A Quasi-Optimal Non-Overlapping Domain Decomposition

Algorithm for the Helmholtz Equation

Y. Boubendir∗, X. Antoine†, C. Geuzaine‡

May 8, 2012

Abstract

This paper presents a new non-overlapping domain decomposition method for the Helmholtzequation, whose effective convergence is quasi-optimal. These improved properties result from acombination of an appropriate choice of transmission conditions and a suitable approximation of theDirichlet to Neumann operator. A convergence theorem of the algorithm is established and numericalresults validating the new approach are presented in both two and three dimensions.

1 Introduction

In this paper, we are interested in non-overlapping Domain Decomposition Methods (DDMs) for theHelmholtz equation. Such methods were introduced by Lions [36] for the Laplace equation and extendedto the Helmholtz equation by Despres [19, 20, 21]. Essentially, the method consists in combining thecontinuity conditions (of the field and its normal derivative) on the artificial interfaces between subdo-mains, in order to obtain Robin boundary conditions and to solve the overall problem by iterating overthe subdomains [39, 47, 45]. Robin conditions (also called absorbing or impedance boundary conditions)are chosen to couple the subdomains because using the natural conditions leads to divergent iterativealgorithms [12].

Improving the convergence properties of the iterative process constitutes the key in designing effectivealgorithms, in particular in mid and high frequency. The optimal convergence is obtained by defining thetransmission conditions, on each interface, using the Dirichlet-to-Neumann (DtN) operator correspondingto the problem sharing the same interface [41, 40]. This however leads to a very expensive procedurein practice. The first contribution following the original method of Despres, using particular non-localtransmission conditions, was made in [17] with a relative success regarding the effective convergence. Agreat variety of techniques based on local transmission conditions have also been proposed to improve theconvergence: these include the class of FETI-H methods [24, 25, 26, 14], the optimized Schwartz approach[27], and the evanescent modes damping algorithm [15, 16, 12]. However, the related impedance operatorsdo not accurately approximate the exact DtN operator on all the modes of the solution, which makes theresulting iterative methods suboptimal.

In this paper, we propose a new square-root based transmission condition, localized using Pade ap-proximants, which accurately approximates the DtN operator and allows to design an algorithm with

∗Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, NJIT, Univ. Heights. 323Dr. M. L. King Jr. Blvd, Newark, NJ 07102, USA.†Institut Elie Cartan Nancy (IECN), Nancy University, INRIA Corida Team, B.P. 239, F-54506 Vandoeuvre-les-Nancy

Cedex, France.‡University of Liege, Department of Electrical Engineering and Computer Science, Montefiore Institute B28, B-4000

Liege, Belgium.

1

Γ

Σij = ΣjiΓ∞

ni

Ωi

nj

Ωj

Figure 1: Example of a two-dimensional non-overlapping domain decomposition method.

quasi-optimal convergence properties. Indeed, we will show that the rate of convergence is optimal onthe evanescent modes and is significantly improved compared to current techniques for the remainingmodes. In the effective convergence, this results in a DDM independent of the wavenumber as well asthe mesh discretization. Moreover, we will see how the resulting approximate DtN operator is easy toimplement in a basic finite element solver.

The paper is organized as follows. In Section 2, we introduce the scattering problem as well as the non-overlapping DDM. We present in the third Section a non-local square-root operator which approximatesthe exact DtN transmission operator. Section 4 develops a convergence analysis for this approximatetransmission condition on a model problem. Section 5 details the complex Pade approximation of thesquare-root operator to get a local representation. Section 6 presents the finite element implementationof the resulting DDM. Numerical results on both two- and three-dimensional problems are presented inSection 7.

2 Scattering Problem and Non-Overlapping Domain Decompo-sition Method

Let us consider the three-dimensional time-harmonic scattering problem of an incident acoustic wave byan obstacle K. We want to compute the scattered field u solution to the exterior Helmholtz equationwith a Dirichlet boundary condition1:

∆u+ k2u = 0 in R3\K,u = f on Γ = ∂K,

lim|x|→∞

|x|(∂|x|u− ıku) = 0.(1)

The boundary data f is fixed by a plane wave: f = −eıkα·x, with x = (x1, x2, x3) ∈ R3 and ı =√−1.

The incidence angle α is normalized on the unit sphere (|α| = 1) and k denotes the wavenumber, related

1The Dirichlet boundary condition models a sound-soft obstacle; Neumann (sound-hard) or Fourier-Robin (impedance)boundary conditions may also be set, which would not fundamentally change the rest of the paper.

2

to the wavelength λ of the incident wave through k = 2π/λ. The last equation of system (1) is theSommerfeld radiation condition at infinity, which imposes that the scattered wave is outgoing.

To solve (1), we combine Absorbing Boundary Conditions (ABCs) with Lions-Despres’ non-overlappingdomain decomposition method. The ABC method consists in truncating the infinite domain by introduc-ing a fictitious boundary Γ∞ to get a bounded computational region. Then, system (1) is approximatedby

∆u+ k2u = 0 in Ω,

u = f on Γ,

∂nu+ Bu = 0 on Γ∞,

(2)

where Ω is the bounded domain enclosed by the fictitious boundary Γ∞ and Γ (see Figure 1) and wherethe operator B represents an approximation of the DtN operator (for example B = −ık) on Γ∞. Thevector n is the outwardly directed unit normal to Γ∞. Let us remark that all what follows can be adaptedto the case of other truncation techniques, like e.g. perfectly matched layers (PMLs) [13, 18, 44, 48].

The first step of the Lions-Despres domain decomposition method [36, 19] consists in splitting Ω intoseveral subdomains Ωi, i = 1, . . . , Ndom, such that (see Figure 1):

• Ω =⋃Ndom

i=1 Ωi (i = 1, . . . , Ndom),

• Ωi ∩ Ωj = ∅, if i 6= j, (i, j = 1, . . . , Ndom),

• ∂Ωi ∩ ∂Ωj = Σij = Σji (i, j = 1, . . . , Ndom) is the artificial interface separating Ωi and Ωj as longas its interior Σij is not empty.

The second step consists in reducing the solution of the initial problem (2) by solving the local transmissionproblems

∆u(n+1)i + k2u

(n+1)i = 0 in Ωi,

u(n+1)i = fi on Γi,

∂niu(n+1)i + Bu(n+1)

i = 0 on Γ∞i ,

(3a)

∂niu(n+1)i + Su(n+1)

i = g(n)ij on Σij , (3b)

and then in forming the quantities to be transmitted through the interfaces

g(n+1)ij = −∂nju(n+1)

j + Su(n+1)j = −g(n)

ij + 2Su(n+1)j on Σij , (4)

where ui = u|Ωi , ni (resp. nj) is the outward unit normal to the boundary of Ωi (resp. Ωj), i =1, . . . , Ndom, j = 1, . . . , Ndom, Γi = ∂Ωi ∩ Γ, Γ∞i = ∂Ωi ∩ Γ∞ and S an invertible operator. Let usnote that the boundary condition on Γi (resp. Γ∞i ) does not take place if the interior of ∂Ωi ∩ Γ (resp.∂Ωi ∩ Γ∞) is the empty set. We will assume in all that follows that the DDM is well-posed, in the sensethat each subproblem (3a)-(3b) is itself well-posed, i.e., away from interior resonances.

Solving at each step all the local transmission problems through (3)-(4) may be recast as one appli-cation of the iteration operator A : ×Ndom

i,j=1L2(Σij)→ ×Ndom

i,j=1L2(Σij) [11] defined by

g(n+1) = Ag(n) + b, (5)

where g(n) is the set of boundary data (g(n)ij )1≤i,j≤Ndom

, and b is given by the Dirichlet boundary condition.Therefore, (3)-(4) can be seen as an iteration of the Jacobi method (or fixed point iteration) applied tothe linear system:

(I −A)g = b, (6)

where I is the identity operator. Following this idea, any Krylov solver could also be applied to solvethis equation. Indeed, we will see that the GMRES algorithm significantly improves the iterative processover the successive approximations procedure (5).

3

3 Square-Root Based Transmission Condition

It is well-known that the convergence of the domain decomposition method for scattering problemsstrongly depends on the choice of the transmission operator S. Indeed, to each choice of operator S cor-responds an iteration operator A with particular spectral properties. Several techniques were developedto improve the convergence by modifying the original algorithm introduced by Despres in [19]. In thislast reference, the low-order approximation of the DtN operator

S0u = −ıku (7)

is used. For this choice of operator S, it can be shown [15] that the resulting iteration operator only actson the part of the spectrum corresponding to the propagating modes while the eigenvalues related to theevanescent modes have unit modulus [15]. This directly impacts the convergence (divergence) propertiesof the resulting iterative scheme.

Two families of techniques have been proposed to overcome this problem. First, algorithms based onthe optimization of the rate of convergence were introduced by Gander et al. [27], where improved localapproximations of the DtN map of order zero or order two are built. For a generic transmitting boundaryΣ the resulting optimized order zero (OO0) and optimized order two (OO2) local transmission operatorsare given by [27]:

SOO0u = au and SOO2u = au− b∆Σu, (8)

where the complex numbers a and b are obtained by solving a min-max optimization problem on the rateof convergence, and ∆Σ is the Laplace-Beltrami operator on the interface Σ. Second, the EvanescentModes Damping Algorithm (EMDA) was introduced by Boubendir in [15, 16], with the explicit aim todamp the evanescent modes:

SXu = −ıku+ Xu, (9)

where X is a self-adjoint positive operator. We only consider here the usual case where X is a real-valuedpositive coefficient.

In this paper we propose a new “square-root” transmission operator [6] that takes the following form:

Ssq,εu = −ık√

1 + divΣ(1

k2ε

∇Σ)u, (10)

wherekε = k + ıε (11)

is a complexified wavenumber, the operator divΣ is the surface divergence of a tangent vector field on Σand ∇Σ is the tangential gradient of a surface field. The square-root

√A of an operator A is classically

defined through the spectral decomposition of A [46]. Furthermore, the notation√z designates the

principal determination of the square-root of a complex number z with branch-cut along the negativereal axis. Note that, as will be seen later, ε can depend on geometrical parameters, like the meancurvature of the surface Σ. We will show in what follows that

1. the nonlocal operator Ssq,ε can be accurately localized using complex Pade approximants, andsuitably combined with finite element methods;

2. the convergence of the resulting DDM is quasi-optimal: the rate of convergence corresponding tothe evanescent modes is zero, and significantly improved for the remaining modes. This results inan effective solution where the iterative procedure is quasi independent of both the wavenumberand the level of accuracy of the spatial discretization (mesh refinement).

4

The Despres, OO0 and EMDA transmission operators (7), (8) (left) and (9) are particular cases ofImpedance Boundary Conditions [43]. The second order (8) (right) and the new square-root operator(10), as well as its Pade localization, fall into the realm of so-called Generalized Impedance BoundaryConditions [43]. In what follows we will refer to the transmission conditions related to the Despresand EMDA algorithms by IBC(0) and IBC(X ), respectively, the second order optimized transmissioncondition by OO2 and to the square-root transmission condition by GIBC(sq,ε).

3.1 Formal Construction of the Square-Root Transmission Operator

To explain the origin of the regularized square-root operator Ssq,ε given by (10), we first consider theparticular case of the half-space, where the following boundary value problem ∆u+ k2u = 0, in R3

+ = x ∈ R3;x1 > 0,u = g, on Σ,u is outgoing,

(12)

is posed. The straight transmitting boundary is Σ := x ∈ R3;x1 = 0. Therefore, x1 gives the normaldirection to Σ while (x2, x3) designate the tangential variables. We can write the Dirichlet-to-Neumann(DtN) operator Λ defined by

Λ : H1/2(Σ) −→ H−1/2(Σ)u|Γ 7−→ ∂nu|Γ = Λ(u|Γ)

(13)

explicitly for the problem (12) by Fourier analysis. Indeed, let us introduce ξ = (ξ2, ξ3) ∈ R2 as the Fouriercovariable of the tangential variable x⊥ = (x2, x3). We denote by Fx⊥ the partial Fourier transform withrespect to (x2, x3) and by F−1

ξ the associated inverse Fourier transform. Applying Fx⊥ to the Helmholtzequation leads to the solution of the ODE

∂2x1u(x1, ξ) + (k2 − |ξ|2)u(x1, ξ) = 0, (14)

for x1 > 0 and |ξ|2 = ξ22 + ξ2

3 . For conciseness, we set: u(x1, ξ) := Fx⊥u(x1, ξ). Let us consider σ1 as thesymbol defined by: σ1(k, ξ) = ık

√1− |ξ|2/k2. Then, the solution to (14) writes down

u(x1, ξ) = A+eσ1(k,ξ)x1 +A−e−σ1(k,ξ)x1 , (15)

for x1 ≥ 0. Since we are looking for a transmitting operator Λ, this implies that the solution u mustbe an outgoing solution and that its L2(R3

+)-norm is finite (to avoid any growing mode at infinity).This also means that u(x1, ξ) has a finite L2(R3

+)-norm by Parseval’s theorem. This can only arise if

A− = 0, resulting in the right traveling solution: u(x1, ξ) = A+eσ1(k,ξ)x1 . Deriving this last expressionwith respect to x1, applying the inverse Fourier transform in ξ and considering the trace of the resultingrelation on the transmitting boundary Σ leads to

∂nu(0,x⊥) = F−1ξ (σ1(k, ξ)u(x1, ξ))|Σ. (16)

In terms of pseudodifferential operators [46], another way of writing this equation is

∂nu(0,x⊥) = Op(σ1)u(0,x⊥), on Σ, (17)

which means that Op(σ1) is the pseudodifferential operator with symbol σ1. This relation provides Λsuch that: Λ := Op(σ1). Since the Helmholtz equation (12) has constant coefficients, it can be provedthat

Λ := Op(σ1) = Op(ık

√1− |ξ|

2

k2) = ık

√1 +

∆Σ

k2, (18)

5

where the Laplace-Beltrami operator over Σ is defined by: ∆Σ := ∂2x2

+ ∂2x3

. For a half-space, thetransmitting operator S is thus simply taken to be equal to

Ssq,0u = −Λu = −ık√

1 +∆Σ

k2u. (19)

Let us now consider a curved surface Σ. Then, if the surface is locally approximated by its tangentplane, we can formally propose a local, but approximate, representation of the surface DtN transmittingboundary condition based on (19), where ∆Σ := divΣ∇Σ is the Laplace-Beltrami operator for the curvedsurface Σ. However, this operator is not an accurate approximation of the DtN operator for the grazingrays [6]. This is a fundamental difference compared to the half-space case where this situation does notarise. Another explanation of this problem is that the symbol σ1 of the operator exhibits a singularity forthe tangential frequencies ξ such that: |ξ| ≈ k (the glancing region in terms of microlocal analysis [2]).The solution proposed in [6] is based on a regularized symmetrical form of the involved Laplace-Beltramioperator, which results in the regularized square-root transmitting operator (10). More details can befound in [2, 6] for a rigorous construction based on pseudodifferential operators and microlocal analysis.

3.2 Relation to Absorbing Boundary Conditions

The construction of the square-root transmission operator is closely related to previous developmentsin the context of Absorbing Boundary Conditions (ABC) for scattering problems, where the goal is tominimize reflection at the fictitious surface Γ∞. In the eighties, the Engquist-Majda [23] and Bayliss-Turkel [9, 2, 10] ABCs were developed, and are written as

∂nu+Mu = 0 on Γ∞, (20)

where M is a local approximation of the DtN operator that takes a form similar to the second ordertransmission operator (8) (right). While these conditions are widely used for practical computationsand most particularly for engineering purposes, it was recently shown in [6] that they also do not modelthe evanescent and grazing modes, resulting in a loss of accuracy of the method. In order to solve thisproblem, a high-order local ABC was introduced in [6, 34], which uses M = Ssq,ε in (20) to model all thescattering modes: propagating, evanescent as well as (in an approximate way) grazing. This ABC can belocalized with complex Pade approximants, and the coefficient ε in (11) can then be chosen to minimizespurious reflections at the boundary.

4 Convergence Analysis for a Model Problem

In order to study the convergence of the proposed DDM with square-root transmission operator, weanalyze the model problem depicted in Figure 2. This model problem couples two subdomains: a disk-shaped bounded subdomain Ω1 of radius R0 and an unbounded domain Ω0 = R2 \ Ω1:

Ω0 := x ∈ R2, |x| > R0, Ω1 := x ∈ R2, |x| < R0, (21)

with ∂Ω0 = ∂Ω1 = Σ. We study the spectral properties of the iteration operator obtained from thedomain decomposition algorithm coupling these two subdomains. Studying the coupling of bounded andunbounded subdomains will allow us to understand the main properties that one could not analyze byconsidering two bounded (e.g. a square domain divided in two) or two unbounded (e.g. two half-planes)subdomains. The considered model problem essentially contains the main difficulties encountered whensolving exterior scattering problems. Note that a closely related problem where the unbounded domain isreplaced by an annulus-shaped bounded domain with an ABC on the exterior boundary could also havebeen selected. However, this would only lead to more complex expressions without providing additionalinsight.

6

n1

Ω0

R0

Σ

Ω1

n0

Figure 2: Model problem with two subdomains and a circular interface.

4.1 Convergence Theorem

For the considered model problem, the domain decomposition method with square-root transmissioncondition consists in solving separately at each iteration the following problems

∆u0 + k2u0 = 0 in Ω0,

∂n0u0 + Ssq,εu0 = g0 on Σ,

lim|x|→∞

|x|1/2(∂|x|u0 − ıku0) = 0,(22)

and ∆u1 + k2u1 = 0 in Ω1,

∂n1u1 + Ssq,εu1 = g1 on Σ,(23)

where g0 and g1 are defined as in equation (4). The convergence analysis can be developed by studyingthe spectral properties of the iteration operator A defined by (5), where

A := ΠT , (24)

and where the two operators T and Π are defined as [15]:

T g(n) :=

( T1 0

0 T0

) g(n)1

g(n)0

and Π :=

(0 II 0

), (25)

withT1g

(n)1 := −g(n)

1 + 2Ssq,εu(n+1)1 , T0g

(n)0 := −g(n)

0 + 2Ssq,εu(n+1)0 . (26)

Since we have a circular geometry, we can analyze the error mode-by-mode by using a Fourier-Hankelseries expansion in the polar coordinates system (r, θ). Let us set

u`(r, θ) =

+∞∑m=−∞

u`,m(r)eımθ and g`(r, θ) =

+∞∑m=−∞

g`,m(r)eımθ, ` = 0, 1. (27)

7

Since eımθm∈Z defines an orthonormal basis of L2(Σ), problems (22) and (23) lead to solve the decoupledproblems

1

r∂r(r∂ru0,m)− m2

r2u0,m + k2u0,m = 0 for r > R0,

−∂ru0,m + Ssq,εm u0,m = g0,m for r = R0,

limr→+∞

r1/2 (∂ru0,m − ıku0,m) = 0,

(28)

and 1

r∂r(r∂ru1,m)− m2

r2u1,m + k2u1,m = 0 for r < R0,

∂ru1,m + Ssq,εm u1,m = g1,m for r = R0.

(29)

In the polar coordinate system the surface divergence and the tangential gradient appearing in (10)reduce to the curvilinear derivative R−1

0 ∂θ. Hence, for a mode m, we get the analytical expression of thesquare-root transmission operator:

Ssq,εm = −ık

√1− m2

k2εR

20

:= −ık(Rm + ıXm). (30)

The solution of the first and last equation of the exterior problem (28) is given by u0,m(r) =

αmH(1)m (kr), where H

(1)m denotes the Hankel function of the first kind and order m. For the interior

problem (29) the solution of the first equation reads u(1)m (r) = βmJm(kr), where Jm is the Bessel function

of order m.Writing the modal decomposition of the operators T0 and T1 as T0 =

∑+∞m=−∞ T0,me

ımθ and T1 =∑+∞m=−∞ T1,me

ımθ, one can show using the second equations of (28) and (29) that:

T`,m =−kZ`,m + Ssq,ε

m

kZ`,m + Ssq,εm

=−Z`,m + ı(Rm + ıXm)

Z`,m + ı(Rm + ıXm), ` = 0, 1, (31)

with

Z0,m =H

(1)′

m (kR0)

H(1)m (kR0)

and Z1,m = −J′m(kR0)

Jm(kR0). (32)

Since the convergence properties of the DDM are strongly related to the spectrum of the iterationoperator A, we study the spectrum of the modal matrices

Am =

(0 T0,m

T1,m 0

)(33)

for each mode, where A =∑+∞m=−∞Ameımθ. Let us begin by considering that the iteration equation (6)

is solved using the Jacobi algorithm. Then, according to Theorem 1 below (see [15]), convergence occursif and only if the spectral radius ρ(A) = maxm∈Zρ(Am) is smaller than one:

Theorem 1 Let s ∈ R be such that (g0, g1) ∈ V := H−s(Σ) × H−s(Σ). The domain decompositionalgorithm converges in V if and only if for all m, ρ(Am) < 1, with ρ(Am) the largest modulus of the twoeigenvalues of the 2× 2 matrix Am.

In our case, the two eigenvalues of Am are: λ±m = ±√T0,mT1,m. Hence, it is sufficient to prove that

|T0,mT1,m| < 1 for each mode m to guarantee the convergence of the iterative method. Before provingthis inequality, let us first establish the following lemma.

8

Lemma 2 Consider the function F defined by

F(R,X ,Z) =−Z + ı(R+ ıX )

Z + ı(R+ ıX ), (34)

where the real coefficients R and X are positive (R > 0, X > 0) and where Z is a complex number, with<(Z) := −x and =(Z) := y. Then, under the conditions x > 0 and y > 0, we have

|F(R,X ,Z)| < 1. (35)

Proof. By writing Z = −x+ ıy, we obtain

|F(R,X ,Z)|2 =(x−X )2 + (R− y)2

(x+ X )2 + (R+ y)2< 1, (36)

since (R− y)2 < (R+ y)2 and (x−X )2 ≤ (x+ X )2.To prove the convergence of the algorithm, we first study separately the two coefficients T0,m and

T1,m, related to the exterior and interior problems respectively. The behavior of the coefficients dependson the choice of the damping parameter ε appearing in the definition of the square-root operator (30)through (11).

Proposition 3 For ε > 0 and any m ∈ Z, we have:

|T0,m| < 1. (37)

If ε = 0 and m 6= ±kR0, then the inequality (37) also holds.

Proof. For any mode m, the coefficient Z0,m can be written as

Z0,m =H

(1)′

m (kR0)H(1)m (kR0)

|H(1)m (kR0)|2

, (38)

with =(H(1)′

m (kR0)H(1)m (kR0)) > 0 and <(H

(1)′

m (kR0)H(1)m (kR0)) < 0 (see [15] for the proof).

Let us first assume that ε = 0. If one has a propagating mode, i.e., |m| < kR0, then Ssq,0m = −ıkRm.

For the evanescent modes such that kR0 < |m|, Ssq,0m = kXm. In these two cases, we deduce that

|T0,m| < 1 by using lemma 2 if there is no mode m such that m = ±kR0.Let us now consider the damped version ε > 0 of the transmission operator Ssq,ε

m . We first recallthat if a and b are two real numbers such that b 6= 0, then

√a+ ıb = p+ ıq, where p and q are two real

numbers given by

p =1√2

√√a2 + b2 + a, q =

sign(b)√2

√√a2 + b2 − a. (39)

Expanding Ssq,εm = −ık(Rm + ıXm) as

Ssq,εm = −ık

√R2

0((k2 − ε2)2 + 4k2ε2)−m2(k2 − ε2) + 2ım2kε

R20((k2 − ε2)2 + 4k2ε2)

, (40)

and knowing that k > 0 and ε > 0, we get that Rm > 0 and Xm > 0. Using (39) and Lemma 2, thisallows us to conclude that |T0,m| < 1, for any m ∈ Z and ε > 0.

Concerning the interior problem, we have the following result.

9

Proposition 4 If ε ≥ 0 and |m| > kR0, we have

|T1,m| < 1. (41)

If ε = 0 and |m| ≤ kR0, we have|T1,m| = 1. (42)

Proof. We observe that Z1,m is real. Let us first consider the case when ε = 0. If |m| ≤ kR0, thenXm = 0 and we trivially have |T1,m| = 1. When |m| > kR0, we study the sign of Z1,m. Since

Z1,m = −J′

m(kR0)

Jm(kR0)= − u′1,m(R0)

ku1,m(R0),

by using the variational formulation of problem (29) we obtain

<(R0u

′1,m(R0)u1,m(R0)

)=

∫ R0

0

r|u′1,m|2 + (

m2

r2− k2)|u1,m|2r

dr.

Hence, for all |m| > kR0, we deduce that <(R0u′1,m(R0)u1,m(R0)) > 0. Furthermore, Rm = 0 but

Xm 6= 0, meaning that |T1,m| < 1.Let us now consider the case ε > 0. If |m| > kR0 the same procedure can be used, in which case

Rm > 0 and |T1,m| < 1.From the preceding propositions we can prove the following result.

Theorem 5 For ε = 0 and if m is not a cut-off mode, i.e. |m| 6= kR0, we have the inequality

|T0,mT1,m| < 1. (43)

If ε > 0, then there exists εmax such that for 0 < ε < εmax the inequality (43) is also satisfied.

Proof. Let ε = 0 and let us assume that there is no cut-off mode. Then, the result is a consequence ofPropositions 3 and 4. If ε > 0, the inequality (43) is true if m corresponds to an evanescent mode. How-ever, we cannot conclude directly for the propagating ones because of the operator T1. These propagatingmodes are in finite number and for ε = 0 we have (43). Therefore, we can then deduce by continuity onε that there exists εmax such that for 0 < ε < εmax, the inequality (43) is satisfied.

The preceding theorem ensures convergence of the Jacobi domain decomposition algorithm with thesquare-root operator when 0 < ε < εmax. The analysis is not valid for the cut-off mode |m| = kR0 whenε = 0: indeed, the operator Ssq,ε is then not invertible, which makes the domain decomposition algorithmill-posed. In the actual implementation of the DDM it is thus important to consider ε > 0 in order toguarantee the well-posedness of the DDM. In addition, as we will show, ε > 0 contributes in improvingthe convergence in the transition zone between the propagating and evanescent modes.

4.2 Quasi-Optimality

Since

H(1)m (t) ≈ −2m(m− 1)!

iπtmand Jm(t) ≈ − tm

2mm!for |m| |t|, (44)

we get, for |m| kR0,

Z0,m ≈ −m

kR0and Z1,m ≈

m

kR0(45)

and thuslim

m−→∞λ±m = ± ıε

−2k − ıε for ε > 0. (46)

10

This shows that there is a clustering of the eigenvalues for |m| kR0. Obviously, the limit is zero ifε = 0. Therefore, the rate of convergence is optimal for the evanescent modes. This property remainstrue for ε > 0 since k ε.

To demonstrate the efficiency of the algorithm on the rest of the modes, let us choose k = 6π, R0 = 1and a maximal number of modes mmax = [10kR0] (where [10kR0] denotes the integer part of 10kR0), andcompare the new algorithm with the Despres, EMDA and OO2 algorithms. Recall that the OO2 methodis based on optimizing the transmission coefficients a and b in relation (8) according to the followingmin-max problem posed in the Fourier space:

mina,b∈iR×R

(max

|ξ|∈[|ξmin|,k−]∪[k+,|ξmax|]|ρ(a, b, |ξ|)|

), (47)

where ρ is the rate of convergence computed in the case of the half-plane (see [27] for the details andnotations). The real positive parameters k− and k+ are introduced to exclude the Fourier modes ξ closeto the cut-off frequency |ξ| = k in the optimization problem. Frequencies ξmin and ξmax are fixed tosolve the min-max problem on a bounded domain. It is clear that ξmin should be fixed to zero and ξmax

should be chosen as the maximal number of modes for the Fourier series expansion: ξmax = mmax. (Ina finite element setting, ξmax is related to the smallest local mesh size h. For the numerical numericalexperiments in Section 7 we chose ξmax = π/h.) It can be proved [27] that the optimal coefficients a andb are given by

a = −α∗β∗ − k2

α∗ + β∗and b = − 1

α∗ + β∗, (48)

withα∗ = i

((k2 − k2

−) (k2 − |ξmin|2

))1/4and β∗ =

((k2

+ − k2) (|ξmax|2 − k2

))1/4. (49)

For an open square cavity with sidelength L the parameters k− and k+ are chosen in [27] as k± = k±∆k,with ∆k = π/L. However, this choice is unclear for general shapes, and a different setting of the min-maxproblem is required. In this work, to fix the exclusion interval size we consider ∆k = π/R0. This choicewill also be done for the numerical experiments in Section 7. Figure 3(b) displays the dependence of therate of convergence with respect to ∆k, showing that this choice is almost optimal for our model problem.Since OO2 essentially depends on ∆k, in what follows we use the notation OO2(∆k).

We report in Figure 3(a) the modal spectral radius ρ(Am) with respect to m for the transmittingboundary conditions IBC(0) (Despres), IBC(k/2) (EMDA), OO2(∆k) and GIBC(sq, 0). As is well-known,IBC(0) has a spectral radius equal to 1 for the evanescent modes, which is improved by EMDA—for whichthe radius of convergence is always less than one. OO2 further improves over EMDA, particularly forlarge spatial modes m. For the new transmission condition with ε = 0 (GIBC(sq,0)), we clearly observean optimal convergence rate for the evanescent part, which is coherent with our previous comment. Wealso observe a significant improvement over the Despres, EMDA and OO2 algorithms on the propagatingmodes.

The damping parameter ε can be optimized to further improve the spectrum of the iteration operatorcorresponding to the modes in the transition zone. We can formulate our optimization problem as a min-max problem: find εopt > 0 such that it minimizes the spectral radius ρ(Am) of the iteration operator(associated with GIBC(sq,ε)) for the mode m ∈ Z where it is maximal. Mathematically, this means thatwe want to solve the problem:

ρsq,εopt = minε∈R+

(maxm∈Z|ρ(Am)|

). (50)

By regularizing the iteration operatorAm thanks to ε, we also regularize the min-max problem, in contrastwith the OO2 approach where an exclusion interval [k−, k+] has to be introduced. By construction, it canthen be shown that the maximal error between the exact DtN operator and its square-root approximation

11

m

ρ(A

m)

m = kR0

IBC(0)

IBC(k/2)

OO2(π)

GIBC(sq,0)GIBC(sq,εopt)

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) Spectral radius of the modal iteration operator Am vs.the mode m.

X or ∆k or ερ(A

) X = k/2

ε = εopt

∆k = π

IBC(X )

OO2(∆k)

GIBC(sq,ε)

0 5 10 15 200.2

0.4

0.6

0.8

1

1.2

1.4

1.6

(b) Spectral radius of the iteration operator A vs. χ (forIBC), ∆k (for OO2) and the damping parameter ε (forGIBC).

Figure 3: Spectral radius of the iteration operators and optimal choice of the damping coefficient ε.

Ssq,ε arises for the modes m satisfying m = ±[kR0] [6]. Therefore,

ρsq,ε := |ρ(A[kR0])| = max|m|∈[0;mmax]

|ρ(Am)|

and the min-max problem (50) becomes a simple minimization problem:

ρsq,εopt = minε∈R+

ρsq,ε. (51)

For such problems it was shown in [6] that the optimal value of the damping parameter is εopt =0.4k1/3H2/3, where H is the mean curvature on Σ. The resulting modal spectral radius for GIBC(sq,εopt)is shown in Figure 3(a), where the improvement around m = kR0 is clearly visible. We can see in Figure3(b) that the spectral radius of the iteration operator is indeed minimum for ε = εopt. From Figure 3(b)one can also observe that Xopt = k/2 (the value chosen for EMDA in [15, 16]) is a fairly good choiceto minimize the spectral radius for IBC(X ). In this example, Xopt = 3k/4 could also be a good choice.Furthermore, OO2(∆k = π/R0) is a suitable choice since it is close to the optimal value (= 4 here). Wealso observe that if ∆k ≥ 3, the spectral radius is only slightly affected.

4.3 Krylov Subspace Solvers

Now, let us analyze the solution of (6) by using a Krylov subspace method like GMRES. It is well-knownthat fast convergence of the GMRES solver is related to the existence of clustering of its eigenvalues. Letus recall that the equation that we solve is given by the operator: (I−A). Therefore, its eigenvalues are:µ±m = 1 ∓ λ±m for each mode m ∈ Z. We report in Figure 4 the spectrum of the iteration operator forIBC(0), IBC(k/2), OO2(π) and GIBC(sq,εopt). We consider again kR0 = 6π for a maximal number ofmodes mmax = [10kR0]. For all transmission operators the spectrum lies in the right half-plane, whichmakes the GMRES converging. However, for IBC(0) many eigenvalues spread out in the complex plane.A slightly better clustering occurs for IBC(k/2) and OO2(π), while there is an excellent clustering of theeigenvalues for GIBC(sq, εopt). In particular, only a few eigenvalues related to the propagating modes

12

Real part

Imaginary

part

IBC(0)

IBC(k/2)

OO2(π)

GIBC(sq,εopt)

0 0.5 1 1.5 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 4: Eigenvalues distribution in the complex plane for (I −A) and different transmission operators.

do not cluster but are very close to (1, 0). Furthermore, the eigenvalues related to the evanescent modesseem to cluster at (1, 0). According to (46), we show that the clustering is at point

C∗ =k + ıεopt

k + ıεopt/2(52)

(in our case C∗ = 1.0008+0.0282ı). Since the eigenvalues cluster for the evanescent modes, we can expectthat in the numerical implementation the GMRES convergence rate will be independent of the densityof discretization points per wavelength nλ [3, 4, 5]. This will be indeed the case as seen in Section 7.

5 Localization of the Square-Root Operator by Complex PadeApproximants

The square-root operator (10) is a nonlocal operator (a pseudo-differential operator of order +1). There-fore, it is impractical in a finite element setting since it would lead to consider full matrices for thetransmission boundaries. Fortunately, a localization process of this operator can be efficiently done byusing partial differential (local) operators to have a sparse matrix representation. In [35, 34, 6], thisis realized by a rotating branch-cut approximation of the square-root and then applying complex Padeapproximants of order Np [38]:√

1 + divΓ∞(1

k2ε

∇Γ∞)u ≈ RαNp(divΓ∞(1

k2ε

∇Γ∞))u

= C0u+

Np∑`=1

A`divΓ∞(k−2ε ∇Γ∞)(1 +B`divΓ∞(k−2

ε ∇Γ∞))−1u,

(53)

13

Real part

Imaginary

part

GIBC(sq,εopt)

GIBC(4,π/4,εopt)

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Real part

Imaginary

part

GIBC(sq,εopt)

GIBC(8,π/4,εopt)

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Figure 5: Eigenvalue distribution in the complex plane for the exact and Pade-localized square-roottransmission operator of order 4 (left) and 8 (right).

which corresponds to the complex Pade approximation

√1 + z ≈ RαNp(z) = C0 +

Np∑`=1

A`z

1 +B`z. (54)

Let us mention some similar ideas in the approximation of the square-root with the approach describedin [29]. The complex coefficients C0, A` and B` are given by

C0 = eıα2 RNp(e−ıα − 1), A` =

e−ıα2 a`

(1 + b`(e−ıα − 1))2, B` =

e−ıαb`1 + b`(e−ıα − 1)

. (55)

In (55) α is the angle of rotation, (a`, b`), ` = 1, ..., Np, are the standard real Pade coefficients

a` =2

2Np + 1sin2(

`π

2Np + 1) , b` = cos2(

`π

2Np + 1), (56)

and RNp is the real Pade approximant of order N :

√1 + z ≈ RNp(z) = 1 +

Np∑`=1

a`z

1 + b`z. (57)

For a variational representation, the approximation of the Pade-localized square-root transmission oper-ator (10)-(57) is realized by using auxiliary coupled functions [35, 34, 6]

SNp,α,εu = −ık(C0u+

Np∑`=1

A`divΓ∞(1

k2ε

∇Γ∞ϕ`)) on Γ∞, (58)

14

where the functions ϕ`, ` = 1, .., Np, are defined on Γ∞ as the solutions of the following surface PDEs:

(1 +B`divΓ∞(1

k2ε

∇Γ∞))ϕ` = u. (59)

The resulting transmitting boundary condition is again a Generalized Impedance Boundary Condition,and is denoted by GIBC(Np, α, ε) for the Pade approximation with Np auxiliary functions, for an angleof rotation α and a damping parameter ε. The lowest-order approximation would be SNp,α,ε = −ıkIwhich also corresponds to the original approximation S0 of S, i.e., IBC(0).

Let us analyze on the model problem of Section 4 the effect of the Pade approximation on the rate ofconvergence. For a mode m, the transmission operator acts as

SNp,α,εm = −ıkRαNp(− m2

k2εR

20

). (60)

In the following the angle of rotation α is always taken equal to π/4, which was found to be the op-timal choice through numerical experiments. We report in Figure 5 the spectrum of GIBC(sq, εopt),GIBC(4, π/4, εopt) and GIBC(8, π/4, εopt). As previously noticed, there is an excellent clustering of theeigenvalues for GIBC(sq, εopt). As expected, the larger Np, the better the approximation of the spectrumof the square-root. Moreover, Np allows to adjust the spectrum accuracy for large modes m (evanescentmodes). Hence, it can be conjectured that Np must be taken with respect to the density of discretizationpoints per wavelength nλ. We will see in the numerical simulations (see Section 7) that relatively smallvalues of Np (Np = 2, 4, 8) give optimal convergence results.

6 Finite Element Implementation

Let us now describe the implementation of the domain decomposition algorithm with Pade-type trans-mission condition in a finite element context. Recall first that the iterative method consists in solvingproblems (3) and computing the transmitted quantities (4). We restrict this description only to a problemposed in a domain, noted here Ω, with no exterior interface nor interface connected to the scatterer. Thissimply implies that Γi = ∅, Γ∞i = ∅ and ∂Ω is an artificial interface. The extention to the other cases

is direct. For the sake of clarity, we abbreviate the notations u(n+1)i , g

(n)ij and g

(n+1)ij , used in problem

(3) and equation (4), by u, gin and gout respectively. An iteration then leads to solving first the interiorboundary value problem in terms of u with coupling auxiliary functions ϕ``=1,...,Np according to theuse of Pade approximation of order Np:

∆u+ k2u = 0 in Ω,

∂nu− ıkC0u− ıkNp∑`=1

A`div∂Ω(1

k2ε

∇∂Ωϕ`) = gin on ∂Ω,

−u+B`div∂Ω(1

k2ε

∇∂Ωϕ`) + ϕ` = 0, ` = 1, ..., Np, on ∂Ω,

(61)

where C0, A` and B` are given by (55). The auxiliary functions ϕ``=1,...,Np are introduced to dealwith the use of the Pade approximation of order Np. Considering now test-functions v ∈ H1(Ω) andv` ∈ H1(∂Ω), ` = 1, ..., Np, we get the coupled variational formulation

15

∫Ωi

(∇u · ∇v − k2uv) dΩ− ıkC0

∫∂Ω

uvd∂Ω

+ık

Np∑`=1

A`

∫∂Ω

1

k2ε

∇∂Ωϕ` · ∇∂Ωv d∂Ω = −∫∂Ω

ginv d∂Ω,

−∫∂Ω

uv` d∂Ω−B`∫∂Ω

1

k2ε

∇∂Ωϕ` · ∇∂Ωv` d∂Ω

+

∫∂Ω

ϕ`v` d∂Ω = 0, ` = 1, ..., Np.

(62)

Following the solution of (62), we update the boundary data through the relation

gout = −gin − 2ıkC0u− 2ık

Np∑`=1

A`div∂Ω(1

k2ε

∇∂Ωϕ`), on ∂Ω. (63)

Let us now consider a covering Ωh of Ω using nt tetrahedral (or triangular in two-dimensions) finiteelements with nv vertices. Parameter h is the usual finite element notation for the maximal side lengthof the tetrahedrals. All the notations are extended with h as subscript for the discrete version of thedomains as well as unknowns. We choose here linear finite element approximations but the extension tohigher-order elements is straightforward. For scattering problems, it is common to introduce the densityof discretization points per wavelength: nλ = λ/h. The local density of discretization points is therefore:nλ = λ/h. Let us denote by SΩh and MΩh respectively the stiffness and mass matrices for linear elementsassociated with the domains Ωh. The matrices have size nv × nv. Furthermore, we introduce S∂Ωh andM∂Ωh as the respective stiffness and mass matrices related to the transmitting surface ∂Ωh. If thesecorrespond to a generalized stiffness matrix for a surface function β, then it is quoted S∂Ωh

β . All these

matrices have a size n∂t × n∂t . Let us denote by u ∈ Cnv the local unknown vector and ϕ` ∈ Cn∂t thesurface unknown auxiliary vectors obtained with linear finite elements. The discrete test-vectors andright hand side are also bold typed. Then, the discretization of the variational problem (61) leads to thesolution of the coupled linear system

(SΩh − k2MΩh − ıkC0M∂Ωh)u + ık

Np∑`=1

A`S∂Ωhk−2εϕ` = −M∂Ωhgin

−M∂Ωhu− (B`S∂Ωhk−2ε−M∂Ωh)ϕ` = 0, ` = 1, ..., Np.

(64)

The size of this linear system is (nv + (1 +Np)n∂t )× (nv + (1 +Np)n

∂t ). Each augmented system is solved

directly by a LU factorization method since each matrix has a moderate size. The boundary data updatecan be done by vector addition using relation (4).

7 Numerical Results

This section is devoted to numerical simulations validating the new domain decomposition algorithmusing the Pade-localized square-root transmission condition. We use the finite element method withlinear (P1) basis functions implemented in GetDP/Gmsh [22, 28] to approximate the local problems (3),as explained in Section 6. The convergence criterion for all the presented examples is identical: theiterations are stopped when the initial residual has decreased by a factor of 10−6. In all the examplesthis stopping criterion leads to an L∞ numerical error between the full and the DDM finite elementsolutions of also about 10−6, independently of the chosen approximate transmission condition (IBC, OO2or GIBC).

16

−1.18 −0.00059 1.18

<(u)

(a) “circle-concentric” decomposition

−1.18 −0.00059 1.18

<(u)

(b) “circle-pie” decomposition

Figure 6: Two-dimensional test cases: reconstruction of the scattered field on the global domain after aDDM computation with k = 4π and Ndom = 5.

7.1 Two-Dimensional Examples

The two-dimensional examples concern the scattering of a plane wave by a unit sound-soft circularcylinder. The second-order Bayliss-Turkel [9, 2, 10] artificial boundary condition is set on a fictitiouscircular boundary with radius 4 (see Figure 6).

7.1.1 Jacobi vs. GMRES

We begin by analyzing the behavior of the iterative methods (successive approximations, i.e. Jacobi, orGMRES [42]) when the domain is partitioned into Ndom concentric subdomains. We call this partitioningthe “circle-concentric” decomposition (see Figure 6(a)). The “size” Li of the subdomain Ωi, defined asthe difference between its exterior and interior radius, is equal to 3/Ndom.

Le us start by considering a wavenumber k = π and a density of discretization points per wavelengthnλ = 12. We report in Table 1 (left) the number of iterations required using Jacobi and GMRES.Concerning the complex Pade approximation, we fix Np = 8 auxiliary equations and θ = π/4. The

damping parameter ε is optimized according to εopt = 0.6k1/3R−2/3j (where Rj is the radius of the j-th

interface). Let us precise that the “size” of the overall domain is L = 3 = 1.5λ. This means that in thecase of Ndom = 10 (resp. Ndom = 15) subdomains, the size of one subdomain is about 0.15λ (resp. about0.1λ) which is very small. For these cases, the EMDA and OO2 algorithms solved by the Jacobi methoddiverge while our approach converges. As it can be seen, the algorithm scales according to Ndom, and asexpected, the GMRES outperforms the Jacobi method.

17

Ndom Jacobi GMRES2 12 95 43 2510 140 5215 281 80

k Jacobi GMRESπ 96 41

2 π 64 363 π 52 394 π 60 38

Table 1: Number of iterations vs. number of subdomains for k = π (left) and Number of iterations vs.wavenumber for Ndom = 8 (right) when using the Jacobi or GMRES algorithm, for the “circle-concentric”decomposition.

Consider now a partition of the initial domain into Ndom = 8. The discretization is again fixed by

nλ = 12. The same parameters (Np = 8, θ = π/4 and εopt = 0.6k1/3R−2/3j ) are kept to approximate the

transmission operator given by (53). We report in Table 1 (right) the number of iterations for convergingaccording to the wavenumber k. As we can see, the convergence is quasi independent of the wavenumberk. This is known to be a very difficult goal to attain when designing iterative schemes for scatteringproblems. In particular, this means that our Pade-type transmission condition with GMRES DDM iswell-suited for high frequency scattering problems—we will analyze this behaviour further in Section 7.1.2.

From these first numerical results we clearly see that the GMRES algorithm outperforms the successiveapproximations (Jacobi), especially when the number of subdomains is large. In what follows we willthus only report results for the GMRES algorithm.

7.1.2 Influence of the Wavenumber, the Discretization Density and the Number of Sub-domains

Let us consider Ndom = 5 subdomains. Figure 7(a) shows the number of iterations to reach convergencewith respect to the wavenumber k, for two densities of discretization points per wavelength nλ. ForIBC(k/2) and OO2(π), the number of iterations increases with respect to k, whereas for GIBC(Np, π/4,εopt) the convergence rate is independent of the wavenumber. Furthermore, in this case, Np = 2 providesalready an optimal convergence compared with Np = 8. Let us recall that one of the important pointshere is that we use complex approximants. Using real Pade approximants would lead to much highervalues of Np, hence penalizing the computational cost of the overall method. Let us also mention thatincreased values of nλ are not only necessary for accuracy purposes but are also imposed to avoid pollutioneffects in the calculation of the solution at high wavenumbers [31, 33, 32, 8]. Figure 7(b) gives the numberof iterations with respect to the density of discretization points per wavelength nλ, for two wavenumbersk. A convergence independent of the mesh size can be achieved provided that Np is sufficiently large,although a very small value (Np = 2) already provides a quasi-optimal result. The number of iterationsfor OO2(π) is also independent of the mesh refinement, whereas the number of iterations for IBC(k/2)grows linearly.

In Figure 7(c) we report the number of iterations of the GMRES DDM with respect to the number ofsubdomains Ndom. We directly see that all the methods scale. Again, for GIBC(Np,π/4,εopt), the scalingdoes not depend on k (and Np), while this is not the case for IBC(k/2) and OO2(π).

As a partial conclusion, we can state that using the GMRES algorithm with GIBC(Np,π/4,εopt) in theDDM leads to a quasi-optimal convergence: the convergence rate is independent of both the wavenumberk and the density nλ. Furthermore, the method scales with the number Ndom of subdomains. Theseconclusions are confirmed by analyzing the behavior of our algorithms for the “circle-pie” decomposition(see Figure 6(b)). Since we have straight interfaces we cannot consider our optimized value of εoptanymore. However, numerical simulations show that εopt = k/4 is a suitable choice. In figures 8(a)-8(c), we represent the same three sets of curves as for the “circle-concentric” decomposition. We get

18

k

GMRESiterations

circle-concentric, Ndom = 5

OO2(π) nλ = 10OO2(π) nλ = 20IBC(k/2) nλ = 10IBC(k/2) nλ = 20GIBC(2,π/4,ǫopt) nλ = 10GIBC(2,π/4,ǫopt) nλ = 20GIBC(8,π/4,ǫopt) nλ = 10GIBC(8,π/4,ǫopt) nλ = 20

0 5 10 15 20 2510

20

30

40

50

60

70

80

90

100

(a) Number of iterations vs. wavenumber

nλ

GMRESiterations

circle-concentric, Ndom = 5

OO2(π) k = 2πOO2(π) k = 6πIBC(k/2) k = 2πIBC(k/2) k = 6πGIBC(2,π/4,ǫopt) k = 2πGIBC(2,π/4,ǫopt) k = 6πGIBC(8,π/4,ǫopt) k = 2πGIBC(8,π/4,ǫopt) k = 6π

5 10 15 20 25 30 35 4020

30

40

50

60

70

80

90

100

(b) Number of iterations vs. mesh density

Ndom

GMRESiterations

circle-concentric, nλ = 10

OO2(π) k = 2πOO2(π) k = 6πIBC(k/2) k = 2πIBC(k/2) k = 6πGIBC(2,π/4,ǫopt) k = 2πGIBC(2,π/4,ǫopt) k = 6πGIBC(8,π/4,ǫopt) k = 2πGIBC(8,π/4,ǫopt) k = 6π

0 5 10 15 200

20

40

60

80

100

120

140

160

180

200

(c) Number of iterations vs. number of subdomains

Figure 7: Convergence for the “circle-concentric” decomposition.

19

k

GMRESiterations

circle-pie, Ndom = 5

OO2(π) nλ = 10OO2(π) nλ = 20IBC(k/2) nλ = 10IBC(k/2) nλ = 20GIBC(2,π/4,k/4) nλ = 10GIBC(2,π/4,k/4) nλ = 20GIBC(8,π/4,k/4) nλ = 10GIBC(8,π/4,k/4) nλ = 20

0 5 10 15 20 2510

15

20

25

30

35

40

45

50

(a) Number of iterations vs. wavenumber

nλ

GMRESiterations

circle-pie, Ndom = 5

OO2(π) k = 2πOO2(π) k = 6πIBC(k/2) k = 2πIBC(k/2) k = 6πGIBC(2,π/4,k/4) k = 2πGIBC(2,π/4,k/4) k = 6πGIBC(8,π/4,k/4) k = 2πGIBC(8,π/4,k/4) k = 6π

5 10 15 20 25 30 35 4010

15

20

25

30

35

40

45

50

55

60

(b) Number of iterations vs. mesh density

Ndom

GMRESiterations

circle-pie, nλ = 10

OO2(π) k = 2πOO2(π) k = 6πIBC(k/2) k = 2πIBC(k/2) k = 6πGIBC(2,π/4,k/4) k = 2πGIBC(2,π/4,k/4) k = 6πGIBC(8,π/4,k/4) k = 2πGIBC(8,π/4,k/4) k = 6π

0 5 10 15 2010

15

20

25

30

35

40

45

50

55

60

(c) Number of iterations vs. number of subdomains

Figure 8: Convergence for the “circle-pie” decomposition.

20

Number of Unknowns Factorization Time (s)IBC(k/2) 20812 0.37

GIBC(1, π/4, k/4) 21053 0.37GIBC(2, π/4, k/4) 21294 0.38GIBC(4, π/4, k/4) 21776 0.39GIBC(8, π/4, k/4) 22740 0.40GIBC(16, π/4, k/4) 24668 0.45GIBC(32, π/4, k/4) 28524 0.53

Table 2: Number of degrees of freedom and computational time for the initial factorization for onesubdomain of the “circle-pie” decomposition, with Ndom = 5, k = 8π and nλ = 10.

a convergence independent of the wave number k and density of discretization nλ (by taking Np largeenough but relatively small). Again, the algorithm scales with respect to Ndom. More generally, theconvergence results are even better for the “circle-pie” than for the “circle-concentric” decomposition.Note that OO2(π) behaves significantly better in the “circle-pie” decomposition compared to the “circle-concentric” case. This can be understood because each subdomain in the “circle-pie” decomposition iscloser to the open cavity problem for which the OO2 parameters have been optimized.

Let us remark here that in the context of the Krylov iterative solution of integral equations for acousticscattering, quasi optimal convergence independent of the wavenumber and density of discretization pointsper wavelength was obtained in [3, 4, 5] by considering Generalized Combined Field Integral Equationsincluding the square-root operator.

7.1.3 Computational Cost

The overall computation time of the DDM depends on the number of iterations and the cost of solving(64) for each subdomain. As explained in Section 6, we use a direct sparse linear solver, and the maincost in solving (64) is thus the initial computation of the LU factorization of the augmented matrices ofsize N = (nv + (1 +Np)n

∂t )× (nv + (1 +Np)n

∂t ). We report in Table 2 the number of unknowns N and

the factorization time for one subdomain in the case of the “circle-pie” decomposition, with Ndom = 5,k = 8π and nλ = 10. The factorization is performed using MUMPS [1] on a 2.66 GHz MacBook Prolaptop computer. For the typical values of Np required to obtain quasi-optimal convergence (Np = 2 orNp = 8), we see that the computational overhead is small (less than 10%).

7.2 Three-Dimensional Example

We now test our algorithm on a more realistic three-dimensional example: the scattering by a submarine.The geometry is represented in Figure 9. For comparison purposes, we choose a unit length submarine.The incident wave is a plane wave with α = (cosπ/8, 0, sinπ/8)T . We consider two high wave numbers,k = 14π and k = 28π, a density of discretization points per wavelength nλ = 10 and we fix Ndom = 5subdomains. The scattered field computed with the GMRES DDM and Pade-type transmission boundarycondition is represented on Figure 9. Figure 10 reports the residual decay for different transmissionconditions and parameter values. Since we have a plane interface here, we cannot consider our optimizedvalue of εopt. As for the “circle-pie” decomposition analyzed in Section 7.1, numerical simulations showthat εopt = k/4 is a suitable choice. We see that GIBC(Np, π/4, εopt) leads to a very good convergencerate of the GMRES, and is clearly better than EMDA (with IBC(k/2)). For this three-dimensional case,an optimal value of Np to get the quasi-optimality is Np = 8.

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−1.57 −1.1 −0.631 −0.16 0.311 0.782 1.25

<(u)x

y

z

Figure 9: Submarine problem with 5 subdomains: iso-surfaces of the real part of the scattered field fork = 14π.

GMRES Iterations

Residual

submarine, Ndom = 5, nλ = 10

GIBC(8,

IBC(k/2) k = 14πGIBC(2,π/4,k/4) k = 14πGIBC(8,π/4,k/4) k = 14πIBC(k/2) k = 28πGIBC(2,π/4,k/4) k = 28πGIBC(8,π/4,k/4) k = 28π

0 10 20 30 40 50 60 70 8010−6

10−5

10−4

10−3

10−2

10−1

100

Figure 10: Convergence of the GMRES DDM solvers for the submarine problem.

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8 Conclusion

We have proposed a quasi-optimal non overlapping domain decomposition algorithm for the Helmholtzequation. It is based on a suitable approximation of the DtN operator. A convergence analysis for amodel problem has been developed, showing the main features of our algorithm. Several numerical testsin both two- and three dimensions validates the numerical quasi-optimality of the proposed algorithm.Extension to Maxwell’s equations is currently being developed.

Acknowledgments

Y. Boubendir gratefully acknowledges support from NSF through grant No. DMS–1016405. X. An-toine gratefully acknowledges support from the Agence Nationale pour la Recherche (Ref : ANR-09-BLAN-0057-01) (http://microwave.math.cnrs.fr/) and the Fondation de Recherche pour l’Aeronautiqueet l’Espace (IPPON Project). C. Geuzaine gratefully acknowledges support from the Belgian Science Pol-icy (IAP P6/21), Belgian French Community (ARC 09/14-02) and Walloon Region (WIST3 No 1017086“ONELAB”).

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