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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

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