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Superalgebraically Convergent Smoothly-Windowed Lattice Sums

for Doubly Periodic Green Functions in Three-Dimensional Space∗

Oscar P. Bruno†

, Stephen P. Shipman‡

, Catalin Turc§

, Stephanos Venakides¶

Abstract

This work, Part I in a two-part series, presents (i) A simple and highly eﬃcient algorithm

for evaluation of quasi-periodic Green functions, as well as (ii) An associated boundary-integral

equation method for the numerical solution of problems of scattering of waves by doubly periodic

arrays of scatterers in three-dimensional space. Except for certain “Wood frequencies” at which

the quasi-periodic Green function ceases to exist, the proposed approach, which is based on

smooth windowing functions, gives rise to tapered lattice sums which converge superalgebraically

fast to the Green function—that is, faster than any power of the number of terms used. This

is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the

absence of smooth windowing. (The Wood-frequency problem is treated in Part II.) This paper

establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety

of numerical results demonstrate the practical eﬃciency of the proposed approach.

Keywords: scattering, periodic Green function, lattice sum, smooth truncation, super-

algebraic convergence, boundary-integral equations.

1 Introduction

The numerical solution of problems of electromagnetic, acoustic and elastic wave scattering by

doubly periodic structures entails signiﬁcant diﬃculties. Assuming harmonic temporal dependence

with frequency ω, the scattered ﬁelds can be obtained by means of numerical methods based

on integral equations—provided that a viable numerical scheme is used to evaluate the classical

radiating quasi-periodic Green function Gqper for the three-dimensional scalar Helmholtz operator

H[u] = ∆u+k2u(k=ω/c where cis the propagation speed). The diﬃculties arise, to a signiﬁcant

extent, from challenges posed by the evaluation of the quasi-periodic Green function.

The quasi-periodic Green function Gqper can be constructed as an inﬁnite sum of free-space

Green functions (Helmholtz monopoles) with doubly periodically distributed monopole singulari-

ties. Let v1and v2denote two independent vectors in R2that characterize the periodicity, and

let v∗

1and v∗

2be the dual vectors, that is v∗

i·vj=δij. The Bloch wavevector will be denoted by

∗This is a preprint of a published article: O. P. Bruno, S. P. Shipman, C. Turc, S. Venakides, Proc. R. Soc. A

2016 472 20160255; DOI: 10.1098/rspa.2016.0255, published 6 July 2016. Please see that publication for the correct,

updated version.

†Applied and Computational Mathematics, Caltech, Pasadena, CA 91125. Email: obruno@caltech.edu

‡Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803. Email: shipman@math.lsu.edu;

ORCID orcid.org/0000-0001-6620-6528

§Dept. of Math. Sciences, New Jersey Inst. of Technology, Newark, NJ 07102. Email: catalin.c.turc@njit.edu

¶Dept. of Mathematics, Duke University, Durham, NC 27708. Email: ven@math.duke.edu

1

k=αv∗

1+βv∗

2, where αand βare the Bloch wavenumbers. With the notation |·|for vector norm

and x= (x, y, z) and ˜

x= (x, y) and

r2

mn =|˜

x+mv1+nv2|2+z2,(1)

the quasi-periodic Green function can be expressed in the form

Gqper (x) = 1

4πX

m,n∈Z

eikrmn

rmn

e−ik·(mv1+nv2).(2)

Notice that k·(mv1+nv2) = αm+βn. The function Gqper(x) possesses the quasi-periodic property

Gqper (˜

x+mv1+nv2, z) = Gqper(˜

x, z)ei(αm+βn).(3)

The series expansion (2) possesses notoriously poor convergence properties. Various methods

to accelerate its convergence, notably the Ewald method [16, 13, 22], have been proposed. A survey

in these regards is given in [18], and a comprehensive discussion of lattice summation techniques

can be found in [3]. A few remarks concerning the computational costs associated with previous

accelerated methods for evaluation of the Green function (2) are made below in this section.

In the approach proposed presently, the inﬁnite sum (2) is evaluated by multiplying its (m, n)-th

term by the value χa(˜rmn) of a slow-rise smooth windowing function χawhich, evaluated at the

cylindrical radius

˜rmn =|˜

x+mv1+nv2|,(4)

restricts the sum to values of mand nsatisfying 0 ≤˜rmn ≤a. (Note that ˜rmn =rmn if and only if

z= 0.) The function χa=χa(˜r) is obtained as a scaled version of an inﬁnitely smooth real-valued

function χ(˜r) that equals zero for ˜r > 1 and equals 1 for ˜r < c, where c < 1 is an adequately

selected positive number. (For the numerical experiments presented in this paper the value c= 0.5

was used.) The function χais then deﬁned by

χa(˜r) = χ(˜r/a).(5)

The function χadecreases from 1 to 0 in a slow and smooth manner: its derivatives tend to zero

as a→ ∞ throughout the region of decrease c·a≤˜r≤a.

The main results in this contribution include (i) A proof, presented in Section 2, establishing

that, as the truncation radius atends to +∞, the smoothly truncated Green function converges

faster than any negative power of a—at least for arrangements of the period, frequency, and Bloch

wavenumbers that lie away from certain “Wood conﬁgurations” (for which the Green function Gqper

ceases to exist); as well as (ii) A new accelerated integral-equation solver presented in Section 3

which, relying on the aforementioned windowed Green function, gives rise to a highly-eﬃcient overall

solution method for the problems at hand. Theorem 2.1 below establishes the super-algebraically

fast convergence of the truncated sum to the three-dimensional quasi-periodic Green function away

from Wood conﬁgurations; a corresponding convergence theorem for 1D-periodic diﬀraction gratings

in R2was presented in [19], cf. also [4]. Figures 1 and 2 in Section 2 demonstrate the convergence

of the windowed series both near and away from Wood conﬁgurations. The numerical methods

presented in Section 3, in turn, integrate the windowed Green function in the context of fast integral-

equation solvers [8, 9]. Interestingly, the structure of the acceleration methodology inherent in these

solvers is exploited to completely avoid evaluation of the windowed Green function at pairs of surface

2

points, using instead a much smaller number of values of the Green function on a certain three-

dimensional Cartesian grid. A variety of numerical results presented in Section 4 demonstrate the

character of the resulting solvers for doubly periodic scattering problems. Green function methods

that are valid even at and around Wood conﬁgurations are presented in [4] for two-dimensional

conﬁgurations, and in Part II for the three-dimensional case.

As is well known, for certain wave numbers kand certain Bloch wave numbers (α, β), the lattice

sum (2) does not converge. This can be seen in the spectral representation of the Green function

that results by applying the Poisson Summation Formula to the series (2). Let A=kv1×v2k.

Then

Gqper (˜

x, z) = i

2AX

j,`∈Z

1

γj`

ei[(2πj v∗

1+ 2π` v∗

2)+k]·˜

xeiγj` |z|,(6)

in which the propagation constants γj` are deﬁned by

v∗

j` = (2πj v∗

1+ 2π` v∗

2) + k, γj` = (k2− kv∗

j`k2)1

2.(7)

(The branch of the square root that deﬁnes γj` is selected in such a way that √1 = 1 and that the

branch cut is the negative imaginary semiaxis.) The lattice sum (2) converges if and only if γj` 6= 0

for all integer pairs (j, `). Conﬁgurations (k, α, β ) for which γj` vanishes for one or more integer

pairs (j, `) are known as Wood conﬁgurations, or Wood anomalies. Clearly the expression (6) is not

meaningful if γj` = 0 for some integer pair (j, `). Wood anomalies were ﬁrst noticed by Wood [26]

and ﬁrst treated mathematically by Rayleigh [23]; a brief discussion concerning historical aspects

can be found in [4, Remark 2.2]. As shown in [4] and Part II, Green function methods can still be

used at Wood anomalies provided appropriately deﬁned Green functions are used.

In view of the branch used in equation (7) for the square root function, Rayleigh waves either

decay as |z|increases (evanescent modes) or are outgoing traveling waves (propagating modes).

There exist ﬁnitely many propagating modes for any given conﬁguration. Wood frequencies are

also called “cutoﬀ frequencies”, since the corresponding Rayleigh wave eiv∗

j` ·˜

xeiγj` |z|switches from

propagating to evanescent as the frequency descends below a Wood value. Rayleigh waves for which

γj` is small impinge upon the periodic structure at “grazing incidence”, and they dominate the sum

(6). In the limit of a particular combination of kand (α, β), at which one or more γj` are zero,

the product of the sum multiplied by any one of the vanishing γj`’s tends to a z-independent linear

combination of exactly grazing waves of the form eiv∗

j` ·˜

x.

Challenges in the calculation of the Green function arise from two main sources, namely

1. The lattice sum (2) does not converge absolutely. This sum does converge conditionally away

from Wood anomalies [10], but its convergence, which results from cancellations amongst

slowly decreasing terms, is too slow to be useful from a computational standpoint.

2. At Wood conﬁgurations the lattice sum (2) does not converge and a denominator in the

Rayleigh-wave expansion (6) exactly vanishes. Additionally, the convergence of the series (2)

increasingly deteriorates as the parameters in the problem are varied in such a way that a

Wood conﬁguration is approached.

The ﬁrst of these challenges is addressed in the present article, and the second is treated in [4] for

the two-dimensional case, and in Part II [11] for three dimensions.

As mentioned above, the proposed approach for summation of the series is based on smooth

windowing of the series (2). A similar windowed-summation technique can be applied to the spectral

3

series (6) with similar super-algebraic convergence. A study of the potential advantages oﬀered by

such a strategy is left for future work.

Previous accelerated procedures based on either or both of the spatial and spectral represen-

tations for the Green function Gqper give rise to signiﬁcantly faster algorithms than does direct

summation of either the expressions (2) or (6). The two-dimensional algorithms (see e.g. [20, Sec-

tion 3.8.2]) and [25]) can be perfectly adequate, but in the three-dimensional context algorithms for

evaluation of quasi-periodic Green functions have remained ineﬃcient. As a signiﬁcant reference

in these regards we mention one of the most advanced hybrid approaches previously put forth for

evaluation of periodic Green functions [17], which is based on use of a combination of spatial and

spectral representations as well as Kummer and Shanks transforms. The hybrid algorithm [17] has

been reported [2] (cf. also [17]) to require several milliseconds per evaluation point. Thus, even for

a small discretization consisting of N= 6 ×16 ×16 points (assuming a total of 6 patches are used

to represent a given scattering surface S, and 6 ×6 discretization points are used in each patch)

the number 2 ×106of evaluations of periodic Green functions which are necessary to evaluate one

matrix-vector product requires a computational time of at least 2 ×103seconds. In contrast, as

it can be seen in Table 2, in the case of periodic two-dimensional arrays of spheres discretized by

means of such a 6×16 ×16 mesh, our solvers require less than 10 seconds per matrix-vector product

(an improvement factor of a least one-hundred)—and can produce full scattering results with an

error of the order of 10−4in a total of 55 seconds.

Boundary-integral equations based on the proposed Green-function methods are presented in

Section 3. In particular, Section 3 describes the numerical methods used to implement the proposed

fast lattice sums and forward maps (matrix-vector products) which, upon use of an iterative linear

algebra solver (GMRES) produces the densities in certain boundary-integral representations of

the scattered ﬁeld. In all numerical examples it was assumed the scatterers satisfy sound-soft

(Dirichlet) boundary conditions. Section 4 demonstrates the resulting method by means of a

variety of numerical results. A few concluding remarks are presented in Section 5.

2 Proof of fast convergence of smoothly truncated lattice sums

Our smooth truncation method proceeds by multiplying the (m, n)-th term of the series (2) by

the scaled cut-oﬀ function χ(˜rmn/a) deﬁned in equation (5); the smoothly truncated series is thus

given by the ﬁnite sum

Ga(x, y, z) := 1

4πX

m,n∈Z

eikrmn

rmn

e−ik·(mv1+nv2)χ(˜rmn/a)≈Gqper(x, y, z),(8)

where of rmn and ˜rmn are given by (1) and (4). The following theorem establishes the super-

algebraic convergence of the truncated lattice sum to the quasi-periodic Green function for triples

(k, α, β) that are not Wood conﬁgurations.

Theorem 2.1 (Windowed Green function at non-Wood frequencies: Super-algebraic convergence).

Let χ(r)be an inﬁnitely smooth truncation function which equals to 1for r≤r1and equals 0for

r≥r2(0< r1< r2). If γj` 6= 0 for all (j, `)∈Z2, then the functions

Ga(x, y, z) = 1

4πX

m,n∈Z

eikrmn

rmn

e−ik·(mv1+nv2)χ(˜rmn/a)

4

converge to the radiating quasi-periodic Green function Gqper(x, y, z)super-algebraically fast as the

truncation radius atends to inﬁnity. In detail, for each posistive integer n, there exist constants

Cn=Cn(k, α, β)such that

|Ga

k(x, y, z)−Gqper(x, y, z)|<Cn(k, α, β)

an(9)

when ais suﬃciently large. The inequality holds uniformly for all points (x, y, z), excluding the

singularities of the Green function for which rmn = 0 for some (m, n)∈Z2. At these points, a

term that is common to Ga

kand Gqper is inﬁnite. If Ga

kand Gqper are modiﬁed by excluding this

term, then the correspondingly modiﬁed version of equation (9) remains valid.

An analogous estimate holds for k∇Ga

k(x, y, z)− ∇Gqper(x, y, z)k.

Proof. Denote by Λ = {mv1+nv2:m, n ∈Z}the lattice of singularities of the Green function,

and denote by Λ∗={jv∗

1+`v∗

2:j, ` ∈Z}the dual lattice. The dual vectors v∗

1and v∗

2are deﬁned

by v∗

i·vj=δij. Initially, we assume that the shift ˜

x= (x, y) from these positions as well as

the Bloch wavenumbers αand βare equal to zero. Setting ε=a−1, we have for the full and the

truncated sums,

4πGqper =X

r∈Λ

exp(ikp|r|2+z2)

p|r|2+z2(10)

and

4πGa=X

r∈Λ

χ(ε|r|)exp(ikp|r|2+z2)

p|r|2+z2.(11)

With the intention of utilizing the Poisson summation formula to calculate the truncated sum we

introduce a smooth function φ(|r|) that vanishes in a neighborhood of |r|= 0 and is equal to 1 for

|r| ≥ r1. For ε < 1, the sum is broken into two pieces,

X

r∈Λ

χ(ε|r|)exp(ikp|r|2+z2)

p|r|2+z2

=X

06=r∈Λ

(1 −φ(|r|))exp(ikp|r|2+z2)

p|r|2+z2+X

r∈Λ

φ(|r|)χ(ε|r|)exp(ikp|r|2+z2)

p|r|2+z2.(12)

In the ﬁrst sum on the right, χis omitted as a factor since it equals unity when φ6= 1. The term

is thus independent of the truncation variable ε. It is easy to check that the fraction in the second

term can be expressed as a product of an exponential function and a Laurent expansion:

exp(ik√r2+z2)

√r2+z2=eikr

rg(r), g(r) = 1 +

∞

X

j=1

ajr−j.(13)

The coeﬃcients ajare functions of zand the expansion is convergent when r > |z|.

We re-express the second sum in (12) by means of the Poisson summation formula:

X

r∈Λ

φ(|r|)χ(ε|r|)g(|r|)eik|r|

|r|=1

AX

ξ∈Λ∗F"φ(|r|)χ(ε|r|)g(|r|)eik|r|

|r|#(ξ),(14)

where A=kv1×v2k. In what follows we re-express the Fourier transform on the right-hand side of

this equation (which, for brevity, we denote by F(ξ)) in terms of suitable contour integrals. To do

5

this, we represent the spatial and Fourier variables in polar coordinates, r= (r, θ) and ξ= (ξ, γ),

and we let f(r) = φ(r)g(r), and we thus obtain

F(ξ) = Z∞

0Zπ

−π

f(r)χ(εr)ei(k−2πξ cos(θ−γ))rdθdr

= 2Z∞

0

f(r)χ(εr)Zπ

0

ei(k−2πξ cos θ)rdθdr = 2Z∞

0

f(r)χ(εr)Z1

−1

ei(k−2πξs)rds

√1−s2dr

= 2 Z∞

0

f(r)χ(εr)Z−1−i∞

−1

ei(k−2πξs)rds

√1−s2dr −Z1−i∞

1

ei(k−2πξs)rds

√1−s2dr.(15)

The last equality is valid by contour integration in the complex s-plane in view of the exponential

decay of the integrand as Im(s)→ −∞. We have thus obtained

F(ξ) = 2 Z−1−i∞

−1

I(s)ds

√1−s2−Z1−i∞

1

I(s)ds

√1−s2(16)

where

I(s) = Z∞

0

f(r)χ(εr)ei(k−2πξs)rdr . (17)

The integrand (17) decays exponentially fast at inﬁnity since Im(s)<0. Thus, integration by parts

(in which the boundary terms vanish because f(r) vanishes near r= 0) yields

I(s) = 1

i(k−2πξs)Z∞

0f0(r) + f0(r)(χ(εr)−1) + εf (r)χ0(εr)ei(k−2πξs)rdr

=I0(s) + Iε(s) (18)

where

I0(s) = 1

i(k−2πξs)Z∞

0

f0(r)ei(k−2πξs)rdr =1

[i(k−2πξs)]nZ∞

0

f(n)(r)ei(k−2πξs)rdr, (19)

and where, noting that {χ= 1}⊇{φ6= 1}for ε < 1, we have χ−1 = 0, χ0= 0 and f(r) = g(r) in

the region {χ= 1}, and, thus

Iε(s) = 1

i(k−2πξs)Z∞

0g0(r)(χ(εr)−1) + εg(r)χ0(εr)ei(k−2πξs)rdr. (20)

Thus, introducing a rescaled version gεof the function g,

gε(ρ) = g(ρ/ε) = ρ

pρ2+ (εz)2exp ikz2ε

ρ+pρ2+ (εz)2!= 1 +

∞

X

j=1

aj

εj

ρj(21)

the integrals Iε(s) become

Iε(s) = ε

i(k−2πξs)Z∞

0g0

ε(εr)(χ(εr)−1) + gε(εr)χ0(εr)ei(k−2πξs)rdr

=1

i(k−2πξs)Z∞

0g0

ε(ρ)(χ(ρ)−1) + gε(ρ)χ0(ρ)ei(k−2πξs)ρ/ε dρ

=(−1)nεn

[i(k−2πξs)]n+1 Z∞

0

dn

dρng0

ε(ρ)(χ(ρ)−1) + gε(ρ)χ0(ρ)ei(k−2πξs)ρ/ε dρ .

6

10

31.6

100

316.2

1

10!2

10!4

10!6

10!8

10!10

10!12

10!14

10

31.6

100

316.2

1

10-2

10-4

10-6

10-8

10-10

10-12

10-14

10

31.6

100

316.2

1

10-2

10-4

10-6

10-8

10-10

10-12

10-14

10

31.6

100

316.2

1

10-2

10-4

10-6

10-8

10-10

10-12

10-14

10

31.6

100

316.2

1

10-2

10-4

10-6

10-8

10-10

10-12

10-14

10

31.6

100

316.2

1

10!2

10!4

10!6

10!8

10!10

10!12

10!14

Figure 1: The error in the approximation of the quasi-periodic Green function by multiplying the lattice

sum (2) (with xreplaced by x−x0) by a smooth truncation function χ(|x+m|/a)χ(|y+n|/a), in which

χ(s) = exp(2e1/(1−s)/(s−2)) for 1 < s < 2; and χ(s) = 1 for s < 1; and χ(s) = 0 for s > 2. The plots show

maxx∈K|Gi+1 −Gi|as a function of aion a log-log scale, in which a truncated lattice sum Giis computed

for a=ai= 1.2i,x0= (00,1) and Kis a grid of evenly spaced points in [00.6] ×[00.6] ×[0.6,1.4], excluding

x=x0. The lattice vectors are v1= (1,00) and v2= (0,1,0), the Bloch wavenumbers are (α= 0, β = 0),

and the frequencies are k= 0.4,0.8,0.95 (ﬁrst row) and k= 0.99,2.24,2.5 (second row). Both k= 1.0 and

k≈2.23607 are Wood frequencies, at which convergence is not available.

In view of (16), the splitting I(s) = I0(s) + Iε(s) eﬀects the splitting

F(ξ) = F0(ξ) + Fε(ξ),(22)

for F(ξ), where letting

S±(ρ) = Z±1−i∞

±1

ei(k−2πξs)ρ/ε

[i(k−2πξs)]n+1

ds

√1−s2=−2iZ∞

0

e(i(k∓2πξ)−2πξt2)ρ/ε

[i(k∓2πξ)−2πξt2]n+1

dt

√±2i+t2(23)

(the last expression of which incorporates the changes of variables s=±1−it2) we have denoted

F0(ξ) = 2 Z∞

0

f(n)(r) Z−1−i∞

−1

ei(k−2πξs)r

[i(k−2πξs)]n

ds

√1−s2dr−Z1−i∞

1

ei(k−2πξs)r

[i(k−2πξs)]n

ds

√1−s2dr!

(24)

and

Fε(ξ) = 2 (−1)nεnZ∞

0

dn

dρng0

ε(ρ)(χ(ρ)−1) + gε(ρ)χ0(ρ)(S−(ρ)−S+(ρ)) dρ . (25)

Assume now that ξ6= 0; the case ξ= 0 will be treated separately. In view of the hypothesis

k−2πξ 6= 0 the integral |S+|admits the ﬁnite upper bound

|S+(ρ)| ≤ Z∞

0

2e−2πξt2ρ/ε dt

[(k−2πξ)2+ (2πξt2)2]n+1

2(4 + t4)1

4≤Z∞

0

√2e−2πξt2ρ/ε dt

|k−2πξ|n+1 (26)

=ε

ρ ξ 1

21

|k−2πξ|n+1 Z∞

0

e−πt2dt =1

2ε

ρ ξ 1

21

|k−2πξ|n+1 .(27)

7

Analogously, in view of the assumption k+ 2πξ 6= 0 we obtain

|S−(ρ)| ≤ 1

2ε

ρ ξ 1

21

|k+ 2πξ|n+1 .(28)

Returning to the expression for Fε(ξ) above, observe that, since χ(ρ) = 1 for ρ≤r1, and χ(ρ) = 0

for ρ≥r2, the integral in ρfrom 0 to ∞in (25) can be re-expressed in the form Fε(ξ) = F1

ε(ξ) +

F2

ε(ξ) where

F1

ε(ξ) = 2(−1)nεn

Zr2

r1

dn

dρng0

ε(ρ)(χ(ρ)−1) + gε(ρ)χ0(ρ)S−(ρ)−S+(ρ)dρ and

F2

ε(ξ) = 2(−1)n+1εnZ∞

r2

g(n+1)

ε(ρ)S−(ρ)−S+(ρ)dρ .

The bounds (26) and (28) thus imply

F1

ε(ξ)≤εn+1

2

ξ1

21

|k−2πξ|n+1 +1

|k+ 2πξ|n+1 Zr2

r1

dn

dρng0

ε(ρ)(χ(ρ)−1) + gε(ρ)χ0(ρ)

dρ

ρ1

2

.

(29)

Clearly, as ε→0 the functions gε(ρ) converge to 1 uniformly over the interval [r1, r2], and

thus the integral (29) integral converges to Rr2

r1χ(n+1)(ρ)ρ−1/2dρ in this limit. In particular these

integrals are bounded by a constant C1

n>0 for all ε < 1 and we have

F1

ε(ξ)≤C1

n

εn+1

2

ξ1

21

|k−2πξ|n+1 +1

|k+ 2πξ|n+1 .(30)

Similarly, for F2

ε(ξ) we have

F2

ε(ξ)≤εn+1

2

ξ1

21

|k−2πξ|n+1 +1

|k+ 2πξ|n+1 Z∞

r2g(n+1)

ε(ρ)dρ

ρ1

2

.(31)

But from (21) we obtain

g(n+1)

ε(ρ) = (−1)n+1

ρn+1

∞

X

j=1

aj

(j+n)!

(j1)!

εj

ρj,(32)

and, we thus see that, for εsuﬃciently small, R∞

r2g(n+1)

ε(ρ)ρ−1

2dρ is bounded by a certain constant

C2

n, so that

F2

ε(ξ)≤C2

n

εn+1

2

ξ1

21

|k−2πξ|n+1 +1

|k+ 2πξ|n+1 .(33)

Combining the estimates F1

ε(ξ) and F2

ε(ξ) we thus ﬁnd that there exists a constant C3

nsuch that

|Fε(ξ)| ≤ εn+1

2C3

n

ξ1

21

|k−2πξ|n+1 +1

|k+ 2πξ|n+1 .(34)

8

10

31.6

100

316.2

1

10!2

10!4

10!6

10!8

10!10

10!12

10!14

10

31.6

100

316.2

1

10-2

10-4

10-6

10-8

10-10

10-12

10-14

10

31.6

100

316.2

1

10-2

10-4

10-6

10-8

10-10

10-12

10-14

Figure 2: These plots are similar to those in Fig. 1 except that the Bloch wavenumbers are (α= 0.4, β =

−0.3), and the frequencies are k= 0.3,0.93,1.1. There is a Wood frequency at k≈0.921954.

For ξ= 0, in turn, we have

F(0) = Z∞

0Zπ

−π

f(r)χ(εr)eikr dθdr = 2πZ∞

0

f(r)χ(εr)eikr dr =

=−2π

ik Z∞

0

f0(r)eikrdr −2π

ik Z∞

0g0

ε(ρ)(χ(ρ)−1) + gε(ρ)χ0(ρ)eikρ/εdρ

=−2π

ik Z∞

0

f0(r)eikrdr + (−1)n2π

ik ε

ik n+1 Z∞

0

dn

dρng0

ε(ρ)(χ(ρ)−1) + gε(ρ)χ0(ρ)eikρ/εdρ

=F0(0) + Fε(0).(35)

Again, F0(0) is independent of εand the integral in Fε(0) has a limit as ε→0. Thus one obtains

constants C0

n>0 such that |Fε(0)| ≤ C0

nεn+1.

The estimates above now allow us to now establish the convergence as ε→0 of the series on

the right-hand side of equation (14). If n≥1, then as long as 2π|ξ| 6=|k|for all ξ∈Λ∗, the sum

of |Fε(ξ)|over all ξ∈Λ∗is convergent, and one obtains

X

ξ∈Λ∗|Fε(ξ)| ≤ Cnεn+1

2.(36)

The Poisson Summation Formula now gives

AX

r∈Λ

φ(|r|)χ(ε|r|)g(|r|)eik|r|

|r|=X

ξ∈Λ∗F0(ξ) + X

ξ∈Λ∗Fε(ξ).(37)

But the ﬁrst term on the right hand side of this equation is independent of ε, and, in view of (36),

the second term on the right hand side tends to zero super-algebraically fast. It follows that the

sum on the left hand side of (37) converges super-algebraically fast, as needed.

Inclusion of the Bloch quasi-periodicity factors in the lattice sum can now be accomplished

by replacing the expression exp(ikp|r|2+z2)/p|r|2+z2by

exp(ikp|r|2+z2)

p|r|2+z2e−ik·r,(38)

where k=αv∗

1+βv∗

2. Equation (37) becomes

AX

r∈Λ

φ(|r|)χ(ε|r|)g(|r|)eik|r|

|r|e−ik·r=X

ξ∈Λ∗F0ξ+k

2π+X

ξ∈Λ∗Fεξ+k

2π,(39)

9

The bound (34), shifted by k/2π, is

Fεξ+k

2π≤εn+1

2C3

n

ξ1

2 1

k−2π|ξ+k/(2π)|n+1 +1

k+ 2π|ξ+k/(2π)|n+1 !,

which is valid whenever

k26=|2πξ+k|2.(40)

The validity of (40) for all ξ∈Z2is exactly the condition that (k, α, β) is not a Wood triple.

Inclusion of a shift in rby a ﬁxed vector r0= (x, y). Consider the lattice sum of the quantities

exp(ikp|r−r0|2+z2)

p|r−r0|2+z2,(41)

in which we have taken k= 0. The case k6= 0 is again treated by shifting the Fourier variable ξ

as shown above. As the cutoﬀ functions φand χare also shifted, there ensues a mere exponential

factor in the Fourier transform, and equation (37) becomes

AX

r∈Λ

φ(|r−r0|)χ(ε|r−r0|)g(|r−r0|)eik|r−r0|

|r−r0|=X

ξ∈Λ∗F0(ξ)e−2πiξ·r0+X

ξ∈Λ∗Fε(ξ)e−2πiξ·r0.(42)

The bound (36) persists, X

ξ∈Λ∗Fε(ξ)e−2πiξ·r0≤Cnεn+1

2,(43)

and one again obtains super-algebraic convergence.

Error bound for the gradient of the Green function. The gradient of the monopole eikr /r

is given by the equations

∂

∂x

eikr

r=ik cos θ−cos θ

reikr

r,∂

∂y

eikr

r=ik sin θ−sin θ

reikr

r.(44)

∂

∂z

eikr

r=ik z

r−z

r2eikr

r.(45)

It suﬃces to show that the error bound proven in the theorem remains true, if the monopole eikr

r

is replaced in the proof by any of the terms of the above equations. These terms are products of

the monopole multiplied by r−1, or by cosθ, or by sin θ, or by a selection of two of these factors.

The bound is clearly preserved when multiplying the monopole by z, since the latter factors out of

the summation that constitutes the Green function.

Multiplying the monopole by r−1or by r−2corresponds to introducing the factor ε/ρ or (ε/ρ)2

respectively in the subsequent integration over ρ, thus enhancing the error bound by one or two or-

ders in ε. The integrand is zero (see explanation following (25)) when ρ<r1, thus the denominator

ρis no cause of concern.

The following observations show that the error bound is preserved in the terms in which the

monopole is multiplied by cos θor by sin θ.

•The ﬁrst double integral of (15) acquires the factors cos θor sin θin its integrand. Thus, the

second double integral in (15) (obtained by the change of the integration variable θ→θ+γ)

exhibits the factors cos(θ+γ) or sin(θ+γ) that can be split into a linear combination of cos θ

and sin θ, with the corresponding splitting of the integral.

10

•The double integral that contains the factor sinθis equal to zero; the integrand of the in-

tegration with respect to θis an exact derivative and the integration is over the closed loop

from −πto π.

•What is left is the second double integral in (15) with the extra factor cos θin the integrand.

The change of the variable of integration cos θ=sleads to having an extra factor sin the

subsequent integrals with respect to s. This introduces the extra factor |s|=| ±1−it2|into

the numerator of the ﬁrst integral in (26).

•Following the change of variable s=±1−it2, the factor |s|is replaced by its upper bound

1 + t2and the integral is split accordingly into a sum of two integrals. The ﬁrst integral is

exactly the one that provides the error bound of the theorem. The extra factor t2in the

second integral provides the extra factor ε

ξρ in the bounds (26) and (28) when ξ6= 0. Thus,

the error bound of the theorem is preserved in this case.

•If ξ= 0, the ﬁrst integral in (35) has the factor cos θor sin θthat integrates to zero.

3 Fast high-order integral solvers for problems of scattering by

doubly-periodic structures

For deﬁniteness, we restrict our treatment to diﬀractive structures consisting of arrays of separated

obstacles arranged in a two-dimensional periodic fashion in three-dimensional space. Denoting

by an open set Ω ⊂R3the region occupied by a “reference obstacle” (which could be given by

the union of a number of connected components) and letting S=∂Ω denote its boundary (the

reference scattering boundary), the overall three-dimensional doubly periodic scattering structure

and its boundary are given by

Ωper =[

m,n∈Z

Ωmn , Sper =[

m,n∈Z

Smn ,(46)

where we have set Ωmn = Ω −mv1−nv2and Smn =S−mv1−nv2,m, n ∈Z. It will be assumed

that the sets Ωmn, as well as their boundaries, are pairwise disjoint. Consider the sound-soft

scattering problem

∆u+k2u= 0 in R3\Ωper

u=−uinc on ∂Ωper,(47)

in which an incident plane wave

uinc(x) = exp[i(k·˜

x−γz)],(48)

with |k|2+γ2=k2and x= (˜

x, z), illuminates the structure from above and thus gives rise to a

scattered ﬁeld u. Owing to the periodicity of the domain Ωper, in the regions Ω+and Ω−above and

below the array (Ω+={(˜

x, z) : z > max z0,(˜

x0, z0)∈Ωper }, and Ω−={(˜

x, z) : z < min z0,(˜

x0, z0)∈

Ωper}) the ﬁelds satisfy radiation conditions expressed in terms of the classical Rayleigh expansions:

11

the scattered ﬁelds u+and u−in the regions Ω+and Ω−must be “outgoing”, that is, they must

admit Rayleigh expansions of the form

u+(x) = X

j,`∈Z

B+

j` exp[i(2πjv∗

1+ 2π`v∗

2+k)·˜

x] exp[iγj`z],x∈Ω+(49)

u−(x) = X

j,`∈Z

B−

j` exp[i(2πjv∗

1+ 2π`v∗

2+k)·˜

x] exp[−iγj`z],x∈Ω−.(50)

wherein no waves in Ω+propagate downward, and no waves in Ω−propagate upward.

Using the outgoing free-space Green function Gk(x) = eik |x|

4π|x|, the scattered ﬁeld uis sought in

the form of a combined-ﬁeld layer potential

u(x) = ZSper

∂Gk(x−x0)

∂n(x0)ϕqper(x0)ds(x0) + iη ZSper

Gk(x−x0)ϕqper(x0)ds(x0) (51)

with unknown surface density ϕqper. Here nis the outer unit normal to Sper and η∈Rdenotes

a coupling constant. The unknown density ϕqper is the solution of the combined-ﬁeld integral

equation

1

2ϕqper(x) + ZSper

∂Gk(x−x0)

∂n(x0)ϕqper(x0)ds(x0) + iη ZSper

Gk(x−x0)ϕqper(x0)ds(x0)

=−exp[i(k·˜

x−γz)],x∈Sper (52)

which enforces the sound-soft boundary condition. The well known term 1

2ϕqper in (52) arises as a

singular contribution of the ﬁrst integral in (51) in the limit as xapproaches the boundary.

Equations (52) can be rewritten in a form that involves integration over the reference boundary S

only. The corresponding integral equations make use of the (α, β)-quasi-periodic Green function (2),

in which xis replaced by the diﬀerence x−x0between source and inﬂuence points,

Gqper

k(x−x0) =

∞

X

m,n=−∞

Gk(˜

x−˜

x0+mv1+nv2, z −z0)e−ik·(mv1+nv2).(53)

The integral equation (52) can equivalently be expressed in the form

1

2ϕqper(x) + X

m,n∈ZZSmn G(x−x0)ϕqper(x0)ds(x0) = −exp[i(k·˜

x−γz)],x∈Sper,(54)

where

G(x−x0) = ∂Gk(x−x0)

∂n(x0)+iηGk(x−x0).(55)

Denoting by ϕthe restriction of ϕqper to the reference boundary Sand taking into account the

quasi-periodicity of the density ϕqper, the integral equation (52) can be re-expressed in the form

ϕ(x)

2+ZS

∂Gper

k(x−x0)

∂n(x0)ϕ(x0)ds(x0) + iη ZS

Gper

k(x−x0)ϕ(x0)ds(x0)

=−exp[i(k·˜

x−γz)],x∈S. (56)

Thus, solution of either equation (54) or (56) produces the density ϕ(x) which, upon insertion

into (51) gives rise to the desired quasi-periodic scattered ﬁeld. Note that, in view of its quasi-

periodicity, the unknown ϕis determined throughout Sper by its values on the unit cell S—and

thus testing on Sshould suﬃce to determine ϕuniquely. Indeed, the uniqueness of the problem

thus posed, which is not pursued here, can be established by using the periodic Green function as

in equation (56) together with a proof similar to the one for the bounded obstacle case [15].

12

3.1 High-order evaluation of quasi-periodic layer potentials

Our Nystr¨om approach relies on use of high-order quadratures for evaluation of the integral oper-

ators

(Kmnϕ)(x) = ZSmn G(x−x0)ϕ(x0)ds(x0)

in equation (54) for x∈S, where ϕ=ϕqper is a quasi-periodic integral density deﬁned on Sper.

As noted in the previous section, testing (and thus operation evaluation) for x∈Ssuﬃces to

determine the solution ϕ. Once such operators have been discretized and evaluated numerically

for a given quasi-periodic function ϕthe solution of the problem can be obtained by means of an

iterative linear algebra solver such as GMRES [24].

We ﬁrst consider a quadrature algorithm for the operator K=K00, which is given by

(Kϕ)(x) = ZSG(x−x0)ϕ(x0)ds(x0),x∈S. (57)

We note that this integral operator coincides with the one introduced in [8] for the problem of

acoustic scattering by a bounded obstacle Sunder sound-soft boundary conditions. In fact, the

algorithm we propose for evaluation of the integral operators in (56) results as an outgrowth of the

fast high-order methods presented in that reference. (Extensions of these methods to sound-hard

and electromagnetic problems can be found in [5] and [6].) Thus, in order to convey the main ideas

underlying our periodic-structure solver, we ﬁrst brieﬂy review the algorithm [8].

The bounded-scatterer algorithm [8] evaluates the integral operator Kin two stages, namely

(a) Evaluation of the adjacent/singular interactions (i.e. integration for x0in areas close to x), and

(b) Accelerated evaluation of nonadjacent interactions (that is, accelerated integration for x0away

from x). The decomposition into adjacent and non-adjacent contributions is eﬀected in this method

by means of ﬂoating partitions of unity—that is, pairs of functions of the form (ηx(x0),1−ηx(x0)),

where ηxis a windowing function with a “small” support, which equals 1 in a neighborhood of x.

Additionally, the approach [8] relies on use of smooth parametrizations of the surface Svia a family

of overlapping two-dimensional parameter patches P`, ` = 1,...P along with smooth mappings P`

from parameter sets H`in two-dimensional space (where actual integrations are performed), as

well as partitions of unity subordinated to the overlapping patch decomposition of the surface. i.e

smooth functions w`supported on P`such that P`w`= 1 throughout S. This framework allows

us to reduce the integration of the density ϕover the surface Sto integration of smooth functions

ϕ`compactly supported in the planar sets H`. The latter calculations require analytic resolution

of weakly singular Green functions (i.e. the order of the singularity is O(|x−x0|−1)) which is

performed via polar changes of variables (whose Jacobian cancels the Green-function singularity)

together with interpolation procedures that facilitate evaluations of the surface density at radial

integration points [8].

3.2 Reference acceleration cell

We construct now a “reference acceleration cell”, associated with the “reference domain” Ω =

Ω00, which equals a cubic domain Cof side length Wthat contains Ω. The cell Cis equipped

with a certain acceleration infrastructure which is based on a corresponding acceleration technique

introduced in [8]. In fact, the reference acceleration cell will be utilized as an element in a method

for FFT acceleration for the problem of scattering by the complete periodic structure Ωper. Here and

through the end of Section 3 the presentation assumes a degree of familiarity with the acceleration

methodology presented in reference [8].

13

The acceleration infrastructure presented in that reference, which is designed to enable eﬃcient

FFT-based acceleration for the numerical evaluation of the integral operator

ZSG(x−x0)ϕ(x0)ds(x0),x∈S, (58)

(the term m=n= 0 in (54) restricted to x∈S) proceeds at ﬁrst by partitioning the cube

Cinto a number L3of identical cubic cells ci, where Ldenotes an integer. The pairs (W, L) of

parameters must be adjusted, if necessary, in order to ensure that the cells cido not admit inner

acoustic resonances (eigenfunctions of the Laplace operator with homogeneous Dirichlet boundary

conditions).

The acceleration algorithm [8] then constructs approximations which are obtained by substitu-

tion of the surface “true” sources within ci(or, more precisely, of the ﬁelds that result from discrete

integration of the product of the kernel Gand the density ϕfor all discretization points within ci)

by “equivalent sources” on a set Π`

i(`= 1,2,3) which equals the union of a pair of parallel circular

domains which contain the faces of cithat are parallel to the plane x`= 0 (with the notation

(x1, x2, x3) = (x, y, z)). There are three diﬀerent such approximations. In all three cases the acous-

tic ﬁelds generated by the ci-equivalent sources approximate with high order accuracy the ﬁelds

produced by the true cisources at all cells cjnon-adjacent to ci. The precise concept of adjacency

in [8] results from a requirement that the approximation corresponding to a given cell cibe valid,

with exponentially small errors, outside a concentric cube Siof side three times larger than that of

ci. For eﬃciency the method relies on use of equivalent sources (acoustic monopoles and dipoles)

as described in what follows.

For a given integral density and for each cell ci, a set of equivalent sources (acoustic monopoles

ξ(m)`

ij Gk(x−x`

ij) and dipoles ξ(d)`

ij ∂Gk(x−x`

ij)/∂x`) are placed at points x`

ij, j = 1,··· , M equiv

contained within the union of two circular domains concentric with and circumscribing the faces

of ci, whose radii are selected in accordance with the prescriptions in [8]. The ﬁelds ψci,true radiated

by the ci-true sources are approximated by ﬁelds ψci,eq radiated by the ciequivalent sources

ψci,eq

00 (x) =

1

2Mequiv

X

j=1 ξ(m)`

ij Gk(x−x`

ij) + ξ(d)`

ij

∂Gk(x−x`

ij)

∂x`!,x6∈ Si.(59)

For a given number Mequiv of equivalent sources (selected so as to maintain a given accuracy),

the unknown monopole and dipole intensities in (59) are chosen so as to minimize in the mean-

square norm the diﬀerences (ψci,eq(x)−ψci,true(x)) as xvaries over a number ncoll collocation

points on ∂Si. Hence, the intensities in (59) are obtained in practice as the least-squares solution

of an overdetermined linear system Aξ=bwhere Ais an ncoll ×Mequiv matrix. As discussed

in Sections 3.2.2 and 3.2.3 below the method is completed via a sequence of steps which include

1) FFTs (which are used to evaluate the Cartesian convolutions that result from use of equivalent

sources); 2) Correction of certain errors that arise per step (1), which are inevitable in the FFT-

based operation of convolution with the Green function, and which result from “incorrect” use of

equivalent sources for near interactions; and ﬁnally, 3) High-order evaluation of surface values from

the values at the FFT grid. But before such discussions, we consider certain specializations of the

methods above to the periodic context which, in conjunction with the windowing methodology used

in this paper, have proven to be especially eﬃcient.

14

3.2.1 Green-function contributions from periodic translates of the reference cell

It is easy to check that the set of equivalent sources for the reference scatterer Ω, as computed per

the methodology described in Section 3.2, can be utilized to produce—by means of simple algegraic

manipulations—the corresponding equivalent sources for any periodic translation of the unit-cell.

Indeed, denoting by (Kmnϕqper)(x) the (m, n)-th term on the left-hand sum in equation (54) and

since for x∈Swe have ϕqper(x−mv1−nv2) = e−ik·(mv1+nv2)ϕ(x), it follows that, for x∈S,

(Kmnϕqper)(x) = e−ik·(mv1+nv2)ZSG(x−(x0−mv1−nv2))ϕ(x0)ds(x0)

=e−ik·(mv1+nv2)ZSG((x+mv1+nv2)−x0)ϕ(x0)ds(x0).

(60)

The integral (58) evaluated at x+mv1+nv2coincides with the last integral in equation (60), and,

therefore, this last integral is approximated closely by the equivalent-source expression ψci,eq

00 (x+

mv1+nv2) where ψci,eq

00 is deﬁned in equation (59). It follows that the quantity (Kmnϕqper)(x) can

in turn be approximated closely by

ψci,eq

mn (x) := e−ik·(mv1+nv2)ψci,eq

00 (x+mv1+nv2) =

1

2Mequiv

X

j=1

e−ik·(mv1+nv2)

× ξ(m)`

ij Gk(x−x`

ij +mv1+nv2) + ξ(d)`

ij

∂Gk(x−x`

ij +mv1+nv2)

∂x`!.

(61)

(Again, (x1, x2, x3) = (x, y, z).) Calling ψci,eq (x) the sum of the quantities ψci,eq

mn (x) over all integers

mand n, in view of equation (61) we have that

ψci,eq(x) :=

∞

X

m,n=−∞

ψci,eq

mn (x) =

1

2Mequiv

X

j=1 ξ(m)`

ij Gqper

k(x−x`

ij) + ξ(d)`

ij

∂Gqper

k(x−x`

ij)

∂x`!

provides a close approximation of the quantity

X

m,n∈ZZSmn G(x−x0)ϕqper (x0)ds(x0),x6∈ Si.(62)

The approximating expression (62) contains the quasi-periodic Green function Gqper

k, and it is at

this point that the proposed accelerated algorithm utilizes the windowed periodic Green function:

Replacing Gqper

kin this expression by its windowed approximation

Ga(x−x0) = 1

4πX

m,n∈Z

eik(|˜

x−˜

x0+mv1+nv2|2+(z−z0)2)1/2

|˜

x−˜

x0+mv1+nv2|2+ (z−z0)21/2

×e−ik·(mv1+nv2)χ(|˜

x−˜

x0+mv1+nv2|

a),(63)

which, as established in Theorem 2.1, gives rise to superalgebraic convergence as a→+∞, we

obtain the corresponding superalgebraically close approximation

ψci,eq(x) :=

1

2Mequiv

X

j=1 ξ(m)`

ij Ga(x−x`

ij) + ξ(d)`

ij

∂Ga(x−x`

ij)

∂x`!for (i, x) such that x6∈ Si.(64)

15

(Note that the kdependence is explicitly displayed in the notation Gkfor the free-space Green

function, but, for notational simplicity, it is suppressed in the notation Gafor the windowed periodic

Green function used in equation (64), for example.) Since for a given `the circular regions Π`

iare

not pairwise disjoint, it is necessary, as indicated in [8], to combine equivalent source intensities

for sources supported at a given point x0that corresponds to two diﬀerent cells, say, crand csfor

which x0=x`

r,p =x`

s,q for some integers pand q. We thus deﬁne the quantities

ψ(∗)`(x) = X

x0∈Π`ξ(m)`

x0Ga(x−x0) + ξ(d)`

x0

∂Ga(x−x0)

∂x0

`(65)

where ξ(m)`

x0and ξ(d)`

x0denote the sum of all intensities of equivalent sources located at a point

x0∈Π`:

ξ(m)`

x0=X

x`

ij =x0

ξ(m)`

ij , ξ(d)`

x0=X

x`

ij =x0

ξ(d)`

ij .

Note that, while the quantity ψ(∗)`contains contributions from cells cifor which the far-ﬁeld

restriction x6∈ Siis not satisﬁed, the algorithmic evaluation of the quantity (64) does proceed by

evaluating ψ(∗)`(by means of an FFT) and then correcting for nearby contributions x∈ Si. These

two steps in the algorithm are discussed in the following subsections.

3.2.2 FFT evaluation of the convolutions and Correction step

As indicated above, the inaccurate quantity ψ(∗)`(x) (equation (65)) plays an important role in

the proposed accelerated quasi-periodic solver. For each `= 1,2,3 the proposed algorithm ﬁrst

evaluates the Cartesian convolutions ψ(∗)`(x) (x∈Π`) by means of the three-dimensional FFT

algorithm. The proposed use of the quasi-periodic Green function, which only occurs in the algo-

rithm as part of the acceleration step, provides the additional advantage that, under the strategies

mentioned in Section 3.3, the Green function needs to be evaluated at a number on the order of

O(N4/3) points only—and not for the O(N2) pairs of discretization points, where O(N) is the

number of grid points that are used to discretize the scatterers in the reference cell. As demon-

strated in Section 4, the combined windowed-Green-function FFT-based algorithm provides a very

eﬃcient quasi-periodic solver—at least away from Wood anomalies.

But, as indicated above, corrections are necessary to the pure FFT-based quantity ψ(∗)`(x):

The incorrect contributions x∈ Simust be subtracted, and corresponding accurate replacements

need to be added. In some detail, the quantity ψ(na,eq)`(x), which equals the sum of the values at

the point x∈Sof all ﬁelds arising from equivalent sources nonadjacent to cican be obtained by

subtracting from ψ(∗)`(x) the ﬁeld arising at xfrom equivalent sources located within Si, where i

is the index for which x∈ci. The “corrections” necessary to produce ψ(na,eq)`(x) from ψ(∗)`(x)

can also be evaluated eﬃciently, by means of a sequence of (small) three-dimensional FFTs, since

they only involve (small) three-dimensional convolutions and free-space Green functions. Once

completed for `= 1, 2, 3, this overall procedure results in accurate values, on a mesh that samples

the boundaries of all cells ci, of the ﬁelds arising from all true sources contained in all cells cjnot

adjacent to ci.

In order to obtain approximations of the nonadjacent interactions ψ(na,true)(x) (that is, the

ﬁelds generated at xby the true discrete surface sources contained outside Si) at surface points

x∈S∩ci, the algorithm employs solutions to the Helmholtz equation within ci, with Dirichlet

boundary conditions given by ψ(na,eq)`,`= 1, 2, 3. These Dirichlet problems can be solved uniquely

16

(in view of our assumption that the wavenumber kis not a resonant frequency), and thus the good

approximation properties of the nonadjacent interactions on the boundary of each cell citranslate

into good approximations for the nonadjacent interactions on the surface S. Following [8], our

algorithm produces the needed solutions of Dirichlet problems by means of approximations of

the form

P(x) =

nw

X

j=1

γjexp(ikuj·x),(66)

valid within ci(in terms of plane wave solutions of the Helmholtz equation), for the ﬁeld ψ(na,true).

Here ujare unit vectors that adequately sample the surface of the unit sphere, and the coeﬃcients

γjare obtained in such a way that the relation P(x) = ψ(na,true)(x) is satisﬁed, in the least-squares

sense, for all xin an adequately chosen collocation mesh on the cubic surface Si.

3.2.3 Adjacent interactions

Having evaluated, by means of FFTs and plane wave expansions, accurate approximations of the

surface values of the ﬁeld ψ(na,true)(x) produced by the non-adjacent surface sources (for all dis-

cretization points x∈S), surface values of the total ﬁeld are then obtained by direct addition

of necessary singular and non-singular adjacent surface sources. Brieﬂy, the ﬁelds that need to

be added to (the approximations just obtained for) the ﬁeld ψ(na,true)(x) (for a point x∈S) in-

clude (i) Adjacent regular sources, that is, trapezoidal-rule contributions to the integral operator

from sources lying outside the support of the ﬂoating POU ηxbut inside Si(none of which are

included in ψ(na,true)(x)), and (ii) Adjacent singular sources, that is, the local contributions to the

integral operator considered in stage (a) of Section 3.1.

3.3 Computational cost

It is easy to estimate the computational cost of the proposed windowed-Green-function/accelerated

algorithm for quasi-periodic scattering problems. The cost of the algorithm is the same as that

of its non-periodic counterpart [8] except that now the use of the equivalent-source intensities

requires values of the quasi-periodic Green function, as shown in equation (65), instead of the free-

space Green function in the former algorithm. (Note that the equivalent sources themselves are

obtained, even in the periodic context, by means of the free-space Green function Gk, as shown in

equation (59).) The operation count now proceeds simply. Let the number of grid points used to

discretize the scatterers in the reference cell be O(N). The algorithm [8] is reported to require a

cost of O(N4/3log N) operations. In addition, the windowed-Green-function accelerated algorithm

requires a precomputation of the Green function Ga(x) and its derivatives along each coordinate

direction and at all points xin the accelerator meshes Π`, ` = 1,2,3. These precomputations are

performed by direct summation at a cost of O(a2N4/3) operations. The overall cost of the algorithm,

including all necessary Green function evaluations, thus amounts to the O(a2N4/3) precomputation

cost plus the necessary number of GMRES iterations at a cost of O(N4/3log N) each.

4 Numerical results

To demonstrate the speed and accuracy of the proposed accelerated N¨ystrom algorithm we present

results of applications of this method to problems of scattering by doubly periodic arrays of

perfectly-conducting obstacles at non-Wood conﬁgurations. For simplicity we consider two-dimensional

17

rectangular lattices of scatterers, that is v1=d1(1,00) and v2=d2(0,1,0). We present two main

accuracy indicators, namely certain convergence studies on one hand, and departure from energy

conservation in the numerical solution, on the other. The latter test, which derives from the en-

ergy conservation satisﬁed by the exact PDE solution for perfectly conducting scatterers—that the

energy ﬂux of the incident ﬁeld must equal the sum of the energy ﬂuxes of the reﬂected ﬁeld and

the transmitted ﬁeld—can be expressed in terms of the Rayleigh coeﬃcients B+

j` of the scattering

problem, X

(j,`)∈P

γj`|B+

j`|2+X

(j,`)∈P

γj`|B−

j` +δ00

j` |2=γ00 ,(67)

in which Pis the set of propagating harmonics P={(j, `) : kv∗

j`k< k2}and γj` are deﬁned

in equation (7). The energy defect for numerically computed Rayleigh coeﬃcients e

B±

r,s is then

deﬁned as

ε=1

γ00 X

(j,`)∈P

γj`|e

B+

j`|2+|e

B−

j` +δ00

j` |2−γ00(68)

(where δ00

j` equals 1 for (j, `) = (0,0) and zero otherwise). Experiments based on fully converged

solutions (as veriﬁed by means of convergence studies), suggest that the energy defect is an excellent

indicator of solution accuracy for the integral solvers under consideration.

All of the numerical examples presented in this section concern problems of scattering by pe-

riodic arrangements of either spherical or bean-shaped scatterers [8], both of which have diameter

equal to 2. In all cases the periods are given by d1=d2= 4, and plane-wave incident ﬁelds with

incidence angles ψ=φ= 0 (that is, normal incidence) and ψ=φ=π/3 (oblique incidence) are

considered. For these experiments we have used the accelerator parameters L= 3, Mequiv = 4,

ncoll = 8, and nw= 4. In all cases the linear systems resulting from our discretization was solved

by means of the GMRES iterative solver with a relative residual tolerance T ol. The tolerance value

T ol = 10−8was used to produce Table 1 while the less restrictive “adequate-accuracy” tolerance

10−4was used for Tables 2 and 3. Table 1 showcases the high-order accuracy achieved by our peri-

odic solvers in the case of doubly periodic arrays of spheres under normal incidence. Tables 2 and 3

present results for periodic arrays of spheres and bean-shaped obstacles for various wavenumbers k

and various values of the window radius a.

The error εpresented in these tables was evaluated in accordance with equation (68). The

error ε1was calculated as the absolute error in the Rayleigh coeﬃcient B+

00 (as estimated by com-

parison with a reference solution obtained by means of a highly-reﬁned discretization, a large value

aand a suﬃciently small tolerance T ol). We also report numbers of iterations and computational

times required by the GMRES solvers to reach the tolerance T ol in each case. The results were

obtained by means of a C++ implementation of our solvers on a single core of a 2.67 GHz Intel

Xeon CPU with 24Gb of RAM.

5 Conclusions

This paper demonstrates that the previous two-dimensional windowed Green-function methodol-

ogy [4] for quasi-periodic scattering problems can successfully be extended to the three-dimensional

context. In particular, this paper presents the ﬁrst rigorous proof of super-algebraic convergence of

the windowed Green-function method in three-dimensional space. An accelerated windowed Green-

function algorithm is presented, which possesses excellent properties. Comparisons, in simple ex-

amples, with one of the most advanced techniques for evaluation of periodic Green functions [17]

18

Scatterer kNa ε ε1Iter

Sphere 1 24,576 25 1.1 ×10−31.9 ×10−311

Sphere 1 24,576 50 1.2 ×10−46.1 ×10−511

Sphere 1 24,576 75 5.0 ×10−62.1 ×10−611

Sphere 1 24,576 150 3.8 ×10−73.5 ×10−811

Table 1: Convergence of the periodic solvers using Gafor increasing values of the truncation

radius afor doubly periodic arrays of spheres under normal incidence. The reference solution for

computing ε1has a= 400 and conservation of energy error ε= 1.2×10−7.

Scatterer kNa ε ε1Iter Computational Times

Set-up Time/It Total

Sphere 0.75 1350 20 5.0 ×10−36.4 ×10−35 14sec 0.4sec 16sec

Sphere 0.75 1350 30 4.7 ×10−41.6 ×10−35 29sec 0.4sec 31sec

Sphere 0.75 1350 40 2.4 ×10−52.2 ×10−45 51sec 0.4sec 53sec

Sphere 9 5766 20 5.0 ×10−33.6 ×10−313 14sec 3.4sec 57sec

Sphere 9 5766 30 1.1 ×10−31.3 ×10−313 29sec 3.4sec 1m14sec

Sphere 9 5766 40 7.0 ×10−52.1 ×10−413 51sec 3.4sec 1m35sec

Bean 0.75 1350 20 3.3 ×10−35.5 ×10−310 14sec 1.2sec 26sec

Bean 0.75 1350 30 1.9 ×10−31.4 ×10−310 29sec 1.2sec 42sec

Bean 0.75 1350 40 3.2 ×10−43.4 ×10−410 51sec 1.2sec 1m5sec

Bean 9 5766 20 6.1 ×10−34.0 ×10−317 14sec 5.35sec 1m45sec

Bean 9 5766 30 1.1 ×10−39.9 ×10−417 29sec 5.35sec 2m0sec

Bean 9 5766 40 3.2 ×10−51.7 ×10−417 51sec 5.35sec 2m30sec

Table 2: Convergence of the periodic solvers using Gafor increasing values of the truncation

radius afor doubly periodic arrays of spherical and bean-shaped scatterers under normal incidence.

The reference solution for computing ε1has a= 120 and conservation of energy error ε≈10−5.

(which is based on a combination of resummation and partitioning techniques) suggests that the

proposed methodology can be orders of magnitude less expensive than former approaches.

Data accessibility. All data applicable to this paper are included in the article.

Competing interests. There are no competing interests relevant to this article.

Authors contributions. All authors are equally considered co-contributors in this article.

Acknowledgements. All acknowledgements are provided in the funding statement below.

Funding statement. The authors gratefully acknowledge support from AFOSR and NSF under

contracts FA9550-15-1-0043 and DMS-1411876 (OB); NSF DMS-0807325 (SPS); NSF DMS-1008076

(CT); and NSF DMS-0707488 and NSF DMS-1211638 (SV).

Ethics statement. No ethics statement applies to this article.

19

Scatterer k N a ε ε1Iter Computational Times

Set-up Time/It Total

Sphere 9 5766 20 8.0 ×10−35.2 ×10−323 14sec 3.4sec 1m31sec

Sphere 9 5766 30 3.7 ×10−38.0 ×10−422 29sec 3.4sec 1m44sec

Sphere 9 5766 50 4.5 ×10−51.7 ×10−422 1m25sec 3.4sec 2m40sec

Bean 9 5766 20 4.4 ×10−37.8 ×10−321 14sec 5.35sec 2m6sec

Bean 9 5766 30 1.2 ×10−33.1 ×10−321 29sec 5.35sec 2m23sec

Bean 9 5766 50 3.0 ×10−52.1 ×10−421 1m25sec 5.35sec 3m17sec

Table 3: Convergence of the periodic solvers using Gafor increasing values of the truncation

radius afor doubly periodic arrays of spherical and bean-shaped scatterers under oblique incidence

φ=ψ=π/3. The reference solution for computing ε1has a= 120 and conservation of energy

error ε≈10−5.

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