Marion<EMAIL_ADDRESS>CNRS UMR 7352 - Université de
Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens cedex 1 Jérémy
<EMAIL_ADDRESS>CNRS UMR 7352 - Université de
Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens cedex 1 Stephanie
<EMAIL_ADDRESS>CNRS UMR 9008 - Université de
Reims Champagne-Ardenne, Moulin de la Housse, 51687 Reims cedex 2 February 14,
2020 February 14, 2020 Inverse Problems and Imaging

# Numerical resolution by the quasi-reversibility method of a data completion
problem for Maxwell’s equations

(July 17, 2020)

###### Abstract

This paper concerns the numerical resolution of a data completion problem for
the time-harmonic Maxwell equations in the electric field. The aim is to
recover the missing data on the inaccessible part of the boundary of a bounded
domain from measured data on the accessible part. The non-iterative quasi-
reversibility method is studied and different mixed variational formulations
are proposed. Well-posedness, convergence and regularity results are proved.
Discretization is performed by means of edge finite elements. Various two- and
three-dimensional numerical simulations attest the efficiency of the method,
in particular for noisy data.

###### keywords:

Maxwell’s equations, inverse problem, Cauchy problem, data completion, quasi-
reversibility method, numerical simulations.

##  1 Introduction

Let $\Omega$ denote a bounded and simply connected domain in $\R^{3}$ of
boundary $\Gamma\coloneqq\partial\Omega$. The unit outward normal to $\Omega$
is denoted by $\bfn$. Assume the medium in $\Omega$ to be inhomogeneous and
isotropic, of constant magnetic permeability $\mu=\mu_{0}$ with $\mu_{0}$ the
magnetic permeability in vacuum. Let $\eps,\sigma$ be non-negative functions
representing the electric permittivity and conductivity respectively. The
refractive index $\kappa$ of the medium in $\Omega$ is defined by
$\kappa(\bfx)=\frac{1}{\eps_{0}}\left(\eps(\bfx)+i\frac{\sigma(\bfx)}{\omega}\right),\bfx\in\Omega$,
where $\eps_{0}$ is the electric permittivity in vacuum. We consider the
Cauchy problem for the time-harmonic Maxwell equations at a frequency
$\omega>0$

$\curl\curl\bfE-k^{2}\kappa\bfE=0,\quad\stext[r]{in}\Omega,$ (1)

with $\bfE$ the electric field intensity and
$k\coloneqq\omega\sqrt{\mu_{0}\eps_{0}}$ the wavenumber. Namely, we assume
that the Cauchy data $\bfE\times\bfn$ and $\curl\bfE\times\bfn$ are known on a
part $\Gamma_{0}$ of the boundary $\Gamma$, called the accessible part. The
aim is to solve the data completion problem which consists in recovering the
missing data on the inaccessible part $\Gamma\setminus\Gamma_{0}$ from
overdetermined data on the accessible part. This covers for instance medical
applications where measurements are taken at electrodes placed at specific
positions on the skin (of the head or the chest). The numerical resolution of
this class of inverse problems is challenging in many application fields such
as non-destructive testing or medical imaging (electroencephalography, brain
stroke detection) since only partial noisy data are available. Inversion
methods are in general more efficient for total data known on the entire
boundary. Data completion algorithms provide interesting preliminary tools to
get an input for the numerical resolution of inverse problems (e.g [12]).

Cauchy problems are well-known to be ill-posed (e.g. [1, 7]). Various
regularization methods have been introduced for reconstructing the missing
data for elliptic equations. There is an extensive literature on the numerical
resolution of the problem. Most of the methods are iterative and are based on
solving one or more well-posed direct problems at each iteration. They can
thus be numerically expensive. Classical least square methods with a Tikhonov
regularization technique are commonly used (see e.g. [24]). There are several
techniques for the optimal choice of the regularization parameter such as the
L-curve method and the Morozov discrepancy principle. Without being
exhaustive, we can cite other approaches: a reconstruction of missing data
based on the minimization of an energy functional is adopted in [2]. In [41],
a boundary control technique is used to obtain an approximation of the missing
Dirichlet boundary data for the Laplace equation. An alternating iterative
method to solve data completion for the Laplace equation and the Lamé system
is originally introduced in [38] and is widely used and studied in the
literature. In this vein, an approach combining the Steklov-Poincaré
variational formulation of elliptic second order equations and the
Lavrentiev’s method has been proposed and validated in [8, 3, 4]. Methods
based on integral equations have been also considered [17, 22, 23].

Very few papers are concerned with the data completion problem for the vector
Maxwell equations. We cite [43, 47] where the Cauchy problem for a steady
state eddy-current model problem is studied and an iterative algorithm based
on a least-square functional is proposed.

In this paper, we focus on the quasi-reversibility method which is a non-
iterative approach. It was introduced by Lattès and Lions in [40]. The idea is
to replace the ill-posed Cauchy problem by a well-posed variational problem
with additional unknowns which depends on a small regularization parameter.
The variational setting is numerically interesting since finite element
methods can be used. The quasi-reversibility method has been successfully
adopted and validated for Laplace’s equation (see e.g. [39, 9, 11, 30]) and
Helmholtz’s equation [13]. We propose to study the quasi-reversibility method
for the vector Maxwell equations. To the best of our knowledge, it’s the first
time that such an approach is studied for solving a data completion problem in
electromagnetics. It is worth noticing that the functional framework of the
vector Maxwell equations is substantially different from the Laplace equation
or even the Lamé system. The natural vector space is given by fields with
square integrable curl and a special attention has to be paid to the
definition of the traces at the boundary. Furthermore, the vector Maxwell
problem is numerically challenging due to the computational complexity,
especially in three dimensions and the large kernel of the discrete
$\curl$-$\curl$-operator. The conditioning of the corresponding matrix
deteriorates with the mesh refinement and preconditioners have to be
prescribed when iterative solvers are used (e.g. [31, 37]).

The article is organized as follows. section 2 is devoted to the setting of
the problem. section 3 presents a first version of the quasi-reversibility
method for solving the data completion problem for Maxwell’s equations in the
electric field. Both theoretical and numerical aspects are addressed. section
4 proposes two relaxed mixed problems with regularization in order to deal
with noisy data. The relaxed problem is shown to be well posed, and
convergence to the solution of the initial Cauchy problem is proven under some
conditions on the involved regularization and relaxation parameters. A
regularity result is given for one of the formulations. Numerical simulations
confirm the theoretical results in two and three dimensions of space and
attest the efficiency of the method. The choice of the different parameters is
discussed. section 5 is a remark about the solutions of the proposed
formulations being seen as the critical points of real-valued cost functions.
Finally, we give some concluding remarks.

##  2 Position of the problem. Notations.

In the sequel, define $\Gamma_{0}$, the accessible part of $\Gamma$ [14],
according to the following

######  Definition 2.1.

Let $\Omega$ be a non-empty, open, bounded connected domain in $\R^{3}$ with
Lipschitz boundary $\Gamma$. Let $\Gamma_{0}$ a smooth non-empty open subset
of $\Gamma$ such that meas $\Gamma_{0}>0$. The part $\Gamma_{0}$ is called the
accessible part of the boundary $\Gamma$ and
$\Gamma_{1}\coloneqq\Gamma\setminus\overline{\Gamma_{0}}$ the inaccessible
part.

######  Remark 2.2.

In this paper, we address the theoretical part in the three-dimensional case.
All the results hold true in two dimensions as well, noticing that two curl-
operators have to be distinguished in this case. Indeed, the curl of vector
fields $\bfE=(E_{1},E_{2})$ with two components is defined by

$\operatorname{curl}\bfE=\partial_{1}E_{2}-\partial_{2}E_{1},$

whereas the vector curl of a scalar function $\varphi$ is given by

$\curl\varphi=\begin{pmatrix}\partial_{2}\varphi\\\
-\partial_{1}\varphi\end{pmatrix}.$

Consequently, the operator of the partial differential equation reads
$\curl\operatorname{curl}$. Numerical results are given for both 2D and 3D
configurations.

Let us make precise the functional setting of the data completion problem. The
fundamental vector space for Maxwell’s equation is given by
$\Hcurl[\Omega]=\set*{\bfu\in L^{2}(\Omega)^{3}}{\curl\bfu\in
L^{2}(\Omega)^{3}}$. For any vector field $\bfu\in\Hcurl[\Omega]$, we define
the tangential trace by continuous extension of the mapping
$\gamma_{t}(\bfu)\coloneqq\restriction{\bfu}{\Gamma}\times\bfn$. We introduce
the trace space

$Y(\Gamma)=\set*{\bff\in
H^{-1/2}(\Gamma)^{3}}{\Exists{\bfu\in\Hcurl[\Omega]}\gamma_{t}(\bfu)=\bff}$
(2)

and its restriction to $\Gamma_{0}$ in the distributional sense
$Y(\Gamma_{0})=\set*{\restriction{\bff}{\Gamma_{0}}}{\bff\in Y(\Gamma)}$. The
following lemma holds.

######  Lemma 2.3.

The partial trace application
$\bfu\mapsto\restriction{\gamma_{t}(\bfu)}{\Gamma_{0}}$ is linear, continuous
and surjective from $\Hcurl[\Omega]$ to $Y(\Gamma_{0})$. Moreover, there
exists a continuous lifting application: we can find a constant $r>0$ such
that, for any $\bfv\in Y(\Gamma_{0})$, there exists $\bfu\in\Hcurl[\Omega]$
with $\restriction{\gamma_{t}(\bfu)}{\Gamma_{0}}=\bfv$ and
$\norm{\bfu}{\Hcurl[\Omega]}\leq r\norm{\bfv}{Y(\Gamma_{0})}$.

2.3 is a direct consequence of the fact that the trace application
$\gamma_{t}$ is linear, continuous and surjective from $\Hcurl[\Omega]$ to
$Y(\Gamma)$ (see [44]). Then, we build an homeomorphism between
$\Hcurl[\Omega]/V_{0}$ (see (4)) and $Y(\Gamma_{0})$.

For further purposes, we also introduce the second trace operator $\gamma_{T}$
that is defined by $\gamma_{T}(\bfu)=(\bfn\times\bfu_{|\Gamma})\times\bfn$ for
smooth vector fields and is well defined as an operator from $\Hcurl[\Omega]$
on the dual space of $Y(\Gamma)$, $Y(\Gamma)^{\prime}$ (see [44] for details).
The trace spaces for fields in $\Hcurl[\Omega]$ can be characterized more
precisely depending on the regularity of the boundary (see [44, 15, 16]).
However, for our study, the above abstract definition is sufficient.

The set of admissible coefficients $\eps$ and $\sigma$ is given in the
following definition.

######  Definition 2.4.

The pair of coefficients $\eps$ and $\sigma$ is admissible if
$\eps,\sigma\in\mcC^{1}(\overline{\Omega})$ such that
$\eps(\bfx)\geq\tilde{\eps}$ and $\sigma(\bfx)\geq 0$ almost everywhere in
$\Omega$ for a strictly positive constant $\tilde{\eps}$.

######  Definition 2.5.

Let $\Omega$ and $\Gamma_{0}$ be as in 2.1. For a pair of admissible
coefficients $(\eps,\sigma)$ defined on $\Omega$ as in 2.4, the corresponding
Cauchy data set $C(\eps,\sigma;\Gamma_{0})$ at a fixed frequency $\omega>0$
consists of pairs $(\bff,\bfg)\in Y(\Gamma_{0})\times Y(\Gamma_{0})$ such that
there exists a field $\bfE\in\Hcurl[\Omega]$ satisfying

$\left\\{\begin{array}[]{rcl@{\hspace{4\tabcolsep}}l}\curl\curl\bfE-k^{2}\kappa\bfE&=&0,\hfil\hskip
24.0pt&\stext[r]{in}\Omega,\\\ \gamma_{t}(\bfE)&=&\bff,\hfil\hskip
24.0pt&\stext[r]{on}\Gamma_{0},\\\ \gamma_{t}(\curl\bfE)&=&\bfg,\hfil\hskip
24.0pt&\stext[r]{on}\Gamma_{0},\end{array}\right.$ (3)

where $\kappa=(\eps+i\sigma/\omega)/\eps_{0}$.

The definition of Cauchy data sets is used for instance in [46, 18, 20]. The
partial boundary data are also given by the admittance map
$\Lambda\colon\restriction{(\bfE\times\bfn)}{\Gamma}\mapsto\restriction{(\curl\bfE\times\bfn)}{\Gamma}$
restricted to $\Gamma_{0}$ if $\omega$ is not a resonant frequency for (1).

The Cauchy problem (3) is known to be ill-posed in the sense that it may not
possess a solution for arbitrary data $(\bff,\bfg)$ in $Y(\Gamma_{0})\times
Y(\Gamma_{0})$ and that the solution is not stable with respect to the data.
On the other hand, if the Cauchy problem (3) admits a solution, this one is
unique. This is an immediate consequence of the following Holmgren-type
theorem (cf. [14], Lemma 5.4):

######  Lemma 2.6.

Let $\Omega$ be a bounded connected Lipschitz domain. Let
$\kappa\in\mathcal{C}^{1}(\overline{\Omega})$ such that $\real{\kappa}\geq c$
for a given constant $c>0$. Let $\Gamma_{0}$ be an open non-empty subset of
$\partial\Omega$. If $\bfu\in\Hcurl[\Omega]$ is such that $\curl\curl\bfu\in
L^{2}(\Omega)^{3}$ and

$\left\\{\begin{array}[]{rcl@{\hspace{4\tabcolsep}}l}\curl\curl\bfu-k^{2}\kappa\bfu&=&0,\hfil\hskip
24.0pt&\stext[r]{in}\Omega,\\\ \gamma_{t}(\bfu)&=&0,\hfil\hskip
24.0pt&\stext[r]{on}\Gamma_{0},\\\ \gamma_{t}(\curl\bfu)&=&0,\hfil\hskip
24.0pt&\stext[r]{on}\Gamma_{0},\end{array}\right.$

then $\bfu\equiv 0$ on $\Omega$.

######  Remark 2.7.

The question whether a solution of the Cauchy problem exists for given data or
not is difficult from a theoretical point of view (see [33, 7]). It is not
addressed in the present paper. From a practical point of view, however, one
can reasonably assume that the given data correspond to (noisy) measurements
and given sources. These data are the consequence of an electric field that
naturally satisfies the Cauchy problem. In the sequel, we therefore assume
that the data belong to the Cauchy set associated with the Cauchy problem.

##  3 The quasi-reversibility method for Maxwell’s equations

From 2.6 and the 2.5 of Cauchy data sets, the Cauchy problem (3) admits a
unique solution for any couple $(\bff,\bfg)$ belonging to
$\mathcal{C}(\eps,\sigma;\Gamma_{0})$. Assuming that the set
$\mathcal{C}(\eps,\sigma;\Gamma_{0})$ is not empty, the rest of the paper is
devoted to the numerical resolution of this data completion problem for
Maxwell’s equations.

###  3.1 Principle for data belonging to the trace space

The method of quasi-reversibility (called hereafter QR method) provides a
regularized solution of Cauchy problems in a bounded domain. An abstract
setting of the method is presented in [13]. Here, we prove that the abstract
setting fits with the functional framework of Maxwell’s equations.

Let us introduce the spaces

$V_{\bff}=\set*{\bfu\in\Hcurl[\Omega]}{\gamma_{t}(\bfu)=\bff\stext{on}\Gamma_{0}}$
(4)

which is well defined for any $\bff\in Y(\Gamma_{0})$, and

$M=\set*{\bfmu\in H(\curl)}{\gamma_{t}(\bfmu)=0\stext{on}\Gamma_{1}}.$

One may notice that a field $\bfmu$ belongs to the vector space $M$ if and
only if

$(\curl\bfmu,\Phi)=(\bfmu,\curl\Phi)\ \forall\Phi\in\
\stackrel{{\scriptstyle\circ\;}}{{\mathcal{C}^{\infty}_{\Gamma_{0}}}}(\Omega)^{3}$

where the space
$\stackrel{{\scriptstyle\circ\;}}{{\mathcal{C}^{\infty}_{\Gamma_{0}}}}(\Omega)$
contains the restriction to $\Omega$ of smooth functions $\phi$ with compact
support in $\R^{3}$ such that ${\rm dist}({\rm supp}(\phi),\Gamma_{0})>0$, in
other words $\phi$ vanishes in a neighborhood of $\Gamma_{0}$.

Assume that $(\bff,\bfg)\in C(\eps,\sigma;\Gamma_{0})$ (see 2.5) where $\eps$
and $\sigma$ are admissible coefficients (see 2.4). We denote by
$a(\cdot,\cdot)$ the sesqui-linear form corresponding to (1):

$a(\bfu,\bfv)=(\curl\bfu,\curl\bfv)-k^{2}(\kappa\bfu,\bfv)$ (5)

with $\bfu,\bfv\in\Hcurl[\Omega]$ and $(\cdot,\cdot)$ the dot-product in
$L^{2}$. On $M$, we introduce the linear form $\ell(\cdot)$ defined by

$\ell(\bfpsi)=\duality{\bfg}{\gamma_{T}(\bfpsi)}{Y(\Gamma_{0}),Y(\Gamma_{0})^{\prime}}.$
(6)

######  Proposition 3.1.

Let $\bfE\in\Hcurl[\Omega]$ be a solution of the Cauchy problem (3) for a
couple $(\bff,\bfg)\in\mathcal{C}(\eps,\sigma;\Gamma_{0})$. Then $\bfE$
belongs to $V_{\bff}$ and satisfies the following variational equation

$a(\bfE,\bfpsi)=\ell(\bfpsi)\ \forall\bfpsi\in M.$ (7)

Reciprocally, any solution of (7) satisfies (3) in the distributional sense.

######  Proof 3.2.

According to the definition of the Cauchy data set, $\bff$ and $\bfg$ belong
to the trace space $Y(\Gamma_{0})$. We deduce from a density result in [5]
that for a given $\bfpsi\in M$, there is a sequence of fields $\bfpsi_{n}\in\
\stackrel{{\scriptstyle\circ\text{\;}}}{{\mathcal{C}^{\infty}_{\Gamma_{1}}}}(\Omega)$
such that $\bfpsi_{n}\longrightarrow\bfpsi$ in $\Hcurl[\Omega]$. Since
$\bfpsi_{n}$ vanishes identically on a neighborhood of $\Gamma_{1}$, its
restriction to $\Gamma_{0}$ belongs to $\mathcal{D}(\Gamma_{0})$. Any solution
$\bfE$ of the Cauchy problem thus satisfies

$\duality{\gamma_{t}(\curl\bfE)}{\restriction{\bfpsi_{n}}{\Gamma_{0}}}{}=\duality{\bfg}{\restriction{\bfpsi_{n}}{\Gamma_{0}}}{}=\duality{\bfg}{\gamma_{T}(\bfpsi_{n})}{\Gamma_{0}}$

since $\bfg\cdot\bfn=0$ by definition of the trace space $Y(\Gamma)$. It
follows from the classical Green’s theorem in $\Hcurl[\Omega]$ that

$\displaystyle\dotprod{\curl\curl\bfE}{\bfpsi_{n}}{}$
$\displaystyle=\dotprod{\curl\bfE}{\curl\bfpsi_{n}}{}-\duality{\gamma_{t}(\curl\bfE)}{\restriction{\bfpsi_{n}}{\Gamma_{0}}}{}$
$\displaystyle=\dotprod{\curl\bfE}{\curl\bfpsi_{n}}{}-\duality{\bfg}{\gamma_{T}(\bfpsi_{n})}{\Gamma_{0}}.$

Taking the limit $n\to\infty$ then yields

$\dotprod{\curl\curl\bfE}{\bfpsi}{}=\dotprod{\curl\bfE}{\curl\bfpsi}{}-\duality{\bfg}{\gamma_{T}(\bfpsi)}{\Gamma_{0}}$

for any $\psi\in M$ since the trace operator $\gamma_{T}$ is continuous on
$\Hcurl[\Omega]$. Inversely, if $\bfE\in V_{\bff}$ is a variational solution
of (7), taking $\psi\in\mathcal{C}^{\infty}_{0}(\Omega)^{3}$ yields
immediately the partial differential equation of (3). The first boundary
condition is included in the variational space $V_{\bff}$. Now, let
$\varphi\in\mathcal{C}^{\infty}_{0}(\Gamma_{0})^{3}$ be a regular vector field
defined on the boundary with support in $\Gamma_{0}$ and such that
$\varphi\cdot\bfn=0$. $\varphi$ is the restriction to $\Gamma_{0}$ of a vector
field $\bfpsi\in\
\stackrel{{\scriptstyle\circ\text{\;}}}{{\mathcal{C}^{\infty}_{\Gamma_{1}}}}(\Omega)^{3}$:
$\restriction{\bfpsi}{\Gamma_{0}}=\varphi$. Obviously, $\bfpsi$ belongs to the
vector space $M$. The variational equation (7) and Green’s formula yield

$\displaystyle\duality{\bfg}{\varphi}{\Gamma_{0}}$
$\displaystyle=\dotprod{\curl\bfE}{\curl\bfpsi}{}-k^{2}\dotprod{\kappa\bfE}{\bfpsi}{}$
$\displaystyle=\duality{\gamma_{t}(\curl\bfE)}{\varphi}{\Gamma_{0}}$

and the second boundary condition follows.

For small $\delta>0$, we now consider the following weak mixed formulation:
Find $(\bfE_{\delta},\bfF_{\delta})\in V_{\bff}\times M$ such that

$\left\\{\begin{array}[]{rcl@{\hspace{4\tabcolsep}}l}\delta\dotprod{\bfE_{\delta}}{\bfphi}{\Hcurl[\Omega]}+a(\bfphi,\bfF_{\delta})&=&0,\hfil\hskip
24.0pt&\forall\bfphi\in V_{0},\\\
a(\bfE_{\delta},\bfpsi)-\dotprod{\bfF_{\delta}}{\bfpsi}{\Hcurl[\Omega]}&=&\ell(\bfpsi),\hfil\hskip
24.0pt&\forall\bfpsi\in M,\end{array}\right.$ (8)

where $\dotprod{}{}{\Hcurl[\Omega]}$ denotes the dot-product in
$\Hcurl[\Omega]$. Theorem 3.3 states that the regularized solution
$(\bfE_{\delta},\bfF_{\delta})$ tends to $(\bfE,0)$ when $\delta$ tends to 0,
where $\bfE$ is solution to (3).

######  Theorem 3.3.

Let $(\bff,\bfg)\in Y(\Gamma_{0})\times Y(\Gamma_{0})$. For any $\delta>0$,
problem (8) admits a unique solution $(\bfE_{\delta},\bfF_{\delta})\in
V_{\bff}\times M$.

If, in addition, $(\bff,\bfg)$ belongs to the Cauchy data set
$C(\eps,\sigma;\Gamma_{0})$, then

$\lim_{\delta\to 0}(\bfE_{\delta},\bfF_{\delta})=(\bfE,0)$ (9)

in $V_{f}\times M$. Here, $\bfE\in V_{\bff}$ is the unique solution of the
Cauchy problem (3).

######  Proof 3.4.

Since the trace operator $\gamma_{t}$ is onto according to 2.3, we can apply
Theorem 2.4 from [13]. Problem (8) can thus be written in a closed form as a
classical variational formulation in the unknowns
$(\bfE_{\delta},\bfF_{\delta})$, involving a continuous and coercive sesqui-
linear form $A_{\delta}\left((\cdot,\cdot)\right)$ defined on
$\Hcurl[\Omega]\times M$. Existence and uniqueness follow from Lax-Milgram’s
Lemma. Assume in addition that the couple $(\bff,\bfg)$ belongs to the Cauchy
set $\mathcal{C}(\eps,\sigma;\Gamma_{0})$. Hence, there exists a field
$\bfE\in V_{\bff}$, solution to the Cauchy problem (3). From 2.6, we infer
that $\bfE$ is unique. The convergence of the sequence
$(\bfE_{\delta},\bfF_{\delta})$ to $(\bfE,0)$ now follows as in [13] since
$\bfE$ is the only solution to the variational formulation.

###  3.2 Discretization by FEM and numerical results

A first series of numerical tests for two dimensional results investigates in
which way the size of the accessible part $\Gamma_{0}$ influences the quality
of the data completion procedure. In view of possible applications of the QR
method in inverse coefficient problems, we are particularly interested in
ring-like computational domains $\Omega$. Indeed, in many configurations, the
material parameters are known in a ring-like neighborhood of the boundary, and
data completion has to be applied to map the available information from part
of the outer boundary to the inner boundary.

The numerical solver for the Maxwell equations has been implemented with
FreeFem++ (see [35]). We consider a regular discretization $\mcT_{h}$ of
$\overline{\Omega}$. For any element $T\in\mcT_{h}$, let $h_{T}$ be its
diameter. Then $h=\max_{T\in\mcT_{h}}h_{T}$ is the mesh parameter of
$\mcT_{h}$. Conforming edge finite elements of order 1 (see [45]) are used to
approximate the solution of the regularized problem (8). The resulting
discrete formulation reads as follows: Find
$(\bfE_{\delta}^{h},\bfF_{\delta}^{h})\in V_{\bff}^{h}\times M^{h}$ such that

$\left\\{\begin{array}[]{rcl@{\hspace{4\tabcolsep}}l}\delta\dotprod{\bfE_{\delta}^{h}}{\bfphi^{h}}{\Hcurl[\Omega]}+a(\bfphi^{h},\bfF_{\delta}^{h})&=&0,\hfil\hskip
24.0pt&\forall\bfphi^{h}\in V_{0}^{h},\\\
a(\bfE_{\delta}^{h},\bfpsi^{h})-\dotprod{\bfF_{\delta}^{h}}{\bfpsi^{h}}{\Hcurl[\Omega]}&=&\ell(\bfpsi^{h}),\hfil\hskip
24.0pt&\forall\bfpsi^{h}\in M^{h}.\end{array}\right.$ (10)

######  Proposition 3.5.

For any $\delta>0$, problem (10) admits a unique solution. If in addition, the
solution fields $\bfE_{\delta}$ and $\bfF_{\delta}$ of problem (8) belong to
$H^{s}(\curl;\Omega)=\set*{\bfu\in H^{s}(\Omega)^{3}}{\curl\bfu\in
H^{s}(\Omega)^{3}}$ with $1/2<s\leq 1$, the following error estimate holds
true

$\norm{(\bfE_{\delta}-\bfE_{\delta}^{h},\bfF_{\delta}-\bfF_{\delta}^{h})}{\Hcurl[\Omega]}^{2}\leq
C\frac{h^{s}}{\min(\delta,1)}\norm{(\bfE_{\delta},\bfF_{\delta})}{H^{s}(\curl;\Omega)}^{2}.$
(11)

######  Proof 3.6.

Since problem (8) can be written as a variational formulation with a
continuous and coercive sesqui-linear form, the discrete problem is well posed
and enters within the framework of Céa’s lemma. We thus have, for all
$(\bfphi^{h},\bfpsi^{h})\in V_{0}^{h}\times M^{h}$,

$\norm{(\bfE_{\delta}-\bfE_{\delta}^{h},\bfF_{\delta}-\bfF_{\delta}^{h})}{\Hcurl[\Omega]}^{2}\leq\frac{C}{\alpha}\norm{(\bfE_{\delta}-\bfphi^{h},\bfF_{\delta}-\bfpsi^{h})}{\Hcurl[\Omega]}^{2}.$

Here, $C$ and $\alpha$ are respectively the continuity and coercivity
constants of the involved sesqui-linear form. Classical estimates for the
considered finite elements yield that the interpolation error is of order
$\mcO(h^{s})$ (see e.g. [44]) under the given regularity assumptions on the
exact solution. However, the coercivity constant of the sesqui-linear form is
given by $\alpha=\min\collection{\delta,1}$. The final discretization error is
thus of order $\mcO\left(\frac{h^{s}}{\min(\delta,1)}\right)$.

3.5 implies that the regularization parameter $\delta$ has to be chosen with
respect to the mesh size and arbitrary small values of $\delta$ are
prohibited.

In all the simulations, we fix $\eps_{0}=\mu_{0}=\omega=1$ and $\kappa=1+i$.
Our reference solution is the plane wave
$\bfE(\bfx)=\bfeta^{\perp}e^{ik\sqrt{\kappa}\bfeta\cdot\bfx}$ where
$\bfeta\in\R^{2}$ is the wave propagation vector and $\bfeta^{\perp}$ is a
unit vector orthogonal to $\bfeta$. The square-root $\sqrt{\kappa}$ stands for
the classical complex square-root with branch-cut along the negative real
axis.

####  3.2.1 Unit disc

In the first example, $\Omega$ is the unit disc discretized with a mesh of
size $h=\scnum{2.26e-2}$, ending up with $43\,256$ triangles and $65\,134$
edges. Two different configurations of accessible/inaccessible parts are
tested (cf. Figure 1):

  1. 1.

In the configuration G34, $\Gamma_{0}$ is the arc of circle starting at angle
0 and ending at angle $\dfrac{3\pi}{2}$. The accessible part then represents
75 of the boundary $\Gamma$.

  2. 2.

In the configuration GE37, $\Gamma_{0}$ represents a set of 37 equally
distributed electrodes of common length $\displaystyle\frac{\pi}{25}$.
Electrodes cover 81.6 of the boundary $\Gamma$.

Figure 1: Choice of the accessible part $\Gamma_{0}$ (grey line). Left:
configuration G34. Right: configuration GE37.

In Figure 2, we show the relative error in the $L^{2}(\Omega)$-norm between
the exact solution $\bfE$ and its approximation obtained by the numerical
resolution of (8), with respect to $\delta$. For both configurations, we
indicate the parameter $\delta$ for which the error is minimal. Notice that
the behaviour of the error for the configuration G34 shown in Figure 2 is in
agreement with the error analysis which claims that the error behaves as
$\mcO\left(\frac{h^{s}}{\delta}\right)$: at fixed mesh size the error
increases for small values of $\delta$.

In Table 1, we report different errors obtained for the value
$\delta=$9.103\text{\times}{10}^{-7}$$. The approximation of the electric
field $\bfE$ is more accurate on the whole domain $\Omega$ than on the
inaccessible part $\Gamma_{1}$. The configuration GE37 yields better results
since the accessible part covers a larger part of the boundary $\Gamma$. In
Figure 3, we illustrate these results by plotting the modulus of the error
$\abs{\bfE-\bfE_{\delta}}$ in the domain $\Omega$. The largest errors are
located on $\Gamma_{1}$, and in particular at the intersection between
$\Gamma_{0}$ and $\Gamma_{1}$. Indeed, the solution
$(\bfE_{\delta},\bfF_{\delta})$ of (8) is the weak solution of a boundary
value problem in $\Omega$ with mixed boundary conditions for $\bfE_{\delta}$
and $\bfF_{\delta}$. Mixed boundary conditions are known to induce
singularities in the fields (see e.g. [32] for the simpler case of the Laplace
operator). This could explain the singular behavior at the intersection
points.

Figure 2: Unit disc. QR method. Relative error $\frac{\norm{\bfE-\bfE_{\delta}}{0,\Omega}}{\norm{\bfE}{0,\Omega}}$ with respect to the regularization parameter $\delta$. Left: configuration G34. Right: configuration GE37. Table 1: Unit disc. QR method. Errors for $\delta=$9.103\text{\times}{10}^{-7}$$. Configuration | $\frac{\norm{\bfE-\bfE_{\delta}}{0,\Omega}}{\norm{\bfE}{0,\Omega}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{1}}}{\norm{\bfE\times\bfn}{0,\Gamma_{1}}}$ | $\norm{\bfF_{\delta}}{0,\Omega}$
---|---|---|---
G34 | 1.1244e-02 | 1.7302e-01 | 1.8374e-04
GE37 | 7.6288e-04 | 1.5185e-02 | 1.8224e-04

Figure 3: Unit disc. QR method. Modulus of the error
$\abs{\bfE-\bfE_{\delta}}$. Left: configuration G34. Right: configuration
GE37.

####  3.2.2 Ring

For the inverse problems we are studying, we are interested in ring-like
domains which, for instance, model the different tissues of the head. The goal
consists in mapping the data measured on part of the exterior boundary to an
inner interface, allowing in a second step to use reconstruction methods in
the interior domain. In this section, we present numerical results on a ring
of internal radius 0.75, discretized with a mesh size $h=\scnum{2.05e-2}$,
$17\,919$ triangles and $27\,316$ edges. Three measurement configurations are
compared: G34 and GE37, both described in the previous section, and GExt where
$\Gamma_{0}$ is the whole external boundary (see Figure 4).

Figure 4: Choice of the accessible part $\Gamma_{0}$ (grey line). Left:
configuration GExt ($\Gamma_{0}$ covers 57 of the ring boundary). Middle:
configuration G34 (43). Right: configuration GE37 (47).

The relative errors with respect to $\delta$ are shown in Figure 5. For each
configuration, we list in Table 2 different errors obtained for the value
$\delta$ realizing the minimum indicated in Figure 5. Notice that for the
ring-like domain we split the inaccessible part into an exterior part
$\Gamma_{1}$ and an interior part $\Gamma_{i}$. Furthermore, the modulus of
the error $\abs{\bfE-\bfE_{\delta}}$ is reported in Figure 6. We observe that
the electric field $\bfE$ is well approximated in the ring by the QR approach
when the data are available on the entire exterior boundary (configuration
GExt). The transmission of the information on the inner boundary $\Gamma_{i}$
is very accurate. The configuration GE37 with electrodes leads also to very
interesting results with errors below $\pc{4}$ on the inaccessible part
$\Gamma_{1}$ and the inner boundary $\Gamma_{i}$. The analysis of the results
obtained with configuration G34 is less obvious: whereas the error of the
auxiliary field $\bfF_{\delta}$ is very satisfying, we observe on $\Omega$,
$\Gamma_{1}$ and $\Gamma_{i}$ errors of 35, 70, and 48, respectively. The next
section will propose a possible improvement of the QR method for such a
configuration.

Figure 5: Ring. QR method. Relative error $\frac{\norm{\bfE-\bfE_{\delta}}{0,\Omega}}{\norm{\bfE}{0,\Omega}}$ with respect to the regularization parameter $\delta$. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37. Table 2: Ring. QR method. Errors. Configuration | $\frac{\norm{\bfE-\bfE_{\delta}}{0,\Omega}}{\norm{\bfE}{0,\Omega}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{1}}}{\norm{\bfE\times\bfn}{0,\Gamma_{1}}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{\text{i}}}}{\norm{\bfE\times\bfn}{0,\Gamma_{\text{i}}}}$ | $\norm{\bfF_{\delta}}{0,\Omega}$
---|---|---|---|---
GExt | 3.9423e-04 | 6.6661e-04 | 6.6661e-04 | 1.7964e-04
G34 | 3.5093e-01 | 6.9598e-01 | 4.8011e-01 | 8.1894e-05
GE37 | 9.9784e-03 | 3.9068e-02 | 3.8737e-02 | 8.4337e-05

Figure 6: Ring. QR method. Modulus of the error $\abs{\bfE-\bfE_{\delta}}$.
Left: configuration GExt. Middle: configuration G34. Right: configuration
GE37.

####  3.2.3 Extension of the computational domain

For similar configurations of the accessible part, the errors obtained on the
disc are more accurate than on the ring. This leads us to the idea of
extending the computational domain to a full disc
$\tilde{\Omega}=\left(\overline{\Omega}\cup\overline{\Omega_{\text{int}}}\right)^{\circ}$
before restricting the solution to the initial ring $\Omega$ through the
algorithm 1.

Input: $(\bff,\bfg)$ on $\Gamma_{0}$

Output: Solution $({\bfE}_{\delta},{\bfF}_{\delta})$ of the QR problem on the
ring $\Omega$

  1. 1.

Extend the coefficients $\eps$ and $\sigma$ defined on the ring $\Omega$ to
admissible coefficients $\tilde{\eps}$ and $\tilde{\sigma}$ defined on
$\tilde{\Omega}$.;

* 2.

Compute $(\tilde{\bfE}_{\delta},\tilde{\bfF}_{\delta})$, solution to the QR
method on $\tilde{\Omega}$. 3. Define the fields
$\bfE_{\delta}\coloneqq\tilde{\bfE}_{\delta|\Omega}$ and
$\bfF_{\delta}\coloneqq\tilde{\bfF}_{\delta|\Omega}$.

Algorithm 1 Extension/restriction method

From a theoretical point of view, if the data $(\bff,\bfg)$ belong to the
Cauchy data set $C(\tilde{\eps},\tilde{\sigma};\Gamma_{0})$ with respect to
the extended domain $\tilde{\Omega}$, the sequence
$(\tilde{\bfE}_{\delta},\tilde{\bfF}_{\delta})$ converges to
$(\tilde{\bfE},0)$ where $\tilde{\bfE}$ is the solution of the Cauchy problem
(3) in $\tilde{\Omega}$ associated with the coefficients $\tilde{\eps}$ and
$\tilde{\sigma}$. The restriction $\bfE$ to the ring $\Omega$ satisfies the
Cauchy problem with data $(\bff,\bfg)$ on $\Omega$. Thanks to the unique
continuation principle, $\bfE$ is the only possible solution and coincides
with the limit of the sequence obtained by the QR method applied on $\Omega$.

Figure 7 shows the error with respect to $\delta$ obtained by the
extension/restriction method for the previous three configurations. The errors
obtained with the values of $\delta$ realizing the minima are reported in
Table 3 and illustrated in Figure 8. We notice a significative improvement of
the approximation on the inaccessible part: 12.5 instead of 70 on
$\Gamma_{1}$, 0.8 instead of 48 on $\Gamma_{i}$. The drawback lies in the
increasing computational cost since the extended domain leads to a larger
number of unknowns.

The numerical results of the previous sections attest the efficiency of the QR
method for Maxwell’s equations in the case of compatible data belonging to the
trace space.

######  Remark 3.7.

It is important to notice that the extension/restriction method is limited to
the case where the belonging of the data $(\bff,\bfg)$ to the Cauchy set with
respect of the extended domain can be guaranteed in addition to the usual
assumption that the data belong to the Cauchy set of the original domain. In
the present simulations, the (synthetic) data on the exterior boundary have
been generated by solving the Maxwell equations on the whole disc. The QR-
method has then been applied to the ring in order to perform data completion
on the interior boundary. Hence, the data belong to the Cauchy set on both the
ring and the extended domain. In a general context, the question of the choice
of the extended domain needs more discussion. While choosing $\tilde{\Omega}$
as the whole disc seems natural, it raises an issue: the Cauchy problem is not
guaranteed to admit a solution in this domain. However, we can see the
extension/restriction method from a different point of view in which the
existence of a solution of the Cauchy problem in the extended domain holds. In
the practical applications evoked in subsubsection 3.2.2 as for instance the
cortical mapping, it is realistic to assume that the refractive index is known
in a neighborhood of the boundary, denoted by $\tilde{\Omega}$ (typically the
scalp layer for medical applications), and also that a solution of the Cauchy
problem in the whole disc exists (corresponding to the measurements $({\bf
f},{\bf g})$). However, the electromagnetic coefficients are unknown in the
inner part of the disc. Extending the coefficients from the known part to the
inner part in an arbitrary way could lead to a situation where the data do no
longer belong to the corresponding Cauchy set, except the (unrealistic)
situation where the extended coefficients coincide exactly with the unknown
parameters. But in our application, it is sufficient to solve the data
completion problem in a ring $\Omega$ which represents only a part of the
known neighborhood $\tilde{\Omega}$, and we apply the QR method for this
issue. The idea is to extend the QR method to the ring $\tilde{\Omega}$, and
to restrict the solution to the thinner ring $\Omega$. In this case, a
solution of the Cauchy problem exists in $\tilde{\Omega}$ and $\Omega$ and the
extension/restriction method still yields better results than the classical
QR-method on $\tilde{\Omega}$. Such a configuration has been tested
successfully in [36, Section 6.7.3, pp. 162–164].

Figure 7: Ring. QR method. Extension/restriction method. Relative error $\frac{\norm{\bfE-\bfE_{\delta}}{0,\Omega}}{\norm{\bfE}{0,\Omega}}$ with respect to the regularization parameter $\delta$. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37. Table 3: Ring. QR method. Extension/restriction method. Errors. Configuration | $\frac{\norm{\bfE-\bfE_{\delta}}{0,\Omega}}{\norm{\bfE}{0,\Omega}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{1}}}{\norm{\bfE\times\bfn}{0,\Gamma_{1}}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{\text{i}}}}{\norm{\bfE\times\bfn}{0,\Gamma_{\text{i}}}}$ | $\norm{\bfF_{\delta}}{0,\Omega}$
---|---|---|---|---
GExt | 3.7745e-04 | 1.5665e-04 | 1.5665e-04 | 3.1125e-04
G34 | 3.4345e-02 | 1.2558e-01 | 8.1477e-03 | 2.6564e-04
GE37 | 1.7728e-03 | 1.1968e-02 | 2.9520e-04 | 9.0131e-05

Figure 8: Ring. QR method. Extension/restriction method. Modulus of the error
$\abs{\bfE-\bfE_{\delta}}$. Left: configuration GExt. Middle: configuration
G34. Right: configuration GE37.

##  4 Quasi-reversibility formulations for noisy data

Noisy data will, in general, not belong to the trace space $Y(\Gamma_{0})$.
Consequently, problem (8) is not well defined since no lifting exists for the
data $\bff$. The boundary condition $\gamma_{t}(\bfE)=\bff$ has thus to be
enforced weakly. Moreover, if $\bfg\not\in Y(\Gamma_{0})$, the definition of
the linear form $\ell(\cdot)$ on the vector space $M$ is no longer
straightforward. To overcome these difficulties, we modify in this section the
basic vector space and assume in the sequel that the boundary data
$(\bff,\bfg)$ belong to $L^{2}(\Gamma_{0})^{3}\times L^{2}(\Gamma_{0})^{3}$.
Notice that this assumption does not imply that the data belong to the trace
space.

###  4.1 A first version

Consider the vector space

$V=\set*{\bfv\in\Hcurl[\Omega]}{\gamma_{t}(\bfv)\in L^{2}(\Gamma_{0})^{3}}$
(12)

with the norm

$\norm{\bfv}{V}=\left(\norm{\bfv}{\Hcurl[\Omega]}^{2}+\norm{\gamma_{t}(\bfv)}{0,\Gamma_{0}}^{2}\right)^{1/2}.$
(13)

The space $M$ will now be defined as the following subspace of $V$,

$M=\set{\bfv\in V}{\gamma_{t}(\bfv)=0\stext{on}\Gamma_{1}},$ (14)

equipped with the norm of $V$, $\norm{}{V}$. Since we cannot assume that there
is a lifting of the boundary data $\bff$ in $V$, the boundary condition will
be imposed weakly through a penalization term. The relaxed version of the
mixed problem then reads

$\left\\{\begin{array}[]{rcl@{\hspace{4\tabcolsep}}l}\lx@intercol\stext[r]{Find}(\bfE_{\alpha},\bfF_{\alpha})\in
V\times M\stext[l]{suchthat}\hfil\lx@intercol\\\
\lx@intercol\delta\dotprod{\bfE_{\alpha}}{\bfphi}{V}+\eta^{2}\dotprod{\gamma_{t}(\bfE_{\alpha})}{\gamma_{t}(\bfphi)}{0,\Gamma_{0}}\hfil\lx@intercol&&\hfil\hskip
24.0pt&\hbox{\multirowsetup$\forall\bfphi\in V,$}\\\
+a(\bfphi,\bfF_{\alpha})&=&\eta^{2}\dotprod{\bff}{\gamma_{t}(\bfphi)}{0,\Gamma_{0}},\hfil\hskip
24.0pt&\\\
a(\bfE_{\alpha},\bfpsi)-\dotprod{\bfF_{\alpha}}{\bfpsi}{V}&=&\ell(\bfpsi),\hfil\hskip
24.0pt&\forall\bfpsi\in M,\end{array}\right.$ (15)

where the subscript $\alpha=(\delta,\eta)$ indicates that the solution of (15)
depends on the regularization parameter $\delta>0$ and the relaxation
parameter $\eta>0$.

######  Theorem 4.1.

Let $(\bff,\bfg)\in L^{2}(\Gamma_{0})\times L^{2}(\Gamma_{0})$. For any
$\alpha=(\delta,\eta)$ such that $\delta>0$ and $\eta>0$, problem (15) admits
a unique solution $(\bfE_{\alpha},\bfF_{\alpha})\in V\times M$.

If, in addition, $(\bff,\bfg)$ belongs to the Cauchy data set
$C(\eps,\sigma;\Gamma_{0})$, then

$\lim_{\delta\to 0}(\bfE_{\alpha},\bfF_{\alpha})=(\bfE,0)$ (16)

in $V\times M$ for any fixed $\eta>0$. Here, $\bfE$ is the unique solution of
the Cauchy problem (3). The following estimates hold true

$\displaystyle\norm{\bfF_{\alpha}}{V}$
$\displaystyle\leq\sqrt{\delta}\norm{\bfE}{V}\ \forall\eta>0,$ (17)
$\displaystyle\norm{\gamma_{t}\bfE_{\alpha}-\bff}{0,\Gamma_{0}}$
$\displaystyle\leq\frac{\sqrt{\delta}}{\eta}\norm{\bfE}{V}.$ (18)

The estimates (17) and (18) in Theorem 4.1 give further insight in the
behavior of the sequence $(\bfE_{\alpha},\bfF_{\alpha})$. Indeed, (17) yields
a convergence order for the convergence of $\bfF_{\alpha}$ to $0$
independently from the choice of $\eta$. Estimate (18) suggests that we should
choose $\eta$ such that $\frac{\sqrt{\delta}}{\eta}\to 0$. Large values of
$\eta$ should accelerate the convergence of the trace of $\bfE_{\alpha}$ on
$\Gamma_{0}$.

######  Proof 4.2.

Problem (15) can be written in the following closed form,

$\left\\{\begin{array}[]{l}\stext[r]{Find}(\bfE_{\alpha},\bfF_{\alpha})\in
V\times M\stext[l]{suchthat}\\\
A_{\alpha}\left((\bfE_{\alpha},\bfF_{\alpha}),(\bfphi,\bfpsi)\right)=L_{\alpha}\left((\bfphi,\bfpsi)\right)\
\forall(\bfphi,\bfpsi)\in V\times M,\end{array}\right.$

where the sesqui-linear form $A_{\alpha}(\cdot,\cdot)$ and the linear form
$L_{\alpha}(\cdot)$ are defined on $V\times M$ by

$A_{\alpha}\left((\bfu,\bfv),(\bfphi,\bfpsi)\right)=\delta\dotprod{\bfu}{\bfphi}{V}+\eta^{2}\dotprod{\gamma_{t}(\bfu)}{\gamma_{t}(\bfphi)}{0,\Gamma_{0}}+a(\bfphi,\bfv)-a(\bfu,\bfpsi)+\dotprod{\bfv}{\bfpsi}{M}$

and

$L_{\alpha}\left((\bfphi,\bfpsi)\right)=\eta^{2}\dotprod{\bff}{\gamma_{t}(\bfphi)}{0,\Gamma_{0}}-\ell(\bfpsi).$

The continuity of $A_{\alpha}(\cdot,\cdot)$ and $L_{\alpha}(\cdot)$ is
obvious, and coercivity follows since

$A_{\alpha}\left((\bfu,\bfv),(\bfu,\bfv)\right)=\delta\norm{\bfu}{V}^{2}+\eta^{2}\norm{\gamma_{t}(\bfu)}{0,\Gamma_{0}}^{2}+\norm{\bfv}{V}^{2}\geq\min(\delta,1)\norm{(\bfu,\bfv)}{V\times
V}^{2}$

taking into account that the scalar product in $M$ is the one of $V$. We thus
can apply Lax-Milgram’s Theorem to prove existence and uniqueness of a
solution of problem (15).

According to 3.1 and since the new space $M$ is a subspace of the former, the
solution $\bfE$ of the Cauchy problem satisfies

$a(\bfE,\bfpsi)=\ell(\bfpsi)\ \forall\bfpsi\in M.$

Together with the second equation of problem (15), we get

$a(\bfE_{\alpha}-\bfE,\bfpsi)-\dotprod{\bfF_{\alpha}}{\bfpsi}{V}=0\
\forall\bfpsi\in M.$ (19)

Now, take $\bfphi=\bfE_{\alpha}-\bfE$ in problem (15) and
$\bfpsi=\bfF_{\alpha}$ in (19) and substract the latter from the first one.
Taking into account that $\gamma_{t}(\bfE)=\bff$, this yields the fundamental
relation

$\delta\dotprod{\bfE_{\alpha}}{\bfE_{\alpha}-\bfE}{V}+\eta^{2}\norm{\gamma_{t}(\bfE_{\alpha})-\bff}{0,\Gamma_{0}}^{2}+\norm{\bfF_{\alpha}}{V}^{2}=0$
(20)

which implies

$\delta\norm{\bfE_{\alpha}}{V}^{2}+\eta^{2}\norm{\gamma_{t}(\bfE_{\alpha})-\bff}{0,\Gamma_{0}}^{2}+\norm{\bfF_{\alpha}}{V}^{2}\leq\delta\norm{\bfE_{\alpha}}{V}\norm{\bfE}{V}.$
(21)

From (21), we see that $\norm{\bfE_{\alpha}}{V}\leq\norm{\bfE}{V}$, i.e. the
sequence $(\bfE_{\alpha})_{\alpha}$ is bounded with respect to the two
parameters $\delta$ and $\eta$. We then deduce from (21) estimation (17),

$\norm{\bfF_{\alpha}}{V}\leq\sqrt{\delta}\norm{\bfE}{V},$

and thus the convergence of $(\bfF_{\alpha})_{\alpha}$ to $0$ whenever the
regularization parameter $\delta$ tends to $0$.

The boundary term can be estimated in a similar way,

$\norm{\gamma_{t}(\bfE_{\alpha})-\bff}{0,\Gamma_{0}}\leq\frac{\sqrt{\delta}}{\eta}\norm{\bfE}{V},$

which yields estimation (18). In particular, the sequence
$(\gamma_{t}(\bfE_{\alpha}))_{\alpha}$ tends to $\bff$ if
$\lim\limits_{\delta\to 0}\left(\frac{\sqrt{\delta}}{\eta}\right)=0$ which is,
for example, the case for any fixed $\eta>0$.

It remains to prove the convergence of the sequence
$(\bfE_{\alpha})_{\alpha}$. Recall that the sequence is bounded in $V$ which
is a Hilbert space. Therefore, there is a subsequence of
$(\bfE_{\alpha})_{\alpha}$ that converges weakly in $V$ to a limit field
$\tilde{\bfE}$.

Passing to the limit in the second equation of (15) yields

$a(\tilde{\bfE},\bfpsi)=\ell(\bfpsi)\ \forall\bfpsi\in M$

if $\delta\to 0$. Moreover, we get on the one hand

$\dotprod{\gamma_{t}(\bfE_{\alpha})}{\xi}{0,\Gamma_{0}}\to\dotprod{\gamma_{t}(\tilde{\bfE})}{\xi}{0,\Gamma_{0}}\
\forall\xi\in L^{2}(\Gamma_{0})^{3},\ \xi\cdot\bfn=0\stext{on}\Gamma_{0}$

from the weak convergence of $(\bfE_{\alpha})_{\alpha}$ in $V$ and the density
of $Y(\Gamma_{0})\cap L^{2}(\Gamma_{0})^{3}$ in the subspace of tangential
fields of $L^{2}(\Gamma_{0})^{3}$, and on the other

$\gamma_{t}(\bfE_{\alpha})\to\bff$

strongly in $L^{2}(\Gamma_{0})^{3}$ according to (18) and the assumptions on
the parameter set $\alpha$. Consequently, the limit field $\tilde{\bfE}$
satisfies $\gamma_{t}(\tilde{\bfE})=\bff$ and is a solution of the weak Cauchy
problem. The uniqueness of this solution yields $\tilde{\bfE}=\bfE$.

We then deduce from (20) that

$\displaystyle\norm{\bfE_{\alpha}-\bfE}{V}^{2}$
$\displaystyle=\real{\dotprod{\bfE_{\alpha}}{\bfE_{\alpha}-\bfE}{V}-\dotprod{\bfE}{\bfE_{\alpha}-\bfE}{V}}$
$\displaystyle=-\frac{\eta^{2}}{\delta}\norm{\gamma_{t}(\bfE_{\alpha}-\bfE)}{0,\Gamma_{0}}^{2}-\frac{1}{\delta}\norm{\bfF_{\alpha}}{V}^{2}-\real{\dotprod{\bfE}{\bfE_{\alpha}-\bfE}{V}}$
$\displaystyle\leq-\real{\dotprod{\bfE}{\bfE_{\alpha}}{V}}+\norm{\bfE}{V}^{2}.$

Since $(\bfE_{\alpha})_{\alpha}$ converges weakly to $\bfE$, the above
inequality implies the strong convergence, at least for a subsequence. It
follows from a standard argument that the whole sequence converges strongly to
$\bfE$ which completes the proof.

######  Remark 4.3.

Under the regularity assumptions of 3.5, we have an a priori error estimate of
order $\frac{h^{s}}{\min(\delta,1)}$ since the coercivity constant of the
bilinear form $A_{\alpha}(\cdot,\cdot)$ is still given by $\min(\delta,1)$ and
the additional boundary term in the $V$-norm can be estimated with help of the
continuity of the trace operator.

###  4.2 A second version

The choice of the space $V$ in the preceding section has been motivated by the
penalization of the boundary condition $\bfE\times\bfn=\bff$ which can no
longer be imposed strongly if the data do not belong to the trace space
$Y(\Gamma_{0})$. One may ask however if it is judicious to restrict the
belonging of the fields trace to $L^{2}$ on the accessible part $\Gamma_{0}$
only. It seems thus natural to investigate another choice for the basic vector
space: let

$W\coloneqq\set*{\bfv\in\Hcurl[\Omega]}{\gamma_{t}(\bfv)\in
L^{2}(\Gamma)^{3}}$ (22)

with the norm

$\norm{\bfv}{W}=\left(\norm{\bfv}{\Hcurl[\Omega]}^{2}+\norm{\gamma_{t}(\bfv)}{0,\Gamma}^{2}\right)^{1/2}.$
(23)

Notice that the space $M$ defined in (14) keeps unchanged since the boundary
condition $\bfv\times\bfn=0$ on $\Gamma_{1}$ implies, together with the
condition $\bfv\in V$ that $M\subset W$ and $\norm{\bfv}{V}=\norm{\bfv}{W}$
for any field in $M$.

In order to obtain the associated relaxed formulation, we have several
choices. We obviously could just replace the space $V$ by $W$ in the
formulation of problem in (15). The existence and uniqueness of a solution to
the mixed problem on $W$ can be proved in the same way as in the proof of
Theorem 4.1. A slight modification occurs in the proof of the convergence of
the sequence $(\bfE_{\alpha},\bfF_{\alpha})$. Indeed, this requires that the
Cauchy problem (3) admits a solution in the modified vector space $W$ and
implies more regularity of the limit field $\bfE$ on $\Gamma_{1}$. The rest of
the proof keeps unchanged.

From a numerical point of view, it seems however appealing to introduce a new
parameter $\nu>0$ that acts as a regularization parameter on the inaccessible
part $\Gamma_{1}$ of the boundary. An appropriate "tuning" of the parameters
should allow to improve the numerical results.

In view of the latter remark, we define the relaxed mixed problem with
regularization on both the accessible and inaccessible parts as follows:

$\left\\{\begin{array}[]{rcl@{\hspace{4\tabcolsep}}l}\lx@intercol\stext[r]{Find}(\bfE_{\beta},\bfF_{\beta})\in
W\times M\stext[l]{suchthat}\hfil\lx@intercol\\\
\lx@intercol\delta\dotprod{\bfE_{\beta}}{\bfphi}{V}+\nu\dotprod{\gamma_{t}(\bfE_{\beta})}{\gamma_{t}(\bfphi)}{0,\Gamma_{1}}\hfil\lx@intercol&&\hfil\hskip
24.0pt&\hbox{\multirowsetup$\forall\bfphi\in W,$}\\\
\qquad+\eta^{2}\dotprod{\gamma_{t}(\bfE_{\beta})}{\gamma_{t}(\bfphi)}{0,\Gamma_{0}}+a(\bfphi,\bfF_{\beta})&=&\eta^{2}\dotprod{\bff}{\gamma_{t}(\bfphi)}{0,\Gamma_{0}},\hfil\hskip
24.0pt&\\\
a(\bfE_{\beta},\bfpsi)-\dotprod{\bfF_{\beta}}{\bfpsi}{W}&=&\ell(\bfpsi),\hfil\hskip
24.0pt&\forall\bfpsi\in M,\end{array}\right.$ (24)

where the subscript $\beta=(\delta,\nu,\eta)$ indicates that the solution of
(15) depends on the regularization parameters $\delta>0$ and $\nu>0$ as well
as on the relaxation parameter $\eta>0$. To our knowledge, this relaxed and
regularized mixed version of the QR method has never been proposed in the
previous studies of the method.

Notice that the first two terms in the first equation are well defined
according to the definition of the space $W$, but that the weight of the
different parts of the norm can be chosen independently.

######  Theorem 4.4.

Let $(\bff,\bfg)\in L^{2}(\Gamma_{0})^{3}\times L^{2}(\Gamma_{0})^{3}$. For
any $\beta=(\delta,\nu,\eta)$ such that $\delta>0$, $\nu>0$ and $\eta>0$,
problem (24) admits a unique solution $(\bfE_{\beta},\bfF_{\beta})\in W\times
M$.

If, in addition, $(\bff,\bfg)\in C(\eps,\sigma;\Gamma_{0})$ and if the
corresponding solution $\bfE$ of the Cauchy problem (3) belongs to the space
$W$, then for any fixed $\eta>0$,

$\lim_{(\delta,\nu)\to 0}(\bfE_{\beta},\bfF_{\beta})=(\bfE,0)$ (25)

in $W\times M$ whenever the parameters $\delta$ and $\nu$ satisfy the relation

$\nu\coloneqq\nu(\delta)=\delta+o(\delta)$ (26)

such that the ratio $\delta/\nu$ tends to $1$ as $\delta\to 0$. The following
estimates hold true

$\displaystyle\norm{\bfF_{\beta}}{W}$
$\displaystyle\leq\sqrt{\max(\delta,\nu)C(\delta,\nu)}\norm{\bfE}{W}$ (27)
$\displaystyle\norm{\gamma_{t}(\bfE_{\beta})-\bff}{0,\Gamma_{0}}$
$\displaystyle\leq\frac{\sqrt{\max(\delta,\nu)C(\delta,\nu)}}{\eta}\norm{\bfE}{W}.$
(28)

where $C(\delta,\nu)\coloneqq\dfrac{\max(\delta,\nu)}{\min(\delta,\nu)}$.

The proof is similar to the one of Theorem 4.1 and we only point out the
influence of the parameter $\nu$ on the different steps of the proof.

######  Proof 4.5.

As before, problem (24) can be written in variational form involving a
continuous and coercive sesqui-linear form on $W\times M$. The coercivity
constant is now given by $\min(\delta,\nu,1)$.

If, in addition, the Cauchy problem (3) admits a solution $\bfE\in W$, the
orthogonality relation reads as follows,

$\delta\dotprod{\bfE_{\beta}}{\bfE_{\beta}-\bfE}{V}+\nu\dotprod{\gamma_{t}(\bfE_{\beta})}{\gamma_{t}(\bfE_{\beta}-\bfE)}{0,\Gamma_{1}}+\eta^{2}\norm{\gamma_{t}(\bfE_{\beta})-\bff}{0,\Gamma_{0}}^{2}+\norm{\bfF_{\beta}}{W}^{2}=0$
(29)

by applying the same method used for the derivation of the relation (19).

Developping the scalar products and applying Cauchy-Schwarz’ inequality on the
real parts yields the following estimation of $\bfE_{\beta}$

$\begin{split}\delta\norm{\bfE_{\beta}}{V}^{2}+\nu\norm{\gamma_{t}(\bfE_{\beta})}{0,\Gamma_{1}}^{2}+\eta^{2}\norm{\gamma_{t}(\bfE_{\beta})-\bff}{0,\Gamma_{0}}^{2}+\norm{\bfF_{\beta}}{W}^{2}\leq\delta\norm{\bfE_{\beta}}{V}\norm{\bfE}{V}+\nu\norm{\gamma_{t}(\bfE_{\beta})}{0,\Gamma_{1}}\norm{\gamma_{t}(\bfE)}{0,\Gamma_{1}}.\end{split}$

In order to get estimates for the $W$-norm, we notice that the left hand side
can be minored by $\min(\delta,\nu)\norm{\bfE_{\beta}}{W}^{2}$, whereas the
right hand side can be majored by
$\max(\delta,\nu)\norm{\bfE_{\beta}}{W}\norm{\bfE}{W}$. Consequently,

$\norm{\bfE_{\beta}}{W}\leq\frac{\max(\delta,\nu)}{\min(\delta,\nu)}\norm{\bfE}{W}.$
(30)

Under the given assumptions, the sequence $(\bfE_{\beta})_{\beta}$ is thus
bounded in $W$ and we obtain as before that $F_{\beta}$ converges to $0$ in
$M$ when $(\delta,\nu)\to 0$ since

$\norm{\bfF_{\beta}}{W}^{2}\leq\max(\delta,\nu)\norm{\bfE_{\beta}}{W}\norm{\bfE}{W}\leq\max(\delta,\nu)C(\delta,\nu)\norm{\bfE}{W}^{2}$

where $C(\delta,\nu)=\dfrac{\max(\delta,\nu)}{\min(\delta,\nu)}$ is bounded.
This yields estimate (27). In the same way, we get (28).

As before, we prove that the sequence $(\bfE_{\beta})_{\beta}$ converges
weakly to the solution $\bfE$ of the Cauchy problem. In order to show strong
convergence, we notice that, according to (29),

$\delta\real{\dotprod{\bfE_{\beta}}{\bfE_{\beta}-\bfE}{V}}+\nu\real{\dotprod{\gamma_{t}(\bfE_{\beta})}{\gamma_{t}(\bfE_{\beta}-\bfE)}{0,\Gamma_{1}}}\leq
0.$

Hence,

$\displaystyle\norm{\bfE_{\beta}-\bfE}{W}^{2}$
$\displaystyle=\real{\dotprod{\bfE_{\beta}}{\bfE_{\beta}-\bfE}{W}}-\real{\dotprod{\bfE}{\bfE_{\beta}-\bfE}{W}}$
$\displaystyle\leq\left(1-\frac{\delta}{\nu}\right)\real{\dotprod{\bfE_{\beta}}{\bfE_{\beta}-\bfE}{V}}-\real{\dotprod{\bfE}{\bfE_{\beta}-\bfE}{W}}$
$\displaystyle\leq\abs*{1-\frac{\delta}{\nu}}\norm{\bfE_{\beta}}{V}\norm{\bfE_{\beta}-\bfE}{V}-\real{\dotprod{\bfE}{\bfE_{\beta}-\bfE}{W}}$

Now, the first term in the last inequality tends to $0$ according to the
assumptions on $\delta$ and $\nu$ and since $(\bfE_{\beta})_{\beta}$ is
bounded in $W$ and thus in $V$. The second term tends to $0$ since
$\bfE_{\beta}$ converges weakly to $\bfE$ in $W$. This completes the proof.

######  Remark 4.6.

For the relaxed regularized version of the QR-method and under the regularity
assumptions of 3.5, the discretization error is now of order
$\frac{h^{s}}{\min(\delta,\nu,1)}$ taking into account the coercivity constant
of the bilinear form. Again, the boundary term in the $W$-norm is controlled
by the $\Hcurl[\Omega]$-norm due to the continuity of the trace operator.

###  4.3 Strong formulations and regularity results

In this section, we give the strong formulation of the relaxed and regularized
QR-method (24). This allows us to get a regularity result which yields further
insight in the numerical behavior of the discrete solution. Taking real-valued
test fields in $\mathcal{D}(\Omega)^{3}$, we see that the solution
$(\bfE_{\beta},\bfF_{\beta})$ satisfies the following partial differential
equations on $\Omega$ in the distributional sense:

$\displaystyle\delta\curl\curl\bfE_{\beta}+\delta\bfE_{\beta}+\curl\curl\overline{\bfF}_{\beta}-k^{2}\kappa\overline{\bfF}_{\beta}$
$\displaystyle=$ $\displaystyle 0\stext{in}\Omega,$ (31a)
$\displaystyle\curl\curl\bfE_{\beta}-k^{2}\kappa\bfE_{\beta}-\curl\curl{\bfF}_{\beta}-{\bfF}_{\beta}$
$\displaystyle=$ $\displaystyle 0\stext{in}\Omega.$ (31b)

System (31) yields

$\displaystyle\curl\curl(\delta\bfE_{\beta}+\overline{\bfF}_{\beta})$
$\displaystyle=\bfG\stext{in}\Omega,$ (32a)
$\displaystyle\curl\curl(\bfE_{\beta}-\bfF_{\beta})$
$\displaystyle=\mathbf{H}\stext{in}\Omega,$ (32b)

with $\bfG=-\delta\bfE_{\beta}+k^{2}\kappa\overline{\bfF}_{\beta}\in
L^{2}(\Omega)^{3}$ and $\mathbf{H}=k^{2}\kappa\bfE_{\beta}+\bfF_{\beta}\in
L^{2}(\Omega)^{3}$. For $\delta\neq 1$, it thus follows that
$\curl\bfE_{\beta}$ and $\curl\bfF_{\beta}$ belong to $\Hcurl[\Omega]$ and the
traces $\gamma_{t}(\curl\bfE_{\beta})$, $\gamma_{t}(\curl\bfF_{\beta})$ are
well defined as elements in $Y(\Gamma)$. Partial integration with appropriated
test fields $\Phi$ and $\Psi$ yields the following boundary conditions on
$\Gamma_{0}$ :

$\displaystyle\delta\gamma_{t}(\curl\bfE_{\beta})+\delta\gamma_{t}(\bfE_{\beta})+\eta^{2}\gamma_{t}(\bfE_{\beta})+\gamma_{t}(\curl\overline{\bfF}_{\beta})$
$\displaystyle=\eta^{2}\bff\stext{on}\Gamma_{0},$ (33a)
$\displaystyle\gamma_{t}(\curl\bfE_{\beta})-\gamma_{t}(\curl\bfF_{\beta})-\gamma_{t}(\bfF_{\beta})$
$\displaystyle=\bfg\stext{on}\Gamma_{0}.$ (33b)

On the inaccessible part $\Gamma_{1}$ of the boundary, we get in a similar way

$\delta\gamma_{t}(\curl\bfE_{\beta})+\nu\gamma_{t}(\bfE_{\beta})+\gamma_{t}(\curl\overline{\bfF}_{\beta})=0\stext{on}\Gamma_{1}.$
(34)

It remains to study the divergence of the fields $\bfE_{\beta}$ and
$\bfF_{\beta}$. Taking test fields that are the gradients of some regular
scalar potentials in $H^{1}_{0}(\Omega)$, we get

$\displaystyle\delta\div\bfE_{\beta}-k^{2}\div(\kappa\overline{\bfF}_{\beta})$
$\displaystyle=0\stext{in}\Omega,$ (35a) $\displaystyle
k^{2}\div(\kappa\bfE_{\beta})+\div\bfF_{\beta}$
$\displaystyle=0\stext{in}\Omega.$ (35b)

The following regularity result in Lipschitz domains is due to Costabel [28].

######  Lemma 4.7.

Let $\bfu$ be a vector field such that $\bfu\in\Hcurl[\Omega]$ and
$\div\bfu\in L^{2}(\Omega)$. Then $\bfu\times\bfn\in L^{2}(\Gamma)^{3}$ if and
only if $\bfu\cdot\bfn\in L^{2}(\Gamma)$. Further, if one of these conditions
is satisfied, $\bfu$ belongs to $H^{1/2}(\Omega)^{3}$.

We are now able to state a regularity result for the solution of the RR-QR
method in the case of a constant coefficient $\kappa$:

######  Theorem 4.8.

Let $(\bfE_{\beta},\bfF_{\beta})\in W\times M$ be the solution of problem (24)
with a constant refractive index $\kappa$ and data $(\bff,\bfg)\in
L^{2}(\Gamma_{0})^{6}$. Assume that the parameter $\delta>0$ is such that
$\delta\neq k^{4}|\kappa|^{2}$. Then, $\div\bfE_{\beta}=\div\bfF_{\beta}=0$ on
$\Omega$ and $(\bfE_{\beta},\bfF_{\beta})\in H^{1/2}(\Omega)^{6}$.

######  Proof 4.9.

For easier reading, denote by $\bfz_{R}$ (resp. $\bfz_{I}$) the real (resp.
imaginary) part of a complex number $\bfz$ and let $\bfz=\div\bfE_{\beta}$ and
$\bfw=\div\bfF_{\beta}$. Then, relations (35) imply that
$\begin{bmatrix}\bfz_{R},\bfz_{I},\bfw_{R},\bfw_{I}\end{bmatrix}^{t}$ is
solution of the linear system $Ax=0$ with

$A=\begin{pmatrix}\delta&0&-k^{2}\kappa_{R}&-k^{2}\kappa_{I}\\\
0&\delta&-k^{2}\kappa_{I}&k^{2}\kappa_{R}\\\
k^{2}\kappa_{R}&-k^{2}\kappa_{I}&1&0\\\
k^{2}\kappa_{I}&k^{2}\kappa_{R}&0&1\end{pmatrix}.$

A straightforward computation yields

${\rm det}(A)=(\delta+k^{4}|\kappa|^{2})(\delta-k^{4}|\kappa|^{2}).$

It follows that $\div\bfE_{\beta}=\div\bfF_{\beta}=0$ in $\Omega$ provided
that $\delta\neq k^{4}|\kappa|^{2}$. Applying 4.7 to the solution of the RR-QR
method shows that has $(\bfE_{\beta},\bfF_{\beta})$ at least
$H^{1/2}$-regularity.

######  Remark 4.10.

If $\kappa$ is a regular function of $W^{1,\infty}(\Omega)$, we can prove in a
similar way that $\div\bfE_{\beta}$ and $\div\bfF_{\beta}$ belong to
$L^{2}(\Omega)$. Indeed, developping the divergence of the products in (35)
yields a system with the same matrix $A$ and a right-hand side in
$L^{2}(\Omega)$. Concerning the particular case of coefficients $\kappa\in
L^{\infty}(\Omega)$ that are piecewise constant with respect to a partition
$\mathcal{P}=\\{\Omega_{p}\\}_{p=1:P}$ into regular subdomains $\Omega_{p}$,
we get in a first step that $\bfE_{\beta}$ and $\bfF_{\beta}$ are divergence
free on each part $\Omega_{p}$ provided $\delta\neq k^{4}|\kappa_{p}|^{2}$ for
all $p$. Then, the transmission conditions along the interfaces $\Sigma_{p}$
read

$\displaystyle\delta[\gamma_{n}(\bfE_{\beta})]-k^{2}[\gamma_{n}(\kappa\overline{\bfF}_{\beta})]$
$\displaystyle=0\stext{on}\Sigma_{p},$ (36a) $\displaystyle
k^{2}[\gamma_{n}(\kappa\bfE_{\beta})]+[\gamma_{n}(\bfF_{\beta})]$
$\displaystyle=0\stext{on}\Sigma_{p}.$ (36b)

For $\delta\neq 1$ we get
$[\gamma_{n}\bfE_{\beta}]=[\gamma_{n}(\kappa\bfE_{\beta})]=0$ which implies
$\gamma_{n}(\bfE_{\beta})=0$ unless $[\kappa]\neq 0$. Again,
$\div\bfE_{\beta}$ belongs to $L^{2}(\Omega)$ and the same result holds for
$\bfF_{\beta}$.

######  Theorem 4.11.

Under the assumptions of Theorem 4.8 and if in addition $\delta\neq 1$, we
have $\curl\bfE_{\beta}\in\Hcurl[\Omega]$ and
$\curl\bfF_{\beta}\in\Hcurl[\Omega]$. Further, for any function
$\xi\in\mathcal{C}^{\infty}(\overline{\Omega})$ such that $\xi\equiv 0$ in a
vicinity $\mathcal{V}$ of $\overline{\Gamma}_{0}\cap\overline{\Gamma}_{1}$,
the fields $\curl(\xi\bfE_{\beta})$ and $\curl(\xi\bfF_{\beta})$ belong to
$\Hcurl[\Omega]\cap H^{1/2}(\Omega)^{3}$.

######  Proof 4.12.

Since the data $\bff$ and $\bfg$ as well as the trace fields
$\gamma_{t}(\bfE_{\beta})$ and $\gamma_{t}(\bfF_{\beta})$ belong to
$L^{2}(\Gamma_{0})^{3}$, we may deduce from (33) that
$\gamma_{t}(\curl\bfE_{\beta})$ and $\gamma_{t}(\curl\bfF_{\beta})$ belong to
$L^{2}(\Gamma_{0})^{3}$ provided that $\delta\neq 1$. On the inaccessible part
$\Gamma_{1}$, we have $\gamma_{t}(\bfF_{\beta})=0$ on $\Gamma_{1}$ since
$\bfF_{\beta}$ belongs to the vector space $M$. According to a classical
differential identity on surfaces, this implies
$\gamma_{n}(\curl\bfF_{\beta})=0$ where $\gamma_{n}$ is the extension of the
normal trace operator. Now, let $\eta$ be a smooth function that vanishes near
the accessible part $\Gamma_{0}$ and let $\bfw=\curl\bfF_{\beta}$. We have
$\gamma_{n}(\eta\bfw)=0$ on $\Gamma$. A straightforward computation shows that
$\curl(\eta\bfw)\in L^{2}(\Omega)^{3}$ and $\div(\eta\bfw)\in L^{2}(\Omega)$.
According to 4.7, the condition $\gamma_{n}(\eta\bfw)\in L^{2}(\Gamma)^{3}$
implies $\gamma_{t}(\eta\bfw)\in L^{2}(\Gamma)^{3}$. Since we already proved
that $\gamma_{t}(\bfw)$ belongs to $L^{2}(\Gamma_{0})^{3}$, we get the same
result for $\xi\bfw$ with a function $\xi$ that vanishes only near the
intersection of $\overline{\Gamma}_{0}$ and $\overline{\Gamma}_{1}$. Applying
once again Costabel’s regularity result yields $\xi\bfw\in
H^{1/2}(\Omega)^{3}$. The result for $\curl\bfE_{\beta}$ can be obtained in a
similar way, noticing that
$\gamma_{t}(\curl\bfE_{\beta})=-\nu\gamma_{t}(\bfE_{\beta})-\gamma_{t}(\curl\overline{\bfF}_{\beta})$
on $\Gamma_{1}$ according to (34).

######  Remark 4.13.

The above regularity results do not yield global $H^{1/2}$-regularity for the
curl of the fields $\bfE_{\beta}$ and $\bfF_{\beta}$. This probably explains
the singular behavior of the discrete solutions near the intersection between
$\overline{\Gamma}_{0}$ and $\overline{\Gamma}_{1}$ observed in the numerical
simulations.

###  4.4 Numerical results with noisy data

We use the same physical parameters as in subsection 3.2. In the sequel, we
describe the generation of synthetic noisy data. To simplify the notation, we
consider here that the input data $\bff$ and $\bfg$ are vectors of degrees of
freedom. They are perturbed as follows. First, two vectors $\bfb_{\bff}$ and
$\bfb_{\bfg}$ are generated, following a standard normal distribution. Then,
the perturbed data $(\bff^{p},\bfg^{p})$ are obtained from

$\bff^{p}=\bff+p\frac{\norm{\bff}{}}{\norm{\bfb_{\bff}}{}}\bfb_{\bff},\quad\bfg^{p}=\bfg+p\frac{\norm{\bfg}{}}{\norm{\bfb_{\bfg}}{}}\bfb_{\bfg},$

where $p>0$ is the applied level of noise and $\|\cdot\|$ the $\ell^{2}$-norm.

Numerical results are presented for the second version of the regularized
relaxed quasi-reversibility (RR-QR) method (24) with $\nu=\delta$ and
$p=\pc{5}$ noise. Different choices for the value of the parameter $\eta$ have
been tested, and for each choice of $\eta$ the error of the QR-method has been
evaluated with respect to $\delta$. It was found that the ratio between $\eta$
and the corresponding optimal $\delta$ was constant. In the sequel, the
parameter $\eta$ is fixed automatically according to the following procedure.
First, we compute the Riesz representative $\bfG^{p}$ of $\bfg^{p}$ in the
Hilbert space $M$ by solving

$\dotprod{\bfG^{p}}{\bfpsi}{W}=\dotprod{\bfg^{p}}{\bfpsi}{0,\Gamma_{0}},\quad\forall\bfpsi\in
M.$

Then, we define

$\eta=\frac{\norm{\bfG^{p}}{{W}}}{\norm{\bff^{p}}{L^{2}(\Gamma_{0})}}.$

As suggested in [13], $\eta$ may then be interpreted as a balance term between
the data $\bfG^{p}$ and $\bff^{p}$ which are given in different measurement
units.

First, let $\Omega$ be the unit disc. In Figure 9 we show the evolution of the
error in $L^{2}(\Omega)$-norm with respect to $\delta$, for the two
configurations of the boundary. For the specific case of $\delta$ minimizing
the error, we show this error over the whole domain in Figure 10 and we list
the different errors in Table 4.

Figure 9: Unit disc. 5 noisy data. RR-QR method. Relative error
$\frac{\norm{\bfE-\bfE_{\beta}}{0,\Omega}}{\norm{\bfE}{0,\Omega}}$ with
respect to the regularization parameter $\delta$ at fixed $\eta$ and
$\nu=\delta$. Left: configuration G34. Right: configuration GE37.

Figure 10: Unit disc. 5 noisy data. RR-QR method. Modulus of the error $\abs{\bfE-\bfE_{\beta}}$. Left: configuration G34. Right: configuration GE37. Table 4: Unit disc. 5 noisy data. RR-QR method. Errors in the approximation of $\bfE$ for $\nu=\delta$ at optimal delta and automatically fixed $\eta$. Configuration | $\frac{\norm{\bfE-\bfE_{\delta}}{0,\Omega}}{\norm{\bfE}{0,\Omega}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{0}}}{\norm{\bfE\times\bfn}{0,\Gamma_{0}}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{1}}}{\norm{\bfE\times\bfn}{0,\Gamma_{1}}}$ | $\norm{\bfF_{\delta}}{0,\Omega}$
---|---|---|---|---
G34 | 3.4698e-02 | 2.9145e-02 | 3.5686e-01 | 7.0667e-03
GE37 | 1.8550e-02 | 3.2848e-02 | 1.7154e-01 | 1.1857e-02

With the same settings, we now test the regularized relaxed quasi-
reversibility method (RR-QR) in the ring with extension/restriction. The
relative error in $\bfE$ with respect to $\delta$ is shown in Figure 11. For
the optimal values of $\delta$, the errors are reported in Table 5 and
illustrated in Figure 12. One notices the good performance of the method for
the three configurations with errors below 8 in all norms except for
configuration G34 on $\Gamma_{1}$ where the error amounts to 15. In
particular, the error on the interior boundary $\Gamma_{i}$ is under $5\%$ for
the three configurations. The method performs better in the ring configuration
than in the unit disc.

Figure 11: Ring. 5 noisy data. RR-QR method with extension/restriction.
Relative error for $\bfE$ in $L^{2}(\Omega)$-norm with respect to $\delta$ for
$\nu=\delta$. $\eta$ automatically fixed from noise level.

Figure 12: Ring. 5 noisy data. RR-QR method with extension/restriction. Modulus of the error in the approximation of $\bfE$ for optimal $\delta$ and $\nu=\delta$. $\eta$ automatically fixed from noise level. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37. Table 5: Ring. 5 noisy data. RR-QR method with extension/restriction. Errors in the approximation of $\bfE$ for optimal $\delta$ and $\nu=\delta$. $\eta$ automatically fixed from noise level. Errors on parts of the boundary refer to the $L^{2}$-norm of the tangential trace. Configuration | $\frac{\norm{\bfE-\bfE_{\delta}}{0,\Omega}}{\norm{\bfE}{0,\Omega}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{0}}}{\norm{\bfE\times\bfn}{0,\Gamma_{0}}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{1}}}{\norm{\bfE\times\bfn}{0,\Gamma_{1}}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{\text{i}}}}{\norm{\bfE\times\bfn}{0,\Gamma_{\text{i}}}}$ | $\norm{\bfF_{\delta}}{0,\Omega}$
---|---|---|---|---|---
GExt | 9.2775e-03 | 3.3871e-02 | 4.2425e-03 | 4.2425e-03 | 1.0602e-02
G34 | 7.9236e-02 | 3.0600e-02 | 1.5695e-01 | 4.2697e-02 | 8.7901e-03
GE37 | 2.6712e-02 | 3.3503e-02 | 7.7043e-02 | 1.6996e-02 | 1.3596e-02

###  4.5 Numerical results in three dimensions

We finally present tests in 3D. The physical settings remain the same as in
the previous section, and we generate a noise of 5 in the data. The parameter
$\eta$ is automatically fixed following the above mentioned procedure, and we
choose $\nu=\delta$.

Here, the domain $\Omega$ is a subset of $\R^{3}$. Two domains are tested. The
first one is the unit ball of $\R^{3}$, discretized with a mesh size
$h=\scnum{1.43e-1}$, ending up with $90\,077$ tetrahedrons and $114\,347$
edges. The accessible part, $\Gamma_{0}$, is defined as the union of 128
electrodes covering approximatively 61 of the whole boundary. This boundary
configuration is illustrated in Figure 13.

Figure 13: Accessible part $\Gamma_{0}$ in 3D configurations. Set of 128
electrodes.

The second domain is a ring of width 0.3, defined as the unit ball of
$\R^{3}$, minus the ball centered at the origin and of radius 0.7. The mesh
size is close to the ball’s one: $h=\scnum{1.41e-1}$, ending up with $70\,420$
tetrahedrons and $96\,686$ edges. The boundary configuration is the same as
for the unit disc: the interior boundary of the ring is part of $\Gamma_{1}$
and $\Gamma_{0}$ represents here about 41 of the whole boundary. The quasi-
reversibility system will be solved in this 3D ring with the
extension/restriction method introduced in algorithm 1.

We show in Figure 14 the evolution of the relative error in $L^{2}$-norm in
the whole domain with respect to $\delta$. As in the two dimensional cases,
the error decreases with $\delta$, reaches a minimum, and then increases.
Table 6 lists the errors obtained with the value of $\delta$ realizing this
minimum. The error $\norm{\bfE_{\delta}-\bfE}{0,\Omega}$ is shown in this case
in Figure 15. One notices that despite the larger mesh size compared to the 2D
configuration, the approximation is still very good, particularly on the inner
boundary in the ring configuration where an error of 5 may be obtained. A
possible way to consider finer meshes would be to implement a domain
decomposition method for 3D Maxwell equations but this is beyond the scope of
this paper.

Figure 14: Relative error $\frac{\norm{\bfE-\bfE_{\delta}}{0,\Omega}}{\norm{\bfE}{0,\Omega}}$ with respect to the regularization parameter $\delta$. Left: Unit ball of $\R^{3}$. Right: 3D ring of internal radius 0.7. Table 6: Errors in the unit ball of $\R^{3}$ and in a three dimensional ring. Errors on parts of the boundary refer to the $L^{2}$-norm of the tangential trace. Domain | $\frac{\norm{\bfE-\bfE_{\delta}}{0,\Omega}}{\norm{\bfE}{0,\Omega}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{0}}}{\norm{\bfE\times\bfn}{0,\Gamma_{0}}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{1}}}{\norm{\bfE\times\bfn}{0,\Gamma_{1}}}$ | $\frac{\norm{(\bfE-\bfE_{\delta})\times\bfn}{0,\Gamma_{\text{i}}}}{\norm{\bfE\times\bfn}{0,\Gamma_{\text{i}}}}$ | $\norm{\bfF_{\delta}}{0,\Omega}$
---|---|---|---|---|---
Ball | 1.0834e-01 | 2.3864e-02 | 4.0949e-01 | $\cdot$ | 2.9336e-03
Ring | 1.2414e-01 | 1.1567e-02 | 2.7969e-01 | 5.6528e-02 | 1.6497e-03

Figure 15: Error in the three-dimensional ring. Left: view of the external
boundary. Right: cut showing the internal boundary.

###  4.6 Choice of the regularization parameter. General case.

From the former numerical results, we can attest that the QR-method in its
classical or regularized and relaxed version is a reliable tool for data
completion and data transmission problems in two and three dimensional
configurations.

The choice of the regularization parameter $\delta$ is crucial for the quality
of the approximation. In the previous tests, the parameter has been fixed to
an optimal value by evaluating the error in the $L^{2}(\Omega)$-norm for each
value in a given range. In practice, when the exact solution is not known,
this procedure is not possible, but classical strategies as the Morozov’s
discrepancy principle or the L-curve method can be applied to retrieve a value
for $\delta$. The L-curve method consists in drawing the graph of
$\norm{\bfE_{\delta}}{0,\Omega}$ with respect to
$\norm{\bfF_{\delta}}{0,\Omega}$ for different values of $\delta$. We then
choose $\delta$ to minimize both norms as much as possible. In Figure 16, we
show the L-curve we obtained in the ring with 5 noise for the configuration
G37 and the three-dimensional ring. Interestingly, the minimum of the error on
$\Gamma_{i}$, highlighted by a circle, is located at the corner of the
L-curve. This indicates that, in the tested cases, the L-curve method yields a
good regularization parameter which can then be automatically computed with,
for example, the triangle method [21]. The convergence of the L-curve method
when noise tends to zero is however not addressed. As it is known that this
method shows limitations in some cases [34], it could be interesting to
compare with other strategies (see for instance [27, 6]) but such comparisons
are beyond the scope of this paper. We refer to [50], Chapter 7, where
different methods are compared in the context of a linear least square
problem.

Figure 16: L-curve corresponding to the RR-QR method with
extension/restriction in the ring. 5 noisy data. Left: configuration G37 (2D).
Right: 3D ring.

The choice of the second regularization parameter $\nu$ should been made in
accordance with Theorem 4.4, i.e. $\nu\coloneqq\nu(\delta)=\delta+o(\delta)$.
However, this condition has only been proven to be sufficient to assure the
convergence of the RR-QR method, and other choices of $\nu$ could possibly
improve the results. Finally, in all our numerical experiments with the RR-QR
method, the relaxation parameter $\eta$ has been fixed according to the
heuristical arguments presented here above. Again, this choice is possibly not
optimal and could be improved.

##  5 A few words on the link with Tikhonov regularization

Before concluding this paper, we make a remark on the link between the quasi-
reversibility method formulated as a mixed problem and standard Tikhonov
regularization, as mentioned in [13]. We shall make precise the corresponding
Tikhonov functionals for the different versions of the QR-method. To this end,
denote by $A\colon\Hcurl[\Omega]\to M$ the unique linear continuous operator
defined by

$a(\bfu,\bfpsi)=\dotprod{A\bfu}{\bfpsi}{\Hcurl[\Omega]}\ \forall\bfpsi\in M$

according to the Riesz representation theorem. In the same way, define by
$\bfG\in M$ the unique Riesz representative of the continuous linear form
$\ell(\cdot)$ such that

$\dotprod{\bfG}{\bfpsi}{\Hcurl[\Omega]}=\ell(\bfpsi)\ \forall\bfpsi\in M.$

Then, the Cauchy problem (3) consists in finding $\bfE\in\Hcurl[\Omega]$ such
that $\gamma_{t}(\bfE)=\bff$ and $A\bfE=\bfG$.

Now, for a given parameter $\delta>0$, introduce the (real-valued) cost
function

$J_{\delta}(\bfv)=\frac{1}{2}\norm{A\bfv-\bfG}{\Hcurl[\Omega]}^{2}+\frac{\delta}{2}\norm{\bfv}{\Hcurl[\Omega]}^{2}$
(37)

defined on $\Hcurl[\Omega]$. The directional derivative of $J_{\delta}$ in the
direction $\bfd\in V_{0}$ is given by

$J_{\delta}^{\prime}(\bfv)\bfd=\real{\dotprod{A\bfv-\bfG}{A\bfd}{\Hcurl[\Omega]}+\delta\dotprod{\bfv}{\bfd}{\Hcurl[\Omega]}}.$
(38)

Now, let $(\bfE_{\delta},\bfF_{\delta})\in V_{\bff}\times M$ be the solution
of the classical QR-problem (8). Then, we get from the second equation of (8)
that $\bfF_{\delta}=A\bfE_{\delta}-\bfG$. Substituting $\bfF_{\delta}$ by this
relation in the first equation and taking the real part, yields
$J_{\delta}^{\prime}(\bfE_{\delta})=0$ according to (38) and the definition of
$A$ and $\bfG$. In conclusion, the unique solution $\bfE_{\delta}$ of the QR-
method is the critical point of the functional $J_{\delta}$ defined by (37).

In the same way, one can show that the solution $\bfE_{\alpha}$ of the relaxed
QR method (15) is the critical point of the functional

$J_{\alpha}(\bfv)=\frac{1}{2}\norm{A\bfv-\bfG}{V}^{2}+\frac{\eta^{2}}{2}\norm{\gamma_{t}(\bfv)-\bff}{0,\Gamma_{0}}^{2}+\frac{\delta}{2}\norm{\bfv}{V}^{2}$

defined on the vector space $V$ for a parameter set $\alpha=(\delta,\eta)$.

Finally, if we let

$J_{\beta}(\bfv)=J_{\alpha}(\bfv)+\frac{\nu}{2}\norm{\gamma_{t}(\bfv)}{0,\Gamma_{1}}^{2},$

for any $\bfv\in W$ and $\beta=(\delta,\eta,\nu)$, the solution $\bfE_{\beta}$
of the relaxed and regularized QR method (24) is the critical point of
$J_{\beta}$.

##  6 Conclusion and future works

In this paper, we have presented a first conclusive study of the applicability
of the quasi-reversibility method to the numerical resolution of a data
completion problem for Maxwell’s equations. The main issue was to retrieve the
missing data on the inaccessible part in a stable way. We have addressed the
theoretical and numerical analysis of the method in vector spaces that are
naturally involved in variational theory for Maxwell’s equations. Different
mixed formulations have been proposed: a classical one, and two regularized
relaxed formulations which are essential to consider noisy data. We have
proved their well-posedness and the convergence of the regularized solution to
the exact solution under some conditions on the regularization and relaxation
parameters. We further showed that the solution of the RR-QR method belongs to
$H^{1/2}(\curl)$ away from the intersection of accessible and inaccessible
parts. Further investigations in the vein of [10] on the involved
singularities could help to improve the results and are part of ongoing work.
A large variety of two-dimensional and three-dimensional numerical results
attest the efficiency of the quasi-reversibility method for data completion
and data transmission problems in electromagnetics. For the case of noisy
data, we have studied in detail the numerical choice of the regularization
parameters.

This non-iterative data completion procedure would be an interesting initial
step for numerical methods to reconstruct the dielectric properties of a
tissue or a material. In [29], we developped for instance an algorithm that is
able to localize a perturbation of small amplitude in the parameters $\eps$
and $\sigma$ from given total data of the perturbed field on the boundary. In
the case of partial data, the results of the present paper could complete the
missing information. We have in mind brain medical applications (e.g.
microwave tomography [48, 49]) and the cortical mapping problem which consists
in reconstructing from electric potential measurements available at part of
the scalp the potential on the inner cortex surface. This question of multi-
layer data completion has been studied for Laplace’s equation in the context
of the EEG (electroencephalography) inverse source localization [25, 26]. To
the best of our knowledge, this problem hasn’t been treated for
electromagnetic inverse medium problems and induces challenging questions from
both the theoretical and numerical point of view. This is part of ongoing
work.

## References

  * [1] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella,  The stability for the Cauchy problem for elliptic equations,  _Inverse problems_ , 25 (2009), 123004.
  * [2] S. Andrieux, T. N. Baranger and A. Ben Abda,  Solving Cauchy problems by minimizing an energy-like functional,  _Inverse Problems_ , 22 (2006), 115–133.
  * [3] M. Azaïez, F. Ben Belgacem and H. El Fekih,  On Cauchy’s problem: II. Completion, regularization and approximation,  _Inverse Problems_ , 22 (2006), 1307–1336.
  * [4] M. Azaïez, F. Ben Belgacem, T. D. Du and F. Jelassi,  A finite element model for the data completion problem: Analysis and assessment,  _Inverse Probl. Sci. Eng._ , 19.8 (2011), 1063–1086.
  * [5] S. Bauer, D. Pauly and M. Schomburg,  The Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions,  _SIAM J. Math. Anal._ , 48 (2016), 2912–2943.
  * [6] F. S. V. Bazán and J. B. Francisco,  An improved fixed-point algorithm for determining a Tikhonov regularization parameter,  _Inverse Problems_ , 25 (2009), 045007.
  * [7] F. Ben Belgacem,  Why is the Cauchy problem severely ill-posed?,  _Inverse Problems_ , 23 (2007), 823–836.
  * [8] F. Ben Belgacem and H. El Fekih,  On Cauchy’s problem: I. A variational Steklov-Poincaré theory,  _Inverse Problems_ , 21 (2005), 1915–1936.
  * [9] L. Bourgeois,  A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation,  _Inverse Problems_ , 21 (2005), 1087–1104.
  * [10] L. Bourgeois and L. Chesnel,  On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: Regularity and error estimates,  _ESAIM Math. Model. Numer. Anal._ , 54 (2020), 493–529.
  * [11] L. Bourgeois and J. Dardé,  A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data,  _Inverse Problems_ , 26.9 (2010), 095016.
  * [12] L. Bourgeois and J. Dardé,  A quasi-reversibility approach to solve the inverse obstacle problem,  _Inverse Probl. Imaging_ , 4.3 (2010), 351–377.
  * [13] L. Bourgeois and A. Recoquillay,  A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems,  _Math. Modell. Numer. Anal._ , 52.1 (2018), 123–145.
  * [14] B. M. Brown, M. Marletta and J. M. Reyes,  Uniqueness for an inverse problem in electromagnetism with partial data,  _J. Differential Equations_ , 260 (2016), 6525–6547.
  * [15] A. Buffa and P. Ciarlet Jr.,  On traces for functional spaces related to Maxwell’s equations: I. An integration by parts formula in Lipschitz polyhedra,  _Math. Meth. Appl. Sci._ , 24 (2001), 9–30.
  * [16] A. Buffa and P. Ciarlet Jr.,  On traces for functional spaces related to Maxwell’s equations: II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications,  _Math. Meth. Appl. Sci._ , 24 (2001), 31–48.
  * [17] F. Cakoni and R. Kress,  Integral equations for inverse problems in corrosion detection from partial Cauchy data,  _Inverse Probl. Imaging_ , 1.2 (2007), 299–245.
  * [18] P. Caro,  On an inverse problem in electromagnetism with local data: stability and uniqueness,  _Inverse Probl. Imaging_ , 5 (2011), 297–322.
  * [19] P. Caro, A. García and J. M. Reyes,  Stability of the Calderón problem for less regular conductivities,  _J. Differential Equations_ , 254.2 (2013), 469–492.
  * [20] P. Caro and T. Zhou,  Global uniqueness for an IBVP for the time-harmonic Maxwell equations,  _Analysis & PDE_, 7.2 (2014), 375–405.
  * [21] J. L. Castellanos, S. Gómez and V. Guerra,  The triangle method for finding the corner of the L-curve,  _Appl. Numer. Math._ , 43.4 (2002), 359–373.
  * [22] R. Chapko and B. T. Johansson,  An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions,  _Inverse Probl. Imaging_ , 2.3 (2008), 317–333.
  * [23] R. Chapko and B. T. Johansson,  On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach,  _Inverse Probl. Imaging_ , 6.1 (2012), 25–38.
  * [24] A. Cimetière, F. Delvare, M. Jaoua and F. Pons,  Solution of the Cauchy problem using iterated Tikhonov regularization,  _Inverse Problems_ , 17 (2001), 553–570.
  * [25] M. Clerc, J. Leblond, J.-P. Marmorat and T. Papadopoulo,  Source localization in EEG using rational approximation on plane sections,  _Inverse Problems_ , 28 (2012), 055018.
  * [26] M. Clerc, J. Leblond, J.-P. Marmorat and C. Papageorgakis,  Uniqueness result for an inverse conductivity recovery problem with application to EEG,  _Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics_ , 48 (2016).
  * [27] D. Colton, M. Piana and R. Potthast,  A simple method using Morozov’s discrepancy principle for solving inverse scattering problems,  _Inverse Problems_ 13 (1999), 1477–1493.
  * [28] M. Costabel,  A remark on the regularity of solutions of Mawxell’s equations on Lipschitz domains,  _Math. Meth. Appl. Sci._ , 12 (1990), 365–368.
  * [29] M. Darbas, J. Heleine et S. Lohrengel,  Sensitivity analysis for 3D Maxwell’s equations and its use in the resolution of an inverse medium problem at fixed frequency,  _Inverse Probl. Sci. Eng._ , 28.4 (2020), 459–496.
  * [30] J. Dardé,  Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems,  _Inverse Probl. Imaging_ , 10.2, (2016), 379–407.
  * [31] C. Greif and D. Schötzau,  Preconditioners for the discretized time-harmonic Maxwell equations in mixed form,  _Numer. Linear Algebra Appl._ , 14 (2007), 281–297.
  * [32] P. Grisvard,  Singularities in Boundary Value Problems,  RMA 22, Masson, Springer-Verlag, 1992.
  * [33] J. Hadamard,  _Lectures on Cauchy’s Problem in Linear Partial Differential Equation_ ,  New York, Dover, 1953.
  * [34] M. Hanke,  Limitations of the L-curve method in ill-posed problems,  _BIT Numer. Math._ 36 (1996), 287–301.
  * [35] F. Hecht,  New Development in FreeFem++,  _J. Numer. Math._ , 20.3-4 (2012), 251–265.
  * [36] J. Heleine,  _Identification de Paramètres électromagnétiques par Imagerie Micro-ondes_ ,  Ph.D thesis, Université de Picardie-Jules Verne in Amiens, France, 2019. http://theses.fr/2019AMIE0013/document
  * [37] R. Hiptmair and J. Xu,  Nodal auxiliary space preconditioning in $H(\curl)$ and $H(\div)$ spaces,  _SIAM J. Numer. Anal._ , 45.6 (2007), 2483–2509.
  * [38] V. A. Kozlov, V. G. Mazya and A. V. Fomin,  An iterative method for solving the Cauchy problem for elliptic equations,  _Comput. Math. Math. Phys._ , 31 (1991), 45–52.
  * [39] M. V. Klibanov and F. Santosa,  A computational quasi-reversibility method for Cauchy problems for Laplace’s equation,  _SIAM J. Appl. Math._ , 51 (1991), 1653–1675.
  * [40] R. Lattès and J.-L. Lions,  _Méthode de Quasi-réversibilité et Applications_ ,  Dunod, Paris, 1967.
  * [41] L. Ling and T. Takeuchi,  Boundary control for inverse Cauchy problems of the Laplace equations,  _Comp. Model. Eng. Sci._ , 29 (2008), 45–54.
  * [42] H. McCann, G. Pisano and L. Beltrachini,  Variation in reported human head tissue electrical conductivity values,  _Brain Topography_ , (2019).
  * [43] V. Melicher and M. Slodička,  Determination of missing boundary data for a steady-state Maxwell problem,  _Inverse Problems_ , 22 (2006), 297–310.
  * [44] P. Monk,  Finite Element Methods for Maxwell’s Equations,  Oxford University Press, 2003.
  * [45] J.-C. Nédélec,  Mixed finite elements in $\R^{3}$,  _Numerische Mathematik_ , 35 (1980), 315–341.
  * [46] P. Ola, L. Päivärinta and E. Somersalo,  Inverse problems for time harmonic electrodynamics,  _Inside Out: Inverse Problems and Applications, Publications of the Research Institute for Mathematical Sciences_ , 47, Cambridge University Press, Cambridge, 2003, 169–191.
  * [47] M. Slodička and V. Melicher,  An iterative algorithm for a Cauchy problem in eddy-current modelling,  _Appl. Math. Comput._ , 207 (2010), 237–246.
  * [48] P.-H. Tournier, et al.,  Microwave imaging of cerebrovascular accidents by using high-performance computing,  _Parallel Computing_ , 85 (2019), 88–97.
  * [49] P.-H. Tournier, et al.,  Numerical modeling and high speed parallel computing: New perspectives for tomographic microwave imaging for brain stroke detection and monitoring,  _IEEE Antennas and Propagation Magazine_ , 59.5 (2017), 98–110.
  * [50] C. R. Vogel,  Computational methods for inverse problems,  SIAM Frontiers in Applied Mathematics, Philadelphia, 2002.