# Boundary integral equation methods for the solution of scattering and
transmission 2D elastodynamic problems

Víctor Domínguez Dep. Ingeniería Matemática e Informática, Universidad Pública
de Navarra. Campus de Tudela 31500 - Tudela, Spain, e-mail:
<EMAIL_ADDRESS>Catalin Turc  Department of Mathematical
Sciences and Center for Applied Mathematics and Statistics, New Jersey
Institute of Technology, Univ. Heights. 323 Dr. M. L. King Jr. Blvd, Newark,
NJ 07102, USA, e-mail<EMAIL_ADDRESS>
(October 11, 2021)

###### Abstract

We introduce and analyze various Regularized Combined Field Integral Equations
(CFIER) formulations of time-harmonic Navier equations in media with piece-
wise constant material properties. These formulations can be derived
systematically starting from suitable coercive approximations of Dirichlet-to-
Neumann operators (DtN), and we present a periodic pseudodifferential calculus
framework within which the well posedness of CIER formulations can be
established. We also use the DtN approximations to derive and analyze
Optimized Schwarz (OS) methods for the solution of elastodynamics transmission
problems. The pseudodifferential calculus we develop in this paper relies on
careful singularity splittings of the kernels of Navier boundary integral
operators which is also the basis of high-order Nyström quadratures for their
discretizations. Based on these high-order discretizations we investigate the
rate of convergence of iterative solvers applied to CFIER and OS formulations
of scattering and transmission problems. We present a variety of numerical
results that illustrate that the CFIER methodology leads to important
computational savings over the classical CFIE one, whenever iterative solvers
are used for the solution of the ensuing discretized boundary integral
equations. Finally, we show that the OS methods are competitive in the high-
frequency high-contrast regime.
Keywords: Time-harmonic Navier scattering and transmission problems, boundary
integral equations, preconditioners, domain decomposition methods.

AMS subject classifications: 65N38, 35J05, 65T40,65F08

##  1 Introduction

Numerical solutions of elastodynamics scattering and transmission problems
based on boundary integral equation (BIE) formulations enjoy certain
attractive features over their volumetric counterparts, mainly on account of
the dimensional reduction and the explicit enforcement of radiation conditions
at infinity. Robust BIE formulations of elastodynamics scattering and
transmission problems can be derived via the combined field strategy.
Furthermore, in the case of impenetrable scattering problems, alternative
robust regularized formulations can be constructed based on the incorporation
of approximations of Dirichlet-to-Neumann operators [10, 12]. These
formulations are intrinsically more suitable for iterative solvers in the
high-frequency regime, where they lead to important computational savings over
the classical combined field formulations.

We pursue in this paper the construction of regularized BIE formulations of
both penetrable and impenetrable elastodynamics scattering problems in two
dimensions. While for the impenetrable case we use similar ideas to construct
regularized formulations as those introduced in [10, 12], in the penetrable
case we follow the blueprint introduced in [3] for Helmholtz problems. In a
nutshell, we use principal symbols of periodic pseudodifferential operators to
construct approximations of DtN operators which are then readily incorporated
in a scheme to construct well posed BIE formulations of scattering and
transmission elastodynamic problems. The DtN approximations we use are square
root Fourier multipliers, and they have the distinct advantage that (a) their
implementation is straightforward in the trigonometric interpolation framework
and (b) the analysis of both impenetrable and penetrable regularized
formulations can be performed in the same vein. In addition, we explore these
DtN approximations to formulate Optimized Schwarz methods for a domain
decomposition approach for transmission elastodynamics problems. Just like in
the Helmholtz case [4], we establish rigorously that the Schwarz iteration
operators are compact perturbations of the identity in the case of smooth
interfaces of material discontinuity, which allows us to prove the well
posedness of the Optimized Schwarz approach.

While the big picture does not differ a great deal from the elastodynamics to
the Helmholtz case with respect to the well posedness of regularized
formulations of scattering and transmission problems, the details are
considerably more involved in the elastodynamics case, notwithstanding the
fact that one has to deal with vector (as opposed to scalar) quantities. A
first notable difference is encountered in the double layer BIOs which are no
longer compact operators in the elastodynamics case, not even for smooth
boundaries. Consequently, the DtN pseudodifferential principal symbol calculus
is more complicated in elastodynamics, as the contributions arising from the
double layer BIOs cannot be any longer ignored. We use in this paper the
periodic pseudodifferential calculus [19] which in conjunction with
logarithmic singularity splittings for the kernels of the four elastodynamics
BIOs allows us to compute their principal symbols in the sense of
pseudodifferential operators. These calculations are the basis on which we
construct our approximations of the DtN operators. However, these kernel
logarithmic singularity splittings are quite cumbersome [13], significantly
more so than their Helmholtz counterparts. Owing to these additional
complications, we chose to present the full details of these calculations in
an appendix to this paper.

We employ Nyström discretizations for the numerical solution of the various
BIEs derived in this paper. We pursue the classical Nyström method based on
trigonometric interpolation, logarithmic kernel singularity splittings and the
classical Kussmaul and Martensen [24, 25] quadratures for the analytic
resolution of periodized logarithmic singularities. This discretization
strategy leads to high-order BIE solvers for elastodynamics scattering and
transmission problems in the case of smooth boundaries. The use of
trigonometric interpolation enables a straightforward implementation of
Fourier multipliers, and thus of the DtN approximations we construct on the
basis of the periodic pseudodifferential calculus.

The paper is organized as follows: in Section 2 we introduce the Navier
equations that govern elastodynamics waves in two dimensions; in Section 3 we
present the boundary layer potentials and integral operators associated with
the Navier equations and we construct their principal parts in the sense of
pseudodifferential operators. An essential part of this construction are
detailed calculations of the kernels of the BIOs involved, with a special
emphasis being given to their splitting into regular and singular parts. An
exhaustive account of these factorizations is presented be found in the
Appendix. In Section 4 we construct and analyze regularized BIE formulations
for both penetrable and impenetrable scattering problems; Nyström
discretizations of the elastodynamics BIOs as well as a variety of numerical
results that illustrate the high-order convergence of these methods and the
iterative behavior of the various formulations considered in this paper are
presented in Section 4.3. The implementation of one Nyström method relies
heavily on the regular-singular part kernel decompositions of the kernels
given in the appendix of this paper.

##  2  Navier equations and boundary integral operators

In this section we present the Navier equations and their associated four
boundary integral operators. We will pay special attentions to the kernels of
such operators for two reasons: (a) to derive the principal part (in the
pseudodifferential sense) of the underlying operators; and (b) to describe
precisely a factorization of these kernels into regular and singular parts
which is subsequently exploited by spectral Kress Nyström methods. Although
the latter methods were introduced in the 80s for the Helmholtz equation and
then extended in the 90s for the elastodynamic equations, the complexity of
the functions involved makes the splitting technique quite difficult in the
elastodynamics case, and many details are unfortunately missing from the
literature. Some references can be found in [13] and [17, Section 4]. In the
former paper, the emphasis is placed on the hypersingular operator, and we
actually follow some of the notations introduced in that work. On the other
hand, we find in the second paper a brief description of all four of the BIO
kernels but the regular and singular factorizations of the kernels is omitted.
Our main objective is to write in a appropriate framework a certain
integration by parts formula for the hypersingular Navier BIO which is an
essential piece in the numerical method object of this paper.

###  2.1 Navier equation

Let us introduce first some notation for geometric quantities. We will use the
following notation for points ${\bm{x}}=(x_{1},x_{2})\in\mathbb{R}^{2}$. Any
vector is understood to be a column vector, in such a way that

${\bm{x}}^{\top}{\bm{y}}={\bm{x}}\cdot{\bm{y}}=x_{1}y_{1}+x_{2}y_{2},\quad{\bm{x}}\>{\bm{y}}^{\top}=\begin{bmatrix}x_{1}y_{1}&x_{1}y_{2}\\\
x_{1}y_{2}&x_{2}y_{2}\end{bmatrix}.$

We will also denote

$\bm{r}:={\bm{x}}-{\bm{y}},\quad r:=|\bm{r}|=|{\bm{x}}-{\bm{y}}|.$

For any sufficiently regular curve $\Gamma$ (Lipschitz is enough), the unit
outward pointing vector is well defined and will be denoted henceforth by
$\bm{n}=(n_{1},n_{2})$. Consider also $\bm{t}=(-n_{2},n_{1})$ the unit tangent
vector obtained by rotating $\bm{n}$ clockwise by ninety degrees. The curve
$\Gamma$ will be assumed to be simply connected unless we state otherwise.

Let ${\bf u}=(u_{1},u_{2}):\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a vector
function. For a linear isotropic and homogeneous elastic medium with Lamé
constants $\lambda$ and $\mu$ such that $\lambda>-\mu$, the strain and the
stress tensor are defined as

$\displaystyle\bm{\epsilon}({\bf u})$ $\displaystyle:=$
$\displaystyle\frac{1}{2}(\nabla{\bf u}+(\nabla{\bf
u})^{\top})=\begin{bmatrix}\partial_{x_{1}}u_{1}&\tfrac{1}{2}\left(\partial_{x_{1}}u_{2}+\partial_{x_{2}}u_{1}\right)\\\
\tfrac{1}{2}\left(\partial_{x_{1}}u_{2}+\partial_{x_{2}}u_{1}\right)&\partial_{x_{2}}u_{2}\end{bmatrix}$
$\displaystyle\bm{\sigma}_{\lambda,\mu}({\bf u})$ $\displaystyle:=$
$\displaystyle 2\mu\bm{\epsilon}({\bf u})+\lambda(\nabla\cdot{\bf u})I_{2}$

where $I_{2}$ is the identity matrix of order 2. The time-harmonic elastic
wave (Navier) equation is given by

${\nabla}\cdot\bm{\sigma}_{\lambda,\mu}({\bf u})+\omega^{2}{\bf
u}=\mu\Delta{\bf u}+(\lambda+\mu)\nabla(\nabla\cdot{\bf u})+\omega^{2}{\bf
u}=0$

where the frequency $\omega\in\mathbb{R}^{+}$ and the divergence operator
${\rm div}$ is applied row-wise. We often alleviate the notation and will
simply write $\bm{\sigma}$ for the stress tensor if the context make clear
which values for the Lamé constants are being considered.

The trace operator on $\Gamma$ is denoted by $\gamma_{\Gamma}{\bf u}$ whereas
the normal stress tensor, or traction operator, is given by

$T_{\Gamma}{\bf u}=\bm{\sigma}({\bf u})\bm{n}=\lambda(\nabla\cdot{\bf
u})\bm{n}+2\mu(\bm{n}\cdot{\nabla}){\bf u}-\mu(\nabla\times{\bf
u})\begin{bmatrix}&-1\\\ 1&\end{bmatrix}{\bm{n}}.$ (2.1)

Let then $\Omega_{-}$ and
$\Omega_{+}=\mathbb{R}^{2}\setminus\overline{\Omega}_{+}$ be the interior and
exterior of $\Gamma$. We will study in this paper BIE formulations for the
following boundary value problems:

  1. (a)

Solution of the exterior Dirichlet/Neumann, unpenetrable domain, time-harmonic
Navier equation:

$\left|\begin{array}[]{rcl}\lx@intercol{\bf u}\in{\bm{H}}_{\rm
loc}^{1}(\Omega_{+}):=H_{\rm loc}^{1}(\Omega_{+})\times H_{\rm
loc}^{1}(\Omega_{+}),\hfil\lx@intercol\\\ {\nabla}\cdot\bm{\sigma}({\bf
u})+\omega^{2}{\bf u}&=&0,\\\ \lx@intercol\text{DC/NC on $\Gamma$
},\hfil\lx@intercol\\\ \lx@intercol\text{+
<EMAIL_ADDRESS>(2.2)

Here $H^{1}_{\rm loc}(\Omega_{+})$ is the space of functions which are locally
in the Sobolev space $H^{1}(\Omega_{+})$, DC stands for Dirichlet condition,
i.e.,

$\gamma_{\Gamma}{\bf u}={\bf f},$

and NC, for Neumann condition,

$T_{\Gamma}{\bf u}=\bm{\lambda}.$

Finally, RC stands for the radiation condition, or Kupradze condition, at
infinity cf. [23] or [1, Ch. 2]. Hence, if ${\bf u}_{p}$ and ${\bf u}_{s}$ are
the longitudinal and transversal wave defined as

${\bf u}_{p}:=-\frac{1}{k_{p}^{2}}\nabla\ \nabla\cdot{\bf u}\qquad{\bf
u}_{s}:={\bf u}-{\bf u}_{p}$ (2.3)

with

$\displaystyle k_{p}^{2}$ $\displaystyle:=$
$\displaystyle\frac{\omega^{2}}{\lambda+2\mu},\quad
k_{s}^{2}:=\frac{\omega^{2}}{\mu},$ (2.4)

the associated wave numbers, we require that uniformly in ${\bm{x}}$, with
$\widehat{\bm{x}}:=\bm{x}/|\bm{{\bm{x}}}|$,

$\frac{\partial{\bf u}_{p}}{\partial\widehat{\bm{x}}}(\bm{x})-ik_{p}{\bf
u}_{p}(\bm{x}),\ \frac{\partial{\bf
u}_{s}}{\partial\widehat{\bm{x}}}(\bm{x})-ik_{s}{\bf
u}_{s}(\bm{x})=o(|{\bm{x}}|^{-1/2}).$ (2.5a) An equivalent formulation is
given by $\bm{\sigma({\bf
u}_{p})}(\bm{x})\cdot\widehat{\bm{x}}-ik_{p}(\lambda+2\mu){\bf
u}_{p}(\bm{x}),\quad\bm{\sigma({\bf
u}_{s})}(\bm{x})\cdot\widehat{\bm{x}}-ik_{s}\mu{\bf
u}_{s}(\bm{x})=o(|{\bm{x}}|^{-1/2}).$ (2.5b) which in turns implies ${\bf
u}_{p}(\bm{x})\cdot{\bf u}_{s}(\bm{x}),\quad(\bm{\sigma({\bf
u}_{s})}(\bm{x})\widehat{\bm{x}})\cdot(\bm{\sigma({\bf
u}_{p})}(\bm{x})\widehat{\bm{x}})=o(|{\bm{x}}|^{-1/2})$ (2.5c)

We refer to [23, Ch. III, §2] for a proof of these results in $\mathbb{R}^{3}$
which can be easily adapted to our two-dimensional case.

Both problems are uniquely solvable (see, for 2D and 3D problems, [7], [23,
Ch. 3] or [1, Ch. 2]).

  2. (b)

Transmission problems: For different material properties
$(\lambda_{+},\mu_{+})$ in $\Omega_{+}$ and $(\lambda_{-},\mu_{-})$ in
$\Omega_{-}$ we seek a solution of

$\left|\begin{array}[]{rcl}\lx@intercol{\bf
u}_{-}\in\bm{H}^{1}(\Omega_{-}),\quad{\bf u}_{+}\in\bm{H}_{\rm
loc}^{1}(\Omega_{+}),\hfil\lx@intercol\\\ {\nabla}\cdot\bm{\sigma}_{-}({\bf
u}_{-})+\omega_{-}^{2}{\bf u}_{-}&=&0\quad\mbox{in }\Omega_{-},\\\
{\nabla}\cdot\bm{\sigma}_{+}({\bf u}_{+})+\omega_{+}^{2}{\bf
u}_{-}&=&0\quad\mbox{in }\Omega_{+},\\\ \gamma_{\Gamma}{\bf
u}_{+}-\gamma_{\Gamma}{\bf u}_{-}&=&-\gamma_{\Gamma}{\bf u}^{\rm inc},\\\
T_{\Gamma}{\bf u}_{+}-T_{\Gamma}{\bf u}_{-}&=&-T_{\Gamma}{\bf u}^{\rm inc},\\\
\lx@intercol\text{+ <EMAIL_ADDRESS>(2.6)

We have used above the $\pm$ signs in the trace ($\gamma_{\pm,\Gamma}$) and
the normal stress tensor ($T_{\pm,\Gamma}$) to clarify the domain from which
they are applied, be it the exterior/interior domain with respect to $\Gamma$
(that is, $\Omega_{+}$ or $\Omega_{-}$). We will keep using this notation
whenever the context is not clear enough to resolve this aspect.

On the other hand, the function ${\bf u}_{\rm inc}$ appearing in the right-
hand-side above is a solution of the exterior Navier problem in a neighborhood
of $\overline{\Omega_{+}}$ (typically a shear or a pressure plane wave). We
will assume in what follows that this problem is uniquely solvable. We refer
to [23] or [1, Ch. 2] for sufficient conditions under which the well posedness
of the boundary value problem under consideration can be established.

We finish this section by introducing the exterior/interior Dirichlet-to-
Neumann operator, which will play an essential role in this work: for
$\bm{g}:\Gamma\to\mathbb{C}$ sufficiently smooth (we will give detailed
conditions later) we define

$Y_{+,\Gamma}{\bf g}:=T_{+,\Gamma}{\bf u}_{+},\quad Y_{-,\Gamma}{\bf
g}:=T_{-,\Gamma}{\bf u}_{-}.$ (2.7)

where ${{\bf u}_{+}}$ satisfies the RC and

$\bm{\sigma}_{\pm}({\bf u}_{\pm})+\omega_{\pm}^{2}{\bf
u}_{\pm}=0,\quad\gamma_{\pm,{\Gamma}}{\bf u}_{\pm}={\bf g}.$

Let us emphasize that the exterior DtN operator $Y_{+,\Gamma}$ is well defined
for all frequencies $\omega$, unlike the interior DtN operator $Y_{-,\Gamma}$,
which fails to be properly defined whenever $\omega^{2}$ is an eigenvalue for
the Dirichlet problem of the Navier differential operator in the bounded
domain $\Omega_{-}$. Notice that the second transmission condition can be
rewritten now as

$Y_{+,\Gamma}\gamma_{+,\Gamma}{\bf u}-Y_{-,\Gamma}\gamma_{-,\Gamma}{\bf
u}=-T_{\Gamma}{\bf u}^{\rm inc}$

assuming that the inner DtN operator is properly defined.

##  3 Boundary integral operators for elastodynamics

###  3.1 Fundamental solution. Boundary layer potentials and integral
operators

The fundamental solution of the time-harmonic elastic wave equation is given
by the $2\times 2$ matrix function

$\displaystyle\Phi({\bm{x}},{\bm{y}})$ $\displaystyle:=$
$\displaystyle\frac{1}{\mu}\phi_{0}(k_{s}r)I_{2}+\frac{1}{\omega^{2}}\nabla_{{\bm{x}}}\nabla_{{\bm{x}}}^{\top}(\phi_{0}(k_{s}r)-\phi_{0}(k_{p}r)),\quad
r=|{\bm{x}}-{\bm{y}}|$

where

$\displaystyle\phi_{j}(z)$ $\displaystyle:=$
$\displaystyle\frac{i}{4}H_{j}^{(1)}(z).$ (3.1)

In the expression above, $H_{j}^{(1)}$ denotes the Hankel function of order
$j$ and first kind, so that for the particular case $j=0$, $\phi_{0}(kz)$ is
just the fundamental solution of the Helmholtz equation $\Delta+k^{2}$.

Using the fundamental solution $\Phi({\bm{x}},{\bm{y}})$ of the Navier
equation we can define the single and double layer potentials. Specifically,
for a given density (vector) function $\bm{\lambda}_{\Gamma}$ defined on
$\Gamma$, the single layer potential is defined as

$(\operatorname{SL}_{\Gamma}{\bm{\lambda}_{\Gamma}})({\bm{z}}):=\int_{\Gamma}\Phi({\bm{z}},{\bm{y}}){\bm{\lambda}}_{\Gamma}({\bm{y}})\,{\rm
d}{\bm{y}},\quad{\bm{z}}\in\mathbb{R}^{2}\setminus\Gamma.$ (3.2)

Analogously, for a given density (vector) function $\bm{g}_{\Gamma}$ defined
on $\Gamma$, the double layer potential is defined as

$(\operatorname{DL}_{\Gamma}\bm{g}_{\Gamma})({\bm{z}}):=\int_{\Gamma}\left[T_{{\bm{y}},\Gamma}\Phi({\bm{z}},{\bm{y}})\right]^{\top}\bm{g}_{\Gamma}({\bm{y}})\,{\rm
d}{\bm{y}},\quad{\bm{z}}\in\mathbb{R}^{2}\setminus\Gamma$ (3.3)

where $T_{{\bm{y}},_{\Gamma}}\Phi({\bm{z}},{\bm{y}})$ is the normal stress
tensor applied column-wise to $\Phi({\bm{z}},{\bm{y}})$ with respect to the
${\bm{y}}$ variable.

The single layer potential is continuous in $\mathbb{R}^{2}$, and thus,
assuming that ${\bm{z}}={\bm{x}}+\varepsilon{\bf n}({\bm{x}}),\
{\bm{x}}\in\Gamma$ and taking the limit as $\varepsilon\to 0$ in equation
(3.2) we can define the single layer boundary integral operator

$(V_{\Gamma}{\bm{\lambda}}_{\Gamma})({\bm{x}}):=\int_{\Gamma}\Phi({\bm{x}},{\bm{y}}){\bm{\lambda}}_{\Gamma}({\bm{y}})\,{\rm
d}{\bm{y}},\quad{\bm{x}}\in\Gamma.$ (3.4)

The application of the traction operator $T_{\Gamma}$ to the single layer
potential $\operatorname{SL}_{\Gamma}\bm{\lambda}_{\Gamma}$ gives rise to jump
discontinuities

$\lim_{\varepsilon\to
0}T_{\Gamma}(\operatorname{SL}_{\Gamma}\bm{\lambda}_{\Gamma})({\bm{x}}\pm\varepsilon{\bf
n}({\bm{x}}))=\mp{\bm{\lambda}}_{\Gamma}({\bm{x}})+(K^{\top}_{\Gamma}\bm{\lambda}_{\Gamma})({{\bm{x}}}),\quad{\bm{x}}\in\Gamma$

where the adjoint double layer operator is defined as

$(K_{\Gamma}^{\top}{\bm{\lambda}}_{\Gamma})({{\bm{x}}}):=\mathrm{p.v.}\,\int_{\Gamma}T_{{\bm{x}},\Gamma}\Phi({\bm{x}},{\bm{y}}){\bm{\lambda}}_{\Gamma}({\bm{y}})\,{\rm
d}{\bm{y}},$

The integral above is singular and it has to be understood in the sense of
Cauchy principal value (which is what $\mathrm{p.v.}\,$ stands for). The
double layer potential $\operatorname{DL}_{\Gamma}$ undergoes a jump
discontinuity across $\Gamma$ so that

$\lim_{\varepsilon\to
0}(\operatorname{DL}_{\Gamma}{\bm{g}_{\Gamma}})({\bm{x}}+\varepsilon{\bf
n}({\bm{x}}))-\lim_{\varepsilon\to
0}(\operatorname{DL}_{\Gamma}{\bm{g}_{\Gamma}})({\bm{x}}-\varepsilon{\bf
n}({\bm{x}}))={\bm{g}_{\Gamma}}({\bm{x}}),\quad{\bm{x}}\in\Gamma$

and

$\lim_{\varepsilon\to
0}(\operatorname{DL}_{\Gamma}{\bm{g}_{\Gamma}})({\bm{x}}+\varepsilon{\bf
n}({\bm{x}}))+\lim_{\varepsilon\to
0}(\operatorname{DL}_{\Gamma}{\bm{g}_{\Gamma}})({\bm{x}}-\varepsilon{\bf
n}({\bm{x}}))=2(K_{\Gamma}\bm{g}_{\Gamma})({{\bm{x}}}),\quad{\bm{x}}\in\Gamma$

where the Double Layer BIO is defined explicitly as

$(K_{\Gamma}\bm{g}_{\Gamma})({{\bm{x}}}):={\mathrm{p.v.}}\int_{\Gamma}\left[T_{{\bm{y}},\Gamma}\Phi({\bm{z}},{\bm{y}})\right]^{\top}\bm{g}_{\Gamma}({\bm{y}})\,{\rm
d}{\bm{y}}.$

Finally, applying the traction operator $T_{\Gamma}$ to the double layer
potential $\operatorname{DL}_{\Gamma}{\bm{g}_{\Gamma}}$ we obtain

$\lim_{\varepsilon\to
0}(T_{\Gamma}\operatorname{DL}_{\Gamma}{\bm{g}_{\Gamma}})({\bm{x}}\pm\varepsilon{\bf
n}({\bm{x}}))=(W_{\Gamma}\bm{g}_{\Gamma})({{\bm{x}}}),\quad{\bm{x}}\in\Gamma$

where the Boundary Integral operator (BIO) $W_{\Gamma}$ is defined as

$(W_{\Gamma}\bm{g}_{\Gamma})({{\bm{x}}}):=\operatorname{f.p.}\int_{\Gamma}T_{{\bm{x}},\Gamma}\big{[}T_{{\bm{y}},\Gamma}\Phi(\bm{r})\big{]}^{\top}{\bm{g}}_{\Gamma}({{\bm{y}}})\,{\rm
d}{\bm{y}}.$

The kernel $W({\bm{x}},{\bm{y}})$ of the BIO $W_{\Gamma}$ defined above is
strongly singular (that is it behaves like
$\mathcal{O}(|{\bm{x}}-{\bm{y}}|^{-2})$ as ${\bm{y}}\to{\bm{x}}$), and as such
the integral in its definition must be interpreted in a Hadamard finite part
sense (which it is what “f.p.” stands for in the expression above).

Remark 3.1. It is a well established result that the matrix boundary integral
operator

${\cal C}_{\Gamma}:\begin{bmatrix}K_{\Gamma}&-V_{\Gamma}\\\
W_{\Gamma}&-K_{\Gamma}^{\top}\end{bmatrix}:\bm{H}^{1/2}(\Gamma)\times\bm{H}^{-1/2}(\Gamma)\to\bm{H}^{1/2}(\Gamma)\times\bm{H}^{-1/2}(\Gamma)$

is continuous. Here $\bm{H}^{\pm 1/2}(\Gamma):=H^{\pm 1/2}(\Gamma)\times
H^{\pm 1/2}(\Gamma)$ are the standard Sobolev spaces in which Dirichlet and
Neumann traces on the boundary are considered. Furthermore,

${\cal C}^{2}_{\Gamma}=\frac{1}{4}{\cal I}$

where ${\cal I}$ is the identity operator. This makes the operators

$\frac{1}{2}{\cal I}\pm{\cal C}_{\Gamma}$

projections in the space in $\bm{H}^{1/2}(\Gamma)\times\bm{H}^{-1/2}(\Gamma)$,
known in the literature as the exterior/interior Calderón operators. Since

${\bf
u}_{\pm}=\pm\operatorname{DL}_{\Gamma}\bm{g}_{\Gamma}\mp\operatorname{SL}_{\Gamma}{\bm{\lambda}}_{\Gamma}\quad{\Leftrightarrow}\quad\left(\frac{1}{2}{\cal
I}\pm{\cal C}_{\Gamma}\right)\begin{bmatrix}{\bm{g}}_{\Gamma}\\\
{\bm{\lambda}}_{\Gamma}\end{bmatrix}=\begin{bmatrix}\gamma_{\pm,\Gamma}{\bf
u}\\\ {T}_{\pm,\Gamma}{\bf u}\end{bmatrix}$

we have

$\left(\frac{1}{2}{\cal I}\pm{\cal
C}_{\Gamma}\right)\begin{bmatrix}\gamma_{\pm,\Gamma}{\bf u}\\\
{T}_{\pm,\Gamma}{\bf u}\end{bmatrix}=\begin{bmatrix}\gamma_{\pm,\Gamma}{\bf
u}\\\ {T}_{\pm,\Gamma}{\bf u}\end{bmatrix},\ \text{or equivalently }\
(\gamma_{\pm,\Gamma}{\bf u},{T}_{\pm,\Gamma}{\bf u})\in\mathop{\rm
Ker}\left(\tfrac{1}{2}{\cal I}\mp{\cal C}_{\Gamma}\right).$

Let us end this remark by stating the convention we have followed so far: We
will use boldface roman letters (as ${\bf f}$ or $\bm{g}$) for functions that
naturally belong to $\bm{H}^{1/2}(\Gamma)$ whereas Greek letters will be used
to represent functions (or distributions) in $\bm{H}^{-1/2}(\Gamma)$ —without
excluding the possibility that for Navier problems with regular enough
boundary data the functions involved also turn out to be more regular even
though $\Gamma$ is only Lipschitz. $\Box$

###  3.2 Parameterized operators and spaces. Principal symbols

In this section we introduce parameterized versions of the kernels of the
Navier layer potentials and their associated boundary integral operators. We
will assume from now on that there exists a $2\pi-$periodic smooth regular
parameterization of the closed curve $\Gamma$, ${\bf x}:\mathbb{R}\to\Gamma$.
We then denote

${\bm{\nu}}(t)=(x^{\prime}_{2}(t),-x^{\prime}_{1}(t))=(\bm{n}\circ{\bf
x})(t)|{\bf x}^{\prime}(t)|,$

the parameterized normal tangent field to $\Gamma$.

We will adopt the following convention:

$\bm{H}^{1/2}(\Gamma)\times\bm{H}^{-1/2}(\Gamma)\ni({\bm{g}}_{\Gamma},\bm{\lambda}_{\Gamma})\quad\rightsquigarrow\quad({\bm{g}},\bm{\lambda}):=({\bm{g}}_{\Gamma}\circ{\bf
x},(\bm{\lambda}_{\Gamma}\circ{\bf x}){|\bf
x^{\prime}}|)\in\bm{H}^{1/2}\times\bm{H}^{-1/2}.$

Here,

$\bm{H}^{s}:=H^{s}\times H^{s},\quad H^{s}:=\left\\{\varphi\in{\cal
D}^{\prime}(\mathbb{R})\ :\
\varphi(\cdot+2\pi)=\varphi,\quad\|\varphi\|_{s}<\infty\right\\}$

with

$\|\varphi\|_{s}^{2}:=|\widehat{\varphi}(0)|^{2}+\sum_{n\neq
0}|n|^{2s}|\widehat{\varphi}(n)|^{2},\qquad\widehat{\varphi}(n):=\int_{0}^{2\pi}\varphi(s)\exp(-ins)\,{\rm
d}s.$

We extend actually such identification for any $s\geq 0$:

$\bm{H}^{s}(\Gamma)\ni\bm{g}_{\Gamma}\ \rightsquigarrow\
{\bm{g}}:={\bm{g}}_{\Gamma}\circ{\bf
x}\in\bm{H}^{s},\qquad\bm{H}^{-s}(\Gamma)\ni\bm{\varphi}_{\Gamma}\
\rightsquigarrow\ {\bm{\varphi}}:=({\bm{\varphi}}_{\Gamma}\circ{\bf x})\>|{\bf
x^{\prime}}|\in\bm{H}^{-s}.\quad$

When $s=0$, i.e. ${\bm{L}}^{2}(\Gamma):=L^{2}(\Gamma)\times L^{2}(\Gamma)$ we
can choose between either of the two identifications above depending on what
is most appropriate for our purposes.

Notice that with this convention

$\int_{\Gamma}\bm{g}_{\Gamma}\cdot\bm{\lambda}_{\Gamma}=\int_{0}^{2\pi}\bm{g}\cdot\bm{\lambda}=:\langle\bm{g},\bm{\varphi}\rangle,\quad\bm{g}=\bm{g}_{\Gamma}\circ{\bf
x},\quad\bm{\lambda}=(\bm{\lambda}_{\Gamma}\circ{\bf x})\>|{\bf x^{\prime}}|.$

For any ${\bf u}:\Omega_{\pm}\to\mathbb{C}^{2}$ sufficiently smooth we follow
this convention and thus define the $2\pi-$periodic vector fields:

$\gamma{\bf u}:={\bf u}\circ{\bf x},\quad T{\bf u}:=(T_{\Gamma}{\bf
u}\circ{\bf x})\,|{\bf x}|=(\bm{\sigma}({\bf u})\circ{\bf x})\bm{\nu}$ (3.5)

which are the (parameterized) trace and normal stress tensor. With these
notations in hand, we then introduce the parameterized version of the Navier
layer potentials

$\displaystyle(\operatorname{SL}{\bm{\lambda}})({\bm{z}})$ $\displaystyle:=$
$\displaystyle\int_{0}^{2\pi}\Phi({\bf z},{\bf x}(t)){\bm{\lambda}}(t)\,{\rm
d}t,\quad{\bm{z}}\in\mathbb{R}^{2}\setminus\Gamma$ (3.6)
$\displaystyle(\operatorname{DL}\bm{g})({\bm{z}})$ $\displaystyle:=$
$\displaystyle\int_{0}^{2\pi}\left.\left[T_{{\bm{y}},\Gamma}\Phi({\bm{z}},{\bm{y}})\right]^{\top}\right|_{{\bf
y}={\bf x}(t)}|{\bf x}^{\prime}(t)|\,\bm{g}(t)\,{\rm d}t$ (3.7)
$\displaystyle=$ $\displaystyle\int_{0}^{2\pi}\left[T_{t}\Phi({\bm{z}},{\bf
x}(t))\right]^{\top}\,\bm{g}(t)\,{\rm
d}t,\quad{\bm{z}}\in\mathbb{R}^{2}\setminus\Gamma\qquad\qquad$

and their associated four boundary integral operators

$\displaystyle(V{\bm{\lambda}})({{t}})$ $\displaystyle:=$
$\displaystyle\int_{0}^{2\pi}V({\tau},t){\bm{\lambda}}(t)\,{\rm
d}t=\int_{0}^{2\pi}\Phi({\bf x}({\tau}),{\bf x}(t)){\bm{\lambda}}(t)\,{\rm
d}t$ (3.8a) $\displaystyle(K\bm{g})({\tau})$ $\displaystyle:=$
$\displaystyle\mathrm{p.v.}\,\int_{0}^{2\pi}K({\tau},t)\,\bm{g}(t)\,{\rm
d}t={\rm p.v}\int_{0}^{2\pi}\left[T_{t}\Phi({\bf x}({\tau}),{\bf
x}(t))\right]^{\top}\,\bm{g}(t)\,{\rm d}t$ (3.8b)
$\displaystyle(K^{\top}{\bm{\lambda}})({\tau})$ $\displaystyle:=$
$\displaystyle\mathrm{p.v.}\,\int_{0}^{2\pi}K^{\top}(t,{\tau})\,\bm{\lambda}(t)\,{\rm
d}t={\rm p.v}\int_{0}^{2\pi}T_{{\tau}}\Phi({\bf x}({\tau}),{\bf
x}(t))\,\bm{\lambda}(t)\,{\rm d}t$ (3.8c) $\displaystyle(W{\bm{g}})({\tau})$
$\displaystyle:=$
$\displaystyle\operatorname{f.p.}\int_{0}^{2\pi}W({\tau},t)\,\bm{g}(t)\,{\rm
d}t=\operatorname{f.p.}\int_{0}^{2\pi}T_{{\tau}}\big{[}T_{t}\Phi({\bf
x}({\tau})-{\bf x}(t))\big{]}^{\top}\,\bm{g}(t)\,{\rm d}t.$ (3.8d)

Observe that with these definitions, $K$ and $K^{\top}$ are effectively
transpose of each other:

$\langle
K\bm{g},\bm{\lambda}\rangle=\langle\bm{g},K^{\top}\bm{\lambda}\rangle,\quad\forall{\bm{g}}\in\bm{H}^{s},\
\forall\bm{\lambda}\in\bm{H}^{-s}.$

As in Remark 3.1 the (parameterized) version

${\cal C}:\begin{bmatrix}K&-V\\\ W&-K^{\top}\end{bmatrix}:\
\bm{H}^{1/2}\times\bm{H}^{-1/2}\to\bm{H}^{1/2}\times\bm{H}^{-1/2}$

can be shown to satisfy the related properties:

${\cal C}^{2}=\frac{1}{4}{\cal I},\quad(\pm\frac{1}{2}{\cal I}+{\cal
C})^{2}=\pm\frac{1}{2}{\cal I}+{\cal C}$

and that a similar relation can be established between functional densities
defined on $\Gamma$ and boundary traces of their associated combined field
layer potentials. Indeed, if we define

${\bf
u}_{\pm}=\pm\operatorname{DL}\bm{g}\mp\operatorname{SL}{\bm{\lambda}},\quad{\Leftrightarrow}\quad\left(\frac{1}{2}{\cal
I}\pm{\cal C}\right)\begin{bmatrix}{\bm{g}}\\\
{\bm{\lambda}}\end{bmatrix}=\begin{bmatrix}\gamma_{\pm}{\bf u}\\\
{T}_{\pm}{\bf u}\end{bmatrix}$

we have

$\left(\frac{1}{2}{\cal I}\pm{\cal C}\right)\begin{bmatrix}\gamma_{\pm}{\bf
u}\\\ {T}_{\pm}{\bf u}\end{bmatrix}=\begin{bmatrix}\gamma_{\pm}{\bf u}\\\
{T}_{\pm}{\bf u}\end{bmatrix},\ \text{or equivalently }\ (\gamma_{\pm}{\bf
u},{T}_{\pm}{\bf u})\in\mathop{\rm Ker}\left(\tfrac{1}{2}{\cal I}\mp{\cal
C}\right).$

In particular, the Somigliana identity looks similar in this framework:

${\bf u}_{\pm}=\pm\operatorname{DL}\gamma_{\pm}{\bf
u}\mp\operatorname{SL}T_{\pm}{\bf u}.$ (3.9)

Our next result establishes the fact that the difference between the kernels
of the four parametrized BIOs associated with the time harmonic Navier
equations (i.e. elastodynamics) and their static counterparts (i.e.elasticity)
exhibit integrable singularities only, and the nature of the latter can be
made explicit:

###### Proposition 3.2

There exist smooth $2\pi-$periodic $2\times 2$ matrix functions

$A_{\rm log},\quad A_{\rm reg},\quad B_{\rm reg},\quad C_{\rm log},\quad
D_{\rm reg}$

so that

$\displaystyle V(\tau,t)$ $\displaystyle=$ $\displaystyle V_{0}(\tau,t)+A_{\rm
log}(\tau,t)\sin^{2}\frac{\tau-t}{2}\log r+A_{\rm reg}(\tau,t)$ $\displaystyle
K(\tau,t)$ $\displaystyle=$ $\displaystyle K_{0}(\tau,t)+B_{\rm
log}(\tau,t)\sin(\tau-t)\log r+B_{\rm reg}(\tau,t)$ $\displaystyle W(\tau,t)$
$\displaystyle=$ $\displaystyle W_{0}(\tau,t)+C_{\rm log}(\tau,t)\log r+D_{\rm
reg}(\tau,t)$

where

$\displaystyle V_{0}(\tau,t)$ $\displaystyle:=$
$\displaystyle-\frac{\lambda+3\mu}{4\pi(\lambda+2\mu)\mu}\log
r\>I_{2}+\frac{\lambda+\mu}{4\pi(\lambda+2\mu)\mu}G(\bm{r})$ $\displaystyle
K_{0}(\tau,t)$ $\displaystyle:=$
$\displaystyle\frac{\mu}{\lambda+2\mu}\left(\frac{\partial}{\partial
t}\frac{1}{2\pi}\log r\right)\>\begin{bmatrix}&-1\\\
1&\end{bmatrix}+\frac{1}{2\pi
r^{2}}\left({\bm{\nu}}(t)\cdot\bm{r}\right)\left(\frac{\mu}{\lambda+2\mu}I_{2}+2\frac{\lambda+\mu}{\lambda+2\mu}G(\bm{r})\right)$
$\displaystyle W_{0}(\tau,t)$ $\displaystyle:=$
$\displaystyle={-\frac{\mu(\lambda+\mu)}{\lambda+2\mu}\frac{\partial^{2}}{\partial\tau\,\partial
t}\frac{1}{{\pi}}\left(-\log r\>I_{2}+G(\bm{r})\right)}.$

with

$\bm{r}:={\bf x}(\tau)-{\bf x}(t),\quad r:=|\bm{r}|=|{\bf x}(\tau)-{\bf
x}(t)|,\quad
G(\bm{r}):=\frac{1}{r^{2}}\bm{r}\bm{r}^{\top}=\begin{bmatrix}r_{1}^{2}&r_{1}r_{2}\\\
r_{1}r_{2}&r_{2}^{2}\end{bmatrix}.$

Proof. We refer the reader to Appendix and Remark A.3. $\Box$
Notice that in the previous results we have used the same symbols ${\bm{r}}$
and $r$ to represent different but very closely related quantities. We believe
that this similarity justifies this slight abuse of notation.

Let us stress some consequences of the previous result. The (matrix) functions
$V_{0}(t,\tau),$ $K_{0}(t,\tau)$, $K^{\top}_{0}(t,\tau)$ and $W_{0}(t,\tau)$
are the kernels of the corresponding boundary integral operators (single
layer, double layer, adjoint double layer and hypersingular operator) for the
elasticity layer operators. Hence, if we introduce the integral operators

$\displaystyle\Lambda\varphi$ $\displaystyle:=$
$\displaystyle-\frac{1}{2\pi}\int_{0}^{2\pi}\log\Big{(}4e^{-1}\sin^{2}\frac{\cdot-t}{2}\Big{)}\varphi(t)\,{\rm
d}t$ $\displaystyle\mathrm{H}g$ $\displaystyle:=$ $\displaystyle-i\Lambda
g^{\prime}+\widehat{g}(0)=\mathrm{p.v.}\,\frac{1}{2\pi
i}\int_{0}^{2\pi}\cot\frac{t-\cdot}{2}g(t)\,{\rm
d}t+\frac{1}{2\pi}\int_{0}^{2\pi}g(t)\,{\rm d}t,$

which are actually Fourier multiplier operators whose action is explicitly
given by

$\Lambda g=\widehat{g}(0)+\sum_{n\neq
0}\frac{1}{|n|}\widehat{g}(n)\exp(in\,\cdot),\quad\mathrm{H}{g}=-\sum_{n<0}\widehat{g}(n)\exp(in\,\cdot)+\sum_{n\geq
0}\widehat{g}(n)\exp(in\,\cdot),$ (3.10)

together with the matrix operators

$\bm{\Lambda}=\begin{bmatrix}\Lambda&\\\
&\Lambda\end{bmatrix},\quad\bm{\Lambda}^{-1}=\begin{bmatrix}\Lambda^{-1}&\\\
&\Lambda^{-1}\end{bmatrix},\quad\bm{H}=\begin{bmatrix}&-H\\\
H&\end{bmatrix}=\bm{H}^{\top}$ (3.11)

the following regularity results can be easily established, cf. Remark A.3

$\displaystyle K-\beta\bm{H}$ $\displaystyle:\bm{H}^{s}\to\bm{H}^{s+2}$
$\displaystyle V-\alpha\bm{\Lambda}$ $\displaystyle:\bm{H}^{s}\to\bm{H}^{s+3}$
(3.12) $\displaystyle W-\beta\bm{\Lambda}^{-1}$
$\displaystyle:\bm{H}^{s}\to\bm{H}^{s+1}$ $\displaystyle K^{\top}-\beta\bm{H}$
$\displaystyle:\bm{H}^{s}\to\bm{H}^{s+2}$

where

$\alpha:=\frac{i\mu}{2(\lambda+2\mu)},\quad\beta:=\frac{\lambda+3\mu}{4\mu(\lambda+2\mu)},\quad\delta:=-\frac{\mu(\lambda+\mu)}{\lambda+2\mu}.$

We note in passing that

$\alpha^{2}+\beta\delta+\frac{1}{4}=0.$ (3.13)

On the other hand, using the 2D Mauer derivative

${\bm{D}}\bm{g}=\begin{bmatrix}&-D\\\
D&\end{bmatrix}\bm{g}=\begin{bmatrix}-g_{2}^{\prime}\\\
g_{1}^{\prime}\end{bmatrix}$

it holds that

$\bm{H}=\bm{\Lambda}\bm{D},\quad\bm{\Lambda}^{-1}=\bm{D}^{\top}\bm{\Lambda}\bm{D}.$

Besides, it is straightforward to derive the following coercivity relations

$\langle\bm{\Lambda}^{-1}\bm{g},\overline{\bm{g}}\rangle\geq\|\bm{g}\|_{\bm{H}^{1/2}}^{2},\quad\langle\bm{\Lambda}\bm{\varphi},\overline{\bm{\varphi}}\rangle\geq\|\bm{\varphi}\|_{\bm{H}^{-1/2}}^{2}.$
(3.14)

Having introduced these notations, we claim that the principal part of the
Calderón operator ${\cal C}$ is given by

${\cal C}_{0}=\begin{bmatrix}\alpha\bm{H}&-\beta\bm{\Lambda}\\\
\delta\bm{\Lambda}^{-1}&-\alpha\bm{H}^{\top}\\\
\end{bmatrix}=\begin{bmatrix}\alpha\bm{H}&-\beta\bm{\Lambda}\\\
\delta\bm{\Lambda}^{-1}&-\alpha\bm{H}\\\ \end{bmatrix},\quad$ (3.15)

that is

###### Proposition 3.3

For any $s\in\mathbb{R}$, ${\cal
C}_{0}:\bm{H}^{s+1}\times\bm{H}^{s}\to\bm{H}^{s+1}\times\bm{H}^{s}$ is
continuous. Moreover,

${\cal C}-{\cal
C}_{0}:\bm{H}^{s+1}\times\bm{H}^{s}\to\bm{H}^{s+3}\times\bm{H}^{s+2}.$

Besides,

${\cal C}_{0}^{2}=\frac{1}{4}{\cal I},\quad\left(\pm\frac{1}{2}{\cal I}+{\cal
C}_{0}\right)^{2}=\pm\frac{1}{2}{\cal I}+{\cal C}_{0}.$

Proof. The first assumption as well as the mapping properties for ${\cal
C}-{\cal C}_{0}$ are consequence of the definition of ${\cal C}_{0}$ and
(3.12). Notice also that

$\bm{H}^{2}=-\bm{I}$

which with the expressions for $\alpha$, $\beta$ and $\delta$, see (3.13),
proves that ${\cal C}^{2}=\frac{1}{4}{\cal I}$. The last result is
straightforward. $\Box$

For a pseudodifferential operator $A$ of order $m$ (i.e.
$A:\bm{H}^{s}\to\bm{H}^{s-m}$) we will denote by $\mathrm{PS}(A)$ its
principal symbol, that is we require that (i) $\mathrm{PS}(A)$ is a Fourier
multiplier and (ii) $A-\mathrm{PS}(A)$ is a pseudodifferential operator of
order $m-1$, i.e. $A-\mathrm{PS}(A):\bm{H}^{s}\to\bm{H}^{s-m+1}$ (the
principal symbol operators are not uniquely defined, and we will use in what
follows various such operators depending on the circumstances). This notation
is readily extended to matrix operators, and, in view of previous result
Proposition 3.3, we have

$\mathrm{PS}({\cal C})={\cal C}_{0}.$

A key ingredient in deriving alternative BIE formulations of scattering and
transmission Navier problems is the incorporation of principal symbols of
Dirichlet-to-Neumann operators. Let us define $Y_{\pm}$ as the usual
parameterization of the Dirichlet-to-Neumann operators $Y_{\pm,\Gamma}$
introduced in (2.7). It is easy to derive from classical results that
$Y_{\pm}:{\bm{H}}^{1/2}\to{\bm{H}}^{-1/2}$. Besides

###### Lemma 3.4

For any $\bm{H}^{1/2}\ni\bm{g}\neq 0$ it holds

$\Im\langle Y_{-}\bm{g},\overline{\bm{g}}\rangle=0,\quad\Im\langle
Y_{+}\bm{g},\overline{\bm{g}}\rangle{>}0.$

Proof. The argument is just an adaptation of classical techniques from the
Helmholtz, which we present here for the sake of completeness. Let ${\bf
v}_{-}\in{\bm{H}}^{1}(\Omega_{-}),\ {\bf v}_{+}\in\bm{H}^{1}_{\rm
loc}(\Omega_{+})$ the solutions of the Navier equations with

$\gamma{\bf v}_{\pm}=\bm{g},\quad T{\bf v}_{\pm}=Y_{\pm}\gamma{\bf
v}_{\pm}=Y_{\pm}\bm{g}.$

Obviously ${\bf v}_{+}$ must satisfy the radiation condition at infinity (2.5)
which easily can be shown to imply

$\lim_{R\to\infty}\int_{\partial B_{R}}\overline{\gamma_{\partial B_{R}}{\bf
v}_{+}}\,T_{\partial B_{R},+}{\bf v}_{+}\in i\,\mathbb{R}^{+}$

where $B_{R}$ is the ball centered at origin and radius $R$.

We are ready to prove the result. For the interior Dirichlet-to-Neumann
operator we just note that by the Green identity to derive

$\displaystyle\langle\overline{\gamma_{-}\bf v}_{-},{T_{-}{\bf v}_{-}}\rangle$
$\displaystyle=$ $\displaystyle\int_{\Gamma}\overline{\gamma_{\Gamma}{\bf
v}}_{-}\cdot{T_{\Gamma,-}{\bf v}_{-}}$ $\displaystyle=$
$\displaystyle\int_{\Omega_{-}}2\mu\varepsilon({\bf
v}_{-}):\overline{\varepsilon({\bf v}_{-})}+\lambda|\nabla\cdot{\bf
v}_{-}|^{2}-\omega^{2}\int_{\Omega_{-}}|{\bf v}_{-}|^{2}\in\mathbb{R}.$

The exterior Dirichlet-to-Neumann operator is only slight more complicated:

$\displaystyle\langle\overline{\gamma_{+}\bf v}_{+},{T_{+}{\bf
v}_{+}}\rangle\\!$ $\displaystyle=$
$\displaystyle\\!\int_{\Gamma}\overline{\gamma_{\Gamma}{\bf
v}_{+}}\cdot{T_{\Gamma,+}{\bf v}_{+}}$ $\displaystyle=-\int_{\Omega_{+}\cap
B_{R}}2\mu\varepsilon({\bf v}_{-}):\overline{\varepsilon({\bf
v}_{-})}-\lambda|\nabla\cdot{\bf v}_{-}|^{2}+\omega^{2}\int_{\Omega_{-}\cap
B_{R}}|{\bf v}_{-}|^{2}+\int_{\partial B_{R}}\overline{\gamma_{\partial
B_{R}}{\bf v}_{+}}\cdot{T_{\partial B_{R},+}{\bf v}_{+}}.$

As consequence of (2.5), and with ${\bf v}_{p}$ and ${\bf v}_{s}$ the
longitudinal and transversal components of ${\bf v}_{+}$,

$\lim_{R\to\infty}\int_{\partial B_{R}}\overline{\gamma_{\partial B_{R}}{\bf
v}_{+}}\cdot{T_{\partial B_{R},+}{\bf
v}_{+}}=\lim_{R\to\infty}ik_{p}(\lambda+2\mu)\int_{\partial B_{R}}|{\bf
v}_{p}|^{2}+ik_{s}\mu\int_{\partial B_{R}}|{\bf v}_{s}|^{2}.$

We stress that, as for Helmholtz equation, the limit above is zero only if so
is ${\bf v}_{+}$. The result is then proven. $\Box$

We are now ready to compute the principal symbols $\mathrm{PS}(Y_{\pm})$. If
$\omega^{2}$ is not an eigenvalue for the Dirichlet problem for the Navier
operator in the interior domain $\Omega_{-}$, which is equivalent to the
invertibility of single layer BIO $V$, cf. Remark 3.1, we derive

$T_{+}{\bf u}=-V^{-1}(\tfrac{1}{2}I-{K}){\bf u}_{+}.$

Similarly, if $\omega^{2}$ is not an eigenvalue for the Neumann problem, which
in turn renders the operator $\tfrac{1}{2}I+{K}$ invertible, we obtain the
alternative formula

$T_{+}{\bf u}=(\tfrac{1}{2}I+{K})^{-1}W{\bf u}_{+}.$

Since $\omega^{2}$ cannot be simultaneously a Neumann and a Dirichlet
eigenvalue in the bounded domain $\Omega_{-}$, we use either of the formulas
above for the exterior Dirichlet-to-Neumann operator $Y_{+}{\bf u}=T_{+}{\bf
u}$ (which is a pseudodifferential operator of order $1$, i.e.,
$Y_{+}:\bm{H}^{s}\to\bm{H}^{s-1}$) together with the principal symbol formulas
for the BIOs involved in those equations and we derive a formula for its
principal symbol

${\rm
PS}(Y_{+})=-\beta^{-1}\bm{\Lambda}^{-1}\left(\frac{1}{2}I_{2}-\alpha\bm{H}\right)=\delta\left(\frac{1}{2}I_{2}+\alpha\bm{H}\right)^{-1}\bm{\Lambda}^{-1}.$
(3.16)

Assuming that the Dirichlet interior problem is well posed, we can proceed in
the same way for the interior Dirichlet-to-Neumann operator $Y_{-}$, and we
get

${\rm
PS}(Y_{-})=\beta^{-1}\bm{\Lambda}^{-1}\left(\frac{1}{2}I_{2}+\alpha\bm{H}\right)=-\delta\left(\frac{1}{2}I_{2}-\alpha\bm{H}\right)^{-1}\bm{\Lambda}^{-1}.$
(3.17)

We note that the matrix operators $\bm{H}$ and $\bm{\Lambda}$ commute, and
furthermore the two choices provided in the right hand sides of equations
(3.16) and respectively (3.17) for the calculation of the principal symbol
operators do produce the same result (this can be easily verified using
equation (3.13)). Finally, it can be easily seen that ${\rm
PS}(Y_{\pm})-Y_{\pm}:\bm{H}^{s}\to\bm{H}^{s+1}$ for all $s$.

We present next some straightforward results that will enable us to establish
certain coercivity properties that the operators ${\rm PS}(Y_{\pm})$ enjoy.

###### Lemma 3.5

It holds,

$\langle\bm{H}\bm{\varphi},\overline{\bm{g}}\rangle=-\langle\bm{\varphi},\overline{\bm{H}\bm{g}}\rangle,\quad\forall\bm{\varphi}\in\bm{H}^{-s},\quad\bm{g}\in\bm{H}^{s}.$

Therefore,

${\Re}\left\langle\bm{H}\bm{\varphi},\overline{\bm{\varphi}}\right\rangle=0,\quad\forall\bm{\varphi}\in\bm{H}^{0}.$

Besides, if $c\in(-1/2,1/2)$ then

$\left\langle\left(\tfrac{1}{2}{\bm{I}}+ci\bm{H}\right)\bm{\varphi},\overline{\bm{\varphi}}\right\rangle\geq\left(\tfrac{1}{2}-c\right)\|{\bm{\varphi}}\|_{\bm{H}^{0}}^{2}.$

Proof. First result follows from the definition and implies trivially the
second result. Finally,

$\left\langle\left(\tfrac{1}{2}{\bm{I}}+ci\bm{H}\right)\bm{\varphi},\overline{\bm{\varphi}}\right\rangle\geq\Big{(}\tfrac{1}{2}-|c|\underbrace{\|\bm{H}\|_{\bm{H}^{0}\to\bm{H}^{0}}}_{=1}\Big{)}\|\bm{\varphi}\|_{\bm{H}^{0}}^{2}$

and the result is proven.

$\Box$

As consequence of the above lemma we can easily derive the following
coercivity result that will be useful in what follows:

###### Proposition 3.6

It holds

$\left\langle\mathrm{PS}(Y_{-})\bm{g},\overline{\bm{g}}\right\rangle\geq\beta^{-1}(1-|\alpha|)\|\bm{g}\|_{\bm{H}^{1/2}}^{2},\quad-\left\langle\mathrm{PS}(Y_{+})\bm{g},\overline{\bm{g}}\right\rangle\geq\beta^{-1}(1-|\alpha|)\|\bm{g}\|_{\bm{H}^{1/2}}^{2}.$

Proof. Since $\bm{\Lambda}^{-1}$ is positive definite we use its square root
$\bm{\Lambda}^{-1/2}$ defined explicitly as

$\bm{\Lambda}^{-1/2}=\begin{bmatrix}\Lambda^{-1/2}\\\
&\Lambda^{-1/2}\end{bmatrix},\quad\Lambda^{-1/2}g=\widehat{g}(0)+\sum_{n\neq
0}|n|^{1/2}\widehat{{g}}(n),$

which turns out to be an isometry from $\bm{H}^{s}$ into $\bm{H}^{s-1/2}$. It
can easily be checked that $\bm{\Lambda}^{-1/2}$ commutes with $\bm{H}$ and
hence

$\displaystyle{\Re}\left\langle\mathrm{PS}(Y_{-})\bm{g},\overline{\bm{g}}\right\rangle$
$\displaystyle=$
$\displaystyle\beta^{-1}{\Re}\left\langle(\tfrac{1}{2}I+\alpha\bm{H})\bm{\Lambda}^{-1/2}\bm{g},\overline{\bm{\Lambda}^{-1/2}\bm{g}}\right\rangle\geq\beta^{-1}(\tfrac{1}{2}-|\alpha|)\|\bm{\Lambda}^{-1/2}\bm{g}\|_{\bm{H}^{0}}^{2}$
$\displaystyle=$
$\displaystyle\beta^{-1}(\tfrac{1}{2}-|\alpha|)\|\bm{g}\|_{\bm{H}^{1/2}}^{2}.$

The second result follows similarly. $\Box$

We will also use an alternative form of the principal symbol operator
$\mathrm{PS}({Y_{\pm}})$ that relies on complexified square roots. To this
end, we select a complex wavenumber  $\kappa$ with $\Re\kappa,\Im\kappa>0$ and
we introduce a complexified version of the Fourier multiplier operator defined
in equation (3.10) in the following form

$\Lambda_{\kappa}=\sum_{n=-\infty}^{\infty}(n^{2}-\kappa^{2})^{-1/2}\widehat{g}(n)\exp(in\,\cdot)$
(3.18)

and then we define the matrix operators
$\bm{\Lambda}_{\kappa}:=\begin{bmatrix}\Lambda_{\kappa}&\\\
&\Lambda_{\kappa}\end{bmatrix}$ and $\bm{\Lambda}^{-1}_{\kappa}$ accordingly.
Clearly, $\bm{\Lambda}_{\kappa}:\bm{H}^{s}\to\bm{H}^{s+1}$. Besides, by
construction

$\Re\langle\bm{\Lambda}_{\kappa}\bm{\varphi},\overline{\bm{\varphi}}\rangle\geq
c\|\bm{\varphi}\|_{\bm{H}^{-1/2}}^{2},\quad\Re\langle\bm{\Lambda}^{-1}_{\kappa}\bm{\varphi},\overline{\bm{\varphi}}\rangle\geq
c\|\bm{g}\|_{\bm{H}^{1/2}}^{2}$ (3.19)

and

$\Im\langle\bm{\Lambda}_{\kappa}\bm{\varphi},\overline{\bm{\varphi}}\rangle\geq
c\|\bm{\varphi}\|_{\bm{H}^{-3/2}}^{2},\quad-\Im\langle\bm{\Lambda}^{-1}_{\kappa}\bm{\varphi},\overline{\bm{\varphi}}\rangle\geq
c\|\bm{g}\|_{\bm{H}^{-1/2}}^{2}.$ (3.20)

The newly defined operators $\bm{\Lambda}_{\kappa}$ enjoy the following
regularity properties

$\bm{\Lambda}_{\kappa}-\bm{\Lambda}:\bm{H}^{s}\to\bm{H}^{s+3},\quad\bm{\Lambda}^{-1}_{\kappa}-\bm{\Lambda}^{-1}:\bm{H}^{s}\to\bm{H}^{s+{1}}.$

We can thus define complexified principal symbol DtN operators in the
following manner

$\displaystyle{\rm PS}_{\kappa}(Y_{-})$ $\displaystyle:=$
$\displaystyle\beta^{-1}\bm{\Lambda}_{\kappa}^{-1}\left(\frac{1}{2}I_{2}+\alpha\bm{H}\right)=-\delta\left(\frac{1}{2}I_{2}-\alpha\bm{H}\right)^{-1}\bm{\Lambda}_{\kappa}^{-1}$
$\displaystyle{\rm PS}_{\kappa}(Y_{+})$ $\displaystyle:=$
$\displaystyle-\beta^{-1}\bm{\Lambda}_{\kappa}^{-1}\left(\frac{1}{2}I_{2}-\alpha\bm{H}\right)=\delta\left(\frac{1}{2}I_{2}+\alpha\bm{H}\right)^{-1}\bm{\Lambda}_{\kappa}^{-1}.$

We establish that the complexified operators defined above are indeed
principal symbol DtN operators that enjoy the same coercivity properties as
the DtN operators themselves

###### Proposition 3.7

It holds ${\rm PS}(Y_{\pm})-{\rm
PS}_{\kappa}(Y_{\pm}):\bm{H}^{s}\to\bm{H}^{s+1}$. Furthermore, there exist
$c_{1}>0$ and, provided that $\Re(n^{2}-\kappa^{2})\neq 0$, $c_{2}>0$ so that

$\displaystyle\Im\left\langle\mathrm{PS}_{\kappa}(Y_{+})\bm{g},\overline{\bm{g}}\right\rangle$
$\displaystyle\geq c_{1}\|\bm{g}\|^{2}_{\bm{H}^{-1/2}},\quad$
$\displaystyle-\Im\left\langle\mathrm{PS}_{\kappa}(Y_{-})\bm{g},\overline{\bm{g}}\right\rangle$
$\displaystyle\geq c_{1}\|\bm{g}\|^{2}_{\bm{H}^{-1/2}},$
$\displaystyle-\Re\left\langle\mathrm{PS}_{\kappa}(Y_{+})\bm{g},\overline{\bm{g}}\right\rangle$
$\displaystyle\geq c_{2}\|\bm{g}\|^{2}_{\bm{H}^{1/2}},\quad$
$\displaystyle\Re\left\langle\mathrm{PS}_{\kappa}(Y_{-})\bm{g},\overline{\bm{g}}\right\rangle$
$\displaystyle\geq c_{2}\|\bm{g}\|^{2}_{\bm{H}^{1/2}}.$

Proof. Using that

$(n^{2}-\kappa^{2})^{1/2}=a_{n}^{2}-ib^{2}_{n},\quad a_{n},b_{n}>0,\
a_{n}\approx n,\quad b_{n}\approx n^{-1},\ \text{as }n\to\infty,$

we can easily show that

$\Lambda_{\kappa}^{-1}={\Theta}^{2}_{r}-i{\Theta}^{2}_{i}$

with

${\Theta}_{r}g:=\sum_{n=-\infty}^{\infty}a_{n}\widehat{g}(n)\exp(i\,n\,\cdot\,),\quad{\Theta}_{i}g:=\sum_{n=-\infty}^{\infty}b_{n}\widehat{g}(n)\exp(i\,n\,\cdot\,).$

Clearly ${\Theta}_{i}:{H}^{s}\to{H}^{s+1/2}$ and, provided that
$\Re(n^{2}-\kappa^{2})\neq 0$, ${\Theta}_{r}:{H}^{s}\to{H}^{s-1/2}$ are
continuous invertible operators.

We then define the matrix operators
$\bm{\Theta}_{r}:=\begin{bmatrix}\Theta_{r}&\\\ &\Theta_{r}\end{bmatrix}$ and
$\bm{\Theta}_{i}:=\begin{bmatrix}\Theta_{i}&\\\ &\Theta_{i}\end{bmatrix}$
accordingly and we notice that

${\rm
PS}_{\kappa}(Y_{\pm})=\mp\beta^{-1}\bm{\Theta}_{r}^{2}\left(\frac{1}{2}I_{2}\mp\alpha\bm{H}\right)\pm
i\beta^{-1}\bm{\Theta}_{i}^{2}\left(\frac{1}{2}I_{2}\mp\alpha\bm{H}\right)$

so that

$\langle{\rm
PS}_{\kappa}(Y_{\pm})\bm{g},\overline{\bm{g}}\rangle=\mp\beta^{-1}\langle\left(\tfrac{1}{2}I_{2}\mp\alpha\bm{H}\right)\bm{\Theta}_{r}\bm{g},\overline{\bm{\Theta}_{r}\bm{g}}\rangle\pm{i}\beta^{-1}\langle\left(\tfrac{1}{2}I_{2}\mp\alpha\bm{H}\right)\bm{\Theta}_{i}\bm{g},\overline{\bm{\Theta}_{i}\bm{g}}\rangle.$

The result follows now from Lemma 3.5.

$\Box$

###  3.3 Duality in product spaces

In this short subsection we present certain properties of matrix operators in
connection to duality in product spaces. We are interested in deriving an
explicit formula for the dual of the the Calderón operators ${\cal
C}:\bm{H}^{s_{1}}\times\bm{H}^{s_{2}}\to\bm{H}^{s_{1}}\times\bm{H}^{s_{2}}$
with an emphasis on the case where $s_{1}=1/2=-s_{2}$. To this end, for
general matrix operators

${\cal A}:\begin{bmatrix}A_{11}&A_{12}\\\ A_{21}&A_{22}\\\
\end{bmatrix}:\bm{H}^{s_{1}}\times{\bm{H}}^{s_{2}}\to\bm{H}^{t_{1}}\times\bm{H}^{t_{2}},\quad\text{(that
is, $A_{ij}:\bm{H}^{s_{j}}\to\bm{H}^{t_{i}}$)}$

we work with the non-standard duality product representation:
$(\bm{H}^{s}\times\bm{H}^{t})^{\prime}=\bm{H}^{-t}\times\bm{H}^{-s}$ via the
non-standard duality product

$[(\bm{g}_{1},\bm{\varphi}_{1}),(\bm{g}_{2},\bm{\varphi}_{2})]=\langle\bm{g}_{1},\bm{\varphi}_{2}\rangle-\langle\bm{g}_{2},\bm{\varphi}_{1}\rangle.$
(3.21)

Notice that, with this convention:

$\left[{\cal A}\begin{bmatrix}\bm{g}_{1}\\\
\bm{\varphi}_{1}\end{bmatrix},\begin{bmatrix}\bm{g}_{2}\\\
\bm{\varphi}_{2}\end{bmatrix}\right]=\left[\begin{bmatrix}\bm{g}_{1}\\\
\bm{\varphi}_{1}\end{bmatrix},{\cal A}^{\top}\begin{bmatrix}\bm{g}_{2}\\\
\bm{\varphi}_{2}\end{bmatrix}\right],\quad\text{with\ }{\cal
A}^{\top}:=\begin{bmatrix}A^{\top}_{22}&-A^{\top}_{12}\\\
-A^{\top}_{21}&A^{\top}_{11}\end{bmatrix}.$

In particular, we find that

${\cal C}^{\top}=-{\cal C},\quad\left(\frac{1}{2}{\cal I}\pm{\cal
C}\right)^{\top}=\left(\frac{1}{2}{\cal I}\mp{\cal C}\right)^{\top}$

i.e., the transpose of the interior Calderon operator is the exterior one and
vice versa. This property will simplify the analysis of boundary integral
formulations for transmission problems as we will see in next section.

##  4 Boundary integral formulations

We present in what follows various strategies to derive BIE formulations of
scattering and transmission elastodynamic problems. Besides the classical
combined field formulations CFIE we derive regularized formulations that rely
on the use of approximations of the DtN operators. The design of the
regularized formulations for elastodynamic scattering and transmission
problems follows the blueprint from the Helmholtz case [5, 16]. We start with
the impenetrable case and then we follow up with the penetrable case.

###  4.1 The impenetrable case: Dirichlet and Neumann boundary conditions

Let us consider $\Omega_{+}$ the exterior of a closed, smooth, simply connect
curve $\Gamma$.

$\left|\begin{array}[]{rcl}{\bf u}\in\bm{H}_{\rm loc}^{1}(\Omega_{+})\\\
{\nabla}\cdot\bm{\sigma}({\bf u})+\omega^{2}{\bf u}&=&0,\quad\text{in
$\Omega_{+}$},\\\ \gamma_{\Gamma}{\bf u}&=&{\bf f}_{\Gamma}\\\
\lx@intercol\text{+ RC<EMAIL_ADDRESS>(4.1)

Just like in the Helmholtz case [6], the classical approach [9, 11] in the
case of Dirichlet boundary conditions is to look for a scattered field in the
form of a Combined Field representation

${\bf
u}({\bm{x}}):=(\operatorname{DL}{\bm{\varphi}})({\bm{x}})-i\eta_{D}(\operatorname{SL}{\bm{\varphi}})({\bm{x}}),\quad{\bm{x}}\in\Omega_{+}$

where the coupling parameter $\eta_{D}\neq 0$ and
$\bm{\varphi}:\mathbb{R}\to\mathbb{C}^{2}$ is $2\pi-$periodic, to the Combined
Field Integral Equation (CFIE)

$\frac{1}{2}{\bm{\varphi}}+K{\bm{\varphi}}-i\eta_{D}V{\bm{\varphi}}={\bf
f}\quad{\rm on}\ \Gamma.$ (4.2)

(With, as taken throughout this paper, ${\bf f}:={\bf f}_{\Gamma}\circ{\bf
x}$). The question of selecting a value of the coupling parameter $\eta_{D}$
that leads to formulations with superior spectral properties (and thus faster
convergence rates for iterative solver solutions) can be settled via
Dirichlet-to-Neumann arguments [11]. We begin with Somigliana’s identities
(3.9) which we rewrite considering as ${\bf u}|_{\Gamma}$ as the primary
unknown boundary density and incorporating the DtN operator in the form

${\bf u}({\bm{x}})=\operatorname{DL}[\gamma{\bf
u}]({\bm{x}})-\operatorname{SL}[Y_{+}\gamma{\bf
u}]({\bm{x}}),\quad{\bm{x}}\in\Omega_{+}.$

The main idea in constructing regularized formulations is to use easily
constructable approximations $\operatorname{R}^{\rm D}$ of the DtN operator
$Y_{+}$ and look for combined field representations in the form

${\bf
u}({\bm{x}}):=(\operatorname{DL}{\bm{g}})({\bm{x}})-(\operatorname{SL}[\operatorname{R}^{\rm
D}{\bm{g}}])({\bm{x}}),\quad{\bm{x}}\in\Omega_{+}$ (4.3)

leading to the Combined Field Regularized Integral Equation (CFIER)

$\frac{1}{2}{\bm{g}}+K{\bm{g}}-V\operatorname{R}^{\rm D}{\bm{g}}={\bf
f}\quad{\rm on}\ \Gamma.$ (4.4)

We show in what follows that under certain assumptions on the regularizing
operator $\operatorname{R}^{\rm D}$, the CFIER is well posed:

###### Theorem 4.1

Consider a preconditioner $\operatorname{R}^{\rm D}$ that satisfies the
following two properties: (a) $\operatorname{R}^{\rm
D}-Y_{+}:\bm{H}^{s}\to\bm{H}^{s}$ is continuous and (b) the non-null condition
holds

$\Im\langle\operatorname{R}^{\rm
D}\bm{\varphi},\overline{\bm{\varphi}}\rangle\neq 0,\quad{\bm{\varphi}\neq
0.}$

Then the CFIER operator $\frac{1}{2}I+K-V[\operatorname{R}^{\rm D}]$ is an
invertible compact perturbation of the identity with continuous inverse.

In particular, ${\rm PS}_{\kappa}(Y_{+})$ is a valid election.

Proof. Notice that by construction:

$\frac{1}{2}I+K-V\operatorname{R}^{\rm
D}-\left(\frac{1}{2}I+\alpha\bm{H}-\beta\bm{\Lambda}\mathrm{PS}(Y_{+})\right):\bm{H}^{s}\to\bm{H}^{s+2}.$

Furthermore, by (3.16)

$\frac{1}{2}I+\alpha\bm{H}-\beta\bm{\Lambda}\,\mathrm{PS}(Y_{+})=\frac{1}{2}I+\alpha\bm{H}-\beta\bm{\Lambda}\left[\beta^{-1}\bm{\Lambda}^{-1}\left(\frac{1}{2}I-\alpha\bm{H}\right)\right]=I$

Consequently, the operator is a compact perturbation of the identity, and, as
such, owing to Fredholm alternative, it suffices to establish the injectivity
in order to complete the proof. To this end, let us assume that
$\bm{g}\in\bm{H}^{s}$ is in the kernel of the operator and define

${\bf
v}({\bm{x}}):=(\operatorname{DL}\bm{g})({\bm{x}})-(\operatorname{SL}[\operatorname{R}^{\rm
D}\bm{g}])({\bm{x}}),\quad{\bm{x}}\in\mathbb{R}^{2}\setminus\Gamma.$

Clearly ${\bf v}_{+}={\bf v}|_{\Omega_{+}}$ is a radiative solution of the
Navier equation in $\Omega_{+}$ with zero Dirichlet boundary conditions on
$\Gamma$, which implies that ${\bf v}_{+}$ is identically zero in
$\Omega_{+}$. Hence,

$\left(\frac{1}{2}{\cal I}+{\cal C}\right)\begin{bmatrix}\bm{g}\\\
\operatorname{R}^{\rm D}\bm{g}\end{bmatrix}={\bf 0},\quad\text{and so
}\quad\left(\frac{1}{2}{\cal I}-{\cal C}\right)\begin{bmatrix}\bm{g}\\\
\operatorname{R}^{\rm D}\bm{g}\end{bmatrix}=\begin{bmatrix}\bm{g}\\\
\operatorname{R}^{\rm D}\bm{g}\end{bmatrix}.\quad$

In other words, ${\bf v}_{-}={\bf v}|_{\Omega_{-}}$ is a solution of the
Navier equation in the domain $\Omega_{-}$ that satisfies

$\gamma_{-}{\bf v}_{-}=\bm{g},\quad T_{-}{\bf v}_{-}=\operatorname{R}^{\rm
D}\bm{g}$

and so

$\displaystyle\langle\operatorname{R}^{\rm
D}{\bm{g}},\overline{{\bm{g}}}\rangle$ $\displaystyle=$ $\displaystyle\langle
T{\bf v}_{-},\overline{{\gamma}{\bf v}_{-}}\rangle\in\mathbb{R}$

by Lemma 3.4. The proof is concluded on the basis of the assumption (b).
$\Box$

The regularizing operator $\operatorname{R}^{\rm D}={\rm PS}_{\kappa}(Y_{+})$
is a pseudodifferential operator of order $1$ whose numerical evaluation is
consequently more involved. Using the high-frequency approximation
$|\kappa|\to\infty$ in the definition of the Fourier multiplier
$\Lambda_{\kappa}^{-1}$, we can construct a simple regularizing operator

$\operatorname{R}^{\rm
D}_{1}=-\frac{2\mu(\lambda+2\mu)}{\lambda+3\mu}i\kappa,$

which, incidentally, can be interpreted as delivering a quasi-optimal choice
for the coupling parameter $\eta_{D}$ in the CFIE formulation

$\eta_{D}^{\rm opt}=\frac{2\mu(\lambda+2\mu)}{\lambda+3\mu}k_{s}$ (4.5)

if we choose $\kappa=k_{s}$. We remark that a similar, easily implementable
low-order approximation of the DtN operator was proposed in [11] as a
regularizing operator in the Dirichlet case.

In the case of Neumann boundary conditions, i.e.,

$\left|\begin{array}[]{rcl}{\bf u}\in\bm{H}_{\rm loc}^{1}(\Omega_{+})\\\
{\nabla}\cdot\bm{\sigma}({\bf u})+\omega^{2}{\bf u}&=&0,\quad\text{in
$\Omega_{+}$},\\\ T_{+,\Gamma}{\bf u}&=&{\bm{\lambda}}_{\Gamma}\\\
\lx@intercol\text{+ RC<EMAIL_ADDRESS>(4.6)

we can look for a scattered field in the form of a Combined Field
representation akin to the Burton-Miller formulation in the Helmholtz case [8]

${\bf
u}({\bm{x}}):=-(\operatorname{SL}\bm{\varphi})({\bm{x}})+i\eta_{N}(\operatorname{DL}\bm{\varphi})({\bm{x}}),\quad{\bm{x}}\in\Omega_{+}$

where the coupling parameter $\eta_{N}\neq 0$, leading to the Combined Field
Integral Equation (CFIE)

$\frac{1}{2}\bm{\varphi}-K^{\top}\bm{\varphi}+i\eta_{N}W\bm{\varphi}=\bm{\lambda}\quad{\rm
on}\ \Gamma.$ (4.7)

Here again we start by recasting the Somigliana’s identities looking at $T{\bf
u}$ as the primary unknown boundary density and making use of the the Neumann-
to-Dirichlet (NtD) operator (which is the inverse of the DtN operator)

${\bf u}({\bm{x}})=(\operatorname{DL}[Y_{+}^{-1}\\{T{\bf
u}\\}])({\bm{x}})-(\operatorname{SL}\\{T{\bf
u}\\})({\bm{x}}),\quad{\bm{x}}\in\Omega_{+}$

The construction of regularized formulations relies again on available
approximations $\operatorname{R}^{\rm N}$ of the NtD operator $Y^{-1}$ via
looking for combined field representations in the form

${\bf u}({\bm{x}}):=(\operatorname{DL}[\operatorname{R}^{\rm
N}\bm{\varphi}])({\bm{x}})-(\operatorname{SL}\bm{\varphi})({\bm{x}}),\quad{\bm{x}}\in\Omega_{+}$
(4.8)

leading to the Combined Field Regularized Integral Equation (CFIER)

$\frac{1}{2}\bm{\varphi}-K^{\top}\bm{\varphi}+W\operatorname{R}^{\rm
N}\bm{\varphi}={\bm{\lambda}}\quad{\rm on}\ \Gamma.$ (4.9)

Again here, the choice $\operatorname{R}^{\rm N}=[{\rm
PS}_{\kappa}(Y_{+})]^{-1}$ leads to well posed CFIER formulations, at least in
the case when the boundary $\Gamma$ is smooth. Indeed, the following result
can be established analogously to that in Theorem 4.1

###### Theorem 4.2

Consider a preconditioner $\operatorname{R}^{\rm N}$ that satisfies the
following two properties: (a) $\operatorname{R}^{\rm
N}-[Y_{+}]^{-1}:\bm{H}^{s}\to\bm{H}^{s+1}$ and (b) the non-null condition

$\Im\langle\operatorname{R}^{\rm
N}\bm{\bm{\varphi}},\overline{\bm{\varphi}}\rangle\neq 0,\quad\bm{\varphi}\neq
0.$

Then the CFIER operator

$\frac{1}{2}I-K^{\top}+W[\operatorname{R}^{\rm N}]$

is an invertible compact perturbation of the identity.

In particular, $\operatorname{R}^{\rm N}=({\rm PS}_{\kappa}(Y_{+}))^{-1}$ is a
valid choice.

Proof. From

$\frac{1}{2}I-\alpha\bm{H}+\delta\bm{\Lambda}^{-1}\left[\delta\Big{(}\frac{1}{2}I_{2}+\alpha\bm{H}\Big{)}^{-1}\bm{\Lambda}^{-1}\right]^{-1}=I$

we can easily see that $\frac{1}{2}I-K^{\top}+W[{\rm
PS}_{\kappa}(Y_{+})]^{-1}$ is a compact perturbation of the identity.

The unicity follows from a similar argument as in Theorem 4.1: take
$\bm{\varphi}$ in the kernel of the operator, define

${\bf w}({\bm{x}}):=(\operatorname{DL}[\operatorname{R}^{\rm
N}\bm{\varphi}])({\bm{x}})-(\operatorname{SL}\bm{\varphi})({\bm{x}}),\quad{\bm{x}}\in\mathbb{R}^{2}\setminus\Gamma$

and derive next ${\bf w}$ vanishes in $\Omega_{+}$. Therefore, with ${\bf
w}_{-}={\bf w}|_{\Omega_{-}}$ and $\bm{g}={\rm
PS}_{\kappa}(Y_{+})^{-1}\bm{\varphi}$

$\langle{\rm
PS}_{\kappa}(Y_{+})\bm{g},\overline{\bm{g}}\rangle=\langle\bm{\varphi},\overline{\operatorname{R}^{\rm
N}\bm{\varphi}}\rangle=\langle T{\bf w}_{-},\overline{\gamma{\bf
w}_{-}}\rangle$

and the result is consequence of Lemma 3.4 and Proposition 3.7.

$\Box$

Similarly, using high-frequency approximations we can construct a simple
regularizing operator

$\operatorname{R}^{\rm
N}_{1}=i\frac{\lambda+3\mu}{2\mu(\lambda+2\mu)}\kappa^{-1},$

which, delivers a quasi-optimal choice for the coupling parameter $\eta_{N}$
in the CFIE formulation

$\eta_{N}^{\rm opt}=\frac{\lambda+3\mu}{2\mu(\lambda+2\mu)}k_{s}^{-1}.$ (4.10)

We remark that similar low-order approximations of NtD operators have been
used in [12] to construct CFIE formulations with superior spectral properties.

Similarly, we can construct _direct_ regularized formulations in the case of
Neumann boundary conditions following the ideas in [2]. Assuming that a smooth
incident field ${\bf u}^{\rm inc}$ (which is a solution of the Navier equation
in the whole $\mathbb{R}^{2}$) impinges on the obstacle $\Omega$, we will
derive these BIEs in terms of unknown boundary quantity $\gamma{\bf u}^{\rm
tot}=\gamma({\bf u}+{\bf u}^{\rm inc})$. We obtain from Somigliana’s
identities by taking into account the fact that $T{\bf u}^{\rm tot}=0$ on
$\Gamma$

${\bf u}({\bm{x}})=(\operatorname{DL}{\bf u}^{\rm
tot})({\bm{x}}),\quad{\bm{x}}\in\Omega_{+}.$ (4.11)

Applying the Dirichlet trace on $\Gamma$ to formula (4.11) we obtain

$\frac{1}{2}\gamma{\bf u}^{\rm tot}-K\gamma{\bf u}^{\rm tot}=\gamma{\bf
u}^{\rm inc}$ (4.12)

while applying the traction operator to formula (4.11) we get

$W\gamma{\bf u}^{\rm tot}=-T{\bf u}^{\rm inc}.$ (4.13)

We combine BIE (4.12) and a preconditioned (on the left) version of the BIE
(4.13) to arrive at the DCFIER

$\frac{1}{2}\gamma{\bf u}^{\rm tot}({\bm{x}})-K\gamma{\bf u}^{\rm
tot}+([\operatorname{R}^{\rm N}W]\gamma{\bf u}^{\rm tot})=\gamma{\bf u}^{\rm
inc}-\operatorname{R}^{\rm N}T{\bf u}^{\rm inc},$ (4.14)

We note that the operators on the left hand side of the DCFIER formulation is
the transpose of the operator in the CFIER formulation, a situation that is
similar to that in the Helmholtz case [2]. Direct formulations can be then
used in the case when $\Omega$ is a Lipschitz domain in order to take
advantage of the increased regularity of $\gamma{\bf u}^{\rm tot}$.

###  4.2 The penetrable case

We consider in this section the penetrable case with transmission conditions,
as formulated in (2.6):

$\left|\begin{array}[]{rcl}\lx@intercol{\bf
u}_{-}\in\bm{H}^{1}(\Omega_{-}),\quad{\bf u}_{+}\in\bm{H}_{\rm
loc}^{1}(\Omega_{+})\hfil\lx@intercol\\\ {\nabla}\cdot\bm{\sigma}_{-}({\bf
u}_{-})+\omega_{-}^{2}{\bf u}_{-}&=&0,\quad\mbox{in }\Omega_{-}\\\
{\nabla}\cdot\bm{\sigma}_{+}({\bf u}_{+})+\omega_{+}^{2}{\bf
u}_{-}&=&0,\quad\mbox{in }\Omega_{+}\\\ \gamma{\bf u}_{+}-\gamma{\bf
u}_{-}&=&-\gamma{\bf u}^{\rm inc}\\\ T_{+}{\bf u}_{+}-T_{-}{\bf
u}_{-}&=&-T_{+}{\bf u}^{\rm inc}\\\ \lx@intercol\text{+ RC
<EMAIL_ADDRESS>(4.15)

In the first approach, we seek to solve for the Cauchy data $(\gamma{\bf
u}_{\pm},T_{\pm}{\bf u})$ and reconstruct the solution via

${\bf u}_{\pm}=\pm\operatorname{DL}_{\pm}\gamma{\bf
u}_{\pm}\mp\operatorname{SL}_{\mp}\gamma{\bf u}_{\pm},$

where the $\pm$ notation is used for denoting the corresponding boundary layer
potential operators for the exterior/interior problem. Hence, denoting by

${\cal C}_{\pm}:=\begin{bmatrix}K_{\pm}&-V_{\pm}\\\
W_{\pm}&-K_{\pm}^{\top}\end{bmatrix}$

we have that

$\left(\frac{1}{2}{\cal I}+{\cal C}_{+}\right)\begin{bmatrix}\gamma{\bf
u}_{+}\\\ T_{+}{\bf u}_{+}\end{bmatrix}=\begin{bmatrix}\gamma{\bf u}_{+}\\\
T_{+}{\bf u}_{+}\end{bmatrix},\quad\left(\frac{1}{2}{\cal I}-{\cal
C}_{-}\right)\begin{bmatrix}\gamma{\bf u}_{-}\\\ T_{-}{\bf
u}_{-}\end{bmatrix}=\begin{bmatrix}\gamma{\bf u}_{-}\\\ T_{-}{\bf
u}_{-}\end{bmatrix}.$

These last two properties together with the fact that

$\left(\frac{1}{2}{\cal I}+{\cal C}_{+}\right)\begin{bmatrix}\gamma{\bf
u}^{\rm inc}\\\ T_{+}{\bf u}^{\rm inc}\end{bmatrix}={\bf 0}$

can be combined to derive a system of BIE equations for the solution of
penetrable Navier problems

${\cal L}_{\rm SC}\begin{bmatrix}\gamma{\bf u}_{-}\\\ T_{-}{\bf
u}_{-}\end{bmatrix}:=-({\cal C}_{+}+{\cal C}_{-})\begin{bmatrix}\gamma{\bf
u}_{-}\\\ T_{-}{\bf u}_{-}\end{bmatrix}=\begin{bmatrix}\gamma{\bf u}^{\rm
inc}\\\ T_{+}{\bf u}^{\rm inc}\end{bmatrix}$ (4.16)

We refer in what follows to the formulation (4.16) as the Stephan-Costabel
formulation as it is the elastodynamics counterpart of the acoustic
formulation introduced by those two authors. Besides, since

$\left(\frac{1}{2}{\cal I}+{\cal C}_{-}\right)\begin{bmatrix}\gamma{\bf
u}_{-}\\\ T_{-}{\bf u}_{-}\end{bmatrix}={\bf 0}$

we find also that

$\left(\frac{1}{2}{\cal I}+{\cal C}_{-}-{\cal R}^{\top}({\cal C}_{+}+{\cal
C}_{-})\right)\begin{bmatrix}\gamma{\bf u}_{-}\\\ T_{-}{\bf
u}_{-}\end{bmatrix}={\cal R}^{\top}\begin{bmatrix}\gamma{\bf u}^{\rm inc}\\\
T_{+}{\bf u}^{\rm inc}\end{bmatrix}$ (4.17)

for any suitable operator

${\cal R}=\begin{bmatrix}R_{11}&R_{12}\\\
R_{21}&R_{22}\end{bmatrix}\quad(\text{recall that }{\cal
R}^{\top}=\begin{bmatrix}R^{\top}_{22}&-R^{\top}_{12}\\\
-R^{\top}_{21}&R^{\top}_{11}\end{bmatrix}).$

Alternatively, we can work with the adjoint of the formulation (4.17), in the
sense discussed in Section 3.3

$\Big{(}\frac{1}{2}{\cal I}-{\cal C}_{-}+({\cal C}_{+}+{\cal C}_{-}){\cal
R}\Big{)}\begin{bmatrix}\bm{g}\\\
\bm{\varphi}\end{bmatrix}=\begin{bmatrix}\gamma{\bf u}^{\rm inc}\\\ T_{+}{\bf
u}^{\rm inc}\end{bmatrix}$ (4.18)

Clearly, owing to duality arguments, the first formulation (4.17) is uniquely
solvable if and only if so is the second (dual) formulation (4.18). The
connection between the boundary densities $(\bm{g},\bm{\varphi})$ which are
the solution of the adjoint formulation (4.18) and the fields ${\bf u}_{\pm}$
is as follows

$\displaystyle{\bf u}_{+}$
$\displaystyle=\begin{bmatrix}\operatorname{DL}_{+}&-\operatorname{SL}_{+}\end{bmatrix}{\cal
R}\begin{bmatrix}\bm{g}\\\ \bm{\varphi}\end{bmatrix}$
$\displaystyle=\operatorname{DL}_{+}(R_{11}\bm{g}+R_{12}\bm{\varphi})-\operatorname{SL}_{+}(R_{21}\bm{g}+R_{22}\bm{\varphi})$
(4.19) $\displaystyle{\bf u}_{-}$
$\displaystyle=\begin{bmatrix}-\operatorname{DL}_{-}&\operatorname{SL}_{-}\end{bmatrix}({\cal
R}-I)\begin{bmatrix}\bm{g}\\\ \bm{\varphi}\end{bmatrix}$
$\displaystyle=\operatorname{DL}_{-}(\bm{g}-R_{11}\bm{g}-R_{12}\bm{\varphi})-\operatorname{SL}_{-}(\bm{\varphi}-R_{21}\bm{g}-R_{22}\bm{\varphi}).$

In other words, formulations (4.17) and respectively (4.18) can be viewed as
regularized direct and indirect BIE formulations of elastodynamic transmission
problems.

In particular, selecting ${\cal R}=\frac{1}{2}{\cal I}$ in either of the
formulations (4.17) or (4.18) we obtain the Kress-Roach type formulation:

${\cal L}_{\rm KR}\begin{bmatrix}\gamma{\bf u}_{-}\\\ T_{-}{\bf
u}_{-}\end{bmatrix}:=({\cal I}+{\cal C}_{-}-{\cal
C}_{+})\begin{bmatrix}\gamma{\bf u}_{-}\\\ T_{-}{\bf
u}_{-}\end{bmatrix}=2\begin{bmatrix}\gamma{\bf u}^{\rm inc}\\\ T_{+}{\bf
u}^{\rm inc}\end{bmatrix}.$ (4.20)

We stress that, unlike the Kress-Roach formulations for the Helmholtz
transmission problems, the BIE formulation (4.20) is not a second kind
formulation since the double layer BIOs are no longer compact in this setting.

We devote the remaining of this subsection to establish that the Stephan-
Constable and the Kress-Roach type formulations are well posed. The proof
follows the ideas used in the seminal work [14, 22] for which the well-
posedness of the transmission problem with reversed material properties

$\left|\begin{array}[]{rcl}\lx@intercol{\bf
v}_{-}\in\bm{H}^{1}(\Omega_{-}),\quad{\bf v}_{+}\in\bm{H}_{\rm
loc}^{1}(\Omega_{+})\hfil\lx@intercol\\\ {\nabla}\cdot\bm{\sigma}_{-}({\bf
v}_{+})+\omega_{-}^{2}{\bf v}_{+}&=&0,\quad\mbox{in }\Omega_{+},\\\
{\nabla}\cdot\bm{\sigma}_{+}({\bf v}_{-})+\omega_{+}^{2}{\bf
v}_{-}&=&0,\quad\mbox{in }\Omega_{-},\\\ \gamma{\bf v}_{+}-\gamma{\bf
v}_{-}&=&-{\bf u}^{\rm inc}\\\ T_{+}{\bf v}_{+}-T_{-}{\bf v}_{-}&=&-T_{+}{\bf
u}^{\rm inc}\\\ \lx@intercol\text{+ RC<EMAIL_ADDRESS>(4.21)

will be essential.

Let us introduce (with $\alpha_{\pm}\in i\mathbb{R}$,
$\beta_{\pm}>0>\delta_{\pm}$ as in (3.15)) operators

$\displaystyle{\cal L}_{0,\rm SC}$ $\displaystyle:=$ $\displaystyle-({\cal
C}_{+,0}+{\cal
C}_{-,0})=-\begin{bmatrix}(\alpha_{+}+\alpha_{-})\bm{H}&-(\beta_{+}+\beta_{-})\bm{\Lambda}\\\
(\delta_{+}+\delta_{-})\bm{\Lambda}^{-1}&-(\alpha_{+}+\alpha_{-})\bm{H}\\\
\end{bmatrix},$ $\displaystyle{\cal L}_{0,{\rm KR}}$ $\displaystyle:=$
$\displaystyle{\cal I}+{\cal C}_{-,0}-{\cal
C}_{+,0}=I+\begin{bmatrix}(\alpha_{-}-\alpha_{+})\bm{H}&-(\beta_{-}-\beta_{+})\bm{\Lambda}\\\
(\delta_{-}-\delta_{+})\bm{\Lambda}^{-1}&-(\alpha_{-}-\alpha_{+})\bm{H}\\\
\end{bmatrix}$

which are simply the principal parts for ${\cal L}_{\rm SC}$ and ${\cal
L}_{\rm KR}$.

We can show easily that

${\cal L}_{0,\rm SC}^{2}=\rho{\cal I}$ (4.22)

where

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle-{(\beta_{+}+\beta_{-})(\delta_{+}+\delta_{-})-(\alpha_{+}+\alpha_{-})^{2}}$
$\displaystyle=$
$\displaystyle\frac{(\lambda_{+}(\mu_{+}+\mu_{-})+\mu_{+}(\mu_{+}+3\mu_{-}))(\lambda_{-}(\mu_{+}+\mu_{-})+\mu_{-}(3\mu_{+}+\mu_{-}))}{4\mu_{+}\mu_{-}(\lambda_{+}+2\mu_{+})(\lambda_{-}+2\mu_{-})}$
$\displaystyle=$ $\displaystyle
F({\lambda_{+}},{\mu_{+}},\frac{{\lambda_{-}}}{{\lambda_{+}}},\frac{{\mu_{-}}}{{\mu_{+}}})=F({\lambda_{-}},{\mu_{-}},\frac{{\lambda_{+}}}{{\lambda_{-}}},\frac{{\mu_{+}}}{{\mu_{-}}}),$

with

$F(x,y,r,s):=\frac{\left((s+1)xy+(3s+1)y^{2}\right)\left(r(s+1)xy+s(s+3)y^{2}\right)}{4sy^{2}(x+2y)(rx+2sy)}.$

We prove in the next lemma that $\rho\geq 1$ with $\rho=1$ if and only if
$\mu_{+}=\mu_{-}$.

###### Lemma 4.3

For all $(x,y,r,s)\in\mathbb{R}^{4}_{+}$, $F(x,y,r,s)\geq 1$ with
$F(x,y,r,s)=1$ if and only if $s=1$.

Proof. Clearly $F(x,y,r,1)=1$. Besides,

$\partial_{r}F(x,y,r,s)=\frac{(s-1)xy(sx+3sy+x+y)}{4(x+2y)(rx+2sy)^{2}}.$

Hence the result follows readily since for $(x,y,r,s)\in\mathbb{R}^{4}_{+}$,
$\partial_{r}F(x,y,r,s)>0$, respectively $\partial_{r}F(x,y,r,s)<0$, if $s>1$,
respectively $s<1$.

$\Box$

###### Proposition 4.4

${\cal L}_{0,\rm SC},{\cal L}_{0,\rm
M}:\bm{H}^{s}\times\bm{H}^{s-1}\to\bm{H}^{s}\times\bm{H}^{s-1}$ are normal and
invertible operators. Moreover, the eigenvalues of ${\cal L}_{0,\rm SC}$ and
${\cal L}_{0,\rm M}$ are $\\{\pm\sqrt{\rho}\\}$ and $\\{1\pm\sqrt{1-\rho}\\}$
respectively where $\rho>1$ if $\mu_{+}\neq\mu_{-}$ and $1$ otherwise.

Proof. It follows from equation (4.22) that ${\cal L}_{0,\rm SC}$ is
invertible and its eigenvalues are $\pm\sqrt{\rho}$. The last claim can be
established by considering the matrix operator

${\cal Q}=\begin{bmatrix}&\varepsilon i\bm{\Lambda}\\\
\varepsilon^{-1}i\bm{\Lambda}^{-1}&\end{bmatrix}\quad\text{where}\
\varepsilon=\left(-\frac{\beta_{+}+\beta-}{\delta_{+}+\delta_{-}}\right)^{1/2}>0$

it can be seen that

${\cal Q}^{-1}={\cal Q},\quad{\cal Q}{\cal L}_{0,\rm SC}{\cal Q}=-{\cal
L}_{0,\rm SC}.$

Thus, if $({\bm{\lambda}},\bm{g})$ is an eigenvector of ${\cal L}_{0,\rm SC}$
associated with the eigenvalue $\pm\sqrt{\rho}$, then ${\cal
Q}({\bm{\lambda}},\bm{g})^{\top}$ is also an eigenvector of ${\cal L}_{0,\rm
SC}$ associated with the eigenvalue $\mp\sqrt{\rho}$.

Regarding ${\cal L}_{0,\rm M}$, we see first, using (4.22),

${\cal L}^{\top}_{0,\rm M}{\cal L}_{0,\rm M}=\big{(}I-({\cal C}_{-,0}-{\cal
C}_{+,0})\big{)}\big{(}I+({\cal C}_{-,0}-{\cal C}_{+,0})\big{)}={\cal
L}^{2}_{0,\rm SC}=\rho^{2}I$

from where the invertibility follows readily. Furthermore, from

$({\cal C}_{-,0}-{\cal C}_{+,0})^{2}=2{\cal C}_{-,0}^{2}+2{\cal
C}_{+,0}^{2}-({\cal C}_{-,0}+{\cal C}_{+,0})^{2}=(1-\rho)I$ (4.24)

we conclude that the eigenvalues of ${\cal L}_{0,\rm M}$ are
$1\pm\sqrt{1-\rho}$. Using a similar argument to the one applied above
establishes the fact that both signs are actually achieved in the formula for
the eigenvalues of the operator ${\cal L}_{0,\rm M}$. $\Box$

Next theorem extends this result to ${\cal L}_{\rm SC}$ and ${\cal L}_{\rm
KR}$

###### Theorem 4.5

The operators ${\cal L}_{\rm SC},{\cal L}_{\rm
KR}:\bm{H}^{s}\times\bm{H}^{s-1}\to\bm{H}^{s}\times\bm{H}^{s-1}$ are
invertible.

Furthermore, the eigenvalues of ${\cal L}_{\rm SC}$ and ${\cal L}_{\rm KR}$
are clustered around $\pm\sqrt{\rho}$ and $1\pm\sqrt{1-\rho}$ respectively.

Proof. From the Proposition 4.4 we conclude that ${\cal L}_{\rm SC}$ and
${\cal L}_{\rm KR}$ are compact perturbations of invertible operators. Hence,
according to the Fredholm theory, their invertibility follows once we
establish their injectivity, which, in turn, will be proved using the same
ideas as for Helmholtz transmission problems.

To this end, take first $(\bm{g},\bm{\varphi})\in\mathop{\rm Ker}{\cal L}_{\rm
SC}$ and define the fields in $\mathbb{R}^{2}\setminus\Gamma$

${\bf
u}_{\pm}=\pm\operatorname{DL}_{\pm}\bm{g}\mp\operatorname{SL}_{\pm}\bm{\varphi}.$

Clearly $({\bf u}_{+}|_{\Omega_{+}},{{\bf u}_{-}}|_{\Omega_{-}})$ are
solutions of the original transmission problem with ${\bf u}^{\rm inc}={0}$.
Therefore ${\bf u}_{+}=0$ in $\Omega_{+}$ and ${\bf u}_{-}=0$ in $\Omega_{-}$.
Since

$\left(\frac{1}{2}{\cal I}-{\cal C}_{+}\right)\begin{bmatrix}\bm{g}\\\
\bm{\varphi}\end{bmatrix}=\begin{bmatrix}\bm{g}\\\
\bm{\varphi}\end{bmatrix}=\left(\frac{1}{2}{\cal I}+{\cal
C}_{-}\right)\begin{bmatrix}\bm{g}\\\ \bm{\varphi}\end{bmatrix}$

we conclude that (${\bf u}_{-}|_{\Omega_{+}},{\bf u}_{+}|_{\Omega_{-}})$ solve
the adjoint problem (4.21), again with ${\bf u}^{\rm inc}=0$. Then by
hypothesis, ${\bf u}_{-}|_{\Omega_{+}}$ and ${\bf u}_{-}|_{\Omega_{+}}$ vanish
too. Hence ${\bf u}^{\pm}=0$ and by the jump properties for the potentials, we
have also $(\bm{g},\bm{\varphi})={\bf 0}$.

Consider now $(\bm{g},\bm{\varphi})\in\mathop{\rm Ker}{\cal L}_{\rm KR}$ and
define in this case

$\displaystyle{\bf
v}^{\pm}=\operatorname{DL}_{\mp}\bm{g}-\operatorname{SL}_{\mp}\bm{\varphi}\Big{|}_{\Omega_{\pm}}.$

Then, ${({\bf v}_{-},{\bf v}_{+})}$ solves the adjoint problem (4.21), as
above with ${\bf u}^{\rm inc}=0$. Thus ${{\bf v}_{\pm}}$ vanish in
$\Omega_{\pm}$. Hence

$\left(\frac{1}{2}{\cal I}+{\cal C}_{-}\right)\begin{bmatrix}\bm{g}\\\
\bm{\varphi}\end{bmatrix}=\left(-\frac{1}{2}I+{\cal
C}_{-}\right)\begin{bmatrix}\bm{g}\\\ \bm{\varphi}\end{bmatrix}={\bf 0}$

which implies $(\bm{g},\bm{\varphi})\in{\rm Ker}({\cal L}_{\rm SC})$ and
therefore, $(\bm{g},\bm{\varphi})={\bf 0}$.

Regarding the eigenvalues distribution, let first $(\lambda_{n})_{n}$ be the
eigenvalues for ${\cal L}_{\rm SC}$. Then, $\\{\lambda^{2}_{n}\\}_{n}$ are
eigenvalues for ${\cal L}^{2}_{\rm SC}={\cal L}_{0,\rm SC}^{2}+{\cal K}=\rho
I+{\cal K}$. Since ${\cal K}$ when acting from $\bm{H}^{s}\times\bm{H}^{s-1}$
into itself is compact, $\rho-\lambda^{2}_{n}$ can only accumulate at zero,
from where this result follows.

The proof for ${\cal L}_{\rm KR}$ follows similarly. Note, as in (4.24),

$({\cal C}_{-}-{\cal C}_{+})^{2}={\cal I}-({\cal C}_{-}+{\cal
C}_{+})^{2}={\cal I}-{\cal L}^{2}_{\rm SC}.$

Hence, if $\\{\mu_{n}\\}_{n}$ are the eigenvalues for ${\cal L}_{\rm KR}$,
$\\{(\mu_{n}-1)^{2}\\}_{n}$ are eigenvalues for ${\cal I}-{\cal L}^{2}_{\rm
SC}$ which can only accumulate at $1-\rho$. Then, $\\{\mu_{n}\\}_{n}$ are
clustered around $1\pm\sqrt{1-\rho}$.

$\Box$

We are ready to analyze the CFIER formulation (4.17) and (4.18) under suitable
assumptions on the regularizing operator ${\cal R}$. Specifically, we assume
that the regularizing operator ${\cal R}$ satisfies

##### Assumption 1

We assume ${\cal R}=\begin{bmatrix}R_{11}&R_{12}\\\
R_{21}&R_{22}\end{bmatrix}$ satisfies the following properties:

  1. (a)

There exists ${\cal
K}:\bm{H}^{s+1}\times\bm{H}^{s}\to\bm{H}^{s+1+\ell}\times\bm{H}^{s+\ell}$ with
$\ell\geq 1$ so that

$\frac{1}{2}{\cal I}-{\cal C}_{-}+({\cal C}_{+}+{\cal C}_{-}){\cal R}={\cal
A}+{\cal K},$ (4.25)

with ${\cal A}:{\bm{H}^{s+1}\times\bm{H}^{s}\to\bm{H}^{s+1}\times\bm{H}^{s}}$
invertible with the eigenvalues clustered around a few accumulation points.

  2. (b)

We have

$\displaystyle\Im\big{(}\langle\bm{g},\overline{\bm{\varphi}}\rangle-\langle\bm{g},\overline{{R_{22}\bm{\varphi}}}\rangle-\langle{R_{12}\bm{\varphi}},\overline{\bm{\varphi}}\rangle-\langle{\bm{g}},\overline{R_{21}\bm{g}}\rangle-\langle
R_{11}\bm{g},\overline{\bm{\varphi}}\rangle\big{)}<0$ (4.26)
$\displaystyle\hskip 142.26378pt\forall{\bf
0}\neq\big{(}\bm{g},\bm{\varphi}\big{)}\in{\bm{H}}^{1/2}\times{\bm{H}}^{-1/2}.$

We establish

###### Theorem 4.6

Under the assumptions (a)–(b) stated above, for the following operators

$\displaystyle{\cal L}_{\rm dir}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}{\cal I}+{\cal C}_{-}-{\cal R}^{\top}({\cal
C}_{+}+{\cal C}_{-}),\quad{\cal L}_{\rm ind}:=\frac{1}{2}{\cal I}-{\cal
C}_{-}+({\cal C}_{+}+{\cal C}_{-}){\cal R}.$ (4.27)

there exists ${\cal
K}:\bm{H}^{s+1}\times\bm{H}^{s}\to\bm{H}^{s+3}\times\bm{H}^{s+2}$ so that
${\cal L}_{\rm ind}={\cal A}+{\cal K}$, ${\cal L}_{\rm dir}={\cal
A}^{\top}+{\cal K}^{\top}$ and both operators ${\cal L}_{\rm dir}$ and ${\cal
L}_{\rm ind}$ are invertible.

Proof. Clearly, (4.25) implies the decomposition stated in the statement of
the theorem. In particular, both operators ${\cal L}_{\rm dir}$ and ${\cal
L}_{\rm ind}$ are also Fredholm, and therefore we just have to prove the
invertibility.

Assume then that

${\cal L}_{\rm ind}\begin{bmatrix}\bm{g}\\\ \bm{\varphi}\end{bmatrix}={\bf
0},$

and define by (4.19) $({\bf u}_{+},{\bf u}_{-})$ in all
$\mathbb{R}^{2}\setminus\Gamma$. Proceeding as in the proof of Proposition of
4.5 we can infer that

$\displaystyle{\bf v}_{+}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}-\operatorname{DL}_{-}&\operatorname{SL}_{-}\end{bmatrix}({\cal
I}-{\cal R})\begin{bmatrix}\bm{g}\\\ \bm{\varphi}\end{bmatrix}$
$\displaystyle{\bf v}_{-}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\operatorname{DL}_{+}&-\operatorname{SL}_{+}\end{bmatrix}{\cal
R}\begin{bmatrix}\bm{g}\\\ \bm{\varphi}\end{bmatrix}$

which, by unicity of solution for the transmission problem, ${\bf
v}_{+}|_{\Omega_{-}}={\bf 0}={\bf v}_{-}|_{\Omega_{+}}$. Then

$\begin{bmatrix}\gamma_{+}{\bf v}_{+}\\\ T_{+}{\bf v}_{+}\end{bmatrix}=({\cal
I}-{\cal R})\begin{bmatrix}\bm{g}\\\
\bm{\varphi}\end{bmatrix},\quad\begin{bmatrix}\gamma_{-}{\bf v}_{-}\\\
T_{-}{\bf v}_{-}\end{bmatrix}=-{\cal R}\begin{bmatrix}\bm{g}\\\
\bm{\varphi}\end{bmatrix}.$

Then, by Lemma 3.4,

$\langle\gamma_{+}{\bf v}_{+},\overline{T_{+}{\bf
v}_{+}}\rangle-\langle\gamma_{-}{\bf v}_{-},\overline{T_{-}{\bf
v}_{-}}\rangle=\langle\bm{g},\overline{\bm{\varphi}}\rangle-\langle\bm{g},\overline{{R_{22}\bm{\varphi}}}\rangle-\langle{R_{12}\bm{\varphi}},\overline{\bm{\varphi}}\rangle-\langle{\bm{g}},\overline{R_{21}\bm{g}}\rangle-\langle
R_{11}\bm{g},\overline{\bm{\varphi}}\rangle$ (4.28)

The result follows then by assumption (b). $\Box$

Obviously, the optimal choice of the regularizing operator

${\cal R}=({\cal C}_{+}+{\cal C}_{-})^{-1}\Big{(}\frac{1}{2}{\cal I}+{\cal
C}_{-}\Big{)}$

which makes

${\cal L}_{\rm dir}={\cal L}_{\rm ind}={\cal I}.$

However, although the optimal operator ${\cal R}$ defined above satisfies
condition (a) in Assumption 1, it fails to satisfy condition (b). Furthermore,
the definition of the regularizing operator involves operator inversion. We
bypass these issues by resorting to the principal part calculus. Indeed, for a
wavenumber $\kappa$ with $\Im\kappa<0$ we consider the principal part
operators

${\cal
C}^{\kappa}_{\pm}=\begin{bmatrix}\alpha_{\pm}\bm{H}&-\beta_{\pm}\bm{\Lambda}_{\kappa}\\\
\delta_{\pm}\bm{\Lambda}_{\kappa}^{-1}&-\alpha_{\pm}\bm{H}\\\
\end{bmatrix},\quad\alpha=\frac{i\mu}{2(\lambda+2\mu)},\quad\beta=\frac{\lambda+3\mu}{4\mu(\lambda+2\mu)},\quad\delta=-\frac{\mu(\lambda+\mu)}{\lambda+2\mu}$

where the Fourier multiplier operator $\bm{\Lambda}_{\kappa}$ was defined in
equation (3.18). Using the result established in equation (4.22)

$\left({\cal C}^{\kappa}_{+}+{\cal C}^{\kappa}_{-}\right)^{2}=\rho{\cal
I},\quad\text{that is, }\left({\cal C}^{\kappa}_{+}+{\cal
C}^{\kappa}_{-}\right)^{-1}=\frac{1}{\rho}\left({\cal C}^{\kappa}_{+}+{\cal
C}^{\kappa}_{-}\right)$

we define instead the regularizing operators

${\cal R}_{\kappa}=\frac{1}{\rho}\left({\cal C}^{\kappa}_{+}+{\cal
C}^{\kappa}_{-}\right)\left(\frac{1}{2}{\cal I}+{\cal
C}_{-}^{\kappa}\right)=\begin{bmatrix}R_{11}^{\kappa}&R_{12}^{\kappa}\\\
R_{21}^{\kappa}&R_{22}^{\kappa}\end{bmatrix}.$ (4.29)

The choice of regularizing operators presented in equation (4.29) still
renders the operators

$\displaystyle{\cal L}^{\kappa}_{\rm dir}$ $\displaystyle:=$
$\displaystyle\left(\frac{1}{2}{\cal I}+{\cal C}_{-}\right)-{\cal
R}_{\kappa}^{\top}({\cal C}_{+}+{\cal C}_{-})$ (4.30) $\displaystyle{\cal
L}^{\kappa}_{\rm ind}$ $\displaystyle:=$ $\displaystyle\left(\frac{1}{2}{\cal
I}-{\cal C}_{-}\right)+({\cal C}_{+}+{\cal C}_{-}){\cal R}_{\kappa}$ (4.31)

compact perturbations of the identity. In what follows we will refer to these
formulations as Direct Regularized Combined Field Integral Equations (DCFIER)
and respectively Indirect Regularized Combined Field Integral Equations
(ICFIER). Indeed, with ${\cal R}_{\kappa}$ selected as in equation (4.29)
above we have

${\cal L}^{\kappa}_{\rm dir}={\cal I}+{\cal K}$

with ${\cal K}:\bm{H}^{s+1}\times\bm{H}^{s}\to\bm{H}^{s+3}\times\bm{H}^{s+2}$
so that (a) in Assumption 1 is satisfied with $\ell=2$. A careful computation
delivers the explicit form of the four entries of the regularizing operator
${\cal R}$ where

$\displaystyle R_{11}^{\kappa}$ $\displaystyle=$
$\displaystyle+\frac{1}{2\rho}(\alpha_{+}+\alpha_{-})\bm{H}-\frac{1}{\rho}(\alpha_{-}(\alpha_{+}+\alpha_{-})+\delta_{-}(\beta_{+}+\beta_{-}))\bm{I}$
$\displaystyle R_{12}^{\kappa}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\rho}(\beta_{+}+\beta_{-})\bm{\Lambda}_{\kappa}-\frac{1}{\rho}(\beta_{-}(\alpha_{+}+\alpha_{-})-\alpha_{-}(\beta_{+}+\beta_{-}))\bm{\Lambda}_{\kappa}\bm{H}$
$\displaystyle=$
$\displaystyle-\frac{1}{2\rho}(\beta_{+}+\beta_{-})\bm{\Lambda}_{\kappa}-\frac{1}{\rho}(\beta_{-}\alpha_{+}-\alpha_{-}\beta_{+})\bm{\Lambda}_{\kappa}\bm{H}$
$\displaystyle=$
$\displaystyle-\frac{\beta_{+}+\beta_{-}}{\rho}\bm{\Lambda}_{\kappa}\left(\frac{1}{2}I+\frac{\alpha_{+}\beta_{-}-\alpha_{-}\beta_{+}}{\beta_{+}+\beta_{-}}\bm{H}\right)$
$\displaystyle R_{21}^{\kappa}$ $\displaystyle=$
$\displaystyle\frac{1}{2\rho}(\delta_{+}+\delta_{-})\bm{\Lambda}_{\kappa}^{-1}+\frac{1}{\rho}(\alpha_{-}(\delta_{+}+\delta_{-})-\delta_{-}(\alpha_{+}+\alpha_{-}))\bm{\Lambda}_{\kappa}^{-1}\bm{H}$
$\displaystyle=$
$\displaystyle\frac{\delta_{+}+\delta_{-}}{\rho}\bm{\Lambda}_{\kappa}^{-1}\left(\frac{1}{2}I-\frac{\alpha_{+}\delta_{-}-\alpha_{-}\delta_{+}}{\delta_{+}+\delta_{-}}\bm{H}\right)$
$\displaystyle R_{22}^{\kappa}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\rho}(\alpha_{+}+\alpha_{-})\bm{H}-\frac{1}{\rho}(\alpha_{-}(\alpha_{+}+\alpha_{-})+\beta_{-}(\delta_{+}+\delta_{-}))\bm{I}.$

###### Lemma 4.7

There exists $c_{1},c_{2}>0$ so that for any $(\bm{\varphi},\
\bm{g})\in\bm{H}^{1/2}\times\bm{H}^{-1/2}$ it holds

$-\Re\langle R^{\kappa}_{12}\bm{\varphi},\overline{\bm{\varphi}}\rangle\geq
c_{1}\|\bm{\varphi}\|^{2}_{\bm{H}^{-1/2}}\quad-\Re\langle
R^{\kappa}_{21}\bm{g},\overline{\bm{g}}\rangle\geq
c_{1}\|\bm{g}\|^{2}_{\bm{H}^{1/2}}.$

and

$-\Im\langle R^{\kappa}_{12}\bm{\varphi},\overline{\bm{\varphi}}\rangle\geq
c_{1}\|\bm{\varphi}\|^{2}_{\bm{H}^{-3/2}}\quad\Im\langle
R^{\kappa}_{21}\bm{g},\overline{\bm{g}}\rangle\geq
c_{1}\|\bm{g}\|^{2}_{\bm{H}^{-1/2}}.$

Proof. The proof it is very similar to that of Proposition 3.7. Notice that we
can take

$\bm{\Lambda}_{\kappa}=\bm{\Theta}^{2}_{r}+i\bm{\Theta}^{2}_{i},\qquad{\bm{\Lambda}^{-1}_{\kappa}}=\bm{\Omega}^{2}_{r}-i\bm{\Omega}^{2}_{i}$

where $\bm{\Theta}_{r}:\bm{H}^{s}\to\bm{H}^{s-1/2}$,
$\bm{\Theta}_{i}:\bm{H}^{s}\to\bm{H}^{s+1/2}$,
$\bm{\Omega}_{r}:\bm{H}^{s}\to\bm{H}^{s+1/2}$,
$\bm{\Omega}_{i}:\bm{H}^{s}\to\bm{H}^{s+3/2}$ are invertible Fourier
multiplier operators that commute with $\bm{H}$. Given that $\bm{\Theta}_{r}$
and $\bm{\Theta}_{i}$ commute with $\bm{H}$ we have

$\displaystyle\langle
R^{\kappa}_{12}\bm{\varphi},\overline{\bm{\varphi}}\rangle$ $\displaystyle=$
$\displaystyle-\frac{\beta_{+}+\beta_{-}}{\rho}\left(\left\langle\left(\tfrac{1}{2}\bm{I}-c_{12}i\bm{H}\right)\bm{\Theta}_{r}\bm{\varphi},\overline{\bm{\Theta}_{r}\bm{\varphi}}\right\rangle+i\left\langle\left(\tfrac{1}{2}\bm{I}-c_{12}i\bm{H}\right)\bm{\Theta}_{i}\bm{\varphi},\overline{\bm{\Theta}_{i}\bm{\varphi}}\right\rangle\right)$
$\displaystyle\langle
R^{\kappa}_{21}\bm{\varphi},\overline{\bm{\varphi}}\rangle$ $\displaystyle=$
$\displaystyle\frac{\delta_{+}+\delta_{-}}{\rho}\left(\left\langle\left(\tfrac{1}{2}\bm{I}+c_{21}i\bm{H}\right)\bm{\Omega}_{r}\bm{\varphi},\overline{\bm{\Omega}_{r}\bm{\varphi}}\right\rangle-i\left\langle\left(\tfrac{1}{2}\bm{I}+c_{21}i\bm{H}\right)\bm{\Omega}_{i}\bm{\varphi},\overline{\bm{\Omega}_{i}\bm{\varphi}}\right\rangle\right)$

where

$c_{12}:=\frac{\alpha_{+}\beta_{-}-\alpha_{-}\beta_{+}}{\beta_{+}+\beta_{-}}i,\quad
c_{21}:=\frac{\alpha_{+}\delta_{-}-\alpha_{-}\delta_{+}}{\delta_{+}+\delta_{-}}i.$

Since $-i\alpha_{\pm}\in(0,1/2)$ and $\beta_{\pm}>0>\delta_{\pm}$,
$c_{12},c_{21}\in(-1/2,1/2)$. The result is therefore a consequence of Lemma
3.5. $\Box$

Finally, we need an additional result

###### Lemma 4.8

For any $(\bm{g},\bm{\varphi})\in\bm{H}^{0}\times\bm{H}^{0}$ it holds

$\langle\bm{g},\overline{\bm{\varphi}}\rangle-\langle\bm{g},\overline{R_{22}^{\kappa}{\bm{\varphi}}}\rangle-\langle
R_{11}^{\kappa}\bm{g},\overline{\bm{\varphi}}\rangle=0.$

Proof. Note that since $\alpha_{\pm}\in i{\mathbb{R}}$

$\displaystyle\langle{R_{11}^{\kappa}\bm{g}},\overline{\bm{\varphi}}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2\rho}(\alpha_{+}+\alpha_{-})\langle\bm{H}\bm{g},\overline{\bm{\varphi}}\rangle-\frac{1}{\rho}(\alpha_{-}(\alpha_{+}+\alpha_{-})+\delta_{-}(\beta_{+}+\beta_{-}))\langle\bm{g},\overline{\bm{\varphi}}\rangle$
$\displaystyle\langle\bm{g},\overline{R_{22}^{\kappa}{\bm{\varphi}}}\rangle$
$\displaystyle=$
$\displaystyle-\frac{1}{2\rho}(\alpha_{+}+\alpha_{-})\langle\bm{H}\bm{g},\overline{\bm{\varphi}}\rangle-\frac{1}{\rho}(\alpha_{-}(\alpha_{+}+\alpha_{-})+\beta_{-}(\delta_{+}+\delta_{-}))\langle{\bm{g}},\overline{\bm{\varphi}}\rangle.$

Therefore

$\langle\bm{g},\overline{R_{22}^{\kappa}{\bm{\varphi}}}\rangle+\langle
R_{11}^{\kappa}\bm{g},\overline{\bm{\varphi}}\rangle=\langle\bm{g},\overline{\bm{\varphi}}\rangle$

because, after some calculations, one can show the following identity (cf
(3.13))

$\displaystyle
2\alpha_{-}(\alpha_{+}+\alpha_{-})+\delta_{-}(\beta_{+}+\beta_{-})+\beta_{-}(\delta_{+}+\delta_{-})$
$\displaystyle\qquad=(\alpha_{+}+\alpha_{-})^{2}+(\beta_{+}+\beta_{-})(\delta_{+}+\delta_{-})+(\underbrace{\alpha_{-}^{2}+\beta_{-}\delta_{-}-\alpha_{+}^{2}-\beta_{+}\delta_{+}}_{=0})=-\rho.$

$\Box$

We are now in the position to prove the main result on the well posedness of
the regularized BIE formulations for transmission problems

###### Theorem 4.9

With ${\cal R}_{\kappa}$ defined in (4.29) there exists ${\cal
K}:\bm{H}^{s+1}\times\bm{H}^{s+2}\to\bm{H}^{s+3}\times\bm{H}^{s+2}$ such that
the operators ${\cal L}^{\kappa}_{\rm ind}=I+{\cal K}$, ${\cal
L}^{\kappa}_{\rm dir}=I+{\cal
K}^{\top}:\bm{H}^{s+1}\times\bm{H}^{s}\to\bm{H}^{s+1}\times\bm{H}^{s}$ are
both invertible.

Proof. We have already seen that (a) in Assumption 1 is verified with ${\cal
A}={\cal I}$. On the other hand, condition (b) for
$(\bm{g},\bm{\varphi})\neq{\bf 0}$ is a consequence of Lemma 4.7, which
implies

$\Im\langle{R_{12}^{\kappa}\bm{\varphi}},\overline{\bm{\varphi}}\rangle,\quad\Im\langle{\bm{g}},\overline{R_{21}^{\kappa}\bm{g}}\rangle<0,$

and Lemma 4.8, which established

$\langle\bm{g},\overline{\bm{\varphi}}\rangle-\langle\bm{g},\overline{R_{22}^{\kappa}{\bm{\varphi}}}\rangle-\langle
R_{11}^{\kappa}\bm{g},\overline{\bm{\varphi}}\rangle=0.$

We invoke the result in Theorem 4.6 to conclude the proof. $\Box$

It is relatively straightforward to refine the regularity property of the
operators ${\cal K}$ in Theorem 4.9:

###### Corollary 4.10

Under the same hypothesis of Theorem 4.9, it holds that ${\cal
K}:\bm{H}^{s}\times\bm{H}^{s}\to\bm{H}^{s+1}\times\bm{H}^{s+1}$ and the
operators ${\cal L}^{\kappa}_{\rm ind}=I+{\cal K}$, ${\cal L}^{\kappa}_{\rm
dir}=I+{\cal
K}^{\top}:\bm{H}^{s}\times\bm{H}^{s}\to\bm{H}^{s}\times\bm{H}^{s}$ are both
invertible.

###  4.3 Optimized Schwarz methods

We finally consider Optimized Schwarz Domain Decomposition Methods [4] for the
solution of the transmission problem whereby we connect the radiative solution
${\bf u}_{+}$ of the time harmonic Navier equation in $\Omega_{+}$ to the
solution ${\bf u}_{-}$ of the time harmonic Navier equation in $\Omega$ via
certain transmission operators $\Upsilon_{\pm}:\bm{H}^{1/2}\to\bm{H}^{-1/2}$

$\displaystyle{\nabla}\cdot\bm{\sigma}_{+}({\bf u}_{+})+\omega^{2}{\bf u}_{+}$
$\displaystyle=$ $\displaystyle 0\quad{\rm in}\ \Omega_{+}$ $\displaystyle
T_{+}[{\bf u}_{+}+{\bf u}^{\rm inc}]+\Upsilon_{+}\gamma_{+}[{\bf u}_{+}+{\bf
u}^{\rm inc}]$ $\displaystyle=$ $\displaystyle T_{+}{\bf
u}_{-}+\Upsilon_{+}\gamma_{-}{\bf u}_{-}$ (4.32)

and

$\displaystyle{\nabla}\cdot\bm{\sigma}_{-}({\bf u}_{-})+\omega^{2}{\bf u}_{-}$
$\displaystyle=$ $\displaystyle 0\quad{\rm in}\ \Omega$ $\displaystyle
T_{-}{\bf u}_{-}+\Upsilon_{-}\gamma_{-}{\bf u}_{-}$ $\displaystyle=$
$\displaystyle T_{+}[{\bf u}_{+}+{\bf u}^{\rm
inc}]+\Upsilon_{-}\gamma_{+}[{\bf u}_{+}+{\bf u}^{\rm inc}].$ (4.33)

First, if the following coercivity conditions on the transmission operators
are satisfied

$\Im{\langle\Upsilon_{+}\bm{g},\overline{\bm{g}}\rangle}>0\qquad-\Im{\langle\Upsilon_{-}\bm{g},\overline{\bm{g}}\rangle}>0$
(4.34)

it can be shown that both Navier problems with generalized Robin conditions
(4.3) and (4.3) are well posed. Furthermore, we note that it suffices that the
operator $\Upsilon_{+}-\Upsilon_{-}$ be injective in order for the boundary
conditions in the DDM system (4.3) and (4.3) to be equivalent to the original
transmission conditions. Clearly, (4.34) implies

$\Im{\langle(\Upsilon_{+}-\Upsilon_{-})\bm{g},\overline{\bm{g}}\rangle}>0,\quad\forall\bm{g}\neq
0,$ (4.35)

and, as byproduct, the invertibility of the operator
$\Upsilon_{+}-\Upsilon_{-}$.

The DDM system (4.3) and (4.3) is typically recast in terms of generalized
elastodynamic Robin data on $\Gamma$

$\bm{\lambda}_{+}:=T_{+}{\bf u}_{+}+\Upsilon_{+}\gamma_{+}{\bf
u}_{+},\quad\bm{\lambda}_{-}:=T_{-}{\bf u}_{-}+\Upsilon_{-}\gamma_{-}{\bf
u}_{-}$ (4.36)

in a form that involves the Schwarz iteration operator

$\begin{bmatrix}I&-\operatorname{S}_{-}\\\
-\operatorname{S}_{+}&I\end{bmatrix}\begin{bmatrix}\bm{\lambda}_{+}\\\
\bm{\lambda}_{-}\end{bmatrix}=\begin{bmatrix}-\left(T{\bf u}^{\rm
inc}+\Upsilon_{+}\gamma_{+}{\bf u}^{\rm inc}\right)\\\ \left(T{\bf u}^{\rm
inc}+\Upsilon_{-}\gamma_{-}{\bf u}^{\rm inc}\right)\end{bmatrix}.$ (4.37)

In equation (4.37) the operators $\operatorname{S}_{\pm}$ are the Robin-to-
Robin (RtR) operators associated to Navier equations with generalized Robin
boundary conditions whose precise definition is given below

$\displaystyle{\nabla}\cdot\bm{\sigma}_{+}({\bf u}_{+})+\omega^{2}{\bf u}_{+}$
$\displaystyle=$ $\displaystyle 0\quad{\rm in}\ \Omega_{+}$ $\displaystyle
T_{+}{\bf u}_{+}+\Upsilon_{+}\gamma_{+}{\bf u}_{+}$ $\displaystyle=$
$\displaystyle\bm{\lambda}_{+}\quad{\rm on}\ \Gamma$
$\operatorname{S}_{+}\bm{\lambda}_{+}:=\left(T_{+}{\bf
u}_{+}+\Upsilon_{-}\gamma_{+}{\bf u}_{+}\right)|_{\Gamma}.$ (4.38)

and respectively

$\displaystyle{\nabla}\cdot\bm{\sigma}_{-}({\bf u}_{-})+\omega^{2}{\bf u}_{-}$
$\displaystyle=$ $\displaystyle 0\quad{\rm in}\ \Omega$ $\displaystyle T{\bf
u}_{-}+\Upsilon_{-}\gamma_{-}{\bf u}_{-}$ $\displaystyle=$
$\displaystyle\bm{\lambda}_{-}\quad{\rm on}\ \Gamma.$
$\operatorname{S}_{-}\bm{\lambda}_{-}:=T{\bf u}_{-}+\Upsilon_{+}\gamma_{-}{\bf
u}_{-}$ (4.39)

The optimal choice of transmission operators with respect to the iterative
solution of the DDM formulation (4.37) is given by $\Upsilon_{+}=-Y_{-}$ and
$\Upsilon_{-}=-Y_{+}$ (that is the DtN operators corresponding to each domain
$\Omega_{\pm}$), a ubiquitous pattern in the case of Optimized Schwarz methods
involving two subdomains [26, 4]. Optimized Schwarz methods employ
approximations of the DtN operators in the formulation above. As such, we use
the following transmission operators

$\Upsilon_{\mp}:=-\mathrm{PS}_{\kappa}(Y^{\pm})=\pm\frac{1}{\beta_{\pm}}\bm{\Lambda}_{\kappa}^{-1}\left(\frac{1}{2}I_{2}\mp\alpha_{\pm}\bm{H}\right)=\mp\gamma_{\pm}\bm{\Lambda}_{\kappa}^{-1}\left(\frac{1}{2}I_{2}\pm\alpha_{\pm}\bm{H}\right)^{-1}.$
(4.40)

We present in what follows: (a) a simple procedures to compute in a robust
manner RtR maps, i.e. the evaluation of $\Upsilon_{\pm}$ (Theorems 4.11-4.14);
(b) a proof of the well-posedness of (4.37) (Theorem 4.15).

We start with the case of the bounded domain $\Omega_{-}$, as the situation is
somewhat simpler. First, we use the Calderón projection in the following way

$\begin{bmatrix}-\tfrac{1}{2}I-K_{-}&V_{-}\\\
W_{-}&\tfrac{1}{2}I-K^{\top}_{-}\end{bmatrix}\begin{bmatrix}\gamma{\bf
u}_{-}\\\ T{\bf u}_{-}\end{bmatrix}=\begin{bmatrix}{\bf 0}\\\ {\bf
0}\end{bmatrix}.$

Secondly, the boundary condition $T{\bf u}_{-}+\Upsilon_{-}\gamma_{-}{\bf
u}_{-}=\bm{\lambda}_{-}$ yields

$\begin{bmatrix}0&0&\\\ -\Upsilon_{-}&-I\end{bmatrix}\begin{bmatrix}\gamma{\bf
u}_{-}\\\ T{\bf u}_{-}\end{bmatrix}=-\begin{bmatrix}{\bf 0}\\\
\bm{\lambda}_{-}\end{bmatrix}.$

Adding these two matrix equations we derive the following direct BIE
formulation of elastodynamic equations with generalized Robin boundary values

${\cal A}_{-}\begin{bmatrix}\gamma{\bf u}_{-}\\\ T{\bf
u}_{-}\end{bmatrix}:=\begin{bmatrix}-\frac{1}{2}I-K_{-}&V_{-}\\\
-\Upsilon_{-}+W_{-}&-\frac{1}{2}I-K_{-}^{\top}\end{bmatrix}\begin{bmatrix}\gamma{\bf
u}_{-}\\\ T{\bf u}_{-}\end{bmatrix}=-\begin{bmatrix}0\\\
\bm{\lambda}_{-}\end{bmatrix}.$ (4.41)

Notice that ${\cal
A}_{-}:\bm{H}^{s}\times\bm{H}^{s+1}\to\bm{H}^{s}\times\bm{H}^{s+1}$. For the
following results, recall the duality product $[\cdot,\cdot]$ cf. (3.21) which
makes possible the realisation
$(\bm{H}^{s}\times\bm{H}^{t})^{\prime}=\bm{H}^{-t}\times\bm{H}^{-s}$.

###### Theorem 4.11

The matrix operator
$\mathcal{A}_{-}:\bm{H}^{s}\times\bm{H}^{s-1}\to\bm{H}^{s}\times\bm{H}^{s-1}$
is a compact perturbation of

$\mathrm{PS}({\cal
A}_{-})=\begin{bmatrix}-\frac{1}{2}I-\alpha_{-}\bm{H}&\beta_{-}\bm{\Lambda}\\\
-\Upsilon_{-}+\gamma_{-}\bm{\Lambda}^{-1}&-\frac{1}{2}I-\alpha_{-}\bm{H}&\\\
\end{bmatrix}$

which is a coercive operator: There exists $c>0$ such that

$\Re[\mathrm{PS}({\cal
A}_{-})(\bm{g},\bm{\psi}),\overline{(\bm{g},\bm{\psi})}]\geq
c\left[\|\bm{g}\|_{\bm{H}^{1/2}}+\|\bm{\varphi}\|_{\bm{H}^{-1/2}}\right].$

Furthermore, the operator $\mathcal{A}_{-}$ is invertible.

Proof. Clearly, ${\cal A}_{-}-\mathrm{PS}({\cal
A}_{-}):\bm{H}^{s}\times\bm{H}^{s-1}\to\bm{H}^{s+2}\times\bm{H}^{s+1}$ is
continuous. Notice also

$\displaystyle[\mathrm{PS}({\cal
A}_{-})(\bm{g},{\bm{\varphi}}),\overline{(\bm{g},{\bm{\varphi}})}]$
$\displaystyle=$
$\displaystyle-\langle\tfrac{1}{2}\bm{g}+\alpha_{-}\bm{H}\bm{g},\overline{\bm{\varphi}}\rangle+\beta_{-}\langle\bm{\Lambda}\bm{\varphi},\overline{\bm{\varphi}}\rangle$
$\displaystyle-\gamma_{-}\langle\bm{\Lambda}^{-1}\bm{g},\overline{\bm{g}}\rangle-\langle\mathrm{PS}_{\kappa}(Y_{+})\bm{g},\overline{\bm{g}}\rangle+\langle\tfrac{1}{2}\bm{\varphi}+\alpha_{-}\bm{H}\bm{\varphi},\overline{\bm{g}}\rangle$

and so

$\Re[\mathrm{PS}({\cal
A}_{-})(\bm{g},{\bm{\varphi}}),\overline{(\bm{g},{\bm{\varphi}})}]=\beta_{-}\Re\langle\bm{\Lambda}\bm{\varphi},\overline{\bm{\varphi}}\rangle-\gamma_{-}\Re\langle\bm{\Lambda}^{-1}\bm{g},\overline{\bm{g}}\rangle-\Re\langle\mathrm{PS}_{\kappa}(Y_{+})\bm{g},\overline{\bm{g}}\rangle.$

On account of relation (3.14) established in Lemma 3.5 (recall that
$\beta_{-}>0>\delta_{-}$) and the result in Proposition 3.7 we conclude the
proof of the first result in the theorem.

Following the Fredholm paradigm, since $\mathrm{PS}({\cal
A}_{-}):\bm{H}^{s}\times\bm{H}^{s-1}\to\bm{H}^{s}\times\bm{H}^{s-1}$ is
invertible we just have to show that
$\mathcal{A}_{-}:\bm{H}^{s}\times\bm{H}^{s-1}\to\bm{H}^{s}\times\bm{H}^{s-1}$
is injective. Let $(\bm{g}_{0},{\bm{\varphi}}_{0})\in\mathop{\rm
Ker}(\mathcal{A}_{-})$ and define the field

${\bf
w}({\bm{x}}):=(\operatorname{DL}_{-}\bm{g}_{0})({\bm{x}})-(\operatorname{SL}_{-}{\bm{\varphi}}_{0})({\bm{x}}),\quad{\bm{x}}\in\mathbb{R}^{2}\setminus\Gamma.$

Clearly ${\bf w}|_{\Omega_{+}}$ is a radiative solution of the Navier equation
in $\Omega_{+}$ with zero Dirichlet boundary conditions on $\Gamma$, and
therefore ${\bf w}$ vanishes in $\Omega_{+}$. Also, ${\bf w}|_{\Omega_{-}}$ is
a solution of the Navier equation in $\Omega_{-}$ whose Cauchy boundary data
on $\Gamma$ is

$\gamma_{-}{\bf w}=-\bm{g}_{0},\qquad T_{-}{\bf w}=-{\bm{\varphi}}_{0}.$

However, we derive from its definition that

$T_{-}{\bf
w}=W_{-}\bm{g}_{0}+\frac{1}{2}{\bm{\varphi}}_{0}-K_{-}^{\top}{\bm{\varphi}}_{0}$

and thus we obtain

$W_{-}\bm{g}_{0}=\frac{1}{2}{\bm{\varphi}}_{0}+K_{-}^{\top}{\bm{\varphi}}_{0}\qquad{\rm
on}\ \Gamma.$

Given that $(\bm{g}_{0},{\bm{\varphi}}_{0})\in\mathop{\rm
Ker}(\mathcal{A}_{-})$, we also have that

$W_{-}\bm{g}_{0}-\Upsilon_{-}\bm{g}_{0}=\frac{1}{2}{\bm{\varphi}}_{0}+K_{-}^{\top}{\bm{\varphi}}_{0}\qquad{\rm
on}\ \Gamma.$

We derive from the last two relations that $\Upsilon_{-}\bm{g}_{0}=0$, which,
in the light of the coercivity properties established in Proposition 3.7
implies that $\bm{g}_{0}=0$. This, in turn, implies that the Cauchy data of
${\bf w}_{-}$ vanishes on $\Gamma$, and thus ${\bf w}_{-}=0$ in $\Omega_{-}$.
Consequently, $\bm{\varphi}_{0}=0$, which completes the proof. $\Box$

For the exterior Robin-to-Robin problem we can derive a similar formulation as
in (4.41) proceeding in the same way, namely

${\cal A}_{+}\begin{bmatrix}\gamma{\bf u}_{-}\\\ T{\bf
u}_{-}\end{bmatrix}:=\begin{bmatrix}\frac{1}{2}I-K_{+}&V_{+}\\\
\Upsilon_{+}+W_{+}&\frac{1}{2}I-K_{+}^{\top}\end{bmatrix}\begin{bmatrix}\gamma{\bf
u}_{-}\\\ T{\bf u}_{-}\end{bmatrix}=\begin{bmatrix}0\\\
\bm{\lambda}_{+}\end{bmatrix}.$ (4.42)

###### Theorem 4.12

The matrix operator
$\mathcal{A}_{+}:\bm{H}^{s}\times\bm{H}^{s-1}\to\bm{H}^{s}\times\bm{H}^{s-1}$
is a compact perturbation of

$\mathrm{PS}({\cal
A}_{+})=\begin{bmatrix}\frac{1}{2}I-\alpha_{+}\bm{H}&\beta_{+}\bm{\Lambda}\\\
\Upsilon_{+}+\gamma_{-}\bm{\Lambda}^{-1}&\frac{1}{2}I-\alpha_{+}\bm{H}&\\\
\end{bmatrix}$

which is a coercive operator: There exists $c>0$ such that.

$\Re[\mathrm{PS}({\cal
A}_{+})(\bm{g},\bm{\psi}),\overline{(\bm{g},\bm{\psi})}]\geq
c\left[\|\bm{g}\|_{\bm{H}^{1/2}}+\|\bm{\varphi}\|_{\bm{H}^{-1/2}}\right],\qquad(\bm{g},\bm{\psi})^{\top}\neq
0.$

Furthermore, if $\omega^{2}$ is not an eigenvalue of the Dirichlet Navier
operator with material parameters $(\mu_{+},\lambda_{+})$ in the domain
$\Omega_{-}$, the operator $\mathcal{A}_{+}$ is invertible.

Proof. The first part of the theorem follows along the same lines as that in
Theorem 4.11. Let ${({\bm{g}_{0}},{\bm{\varphi}_{0}})}\in\mathop{\rm
Ker}(\mathcal{A}_{+})$ and define the field

${\bf
w}({\bm{x}}):=-(\operatorname{DL}_{+}\bm{g}_{0})({\bm{x}})+(\operatorname{SL}_{+}\bm{\varphi}_{0})({\bm{x}}),\quad{\bm{x}}\in\mathbb{R}^{2}\setminus\Gamma.$

It follows immediately that $\gamma_{-}{\bf w}=0$ and $T_{+}{\bf
w}=\Upsilon_{+}\bm{g}_{0}$. Given the jump relations of Navier layer
potentials it follows that $\gamma_{+}{\bf w}=-\bm{g}_{0}$ and thus

$T_{+}{\bf w}+\Upsilon_{+}\gamma_{+}{\bf w}=0\qquad{\rm on}\ \Gamma.$

Now, ${{\bf w}|_{\Omega_{+}}}$ is a radiative solution of Navier equations in
$\Omega_{+}$, and given that
$\Im{\langle\Upsilon_{+}{\bm{\varphi}},{\bm{\varphi}}\rangle}>0$, it follows
that ${{\bf w}|_{\Omega_{+}}}=0$ in $\Omega_{+}$, and hence $\bm{g}_{0}=0$.
Under the assumption in the theorem, we also have that ${{\bf
w}|_{\Omega_{-}}}=0$, and thus ${\bm{\varphi}_{0}=T_{-}[{\bf
w}|_{\Omega_{-}}]}=0$ as well.

$\Box$

It is possible to derive BIE formulations to solve exterior Navier equations
with generalized Robin boundary conditions that are uniquely solvable for all
frequencies $\omega$. Indeed, let

$\widetilde{V}=\beta_{-}\bm{\Lambda}_{\kappa}\left(\frac{1}{2}I-\alpha_{-}\bm{H}\right),$
(4.43)

and consider alternative formulations obtained by adding the regularized
boundary condition $\varepsilon\widetilde{V}(T{\bf
u}_{+})+\varepsilon\widetilde{V}\Upsilon_{+}{\bf
u}_{+}=\varepsilon\widetilde{V}\bm{\lambda}_{+}$ to the second equation in
formulation (4.42)

$\mathcal{A}_{+}^{\varepsilon}:=\begin{bmatrix}\frac{1}{2}I-K_{+}+\varepsilon\widetilde{V}\Upsilon_{+}&V_{+}+\varepsilon\widetilde{V}\\\
\Upsilon_{+}+W_{+}&\frac{1}{2}I-K_{+}^{\top}\end{bmatrix}\begin{bmatrix}\gamma{\bf
u}_{-}\\\ T{\bf
u}_{-}\end{bmatrix},\qquad\mathcal{A}_{+}^{\varepsilon}\begin{bmatrix}\gamma{\bf
u}_{-}\\\ T{\bf
u}_{-}\end{bmatrix}=\begin{bmatrix}\varepsilon\widetilde{V}{\bm{\lambda}_{+}}\\\
\bm{\lambda}_{+}\end{bmatrix}.$ (4.44)

We establish

###### Theorem 4.13

The matrix operator
$\mathcal{A}_{+}^{\varepsilon}:\bm{H}^{s}\times\bm{H}^{s-1}\to\bm{H}^{s}\times\bm{H}^{s-1}$
in the left hand side of equation (4.44) is a compact perturbation of a
coercive operator in the space $\bm{H}^{1/2}\times\bm{H}^{-1/2}$ provided
$\varepsilon>0$ is small enough. Furthermore, under the same assumption, the
operator $\mathcal{A}_{+}^{\varepsilon}$ is invertible.

Proof. Clearly

$\mathrm{PS}({\cal A}_{+}^{\varepsilon})=\mathrm{PS}({\cal
A}_{+})+{\varepsilon}\begin{bmatrix}\widetilde{V}\Upsilon_{+}&\widetilde{V}\\\
0&0\end{bmatrix}$

is coervice, in the sense we have stated in previous results, for
$\varepsilon$ small enough since so is $\mathrm{PS}({\cal A}_{+})$.

Now let $(\bm{g}_{0},\bm{\varphi}_{0})\in\mathop{\rm
Ker}(\mathcal{A}_{+}^{\varepsilon})$ and define the field

${\bf
w}({\bm{x}}):=-(\operatorname{DL}_{+}\bm{g}_{0})({\bm{x}})+(\operatorname{SL}_{+}\bm{\varphi}_{0})({\bm{x}}),\quad{\bm{x}}\in\mathbb{R}^{2}\setminus\Gamma.$

It follows immediately that $\gamma_{-}{\bf
w}=-\varepsilon\widetilde{V}(\Upsilon_{+}\bm{g}_{0}+\bm{\varphi}_{0})$ and
$T_{-}{\bf w}_{-}=\Upsilon_{+}\bm{g}_{0}+\bm{\varphi}_{0}$. Therefore ${\bf
w}|_{\Omega_{-}}$ is a solution of Navier equations in $\Omega_{-}$ with
generalized Robin boundary conditions

$T_{-}{\bf w}+(\varepsilon\widetilde{V})^{-1}\gamma_{-}{\bf w}=0,$

and therefore

$\langle T_{-}{\bf w},\overline{\gamma{\bf
w}}_{-}\rangle+\langle(\varepsilon\widetilde{V})^{-1}{\gamma_{-}{\bf
w}},\overline{\gamma_{-}{\bf w}}\rangle=0$

By Lemma 3.4, the first term is a pure imaginary number, whereas the second
term can only vanish if $\gamma_{-}{\bf w}=0$. Therefore,

$\bm{\varphi}_{0}=-\Upsilon_{+}\bm{g}_{0}.$

The jump conditions of Navier layer potentials yield $\gamma_{+}{\bf
w}=-\bm{g}_{0}$ and $T_{+}{\bf w}=-\bm{\varphi}_{0}=\Upsilon_{+}\bm{g}_{0}$.
Consequently, ${\bf w}|_{\Omega_{+}}$ is a radiative solution of Navier
equations in $\Omega_{+}$ with generalized Robin boundary conditions

$T_{+}{\bf w}+\Upsilon_{+}\gamma_{+}{\bf w}=0.$

Thus, from Lemma 3.4

$0=\Im\langle{T}_{+}{\bf w},\overline{\gamma_{+}{\bf
w}}\rangle+\Im{\langle\Upsilon_{+}\gamma_{+}{\bf w},\overline{\gamma_{+}{\bf
w}}\rangle}\geq-\Im{\langle{\rm PS}_{\kappa}(Y_{-})\gamma_{+}{\bf
w},\overline{\gamma_{+}{\bf w}}\rangle}$

from where we infer, Proposition 3.7, that $\gamma_{+}{\bf w}=0$ and thus
${\bm{\varphi}}_{0}=\bm{g}_{0}=0$. $\Box$

Alternatively, instead of working with a full system of BIEs , it is also
possible to construct a single robust regularized BIE formulations for the
solution of exterior Navier equations with generalized Robin boundary
conditions. Defining first the operator

$\operatorname{R}^{\rm os}:=({\rm PS}_{\kappa}(Y_{+})-{\rm
PS}_{\kappa}(Y_{-}))^{-1}$

we look for a radiative solution of the Navier equation in $\Omega_{+}$ of the
form

${\bf u}_{+}({\bm{x}}):=(\operatorname{DL}_{+}[\operatorname{R}^{\rm
os}{\bm{\varphi}]})({\bm{x}})-(\operatorname{SL}_{+}[{\rm
PS}_{\kappa}(Y_{+})\operatorname{R}^{\rm
os}{\bm{\varphi}}])({\bm{x}}),\quad{\bm{x}}\in\Omega_{+}.$ (4.45)

We have

$\displaystyle\gamma{\bf u}_{+}$ $\displaystyle=$
$\displaystyle\left[\left(\frac{1}{2}+K_{+}-V_{+}{\rm
PS}_{\kappa}(Y_{+})\right)\operatorname{R}^{\rm os}\right]{\bm{\varphi}}$
$\displaystyle T_{+}{\bf u}_{+}$ $\displaystyle=$
$\displaystyle\left[\left(W_{+}+\frac{1}{2}{\rm
PS}_{\kappa}(Y_{+})-K^{\top}_{+}{\rm
PS}_{\kappa}(Y_{+})\right)\operatorname{R}^{\rm os}\right]{\bm{\varphi}}.$

It follows than that ${\bf u}_{+}$ satisfies the boundary condition $T{\bf
u}_{+}+\Upsilon_{+}{\bf u}_{+}=T{\bf u}_{+}-{\rm PS}_{\kappa}(Y_{-}){\bf
u}_{+}=\bm{\lambda}_{+}$ if and only if ${\bm{\varphi}}$ satisfies the BIE

$\mathcal{B}_{+}{\bm{\varphi}}=\bm{\lambda}_{+},\quad\mathcal{B}_{+}:=\left[\left(W_{+}+\frac{1}{2}{\rm
PS}_{\kappa}(Y_{+})-K^{\top}_{+}{\rm PS}_{\kappa}(Y_{+})\right)-{\rm
PS}_{\kappa}(Y_{-})\left(\frac{1}{2}+K_{+}-V_{+}{\rm
PS}_{\kappa}(Y_{+})\right)\right]\operatorname{R}^{\rm os}.$ (4.46)

###### Theorem 4.14

The operator $\mathcal{B}_{+}:\bm{H}^{s}\to\bm{H}^{s}$ is a compact
perturbation of the identity. Furthermore, the operator $\mathcal{B}_{+}$ is
invertible.

Proof. Notice that

${\cal R}^{-1}=-\left(\frac{\beta_{+}+\beta_{-}}{2\beta_{+}\beta_{-}}{\cal
I}-\frac{\alpha_{+}\beta_{-}-\alpha_{-}\beta_{+}}{\beta_{-}\beta_{+}}\bm{H}\right)\bm{\Lambda}^{-1}_{\kappa},$
(4.47)

that

${\cal B}_{+}={\cal B}_{+,0}+{\cal K}$

with ${\cal K}:\bm{H}^{s}\to\bm{H}^{s+2}$

$\displaystyle{\cal B}_{+,0}$ $\displaystyle=$
$\displaystyle\bigg{[}\delta_{+}\Lambda_{\kappa}^{-1}-\left(\tfrac{1}{2}I-\alpha_{+}\bm{H}\right)\frac{1}{\beta_{+}}\bm{\Lambda}_{\kappa}^{-1}\left(\tfrac{1}{2}I-\alpha_{+}\bm{H}\right)$
$\displaystyle-\frac{1}{\beta_{-}}\bm{\Lambda}_{\kappa}^{-1}(\tfrac{1}{2}I+\alpha_{-}\bm{H})\left(\tfrac{1}{2}I+\alpha_{+}\bm{H}-\beta_{+}\bm{\Lambda}_{\kappa}\left(-\tfrac{1}{\beta_{+}}\right)\bm{\Lambda}_{\kappa}^{-1}\left(\tfrac{1}{2}I-\alpha_{+}\bm{H}\right)\right)\bigg{]}{\cal
R}$ $\displaystyle=$
$\displaystyle\Big{[}\left(\delta_{+}-\tfrac{1}{\beta_{+}}\left(\tfrac{1}{4}-\alpha_{+}^{2}\right)-\tfrac{1}{2\beta_{-}}\right)I+\left(\tfrac{\alpha_{+}}{\beta_{+}}-\tfrac{\alpha_{-}}{\beta_{-}}\right)\bm{H}\Big{]}\bm{\Lambda}_{\kappa}^{-1}\bigg{]}{\cal
R}$

But since

$\delta_{+}-\frac{1}{\beta_{+}}\left(\frac{1}{4}-\alpha_{+}^{2}\right)=-\frac{1}{2\beta_{+}}$

we can conclude

${\cal
B}_{+,0}=-\Big{[}\left(\frac{\beta_{+}+\beta_{-}}{2\beta_{-}\beta_{+}}\right)I-\left(\frac{\alpha_{+}\beta_{-}-\alpha_{-}\beta_{+}}{\beta_{+}\beta_{-}}\right)\bm{H}\Big{]}\bm{\Lambda}_{\kappa}^{-1}{\cal
R}=I$

in view of (4.47). Clearly, the invertibility of the operator
$\mathcal{B}_{+}$ is then equivalent to its injectivity. Let
${\bm{\varphi}}_{0}\in\mathop{\rm Ker}(\mathcal{B}_{+})$ and define the field

${\bf w}({\bm{x}}):=(\operatorname{DL}_{+}[\operatorname{R}^{\rm
os}{\bm{\varphi}_{0}]})({\bm{x}})-(\operatorname{SL}_{+}[{\rm
PS}_{\kappa}(Y_{+})\operatorname{R}^{\rm
os}{\bm{\varphi}_{0}}])({\bm{x}}),\quad{\bm{x}}\in\mathbb{R}^{2}\setminus\Gamma.$
(4.48)

It follows that ${\bf w}|_{\Omega_{+}}$ is a radiative solution of the Navier
equations in $\Omega_{+}$ with generalized Robin conditions $T_{+}{\bf
w}+\Upsilon_{+}\gamma_{+}{\bf w}=0$ on $\Gamma$, and thus ${\bf
w}|_{\Omega_{+}}=0$ in $\Omega_{+}$. Given the jump properties of Navier layer
potentials, we obtain

$\gamma_{-}{\bf w}=-\operatorname{R}^{\rm os}{\bm{\varphi}_{0}}\qquad
T_{-}{\bf w}=-{\rm PS}_{\kappa}(Y_{+})\operatorname{R}^{\rm
os}{\bm{\varphi}_{0}}=\Upsilon_{-}\operatorname{R}^{\rm
os}{\bm{\varphi}_{0}}.$

We have that ${\bf w}|_{\Omega_{-}}$ is a solution of the Navier equation with
material parameters $(\mu_{+},\lambda_{+})$ in $\Omega$ that satisfies the
generalized Robin boundary condition on $\Gamma$

$T_{-}{\bf w}+\Upsilon_{-}\gamma_{-}{\bf w}_{-}=0.$

We derive then that ${\bf w}|_{\Omega_{-}}=0$ in $\Omega$ which in turn
implies that $\operatorname{R}^{\rm os}{\bm{\varphi}_{0}}=0$ on $\Gamma$.
Given that the operator $\operatorname{R}^{\rm os}$ is invertible, we get that
${\bm{\varphi}_{0}}=0$, which was to be proved. $\Box$

Having show some different formulations to solve (4.38) and (4.39) (and
therefore, to compute $\operatorname{S}_{\pm}$), we are now in the position to
prove the main result concerning Schwarz iteration operators:

###### Theorem 4.15

The RtR operators $\operatorname{S}_{\pm}:\bm{H}^{s}\to\bm{H}^{s+2}$ are
continuous, and so compact when viewed acting in $\bm{H}^{s}$ into itself.
Furthermore, the operator
$I-\operatorname{S}_{-}\operatorname{S}_{+}:\bm{H}^{s}\to\bm{H}^{s}$ is
invertible with continuous inverse. As a consequence, the underlying Schwarz
iteration operator in the left hand side of equation (4.37) is a invertible
compact perturbation of the identity in $\bm{H}^{s}\times\bm{H}^{s}$.

Proof. Notice that $I-\operatorname{S}_{-}\operatorname{S}_{+}$ is the Schur
complement of the matrix operator in the left hand side of equation (4.37).
Hence, the last result stated in the theorem is a consequence of the previous
statements. Let us recall that in view of (4.40) we have

$\operatorname{S}_{\pm}{\bf f}=T_{\pm}{\bf
u}_{\pm}-{\Upsilon}_{\pm}\gamma_{\pm}{\bf u}_{\pm},$

where ${\bf u}_{\pm}$ are the solutions of (4.38) and respectively (4.39). The
following estimate basically follows from the definition of the operators
involved

$\|T_{\pm}{\bf u}_{\pm}-\Upsilon_{\pm}\gamma_{\pm}{\bf
u}_{\pm}\|_{\bm{H}^{s+1}}\leq C\|\gamma_{\pm}{\bf u}_{\pm}\|_{\bm{H}^{s}}$

for some $C>0$ independent of ${\bf u}_{\pm}$. Besides, from (4.41) and (4.42)
(or (4.44)) we derive the estimate

$\|\gamma_{\pm}{\bf u}_{\pm}\|_{\bm{H}^{s}}\leq
C^{\prime}\|\bm{\lambda}_{\pm}\|_{\bm{H}^{s-1}}$

with $C^{\prime}>0$ independent of $\bm{g}$. Gathering these properties we can
conclude that $\operatorname{S}_{\pm}:\bm{H}^{s}\to\bm{H}^{s+2}$ is
continuous.

In order to establish the invertibility of the operator
$I-\operatorname{S}_{-}\operatorname{S}_{+}$ it suffices to prove its
injectivity. Assume that ${\bm{\varphi}}$ such as
${\bm{\varphi}}=\operatorname{S}_{-}\operatorname{S}_{+}{\bm{\varphi}}$ and
solve the Navier problem in the exterior domain $\Omega_{+}$

$\displaystyle{\nabla}\cdot\bm{\sigma}_{+}({\bf w}_{+})+\omega^{2}{\bf w}_{+}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle\quad{\rm in}\ \Omega_{+},$
(4.49) $\displaystyle T_{+}{\bf w}_{+}+\Upsilon_{+}\gamma_{+}{\bf w}_{+}$
$\displaystyle=$ $\displaystyle{\bm{\varphi}}$ $\displaystyle\quad{\rm on}\
\Gamma$

where ${\bf w}_{+}$ is radiative at infinity, so that

$\operatorname{S}_{+}{\bm{\varphi}}=\left(T_{+}{\bf
w}_{+}+\Upsilon_{-}\gamma_{+}{\bf w}\right)|_{\Gamma}.$

We also consider the Navier problem in the bounded domain $\Omega$

$\displaystyle{\nabla}\cdot\bm{\sigma}_{-}({\bf w}_{-})+\omega^{2}{\bf w}_{-}$
$\displaystyle=$ $\displaystyle 0\quad{\rm in}\ \Omega_{-}$ $\displaystyle
T_{-}{\bf w}_{-}+\Upsilon_{-}\gamma_{-}{\bf w}$ $\displaystyle=$
$\displaystyle\operatorname{S}_{+}{\bm{\varphi}}\quad{\rm on}\ \Gamma$

so that

$\operatorname{S}_{-}\operatorname{S}_{+}{\bm{\varphi}}=\left(T_{-}{\bf
w}_{-}+\Upsilon_{+}\gamma_{-}{\bf w}\right)|_{\Gamma}.$ (4.50)

Given that
${\bm{\varphi}}=\operatorname{S}_{-}\operatorname{S}_{+}{\bm{\varphi}}$ we
obtain

$T_{+}{\bf w}_{+}+\Upsilon_{+}\gamma_{+}{\bf w}_{+}=T_{-}{\bf
w}_{-}+\Upsilon_{+}\gamma_{-}{\bf w}_{-}.$ (4.51)

Furthermore, the boundary condition of ${\mathbf{w}_{-}}$ on $\Gamma$ leads to

$T_{+}{\bf w}_{+}+\Upsilon_{-}\gamma_{+}{\bf w}=T_{-}{\bf
w}_{-}+\Upsilon_{-}\gamma_{-}{\bf w}.$ (4.52)

We note that it follows from equations (4.51) and (4.52) that

$(\Upsilon_{+}-\Upsilon_{-})[\gamma_{+}{\bf w}_{+}-\gamma_{-}{\bf w}_{-}]=0.$

which since the operator $\Upsilon_{+}-\Upsilon_{-}$ is injective, in turn
implies

$\gamma_{+}{\bf w}_{+}=\gamma_{-}{\bf w}_{-}$

and then, from (4.51) ,

$T_{+}{\bf w}_{+}=T_{-}{\bf w}_{-}$

which implies

$\Im\langle T_{+}{\bf w}_{+},\overline{\gamma{\bf w}}_{+}\rangle=\Im\langle
T_{-}{\bf w}_{-},\overline{\gamma_{-}{\bf w}}_{-}\rangle=0$

and hence, by Lemma 3.4, $\gamma_{+}{\bf w}_{+}=0=T_{+}{\bf w}_{+}$ which
yields, via (4.49), that $\bm{\varphi}={\bf 0}$ $\Box$

Having presented BIE formulations for the various elastodynamic scattering and
transmission problems considered in this paper, we turn next to describing
Nyström discretizations of those BIEs.

##  5 Nyström discretizations

We use Nyström discretizations for the numerical solution of the various
elastodynamic BIE that rely on global trigonometric interpolation with $2n$
nodes

$t_{j}=\frac{j\pi}{n},\ j=0,1,\ldots,2n-1$

onto the space of trigonometric polynomials

$T_{n}=\left\\{\varphi(t)=\sum_{m=0}^{n}a_{m}\cos{mt}+\sum_{m=1}^{n-1}b_{m}\sin{mt}\
:{\ a_{m},b_{m}\in\mathbb{C}}\right\\}.$

Our BIE formulations works with two types of operators: (a) BIO and (b)
Fourier multipliers. The latter are discretized in a straightforward manner
using trigonometric interpolation and FFTs. With regards to (a), the majority
of the kernels of the BIOs discussed in this paper exhibit a logarithmic
singularity which is resolved via the Kusmaul-Martensen quadrature [24, 25]

$\frac{1}{2\pi}\int_{0}^{2\pi}H({\tau},{t})\log\left(4\sin^{2}\frac{\tau-t}{2}\right)\varphi({t})\,{\rm
d}t\approx\sum_{m=0}^{2n-1}R_{m}({\tau})H({\tau},{t}_{m})\varphi({t}_{m})$

where

$R_{m}({\tau}):=-\frac{1}{n}\sum_{j=1}^{n-1}\frac{1}{j}\cos{j({\tau}-t_{m})}-\frac{1}{2n^{2}}\cos{n({\tau}-t_{m})},\
0\leq m\leq 2n-1.$

This turns to be equivalent to consider the approximation for the underlying
integral operator given by

$\frac{1}{2\pi}\int_{0}^{2\pi}\log\left(4\sin^{2}\frac{{t}-{\tau}}{2}\right)P_{n}(H({\tau},{\cdot})\varphi)(t)\,{\rm
d}t$

$P_{n}:C[0,2\pi]\to T_{n}$ appearing above is the trigonometric interpolation
operator, since this integral can be computed analytically.

In the case of smooth integrands in the definition of BIE, we use the
trapezoidal rule for their discretizations. The hypersingular part of the
operator $W$, on the other hand, is treated via the Kress quadrature [20] for
the evaluation of $2\pi$ periodic Hilbert transforms

$\frac{1}{2\pi}{\mathrm{p.v.}\,}\int_{0}^{2\pi}\cot{\frac{{t-\tau}}{2}}\varphi^{\prime}({t}){\rm
d}t\approx\sum_{m=0}^{2n-1}T_{m}({\tau})\varphi(t_{m})$

where

$T_{m}({\tau}):=-\frac{1}{n}\sum_{j=1}^{n-1}j\cos{j({\tau}-t_{m})}-\frac{1}{2}\cos{n({\tau}-t_{m})},\
0\leq m\leq 2n-1.$

Finally, the discretization of the Cauchy principal value integral that enters
the definition of the principal part of the double layer operators $K$ and
$K^{\top}$, requires more care. Indeed, as discussed in [21], the use of
trigonometric interpolation in the evaluation of Cauchy principal value
integrals

$\frac{1}{2\pi}{\mathrm{p.v.}\,}\int_{0}^{2\pi}\cot{\frac{{t}-{\tau}}{2}}\varphi(t){\rm
d}t\approx\frac{1}{2\pi}{\mathrm{p.v.}\,}\int_{0}^{2\pi}\cot{\frac{{t}-{\tau}}{2}}[P_{n}\varphi](t){\rm
d}t$

leads to

$\frac{1}{2\pi}{\mathrm{p.v.}\,}\int_{0}^{2\pi}\cot{\frac{{t}-{\tau}}{2}}\varphi({t}){\rm
d}t\approx\sum_{m=0}^{2n-1}Q_{m}(t)\varphi(t_{m})$

with

$Q_{m}(\tau):=\frac{1}{n}\sum_{j=1}^{n-1}\sin{j(\tau-
t_{m})}+\frac{1}{2n}\sin{n(\tau-t_{m})},\ 0\leq m\leq 2n-1.$

However, cf. (3.10),

$\frac{1}{2\pi}{\mathrm{p.v.}\,}\int_{0}^{2\pi}\cot{\frac{{t}-{\tau}}{2}}\cos
nt\,{\rm d}t=\sin n\tau$

and thus the Nyström discretization of the Cauchy principal value operator is
not a bijection from $P_{n}$ to itself. For these reasons we use the half grid
size shifted quadrature method [21]

$\frac{1}{2\pi}{\mathrm{p.v.}\,}\int_{0}^{2\pi}\cot{\frac{t-\tau}{2}}\varphi({t}){\rm
d}t\approx\frac{\pi}{n}\sum_{m=0}^{2n-1}\cot\frac{t_{m+\frac{1}{2}}-\tau}{2}\varphi\left(t_{m+\frac{1}{2}}\right),\quad
t_{m+\frac{1}{2}}:=t_{m}+\frac{\pi}{2n}.$

The evaluation of the density function at the shifted grid points
$t_{m+\frac{1}{2}}$ is readily achieved via trigonometric interpolation. Given
that the Nyström discretizations of the principal parts of the elastodynamic
operators are available in the literature, the main difficulty of the overall
collocation schemes resides in the logarithmic splitting of various kernels.
We present in what follows numerical results concerning the various
formulations considered in this paper.

###  5.1 Numerical results

In this section we present various results concerning the accuracy of
elastodynamic BIE solvers based on Nyström discretizations as well as the
rates of convergence of iterative solvers (e.g. GMRES [27]) for the solution
of the linear systems ensuing from the various BIE formulations considered in
this text. With regards to the accuracy of our discretizations, we present
results concerning far field data. For a scattered elastic field ${\bf u}$ we
can define the associated longitudinal wave ${\bf u}_{p}$ and the transversal
wave ${\bf u}_{s}$ (recall [23] or [1, Ch. 2]) as

${\bf u}_{p}:=-\frac{1}{k_{p}^{2}}\nabla\ \nabla\cdot{\bf u}\qquad{\bf
u}_{s}={\bf u}-{\bf u}_{p}.$

The Kupradze radiation conditions [23] take on the form

${\bf
u}_{p}({\bm{x}})=\frac{e^{ik_{p}|{\bm{x}}|}}{\sqrt{|{\bm{x}}|}}\left({\bf
u}_{p,\infty}(\hat{{{\bm{x}}}})+\mathcal{O}\left(\frac{1}{|{\bm{x}}|}\right)\right)\qquad{\bf
u}_{s}({\bm{x}})=\frac{e^{ik_{s}|{\bm{x}}|}}{\sqrt{|{\bm{x}}|}}\left({\bf
u}_{s,\infty}(\hat{{{\bm{x}}}})+\mathcal{O}\left(\frac{1}{|{\bm{x}}|}\right)\right)$

as $|{\bm{x}}|\to\infty$ where $\hat{{{\bm{x}}}}={\bm{x}}/|{\bm{x}}|$. We
present in this section the maximum errors $\varepsilon_{\infty}$ achieved
when computing the quantities ${\bf u}_{p,\infty}$ and ${\bf u}_{s,\infty}$
evaluated at fine enough meshes on the unit circle $|\hat{{{\bm{x}}}}|=1$. We
consider two types of incident fields: (a) elastodynamic point sources (in the
method of manufactured solutions); and (b) plane waves. In case (b) the far
field errors are computed with respect to reference solutions produced through
very fine discretizations of the BIE. We also report the size $n$ of the $2n$
dimensional trigonometric polynomial space $T_{n}$. Therefore, in the
impenetrable case, the size of the linear systems Nyström discretization
linear systems of the various (vectorial) BIE is $(4n)\times(4n)$; in the
penetrable case, the size increases to $(8n)\times(8n)$. We start with an
illustration of the accuracy that can be achieved in the far field when the
four elastodynamic BIOs are employed in the solution of elastodynamic BIE.

The domains considered in this numerical section are: (i) the unit circle
centered at origin; (ii) the starfish obstacle [18] whose paramaterization is
given by

${\bm{x}}(t)=\left(1+\tfrac{1}{4}\sin{5t}\right)(\cos{t},\sin{t}),\ 0\leq
t<2\pi;$

(iii) the cavity-like geometry given by

$\displaystyle{\bm{x}}(t)$ $\displaystyle=$
$\displaystyle(x_{1}(t),x_{2}(t)),\qquad\ 0\leq t<2\pi;$ $\displaystyle
x_{1}(t)$ $\displaystyle=$ $\displaystyle\tfrac{2}{5}(\cos(t)+2\cos(2t))$
$\displaystyle x_{2}(t)$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\sin(t)+\tfrac{1}{2}\sin(2t)+\tfrac{1}{4}\sin(3t)+\tfrac{1}{48}(4\sin(t)-7\sin(2t)+6\sin(3t)-2\sin(4t)),$

We depict in Figure 1 a sketch of these domains. Notice that the diameters of
all of the scatterers considered in this paper are equal to 2.

###  5.2 The method of manufactured solutions

The method of manufactured solutions is a reliable test of the accuracy of PDE
solvers. In the Dirichlet case, we consider the time-harmonic Navier equation
with boundary value data produced by a point source ${\bm{x}}_{0}$ placed
inside the scatter $\Omega_{-}$

${\bf
f}_{\Gamma}({\bm{x}})=\Phi({\bm{x}},{\bm{x}}_{0}),\quad{\bm{x}}\in\Gamma,\
{\bm{x}}_{0}\in\Omega_{-}.$

In this case, the radiative solution of the Dirichlet scattering problem in
the exterior domain ${\Omega_{+}}$ is the point source itself, that is ${\bf
u}({\bm{x}})=\Phi({\bm{x}},{\bm{x}}_{0})$ for all ${\bm{x}}\in{\Omega_{+}}$.
We look for the solution ${\bf u}$ in the form of either a single or double
layer potential

${\bf u}({\bm{x}})=(\operatorname{SL}\bm{\varphi})({\bm{x}})\quad{\rm
or}\quad{\bf u}({\bm{x}})=(\operatorname{DL}\bm{g})({\bm{x}}),\
{\bm{x}}\in{\Omega_{+}}$

and we solve the corresponding BIEs (with, as usual, ${\bf f}={\bf
f}_{\Gamma}\circ{\bf x}$)

$V\bm{\varphi}={\bf f}\quad{\rm and}\quad\frac{1}{2}\bm{g}+K\bm{g}={\bf f}$

Assuming that the frequency $\omega$ is selected such as both BIE formulations
above are uniquely solvable, we use Nyström discretizations to solve
numerically these BIEs and we compare in the far-field the computed solutions
with the exact point source solution. We repeat the experiments in the case of
Neumann boundary conditions, that is we choose a boundary data in the form

$\bm{\lambda}_{\Gamma}({\bm{x}})=T_{\Gamma}\Phi({\bm{x}},{\bm{x}}_{0}),\quad{\bm{x}}\in\Gamma,\
{\bm{x}}_{0}\in\Omega_{-}$

and we solve the BIEs

$-\frac{1}{2}\bm{\varphi}+K^{\top}\bm{\varphi}=\bm{\lambda}\quad{\rm and}\quad
W\bm{g}=\bm{\lambda},\ $

Again here, we compare the numerical solutions in the far field against the
exact point source solution.

We report errors corresponding to (a) the Dirichlet single layer formulation
(under the heading $V$, given that we used the single layer $V$ BIO to solve
the time-harmonic Navier equation), (b) Dirichlet double layer formulation
(under the heading $K$), and (c) Neumann double layer formulation (under the
heading $W$). Given that the BIO $K^{\top}$ is the (real) $L^{2}$ adjoint of
the BIO $K$, its discretization produces identical levels of accuracy to those
of $K$, and therefore we chose not to present them.

$-1$$0$$1$$-1$$0$$1$

$-1$$0$$1$$-1$$0$$1$

$-1$$0$$1$$-1$$0$$1$

Figure 1:  Domains for the experiments: from left to right, the unit circle, the starfish domain, and the cavity problem. $\omega$ | $n$ | $V$ | $K$ | $W$
---|---|---|---|---
|  | $\varepsilon_{\infty}$ | $\varepsilon_{\infty}$ | $\varepsilon_{\infty}$
16 | 32 | 3.0 $\times$ $10^{-2}$ | 9.9$\times$ $10^{-2}$ | 1.3 $\times$ $10^{-1}$
16 | 64 | 7.1 $\times$ $10^{-7}$ | 1.1 $\times$ $10^{-3}$ | 1.6 $\times$ $10^{-6}$
16 | 128 | 5.5 $\times$ $10^{-14}$ | 4.2 $\times$ $10^{-13}$ | 1.6 $\times$ $10^{-9}$
32 | 64 | 3.0 $\times$ $10^{-2}$ | 2.5 $\times$ $10^{-1}$ | 3.4 $\times$ $10^{-1}$
32 | 128 | 2.2 $\times$ $10^{-7}$ | 2.4 $\times$ $10^{-4}$ | 5.0 $\times$ $10^{-8}$
32 | 256 | 1.0 $\times$ $10^{-15}$ | 8.3 $\times$ $10^{-13}$ | 5.4 $\times 10^{-12}$
Table 1: Errors in the method of manufactered solution for the smooth starfish
geometry for different values of the frequency $\omega$ and parameter values
$\lambda=1$, $\mu=1$, at various levels of discretization.

###  5.3 Iterative behavior of BIE formulations in the impenetrable case

We present in this section various numerical experiments regarding the
iterative behavior of the various BIE formulations for the solution of
elastodynamic impenetrable scattering problems. We consider incident plane
waves of the form

${\bf u}^{\rm
inc}({\bm{x}})=\frac{1}{\mu}e^{ik_{s}{\bm{x}}\cdot{\bm{d}}}({\bm{d}}\times{\bm{p}})\times{\bm{d}}+\frac{1}{\lambda+2\mu}e^{ik_{p}{\bm{x}}\cdot{\bm{d}}}({\bm{d}}\cdot{\bm{p}}){\bm{d}}$
(5.1)

where the direction ${\bm{d}}$ has unit length $|{\bm{d}}|=1$. If the vector
${\bm{p}}$ is chosen such as ${\bm{p}}=\pm{\bm{d}}$, the incident plane is a
pressure wave or P-wave. In the case when ${\bm{p}}$ is orthogonal to the
direction of propagation ${\bm{p}}$, the incident plane wave is referred to as
a shear wave or S-wave. We considered plane waves of direction
${\bm{d}}=\begin{bmatrix}0&-1\end{bmatrix}^{\top}$ in all of our numerical
experiments; in the case of S-wave incidence we selected
${\bm{p}}=\begin{bmatrix}1&0\end{bmatrix}^{\top}$. We observed that other
choices of the direction ${\bm{d}}$ and of the vector ${\bm{p}}$ lead to
qualitatively similar results.

We investigated the iterative behavior of Dirichlet integral solvers based on
three formulations: (1) the CFIE formulation (4.4) with the coupling constant
$\eta_{D}=1$; (2) the CFIE formulation with the optimal coupling constant
$\eta_{D}$ given in equation (4.5)—which we refer to by the acronym “CFIE
$\eta_{D}$ opt”; and (3) the CFIER formulation (4.4) with the choice
$\operatorname{R}^{\rm D}={\rm PS}_{\kappa}(Y_{+})$ with $\kappa=k_{s}+0.4\ i\
k_{s}^{1/3}$. We report the number of GMRES iterations required by each of
these three BIE formulations to reach relative residuals of $10^{-8}$ and
corresponding discretizations that deliver results with accuracies at the
$10^{-7}$ level in the case of smooth scatterers. As it can be seen from the
results presented in Tables 2-4, the CFIE formulation with the optimal
coupling parameter $\eta_{D}$ exhibits the best iterative behavior in the
high-frequency regime. We note that although the double layer operator $K$ is
not compact, all of the three BIE formulations considered in the numerical
experiments behave like integral equations of the second kind, that is, the
numbers of GMRES iterations required to reach a certain residual do not
increase with more refined discretizations.

$\omega$ | $n$ | # iter CFIE (4.4) $\eta_{D}=1$ | # iter CFIE $\eta_{D}$ opt | # iter CFIER $\operatorname{R}^{\rm D}={\rm PS}_{\kappa}(Y_{+})$
---|---|---|---|---
10 | 64 | 31 | 22 | 21
20 | 128 | 50 | 27 | 30
40 | 256 | 97 | 28 | 38
80 | 512 | 231 | 27 | 39
160 | 1024 | 438 | 32 | 43
Table 2: Numbers of GMRES iterations of various formulations to reach residuals of $10^{-8}$ for various BIE formulations of Dirichlet elastodynamic scattering problems at high frequencies in the case when $\Gamma$ is a unit circle. The material parameters are $\lambda=2$ and $\mu=1$, and the incidence was S-wave. The discretizations used in these numerical experiments delivered results accurate at the level of $10^{-7}$. $\omega$ | $n$ | # iter CFIE (4.4) $\eta_{D}=1$ | # iter CFIE $\eta_{D}$ opt | # iter CFIER $\operatorname{R}^{\rm D}={\rm PS}_{\kappa}(Y_{+})$
---|---|---|---|---
10 | 64 | 41 | 32 | 26
20 | 128 | 85 | 39 | 36
40 | 256 | 166 | 46 | 56
80 | 512 | 353 | 51 | 76
160 | 1024 | 782 | 57 | 100
Table 3: Numbers of GMRES iterations of various formulations to reach residuals of $10^{-8}$ for various BIE formulations of Dirichlet elastodynamic scattering problems at high frequencies in the case when $\Gamma$ is the starfish contour. The material parameters are $\lambda=2$ and $\mu=1$, and the incidence was S-wave. The discretizations used in these numerical experiments delivered results accurate at the level of $10^{-7}$. $\omega$ | $N$ | # iter CFIE (4.4) $\eta_{D}=1$ | # iter CFIE $\eta_{D}$ opt | # iter CFIER $\operatorname{R}^{\rm D}={\rm PS}_{\kappa}(Y_{+})$
---|---|---|---|---
10 | 64 | 73 | 50 | 52
20 | 128 | 137 | 80 | 89
40 | 256 | 263 | 144 | 116
80 | 512 | 511 | 185 | 235
160 | 1024 | 1008 | 320 | 406
Table 4: Numbers of GMRES iterations of various formulations to reach
residuals of $10^{-8}$ for various BIE formulations of Dirichlet elastodynamic
scattering problems at high frequencies in the case when $\Gamma$ is the
cavity contour. The material parameters are $\lambda=2$ and $\mu=1$, and the
incidence was an S-wave. The discretizations used in these numerical
experiments delivered results accurate at the level of $10^{-7}$.

####  5.3.1 Neumann boundary conditions

We present next in Tables 5 and 6 results related to the iterative behavior of
the BIE formulations considered in this paper for the solution of
elastodynamic problems with Neumann boundary conditions in the case of smooth
scatterers. Specifically, we consider (a) the classical CFIE formulation (4.7)
with the coupling parameter $\eta_{N}=1$, (b) the same formulation with the
optimal choice of the coupling constant $\eta_{N}$ given in equation (4.10),
and (b) the CFIER formulation with the regularizing operator
$\operatorname{R}^{\rm N}=({\rm PS}_{\kappa}(Y_{+}))^{-1}$. We present results
for two level of discretizations, a coarser one that delivers results within
the $10^{-6}$ range accuracy, and a finer one that produces results in the
$10^{-7}$ accuracy range. We observe that given that the BIO $W$ is a
pseudodifferential operator of order one, the CFIE formulations require more
GMRES iterations to reach the same residual for finer discretizations, whereas
the CFIER formulations appear to behave like an integral equation of the
second kind. The use of an optimized coupling constant $\eta_{N}$ is
beneficial for CFIE formulations, and the use of the regularizing operator
$\operatorname{R}^{\rm N}=({\rm PS}_{\kappa}(Y_{+}))^{-1}$ in the CFIER
formulations gives rise to savings of a factor of $1.7$ in the numbers of
GMRES iterations when finer discretizations are applied. Given the fact that
the Fourier multiplier $({\rm PS}_{\kappa}(Y_{+}))^{-1}$ can be effected
efficiently with FFTs, the savings in computational times are of the same
magnitude.

$\omega$ | $n$ | # iter CFIE (4.7) $\eta_{N}=1$ | # iter CFIE $\eta_{N}$ opt | # iter CFIER $\operatorname{R}^{\rm N}=({\rm PS}_{\kappa}(Y_{+}))^{-1}$
---|---|---|---|---
10 | 64/128 | 86/114 | 49/63 | 29/29