{\tau}\|^{2}_{L^{2}(C)}-\frac{|q|}{4|a_{0}|^{2}}\|d_{l}M_{\nu}\|^{2}_{L^{2}(C)}.$

Sufficient uniqueness conditions (24) says that $\Re(a_{1}-b^{*}_{1}a_{0})=0$
or $\Re(a_{1})=\Re(b_{1}a^{*}_{0})$. So the sum of operators $A_{2}$ and
$A_{3}$ get

$\Re(A_{2})+\Re(A_{3})\geq$
$-\frac{1}{2}\left(\Re(a_{1})+\frac{|q|}{2}\right)\|d_{l}J_{\tau}\|^{2}_{L^{2}(C)}-\frac{1}{2|a_{0}|^{2}}\left(\Re(a_{1})+\frac{|q|}{2}\right)\|d_{l}M_{\nu}\|^{2}_{L^{2}(C)}$

So, if $\Re(a_{1})+\frac{|q|}{2}=0$, we get that

$\Re(A)=\Re(A_{1})+\Re(A_{2})+\Re(A_{3})\geq$
$\geq\gamma_{1}\|U\|^{2}_{H^{1}(C)}-c\|U\|^{2}_{L^{2}(C)}$

That gives us coercivity of bilinear form $A(U,\Psi)$. ∎

Therefore, we have the result.

######  Theorem 7.1.

The problem (7.1) admits a unique solution $U\in[H^{1}(C)]^{2}$ for any
$\Psi\in[H^{1}(C)]^{2}$, if coefficients satisfy

$\Re(a_{1})+\frac{|a_{0}||b_{1}+a^{*}_{1}/a^{*}_{0}|}{2}=0.$ (46)

###### Proof.

Lemmas 7.1-7.2 give us that the bilinear form $A(U,\Psi)$ verifies hypothesis
of the theorem 5.6.1 (see [14]). ∎

###  7.4 Two dimensional discretization

We again recognize that the two dimensional case is invariant in one direction
and a two dimensional object has a unidimensional boundary. So we will use
finite element method on a curvilinear contour. We use auxiliary variables
$X,Y$ that were mentioned in [7], to avoid integration by parts. So we take
formulation of bilinear form $A(U,\Psi)$ before carrying out integration by
parts And finally we get bilinear form for
$U=(J_{\tau},M_{\nu},X,Y)\in[H^{1}(C)]^{4}$

$A(U,\Psi)=iZ_{0}\iint_{C}kG(l,l^{\prime})\ J_{\tau}(l^{\prime})\
\Psi_{J\tau}\
[\bm{\tau}(l^{\prime})\cdot\bm{\tau}(l)]-\frac{1}{k}G(l,l^{\prime})\
d^{\prime}_{l}J_{\tau}(l^{\prime})\ d_{l}\Psi_{J\tau}(l)dl^{\prime}dl$
$+\iint_{C}\Psi_{J\tau}(l)\ M_{\nu}(l^{\prime})\
\mathbf{n}(l)\cdot\nabla_{\Gamma}G(l,l^{\prime})dl^{\prime}dl+\iint_{C}\Psi_{M\nu}\
J_{\tau}\
\mathbf{n}(l^{\prime})\cdot\nabla_{\Gamma}G(l,l^{\prime})dl^{\prime}dl$
$+\frac{i}{Z_{0}}\iint_{C}kG(l,l^{\prime})\ M_{\nu}(l^{\prime})\
\Psi_{M\nu}(l)dl^{\prime}dl+\frac{a_{0}}{2}\int_{C}J_{\tau}\
\Psi_{J\tau}dl+\frac{1}{2a_{0}}\int_{C}M_{\nu}\ \Psi_{M\nu}dl$
$+\frac{a_{1}}{2}\int_{C}d_{l}X\ \Psi_{J\tau}dl+\frac{b_{1}}{2}\int_{C}d_{l}Y\
\Psi_{J\tau}dl+\frac{b_{1}}{2a_{0}}\int_{C}d_{l}Y\
\Psi_{M\nu}dl+\frac{a_{1}}{2a_{0}}\int_{C}d_{l}X\ \Psi_{M\nu}dl$
$+c_{1}\int_{C}X\ X^{\prime}dl-c_{1}\int_{C}d_{l}J_{\tau}\
X^{\prime}dl+d_{1}\int_{C}Y\ Y^{\prime}dl-d_{1}\int_{C}d_{l}M_{\nu}\
Y^{\prime}dl$ (47)

for all
$\Psi=(\Psi_{J\tau},\Psi_{M\nu},X^{\prime},Y^{\prime})\in[H^{1}(C)]^{4}$.

Here, we study only TE polarization case. We assume that TM polarization has
analogous results.

We approximate the curve $C$ by means of $N$ straight line segments $C_{i}$,
satisfying the general overlapping conditions for a finite element method. The
line segments are also called elements and provide a contour piecewise linear
approximation $C^{h}=\cup^{N}_{i=1}C_{i}$, where $h$ is a positive parameter
such that $\lim_{h\rightarrow 0}N(h)=+\infty$. Curvilinear structures include
geometry modeling errors in this approach. These errors can only be reduced by
decreasing the segments lengths; i.e. by increasing the number of segments
$N$. We denote nodes from $1$ to $N$. We consider $V_{h}$ a finite dimensional
subspace of $H^{1}(C^{h})$

$V_{h}=\left\\{v_{h}:C^{h}\rightarrow\mbox{\bf R},\ v_{h}\in H^{1}(C^{h}),\
v_{h}|_{C_{i}}\in P_{1},\ \forall i\in{1,...,N}\right\\}\subset H^{1}(C^{h})$

where $P_{1}$ is the space of first degree polynomials, and

$W_{h}=\left\\{w_{h}:C^{h}\rightarrow\mbox{\bf R},\ w_{h}\in H^{1}(C^{h}),\
w_{h}|_{C_{i}}\in P_{0},\ \forall i\in{1,...,N}\right\\}\subset L^{2}(C^{h})$

where $P_{0}$ is the space of constant functions.

We observe discretization of the unknowns by basis functions

$J_{\tau}\approx J^{h}_{\tau}(l)=\sum^{N}_{i=1}J_{\tau i}\phi_{i}(l)\ \in
V_{h}$ (48) $M_{\nu}\approx M^{h}_{\nu}(l)=\sum^{N}_{i=1}M_{\nu i}\psi_{i}(l)\
\in W_{h}$ (49) $X\approx X^{h}(l)=\sum^{N}_{i=1}X_{i}\psi_{i}(l)\ \in W_{h}$
(50)

and

$Y\approx Y^{h}(l)=\sum^{N}_{i=1}Y_{i}\psi_{i}(l)\ \in W_{h}.$ (51)

where $\phi_{i}\in V_{h}$ and $\psi_{i}\in W_{h}$

So the bilinear form $A(U,\Psi)$ in (47) can be written as

$A(U^{h},\Psi^{h})=iZ_{0}\sum_{i,j=1}^{N}\left(\iint_{C^{h}}kG\ \phi_{j}\
\phi_{i}\ [\vec{\tau}_{j}(l^{\prime})\cdot\vec{\tau}_{i}(l)]-\frac{1}{k}G\
d^{\prime}_{l}\phi_{j}\ d_{l}\phi_{i}\ dl^{\prime}dl\right)J^{h}_{\tau j}$
$+\sum_{i,j=1}^{N}\left(\iint_{C^{h}}\psi_{j}\ \phi_{i}\
\mathbf{n}_{i}\cdot\nabla_{\Gamma}Gdl^{\prime}dl\right)M^{h}_{\nu j}$
$+\sum_{i,j=1}^{N}\left(\iint_{C^{h}}\phi_{j}\ \psi_{i}\
\mathbf{n}_{j}\cdot\nabla_{\Gamma}Gdl^{\prime}dl\right)J^{h}_{\tau
j}+\frac{i}{Z_{0}}\sum_{i,j=1}^{N}\left(\iint_{C^{h}}kG\ \psi_{j}\ \psi_{i}\
dl^{\prime}dl\right)M^{h}_{\nu j}$
$+\frac{a_{0}}{2}\sum_{i,j=1}^{N}\left(\int_{C^{h}}\phi_{j}\
\phi_{i}dl\right)J^{h}_{\tau
j}+\frac{1}{2a_{0}}\sum_{i,j=1}^{N}\left(\int_{C^{h}}\psi_{j}\
\psi_{i}dl\right)M^{h}_{\nu j}$
$+\frac{a_{1}}{2}\sum_{i,j=1}^{N}\left(\int_{C^{h}}d_{l}\psi_{j}\
\phi_{i}dl\right)X^{h}_{j}+\frac{b_{1}}{2}\sum_{i,j=1}^{N}\left(\int_{C^{h}}d_{l}\psi_{j}\
\phi_{i}dl\right)Y^{h}_{j}$
$+\frac{b_{1}}{2a_{0}}\sum_{i,j=1}^{N}\left(\int_{C^{h}}d_{l}\psi_{j}\
\psi_{i}dl\right)Y^{h}_{j}+\frac{a_{1}}{2a_{0}}\sum_{i,j=1}^{N}\left(\int_{C^{h}}d_{l}\psi_{j}\
\psi_{i}dl\right)X^{h}_{j}$ $+\sum_{i,j=1}^{N}\left(\int_{C^{h}}\psi_{j}\
\psi_{i}dl\right)X^{h}_{j}-\sum_{i,j=1}^{N}\left(\int_{C^{h}}d_{l}\psi_{j}\
\psi_{i}dl\right)J^{h}_{\tau j}$ $+\sum_{i,j=1}^{N}\left(\int_{C^{h}}\psi_{j}\
\psi_{i}dl\right)Y^{h}_{j}-\sum_{i,j=1}^{N}\left(\int_{C^{h}}d_{l}\psi_{j}\
\psi_{i}dl\right)M^{h}_{\nu j}$

In order to simplify the equations in matrix form we define the following
matrices

$(B-S)_{ij}=i\iint_{C^{h}}kG(l,l^{\prime})\ \phi_{j}(l^{\prime})\ \phi_{i}\
[\bm{\tau}_{j}\cdot\bm{\tau}_{i}]-\frac{1}{k}G(l,l^{\prime})\
d^{\prime}_{l}\phi_{j}(l^{\prime})\ d_{l}\phi_{i}(l)\ dl^{\prime}dl$
$Q_{ij}=\iint_{C^{h}}\phi_{i}(l)\ \psi_{j}(l^{\prime})\
\mathbf{n}_{i}\cdot\nabla_{\Gamma}G(l,l^{\prime})\ dl^{\prime}dl$
$B_{ij}=i\iint_{C^{h}}kG(l,l^{\prime})\ \psi_{j}(l^{\prime})\ \psi_{i}(l)\
dl^{\prime}dl$ $I1_{ij}=\int_{C^{h}}\phi_{i}(l)\ \phi_{j}(l)\ dl$
$I2_{ij}=\int_{C^{h}}\psi_{i}(l)\ \psi_{j}(l)\ dl$
$D1_{ij}=\int_{C^{h}}\phi_{i}(l)\ d_{l}\psi_{j}(l)\ dl$
$D3_{ij}=\int_{C^{h}}\psi_{i}(l)\ d_{l}\psi_{j}(l)\ dl$
$D5_{ij}=\int_{C^{h}}\psi_{i}(l)\ d_{l}\phi_{j}(l)\ dl$

Note: $[I2]$ is the nonsingular diagonal matrix, therefore it is invertible.

So we can write our system in a matrix form:

$\left[\begin{array}[]{cccc}Z_{0}[B-S]+\frac{a_{0}}{2}[I1]&[Q]&\frac{a_{1}}{2}[D1]&\frac{b_{1}}{2}[D1]\\\
{[Q]}^{T}&\frac{1}{Z_{0}}[B]+\frac{1}{2a_{0}}[I2]&\frac{a_{1}}{2a_{0}}[D3]&\frac{b_{1}}{2a_{0}}[D3]\\\
-[D5]&0&[I2]&0\\\
0&-[D3]&0&[I2]\end{array}\right]\left(\begin{array}[]{cccc}\overline{J}^{h}\\\
\overline{M}^{h}\\\ \overline{X}^{h}\\\
\overline{Y}^{h}\end{array}\right)=\left(\begin{array}[]{cccc}\overline{E}^{h}\\\
\overline{H}^{h}\\\ 0\\\ 0\end{array}\right)$ (52)

Where right-side vectors $\overline{E}^{h}$, $\overline{H}^{h}$ are defined as
follows:

$E_{i}^{h}=\int_{C^{h}}\mathbf{E}^{inc}\cdot\bm{\phi}_{i}dl;\,\,\,\,\,\,\,\,\,H_{i}^{h}=\int_{C^{h}}\mathbf{H}^{inc}\cdot\bm{\psi}_{i}dl.$

And vectors $\overline{J}^{h}$ and $\overline{M}^{h}$ are unknowns. Now, we
are interested in elimination of auxiliary variable. From the last two lines
in (52), we get

$-[D5]\ \overline{J}^{h}+[I2]\ \overline{X}^{h}=0\ \rightarrow\
\overline{X}^{h}={[I2]}^{-1}\ [D5]\ \overline{J}^{h};$ $-[D3]\
\overline{M}^{h}+[I2]\ \overline{Y}^{h}=0\ \rightarrow\
\overline{Y}^{h}={[I2]}^{-1}\ [D3]\ \overline{M}^{h}.$

We put $X,Y$ in first two equations.

$\left[\begin{array}[]{cc}[A1]&[A2]\\\
{[A3]}&[A4]\end{array}\right]\left(\begin{array}[]{cc}\overline{J}^{h}\\\
\overline{M}^{h}\end{array}\right)=\left(\begin{array}[]{cc}\overline{E}^{h}\\\
\overline{H}^{h}\end{array}\right)$ (53)

where matrices are defined as

$[A1]=Z_{0}[B-S]+\frac{a_{0}}{2}[I1]+\frac{a_{1}}{2}[D1]\ {[I2]}^{-1}\ [D5]$
$[A2]=[Q]+\frac{b_{1}}{2}[D1]\ {[I2]}^{-1}\ [D3]$
$[A3]={[Q]}^{T}+\frac{a_{1}}{2a_{0}}[D3]\ {[I2]}^{-1}\ [D5]$
$[A4]=\frac{1}{Z_{0}}[B]+\frac{1}{2a_{0}}[I2]+\frac{b_{1}}{2a_{0}}[D3]\
{[I2]}^{-1}\ [D3].$

###  7.5 Numerical tests 2D

Figure 5: Cylinder Figure 6: Plate with thin layer

In order to illustrate these points scattering by three typical coated
cylinders (see Figure 5) will be considered next. Figure 7 show the monostatic
RCS for a coated conducting cylinder with inner radius $\lambda_{0}$, coating
thickness $d=0.1\lambda_{0}$, and coating parameters $\epsilon_{r}=4.0-0.5i$
and $\mu_{r}=1.0$. The exact series solution is presented along with the HOIBC
and SIBC solutions. We computed monostatic RCS for different frequencies to
see how do results depend on frequency. In TE-polarization we can see that
results of SIBC jumps in range between $6GHz$ and $8GHz$ (see
LABEL:cyl_ep5_TE). Much bigger difference, we can see in TM-polarization
between $7GHz$ and $9GHz$ (see fig. LABEL:cyl_ep5_TM).

(a) TE polarization

(b) TM polarization

Figure 7: Monostatic RCS for a coated circular cylinder

Next, we consider bistatic RCS for different scattering angles. Figure 8 show
the bistatic radar cross section for a coated conducting cylinder with inner
radius $\lambda_{0}$, coating thickness $d=1.5mm$, and coating parameters
$\epsilon_{r}=10-5i$ and $\mu_{r}=1$, for fixed frequency $f=6.8GHz$ in TE and
TM polarizations. The exact series solution is presented along with the IBC0
and HOIBC order 1 and order 2 solutions.

(a) TE polarization

(b) TM polarization

Figure 8: Bistatic RCS for a coated circular cylinder with $d=1.5mm$,
$\epsilon_{r}=10-5i$, $\mu_{r}=1$ and $f=6.8GHz$

After we increase thickness of a boundary and decrease frequency, so we
considered bistatic RCS for different scattering angles. Figure 9 shows the
bistatic radar cross section for a coated conducting cylinder with inner
radius $\lambda_{0}$, coating thickness $d=3mm$ and frequency $f=3.4GHz$,
coating parameters $\epsilon_{r}=10-5i$ and $\mu_{r}=1.0$, in TE and TM
polarizations. The exact series solution is presented along with the SIBC and
HOIBC order 1 and order 2 solutions.

(a) TE polarization

(b) TM polarization

Figure 9: Bistatic RCS for a coated circular cylinder with TM polarization

Here we comput bistatic RCS for coated circular cylinder with parameters,
$d=0.1\lambda_{0},\epsilon_{r}=4-0.5i$ and $\mu_{r}=1$. And we compare to
Rahmat-Samii results for same test. The backscatter direction is
$\phi=180^{\circ}$. Results for exact formulation, IBC0 or Leontovich IBC
formulation and the formulation based on the planar higher order IBC are
presented in the figure 10. As can be seen in the figure, the results using
the planar HOIBC are in excellent agreement with the exact solution over most
of the angular range, while IBC0 solutions give only the average behavior of
the scattered field.

Figure 10: Bistatic RCS for a coated circular cylinder, when
$d=0.1\lambda_{0}$, $\epsilon_{r}=4-0.5j$, $\mu_{r}=1$ with TE polarization

Next we consider conducting plate with open boundary thin dielectric layer
(see fig. 6). Figures 10(a)-10(b) show the bistatic RCS for layer thickness
$d=4mm$ and frequency $f=6.8GHz$. This example is interesting because it shows
that method works even for open boundaries. And we can see that it solves
problem much better than with Leontovich IBC. But it is difficult to see
difference between first order and second order IBCs.

(a) TE polarization

(b) TM polarization

Figure 11: Bistatic RCS for a coated 2D plate

Then we propose a new variational formulation where electric and magnetic
currents are the principal unknowns and in which we sum the weak form of the
HOIBC with the Electric Field Integral Equation (EFIE) and use the Magnetic
Field Integral Equation (MFIE) as a second equation.

##  8 Variational formulation for 3D problem

Moreover, we have the boundary condition on $\Gamma$:

$(I+b_{1}L_{D}-b_{2}L_{R})(\mathbf{n}\times\textbf{M})=(a_{0}I+a_{1}L_{D}-a_{2}L_{R})\mathbf{J}.$
(54)

We sum the weak form of the HOIBC with the EFIE and use the MFIE as in (33)
and (34) and using properties of vector analysis [12], we get

$A((\mathbf{J},\mathbf{M}),(\bm{\Psi}_{J},\bm{\Psi}_{M}))=F((\bm{\Psi}_{J},\bm{\Psi}_{M}))$

where

$F((\bm{\Psi}_{J},\bm{\Psi}_{M}))=\int_{\Gamma}\mathbf{E}^{inc}\cdot\bm{\Psi}_{J}ds+\int_{\Gamma}\mathbf{H}^{inc}\cdot\bm{\Psi}_{M}ds$

and

$A((\mathbf{J},\mathbf{M}),(\bm{\Psi}_{J},\bm{\Psi}_{M}))=<Z_{0}(B-S)\textbf{J},\bm{\Psi}_{J}>+\frac{1}{Z_{0}}<(B-S)\textbf{M},\bm{\Psi}_{M}>$
$+<Q\textbf{M},\bm{\Psi}_{J}>-<Q\textbf{J},\bm{\Psi}_{M}>+\frac{a_{0}}{2}<\textbf{J},\bm{\Psi}_{J}>+\frac{1}{2a_{0}}<\textbf{M},\bm{\Psi}_{M}>$
$-\frac{a_{1}}{2}<\mathrm{div}_{\Gamma}\textbf{J},\mathrm{div}_{\Gamma}\bm{\Psi}_{J}>-\frac{a_{2}}{2}<\mathrm{div}_{\Gamma}(\mathbf{n}\times\textbf{J}),\mathrm{div}_{\Gamma}(\mathbf{n}\times\bm{\Psi}_{J})>$
$+\frac{b_{1}}{2}<\mathrm{div}_{\Gamma}(\mathbf{n}\times\textbf{M}),\mathrm{div}_{\Gamma}\bm{\Psi}_{J}>-\frac{b_{2}}{2}<\mathrm{div}_{\Gamma}\textbf{M},\mathrm{div}_{\Gamma}(\mathbf{n}\times\bm{\Psi}_{J})>$
$-\frac{b_{1}}{2a_{0}}<\mathrm{div}_{\Gamma}(\mathbf{n}\times\textbf{M}),\mathrm{div}_{\Gamma}(\mathbf{n}\times\bm{\Psi}_{M})>-\frac{b_{2}}{2a_{0}}<\mathrm{div}_{\Gamma}\textbf{M},\mathrm{div}_{\Gamma}\bm{\Psi}_{M}>$
$+\frac{a_{1}}{2a_{0}}<\mathrm{div}_{\Gamma}\textbf{J},\mathrm{div}_{\Gamma}(\mathbf{n}\times\bm{\Psi}_{M})>-\frac{a_{2}}{2a_{0}}<\mathrm{div}_{\Gamma}(\mathbf{n}\times\textbf{J}),\mathrm{div}_{\Gamma}\bm{\Psi}_{M}>$

As $\mathbf{A}\in H^{-1/2}(\mathrm{div},\Gamma)$, the
$\mathrm{rot}_{\Gamma}\mathbf{A}$ and
$\mathrm{div}_{\Gamma}(\mathbf{n}\times\mathbf{A})$ is problematic. To
overcome this difficulty we introduce Lagrange multipliers $\bm{\lambda}_{J}$
and $\bm{\lambda}_{M}$ as in [13]:

$\int_{\Gamma}\bm{\lambda}_{J}\cdot(\tilde{\bm{\Psi}}_{J}-\mathbf{n}\times\bm{\Psi}_{J})ds=0\;\;\forall(\tilde{\bm{\Psi}}_{J},(\bm{\Psi}_{J}),$
$\int_{\Gamma}\bm{\lambda}_{M}\cdot(\tilde{\bm{\Psi}}_{M}-\mathbf{n}\times\bm{\Psi}_{M})ds=0,\;\;\forall(\tilde{\bm{\Psi}}_{M},(\bm{\Psi}_{M}),$

and we have:

$\int_{\Gamma}\bm{\lambda}^{\prime}_{J}\cdot(\tilde{\mathbf{J}}-\mathbf{n}\times\mathbf{J})ds=0\;\;\bm{\lambda}^{\prime}_{J}$
$\int_{\Gamma}\bm{\lambda}^{\prime}_{M}\cdot(\tilde{\mathbf{M}}-\mathbf{n}\times\mathbf{M})ds=0\;\;\forall\bm{\lambda}^{\prime}_{M}$

Finally, we have the following problem:

######  Problem 8.1.

Find $U=(\mathbf{J},\mathbf{M},\tilde{\mathbf{J}},\tilde{\mathbf{M}})\in
V=[H^{-1/2}(div,\Gamma)\cap L^{2}(\Gamma)]^{4}$ and
$\lambda=(\bm{\lambda}_{J},\bm{\lambda}_{M})\in[H^{-1/2}(\Gamma)]^{2}$ such
that

$\begin{cases}A(U,\Psi)+B^{T}(\lambda,\Psi)=F(\Psi)\\\
B(U,\lambda^{\prime})=0\end{cases}$ (55)

for all
$\Psi=(\bm{\Psi}_{J},\bm{\Psi}_{M},\tilde{\bm{\Psi}}_{J},\tilde{\bm{\Psi}}_{M})\in
V=[H^{-1/2}(div,\Gamma)\cap L^{2}(\Gamma)]^{4}$ and
$\lambda^{\prime}=(\bm{\lambda}^{\prime}_{J},\bm{\lambda}^{\prime}_{M})\in
W=[H^{-1/2}(\Gamma)]^{2}$.

The bilinear forms are defined by:

$B(U,\lambda^{\prime})=\int_{\Gamma}\bm{\lambda}^{\prime}_{J}\cdot(\tilde{\mathbf{J}}-\mathbf{n}\times\mathbf{J})ds+\int_{\Gamma}\bm{\lambda}^{\prime}_{M}\cdot(\tilde{\mathbf{M}}-\mathbf{n}\times\mathbf{M})ds$

and

$A(U,\Psi)=iZ_{0}\iint_{\Gamma}kG\
(\textbf{J}\cdot\bm{\Psi}_{J})-\frac{1}{k}G\ \mathrm{div}{\bm{\Psi}_{J}}\
\mathrm{div}{\textbf{J}}dsds^{\prime}$ (56) $+\frac{i}{Z_{0}}\iint_{\Gamma}kG\
(\bm{\Psi}_{M}\cdot\textbf{M})-\frac{1}{k}G\ \mathrm{div}{\bm{\Psi}_{M}}\
\mathrm{div}{\textbf{M}}dsds^{\prime}$
$+\iint_{\Gamma}\nabla^{\prime}G\cdot(\bm{\Psi}_{J}\times\textbf{M})dsds^{\prime}-i\iint_{\Gamma}\nabla^{\prime}G\cdot(\bm{\Psi}_{M}\times\textbf{J})dsds^{\prime}$
$+\frac{a_{0}}{2}\int_{\Gamma}\textbf{J}\cdot\bm{\Psi}_{J}ds+\frac{1}{2a_{0}}\int_{\Gamma}\textbf{M}\cdot\bm{\Psi}_{M}ds$
$-\frac{a_{1}}{2}\int_{\Gamma}\mathrm{div}_{\Gamma}\textbf{J}\
\mathrm{div}_{\Gamma}\bm{\Psi}_{J}ds-\frac{a_{2}}{2}\int_{\Gamma}\mathrm{div}_{\Gamma}\tilde{\textbf{J}}\
\mathrm{div}_{\Gamma}\tilde{\bm{\Psi}}_{J}ds$
$+\frac{b_{1}}{2}\int_{\Gamma}\mathrm{div}_{\Gamma}\tilde{\textbf{M}}\
\mathrm{div}_{\Gamma}\bm{\Psi}_{J}ds-\frac{b_{2}}{2}\int_{\Gamma}\mathrm{div}_{\Gamma}\textbf{M}\
\mathrm{div}_{\Gamma}\tilde{\bm{\Psi}}_{J}ds$
$-\frac{b_{1}}{2a_{0}}\int_{\Gamma}\mathrm{div}_{\Gamma}\tilde{\textbf{M}}\
\mathrm{div}_{\Gamma}\tilde{\bm{\Psi}}_{M}ds-\frac{b_{2}}{2a_{0}}\int_{\Gamma}\mathrm{div}_{\Gamma}\textbf{M}\
\mathrm{div}_{\Gamma}\bm{\Psi}_{M}ds$
$+\frac{a_{1}}{2a_{0}}\int_{\Gamma}\mathrm{div}_{\Gamma}\textbf{J}\
\mathrm{div}_{\Gamma}\tilde{\bm{\Psi}}_{M}ds-\frac{a_{2}}{2a_{0}}\int_{\Gamma}\mathrm{div}_{\Gamma}\tilde{\textbf{J}}\
\mathrm{div}_{\Gamma}\bm{\Psi}_{M}ds.$

In the next section, we study mathematically this problem.

###  8.1 Existence and uniqueness theorem

We use a theorem from [14] to prove existence and uniqueness of a solution to
the problem 8.1. In the fisrt time we prove the following result.

######  Lemma 8.1.

The operator $A$ is continuous on $V\times V$ for all $\Psi\in V$ and we have
that

$|A(U,\Psi)|\leq C\|U\|_{V}\|\Psi\|_{V}$

###### Proof.

: In order to prove this lemma, it is convenient to introduce a decomposition
of $A$ as $A=A_{1}+A_{2}+A_{3}$
where

$A_{1}(U,\Psi)=<Z_{0}(B-S)\textbf{J},\bm{\Psi}_{J}>+\frac{1}{Z_{0}}<(B-S)\textbf{M},\bm{\Psi}_{M}>$
$+<Q\textbf{M},\bm{\Psi}_{J}>-<Q\textbf{J},\bm{\Psi}_{M}>+\frac{a_{0}}{2}<\textbf{J},\bm{\Psi}_{J}>+\frac{1}{2a_{0}}<\textbf{M},\bm{\Psi}_{M}>$
$A_{2}(U,\Psi)=-\frac{a_{1}}{2}<\mathrm{div}_{\Gamma}\textbf{J},\mathrm{div}_{\Gamma}\bm{\Psi}_{J}>-\frac{a_{2}}{2}<\mathrm{div}_{\Gamma}\tilde{\textbf{J}},\mathrm{div}_{\Gamma}\tilde{\bm{\Psi}}_{J}>$
$-\frac{b_{1}}{2a_{0}}<\mathrm{div}_{\Gamma}\tilde{\textbf{M}},\mathrm{div}_{\Gamma}\tilde{\bm{\Psi}}_{M}>-\frac{b_{2}}{2a_{0}}<\mathrm{div}_{\Gamma}\textbf{M},\mathrm{div}_{\Gamma}\bm{\Psi}_{M}>$

and

$A_{3}(U,\Psi)=\frac{b_{1}}{2}<\mathrm{div}_{\Gamma}\tilde{\textbf{M}},\mathrm{div}_{\Gamma}\bm{\Psi}_{J}>-\frac{b_{2}}{2}<\mathrm{div}_{\Gamma}\textbf{M},\mathrm{div}_{\Gamma}\tilde{\bm{\Psi}}_{J}>$
$+\frac{a_{1}}{2a_{0}}<\mathrm{div}_{\Gamma}\textbf{J},\mathrm{div}_{\Gamma}\tilde{\bm{\Psi}}_{M}>-\frac{a_{2}}{2a_{0}}<\mathrm{div}_{\Gamma}\tilde{\textbf{J}},\mathrm{div}_{\Gamma}\bm{\Psi}_{M}>$

Hence, we get by theorems $2.2$ and $4.6$ in [15] that there exists constants
$C_{1}$ and $C_{2}$ such that

$|A_{1}(U,\Psi)|\leq C_{1}\|U\|_{V}\|\Psi\|_{V},$ (57)

and

$|A_{2}(U,\Psi)+A_{3}(U,\Psi)|\leq C_{2}\|U\|_{V}\|\Psi\|_{V}$ (58)

Combining (57) and (58) we obtain:

$|A(U,\Psi)|=|A_{1}+A_{2}+A_{3}|\leq|A_{1}(U,\Psi)|+|A_{2}(U,\Psi)+A_{3}(U,\Psi)|\leq
C\|U\|_{V}\|\Psi\|_{V}$

where $C=C_{1}+C_{2}$. ∎

We have also the lemma.

######  Lemma 8.2.

The operator $A$ is coercive on $V$. We have to show that there exist
$\alpha>0$ such that

$\Re[A(U,U^{*})]\geq\alpha\|U\|^{2}_{V}-C\|U\|^{2}_{V},\ \forall U\in V.$

###### Proof.

: From [15], it follow that there exists $\alpha$ such that

$\Re(A_{1})=\Re(<Z_{0}(B-S)\mathbf{J},\mathbf{J}^{*}>)+\Re(<Z^{-1}_{0}(B-S)\mathbf{M},\mathbf{M}^{*}>)+\Re(<Q\mathbf{M},\mathbf{J}^{*}>)$
$-\Re(<Q\mathbf{J},\mathbf{M}^{*}>)+\Re(\frac{a_{0}}{2}\int_{\Gamma}\mathbf{J}\cdot\mathbf{J}^{*}ds)+\Re(\frac{1}{2a_{0}}\int_{\Gamma}\mathbf{M}\cdot\mathbf{M}^{*}ds)$
$\geq\alpha\left(\|\mathbf{J}\|^{2}_{-1/2,\mathrm{div}_{\Gamma}}+\|\mathbf{M}\|^{2}_{-1/2,\mathrm{div}_{\Gamma}}\right)+\frac{\Re(a_{0})}{2}\|\mathbf{J}\|^{2}_{L^{2}(\Gamma)}+\frac{\Re(a_{0})}{2|a_{0}|^{2}}\|\mathbf{M}\|^{2}_{L^{2}(\Gamma)}.$

We can easily show that

$\Re(A_{2})=-\frac{\Re(a_{1})}{2}\|\mathrm{div}_{\Gamma}\mathbf{J}\|^{2}_{L^{2}(\Gamma)}-\frac{\Re(a_{2})}{2}\|\mathrm{div}_{\Gamma}\tilde{\mathbf{J}}\|^{2}_{L^{2}(\Gamma)}$
$-\frac{\Re(b_{1}a^{*}_{0})}{2|a_{0}|^{2}}\|\mathrm{div}_{\Gamma}\tilde{\mathbf{M}}\|^{2}_{L^{2}(\Gamma)}-\frac{\Re(b_{2}a^{*}_{0})}{2|a_{0}|^{2}}\|\mathrm{div}_{\Gamma}\mathbf{M}\|^{2}_{L^{2}(\Gamma)}$

For the rest, we do as follows:

$\Re(A_{3})=\Re(\frac{b_{1}}{2}\int_{\Gamma}\mathrm{div}_{\Gamma}\tilde{\mathbf{M}}\
\mathrm{div}_{\Gamma}\mathbf{J}^{*}ds)-\Re(\frac{b_{2}}{2}\int_{\Gamma}\mathrm{div}_{\Gamma}\mathbf{M}\
\mathrm{div}_{\Gamma}\tilde{\mathbf{J}}^{*}ds)$
$+\Re(\frac{a_{1}}{2a_{0}}\int_{\Gamma}\mathrm{div}_{\Gamma}\mathbf{J}\
\mathrm{div}_{\Gamma}\tilde{\mathbf{M}}^{*}ds)-\Re(\frac{a_{2}}{2a_{0}}\int_{\Gamma}\mathrm{div}_{\Gamma}\tilde{\mathbf{J}}\
\mathrm{div}_{\Gamma}\mathbf{M}^{*}ds)$
$=\Re\left\\{\left(\frac{b_{1}}{2}+\frac{a^{*}_{1}}{2a^{*}_{0}}\right)\int_{\Gamma}\mathrm{div}_{\Gamma}\tilde{\mathbf{M}}\
\mathrm{div}_{\Gamma}\mathbf{J}^{*}ds\right\\}-\Re\left\\{\left(\frac{b_{2}}{2}+\frac{a^{*}_{2}}{2a^{*}_{0}}\right)\int_{\Gamma}\mathrm{div}_{\Gamma}\tilde{\mathbf{J}}^{*}\
\mathrm{div}_{\Gamma}\mathbf{M}ds\right\\}$
$=\Re\left\\{\int_{\Gamma}\frac{1}{|a_{0}|^{1/2}}\left(\frac{b_{1}}{2}+\frac{a^{*}_{1}a_{0}}{2|a_{0}|^{2}}\right)^{1/2}\mathrm{div}_{\Gamma}\tilde{\mathbf{M}}\
\cdot|a_{0}|^{1/2}\left(\frac{b_{1}}{2}+\frac{a^{*}_{1}a_{0}}{2|a_{0}|^{2}}\right)^{1/2}\mathrm{div}_{\Gamma}\mathbf{J}^{*}ds\right\\}$
$-\Re\left\\{\int_{\Gamma}|a_{0}|^{1/2}\left(\frac{b_{2}}{2}+\frac{a^{*}_{2}a_{0}}{2|a_{0}|^{2}}\right)^{1/2}\mathrm{div}_{\Gamma}\tilde{\mathbf{J}}^{*}\cdot\frac{1}{|a_{0}|^{1/2}}\left(\frac{b_{2}}{2}+\frac{a^{*}_{2}a_{0}}{2|a_{0}|^{2}}\right)^{1/2}\mathrm{div}_{\Gamma}\mathbf{M}ds\right\\}.$

We also define $q_{1}$ and $q_{2}$ by

$q_{1}=b_{1}|a_{0}|+a^{*}_{1}a_{0}/|a_{0}|\,\,\,\,\,\,\,\,\,q_{2}=b_{2}|a_{0}|+a^{*}_{2}a_{0}/|a_{0}|,$

we find that

$\Re(A_{3})\geq-\frac{|q_{1}|}{4}\|\mathrm{div}_{\Gamma}\mathbf{J}\|^{2}_{L^{2}(\Gamma)}-\frac{|q_{1}|}{4|a_{0}|^{2}}\|\mathrm{div}_{\Gamma}\tilde{\mathbf{M}}\|^{2}_{L^{2}(\Gamma)}$
$-\frac{|q_{2}|}{4}\|\mathrm{div}_{\Gamma}\tilde{\mathbf{J}}\|^{2}_{L^{2}(\Gamma)}-\frac{|q_{2}|}{4|a_{0}|^{2}}\|\mathrm{div}_{\Gamma}\mathbf{M}\|^{2}_{L^{2}(\Gamma)}.$

Using the conditions on coefficients

$\Re(a_{j})+\frac{|q_{j}|}{2}=0,\,\,\,\,\,\,\,\text{where}\,\,\,j=1,2$

and from the sufficient uniqueness conditions we have that
$\Re(a_{j})=\Re(b_{j}^{*}a_{0})$.
Thus, we obtain

$\Re(A_{2})+\Re(A_{3})\geq 0.$

Finally we have for operator $A$ that

$\Re(A)\geq\alpha\left(\|\mathbf{J}\|^{2}_{-1/2,\mathrm{div}_{\Gamma}}+\|\mathbf{M}\|^{2}_{-1/2,\mathrm{div}_{\Gamma}}\right)+\frac{\Re(a_{0})}{2}\|\mathbf{J}\|^{2}_{L^{2}(\Gamma)}+\frac{\Re(a_{0})}{2|a_{0}|^{2}}\|\mathbf{M}\|^{2}_{L^{2}(\Gamma)}$

∎

######  Lemma 8.3.

The operator $B$ verifies the inequality

$\sup_{\|U\|_{V}=1}|B(U,\lambda)|\geq\beta\|\lambda\|_{W},\ \forall\lambda\in
W=[H^{-1/2}(\Gamma)]^{3}\times[H^{-1/2}(\Gamma)]^{3}$

where $U\in V=[H^{-1/2}(div,\Gamma)\cap L^{2}(\Gamma)]^{4}$ and $\beta>0$.

###### Proof.

: Here we have to show that there exists $\beta>0$ such that

$\sup_{\|U\|_{V}=1}\left|\int_{\Gamma}\bm{\lambda}_{J}\cdot(\tilde{\mathbf{J}}-n\times\mathbf{J})+\bm{\lambda}_{M}\cdot(\tilde{\mathbf{M}}-n\times\mathbf{M})ds\right|\geq\beta\|\lambda\|_{W}$

First, we take

$\mathbf{J}=0;\ \mathbf{M}=0;\
\tilde{\mathbf{J}}=\frac{\hat{\mathbf{J}}}{\|\hat{\mathbf{J}}\|_{V}}\ \ and\ \
\tilde{\mathbf{J}}=\frac{\hat{\mathbf{M}}}{\|\hat{\mathbf{M}}\|_{V}}$
$\hat{\mathbf{J}}=\int_{\Gamma\setminus
x}\frac{\bm{\lambda}_{J}}{|x-y|}ds_{y}\ \ \ and\ \ \
\hat{\mathbf{M}}=\int_{\Gamma\setminus x}\frac{\bm{\lambda}_{M}}{|x-y|}ds_{y}$

so, we get following inequality

$\sup_{\|U\|_{V}=1}\left|\int_{\Gamma}\bm{\lambda}_{J}\cdot(\tilde{\mathbf{J}}-n\times\mathbf{J})+\bm{\lambda}_{M}\cdot(\tilde{\mathbf{M}}-n\times\mathbf{M})ds\right|\geq$
$\frac{1}{\|\hat{\mathbf{J}}\|_{-1/2,\mathrm{div}_{\Gamma}}}\iint_{\Gamma\Gamma}\frac{\bm{\lambda}_{J}(x)\bm{\lambda}_{J}(y)}{|x-y|}ds_{y}ds_{x}+\frac{1}{\|\hat{\mathbf{M}}\|_{-1/2,\mathrm{div}_{\Gamma}}}\iint_{\Gamma\Gamma}\frac{\bm{\lambda}_{M}(x)\bm{\lambda}_{M}(y)}{|x-y|}ds_{y}ds_{x}$
(59)

Using the Planchard-Nédélec inequality [16], we get

$\iint_{\Gamma\Gamma}\frac{\mathbf{\lambda}(x)\mathbf{\lambda}(y)}{|x-y|}ds_{y}ds_{x}\geq\beta\|\mathbf{\lambda}\|^{2}_{-1/2,\Gamma}$
(60)

and therefore, by (59) and (60), it follows that

$\sup_{\|U\|_{V}=1}\left|\int_{\Gamma}\bm{\lambda}_{J}\cdot(\tilde{\mathbf{J}}-n\times\mathbf{J})+\bm{\lambda}_{M}\cdot(\tilde{\mathbf{M}}-n\times\mathbf{M})ds\right|\geq$
$\frac{1}{\|\hat{\mathbf{J}}\|_{-1/2,\mathrm{div}_{\Gamma}}}\beta_{J}\|\mathbf{\lambda}_{J}\|^{2}_{-1/2,\Gamma}+\frac{1}{\|\hat{\mathbf{M}}\|_{-1/2,\mathrm{div}_{\Gamma}}}\beta_{M}\|\mathbf{\lambda}_{M}\|^{2}_{-1/2,\Gamma}$
(61)

furthermore, there exist $C_{J}>0$ and $C_{M}>0$ such that

$\|\hat{\mathbf{J}}\|_{-1/2,\mathrm{div}_{\Gamma}}\leq
C_{J}\|\mathbf{\lambda}_{J}\|_{-1/2,\Gamma}\ \ and\ \
\|\hat{\mathbf{M}}\|_{-1/2,\mathrm{div}_{\Gamma}}\leq
C_{M}\|\mathbf{\lambda}_{M}\|_{-1/2,\Gamma}$ (62)

Consequently, we obtain that

$\displaystyle\sup_{\|U\|_{V}=1}\left|\int_{\Gamma}\bm{\lambda}_{J}\cdot(\tilde{\mathbf{J}}-n\times\mathbf{J})+\bm{\lambda}_{M}\cdot(\tilde{\mathbf{M}}-n\times\mathbf{M})ds\right|$
$\displaystyle\geq$
$\displaystyle\frac{\beta_{J}}{C_{J}}\|\mathbf{\lambda}_{J}\|_{-1/2,\Gamma}+\frac{\beta_{M}}{C_{M}}\|\mathbf{\lambda}_{M}\|_{-1/2,\Gamma}$
$\displaystyle\geq$ $\displaystyle\beta\|\lambda\|_{W}$

where $\beta=\min(\beta_{J}/C_{J};\beta_{M}/C_{M})$. ∎

######  Theorem 8.1.

The problem (8.1) admits a unique solution $U\in V=[H^{-1/2}(div,\Gamma)\cap
L^{2}(\Gamma)]^{4}$ and $\lambda\in[H^{-1/2}(\Gamma)]^{2}$, if coefficients
satisfy

$\Re(a_{j})+\frac{|a_{0}||b_{j}+a^{*}_{j}/a^{*}_{0}|}{2}=0\ \ for\ \ j=1,2.$
(63)

###### Proof.

Using the above lemma, we can show as in [14] that the variational problem 8.1
has a unique solution. ∎

In the next, a discretization of the problem 8.1 is done with Reo-Wilton
functions.

###  8.2 Discretization of the variational problem with HOIBC

The first step is to approach the surface of the obstacle by a surface
$\Gamma_{h}$ composed of finite number of two dimensional elements. These
elements are triangular facets denoted by $T_{i}$ for $i=1$ to $N_{T}$:

$\Gamma_{h}=\bigcup_{i=1}^{N_{T}}T_{i}.$

We denote by $N_{e}$ the total number of edges of the mesh component
$\Gamma_{h}$. Let $\\{\mathbf{f}_{i}\\}_{i=1,N_{e}}$ be a function of Rao-
Wilton-Glisson functions,[17, 18]. We decompose the electric and magnetic
currents with thes basis functions:

$\mathbf{J}(y)=\sum^{N_{e}}_{l=1}J_{l}\mathbf{f}_{l}(y),\ \
\mathbf{M}(y)=\sum^{N_{e}}_{l=1}M_{l}\mathbf{f}_{l}(y),$

as well as auxiliary unknowns

$\tilde{\mathbf{J}}(x)=\sum^{N_{e}}_{l=1}\tilde{J}_{l}\mathbf{f}_{l}(x),\ \
\tilde{\mathbf{M}}(x)=\sum^{N_{e}}_{l=1}\tilde{M}_{l}\mathbf{f}_{l}(x).$

For the Lagrange multipliers, we use basis functions proposed in [13]:

$\lambda_{J}=\sum^{N_{e}}_{k=1}\lambda_{Jk}\mathbf{g}_{k},\ \
\lambda_{M}=\sum^{N_{e}}_{k=1}\lambda_{Mk}\mathbf{g}_{k},$ (64)

where $\mathbf{g_{k}}$ is defined as follows:

$\mathbf{g}_{n}(x)=\left\\{\begin{array}[]{cc}(1-2\omega^{+}_{i+2}(x))(\nu_{n}\times\mathbf{n}^{+}),\
\ \ \ x\in T^{+}_{n}\\\
(1-2\omega^{-}_{i+2}(x))(\nu_{n}\times\mathbf{n}^{-}),\ \ \ \ x\in
T^{-}_{n}\end{array}\right.$ (65)

with $\nu_{n}$ is a direction vector of the edge $\mathbf{n}$ and
$\\{\omega_{i}\\}_{i=1,3}$ are barycentric coordinates of $x$ relative to
triangles $T^{+}_{n}$ or $T^{+}_{n}$ (see [21]).

Then, the discretized problem of (55) is:

$\begin{cases}A^{h}(U_{h},\Psi_{h})+B^{h}(\lambda_{h},\Psi_{h})=\sum_{i=1}^{N_{e}}<\mathbf{E}^{inc},\mathbf{f}_{i}>+\sum_{i=1}^{N_{e}}<\mathbf{H}^{inc},\mathbf{f}_{i}>\\\
B^{h}(U_{h},\lambda_{h})=0\end{cases}$ (66)

where

$A^{h}(U_{h},\Psi_{h})=\sum^{N_{e}}_{i,j=1}<Z_{0}(B-S)\textbf{f}_{j},\mathbf{f}_{i}>J_{j}+Z^{-1}_{0}\sum^{N_{e}}_{i,j=1}<(B-S)\textbf{f}_{j},\mathbf{f}_{i}>M_{j}$
$+\sum^{N_{e}}_{i,j=1}<Q\textbf{f}_{j},\mathbf{f}_{i}>M_{j}-\sum^{N_{e}}_{i,j=1}<Q\textbf{f}_{j},\mathbf{f}_{i}>J_{j}+\frac{a_{0}}{2}\sum^{N_{e}}_{i,j=1}<\textbf{f}_{j},\mathbf{f}_{i}>J_{j}+\frac{1}{2a_{0}}\sum^{N_{e}}_{i,j=1}<\textbf{f}_{j},\mathbf{f}_{i}>M_{j}$
$-\frac{a_{1}}{2}\sum^{N_{e}}_{i,j=1}<\mathbf{div}_{\Gamma}\textbf{f}_{j},\mathbf{div}_{\Gamma}\mathbf{f}_{i}>J_{j}-\frac{a_{2}}{2}\sum^{N_{e}}_{i,j=1}<\mathbf{div}_{\Gamma}\textbf{f}_{j},\mathbf{div}_{\Gamma}\mathbf{f}_{i}>\tilde{J}_{j}$
$+\frac{b_{1}}{2}\sum^{N_{e}}_{i,j=1}<\mathbf{div}_{\Gamma}\textbf{f}_{j},\mathbf{div}_{\Gamma}\mathbf{f}_{i}>\tilde{M}_{j}-\frac{b_{2}}{2}\sum^{N_{e}}_{i,j=1}<\mathbf{div}_{\Gamma}\textbf{f}_{j},\mathbf{div}_{\Gamma}\mathbf{f}_{i}>M_{j}$
$-\frac{b_{1}}{2a_{0}}\sum^{N_{e}}_{i,j=1}<\mathbf{div}_{\Gamma}\textbf{f}_{j},\mathbf{div}_{\Gamma}\mathbf{f}_{i}>\tilde{M}_{j}-\frac{b_{2}}{2a_{0}}\sum^{N_{e}}_{i,j=1}<\mathbf{div}_{\Gamma}\textbf{f}_{j},\mathbf{div}_{\Gamma}\mathbf{f}_{i}>M_{j}$
$+\frac{a_{1}}{2a_{0}}\sum^{N_{e}}_{i,j=1}<\mathbf{div}_{\Gamma}\textbf{f}_{j},\mathbf{div}_{\Gamma}\mathbf{f}_{i}>J_{j}-\frac{a_{2}}{2a_{0}}\sum^{N_{e}}_{i,j=1}<\mathbf{div}_{\Gamma}\textbf{f}_{j},\mathbf{div}_{\Gamma}\mathbf{f}_{i}>\tilde{J}_{j};$

and

$B^{h}(U_{h},\lambda^{\prime}_{h})=\sum^{N_{e}}_{i,k=1}<\mathbf{g}_{k},\mathbf{f}_{i}>\tilde{J}_{i}-\sum^{N_{e}}_{i,k=1}<\mathbf{g}_{k},\mathbf{n}\times\mathbf{f}_{i}>J_{i}$
$+\sum^{N_{e}}_{i,k=1}<\mathbf{g}_{k},\mathbf{f}_{i}>\tilde{M}_{i}-\sum^{N_{e}}_{i,k=1}<\mathbf{g}_{k},\mathbf{n}\times\mathbf{f}_{i}>M_{i}$

Obviously, this problem has a unique solution since we use conforming finite
element method.

Now, we express the problem in matrix form:

$(B-S)_{i,j}=i\int\int_{\Gamma_{h}}kG(s,s^{\prime})\mathbf{f}_{j}(s^{\prime})\cdot\mathbf{f}_{i}(s)-\frac{1}{k}G(s,s^{\prime})(\mathbf{div}_{\Gamma}\mathbf{f}_{i})(\mathbf{div}_{\Gamma}^{\prime}\mathbf{f}_{j})dsds^{\prime}$
$Q_{i,j}=-i\int\int_{\Gamma_{h}}[\mathbf{f}_{i}(s)\times\mathbf{f}_{j}(s^{\prime})]\cdot\nabla^{\prime}_{\Gamma}G(s,s^{\prime})dsds^{\prime}$
$I_{i,j}=\int_{\Gamma_{h}}\mathbf{f}_{i}\cdot\mathbf{f}_{j}ds;\ \ \ \
D_{i,j}=\int_{\Gamma_{h}}(\mathbf{div}_{\Gamma}\mathbf{f}_{j})(\mathbf{div}_{\Gamma}\mathbf{f}_{i})ds$
$C_{Hi,j}=\int_{\Gamma_{h}}\mathbf{g}_{i}\cdot\mathbf{f}_{j}ds;\ \ \ \
C_{Ki,j}=\int_{\Gamma_{h}}\mathbf{g}_{i}\cdot(\mathbf{n}\times\mathbf{f}_{j})ds,$
(67)

where $[C_{H}]$ is a nonsingular diagonal matrix. Therefore it is invertible.
Then, we define $[A1]$ and $[A2]$ by

$[A1]=[(B-S)]+\frac{a_{0}}{2}[I]-\frac{a_{1}}{2}[D],\ \
[A2]=[(B-S)]+\frac{1}{2a_{0}}[I]-\frac{b_{2}}{2a_{0}}[D].$ (68)

Using (67) and (68), we now present (66) in the following matrix form

$\left(\begin{array}[]{cccccc}[A1]&[Q]&0&\frac{b_{1}}{2}[D]&[C_{K}]^{T}&0\\\
{[Q]}^{T}&[A2]&-\frac{a_{2}}{2a_{0}}[D]&0&0&[C_{K}]^{T}\\\
0&-\frac{b_{2}}{2}[D]&-\frac{a_{2}}{2}[D]&0&[C_{H}]^{T}&0\\\
\frac{a_{1}}{2a_{0}}[D]&0&0&-\frac{b_{1}}{2a_{0}}[D]&0&[C_{H}]^{T}\\\
{[C_{K}]}&0&[C_{H}]&0&0&0\\\
0&[C_{K}]&0&[C_{H}]&0&0\end{array}\right)\left(\begin{array}[]{cccccc}\overline{J}\\\
\overline{M}\\\ \overline{\tilde{J}}\\\ \overline{\tilde{M}}\\\
\overline{\lambda_{J}}\\\
\overline{\lambda_{M}}\end{array}\right)=\left(\begin{array}[]{cccccc}\overline{E}\\\
\overline{H}\\\ 0\\\ 0\\\ 0\\\ 0\end{array}\right)$ (69)

where right-side vectors $\overline{E}$, $\overline{H}$ are defined as
follows:

$E_{i}=\int_{\Gamma_{h}}\mathbf{E}^{inc}\cdot\mathbf{f}_{i}ds;\ \ \ \
H_{i}=\int_{\Gamma_{h}}\mathbf{H}^{inc}\cdot\mathbf{f}_{i}ds.$

In the second step, we focus ours attention on elimination auxiliary currents
and the Lagrange multiplier. Using the definition of the basis function (65),
we obtain that

$\int_{\Gamma}\mathbf{g}_{n}(s)\cdot\tilde{\mathbf{J}}ds=\frac{|T^{+}_{n}|+|T^{-}_{n}|}{3}\tilde{J}_{n}.$

On the other side, we have

$\int_{\Gamma}\mathbf{g}_{n}(s)\cdot(\mathbf{n}\times\mathbf{J})ds=\int_{T^{+}_{n}\cup
T^{-}_{n}}\mathbf{g}_{n}(s)\cdot(\mathbf{n}\times\mathbf{J})ds,$

that is calculated with help of Gaussian quadrature. And we get next equation

$\frac{|T^{+}_{n}|+|T^{-}_{n}|}{3}\tilde{J}_{n}=\int_{T^{+}_{n}\cup
T^{-}_{n}}\mathbf{g}_{n}(s)\cdot(\mathbf{n}\times\mathbf{J})ds$ (70)

Thus, we obtain

$\tilde{J}_{n}=\frac{3}{|T^{+}_{n}|+|T^{-}_{n}|}\int_{T^{+}_{n}\cup
T^{-}_{n}}\mathbf{g}_{n}(s)\cdot(\mathbf{n}\times\mathbf{J})ds$ (71)

on each edge of the mesh. Consequently, we conclude that the auxiliary
currents can be rewritten as

$\begin{array}[]{cc}\overline{\tilde{J}}=-[C_{H}]^{-1}[C_{K}]\overline{J}\,\,\,\,\text{and}\,\,\,\,\overline{\tilde{M}}=-[C_{H}]^{-1}[C_{K}]\overline{M}\end{array}$
(72)

So (69) together with (72) yields the following expression of Lagrange
multipliers in terms of $\overline{J}$ and $\overline{M}$

$\overline{\lambda_{J}}=\frac{b_{2}}{2}[C_{H}]^{-T}[D]\overline{M}-\frac{a_{2}}{2}[C_{H}]^{-T}[D][C_{H}]^{-1}[C_{K}]\overline{J}$
(73)
$\overline{\lambda_{M}}=-\frac{a_{2}}{2a_{0}}[C_{H}]^{-T}[D]\overline{J}-\frac{b_{1}}{2a_{0}}[C_{H}]^{-T}[D][C_{H}]^{-1}[C_{K}]\overline{M}$
(74)

Now we define matrixes:

$[C_{KH}]=[C_{H}]^{-1}[C_{K}]\,\,\,\,\text{and}\,\,\,\,[C_{KH}]^{T}=[C_{K}]^{T}[C_{H}]^{-T}$
(75)

Then, by (73),(74) and (75), we obtain the final system

$\left(\begin{array}[]{cc}[A1]-\frac{a_{2}}{2}[C_{KH}]^{T}[D][C_{KH}]&[Q]-\frac{b_{1}}{2}[D][C_{KH}]+\frac{b_{2}}{2}[C_{KH}]^{T}[D]\\\
\\\
{[Q]}^{T}+\frac{a_{2}}{2a_{0}}[D][C_{KH}]-\frac{a_{1}}{2a_{0}}[C_{KH}]^{T}[D]&[A2]-\frac{b_{1}}{2a_{0}}[C_{KH}]^{T}[D][C_{KH}]\end{array}\right)\left(\begin{array}[]{cc}\overline{J}\\\
\\\ \overline{M}\end{array}\right)=\left(\begin{array}[]{cc}\overline{E}\\\
\\\ \overline{H}\end{array}\right)$

###  8.3 Numerical results

We present now some numerical results obtained. As a first example we consider
the case of a coated conducting sphere with a conductor radius of
$1.5\lambda_{0}$ and coating thickness of $0.0075\lambda_{0}$, with
$\varepsilon_{r}=5$ and $\mu_{r}=1.0$. Figure 12 show the $\theta\theta$
components of the bistatic RCS for a plane wave incident from $\theta=0$.
Three solutions are included: the exact serie solution (MIE) and the solutions
of the methods of moment studied below with SIBC and the HOIBC. The Figure
clearly shows the increased accuracy of the HOIBC solution relative to the
SIBC solution. The SIBC gives only the average behavior of the scattered field
while the HOIBC accurately predicts the sidelobe behavior.

Figure 12: $\theta\theta$ component of the bistatic RCS for a coated
conducting sphere with frequency $f=0.45GHz$, layer thickness $d=0.05m$,
$\varepsilon_{r}=5$ and $\mu_{r}=1$. Exact serie solution and HOIBC solution.

The second test is a coated conducting spheroid whose the radii are 0.5m and
1m with a coating thickness of $0.17\lambda_{0}$, with $\varepsilon_{r}=5$ and
$\mu_{r}=1.0$. Figure 13 and Figure 14 show the $\theta\theta$ and $\phi\phi$
components of the monostatic RCS. Three solutions are included : a method of
moments solution called PMCHWT [25], the HOIBC solution and the SIBC solution.
It does not exits exact solution for this case. We note that a slight
difference between the PMCHWT solution and the HOIBC solution and the SIBC
solution is very poor. In both cases, excellent results are obtained for all
angles of incidence. But, the finite radii of curvature on the spheroid
contribute to the inaccuracy of the HOIBC result.

Figure 13: $\theta\theta$ component of the monostatic RCS for a coated
conducting spheroid, PMCHWT solution, SIBC solution and HOIBC solution. Figure
14: $\phi\phi$ component of the monostatic RCS for a coated conducting
spheroid, PMCHWT solution, SIBC solution and HOIBC solution.

##  9 Conclusion

In this paper, we proposed a new formulation of electromagnetic scattering
problem with higher order impedance boundary condition. We prove the existence
and uniqueness of the solution of the variational formulation. Then we
discretize the formulation with RWG basis functions. The validations are shown
an important accuracy improvement of the HOIBC model over the SIBC model in
the case of materials of lower index and/or with small losses.

## References

  * [1] R. F. Harrington, _Field Computation by Moment Methods_ , Wiley-IEEE Press,(1993).
  * [2] , A. Bendali and K. Lemrabet, _The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation_ , SIAM J. Appl. Math.,pp. 1664-1693,(1996).
  * [3] Y. Rahmat-Samii and J. H. Daniel, _Scattering by Superquadric Dielectric-Coated Cylinders Using Higher Order Impedance Boundary Conditions_ , IEEE Trans. Antennas Propagat.,vol. 40,No. 12,pp. 1513-1522,Dec (1992).
  * [4] O. Mareaux, B. Stupfel, _High-order impedance boundary conditions for multilayer coated 3-D Ojects_ , IEEE Transactions on Antennas and Propagations,Vol. 48,No. 3,pp. 429-436,March(2000).
  * [5] B. Stupfel, _Impedance Boundary Conditions for Finite Planar or Curved Frequency Selective Surfaces Embedded in Dielectric Layers_ , IEEE Trans. Antennas Propagat,Vol. 53,pp. 3654–3662,Nov (2005).
  * [6] , B. Stupfel and O. Marceaux, _High-Order Impedance Boundary Conditions for Multilayer Coated 3-D Objects_ IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, Vol. 48,N. 3 pp. 429-436, March(2000).
  * [7] B. Stupfel and Y. Pion, _Impedance Boundary Conditions for Finite Planar or Curved Frequency Selective Surfaces_ , IEEE Trans. Antennas Propagat,Vol. 53,pp. 1415–1424,April (2005).
  * [8] B. Stupfel, D. Poget and J. C. Nédélec, _Sufficient uniqueness conditions for the solution of the time harmonic Maxwell’s equations associated with surface impedance boundary conditions_ , Journal of Computational Physics,Vol. 230, No 12,pp. 4571–4587, (2011).
  * [9] B. Stupfel, _Implementation of High-Order Impedance Boundary Conditions in Some Integral Equation Formulations_ , IEEE Trans. Antennas Propagat, Vol. 63, NO. 4,April (2015).
  * [10] B. Stupfel, _One-way domain decomposition method with adaptive absorbing boundary condition for the solution of Maxwell’s equations_ , IEEE Trans. Antennas Propagat, Vol. 61, NO. 10,pp. 5100–5108, Aug (2015).
  * [11] S. M. Rytov, _Calcul du skin-effet par la méthode des perturbations_ , J. Phys. USSR,Vol. 2,pp. 233–242, Oct (1940).
  * [12] J. G. Van Bladel, _Electromagnetic Fields_ , Wiley, Second, (2007).
  * [13] A. Bendali, M. Fares and J. Gay, _A Boundary-Element Solution of the Leontovich Problem_ , IEEE Trans. Antennas Propagat., Vol. 47, pp. 1597–1605, October, (1999).
  * [14] J. C. Nédélec, _Acoustic and electromagnetic equations: Integral representations for harmonic problems_ , Springer édition, Vol. 144, (2000).
  * [15] V. Lange, _Equations intégrales espace-temps pour les équations de Maxwell: calcul du champ diffracté par un obstacle dissipatif_ , PhD thesis, Mathématiques appliquées, Bordeaux 1, (1995).
  * [16] C. Daveau and J. Laminie, _Mixed and Hybrid Formulations For The Three-Dimensional Magnetic Problem_ , Numerical Methods for Partial Differential Equations, vol. 18, no. 1, pp. 85-104, January,(2002).
  * [17] S. M. Rao,D. R. Wilton and A. W. Glisson, _Electromagnetic scattering by surfaces of arbitrary shapes_ , IEEE Trans. Antennas Propag, Vol. 30, No. 3, pp. 409–418, (1982).
  * [18] A. W. Glisson, _Electromagnetic scattering by arbitrary shapes surfaces with impedance boundary conditions_ , Radio Sci, Vol. 27, No. 6, pp. 935–943, May, (1992).
  * [19] D. S. Wang, _Limits and validity of the impedance boundary conditions on penetrable surfaces_ , IEEE Trans. Antennas Propag, Vol. AP-35, No. 6, pp. 453–457, (1987).
  * [20] R. Cicchetti, _A class of exact and higher-order surface boundary conditions for layered st_ , IEEE Trans. Antennas Propag, Vol. AP-35, No. 6, pp. 453–457, (1987).
  * [21] J. Jin, _The Finite Element Method in Electromagnetics_ , New York: Wiley, (1993).
  * [22] S. N. Karp and F. C. Karal Jr, _Generalized impedance boundary conditions with applications to surface wave structures_ , Electromagnetic Wave Theory, J. Brown, Ed. New York: Pergamon, pp. 479–483, (1967).
  * [23] T. B. A. Senior and J. L. Volakis, _Approximate boundary conditions in electromagnetics_ , Inst. Elect. Eng. Electromagn. Waves Series 41, (1995).
  * [24] A. Aubakirov, _Electromagnetic scattering problem with hight order impedance boundary condition and integral methods_ , PHD Thesis, Applied Mathematics, University of Cergy-Pontoise, January (2014).
  * [25] Medgyiesi-Metschang, L. N., Putnam J.M. and Gedera M. B., _Generalized method of moments for three-dimensional penetrable scatterers_ , J. Opt. Soc. Am. A , Vol. 11,No. 4, (1994).
  * [26] Y. Rahmat-Samii and J.H. Daniel, _Impedance Boundary Conditions in Electromagnetics_ , Taylor & Francis, (1995).
  * [27] J. R. Rogers, _Moment method scattering solutions to impedance boundary condition integral equations_ , IEEE AP-S Int. Symp., Boston, MA, pp. 347–350, (1984).
  * [28] T.B.A. Senior and J.L. Volakis, _Approximate boundary conditions in electromagnetics_ , IEE Electromagnetic Waves Series 41, (1995).
  * [29] V. Lange,_Equations intégrales espace-temps pour les équations de Maxwell : calcul du champ diffracté par un obstacle dissipatif_ , PHD thesis Mathématiques appliquées, Bordeaux, 1, (1995).
  * [30] G. C. Hsiao and R. F. Kleinman, _Mathematical foundations for error estimations in numerical solutions of integral equations in electromagnetics_ , IEEE Trans. Antennas Propagat., vol.45, pp.316-328, Mar.(1997).
  * [31] I. Terrasse, _Résolution mathémathique et numérique des équations de Maxwell instationnaires par une méthode de potentiels retardés_ , L’école Polytechnique, .(1993).