Content uploaded by Carlos Pérez-Arancibia
Author content
All content in this area was uploaded by Carlos Pérez-Arancibia on Mar 01, 2018
Content may be subject to copyright.
Domain Decomposition Methods based on quasi-optimal
transmission operators for the solution of Helmholtz transmission
problems
Yassine Boubendir∗
, Carlos Jerez-Hanckes†
, Carlos P´erez-Arancibia‡
, Catalin Turc§
Abstract
We present non-overlapping Domain Decomposition Methods (DDM) based on quasi-optimal
transmission operators for the solution of Helmholtz transmission problems with piece-wise
constant material properties. The quasi-optimal transmission boundary conditions incorporate
readily available approximations of Dirichlet-to-Neumann operators. These approximations con-
sist of either complexified hypersingular boundary integral operators for the Helmholtz equation
or square root Fourier multipliers with complex wavenumbers. We show that under certain regu-
larity assumptions on the closed interface of material discontinuity, the DDM with quasi-optimal
transmission conditions are well-posed. We present a DDM framework based on Robin-to-Robin
(RtR) operators that can be computed robustly via boundary integral formulations. More im-
portantly, the use of quasi-optimal transmission operators results in DDM that converge in small
numbers of iterations even in the challenging high-contrast, high-frequency regime of Helmholtz
transmission problems. Furthermore, the DDM presented in this text require only minor mod-
ifications to handle the case of transmission problems in partially coated domains, while still
maintaining excellent convergence properties. We also investigate the dependence of the DDM
iterative performance on the number of subdomains.
Keywords: Helmholtz transmission problems, domain decomposition methods, partial coat-
ings.
AMS subject classifications: 65N38, 35J05, 65T40,65F08
1 Introduction
The phenomenon of electromagnetic wave scattering by bounded penetrable objects with constant
electric permittivities is relevant for numerous applications in antenna design, diffraction gratings,
to name but a few. Numerical methods based on Boundary Integral Equations (BIE) are well-suited
for simulation of these types of applications owing to the dimension reduction, explicit enforcement
of radiation conditions, and lack of numerical dispersion that they enjoy. There is a wide array of
well-conditioned BIE of the second kind for the solution of Helmholtz transmission problems, see
references [5, 6] for in-depth discussions. These formulations rely on regularization techniques that
∗Department of Mathematical Sciences, NJIT, e-mail:boubendi@njit.edu
†Institute for Mathematical and Computational Engineering, School of Engineering, Pontificia Universidad
Cat´olica de Chile, Av. Vicuna Mackenna 4860, Macul, Santiago, Chile, e-mail: cjerez@ing.puc.cl
‡Department of Mathematics, Massachusetts Institute of Technology, e-mail: cperezar@mit.edu
§Department of Mathematical Sciences, NJIT, e-mail:catalin.c.turc@njit.edu
1
incorporate readily available approximations of Dirichlet-to-Neumann (DtN) operators. Neverthe-
less, these formulations require relative large numbers of Krylov subspace iterations for convergence
in the high-contrast, high-frequency regime. This situation can be attributed to the lack of easily
computable approximations of DtN operators for bounded domains [6]. Furthermore, the regu-
larization strategy becomes cumbersome in the case of more complicated material properties, e.g.,
perfectly conducting coatings, multiple junctions. A different alternative is to resort to matrix com-
pression techniques to produce direct solvers for the solution of Helmholtz transmission problems in
the computationally challenging high-contrast, high-frequency regime [16, 18]. However, the direct
solvers proposed in [16, 18] require large memory consumption and are difficult to parallelize.
Domain Decomposition Methods (DDM) are natural candidates for the solution of scattering
problems involving composite scatterers. DDM are divide-and-conquer strategies whereby the com-
putational domain is divided into smaller subdomains for which solutions are matched via trans-
mission conditions on subdomain interfaces. The convergence of DDM for time-harmonic wave
scattering applications depends a great deal on the choice of the transmission conditions that allow
the exchange of information between adjacent subdomains. These interface transmission conditions
should ideally allow information to flow out of a subdomain with as little as possible information
being reflected back into the subdomain. Thus, the interface transmission conditions fall into the
category of Absorbing Boundary Conditions (ABC). From this perspective, the ideal choice of trans-
mission conditions on an interface between two subdomains is such that the impedance/transmission
operator is the restriction to the common interface of the DtN operator corresponding to the ad-
jacent subdomain. Traditionally, the interface transmission conditions were chosen as the classical
first order ABC outgoing Robin/impedance boundary conditions [11, 15]. The convergence of DDM
with classical Robin interface boundary conditions is slow and is adversely affected by the num-
ber of subdomains. Fortunately, the convergence of DDM can be considerably improved through
incorporation of ABC that constitute higher order approximations of DtN operators in the form
of second order approximations with optimized tangential derivative coefficients [14], square root
approximations [3], or other types of non-local transmission conditions [15, 26]. Alternatively,
so-called Perfectly Matched Layers can be used at subdomain interfaces [27].
We devote our attention to DDM for Helmholtz transmission problems that use transmission op-
erators given by approximations of DtN operators in the form of either (a) hypersingular Helmholtz
BIO, (b) square root Fourier multipliers, and (c) scalar multiples of identity operators. A funda-
mental requirement in DDM is that the subdomain Helmholtz problems with Robin/impedance
boundary conditions that incorporate the aforementioned transmission operators are well posed.
This entails the complexification of the wavenumbers in the definition of the three types of trans-
mission operators considered above; this modification ensures that the transmission operators enjoy
certain coercivity property that suffices for the well posedness of subdomain Helmholtz problems
with Robin boundary conditions. In practice, these complex wavenumbers in the definition of the
transmission operators are chosen to optimize the rate of convergence of the ensuing DDM in the
case of simple geometries (e.g circles, infinite waveguides) amenable to analytical modal analysis [3].
A judicious choice of the complex wavenumbers in the definition of transmission operators gives
rise to DDM whose rate of convergence is virtually independent of frequency; for this reason the
transmission operators described above with appropriate complex wavenumbers are referred to as
quasi-optimal transmission operators [3]. The DDM can be easily recast in operator form using
certain subdomain Robin-to-Robin (RtR) operators that map outgoing Robin data to incoming
Robin data defined in terms of transmission operators. In the case when the transmission opera-
tors correspond to the classical first order ABC, the ensuing RtR operators turn out to be unitary,
2
a key ingredient in establishing the well posedness of DDM [11, 15]. However, the transmission
operators of type (a), (b), and (c) considered above do not give rise to unitary RtR maps, and
the wellposedness of the ensuing DDM is more complicated. Nevertheless, we investigate in more
detail the RtR operators by expressing them in terms of Boundary Integral Operators (BIO), and
we are able to establish the well posedness of DDM for Helmholtz transmission problems with one
closed interface of discontinuity under various assumptions on the regularity of that interface.
The key computational ingredient in the implementation of DDM is the computation of sub-
domain RtR maps. We provide several robust representations of RtR maps in terms of BIOs and
their inverses. We present high-order discretizations of the RtR operators based on Nystr¨om dis-
cretizations of the BIO that enter in their representations. We provide ample evidence that the
DDM based on the quasi-optimal transmission operators considered in this text give rise to small
numbers of Krylov subspace iterations for Helmholtz transmission problems in the high-frequency,
high-contrast regime. Furthermore, these numbers of iterations depend very mildly on the fre-
quency, which is in stark contrast with solvers based on BIE [5]. Nevertheless, the computation
of RtR maps becomes more problematic in the high-frequency regime, where matrices of large size
need be inverted. Thus, it is customary to resort to divide the penetrable scatterer in a collection
of non-overlapping subdomains and formulate a DDM that takes these further subdivisions into
account. In particular, the quasi-optimal transmission operators need be restricted to subdomain
interfaces that are open arcs. This is slighly delicate, given that those operators are global operators,
i.e. their definition requires an integration boundary that is a closed curve. These restrictions can
be effected by localizations via smooth cut-off functions supported on subdomain interfaces [19].
This strategy allows for an extension of DDM with quasi-optimal transmission operators in the
presence of cross points between subdomains, i.e. points where three or more domains with differ-
ent material properties meet. In particular, we can use the cut-off methodology to formulate DDM
with quasi-optimal transmission operator for the case of transmission problems in partially coated
domains. Again, the DDM that incorporate quasi-optimal transmission operators perform well in
terms of numbers of iterations in the case of penetrable scatterers that are partially coated.
Although the use of quasi-optimal transmission operators, as those recounted above, accelerates
a great deal the convergence of DDM, the number of iterations required for convergence still grows
with the number of subdomains. This is not entirely surprising since the transmission operators are
chosen to optimize the local exchange of information between adjacent subdomains, and affect to a
lesser degree the global exchange of information between distant subdomains. Recent efforts have
been directed to construct “double sweep”-type preconditioners that address the latter issue [30, 32].
The resulting preconditioned DDM scale favorably with frequency and number of subdomains, but
appear to be somewhat less effective for wave propagation problems in composite media that exhibit
sharp high-contrast interfaces. The incorporation of DDM preconditioners is currently underway.
The structure of this paper is as follows. Section 2 describes the Helmholtz transmission prob-
lem. In Section 3 we review the BIOs associated with the Helmholtz equation and their mapping
properties, as well as the classical boundary integral equations of the second kind for the solution
of transmission problems. The DDM approach with three choices of transmission operators is then
introduced and analyzed in Section 4. We present in Section 5 the transmission problem in par-
tially coated obstacles as well as BIE and DDM formulations of such problems. Section 6 discusses
high-order Nystr¨om discretizations of the Robin-to-Robin maps that are central to DDM, while
a variety of numerical results are shown in Section 7. Finally, the conclusions of this work are
presented in Section 8.
3
2 Scalar transmission problems
We consider the problem of two dimensional scattering by penetrable homogeneous scatterers. Let
Ω1denote a bounded domain in R2whose boundary Γ := ∂Ω1is a closed curve, and let Ω0:= R2\Ω1.
We seek to find fields u0and u1that are solutions of the following scalar Helmholtz transmission
problem:
∆uj+k2
juj= 0 in Ωj, j = 0,1
u0+uinc =u1on Γ,
α0(∂n0u0+∂n0uinc) = −α1∂n1u1on Γ,
lim
r→∞ r1/2(∂u0/∂r −ik0u0)=0.
(2.1)
We assume that the wavenumbers kjand the quantities αjin the subdomains Ωjare positive real
numbers. The unit normal to the boundary ∂Ωjis here denoted by njand is assumed to point
to the exterior of the subdomain Ωj. The incident field uinc, on the other hand, is assumed to
satisfy the homogeneous Helmholtz equation with wavenumber k0in the unbounded domain Ω0.
Finally, we assume that the parameters αjare positive so that the transmission problem (2.1) is
well posed under the assumption that Γ is given locally by the graph of a Lipschitz function. The
well posedness remains valid in the case when Γ is more regular.
In what follows, we review two main formulations of the transmission problem (2.1). One
formulation relies on BIE, while the other is is a DDM.
3 Boundary integral equation formulations
There is a wide variety of possibilities in which equations (2.1) can be reformulated via robust BIE,
see contribution [12] for an in-depth discussion. We will present here a BIE of the second kind.
To this end, we make use of the four BIO associated with the Calder´on calculus. Let D⊂R2be
a bounded domain whose boundary ∂D = Γ is a closed curve. In what follows we will focus on
two cases: (1) Γ is a C2curve (or smoother), and (2) Γ is given locally by the graph of a Lipschitz
function. Given a wavenumber k > 0, and a density ϕ: Γ →C, we recall the definitions of the
single layer potential
[SLΓ,k(ϕ)](z) := ZΓ
Gk(z−y)ϕ(y)ds(y),z∈R2\Γ,
and the double layer potential
[DLΓ,k(ϕ)](z) := ZΓ
∂Gk(z−y)
∂n(y)ϕ(y)ds(y),z∈R2\Γ,
where Gk(x) = i
4H(1)
0(k|x|) denotes the free-space two-dimensional Green’s function of the Helmholtz
equation with wavenumber k, and ndenotes the unit normal pointing outside the domain D. Ap-
plying exterior (resp. interior) Dirichlet and Neumann traces on Γ, which are denoted by γD,ext
Γ
and γN,ext
Γ(resp. γD,int
Γand γN,int
Γ), respectively, to the single and double layer potentials, we define
the four Helmholtz BIO: single layer (SΓ,k), double layer (KΓ,k), adjoint double layer (K>
Γ,k) and
4
hypersingular (NΓ,k) operators, which satisfy
γD,ext
ΓSLΓ,k(ϕ) = γD,int
ΓSLΓ,k (ϕ) = SΓ,k ϕ, γN,ext
ΓDLΓ,k(ϕ) = γN ,int
ΓDLk(ϕ) = NΓ,k ϕ,
γN,ext
ΓSLΓ,k (ϕ) = −ϕ
2+K>
Γ,kϕ, γD,ext
ΓDLΓ,k (ϕ) = ϕ
2+KΓ,kϕ,
γN,int
ΓSLΓ,k (ϕ) = ϕ
2+K>
Γ,kϕ, γD,int
ΓDLΓ,k (ϕ) = −ϕ
2+KΓ,kϕ.
(3.1)
Next, we replace the subindex kin the definition of the layer potentials and BIO by the subindex
jof the wavenumber kjcorresponding to the Ωjsubdomain. We also denote the BIO associated
with the Laplace equation—wavenumber equal to zero—by using the subindex L.
For any D⊂R2domain with bounded boundary Γ, we denote by Hs(D) the classical Sobolev
space of order son D(cf. [23, Ch. 3] or [1, Ch. 2]). If Γ is of regularity C2, the Sobolev
spaces defined on the boundary Γ, Hs(Γ) are well defined for any s∈[−3,3]. If Γ is a Lipschitz
boundary, Hs(Γ) is well defined for any s∈[−1,1]. We recall that for any s>t,Hs(Σ) ⊂Ht(Σ),
Σ∈ {D, Γ}with compact support. Moreover, and Ht(Γ)0=H−t(Γ) when the inner product of
H0(Γ) = L2(Γ) is used as duality product. Let Γ0⊂Γ such that meas(Γ0)>0. For 0 < s ≤1/2
we define by Hs(Γ0) be the space of distributions that are restrictions to Γ0of functions in Hs(Γ).
The space e
Hs(Γ0) is defined as the closed subspace of Hs(Γ0)
e
Hs(Γ0) := {u∈Hs(Γ0) : eu∈Hs(Γ)},0< s ≤1/2,
where
eu:= (u, on Γ,
0,on Γ \Γ0.
We define then Ht(Γ0) to be the dual of e
H−t(Γ0) for −1/2≤t < 0, and e
Ht(Γ0) the dual of H−t(Γ0)
for −1/2≤t < 0.
We recount next several important results related to the mapping properties of the four BIO of
the Calder´on calculus [12]. These mapping properties depend a great deal on the regularity of Γ.
In the case when Γ is a C2closed curve we have
Theorem 3.1 Let Dbe a bounded domain in R2, with a boundary Γthat is C2. The following
mappings
•Sk:Hs(Γ) →Hs+1(Γ)
•Kk:Hs(Γ) →Hs+3(Γ)
•K>
k:Hs(Γ) →Hs+3(Γ)
•Nk:Hs+1(Γ) →Hs(Γ)
are continuous for s∈[−3,0]. Furthermore, if k16=k2we have that
•Sk1−Sk2:H0(Γ) →H3(Γ)
•Nk1−Nk2:H0(Γ) →H1(Γ).
are continuous.
5
In the case of Lipschitz boundaries Γ, we will make use of the following mapping properties [12]:
Theorem 3.2 Let Dbe a bounded domain in R2, with Lipschitz boundary Γ. The following map-
pings
•Sk:Hs(Γ) →Hs+1(Γ)
•Kk:Hs+1(Γ) →Hs+1(Γ)
•K>
k:Hs(Γ) →Hs(Γ)
•Nk:Hs+1(Γ) →Hs(Γ)
are continuous for s∈[−1,0]. Furthermore, if k16=k2we have that
•Sk1−Sk2:H−1(Γ) →H1(Γ)
•Kk1−Kk2:H0(Γ) →H1(Γ)
•K>
k1−K>
k2:H−1(Γ) →H0(Γ)
•Nk1−Nk2:H0(Γ) →H0(Γ).
are continuous and compact.
We also recount a result due to Escauriaza, Fabes and Verchota [13]. In this result, KL,K>
L
are the double and adjoint double layer operator for Laplace equation (which obviously correspond
to k= 0).
Theorem 3.3 For any Lipschitz curve Γand λ6∈ [−1/2,1/2), the mappings
λI +KL:Hs(Γ) →Hs(Γ)
are invertible for s∈[−1,1]. Furthermore, the mappings
1
2I±KL:Hs(Γ) →Hs(Γ)
are Fredholm of index zero for s∈[−1,1].
BIE formulations of the transmission problem (2.1) can be derived using layer potentials defined
on Γ: the solutions uj, j = 0,1,of the transmission problem are sought in the form:
uj(x) := SLΓ,j v+ (−1)jα−1
jDLΓ,j p, x∈Ωj,(3.2)
where vand pare densities defined on the Γ and the double layer operators are defined with respect
to exterior unit normals ncorresponding to each domain Ωj. Applying Dirichlet and Neumann
traces followed by transmission conditions, we arrive at the the following pair of integral equations:
α−1
0+α−1
1
2p−(α−1
0K0+α−1
1K1)p+ (S1−S0)v=uinc
α0+α1
2v+ (N0−N1)p+ (α0K>
0+α1K>
1)v=−α0∂n0uinc (3.3)
6
Note that the combination N0−N1occurs, this is an integral operator with a weakly-singular
kernel. In what follows we refer to the integral equations (3.3) by CFIESK. The well posedness of
the CFIESK formulation in the space (p, v)∈H0(Γ)×H0(Γ) was established in [9] in the case when
Γ is C2. The well posedness of the CFIESK formulation in the space (p, v)∈H1/2(Γ) ×H−1/2(Γ)
was established in [28] in the case when Γ is Lipschitz. There are several other possibilities to
reformulate the transmission equation (2.1) in terms of well-posed BIE [5, 12]. We chose to focus
on CFIESK formulations in this text, as these can be readily extended to more complex scenarios
such as transmission problems in piece-wise constant composite domains that feature multiple
junctions [8, 16] or transmission problems in partially coated domains—see Section 5.
4 Domain decomposition approach
DDM are natural candidates for numerical solution of transmission problems (2.1). A non-overlapping
domain decomposition approach for the solution of equations (2.1) consists of solving subdomain
problems in Ωj, j = 0,1 with matching Robin transmission boundary conditions on the common
subdomain interface Γ. Indeed, this procedure amounts to computing the subdomain solutions:
∆uj+k2
juj= 0 in Ωj,(4.1)
αj(∂njuj+δ0
j∂njuinc) + Zj(uj+δ0
juinc) = −α`(∂n`u`+δ0
`∂n`uinc) + Zj(u`+δ0
`uinc) on Γ,
where {j, `}={0,1}and δ0
jstands for the Kronecker symbol, and Zj, Z`are transmission operators
with the following mapping property Zj,` :H1/2(Γ) →H−1/2(Γ). The choice of the operators Zj, Z`
should be such that the following PDEs are well posed
∆uj+k2
juj= 0 in Ωj,
αj∂njuj+Zjuj=ψjon Γ,(4.2)
for j= 0,1, where we require in addition that u0be radiative at infinity. A sufficient condition for
the well-posedness of these problems is given by
±=ZΓ
Z1ϕ ϕds > 0 and =ZΓ
Z0ϕ ϕds < 0,for all ϕ∈H1/2(Γ),(4.3)
under the assumption that αjare positive numbers (cf. [9, Theorem 3.37]). In addition, Z0+Z1:
H1/2(Γ) →H−1/2(Γ) must be a bijective operator in order to guarantee that the solution of the
DDM system (4.1) is also a solution of the original transmission problem (2.1) (see Theorem 4.6). In
order to describe the DDM method more concisely we introduce subdomain Robin-to-Robin (RtR)
maps [15]. For each subdomain Ωj,j= 0,1, we define RtR maps Sj,j= 0,1, in the following
manner:
S0(ψ0) := (α0∂n0u0−Z1u0)|Γ,S1(ψ1) := (α1∂n1u1−Z0u1)|Γ(4.4)
where uj,j= 0,1, are solutions of equations (4.2). The DDM (4.1) can be recast in terms of
computing the global Robin data f= [f0f1]>with
fj:= (αj∂njuj+Zjuj)|Γ, j = 0,1,
7
as the solution of the following linear system that incorporates the subdomain RtR maps Sj, j = 0,1,
previously defined
(I+S)f=g, S:= 0S1
S00(4.5)
with right-hand side g= [g0g1]>wherein
g0=−(α0∂n0uinc +Z0uinc)|Γ
g1= (−α0∂n0uinc +Z1uinc)|Γ.
We note that due to its possibly large size, the DDM linear system (4.5) is typically solved in practice
via iterative methods. The behavior of iterative solvers of equations (4.5) depends a great deal on
the choice of transmission operators Zj,j= 0,1. Ideally, these transmission operators should be
chosen so that information flows out of the subdomain and no information is reflected back into the
subdomain. This can be achieved if the operator Z0is the Dirichlet-to-Neumann (DtN) operator
corresponding to the Helmholtz equation (4.2) posed in the domain Ω1and viceversa [24, 17]. Since
such DtN operators are not well defined for all wavenumbers k0and k1, and expensive to calculate
even when properly defined, easily computable approximations of DtN maps can be employed
effectively to lead to faster convergence rates of GMRES solvers for DDM algorithms [3]. For
instance, the transmission operators can be chosen in the following manner [29]:
Z0=−2α1NΓ,k1+iσ1, Z1=−2α0NΓ,k0+iσ0, σj>0.(4.6)
Given that amongst Helmholtz BIOs, hypersingular operators are more expensive to compute,
we proceed to replace the hypersingular operators in equation (4.6) by principal symbol Fourier
multiplier operators. The latter principal symbols are defined as
pN(ξ, k0+iσ0) = −1
2p|ξ|2−(k0+iσ0)2and pN(ξ, k2+iσ2) = −1
2p|ξ|2−(k1+iσ1)2,(4.7)
where the square root branches are chosen such that the imaginary parts of the principal symbols
are positive. The principal symbol Fourier multipliers are defined in the Fourier space T M (Γ) [2]
as
[P S(NΓ,kj+iσj)ϕ1]ˆ(ξ) = pN(ξ, kj+iσj) ˆϕ1(ξ) (4.8)
for a density ϕ1defined on ∂Ω1. We define accordingly
ZP S
0=−2α1P S(NΓ,k1+iσ1), Z P S
1=−2α0P S(NΓ,k0+iσ0), σj>0,(4.9)
and use the operators in equation (4.9) as transmission operators in the DDM formulation. We refer
to the ensuing DDM with transmission operators defined in (4.9) as Optimized DDM (DDMO).
We note that given that both operators Zjand ZP S
jsatisfy a G˚arding inequality for j= 0,1, it
follows that Z0+Z1as well as ZPS
0+ZP S
1also satisfy G˚arding inequalities, and thus the latter
operators are also invertible as operators from H1/2(Γ) to H−1/2(Γ). In addition, a high-frequency
approximation as kj→ ∞ of the square root expressions defined in equations (4.7) results in yet
another possible choice of transmission operators
Za
0=−iα1(k1+iσ1)I Za
1=−iα0(k0+iσ0)I, (4.10)
where Idenotes the identity operator. The transmission operators defined in equation (4.10)
were originally introduced in a DDM setting in [4]. We study in this paper the well-posedness
8
of the DDM system (4.5) with the aforementioned choices of transmission operators (4.6), (4.9),
and (4.10). To the best of our knowledge, the first proof regarding the well-posedness of DDM with
Robin transmission for Helmholtz problems condition was provided in [15] with Zj=iη, η < 0.
In that case the RtR operators turn out to be unitary, a property that plays a crucial role in the
well-posedness proof. In our case, neither of the choices presented above –i.e. equations (4.6),(4.9),
and (4.10)– leads to unitary RtR operators, and thus the proof of well-posedness of the DDM
system (4.5) should rely on different arguments. To this end, we look closer into the nature of the
RtR operators by deriving exact representations of those in terms of boundary integral operators.
4.1 Calculations of RtR operators in terms of boundary integral operators
The RtR operators S0and S1can be expressed in terms of solutions of the following Helmholtz
problems
∆uj+k2
juj= 0 in Ωj,
∂njuj+α−1
jZjuj=ϕjon Γ,
for j= 0,1, and with u0radiative at infinity, for which
S0(ϕ0) := (∂n0u0−α−1
0Z1u0)|Γand S1(ϕ1) := (∂n1u1−α−1
1Z0u1)|Γ.
It turns out that the operators S1can be computed robustly in a straightforward manner. Indeed,
we start with Green’s identity
u1=−DL1(u1|Γ) + SL1(∂n1u1|Γ),in Ω1
to which we apply the Dirichlet trace on Γ to derive another direct boundary integral equation
B1u1|Γ=S1ϕ1,on Γ where B1u1|Γ:= 1
2I+K1+α−1
1S1Z1u1|Γ.(4.11)
We establish the following result
Theorem 4.1 The operator B1defined in equation (4.11) with Z1=−2α0Nk0+iσ0is invertible
with continuous inverse in the spaces Hs(Γ) for all s∈[−3,3] in the case when Γis C2. In the
case of Lipschitz Γ, the operator B1defined in equation (4.11) is invertible with continuous inverse
in the spaces Hs(Γ) for all s∈[−1,1].
Proof. We will start by establishing the Fredholm property of B1in the case of Lipschitz Γ as
the arguments are slightly more involved in this case. From Calder´on identities we have that B1
can be expressed as
B1=1
2I+K1−2α0
α1
S1Nk0+iσ0
=1
2I+KL+α0
2α1
I−2α0
α1
K2
L+e
B1,
where in what follows the BIOs with subscript Ldenote the BIO corresponding to the Laplace
equation, and
e
B1:= (K1−KL)−2α0
α1
S1(Nk0+iσ0−NL)+2α0
α1
(S1−SL)NL.
9
Using the mapping properties recounted in Theorem 3.2 it follows immediately that the operator
e
B1:L2(Γ) →H1(Γ) and thus it is a compact operator in L2(Γ). On the other hand, we can establish
the following identity
e
B2:= 1
2I+KL+α0
2α1
I−2α0
α1
K2
L=−2α0
α11
2I+KL−α0+α1
2α0
+KL
and thus the operator e
B2is the product of an operator that is Fredholm of index zero and an
invertible operator (indeed, since α0+α1
2α0>1
2, we can apply the results in Theorem 3.3), and
hence e
B2is itself Fredholm of index zero in L2(Γ). Consequently, the operator B1is a compact
perturbation of a Fredholm operator of index zero in L2(Γ). In the case when Γ is C2, we use
the decompositions above and we take advantage of the increased regularity of the double layer
operators KLrecounted in Theorem 3.1 to establish
B1=α0+α1
2α1
I+f
B3(4.12)
where the operator e
B3:L2(Γ) →H1(Γ), and thus it is compact in L2(Γ). Hence, the operator
B1is a compact perturbation of a multiple of the identity operator in the space L2(Γ) in the case
when Γ is C2.
The conclusion of the theorem follows once we establish the injectivity of the operator B1. The
arguments are identical for both cases of boundary Γ considered. Let ψ∈Ker(B1) and let us
define
w:= DL1ψ−2α0
α1
SL1[Nk0+iσ0]ψ, in R2\Γ.
It follows that γD,ext
Γw= 0 and hence, from the uniqueness results for the exterior Dirichlet problem
(cf. [9, Theorem 3.21]), we obtain that w= 0 in Ω0. Using relations (3.1) we derive
γD,int
Γw=−ψ γN,int
Γw=−2α0
α1
Nk0+iσ0ψ.
Using Green’s identities we obtain
ZΩ1
(|∇w|2−k2
1w)dx = 2α0
α1ZΓ
(Nk0+iσ0ψ)ψ ds.
Using the fact that [5]
=ZΓ
(Nk0+iσ0ψ)ψ ds > 0, ψ 6= 0
we obtain that ψ= 0 which conclude the proof of the theorem in the space L2(Γ) = H0(Γ). Clearly,
the arguments of the proof can be repeated verbatim in the Sobolev spaces Hs(Γ) for all s∈[−1,0)
in the case when Γ is Lipschitz. The result in the remaining Sobolev spaces Hs(Γ), s ∈(0,1] follows
then from duality arguments. Similar arguments hold in the case when Γ is C2.
Once the invertibility of the operator B1is established, we immediately obtain a representation
of the corresponding RtR operator
S1=I−α−1
1(Z0+Z1)B−1
1S1.(4.13)
The result established in Theorem 4.1 remains valid in the case of impedance operators Za
1, yet
there are certain differences that we will comment on in the proof of Theorem 4.7. In the case
10
when Γ is C2, one can establish the compactness of the difference operator Nk0+iσ0−P S (Nk0+iσ0)
in the space H1(Γ) [5], and the conclusion of Theorem 4.1 remains valid in the case of impedance
operator ZP S
1. Whether the aforementioned compactness property of the difference operator holds
in the case of Lipschitz curves Γ is an open question. We note that the arguments in the proof of
Theorem 4.1 go through in the case of the exterior domain Ω0provided that k0is not an eigenvalue
of the Laplacean with Dirichlet boundary conditions in the domain Ω1. However, the well-posedness
of the formulation in Theorem 4.1 cannot be establish for all positive wavenumbers k0. This is not
altogether surprising, as we have applied only Dirichlet traces to the Green’s identities in order
to derive formulations (4.11). If we combine the application of Dirichlet and Neumann traces to
Green’s identities, the latter preconditioned on the left by suitable regularizing operators [29] we
can derive a well-posed direct boundary integral equation of the second kind for the solution of
both interior and exterior impedance boundary value Helmholtz problems. These formulations are
expressed in the form
Aj(uj|Γ)=(Sj+Sκj−2SκjK>
j)ϕj, κj=kj+iσj, σj>0,
Aj:= 1
2I−2SκjNj+α−1
jSκjZj−2α−1
jSκjK>
jZj+Kj+α−1
jSjZj.(4.14)
It is a straightforward matter [29] to show that in the case when Γ is Lipschitz one can use the
decomposition:
Aj=α−1
j(α0+α1)I+α−1
j(αj−αj+1)KL−2α−1
j(αj+ 2αj+1)K2
L+ 4α−1
jαj+1K3
L+f
Aj(4.15)
where the operators f
Aj:L2(Γ) →H1(Γ), and hence f
Aj:L2(Γ) →L2(Γ) are compact for j= 0,1,
and j+ 1 = j+ 1( mod 2). In the case when Γ is C2, the decomposition can be simplified in the
form
Aj=α−1
j(α0+α1)I+g
Areg
j,(4.16)
where the operators g
Areg
j:L2(Γ) →H1(Γ), and thus are compact in L2(Γ) for j= 0,1. In both
instances the RtR operators Sjcan be expressed as
Sj=I−α−1
j(Z0+Z1)A−1
j(Sj+Sκj−2SκjK>
j), j = 0,1.(4.17)
Another possibility to derive robust BIE formulations for the solution of impedance boundary
value problems with impedance operators Zj,ZP S
j, and Za
jwas proposed in [26]. This approach
consists of applying both Dirichlet and Neumann traces to Green’s identities in order to derive a
system of boundary integral equations whose unknowns are the Cauchy data on the boundary Γ.
Besides their simplicity, these formulations have the advantage of being well-posed for all three
choices of impedance operators above and Lipschitz Γ. We start our presentation with the case of
the bounded domain Ω1. Applying the interior Dirichlet and Neumann traces to Green’s identity
in the domain Ω1we obtain 1
2I+K1u1|Γ−S1∂n1u1|Γ= 0,
−N1u1|Γ+−1
2I+K>
1∂n1u1|Γ= 0.
Adding to the second equation above the impedance boundary condition we derive the following
system of BIE −α−1
1Z1+N1−1
2I−K>
1
−1
2I−K1S1 u1|Γ
∂n1u1|Γ=ϕ1
0.(4.18)
11
The well-posedness of the formulation (4.18) can be established by making use of the bilinear form
h(f, ϕ),(g, ψ)i:= ZΓ
fg +ZΓ
ϕψ, (f, ϕ)∈H1/2(Γ) ×H−1/2(Γ),(g, ψ)∈H−1/2(Γ) ×H1/2(Γ)
and following the same arguments presented in [26]. We thus arrive at the following result whose
proof can be obtained from a simple adaptation of the proof of Theorem 5.25 in [31]:
Theorem 4.2 The operator
C1:= −α−1
1Z1+N1−1
2I−K>
1
−1
2I−K1S1,C1:H1/2(Γ) ×H−1/2(Γ) →H−1/2(Γ) ×H1/2(Γ)
is invertible and its inverse is continous when Γis Lipschitz.
The equivalent formulation (4.18) cannot be shown to be well-posed in the case of the analogous
impedance boundary value problem in the the exterior domain Ω0, unless k0is not an eigenvalue of
the Laplacean with Dirichlet boundary conditions in Ω1. The remedy is to consider the following
system of integral equations
−α−1
0Z0+N0−1
2I−K>
0
α−1
0Sk0+iσ0Z0−1
2I−K0S0+Sk0+iσ0 u0|Γ
∂n0u0|Γ=ϕ0
Sk0+iσ0ϕ0(4.19)
whose derivation is absolutely similar to that of equations (4.18) except that we add to both sides
of the second equation in (4.18) the identity
α−1
0Sk0+iσ0Z0u0+Sk0+iσ0∂n0u0=Sk0+iσ0ϕ0.
In that case we have the following result whose proof follows from the same arguments as in the
proof of Lemma 5.29 in [31]:
Theorem 4.3 The operator
C0:= −α−1
0Z0+N0−1
2I−K>
0
α−1
0Sk0+iσ0Z0−1
2I−K0S0+Sk0+iσ0,
with the mapping property C0:H1/2(Γ) ×H−1/2(Γ) →H−1/2(Γ) ×H1/2(Γ) is invertible with
continuous inverse when Γis Lipschitz.
Again, the result established in Theorem 4.2 remains in the case of impedance operators Za
1
as well as ZP S
1. Indeed, the key ingredient in establishing that result is the coercivity of the
principal part of the operators C1, coercivity which is enjoyed by both operators Za
1and ZP S
1.
On the other hand, the result in Theorem 4.3 remains valid in the case of impedance operators
Za
0and a simple modification renders it valid in the case of impedance operators ZP S
0as well.
This modification consists of replacing the single layer operators Sk0+iσ0by Fourier multipliers
whose principal symbols are the reciprocal of pN(ξ, k0+iσ0). Clearly, the results established in
Theorem 4.2 and Theorem 4.3 remain valid in the case when Γ is C2.
Having discussed various strategies to derive robust BIE formulations of RtR operators Sj,
j= 0,1, we next turn our attention to the well-posedness of the DDM formulation (4.5).
12
4.2 Well-posedness of the DDM formulation (4.5)
The well-posedness of the DDM formulation (4.5) in the space L2(Γ) (and all Hs(Γ), s ∈[−1,1]
in the case when Γ is Lipschitz and all Hs(Γ), s ∈[−3,3] in the case when Γ is C2) hinges on the
invertibility of the operator
I− S0S1:L2(Γ) →L2(Γ)
via the formula
(I+S)−1=I+S1(I− S0S1)−1S0−S1(I− S0S1)−1
−(I− S0S1)−1S0(I− S0S1)−1.(4.20)
The invertibility of the operator I− S0S1, in turn, can be established via Fredholm arguments. In
the case of more regular boundaries Γ, the situation is somewhat simpler, since
Lemma 4.4 The RtR operators Sj:L2(Γ) →L2(Γ) corresponding to the impedance operators Zj
and ZP S
j,j= 0,1, are compact when the boundary Γis C2.
Proof. We start from formula (4.17) and we get
Sj=I−α−1
j(Z0+Z1)A−1
j(Sj+Sκj+ 2SκjK>
j)=(Z0+Z1)A−1
jA1
j(Z0+Z1)−1,
A1
j:= Aj−α−1
j(Sj+Sκj+ 2SκjK>
j)(Z0+Z1).
A closer look into the operator A1
jreveals via the decomposition (4.16)
A1
j=α−1
j(α0+α1)I+ 2α−1
j(Sj−SL+Sκj−SL)(α1Nκ1+α0Nκ2)
+ 4α−1
j(α0+α1)SLNL+ 4α−1
jSL(α1(Nκ1−NL) + α0(Nκ0−NL))
−2α−1
jSκjK>
j(Z0+Z1) + g
Areg
j,
which can be further simplified after using Calder´on’s identities
A1
j= 2α−1
j(Sj−SL+Sκj−SL)(α1Nκ1+α0Nκ2)
+ 4α−1
j(α0+α1)K2
L+ 4α−1
jSL(α1(Nκ1−NL) + α0(Nκ0−NL))
−2α−1
jSκjK>
j(Z0+Z1) + g
Areg
j.
Clearly the operators A1
jenjoy the mapping property A1
j:L2(Γ) →H1(Γ) and thus it can be
seen that Sj:L2(Γ) →H1(Γ), from which the claim of the lemma follows. Under the regularity
assumption of the interface Γ, the arguments in the proof of the lemma carry over in the case of
RtR operators corresponding to the impedance operators ZP S
j,j= 0,1.
The result in Lemma 4.4 is no longer valid in the case of Lipschitz interfaces Γ. To see this, we
start from formula (4.13) and we get
S1=I−α−1
1(Z0+Z1)B−1
1S1= (Z0+Z1)B−1
1B1
1(Z0+Z1)−1,
B1
1:= B1−α−1
1S1(Z0+Z1) = B1+ 2α−1
1S1(α1Nk1+iσ1+α0Nk0+iσ0)
=B1−1
2α−1
1(α0+α1)I+ 2α−1
1(α0+α1)K2
L+B2
1,
B2
1:= 2α−1
1(α0+α1)(S1−SL)NL+ 2α−1
1S1(α1(Nk1+iσ1−NL) + α0(Nk0+iσ0−NL)).
13
We recall from the proof of Theorem 4.1 that the operator B1was expressed in the form:
B1=B1,P +e
B1(4.21)
in terms of
B1,P := 1
2α−1
1(α0+α1)I+KL−2α0
α1
K2
L(4.22)
and the operator f
B1:L2(Γ) →H1(Γ). Putting together these two representations we obtain
B1
1=B1
1,P +B2
1+e
B1where B1
1,P := KL+ 2K2
L.(4.23)
Using the mapping properties recounted in Theorem 3.2 we see immediately that B2
1:L2(Γ) →
H1(Γ). However, although both operators B2
1and e
B1are compact in L2(Γ), the operator B1
1,P is
no longer compact in the same space, and hence the operator B1
1is no longer compact in L2(Γ).
Consequently, the RtR operator S1is not compact either. Thus, one has to look deeper into the
properties of the iteration operator I− S0S1. We thus have the following results:
Theorem 4.5 In the case of Lipschitz interfaces Γ, the operators I−S0S1are Fredholm of index
zero in the space L2(Γ).
Proof. We start with the assumption that k0is not a Laplace eigenvalue with Dirichlet
boundary conditions in the domain Ω1. This allows us to use the representation of the RtR operators
based on the operators Bj, j = 0,1, in which case the calculations are simpler. Using the splittings
presented above we derive
I− S0S1= (Z0+Z1)B−1
0(B0− B1
0B−1
1B1
1)(Z0+Z1)−1,(4.24)
where the operators B1
0are defined analogously to the operators B1
1by effecting similar decompo-
sitions to the operators S0. Clearly, we have that
B1B1
1=B1,P B1
1,P +BRwhere BR:= B1,P (B2
1+e
B1) + e
B1B1
1,P ,
and
B1
1B1=B1
1,P B1,P +BLwhere BL:= B1
1,P e
B1+ (B2
1+e
B1)e
B1.
Since the operators B1,P and B1
1,P commute, it follows that
B1B1
1− B1
1B1=BR− BL:L2(Γ) →H1(Γ).
From the last identity we derive immediately
B−1
1B1
1− B1
1B−1
1=B−1
1(BL− BR)B−1
1:L2(Γ) →H1(Γ).
Using the last identity in equation (4.24) we get that
I− S0S1= (Z0+Z1)B−1
0(B0B1− B1
0B1
1)B−1
1(Z0+Z1)−1+SR,
14
where SR:L2(Γ) →H1(Γ). Using similar decompositions for the operator B0, a simple calculation
delivers
B0B1− B1
0B1
1=B0,P B1,P − B0,P B1,P +Breg ,
B0,P B1,P − B0,P B1,P =2(α0+α1)2
α0α11
2I+KL21
2I−KL
and Breg :L2(Γ) →H1(Γ). Given that 1
2I+KLis invertible in L2(Γ) and 1
2I−KLis Fredholm of
index zero in L2(Γ), it follows that I− S0S1is also Fredholm of index zero in L2(Γ).
The mechanics of the calculations above can be adapted to the case when the RtR operators
Sjare represented via the operators Aj. In this case, we make use of the splittings put forth in
equations (4.15) in the form:
Aj=Aj,P +e
Aj
Aj,P := α−1
j(αj+αj+1)I+α−1
j(αj−αj+1)KL−2α−1
j(αj+ 2αj+1)K2
L+ 4α−1
jαj+1K3
L
as well as
Sj= (Z0+Z1)A−1
jA1
j(Z0+Z1)−1,
A1
j:= Aj−α−1
j(Sj+Sκj−2SκjK>
j)(Z0+Z1) = Aj,P +Areg
j,
Aj,P := 2KL+ 4K2
L−4K3
L,
and Areg
j:L2(Γ) →H1(Γ). Just in the case of the calculations above pertaining to the use of
operators Bj, we can establish that
I− S0S1= (Z0+Z1)A−1
0(A0,P A1,P − A1
0,P A1
1,P )A−1
1(Z0+Z1)−1+Sreg
where Sreg :L2(Γ) →H1(Γ). Now, we have that
A0,P A1,P − A1
0,P A1
1,P = 4 α0+α1
α0α1(I−KL)1
2I+KL,
from which it follows that I− S0S1is Fredholm of index zero in L2(Γ) for all real wavenumbers
kj,j= 0,1.
We are now in the position to prove the main result:
Theorem 4.6 The DDM operators I− S0S1:L2(Γ) →L2(Γ) corresponding to the impedance
operators Zjare invertible with continous inverses when the boundary Γis Lipschitz. In the case
when the boundary Γis C2, the DDM operators I− S0S1:L2(Γ) →L2(Γ) corresponding to the
impedance operators ZP S
j,j= 0,1are invertible with continous inverses.
Proof. Given the results in Lemma 4.4 and Theorem 4.5, it suffices to establish the injectivity
of the DDM operator I−S0S1. The arguments in the proof hold for the regularity of the boundary
Γ stated in the hypothesis. Let ϕ∈Ker(I− S0S1) and we consider the following Helmholtz
equation
∆w1+k2
1w1= 0 in Ω1,
∂n1w1+α−1
1Z1w1=ϕon Γ.
15
Then, we have that
S1ϕ=∂n1w1−α−1
1Z0w1.
Consider also the following Helmholtz problem:
∆w0+k2
0w0= 0 in Ω0,
∂n0w0+α−1
0Z0w0=S1ϕon Γ.
and w0radiative at infinity. Using the fact that S0S1ϕ=ϕit follows that
S0S1ϕ=∂n0w0−α−1
0Z1w0=∂n1w1+α−1
1Z1w1.
Thus, we have derived the following system of equation on Γ
∂n0w0−α−1
0Z1w0=∂n1w1+α−1
1Z1w1,
∂n0w0+α−1
0Z0w0=∂n1w1−α−1
1Z0w1,
from which we get that
(Z0+Z1)(α−1
0w0+α−1
1w1) = 0 on Γ.
Given the invertibility of the operator Z0+Z1we obtain
w0|Γ=−α−1
1α0w1|Γ,
and then
∂n1w0|Γ=−∂n1w1|Γ.
Using the last two identities we derive
=ZΓ
∂n1w0w0ds =α−1
1α0=ZΓ
∂n1w1w1ds =α−1
1α0=ZΩ1
(|∇w1|2−k2
1w1)dx = 0.
The last relation implies that w0= 0 identically in Ω0, from which follows immediately that w1= 0
in Ω1, and hence ϕ= 0.
We turn next to the case of DDM formulations with impedance operators Za
j, j = 0,1. The
situation is quite different in this case due to the entirely different mapping properties of the
operators Za
j, j = 0,1. Regarding this case we present the following result:
Theorem 4.7 The DDM operators I− S0S1:L2(Γ) →H1(Γ) corresponding to the impedance
operators Za
j,j= 0,1, are invertible with continous inverse when the boundary Γis C2.
Proof. We note that is suffices to establish the Fredholmness of the operators I− S0S1:
L2(Γ) →H1(Γ). A key ingredient is to revisit the result established in formula (4.16), which in the
case when the boundary Γ is C2implies that
Aj=I+ 2α−1
jZa
jSL+g
Areg
j,a , j = 0,1,
where the operators g
Areg
j,a :L2(Γ) →H2(Γ), and thus g
Areg
j,a :L2(Γ) →H1(Γ) are compact for
j= 0,1. In the light of this fact, we obtain from formula (4.17)
Sj=A−1
j(I−2α−1
jZa
j+1SL) + e
Sj, j = 0,1,
16
where e
Sj:L2(Γ) →H2(Γ). Following similar calculations to those in the proof of Theorem 4.5 we
arrive at
I− S0S1= 2(α−1
0+α−1
1)(Za
0+Za
1)SL+D,
where D:L2(Γ) →H2(Γ) and thus D:L2(Γ) →H1(Γ) is a compact operator. Clearly, since
<(Za
j)>0, j = 0,1, the operator I−S0S1satisfies a G˚arding inequality given that <RΓSLϕ ϕ ds ≥
ckϕk2
H−1/2(Γ), and thus the operator I− S0S1:L2(Γ) →H1(Γ) is Fredholm of index zero. Its
injectivity can be established by the same arguments as in the proof of Theorem 4.6.
To summarize, the well-posedness of the DDM formulations was established for all three choices
of impedance operators in the case of C2boundaries Γ, and for the impedance operators Zj, j = 0,1
in the case of Lipschitz domains.
4.3 Domain decomposition approach with further subdomain divisions
As it clear from Section 4.1, the calculation of the RtR maps Sj, j = 0,1 needed in the DDM
system (5.4) requires operator inversions. In the high-frequency regime, the computation of the RtR
maps requires inversion of large matrices, which becomes expensive if direct linear algebra solvers
are employed. Furthermore, iterative Krylov subspace solvers require increasingly larger numbers
of iterations for computation of interior RtR maps as the frequency increases, regardless of the BIE
formulation used for these computations. Thus, one possibility to reduce the computational costs
incurred by the computation of RtR maps is to further subdivide the interior domain into a union
of non-overlapping subdomains Ω1=∪J
j=1Ω1j. We assume that the decomposition is such that (1)
each of the subdomains Ω1jis simply connected/convex and (2) there are always cross points that
belong to more than three subdomains—see Figure 1 for an illustration of such subdivisions of an
L-shaped domain Ω1. We define Γj` := ∂Ω1j∩∂Ω1`,1≤j, ` in the case when the subdomains Ω1j
and Ω1`share an edge in common, and Γj0:= ∂Ω1j∩∂Ω0,1≤jin the case when the subdomain
Ω1jand ∂Ω0= Γ share an edge in common. With these additional notations in place, the DDM
system is written in the form
∆u1j+k2
1u1j= 0 in Ω1j,(4.25)
α1∂nju1j+Z1ju1j=−α1∂n`u1`+Z1ju1`on Γj`,
α1∂nju1j+Z1ju1j=−α0(∂n0u0+∂n0uinc) + Z1j(u0+uinc) on Γj0,
α0(∂n0u0+∂n0uinc) + Z0j(u0+uinc ) = −α1∂nju1j+Z0ju1jon Γj0,
where njdenotes the unit normal on ∂Ω1jpointing to the exterior of the subdomain Ω1j. The
transmission operators Z0jand Z1jin equations (4.25) can be defined in the following way:
Z0j:= −2α1χ0jN∂Ω1j,k1+iσ1χ0j,(4.26)
where χ0jis a smooth cutoff function supported on ∂Ω1j∩∂Ω0(again, assumed to have non-zero
one-dimensional measure), and respectively
Z1j:= −2α0χ1j,0NΓ,k0+iσ0χ1j,0−2α1X
`
χ1j,1`N∂Ω`,k1+iσ1χ1j,1`,(4.27)
where χ1j,0is a smooth cutoff function supported on ∂Ω1j∩Γ (in the case when the one-dimensional
measure of the intersection is non-zero), and χ1j,1`is a smooth cutoff function supported on Γj` =
17
Ω0
Ω11
Ω12
Ω13
Figure 1: Typical subdomain decomposition.
∂Ω1j∩∂Ω1`for all interior subdomains Ω1`, ` 6=jthat share an edge with the given subdomain Ω1j.
This type of localization was previously discussed in [29]: the role of cutoffs is to blend the various
operators in a manner that (1) is consistent for open arcs Γj` , and (2) gives rise to well-posed
local Helmholtz problems. The cutoff technique also allows for blending of the Fourier multiplier
operators ZP S
j, j = 0,1. In the case of transmission operators Za
j, j = 0,1, the blending is not
necessary, since piece-wise constant impedances pose no difficulties. The DDM formulation (4.25)
can be recast in terms of computing Robin data:
f1j:= (α1∂njuj+Z1juj)|∂Ω1j, j = 1, . . . , J
and
f0j:= (α0∂n0u0+Z0ju0)|∂Ω1j∩Γ, meas(∂Ω1j∩Γ) 6= 0,
via suitably defined RtR maps that take into account adjacent subdomains and their corresponding
transmission operators. While the choice of blended transmission operators presented above gives
rise to well-posed subdomain problems, the well-posedness of the DDM formulation (4.25) remains
an open question. Owing to the different nature of the transmission operators incorporated in
the DDM formulation (4.25), the arguments used in the proof of the well-posedness of the DDM
formulation (4.1) that we presented in Section 4.2 cannot be readily translated to the new setting.
5 Scalar transmission problems in partially coated domains
We consider next the problem of two dimensional transmission by structures that feature partial
coatings, i.e. penetrable scattering problems when parts of the boundary of the scatterer are per-
fectly conducting/impenetrable. Let Ω1denote a bounded domain in R2whose boundary Γ := ∂Ω1
is given locally by the graph of a Lipschitz function, and let Ω0:= R2\Ω1. We seek to find fields
u0and u1that are solutions of the following scalar Helmholtz transmission problem:
18
Ω0
Ω1
ΓPEC
ΓT
Figure 2: Partially coated domain.
∆uj+k2
juj= 0 in Ωj, j = 0,1,
u0+uinc =u1on ΓT,
α0(∂n0u0+∂n0uinc) = −α1∂n1u1on ΓT,
∂n0(u0+uinc) = 0 on ΓPEC ,
∂n1u1= 0 on ΓPEC,
lim
r→∞ r1/2(∂u0/∂r −ik0u0) = 0,
(5.1)
where ΓT∩ΓPEC = Γ, and |ΓPEC |>0, where |ΓP E C |denotes the one dimensional measure of the
set ΓPEC. We assume that the wavenumbers kjand the quantities αjin the subdomains Ωjare
positive real numbers. The unit normal to the boundary ∂Ωjis here denoted by njand is assumed
to point to the exterior of the subdomain Ωj. The incident field uinc, on the other hand, is assumed
to satisfy the Helmholtz equation with wavenumber k0in the unbounded domain Ω0. Finally, we
assume that the parameters αjare such that the transmission problem (5.1) is well posed.
In what follows, we present again two formulations of the transmission problem (5.1): a BIE
and a DDM approach.
5.1 Boundary integral equation formulations
Again, boundary integral formulations of the transmission problem (5.1) can be derived using layer
potentials defined on Γ: the solutions uj, j = 0,1,of the transmission problem are sought in the
form:
uj(x) := SLΓ,j v+ (−1)jα−1
jDLΓ,j p, x∈Ωj,(5.2)
where vand pare densities defined on the Γ and the double layer operators are defined with respect
to exterior unit normals ncorresponding to each domain Ωj. Here we used the same convention
that the index jin equation (5.2) refers to the wavenumber kjfor j= 0,1. The enforcement of the
transmission conditions on the interface ΓTas well as the enforcement of the PEC conditions on
the interface ΓPEC lead to the following system of boundary integral equations:
α−1
0+α−1
1
2ΠTp−ΠT(α−1
0K0+α−1
1K1)p+ ΠT(S1−S0)v=uinc on Γ,
α0+α1
2ΠTv+ ΠT(α0K>
0+α1K>
1)v+ ΠT(N0−N1)p=−α0ΠT∂n0uinc on Γ,
1
2ΠPECv+ ΠPEC K>
0v+α−1
0ΠPECN0p=−ΠPEC∂n0uinc on Γ,
1
2ΠPECv+ ΠPEC K>
1v−α−1
1ΠPECN1p= 0 on Γ,
(5.3)
19
where the restriction operators ΠTand ΠPEC are defined as ΠTψ:= ψ|ΓTand respectively ΠPECψ:=
ψ|ΓPEC , for functions ψdefined on Γ. The restriction operators can be extended to distributions.
In what follows we refer to the formulation (5.3) by the acronym CFIESK. To the best of our
knowledge, the well-posedness of the CFIESK formulations (5.3) has not been established in the
literature.
5.2 Domain decomposition approach
DDM are natural candidates for numerical solution of transmission problems (5.1). A non-overlapping
domain decomposition approach for the solution of equations (5.1) consists of solving subdomain
problems in Ωj, j = 0,1 with matching Robin transmission boundary conditions on the common
subdomain interface ΓT. Indeed, this procedure amounts to computing the subdomain solutions:
∆uj+k2
juj= 0 in Ωj,(5.4)
αj(∂njuj+δ0
j∂njuinc) + Zj(uj+δ0
juinc) = −α`(∂n`u`+δ0
`∂n`uinc) + Zj(u`+δ0
`uinc) on ΓT,
∂njuj+δ0
j∂njuinc = 0 on ΓPEC,
where {j, `}={0,1}and δ0
jstands for the Kronecker symbol, and Zj, Z`are transmission operators
that will be defined in what follows. In order to describe the DDM method more concisely we
introduce subdomain RtR maps [15]. For each subdomain Ωj, j = 01,1 we define RtR maps
Sj, j = 0,1 in the following manner:
S0(ψ0) := (α0∂n0u0−Z1u0)|ΓTand S1(ψ1) := (α1∂n1u1−Z0u1)|ΓT.(5.5)
The DDM (5.4) can be recast in terms of computing the global Robin data f= [f1f0]>with
fj:= (αj∂njuj+Zjuj)|ΓT, j = 0,1,
as the solution of the following linear system that incorporates the subdomain RtR maps Sj, j = 0,1,
previously defined
(I+S)f=gwhere S:= 0S0
S10,(5.6)
with right-hand side g= [g1g0]>wherein
g1= (−α0∂n0uinc +Z1uinc)|ΓT,
g0=−(α0∂n0uinc +Z0uinc)|ΓT.
Ideally, the operator Z0should be the restriction to ΓTof Dirichlet-to-Neumann (DtN) operator
corresponding to the Helmholtz equation posed in the domain Ω1with generalized Robin boundary
conditions on ΓTand zero Neumann boundary conditions on ΓPEC. Following the methodology
presented in the context of classical transmission problems, we employ a smooth cutoff function χT
supported on ΓTin order to define the following the transmission operators:
Z0=−2α0χTNΓ,k1+iσ1χTand Z1 = −2α1χTNΓ,k0+iσ0χT, σj>0.(5.7)
We can also use principal symbol transmission operators define accordingly
ZP S
0=−2α0χTP S(NΓ,k1+iσ1)χTand ZP S
1=−2α1χTP S(NΓ,k0+iσ0)χT, σj>0,(5.8)
20
as well as the simpler transmission operators
Za
0=−iα0(k1+iσ1)ΠTand Za
1=−iα1(k0+iσ0)ΠT.(5.9)
The RtR maps corresponding to the new transmission operators defined in equations (5.7), (5.8),
and (5.9) can be computed by readily incorporating in the methodology presented in Section 4.1
the additional requirement of zero Neumann traces on the portion ΓPEC of the boundaries.
6 High-order Nystr¨om discretizations
We use Nystr¨om discretizations of the CFIESK equation (5.3), as well as the RtR maps associated
with the various DDM formulations. The key ingredient is the Nystr¨om discretization of the four
BIO in the Calder´on calculus for piecewise smooth boundaries. These discretizations were intro-
duced in [12] where this methodology was presented in full detail. In particular, the discretization
of the CFIESK equation (5.3) was described in the aforementioned contribution. Therefore, we
present here the discretization of the DDM formulations that relies, in turn, on discretizations
of the corresponding RtR maps. Specifically, graded meshes produced by means sigmoid trans-
forms [20] that accumulate points polynomially toward corner and junction points (where ΓPE C
and ΓTmeet) are utilized on the closed curve Γ. For each of the subdomains Ωj,j= 0,1, we
employ graded meshes denoted by
Lj:= {xj
m, m = 0, . . . , Nj−1}on ∂Ωj= Γ,
with the same polynomial degree of the sigmoid transforms on all subdomains. All meshes in the
parameter space [0,2π] are shifted by the same amount so that none of the grid points on the
skeleton corresponds to a triple/multiple junction or a corner point. We allow for non-conforming
meshes, that is N1may not be equal to N0; the size Njof the mesh Ljis chosen to resolve the
wavenumber kjcorresponding to the domain Ωj.
Using graded meshes that avoid corner points, trigonometric interpolation, and the classical sin-
gular quadratures of Kusmaul and Martensen [21, 22], we perform the Nystr¨om discretization
presented in [12] to produce high-order Nj×Njcollocation matrix approximations of the four BIO
described in equations (3.1). We note that discretizations of the Fourier multiplier operators ZP S
j,
j= 0,1 is straightforward via trigonometric interpolation [12]. Based on these, the DDM algo-
rithm proceeds with a precomputational stage whereby matrix approximations of all the RtR maps
needed are produced. The precomputational stage is computationally expensive on account of the
matrix inversions needed for the computation of discrete RtR matrices. Nevertheless, this stage
is highly parallelizable since the computation of the RtR matrix corresponding to a subdomain
does not require information from adjacent subdomains. In order to avoid complications related to
singularities at junction/cross points, we replace in the DDM algorithm the RtR maps by weighted
parametrized counterparts
Sj,w(αj|x0
j|∂njuj+Zjuj) := αj|x0
j|∂njuj−Zj+1 uj.
Collocated discretizations of the latter weighted RtR maps can be easily computed through a
simple modification of the methodology introduced in [29] and recounted above. Nevertheless, the
representation of RtR maps in terms of BIO requires use of inverses of matrices corresponding to
21
Nystr¨om discretizations of either operators Bj,cf. (4.13), Aj,cf. (4.17), or Cj,cf. (4.18). The
inversion of these matrices can be performed via direct or iterative linear algebra methods. In the
former case, the discretization of the weighted RtR maps corresponding to each domain ∂Ωjis
constructed as Nj×Njcollocation matrices Sj
Nj. For bounded (interior) domains, the formulations
based on the use of the simpler operators B1are the most efficient for use of direct linear algebra
solvers; the ones based on operators A1are more complex, and the ones based on the operators C1
require inversions of matrices twice as large. For the unbounded domain Ω0, the formulations based
on the use of operators A0are preferred owing to their stability valid for all real wavenumbers
k0. However, the use of direct linear algebra solvers at this stage imposes limitations on the
discretization size Nj. This size can be further reduced by employing subdivisions of the interior
domain Ω1as described in Section 4.3. For the examples considered in this text, such subdivisions
are straightforward. Alternatively, when iterative linear algebra methods are employed for the
calculation of RtR maps, the latter are an inner iteration in the iterative solution of the DDM
linear system (4.5). Our numerical experiments presented in the next Section suggest that the use
of BIE formulations based on the operators A0(4.17) for the calculation of exterior RtR maps S0
result in small numbers of iterations that grow slowly as the frequency increases. The situation is
entirely different in the case of interior RtR maps S1: all three BIE formulations considered in this
text give rise to numbers of iterations that grow significantly with the frequency. Again, a remedy
for this issue is employing subdivisions of the interior domain Ω1and thus effectively reducing the
acoustic/electric size of the subdomains.
Once the discretized RtR matrices S0
N0∈RN0×N0and respectively S1
N1∈RN1×N1are computed
(we assume in what follows that k0< k1and thus N0≤N1) the discretization of the DDM linear
system (4.5) is easily set up in the form
fN0
0+PN1→N0S1
N1fN1
1=gN0
0,
fN1
1+EN0→N1S0
N0fN0
0=gN1
1,(6.1)
where fNj
jare approximations of the Robin data fjtrigonometrically collocated on the grids Ljfor
j= 0,1, and the projection operator PN1→N0and the extension operator EN0→N1allow for transfer
of information via Fourier space from the two grids L0and L1. Specifically, the extension operator
EN0→N1is realized via zero padding in the Fourier space, while the projection operator PN1→N0
is a cutoff operator in the Fourier space. The right hand-side in equation (6.1) are obtained by
simply evaluating gjon the grids Ljfor j= 0,1. In order to further reduce the size of the linear
system that we solve, we further eliminate the data fN1
1from the linear system (6.1) and solve the
reduced linear system
fN0
0−PN1→N0S1
N1EN0→N1S0
N0fN0
0=gN0
0−PN1→N0S1
N1gN1
1.(6.2)
Once the exterior Robin data fN0
0is computed by solving the linear system (6.2), the exterior
Cauchy data on Γ can be immediately retrieved via the RtR operator S0. The interior Cauchy
data on Γ is then readily computed from the continuity conditions. In what follows, we present
a concise algorithmic description of the DDM formulation (4.5). The modifications needed to
cover the DDM with further domain subdivisions (4.25) or the DDM for partial coatings (5.6) are
straightforward.
22
1Offline: For each subdomain Ωj, discretize all the BIO that feature in formulas (4.11)
and (4.14) corresponding to each boundary ∂Ωjusing Nystr¨om discretizations. The
discretization of each BIO results in a collocation matrix of size Nj×Nj, whose
computational cost is O(N2
j);
2Offline: Compute all the collocated subdomain RtR matrices Sj
Njusing formulation (4.11)
for the interior domain and the formulation (4.14) for the exterior domain. We compute
discretizations of the RtR maps via LU factorizations, and thus the cost of evaluating each
subdomain RtR map is O(N3
j);
3Solution: Set up the DDM linear system according to formula (6.2) and solve for the Robin
data fN0
0using GMRES;
4Post-processing: Use the Robin data fN0
0computed in the previous step and the RtR matrix
S0
N0to compute Cauchy data on Γ.
Algorithm 1: Description of the DDM algorithm
7 Numerical results
In this section we present numerical experiments concerning the iterative behavior of various DDM
solvers considered in this text. We also document the iterative behavior of the CFIESK solvers. We
mention that a comprehensive comparison between various integral formulations for transmission
problems was pursued in [5, 12, 19]. While the CFIESK formulations are not the most performant
formulations vis-a-vis iterative solvers, they are the simplest and most widely used in the collo-
cation discretization community [25, 16]. Also, and as mentioned previously, the CFIESK can be
relatively easily extended to more complex boundary conditions scenarios. It is not our goal to
carry in this text a detailed computational efficiency comparison between BIE formulations and
DDM formulations of Helmholtz transmission problems. On the one hand, there is a relatively large
body of work in which fast methods and matrix compression techniques are used to accelerate the
performance of BIE based solvers [16, 7]. On the other hand, DDM with quasi-optimal transmission
operators for transmission Helmholtz equations have been studied to a very limited extent; it is our
intent to highlight in this paper the remarkable iterative properties that these solvers enjoy, and
to point out several challenges that they face related to efficient computations of RtR maps. It is
important to bear in mind that one of the main attractive feature of DDM is their embarassing
parallelism, which is much harder to achieve by BIE solvers. We plan to pursue elsewhere an in
depth comparison between the computational efficiency of BIE solvers and DDM solvers for three
dimensional transmission problems.
All of the formulations considered were discretized following the prescription in Section 6. In all
the numerical experiments we used meshes that rely on sigmoid transforms of polynomial degree
3. Also, following the optimality prescriptions in [3], we selected σj=k1/3
jin the definition of
the complex wavenumbers that enter the definition of the corresponding tranmsission operators.
Unless specified otherwise, in all the numerical experiments we present numbers of GMRES itera-
tions for various solvers to reach a relative residual of 10−4and present errors in the far-field for
1024 equi-spaced far-field directions. In all the numerical results presented, the reference solutions
were computed using highly refined discretizations of CFIESK solvers. We start in Table 1 with
an illustration of the accuracy of the Nystr¨om discretizations of the CFIESK and various DDM
23
Unknowns CFIESK DDM Zj, j = 0,1 DDM ZP S
j, j = 0,1 DDM Za
j, j = 0,1
It ε∞It ε∞It ε∞It ε∞
72 51 9.2 ×10−426 4.3 ×10−330 4.3 ×10−354 4.3 ×10−3
144 51 5.6 ×10−626 3.4 ×10−430 3.4 ×10−466 3.4 ×10−4
288 51 3.9 ×10−726 3.9 ×10−530 3.9 ×10−574 3.9 ×10−5
572 51 2.5 ×10−825 4.1 ×10−630 4.1 ×10−687 4.1 ×10−6
1144 51 1.6 ×10−925 2.6 ×10−730 2.6 ×10−7104 2.6 ×10−7
Table 1: Far-field errors ε∞computed using various formulations considered in this text in the case
of scattering from an L-shaped domain with ω= 2, ε0= 1, and ε1= 4 with αj= 1, j = 0,1. We
considered a GMRES residual of 10−12 in all the tests presented in the Table. CFIESK formulations
uses twice as many unknowns as the DDM formulations.
ωCFIESK DDM Zj,j = 0,1 DDM ZP S
j, j = 0,1 DDM Za
j, j = 0,1
It ε∞It ε∞It ε∞It ε∞
1 24 3.1 ×10−410 5.2 ×10−310 5.1 ×10−320 5.0 ×10−3
2 39 8.2 ×10−411 1.0 ×10−312 9.9 ×10−428 1.1 ×10−3
4 93 2.3 ×10−312 1.2 ×10−317 1.4 ×10−346 1.3 ×10−3
8 162 6.3 ×10−310 2.1 ×10−319 2.2 ×10−384 2.1 ×10−3
16 333 7.6 ×10−311 4.5 ×10−329 4.2 ×10−3151 4.1 ×10−3
32 565 1.2 ×10−213 2.9 ×10−356 2.8 ×10−3253 2.9 ×10−3
Table 2: Far-field errors ε∞computed using various formulations considered in this text in the case
of scattering from a square of size 4 with ε0= 1 and ε1= 16 with αj= 1, j = 0,1. The DDM
discretization used conforming meshes, that is N0=N1, and 64,128,256,512,1024 and respectively
2048 unknonws (these are the values of N0); CFIESK formulations used twice as many unknowns.
The numbers of iterations required by the DDM solvers with transmission operators Zj,j= 0,1,
were 13, 15, 14, 19, 23, and respectively 31 in the case when αj=ε−1
j, j = 0,1.
formulations of the transmission problem (2.1) that used conforming meshes, that is N0=N1.
We note that the CFIESK and DDM with transmission operators Zjand ZP S
jexhibit iterative
behaviors corresponding to second kind formulations, while the DDM with transmission operators
Za
jbehave like first kind formulations. Also, the solvers based on CFIESK formulations are more
accurate than the DDM solvers, and the accuracy of the latter formulations is virtually independent
of the choice of transmission operators.
We present in Tables 2 and 3 the behavior of the various formulations for the transmission prob-
lem (2.1) as a function of frequency in the case of high-contrast material properties, that is ε0= 1
and ε1= 16 and two scatterers: a square of size 4 in Table 2 and an L-shaped domain of size 4
in Table 3. We used conforming meshes, i.e. N0=N1for the DDM solvers. As it can be seen
from the results in Tables 2 and 3, the numbers of iterations required by the DDM solvers with
transmission operators Zj,j= 0,1 are small and depend very mildly on the increasing frequency.
Also, the iterative behavior of the DDM based on the transmission operators ZP S
j,j= 0,1, deteri-
orates somewhat with respect to that of DDM solvers with transmission operators Zj,j= 0,1. In
contrast, the iterative behavior of DDM based on the simplest transmission operators Za
j, j = 0,1
is quite poor in the high-frequency, high-contrast case.
The superior iterative performance of the DDM formulations that rely on transmission operators
Zj, j = 0,1 can be inferred from the clustering of the eigenvalues of the iteration operator I−S1S0
around one. We present in Figure 3 the remarkable eigenvalue clustering in the case of the L-shaped
scatterer for high-frequencies. It is important to note from the evidence presented in Figure 3
24
ωCFIESK DDM Zj,j = 0,1 DDM ZP S
j, j = 0,1 DDM Za
j, j = 0,1
It ε∞It ε∞It ε∞It ε∞
1 43 1.0 ×10−315 4.7 ×10−316 4.6 ×10−331 4.6 ×10−3
2 72 1.1 ×10−315 9.0 ×10−417 1.2 ×10−346 8.3 ×10−4
4 135 2.1 ×10−316 2.4 ×10−324 2.4 ×10−381 2.3 ×10−3
8 208 2.4 ×10−315 4.0 ×10−329 4.0 ×10−3112 4.1 ×10−3
16 493 8.8 ×10−321 8.1 ×10−356 8.1 ×10−3276 8.0 ×10−3
32 887 1.2 ×10−222 9.6 ×10−387 9.6 ×10−3488 9.6 ×10−3
Table 3: Far-field errors computed using various formulations considered in this text in the case
of scattering from a L-shaped domain of size 4 with ε0= 1 and ε1= 16 with αj= 1, j = 0,1.
The DDM discretization used conforming meshes, that is N0=N1and 64,128,256,512,1024 and
respectively 2048 unknonws (these are the values of N0); CFIESK formulations used twice as many
unknowns. The numbers of iterations required by the DDM solvers with transmission operators
Zj, j = 0,1 were 21, 23, 21, 23, 29, and respectively 37 in the case when αj=ε−1
j, j = 0,1.
ωDDM (1) Zj, j = 0,1 Square DDM (2) Zj, j = 0,1 Square DDM (1) Zj, j = 0,1 L-shape DDM (1) Zj, j = 0,1 L-shape
N0=N1It ε∞N0It ε∞N0=N1It ε∞N0It ε∞
4 256 10 1.2 ×10−3192 10 1.2 ×10−3256 16 2.4 ×10−3192 14 6.0 ×10−3
8 512 10 2.1 ×10−3384 14 6.1 ×10−3512 15 4.0 ×10−3384 12 3.1 ×10−3
16 1024 11 4.5 ×10−3768 16 6.7 ×10−31024 21 8.1 ×10−3768 22 1.2 ×10−2
32 2048 13 2.9 ×10−31536 15 4.9 ×10−32048 22 9.6 ×10−31536 27 1.3 ×10−2
Table 4: Comparison between the conforming (N0=N1) and non-conforming (N0< N1) DDM
with transmission operators Zj, j = 0,1 for high-contrast transmission problems with ε0= 1 and
ε1= 16 with αj= 1, j = 0,1. In the non-conforming case, the values of N1are equal to those in
the conforming case for the same frequency.
that although the eigenvalues of the iteration operator I− S1S0corresponding to high-frequency
eigenmodes are tighlty clustered around one, the operator S1S0is not a contraction.
Clearly, in the case of high-frequency, high-contrast transmission problems, DDM that use conform-
ing meshes are not the most advantageous computationally. Rather, the use of non-conforming
meshes that resolve the wavenumber corresponding to each subdomain are more favorable. We
present in Table 4 results corresponding to use of non-conforming meshes in the DDM with trans-
mission operators Zj,j= 0,1. We note that the iterative behavior of the non-conforming DDM is
very similar to that of conforming DDM, without major compromise on accuracy.
The use of optimized transmission operators Zjand ZP S
jfor j= 0,1 gives rise to superior DDM
iterative performance. However, given that the transmission operators Zj,j= 0,1, and ZP S
j,j=
0,1 are non-local operators, their implementation favors boundary integral equation solvers, while
posing challenges to finite difference/finite element discretizations. Therefore, approximations of
the square root Fourier multiplier operators ZP S
jmore amenable to the latter types of discretizations
were proposed in the literature. There are two classes of such approximations that were widely
used: local second order approximations with optimized coefficients [14] and Pad´e approximations.
Reference [3] provides numerical evidence that the incorporation of Pade´e approximations of square
root operators results in DDM with faster rates of convergence than the use of local second order
approximations. In what follows, we explain briefly the Pad´e approximations used in [3]; we start
from formulas √1 + X≈eiθ/2Rp(e−iθX) = A0+
p
X
j=1
AjX
1 + BjX
25
Figure 3: Eigenvalue distributions of the DDM iteration operator I−S1S0with the choice of trans-
mission operators Zj, j = 0,1 for the L-shaped scatterer and high-contrast transmission problems
with ε0= 1, ε1= 16, αj= 1, j = 0,1 and ω= 16 (top) and ω= 32 (bottom).
26
Figure 4: The numbers of iterations required by the DDM solvers with transmission operators
ZP S
j, j = 0,1 as well as Pad´e approximations ZP ade,p
j, j = 0,1 for various values of p, square
scatterer and the same material parameters as those in Table 2.
where the complex numbers A0,Ajand Bjare given by
A0=eiθ/2Rp(e−iθ −1), Aj=e−iθ/2aj
(1 + bj(e−iθ −1))2, Bj=e−iθbj
1 + bj(e−iθ −1)
and
Rp(z) = 1 +
p
X
j=1
ajz
1 + bjz
with
aj=2
2p+ 1 sin2(jπ
2p+ 1)bj= cos2(jπ
2p+ 1).
These Pad´e approximations of square roots above give rise to the following transmission operators
ZP ade,p
j=−i
2(kj+iσj)
A0I−
p
X
j=1
Aj∂2
s
(kj+iσj)2I−Bj∂2
s
(kj+iσj)2−1
,(7.1)
where ∂sis the tangential derivative on Γ. We note that the discretizations of the operators
ZP ade,p
j, j = 0,1 defined in equation (7.1) is relatively straightforward using trigonometirc inter-
polants. However, their discretization requires pmatrix inverses per wavenumber. We present in
Figure 4 a comparison between the DDM iterations as a function of the Pad´e parameter pin the case
of a L-shaped scatterer and the same material parameters as those in Table 2. For the configuration
presented in Figure 4, we have found in practice that the value p= 16 leads to optimal iterative
behavior of the DDM, but this behavior is sensitive to the values of pin the high-frequency regime.
Albeit smaller values of the Pad´e parameter prequire less expensive evaluations of the transmission
operators ZP ade,p
j, j = 0,1, they lead to larger numbers of DDM iterations in the high-frequency
regime.
As it can be seen from the results in Tables 2 and 3, the DDM solvers based on optimized transmis-
sion operators Zjand ZP S
jexhibit superior iterative Krylov subspace performance. Nevertheless,
27
k0Ω0k1Ω1
A0(4.15) B0(4.11) C0(4.19) A1(4.15) B1(4.11) C1(4.18)
1 13 16 37 4 18 21 49
2 17 21 49 8 26 29 70
4 24 36 84 16 51 56 131
8 31 49 104 32 83 79 217
16 35 75 143 64 170 142 431
32 42 125 228 128 263 214 793
Table 5: Numbers of iterations required for the calculation of the RtR operators Sj, j = 0,1
corresponding to the transmission operators Zj, j = 0,1 in the case of the square scatterer Ω1using
various boundary integral equation formulations discussed in this text .
DDM formulations rely on discretization of RtR operators Sj, which, in turn, require matrix inver-
sions. As the frequency increases, the size of the matrices that need be inverted grows commensu-
rably; furthermore, for three dimensional applications, the numbers of unknowns quoted in Tables 2
and 3 ought to be squared for the same acoustical/electrical size of domains. Clearly, a straightfor-
ward use of direct linear algebra solvers for computations of RtR operators is not possible in the high
frequency regime. Therefore, we turn our attention in Tables 5 and 6 to the numbers of iterations
required for computation of Sjcorresponding to the transmission operators Zj, j = 0,1 based on
the three formulations discussed in this text. Specifically, we used (1) interior/exterior formulations
that require inversion of the operators Aj, j = 0,1 defined in equation (4.15); (2) interior/exterior
formulations that require inversion of the operators Bj, j = 0,1 defined in equation (4.11); and (3)
interior formulations that require inversion of the operators C1defined in equation (4.18) and ex-
terior formulations that require inversion of the operators C0defined in equation (4.19). Although
there is no theory in place for the well-posedness of boundary integral equations that involve in-
version of the operators B0defined in equation (4.11), our numerical experiments suggest that it is
possible to invert discretizations of those operators. As it can be seen from the results presented
in Tables 5 and 6, while the numbers of iterations required to solve exterior impedance problems
do not increase significantly with frequency provided that carefully defined formulations A0(4.15)
are used, this is no longer the case for interior impedance problems, regardless of formulation used.
Similar scenarios occur for the other choices of transmission operators discussed in this text. As
it can be seen from the results in Tables 5 and 6, the numbers of iterations required for the com-
putation of the interior RtR map S1cannot be controlled as the frequency increases, regardless of
the use of any of the three BIE formulations considered in this text. We submit that this is related
to the fact that easily computable approximations of DtN maps for interior domains (even when
properly defined) are simply not available for high-frequencies.
Given the large computational costs required to compute the RtR operators S1at high frequencies,
it is preferrable that the interior domain Ω1is decomposed in smaller non-overlapping subdomains
giving rise to DDM formulations (4.25), in which case direct solvers such as LU can be used for the
calculation of all the RtR maps required. However, as shown in Figure 5, the numbers of iterations
grow considerally with the number of subdomains, albeit the computation of RtR maps becomes
much more efficient since the electric size of interior subdomains has been decreased. This increase
of number of iterations as the number of subdomains increases and the adjacency graph becomes
more complex can be attributed to the global communication flow between subdomains, regardless
of choice of transmission operators. This increase is more dramatic for the transmission operators
28
k0Ω0k1Ω1
A0(4.15) B0(4.11) C0(4.19) A1(4.15) B1(4.11) C1(4.18)
1 17 22 44 4 24 26 67
2 22 27 58 8 38 42 92
4 31 39 80 16 66 65 160
8 34 63 131 32 106 94 247
16 38 104 188 64 218 195 473
32 45 168 309 128 405 333 890
Table 6: Numbers of iterations required for the calculation of the RtR operators Sj, j = 0,1
corresponding to the transmission operators Zj, j = 0,1 in the case of the L-shaped scatterer Ω1
using various boundary integral equation formulations discussed in this text.
ωCFIESK DDM Zj,j = 0,1 DDM ZP S
j, j = 0,1 DDM Za
j, j = 0,1
It ε∞It ε∞It ε∞It ε∞
1 85 4.7 ×10−313 6.2 ×10−312 6.2 ×10−321 6.3 ×10−3
2 165 5.2 ×10−317 6.8 ×10−315 6.9 ×10−334 6.8 ×10−3
4 315 5.8 ×10−317 7.4 ×10−318 7.4 ×10−337 7.3 ×10−3
8 617 6.1 ×10−319 7.6 ×10−323 7.6 ×10−352 7.5 ×10−3
16 1225 6.8 ×10−321 7.8 ×10−329 7.8 ×10−3118 7.9 ×10−3
32 2271 7.2 ×10−323 8.5 ×10−344 8.5 ×10−3265 8.4 ×10−3
Table 7: Far-field errors computed using various formulations considered in this text in the case
of scattering from a circle of radius one with ε0= 1 and ε1= 16 with αj= 1, j = 0,1, and the
lower semi-circle is PEC. The DDM discretization used 64,128,256,512,1024 and respectively 2048
unknonws; CFIESK formulations used twice as many unknowns. In the case when the domain
Ω1is further subdivided into two subdomains Ω11 and Ω12 the numbers of DDM iterations are (i)
24,33,39,56,95,173 for transmission operators Zj, (ii) 22,31,43,69,135,256 for transmission oper-
ators ZP S
j, and (iii) 34,63,73,125,251,529 for transmission operators Za
jfor the same frequencies
and material parameters.
Zjand ZP S
j, and less so for the transmission operators Za
j. Still, the numbers of iterations required
by the DDM with interior domain subdivisions (4.25) and transmission operators Zjreported
in Figure 5 are smaller than those corresponding to transmission operators ZP S
jand Za
j. Thus,
even though the exchange of information between adjacent subdomains can be optimized, the
number of DDM iterations does not scale with the number of subdomains, and preconditioners are
needed to stabilize this phenomenon. The design of effective preconditioners for the DDM (4.25)
for Helmholtz equation with large numbers of subdomains that control the global interdomain
communication is an active area of research. The most promising directions are (a) the use of coarse
grid preconditioners [27, 10] and (b) the use of sweeping preconditioners [30]. The incorporation of
these preconditioning strategies in the case of DDM (4.25) is subject of ongoing investigation.
We conclude with numerical experiments concerning transmission problems with partially coated
boundaries. Specifically, we present in Table 7 numbers of iterations required by the CFIESK
formulation (5.3) and DDM with various transmission operators considered in this text. The
domain Ω1is a circle of radius one whose lower semicircle is coated. In this case, the reductions
in numbers of iterations that can be garnered from use of DDM over the use of CFIESK is more
pronounced. Finally, we plot in Figure 6 fields scattered by a penetrable scatterer whose boundary
is partially coated under plane wave incidence of various directions and frequencies.
29
Figure 5: The numbers of iterations required by the DDM solvers with transmission operators
Zj, j = 0,1 in the case when the interior domain Ω1is a circle of radius one that is divided into (a)
two interior half-circle subdomains Ω1= Ω11 ∪Ω12, and (b) four quater-circle interior subdomains
Ω1=∪4
j=1Ω1j. We used ε0= 1, ε1= 16, αj= 1, j = 0,1, and ω= 1,2,4,8,16,32,64. In the case
when the L-shaped subdomain of size 4 is divided into three subdomains as depicted in Figure 1,
the number of iterations of the DDM algorithm (4.25) are (i) 58, 72, 106, 171, 306, and respectively
447 for transmission operators Zj, (ii) 65, 83, 118, 179, 330, and respectively 509 for transmission
operators ZP S
j, and (iii) 103, 133, 179, 286, 533, and respectively 740 for transmission operators
Za
j, for frequencies ω= 1,2,4,8,16,32 and material properties described in Table 3.
8 Conclusions
We presented analysis and numerical experiments concerning DDM based on quasi-optimal trans-
mission operators for the solution of Helmholtz transmission problems in two dimensions. The
quasi-optimal transmission operators that we used are readily computable approximations of DtN
operators. Under certain assumptions on the regularity of the of the (closed) boundary of material
discontinuity we established the well posedness of the DDM with the transmission operators consid-
ered. We provided ample numerical evidence that the incorporation of quasi-optimal transmission
operators within DDM gives rise to small numbers of Krylov subspace iterations for convergence
that depend very mildly on the frequency or contrast. However, the numers of iterations do not
scale with the number of subdomains involved in the DDM. Extensions to three-dimensional con-
figurations are currently underway.
Acknowledgments
Yassine Boubendir gratefully acknowledges support from NSF through contracts DMS-1720014.
Catalin Turc gratefully acknowledges support from NSF through contracts DMS-1614270. Carlos
Jerez-Hanckes thanks partial support from Conicyt Anillo ACT1417 and Fondecyt Regular 1171491.
30
Figure 6: Fields scattered by a circular structure filled with a material with ε1= 16 and PEC
lower semicircle in the case of ω=k0= 16 (top left), ω=k0= 32 (top and bottom right), and
ω=k0= 64 (bottom left) and various plane wave incident fields.
31
References
[1] R.A. Adams and J.J.F. Fournier. Sobolev spaces, volume 140 of Pure and Applied Mathematics
(Amsterdam). Elsevier/Academic Press, Amsterdam, second edition, 2003.
[2] X. Antoine and M. Darbas. Alternative integral equations for the iterative solution of acoustic
scattering problems. Quart. J. Mech. Appl. Math., 58(1):107–128, 2005.
[3] Y. Boubendir, X. Antoine, and C. Geuzaine. A quasi-optimal non-overlapping domain decom-
position algorithm for the Helmholtz equation. J. Comput. Phys., 231(2):262–280, 2012.
[4] Y Boubendir, A Bendali, and F Collino. Domain decomposition methods and integral equations
for solving helmholtz diffraction problem. In Fifth International Conference on Mathematical
and Numerical Aspects of Wave Propagation, Philadelphia, PA, pages 760–4, 2000.
[5] Y. Boubendir, O. Bruno, C. Levadoux, and C. Turc. Integral equations requiring small num-
bers of Krylov-subspace iterations for two-dimensional smooth penetrable scattering problems.
Appl. Numer. Math., 95:82–98, 2015.
[6] Yassine Boubendir, Victor Dominguez, David Levadoux, and Catalin Turc. Regularized com-
bined field integral equations for acoustic transmission problems. SIAM Journal on Applied
Mathematics, 75(3):929–952, 2015.
[7] O.P. Bruno, T. Elling, and C. Turc. Regularized integral equations and fast high-order solvers
for sound-hard acoustic scattering problems. Internat. J. Numer. Methods Engrg., 91(10):1045–
1072, 2012.
[8] Xavier Claeys, Ralf Hiptmair, and Elke Spindler. A second-kind galerkin boundary element
method for scattering at composite objects. BIT Numerical Mathematics, 55(1):33–57, 2015.
[9] D. Colton and R. Kress. Integral equation methods in scattering theory. Pure and Applied
Mathematics (New York). John Wiley & Sons Inc., New York, 1983. A Wiley-Interscience
Publication.
[10] Lea Conen, Victorita Dolean, Rolf Krause, and Fr´ed´eric Nataf. A coarse space for heteroge-
neous helmholtz problems based on the dirichlet-to-neumann operator. Journal of Computa-
tional and Applied Mathematics, 271:83–99, 2014.
[11] Bruno Despr´es. D´ecomposition de domaine et probl`eme de Helmholtz. C. R. Acad. Sci. Paris
S´er. I Math., 311(6):313–316, 1990.
[12] Victor Dominguez, Mark Lyon, Catalin Turc, et al. Well-posed boundary integral equation
formulations and nystr¨om discretizations for the solution of helmholtz transmission problems in
two-dimensional lipschitz domains. Journal of Integral Equations and Applications, 28(3):395–
440, 2016.
[13] L. Escauriaza, E. B. Fabes, and G. Verchota. On a regularity theorem for weak solu-
tions to transmission problems with internal Lipschitz boundaries. Proc. Amer. Math. Soc.,
115(4):1069–1076, 1992.
[14] Martin J. Gander, Fr´ed´eric Magoul`es, and Fr´ed´eric Nataf. Optimized Schwarz methods without
overlap for the Helmholtz equation. SIAM J. Sci. Comput., 24(1):38–60 (electronic), 2002.
32
[15] S. Ghanemi, F. Collino, and P. Joly. Domain decomposition method for harmonic wave equa-
tions. In Mathematical and numerical aspects of wave propagation (Mandelieu-La Napoule,
1995), pages 663–672. SIAM, Philadelphia, PA, 1995.
[16] L Greengard, Kenneth L. Ho, and J.-Y. Lee. A fast direct solver for scattering from periodic
structures with multiple material interfaces in two dimensions. J. Comput. Phys., 258:738–751,
2014.
[17] R. Hiptmair, C. Jerez-Hanckes, J.-F. Lee, and Z. Peng. Domain decomposition for boundary
integral equations via local multi-trace formulations. In J. Erhel, M.J. Gander, L. Halpern,
G. Pichot, T. Sassi, and O. Widlund, editors, Domain Decomposition Methods in Science and
Engineering XXI., volume 98 of Lecture Notes in Computational Science and Engineering,
pages 43–58, Berlin, 2014. Springer.
[18] K. L. Ho and L. Greengard. A fast semidirect least squares algorithm for hierarchically block
separable matrices. SIAM J. Matrix Anal. Appl., 35(2):725–748, 2014.
[19] Carlos Jerez-Hanckes, Carlos P´erez-Arancibia, and Catalin Turc. Multitrace/singletrace for-
mulations and domain decomposition methods for the solution of helmholtz transmission prob-
lems for bounded composite scatterers. Journal of Computational Physics, 2017.
[20] R. Kress. A Nystr¨om method for boundary integral equations in domains with corners. Numer.
Math., 58(2):145–161, 1990.
[21] R. Kussmaul. Ein numerisches Verfahren zur L¨osung des Neumannschen Aussenraumproblems
f¨ur die Helmholtzsche Schwingungsgleichung. Computing (Arch. Elektron. Rechnen), 4:246–
273, 1969.
[22] E. Martensen. ¨
Uber eine Methode zum r¨aumlichen Neumannschen Problem mit einer Anwen-
dung f¨ur torusartige Berandungen. Acta Math., 109:75–135, 1963.
[23] W. McLean. Strongly elliptic systems and boundary integral equations. Cambridge University
Press, Cambridge, 2000.
[24] Fr´ed´eric Nataf. Interface connections in domain decomposition methods. In Modern methods
in scientific computing and applications (Montr´eal, QC, 2001), volume 75 of NATO Sci. Ser.
II Math. Phys. Chem., pages 323–364. Kluwer Acad. Publ., Dordrecht, 2002.
[25] V. Rokhlin. Solution of acoustic scattering problems by means of second kind integral equa-
tions. Wave Motion, 5(3):257 – 272, 1983.
[26] O Steinbach and M Windisch. Stable boundary element domain decomposition methods for
the helmholtz equation. Numerische Mathematik, 118(1):171–195, 2011.
[27] Christiaan C Stolk. A rapidly converging domain decomposition method for the helmholtz
equation. Journal of Computational Physics, 241:240–252, 2013.
[28] R. H. Torres and Grant V. Welland. The Helmholtz equation and transmission problems with
Lipschitz interfaces. Indiana Univ. Math. J., 42(4):1457–1485, 1993.
33
[29] Catalin Turc, Yassine Boubendir, Mohamed Kamel Riahi, et al. Well-conditioned boundary
integral equation formulations and nystr¨om discretizations for the solution of helmholtz prob-
lems with impedance boundary conditions in two-dimensional lipschitz domains. Journal of
Integral Equations and Applications, 29(3):441–472, 2017.
[30] Alexandre Vion and Christophe Geuzaine. Double sweep preconditioner for optimized schwarz
methods applied to the helmholtz problem. Journal of Computational Physics, 266:171–190,
2014.
[31] Markus Windisch. Boundary element tearing and interconnecting methods for acoustic and
electromagnetic scattering. Verlag der Techn. Univ. Graz, 2011.
[32] Leonardo Zepeda-N´unez and Laurent Demanet. The method of polarized traces for the 2d
helmholtz equation. Journal of Computational Physics, 308:347–388, 2016.
34
