AN ADAPTIVE PERFECTLY MATCHED LAYER TECHNIQUEFOR TIME-HARMONIC SCATTERING PROBLEMS

ZHIMING CHEN∗ AND XUEZHE LIU†

Abstract. We develop an adaptive perfectly matched layer (PML) technique for solving thetime harmonic scattering problems. The PML parameters such as the thickness of the layer andthe fictitious medium property are determined through sharp a posteriori error estimates. Thederived finite element a posteriori estimate for adapting meshes has the nice feature that it decaysexponentially away from the boundary of the fixed domain where the PML layer is placed. Thisproperty makes the total computational costs insensitive to the thickness of the PML absorbinglayers. Numerical experiments are included to illustrate the competitive behavior of the proposedadaptive method.

Key words. Adaptivity, perfectly matched layer, a posteriori error analysis, scattering problems.

AMS subject classifications. 65N30, 78A45, 35Q60

1. Introduction. We propose and study an adaptive perfectly matched layer(PML) technique for solving Helmholtz-type scattering problems with perfectly con-ducting boundary:

∆u+ k2u = 0 in R2\D, (1.1a)

∂u

∂n= −g on ΓD , (1.1b)

√r

(

∂u

∂r− iku

)

→ 0 as r = |x| → ∞. (1.1c)

Here D ⊂ R2 is a bounded domain with Lipschitz boundary ΓD, g ∈ H−1/2(ΓD)is determined by the incoming wave, and n is the unit outer normal to ΓD. We assumethe wave number k ∈ R is a constant. We remark that the results in this paper canbe easily extended to solve the scattering problems with other boundary conditionssuch as Dirichlet or the impedance boundary condition on ΓD, or to solve the acousticwave propagation through inhomogeneous media with a variable wave number k2(x)inside some bounded domain.

Since the work of Berenger [3] which proposed a PML technique for solving withthe time dependent Maxwell equations, various constructions of PML absorbing layershave been proposed and studied in the literature (cf. e.g. Turkel and Yefet [20],Teixeira and Chew [19] for the reviews). Under the assumption that the exteriorsolution is composed of outgoing waves only, the basic idea of the PML technique isto surround the computational domain by a layer of finite thickness with speciallydesigned model medium that would either slow down or attenuate all the waves thatpropagate from inside the computational domain. The PML equation for the time-harmonic scattering problem (1.1a) is derived in Collino and Monk [10] by a complex

∗LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences,Chinese Academy of Sciences, Beijing 100080, People’s Republic of China. This author was supportedin part by China NSF under the grant 10025102 and by China MOST under the grant G1999032802.(zmchen@lsec.cc.ac.cn).

†Institute of Computational Mathematics, Academy of Mathematics and System Sciences, ChineseAcademy of Sciences, Beijing 100080, People’s Republic of China. (liuxz@lsec.cc.ac.cn).

1

extension of the solution u in the exterior domain. It is proved in Lassas and Somersalo[13], Hohage, Schmidt and Zschiedrich [12] that the resultant PML solution convergesexponentially to the solution of the original scattering problem as the thickness ofthe PML layer tends to infinity. We remark that in practical applications involvingPML techniques, one cannot afford to use a very thick PML layer if uniform finiteelement meshes are used because it requires excessive grid points and hence morecomputer time and more storage. On the other hand, a thin PML layer requires arapid variation of the artificial material property which deteriorates the accuracy iftoo coarse mesh is used in the PML layer.

A posteriori error estimates are computable quantities in terms of the discretesolution and data that measure the actual discrete errors without the knowledge ofexact solutions. They are essential in designing algorithms for mesh modificationwhich equi-distribute the computational effort and optimize the computation. Eversince the pioneering work of Babuska and Rheinboldt [2], the adaptive finite ele-ment methods based on a posteriori error estimates have become a central theme inscientific and engineering computations. The ability of error control and the asymp-totically optimal approximation property (see e.g. Morin, Nochetto and Siebert [17],Chen and Dai [5]) make the adaptive finite element method attractive for compli-cated physical and industrial processes (cf. e.g. Chen and Dai [4], Chen, Nochettoand Schmidt [7]). For the efforts to solve scattering problems using adaptive methodsbased on a posterior error estimate, we refer to the recent work Monk [15], Monk andSuli [16].

It is proposed in Chen and Wu [8] for scattering problem by periodic structures(the grating problem) that one can use the a posteriori error estimate to determinethe PML parameters. Moreover, the derived a posteriori error estimate in [8] has thenice feature of exponential decay in terms of the distance to the boundary of the fixeddomain where the PML layer is placed. This property leads to coarse mesh size awayfrom the fixed domain and thus makes the total computational costs insensitive tothe thickness of the PML absorbing layer.

In this paper we extend the idea of using a posteriori error estimates to determinethe PML parameters and propose an adaptive PML technique for solving the scat-tering problem (1.1a)-(1.1c). The main difficulty of the analysis is that in contrastto the grating problems in which there are only finite number of outgoing modes [8],now there are infinite number of outgoing modes expressed in terms of Hankel func-tions. We overcome this difficulty by exploiting the following uniform estimate forthe Hankel functions H1

ν , ν ∈ R,:

|H (1)

ν (z)| ≤ e−Im (z)

(

1− Θ2

|z|2

)1/2

|H (1)

ν (Θ)|, (1.2)

for any z ∈ C++,Θ ∈ R such that 0 < Θ ≤ |z|, where C++ = z ∈ C : Im (z) ≥0,Re (z) ≥ 0. To our knowledge this sharp estimate is new and allows us to provethe exponentially decaying property of the PML solution without resorting to theintegral equation technique in [13] or the representation formula in [12]. We remarkthat in [13], [12], it is required that the fictitious absorbing coefficient must be linearafter certain distance away from the boundary where the PML layer is placed. Theestimate (1.2) is proved in Lemma 2.2 which depends on the Macdonald formula forthe modified Bessel functions. We also remark that since (1.2) is valid for all realorder ν, the results of this paper can be extended directly to study three dimensionalHelmholtz-type scattering problems. We will report progress in this direction as wellas the study of the electromagnetic scattering problems elsewhere in future.

2

Let ΩPML = Bρ\BR, where 0 < R < ρ and Ba denotes the circle of radius a > 0.Let α(r) = 1 + iσ(r) be the fictitious medium property. In practical applications, σis usually taken as power functions:

σ = σ(r) = σ0

(

r −R

ρ−R

)m

for some constant σ0 > 0 and integer m ≥ 1.

Under the assumption that the Dirichlet problem of the PML equation in the PMLlayer is uniquely solvable, we prove the following key estimate between the Dirichlet-to-Neumann mapping for the original scattering problem T : H1/2(ΓR) → H−1/2(ΓR)and the PML problem T (cf. Lemma 2.5), where ΓR = ∂BR,

‖T − T ‖L(H1/2(ΓR),H−1/2(ΓR)) ≤ C(1 + kR)2|α0|2e−kIm (ρ)(

1− R2

|ρ|2

)1/2

,

where α0 = 1 + iσ0, and ρ =∫ ρ

0 α(t)dt is the complex radius corresponding to ρ. Weremark that the assumption of the unique solvability of the PML Dirichlet problem inthe PML layer is rather mild in practical applications because standard Fredholm al-ternative theory implies that the PML Dirichlet problem in the PML layer is uniquelysolvable for all but a discrete number of real k. Moreover, in the appendix of thispaper, we show that for any given ρ,R, the Dirichlet PML problem in the PML layeris uniquely solvable for sufficiently large σ0 > 0.

The layout of the paper is as follows. In section 2 we recall the PML formulationfor (1.1a)-(1.1c), derive the key estimates for Hankel functions, and study the prop-erties of the PML equation in the PML layer. Existence, uniqueness and convergenceof the PML formulation are considered. In section 3 we introduce the finite elementdiscretization. In section 4 we derive the sharp a posteriori error estimate which laysdown the basis of the combined adaptive PML and finite element methods. In section5 we discuss the implementation of the adaptive method and present several numer-ical examples to illustrate the competitive behavior of the method. Finally in theappendix we show the unique solvability of the Dirichlet PML problem in the PMLlayer for sufficiently large σ0.

2. The PML formulation. Let D be contained in the interior of the circleBR = x ∈ R2 : |x| < R. We start by introducing an equivalent variational formula-tion of (1.1a)-(1.1c) in the bounded domain ΩR = BR\D. In the domain R2\BR, thesolution u of (1.1a)-(1.1c) can be written under the polar coordinates as follows:

u(r, θ) =∑

n∈Z

H (1)n (kr)

H (1)n (kR)

uneinθ, un =

1

2π

∫ 2π

0

u(R, θ)e−inθdθ. (2.1)

where H (1)n is the Hankel function of the first kind and order n. The series in (2.1)

converges uniformly for r > R (cf. e.g. Colten and Kress [11]). Let T : H1/2(ΓR) →H−1/2(ΓR), where ΓR = ∂BR, be the Dirichlet-to-Neumann operator defined as fol-lows: for any f ∈ H1/2(ΓR),

Tf =∑

n∈Z

kH (1)n

′(kR)

H (1)n (kR)

fneinθ, fn =

1

2π

∫ 2π

0

fe−inθdθ. (2.2)

It is known that T is well-defined and the solution u written as in (2.1) satisfies

∂u

∂n

∣

∣

∣

ΓR

= Tu.

3

Let a : H1(ΩR) ×H1(ΩR) → C be the sesquilinear form:

a(ϕ, ψ) =

∫

ΩR

(

∇ϕ · ∇ψ − k2ϕψ)

dx− 〈Tϕ, ψ〉ΓR , (2.3)

where 〈·, ·〉ΓR stands for the inner product on L2(ΓR) or the duality pairing betweenH−1/2(ΓR) andH1/2(ΓR). Similar notation applies for 〈·, ·〉ΓD , 〈·, ·〉Γρ . The scatteringproblem (1.1a)-(1.1c) is equivalent to the following weak formulation (cf. e.g. [11]):Given g ∈ H−1/2(ΓD), find u ∈ H1(ΩR) such that

a(u, ψ) = 〈g, ψ〉ΓD ∀ψ ∈ H1(ΩR). (2.4)

The existence of a unique solution of the variational problem (2.4) is known (cf. e.g.[11], McLean [14]). Then the general theory in Babuska and Aziz [1, Chapter 5]implies that there exists a constant µ > 0 such that the following inf-sup conditionholds:

sup06=ψ∈H1(ΩR)

|a(ϕ, ψ)|‖ψ ‖H1(ΩR)

≥ µ‖ϕ ‖H1(ΩR) ∀ϕ ∈ H1(ΩR). (2.5)

Fig. 2.1. Setting of the scattering problem with the PML layer.

Now we turn to the introduction of the absorbing PML layer. We surround thedomain ΩR with a PML layer ΩPML = x ∈ R2 : R < |x| < ρ. The specially designedmodel medium in the PML layer should basically be so chosen that either the wavenever reaches its external boundary or the amplitude of the reflected wave is so smallthat it does not essentially contaminate the solution in ΩR. Throughout the paperwe assume ρ ≤ CR for some generic fixed constant C > 0.

Let α(r) = 1 + iσ(r) be the model medium property which satisfies

σ ∈ C(R), σ ≥ 0, and σ = 0 for r ≤ R.

Denote by r the complex radius defined by

r = r(r) =

r if r ≤ R,∫ r

0 α(t)dt = rβ(r) if r ≥ R.(2.6)

4

Following [10], we introduce the PML equation

∇ · (A∇w) + αβk2w = 0 in ΩPML, (2.7)

where A = A(x) is a matrix which satisfies, in polar coordinates,

∇ · (A∇) =1

r

∂

∂r

(

βr

α

∂

∂r

)

+α

βr2∂2

∂θ2. (2.8)

The PML solution u in Ωρ = Bρ\D is defined as the solution of the following system

∇ · (A∇u) + αβk2u = 0 in Ωρ, (2.9a)

∂u

∂n= −g on ΓD, u = 0 on Γρ. (2.9b)

This problem can be reformulated in the bounded domain ΩR by imposing theboundary condition

∂u

∂n

∣

∣

∣

ΓR

= T u,

where the operator T : H1/2(ΓR) → H−1/2(ΓR) is defined as follows: Given f ∈H1/2(ΓR),

T f =∂ζ

∂n

∣

∣

∣

ΓR

,

where ζ ∈ H1(ΩPML) satisfies

∇ · (A∇ζ) + αβk2ζ = 0 in ΩPML, (2.10a)

ζ = f on ΓR, ζ = 0 on Γρ. (2.10b)

The existence and uniqueness of the solutions of the PML problem (2.10a)-(2.10b)will be studied in the subsection 2.2 below.

Based on the operator T , we introduce the sesquilinear form a : H1(ΩR) ×H1(ΩR) → C by

a(ϕ, ψ) =

∫

ΩR

(

A∇ϕ · ∇ψ − k2αβϕψ)

dx− 〈Tϕ, ψ〉ΓR . (2.11)

Then the weak formulation for (2.9a)-(2.9b) is: Given g ∈ H−1/2(ΓD), find u ∈H1(ΩR) such that

a(u, ψ) = 〈g, ψ〉ΓD ∀ψ ∈ H1(ΩR). (2.12)

The well-posedness of the PML problem (2.12) and the convergence of its solution tothe solution of the original scattering problem (2.4) will be studied in the subsection2.3. In the following we first derive some basic estimates for the Hankel function H (1)

n

which play a key role in the analysis in this paper.

5

2.1. Hankel functions. For ν ∈ C, the two Hankel functions H (1)ν (z), H (2)

ν (z),where z ∈ C, are two fundamental solutions of the Bessel equation for functions oforder ν:

z2 d2y

dz2+ z

dy

dz+ (z2 − ν2)y = 0, (2.13)

which satisfy the following asymptotic behaviors as |z| → ∞:

H (1)

ν (z) ∼(

2

πz

)1/2

ei(z−12 νπ−

14π), H (2)

ν (z) ∼(

2

πz

)1/2

e−i(z− 12 νπ−

14π). (2.14)

We also need the Bessel functions of purely imaginary argument Kν(z), also calledthe modified Bessel functions, which is the solution of the differential equation

z2 d2y

dz2+ z

dy

dz− (z2 + ν2)y = 0. (2.15)

It is connected with H (1)ν (z) through the relation

Kν(z) =1

2πie

12 νπiH (1)

ν (iz). (2.16)

The importance of the function Kν(z) in mathematical physics lies in the fact that itis a solution of (2.15) which tends to zero exponentially as z → ∞ through positivevalues. We refer to the treatise Watson [21] for extensive studies on the functionsH (1)ν (z), H (2)

ν (z) and Kν(z).The following lemma is proved in [21, P.439].Lemma 2.1 (Macdonald formula). For any ν ∈ C and z1, z2 ∈ C satisfying

| arg z1| < π, | arg z2| < π and | arg(z1 + z2)| <1

4π,

we have

Kν(z1)Kν(z2) =1

2

∫ ∞

0

e−v2−

z21+z2

22v Kν

(z1z2v

) dv

v.

An important consequence of this lemma is that for real ν, Kν(z) has no zerosif | arg z| ≤ 1

2π [21, P.511], which, by (2.16), implies that H (1)ν (z) has no zeros when

Im (z) ≤ 0 . In particular, we have H (1)n (kR) 6= 0 for any n ∈ Z, R > 0. This justifies

the writing of H (1)n (kR) in the denominator in (2.1), (2.2).

Lemma 2.2. For any ν ∈ R, z ∈ C++ = z ∈ C : Im (z) ≥ 0,Re (z) ≥ 0, andΘ ∈ R such that 0 < Θ ≤ |z|, we have

|H (1)

ν (z)| ≤ e−Im (z)

(

1− Θ2

|z|2

)1/2

|H (1)

ν (Θ)|. (2.17)

This estimate, which to our knowledge is new, will play an important role in theanalysis of this paper. The importance of the estimate (2.17) lies in the fact that it isuniform with respect to ν. We remark that the large argument asymptotic expansionssuch as (2.14) in the literature usually depend on ν and thus are insufficient for ourpurpose.

6

Proof. By (2.16) we know that

|H (1)

ν (z)|2 = H (1)

ν (z)H (1)ν (z) =

4

π2Kν(−iz)Kν(−iz) =

4

π2Kν(−iz)Kν(iz),

where we have used the formula Kν(z) = Kν(z) for real ν. Since z ∈ C++, we knowthat | arg(−iz)| < π, | arg(iz)| < π and | arg(−iz + iz)| = 0 < π

4 . Thus by Lemma 2.1we obtain

|H (1)

ν (z)|2 =2

π2

∫ ∞

0

e−v2−

−z2−z2

2v Kν

( |z|2v

)

dv

v.

After the change of variable w = |z|2/v, we get

|H (1)

ν (z)|2 =2

π2

∫ ∞

0

e− |z|2

2w + z2+z2

2|z|2wKν(w)

dw

w,

which, for any Θ > 0, we rewrite as

|H (1)

ν (z)|2 =2

π2

∫ ∞

0

e−

|z|2−Θ2

2w −2|z|2−z2−z2

2|z|2w · e−Θ2

2w +wKν(w)dw

w.

Now for 0 < Θ ≤ |z|, by Cauchy-Schwarz inequality, we deduce that

e− |z|2−Θ2

2w − 2|z|2−z2−z2

2|z|2w

= e− |z|2−Θ2

2w − 2Im (z)2

|z|2w ≤ e

−2Im(z)(

1− Θ2

|z|2

)1/2

.

Therefore

|H (1)

ν (z)|2 ≤ e−2Im (z)

(

1− Θ2

|z|2

)1/2 2

π2

∫ ∞

0

e−Θ2

2w +wKν(w)dw

w

= e−2Im (z)

(

1− Θ2

|z|2

)1/2

|H (1)

ν (Θ)|2.

This completes the proof.To proceed further, we recall the following Nicholson integral [21, P.441]:

J2ν (z) + Y 2

ν (z) =8

π2

∫ ∞

0

K0(2z sinh t) cosh(2νt)dt for z ∈ C,Re (z) > 0.

Here K0(z) is the modified Bessel function of order zero in (2.16). Since cosh(t) =(et + e−t)/2 is an increasing function in R

+, we have, for Θ > 0, n ≥ 1 that

J2n−1(Θ) + Y 2

n−1(Θ) =8

π2

∫ ∞

0

K0(2Θ sinh t) cosh(2(n− 1)t)dt

≤ 8

π2

∫ ∞

0

K0(2Θ sinh t) cosh(2nt)dt

= J2n(Θ) + Y 2

n (Θ).

Thus

|H (1)

n−1(Θ)| ≤ |H (1)

n (Θ)| for any Θ > 0, n ≥ 1. (2.18)

7

Lemma 2.3. For any z ∈ C++ and Θ ∈ R such that 0 < Θ ≤ |z|, we have

|H (1)

n′(z)| ≤ e

−Im (z)(

1− Θ2

|z|2

)1/2(

1 +|n||z|

)

|H (1)

n (Θ)| for n ∈ Z, |n| ≥ 1, (2.19)

|H (1)

0′(z)| ≤ e

−Im (z)(

1− Θ2

|z|2

)1/2

|H (1)

0′(Θ)|. (2.20)

Proof. Since H (1)

−n = einπH (1)n (z), we only need to prove (2.19) for n ∈ Z, n ≥ 1.

By the formula

zdH (1)

n (z)

dz+ nH (1)

n (z) = zH (1)

n−1(z),

Lemma 2.2, and (2.18), we know that

|H (1)

n′(z)| ≤ |H (1)

n−1(z)| +n

|z| |H(1)

n (z)|

≤ e−Im (z)

(

1− Θ2

|z|2

)1/2(

|H (1)

n−1(Θ)| + n

|z| |H(1)

n (Θ)|)

≤ e−Im (z)

(

1− Θ2

|z|2

)1/2(

1 +n

|z|

)

|H (1)

n (Θ)|.

This proves (2.19). The estimate (2.20) can be proved similarly by using the formuladH (1)

0 (z)/dz = −H (1)

1 (z). This completes the proof.

2.2. The PML equation in the layer. In this subsection we consider theDirichlet problem of the PML equation in the layer ΩPML:

∇ · (A∇w) + αβk2w = 0 in ΩPML, (2.21a)

w = 0 on ΓR, w = q on Γρ. (2.21b)

where q ∈ H1/2(Γρ). Let b : H1(ΩPML) ×H1(ΩPML) → C be the sesquilinear form:

b(ϕ, ψ) =

∫ ρ

R

∫ 2π

0

(

βr

α

∂ϕ

∂r

∂ψ

∂r+

α

βr

∂ϕ

∂θ

∂ψ

∂θ− αβk2rϕψ

)

dr dθ. (2.22)

Then from (2.8) we know that the weak formulation for (2.21a)-(2.21b) is: Givenq ∈ H1/2(Γρ), find w ∈ H1(ΩPML) such that w = 0 on ΓR, w = q on Γρ, and

b(w,ϕ) = 0 ∀ϕ ∈ H10 (ΩPML). (2.23)

We make the following assumption on the fictitious medium property σ, which israther mild in the practical application of the PML techniques:

(H1) σ = σ0

(

r −R

ρ−R

)m

for some constant σ0 > 0 and some integer m ≥ 1.

From (H1) we know that β(r) = 1 + iσ(r), where

σ(r) =1

r

∫ r

R

σ(t)dt =σ0

m+ 1

r −R

r

(

r −R

ρ−R

)m

.

8

Thus σ ≤ σ for all r ≥ R. Notice that for α = 1 + iσ, β = 1 + iσ, we have

Re

(

β

α

)

=1 + σσ

1 + σ2, Re

(

α

β

)

=1 + σσ

1 + σ2, Re (αβ) = 1 − σσ,

and, consequently,

Re [b(v, v)] =

∫ ρ

R

∫ 2π

0

[

1 + σσ

1 + σ2r

∣

∣

∣

∣

∂v

∂r

∣

∣

∣

∣

2

+1 + σσ

1 + σ2

1

r

∣

∣

∣

∣

∂v

∂θ

∣

∣

∣

∣

2

+ (σσ − 1)k2r|v|2]

dr dθ.

Since

1 + σσ

1 + σ2≥ 1

1 + σ2≥ |α0|−2,

1 + σσ

1 + σ2≥ 1 ≥ |α0|−2, (2.24)

where α0 = 1 + iσ0, by using the analytic Fredholm alternative theorem we knowthat the PML problem in the layer (2.23) exists a unique solution for every real kexcept possibly for a discrete set of values of k (cf. e.g. the argument in [10, Theorem2]). In this paper we will not elaborate on this issue and simply make the followingassumption:

(H2) There exists a unique solution to the Dirichlet PML problem (2.23) in the layer.

For any ϕ ∈ H1(ΩPML), define

‖ϕ ‖∗,ΩPML =

[

∫ ρ

R

∫ 2π

0

(

1 + σσ

1 + σ2r

∣

∣

∣

∣

∂ϕ

∂r

∣

∣

∣

∣

2

+1 + σσ

1 + σ2

1

r

∣

∣

∣

∣

∂ϕ

∂θ

∣

∣

∣

∣

2

+ (1 + σσ)k2r|ϕ|2)]1/2

.

It is easy to see that ‖ · ‖∗,ΩPML is an equivalent norm on H1(ΩPML). By using the

general theory in [1, Chapter 5], (H2) implies that there exists a constant C > 0 suchthat

sup06=ψ∈H1

0 (ΩPML)

|b(ϕ, ψ)|‖ψ ‖∗,ΩPML

≥ C‖ϕ ‖∗,ΩPML ∀ϕ ∈ H10 (ΩPML). (2.25)

The constant C depends in general on the domain ΩPML and the wave number k. Inthe appendix of the paper, however, we will show that for sufficiently large σ0, (H2)can be proved and C can be chosen as independent of ΩPML and k. Without loss ofgenerality we assume C ≤ 1.

To proceed, we introduce the following notation. For any function ξ defined on acircle Γa = x ∈ R2 : |x| = a having the Fourier expansion:

ξ =∑

n∈Z

ξneinθ, ξn =

1

2π

∫ 2π

0

ξe−inθdθ,

we define

‖ ξ ‖2H1/2(Γa) = 2π

∑

n∈Z

(1 + n2)1/2|ξn|2, ‖ ξ ‖2H−1/2(Γa) = 2π

∑

n∈Z

(1 + n2)−1/2|ξn|2.

The following theorem is the main objective of this subsection.

9

Theorem 2.4. Let (H1)-(H2) be satisfied. There exists a constant C > 0 inde-pendent of k,R, ρ, and σ0 such that the following estimates are satisfied

‖ |α|−1∇w ‖L2(ΩPML) ≤ CC−1(1 + kR)|α0|‖ q ‖H1/2(Γρ), (2.26)∥

∥

∥

∥

∂w

∂n

∥

∥

∥

∥

H−1/2(ΓR)

≤ CC−1(1 + kR)2|α0|2‖ q ‖H1/2(Γρ), (2.27)

where α0 = 1 + iσ0.Proof. We first show that there exists a constant C independent of k, ρ,R and σ0

such that

|b(ϕ, ψ)| ≤ C(1 + kR)|α0|‖ψ ‖∗,ΩPML‖|ϕ‖|H1(ΩPML), (2.28)

where ‖|ϕ‖|H1(ΩPML) = (‖∇ϕ ‖2L2(ΩPML) + R−2‖ϕ ‖2

L2(ΩPML))1/2 is the weighted H1-

norm. In fact, since σ ≤ σ ≤ σ0, we have

∣

∣

∣

∣

∫ ρ

R

∫ 2π

0

(

β

αr∂ϕ

∂r

∂ψ

∂r+

α

βr

∂ϕ

∂θ

∂ψ

∂θ− αβk2rϕψ

)

dr dθ

∣

∣

∣

∣

≤(

∫ ρ

R

∫ 2π

0

1 + σσ

1 + σ2r

∣

∣

∣

∣

∂ψ

∂r

∣

∣

∣

∣

2)1/2(

∫ ρ

R

∫ 2π

0

1 + σ2

1 + σσr

∣

∣

∣

∣

∂ϕ

∂r

∣

∣

∣

∣

2)1/2

+

(

∫ ρ

R

∫ 2π

0

1 + σσ

1 + σ2

1

r

∣

∣

∣

∣

∂ψ

∂θ

∣

∣

∣

∣

2)1/2(

∫ ρ

R

∫ 2π

0

1 + σ2

1 + σσ

1

r

∣

∣

∣

∣

∂ϕ

∂θ

∣

∣

∣

∣

2)1/2

+

(∫ ρ

R

∫ 2π

0

k2(1 + σσ)r|ψ|2)1/2 (∫ ρ

R

∫ 2π

0

k2r|αβ|2

1 + σσ|ϕ|2

)1/2

≤ C(1 + kR)|α0|‖ψ ‖∗,ΩPML‖|ϕ‖|H1(ΩPML).

This implies the estimate (2.28).Now we turn to the proof the estimate (2.26). Let ψ ∈ H1(ΩPML) such that

ψ = 0 on ΓR and ψ = q on Γρ. By taking ϕ = w−ψ ∈ H10 (ΩPML) in (2.23), we know

from (2.28) that

|b(ϕ, ϕ)| = |b(w − ψ, ϕ)| = |b(ψ, ϕ)| ≤ C(1 + kR)|α0|‖ϕ ‖∗,ΩPML‖|ψ‖|H1(ΩPML),

which implies by (2.25) that

‖ϕ ‖∗,ΩPML ≤ CC−1(1 + kR)|α0|‖|ψ‖|H1(ΩPML).

Notice that

‖ψ ‖∗,ΩPML ≤ C(1 + kR)|α0|‖|ψ‖|H1(ΩPML),

we get

‖w ‖∗,ΩPML = ‖ϕ+ ψ ‖∗,ΩPML ≤ CC−1(1 + kR)|α0|‖|ψ‖|H1(ΩPML).

Since the above estimate is valid for any ψ ∈ H1(ΩPML) such that ψ = 0 on ΓR,ψ = q on Γρ, we deduce by standard scaling argument using the assumption ρ ≤ CRthat

‖w ‖∗,ΩPML ≤ CC−1(1 + kR)|α0|‖ q ‖H1/2(Γρ). (2.29)

10

This shows the estimate (2.26) upon using (2.24).

To show (2.27) we multiply the equation (2.21a) by any function ϕ ∈ H1(ΩPML)such that ϕ = 0 on Γρ and integrate over ΩPML to obtain

−∫

ΩPML

A∇w · ∇ϕdx −∫

ΓR

∂w

∂rϕds+

∫

ΩPML

αβk2wϕdx = 0.

Thus∣

∣

∣

∣

∫

ΓR

∂w

∂rϕds

∣

∣

∣

∣

= |b(w, ϕ)| ≤ C(1 + kR)|α0|‖w ‖∗,ΩPML‖|ϕ‖|H1(ΩPML),

for any ϕ ∈ H1(ΩPML) such that ϕ = 0 on Γρ. This implies by (2.29) that

∣

∣

∣

∣

∫

ΓR

∂w

∂rϕds

∣

∣

∣

∣

≤ CC−1(1 + kR)2|α0|2‖ q ‖H1/2(Γρ)‖ϕ‖H1/2(ΓR) ∀ϕ ∈ H1/2(ΓR).

This completes the proof of the theorem.

2.3. Convergence of the PML problem. In this subsection we consider theconvergence of the PML problem (2.12) to the original scattering problem (2.4). Fol-lowing an idea in [13], for any function f ∈ H1/2(ΓR), we introduce the propagationoperator P : H1/2(ΓR) → H1/2(Γρ):

P (f) =∑

n∈Z

H (1)n (kρ)

H (1)n (kR)

fneinθ, fn =

1

2π

∫ 2π

0

fe−inθdθ. (2.30)

By Lemma 2.2, it is easy to see that P : H1/2(ΓR) → H1/2(Γρ) is well-defined, and

‖P (f) ‖H1/2(Γρ) ≤ e−kIm (ρ)

(

1− R2

|ρ|2

)1/2

‖ f ‖H1/2(ΓR) ∀r ≥ R. (2.31)

Moreover, by Theorem 2.4, under the assumptions (H1)-(H2), the operator T :H1/2(ΓR) → H−1/2(ΓR), which is defined through the Dirichlet problem of the PMLequation in the layer, is also well-defined. Furthermore, we have the following esti-mate.

Lemma 2.5. Let (H1)-(H2) be satisfied. We have

‖Tf − T f ‖H−1/2(ΓR) ≤ CC−1(1 + kR)2|α0|2e−kIm (ρ)(

1− R2

|ρ|2

)1/2

‖ f ‖H1/2(ΓR).

Proof. For any f ∈ H1/2(ΓR), we know that

Tf − T f =∂w

∂n

∣

∣

∣

ΓR

,

where w ∈ H1(ΩPML) satisfies

∇ · (A∇w) + αβk2w = 0 in ΩPML,

w = 0 on ΓR, w = P (f) on Γρ.

11

By (2.27) and (2.31) we then have

∥

∥

∥

∥

∂w

∂n

∥

∥

∥

∥

H−1/2(ΓR)

≤ CC−1(1 + kR)2|α0|2‖P (f) ‖H1/2(Γρ)

≤ CC−1(1 + kR)2|α0|2e−kIm (ρ)(

1− R2

|ρ|2

)1/2

‖ f ‖H1/2(ΓR).

This completes the proof.

The following theorem is the main results of this section.Theorem 2.6. Let (H1)-(H2) be satisfied. Then for sufficiently large σ0 > 0,

the PML problem (2.12) has a unique solution u ∈ H1(Ωρ). Moreover, we have thefollowing estimate

‖u− u ‖H1(ΩR) ≤ CC−1(1 + kR)2|α0|2e−kIm (ρ)(

1− R2

|ρ|2

)1/2

‖ u ‖H1/2(ΓR). (2.32)

Proof. The existence of a unique solution for (2.12) follows from Lemma 2.5 byusing the same argument as in [8, Theorem 2.4]. Next, by (2.4) and (2.12), we have

a(u− u, ϕ) = a(u, ϕ) − a(u, ϕ) = 〈T u− T u, ϕ〉ΓR ∀ϕ ∈ H1(ΩPML).

This implies the desired estimate (2.32) upon using Lemma 2.5 and (2.5).

3. Finite element approximations. In this section we introduce the finiteelement approximations of the PML problems (2.9a)-(2.9b). From now on we assumeg ∈ L2(ΓD). Let b : H1(Ωρ) ×H1(Ωρ) → C be the sesquilinear form given by

b(ϕ, ψ) =

∫

Ωρ

(

A∇ϕ · ∇ψ − αβk2ϕψ)

dx. (3.1)

Denote by H1(0)(Ωρ) = v ∈ H1(Ωρ) : v = 0 on Γρ. Then the weak formulation of

(2.9a)-(2.9b) is: Given g ∈ L2(ΓD), find u ∈ H1(0)(Ωρ) such that

b(u, ψ) =

∫

ΓD

gψds ∀ψ ∈ H1(0)(Ωρ). (3.2)

Let Γhρ , which consists of piecewise segments whose vertices lie on Γρ, be an

approximation of Γρ. Let Ωhρ be the subdomain of Ωρ bounded by ΓD and Γhρ . Let

Mh be a regular triangulation of the domain Ωhρ . We assume the elements K ∈ Mh

may have one curved edge align with ΓD so that Ωhρ = ∪K∈MhK.

Let Vh ⊂ H1(Ωhρ) be the conforming linear finite element space over Ωhρ , and

V h = vh ∈ Vh : vh = 0 on Γhρ. In the following we will always assume that the

functions in

V h are extended to the domain Ωρ by zero so that any function vh ∈

V his also a function in H1

(0)(Ωρ). The finite element approximation to the PML problem

(2.9a)-(2.9b) reads as follows: Find uh ∈

V h such that

b(uh, ψh) =

∫

ΓD

gψhds ∀ ψh ∈

V h. (3.3)

12

Following the general theory in [1, Chap. 5], the existence of unique solution of thediscrete problem (3.3) and the finite element convergence analysis depend on thefollowing discrete inf-sup condition

sup

06=ψh∈V h

|b(ϕh, ψh)|‖ψh ‖H1(Ωρ)

≥ µ ‖ϕh ‖H1(Ωρ) ∀ ϕh ∈

V h, (3.4)

where the constant µ > 0 is independent of the finite element mesh size. Since thecontinuous problem (3.2) has a unique solution by Theorem 2.6, the sesquilinear formb : H1

(0)(Ωρ) × H1(0)(Ωρ) → C satisfies the continuous inf-sup condition. Then a

general argument of Schatz [18] implies (3.4) is valid for sufficiently small mesh sizeh < h∗. Based on (3.4), appropriate a priori error estimate can also be derived whichdepends on the regularity of the PML solution u. In this paper, we are interestedin a posterior error estimates and the associated adaptive algorithm. Thus in the

following we simply assume the discrete problem (3.3) has a unique solution uh ∈

V h.For any K ∈ Mh, we denote by hK its diameter. Let Bh denote the set of all

sides that do not lie on ΓD and Γhρ . For any e ∈ Bh, he stands for its length. For anyK ∈ Mh, we introduce the residual:

Rh := ∇ · (A∇uh|K) + αβk2uh|K . (3.5)

For any interior side e ∈ Bh which is the common side of K1 and K2 ∈ Mh, we definethe jump residual across e:

Je := (A∇uh|K1 −A∇uh|K2) · νe, (3.6)

using the convention that the unit normal vector νe to e points from K2 to K1. Ife = ΓD ∩ ∂K for some element K ∈ Mh, then we define the jump residual

Je := 2(∇uh|K · n + g) (3.7)

For any K ∈ Mh, denote by ηK

the local error estimator which is defined by

ηK = maxx∈K

ω(x) ·(

‖hKRh‖2L2(K) +

1

2

∑

e⊂∂K

he‖ Je ‖2L2(e)

)1/2

, (3.8)

where K is the union of all elements having nonempty intersection with K, and

ω(x) =

1 if x ∈ ΩR,

|α0α|e−kIm (r)(

1− r2

|r|2

)1/2

if x ∈ ΩPML,

The following theorem is the main result of this paper.Theorem 3.1. There exists a constant C depending only on the minimum angle

of the mesh Mh such that the following a posterior error estimate is valid

‖u− uh ‖H1(ΩR) ≤ CC−1Λ(kR)1/2(1 + kR)

(

∑

K∈Mh

η2K

)1/2

+CC−1(1 + kR)2|α0|2e−kIm (ρ)(

1− R2

|ρ|2

)1/2

‖uh ‖H1/2(ΓR).(3.9)

13

Here Λ(kR) is defined in Lemma 4.3 below.The proof of this theorem will be given in §4. The important exponentially

decaying factor e−kIm (r)

(

1− r2

|r|2

)1/2

in the PML region ΩPML allows us to take thickerPML layers without introducing unnecessary fine meshes away from the fixed domainΩR. Recall that thicker PML layers allow smaller PML medium property, whichenhances numerical stability.

4. A posteriori error estimates. In this section we prove the a posteriori errorestimates in Theorem 3.1.

4.1. Error representation formula. For any ϕ ∈ H1(ΩR), let ϕ be its exten-sion in ΩPML such that

∇ · (A∇ϕ) + αβk2ϕ = 0 in ΩPML, (4.1a)

ϕ = ϕ on ΓR, ϕ = 0 on Γρ. (4.1b)

Lemma 4.1. Let (H2) be satisfied. For any ϕ, ψ ∈ H1(ΩPML), we have

〈Tϕ, ψ〉ΓR = 〈T ψ, ϕ〉ΓR .

Proof. By definition, T ϕ = ∂w/∂n on ΓR, where w satisfies

∇ · (A∇w) + αβk2w = 0 in ΩPML,

w = ϕ on ΓR, w = 0 on Γρ.

Thus

w(x) =∑

n∈Z

(anH(1)

n (kr) + bnH(2)

n (kr)) einθ

with the coefficients an, bn being determined by the boundary conditions in (4.1b)

anH(1)

n (kR) + bnH(2)

n (kR) = ϕn, anH(1)

n (kρ) + bnH(2)

n (kρ) = 0,

where ϕn = 12π

∫ 2π

0 ϕ(R, θ)e−inθdθ is the n-th Fourier coefficient of ϕ|ΓR . Denote by

Hn(kr) = H (1)

n (kr)H (2)

n (kρ) −H (2)

n (kr)H (1)

n (kρ).

Then since by (H2) the Dirichlet PML problem in the layer has a unique solution, weget Hn(kR) 6= 0, and

an =H (2)n (kρ)

Hn(kR)ϕn, bn = − H (1)

n (kρ)

Hn(kR)ϕn.

Thus

w = w(r, θ) =∑

n∈Z

Hn(kr)

Hn(kR)ϕne

inθ,

which, since r′(R) = α(R) = 1 and R = R, implies

Tϕ|ΓR =∑

n∈Z

kH ′n(kR)

Hn(kR)ϕne

inθ.

14

Therefore

〈Tϕ, ψ〉ΓR =∑

n∈Z

kH ′n(kR)

Hn(kR)ϕn

¯ψn ∀ϕ, ψ ∈ H1(ΩPML).

This completes the proof.Whenever no confusion of the notation incurred, we shall write in the following

ϕ as ϕ in ΩPML.Lemma 4.2 (Error representational formula). For any ϕ ∈ H1(ΩR), which is

extended to be a function in H1(Ωρ) according to (4.1a)-(4.1b), and ϕh ∈

V h, wehave

a(u− uh, ϕ) =

∫

ΓD

g(ϕ− ϕh) − b(uh, ϕ− ϕh) + 〈Tuh − T uh, ϕ〉ΓR . (4.2)

Proof. By (2.4) and the definitions (2.3) and (3.1),

a(u− uh, ϕ)

=

∫

ΓD

gϕ−∫

ΩR

(A∇uh · ∇ϕ− αβk2uhϕ) + 〈Tuh, ϕ〉ΓR

=

∫

ΓD

gϕ− b(uh, ϕ) +

∫

ΩPML

(A∇uh · ∇ ¯ϕ− αβk2uh ¯ϕ) + 〈Tuh, ϕ〉ΓR . (4.3)

On the other hand, by multiplying (4.1a) by uh, integrating by parts, and recallingthat n is the unit outer normal to ΓR which points outside ΩR, we deduce that

−∫

ΩPML

(A∇ϕ · ∇uh − αβk2ϕuh) −⟨

∂ϕ

∂n, uh

⟩

ΓR

= 0,

which is equivalent to

∫

ΩPML

(A∇uh · ∇ ¯ϕ− αβk2uh ¯ϕ) = −⟨

∂ ¯ϕ

∂n, uh

⟩

ΓR

. (4.4)

Since by the definition of T : H1/2(ΓR) → H−1/2(ΓR),

∂ ¯ϕ

∂n

∣

∣

∣

ΓR

= T ϕ,

we obtain by substituting (4.4) into (4.3) that

a(u− uh, ϕ) =

∫

ΓD

gϕ− b(uh, ϕ) + 〈Tuh, ϕ〉 − 〈T ϕ, uh〉.

This completes the proof upon using Lemma 4.1 and (3.3).

4.2. Estimates for the extension. For any ϕ ∈ H1(ΩR), we define, for r ≥ R,

φ = φ(r, θ) =∑

n∈Z

H (1)n (kr)

H (1)n (kR)

¯ϕneinθ, ϕn =

1

2π

∫ 2π

0

ϕ(R, θ)e−inθdθ. (4.5)

15

The function φ satisfies

∇ · (A∇φ) + αβk2φ = 0 in R2\BR, (4.6a)

φ = ϕ on ΓR, (4.6b)

|φ| is uniformly bounded as r = |x| → ∞. (4.6c)

By Lemma 2.2, it is easy to see that

‖φ ‖H1/2(Γρ) ≤ e−kIm (ρ)

(

1− R2

|ρ|2

)1/2

‖ϕ ‖H1/2(ΓR). (4.7)

Set

γ(r) = ekIm (r)

(

1− r2

|r|2

)1/2

.

Since r = r(1 + iσ), we obatin by simple calculation that

γ′(r) = γ(r) · k(

σσ

(1 + σ2)1/2+

rσσ′

(1 + σ2)3/2

)

,

which, together with rσ′ = σ − σ ≤ σ, implies

0 ≤ γ′(r) ≤ 2σkγ(r) ∀r ≥ R. (4.8)

Lemma 4.3. Let Λ(kR) = max

(

1,|H

(1)0

′(kR)|

|H(1)0 (kR)|

)

. Then there exists a constant

C > 0 independent of k,R, ρ, and σ0 such that

‖ |α|−1γ∇φ ‖L2(ΩPML) ≤ CΛ(kR)1/2(1 + kR)|α0|‖ϕ ‖H1/2(ΓR).

Proof. We multiply (4.6a) by γ2φ and integrate over ΩPML to obtain

∫ ρ

R

∫ 2π

0

γ2

(

βr

α

∣

∣

∣

∣

∂φ

∂r

∣

∣

∣

∣

2

+α

βr

∣

∣

∣

∣

∂φ

∂θ

∣

∣

∣

∣

2)

dr dθ

= −∫ ρ

R

∫ 2π

0

(

βr

α

∂φ

∂r(γ2)′φ− αβk2rγ2|φ|2

)

dr dθ

+

∫ 2π

0

[

βr

αγ2 ∂φ

∂rφ

]

(ρ)dθ −∫ 2π

0

[

βr

αγ2 ∂φ

∂rφ

]

(R)dθ.

Taking the real part of the equation we get

∫ ρ

R

∫ 2π

0

γ2

(

1 + σσ

1 + σ2r

∣

∣

∣

∣

∂φ

∂r

∣

∣

∣

∣

2

+1 + σσ

1 + σ2

1

r

∣

∣

∣

∣

∂φ

∂θ

∣

∣

∣

∣

2)

dr dθ

≤∫ ρ

R

∫ 2π

0

∣

∣

∣

∣

βr

α

∂φ

∂r2γγ′φ

∣

∣

∣

∣

dr dθ +

∫ ρ

R

∫ 2π

0

|αβ|k2rγ2|φ|2dr dθ

+

∫ 2π

0

∣

∣

∣

∣

[

βr

αγ2 ∂φ

∂rφ

]

(ρ)

∣

∣

∣

∣

dθ +

∫ 2π

0

∣

∣

∣

∣

[

βr

αγ2 ∂φ

∂rφ

]

(R)

∣

∣

∣

∣

dθ.

:= I1 + · · · + I4. (4.9)

16

Since γ′ ≤ 2kσγ by (4.8), we obtain by Cauchy-Schwarz inequality and the fact σ ≤ σthat

I1 ≤(

∫ ρ

R

∫ 2π

0

γ2 1 + σσ

1 + σ2r

∣

∣

∣

∣

∂φ

∂r

∣

∣

∣

∣

2)1/2(

∫ ρ

R

∫ 2π

0

16k2σ2γ2

∣

∣

∣

∣

β

α

∣

∣

∣

∣

21 + σ2

1 + σσr|φ|2

)1/2

≤ 4

(

∫ ρ

R

∫ 2π

0

γ2 1 + σσ

1 + σ2r

∣

∣

∣

∣

∂φ

∂r

∣

∣

∣

∣

2

dr dθ

)1/2(∫ ρ

R

∫ 2π

0

k2σ2γ2r|φ|2dr dθ)1/2

.

On the other hand, by (4.5) and Lemma 2.2, we know that

∫ ρ

R

∫ 2π

0

k2σ2γ2r|φ|2dr dθ = 2π∑

n∈Z

∫ ρ

R

k2σ2γ2r

∣

∣

∣

∣

H (1)n (kr)

H (1)n (kR)

∣

∣

∣

∣

2

dr · |φn|2

≤ 2π∑

n∈Z

∫ ρ

R

k2σ2rdr · |φn|2

≤ C(1 + kR)2|α0|2‖φ ‖2L2(ΓR). (4.10)

Hence

I1 ≤ 1

2

∫ ρ

R

∫ 2π

0

γ2 1 + σσ

1 + σ2r

∣

∣

∣

∣

∂φ

∂r

∣

∣

∣

∣

2

dr dθ + C(1 + kR)2|α0|2‖φ ‖2L2(ΓR).

By (4.10) we also have

I2 ≤ C(1 + kR)2|α0|2‖φ ‖2L2(ΓR).

Next, since r′(r) = α(r), by (4.5) and Lemma 2.3, we have

I3 ≤ 2π∑

n∈Z

∣

∣

∣

∣

kρβ(ρ)γ(ρ)2H (1)n

′(kρ)

H (1)n (kR)

H (1)n (kρ)

H (1)n (kR)

∣

∣

∣

∣

· |φn|2

≤ 2π|α0|∑

n6=0

kρ

(

1 +|n||kρ|

)

|φn|2 + 2π|α0|kρ∣

∣

∣

∣

H (1)

0′(kR)

H (1)

0 (kR)

∣

∣

∣

∣

· |φ0|2

≤ 2π|α0|∑

n6=0

(kρ+ |n|)|φn|2 + 2π|α0|kρΛ(kR)|φ0|2,

where in the last inequality we have used the relation ρ ≤ |ρ|. Since kρ + |n| ≤(1 + kρ)(1 + n2)1/2, we deduce finally

I3 ≤ CΛ(kR)(1 + kρ)|α0|‖φ ‖2H1/2(ΓR) ≤ CΛ(kR)(1 + kR)|α0|‖φ ‖2

H1/2(ΓR).

Similarly, we can prove

I4 ≤ CΛ(kR)(1 + kR)‖φ ‖2H1/2(ΓR).

Substituting the estimates for I1, · · · , I4 into (4.9), we conclude that

∫ ρ

R

∫ 2π

0

γ2

(

1 + σσ

1 + σ2r

∣

∣

∣

∣

∂φ

∂r

∣

∣

∣

∣

2

+1 + σσ

1 + σ2

1

r

∣

∣

∣

∣

∂φ

∂θ

∣

∣

∣

∣

2)

dr dθ

≤ CΛ(kR)(1 + kR)2|α0|2‖φ ‖2H1/2(ΓR).

17

This completes the proof.The following lemma is the main objective of this subsection.Lemma 4.4. For any ϕ ∈ H1(ΩR), which is extended to be a function ϕ ∈

H1(ΩPML) according to (4.1a)-(4.1b), we have the following estimate

‖ |α|−1γ∇ϕ ‖L2(ΩPML) ≤ CC−1Λ(kR)1/2(1 + kR)|α0|‖ϕ ‖H1/2(ΓR).

Proof. Let w = ϕ− φ, then from (4.1a)-(4.1b) and (4.6a)-(4.6b) we know that wsatisfies

∇ · (A∇w) + αβk2w = 0 in ΩPML,

w = 0 on ΓR, w = −φ on Γρ.

By Theorem 2.4 and (4.7) we have

‖ |α|−1∇w ‖L2(ΩPML) ≤ CC−1(1 + kR)|α0|‖w ‖H1/2(Γρ)

= CC−1(1 + kR)|α0|‖φ ‖H1/2(Γρ)

≤ CC−1(1 + kR)|α0|e−kIm (ρ)(

1− R2

|ρ|2

)1/2

‖ϕ ‖H1/2(ΓR).

By (4.8), γ is a increasing function, we know that, for r ≤ ρ,

γ(r)e−kIm (ρ)

(

1− R2

|ρ|2

)1/2

≤ γ(ρ)e−kIm (ρ)

(

1− R2

|ρ|2

)1/2

≤ 1.

Hence

‖ |α|−1γ∇w ‖L2(ΩPML) ≤ CC−1(1 + kR)|α0|‖ϕ ‖H1/2(ΓR).

This completes the proof upon using Lemma 4.3.To conclude this subsection we remark that a direct consequence of this lemma

is that

‖ω−1∇ϕ ‖L2(ΩPML) ≤ CC−1Λ(kR)1/2(1 + kR)‖ϕ ‖H1/2(ΓR). (4.11)

4.3. Proof of Theorem 3.1. Since we are going to interpolate nonsmooth func-

tions, we resort to an interpolation operator Πh : H1(0)(Ωρ) →

V h of Clement type [9],

where H1(0)(Ωρ) = v ∈ H1(Ωρ) : v = 0 on Γρ. Let Nh = aiNi=1 be the set of

the nodes of Mh which is interior to Ωhρ or on the boundary ΓD, and φiNi=1 be thecorresponding nodal basis of Vh. Define ∆i = suppφi ∩ Ωρ. Then the interpolationoperator Πh : H1

(0)(Ωρ) → Vh is defined by

Πhv(x) =

N∑

i=1

(

1

|∆i|

∫

∆i

v(x)dx

)

φi(x).

Since the nodes on Γhρ are not included in the definition of Πh, we know that Πhv ∈

V h.Moreover, by slightly modifying the argument in [6, Lemma 3.1 and Lemma 3.2], onecan show that the operator Πh enjoys the following interpolation estimates, for anyv ∈ H1

(0)(Ωρ),

‖ v − Πhv ‖L2(K) ≤ ChK‖∇v ‖L2(K), ‖ v − Πhv ‖L2(e) ≤ Ch1/2e ‖∇v ‖L2(e),(4.12)

18

where K and e are the union of all elements in Mh having non-empty intersectionwith K ∈ Mh and the side e, respectively.

Now we take ϕh = Πhϕ ∈

V h in the error representation formular (4.2) to get

a(u− uh, ϕ) =

∫

ΓD

g(ϕ− Πhϕ) − b(uh, ϕ− Πhϕ) + 〈Tuh − T uh, ϕ〉ΓR

:= II1 + II2 + II3. (4.13)

We observe that, by integration by parts and using (3.5)-(3.7),

II1 + II2 =∑

K∈Mh

(

∫

K

Rh(ϕ− Πhϕ)dx +∑

e⊂∂K

1

2

∫

e

Je(ϕ − Πhϕ)ds

)

.

Standard argument in the a posteriori error analysis using (4.12) and (4.11) implies

|II1 + II2| ≤ C∑

K∈Mh

(

‖hKRh ‖2L2(K) +

1

2

∑

e⊂∂K

‖h1/2e Je ‖2

L2(e)

)1/2

‖∇ϕ ‖L2(K)

≤ C∑

K∈Mh

ηK‖ω−1∇ϕ ‖L2(K)

≤ CC−1Λ(kR)1/2(1 + kR)

(

∑

K∈Mh

η2K

)1/2

‖ϕ ‖H1/2(ΓR).

By Lemma 2.5, we obtain

|II3| ≤ CC−1(1 + kR)2|α0|2e−kIm (ρ)(

1− R2

|ρ|2

)1/2

‖uh ‖H1/2(ΓR)‖ϕ ‖H1/2(ΓR).

Therefore, by the inf-sup condition (2.5), we finally get

‖u− uh ‖H1(ΩR) ≤ C sup06=ϕ∈H1(ΩR)

|a(u− uh, ϕ)|‖ϕ ‖H1(ΩR)

≤ CC−1Λ(kR)1/2(1 + kR)

(

∑

K∈Mh

η2K

)1/2

+CC−1(1 + kR)2|α0|2e−kIm (ρ)(

1− R2

|ρ|2

)1/2

‖uh ‖H1/2(ΓR).

This completes the proof.

5. Implementation and numerical examples. The implementation of theadaptive algorithm in this section is based on the PDE toolbox of MATLAB. We usethe a posteriori error estimate in Theorem 3.1 to determine the PML parameters.According to the discussion in §2, we choose the PML medium property as the powerfunction and thus we need only to specify the thickness ρ − R of the layer and themedium parameter σ0. Recall from Theorem 3.1 that the a posteriori error estimateconsists of two parts: the PML error and the finite element discretization error. Inour implementation we first choose ρ and σ0 such that the exponentially decayingfactor:

ω = e−kIm (ρ)(1− R2

|ρ|2)1/2

≤ 10−8, (5.1)

19

which makes the PML error negligible compared with the finite element discretizationerrors. Once the PML region and the medium property are fixed, we use the standardfinite element adaptive strategy to modify the mesh according to the a posteriori errorestimate. Now we describe the adaptive algorithm we used in the paper.

Algorithm 5.1. Given tolerance TOL > 0. Let m = 2.

• Choose ρ and σ0 such that the exponentially decaying factor ω ≤ 10−8;• Set the computational domain Ωρ = Bρ\ΓD and generate an initial mesh Mh over

Ωρ;

• While EFEM =(∑

K∈Mhη2K

)1/2> TOL do

- refine the mesh Mh according to the strategy:

if ηK > 12 maxK∈Mh

ηK , refine the element K ∈ Mh

- solve the discrete problem (3.3) on Mh

- compute error estimators on Mh

end while

In the following we report two numerical examples to demonstrate the competitivebehavior of the proposed algorithm. In the computations we first prescribe ρ and thendetermine σ0 according to (5.1). We scale the error estimator for determining finiteelement meshes by a factor 0.15 as in the PDE toolbox of MATLAB.

Example 1. Let the scatter D be unit circle. We consider the scattering problem

whose exact solution is known: u = H(1)0 (kr), where r = |x|. We take R = 2,

and k = 1. Table 5.1 shows the different choices of the PML parameters ρ and σ0

determined by the relation (5.1).

Table 5.1The PML parameters for Example 1 and Example 2.

Example 1 Example 2ρ σ0 ρ σ0

2R 30 2R 43R 15 3R 24R 10 4R 18R 5

Figure 5.1 shows the logNk-log ‖∇(u−uk) ‖L2(ΩR) curves, whereNk is the numberof nodes of the mesh Mk and uk is the finite element solution of (3.3) over the meshMk. It indicates that the meshes and the associated numerical complexity are quasi-

optimal: ‖∇(u− uk) ‖L2(ΩR) = CN−1/2k is valid asymptotically.

One of the important quantities in the scattering problems is the far field pattern:

u∞(x) =ei

π4√

8πk

∫

∂D

(

u(y)∂e−ikx·y

∂υ(y)− ∂u(y)

∂υ(y)e−ikx·y

)

ds(y), x =x

|x| .

We compute the far field u∞(x), x = (cos(θ), sin(θ))T in the observation directionθ = π/4. Figures 5.2 shows the far fields for different choices of PML parameters ρ

20

and σ0. We observe that our adaptive algorithm is robust with respect to the choiceof the thickness of PML layer: the far fields of the scattering solutions are insensitiveto the choices of the PML parameters.

102 103 104 10510−2

10−1

100

101

Number of nodal points

H1 E

rror

ρ=2Rρ=3Rρ=4Rρ=8Ra line with slope −1/2

Fig. 5.1. Quasi-optimality of the adaptive mesh refinements of the error ‖∇(u − uh ‖L2(ΩR)

for Example 1.

Example 2. This example is taken from [10] which concerns the scattering of theplane wave uI = eikx1 from a perfectly conducting metal. The scatter D is containedin the box x ∈ R : −2 < x1 < 2.2,−0.7 < x2 < 0.7 as plotted in Figure 5.3. W takeR = 3 and k = 2π. The different choices of PML parameters ρ and σ0 determined bythe relation (5.1) are shown in Table 5.1.

Figure 5.4 shows the logNk-log Ek curves, where Nk is the number of nodes ofthe mesh Mk and the Ek = (

∑

K∈Mkη2K)1/2 is the associated a posteriori error

estimate. It indicates that the meshes and the associated numerical complexity are

quasi-optimal: Ek = CN−1/2k is valid asymptotically.

Figures 5.5 and 5.6 show the far fields in the incident direction θ = 0 and thereflective direction θ = π. Again we observe that the far fields are insensitive to thechoices of PML parameters.

In Figure 5.7 we show the mesh after 13 adaptive iterations when ρ = 3R. Weobserve that the mesh near the boundary Γρ is rather coarse, as a consequence of theexponentially decaying factor in our finite element a posteriori error estimator.

Appendix: The PML equation in the layer for large σ0. The purpose ofthe appendix is to show that for sufficiently large σ0, the PML problem in the layer(2.23) has a unique solution w. Moreover, the constant C in (2.25) can be chosenindependent of ΩPML and k.

21

0 1 2 3 4 5 6 7 8

x 104

−0.27

−0.265

−0.26

−0.255

−0.25

−0.245

−0.24

−0.235

−0.23

−0.225

Number of nodal points

The

far−

field

pat

tern

u∞

ρ=2Rρ=3Rρ=4Rρ=8R

Fig. 5.2. The real part of the far fields when the observing angle θ = π/4 for Example 1.

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Fig. 5.3. The geometry of the scatter for Example 2.

From (H1) we know that β(r) = 1 + iσ(r), where

σ(r) =1

r

∫ r

R

σ(t)dt =σ0

m+ 1

r −R

r

(

r −R

ρ−R

)m

.

Define

ζ(r) :=2σ2

0

σσ(r)=

2(m+ 1)r(ρ−R)2m

(r −R)2m+1, ∀r > R.

22

102 103 104 105100

101

102

Number of nodal points

A P

oste

riori

Err

or In

dica

tor

ρ=2Rρ=3Rρ=4Ra line with slope −1/2

Fig. 5.4. Quasi-optimality of the adaptive mesh refinements of the a posteriori error estimator

for Example 2.

0 1 2 3 4 5 6

x 104

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Number of nodal points

The

far−

field

pat

tern

u∞

ρ=2Rρ=3Rρ=4R

Fig. 5.5. The real part of the far fields in the incident direction for Example 2.

23

0 1 2 3 4 5 6

x 104

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Number of nodal points

The

far−

field

pat

tern

u∞

ρ=2Rρ=3Rρ=4R

Fig. 5.6. The real part of the far fields in the reflective direction for Example 2.

It is clear that ζ : (R,∞) → R is strictly monotone decreasing and ζ(r) → ∞as r → R, ζ(r) → 0 as r → ∞. Thus, for any σ0 > 0, there exists a uniqueR = R(σ0) > R such that

σ20 = ζ(R) =

2(m+ 1)R(ρ−R)2m

(R−R)2m+1. (5.1)

Hence, since σσ : (R,∞) → R is increasing, we have

σσ(r) ≥ σσ(R) =2σ2

0

ζ(R)= 2 for r ≥ R. (5.2)

In this appendix we make the following assumption on the choice of σ0:

(H3) σ20 ≥ ζ(Rmax), where Rmax := maxr ∈ (R, ρ) : θ(r) ≤ 1 with

θ(r) = k2R2

[(

r2

R2− 1

)

lnr

R+

2r2(r −R)

(2m+ 1)R3

]

∀r ≥ R.

Since the function θ : (R, ρ) → R is strictly monotone increasing and θ(R) = 0,Rmax is well-defined.

Lemma 5.1. Let (H1) and (H3) be satisfied. Then

∫ R

R

k2r

(∫ r

R

1 + σ2(t)

tdt

)

dr ≤ 1

2.

24

−9 −6 −3 0 3 6 9−9

−6

−3

0

3

6

9

Fig. 5.7. The mesh of 7048 nodes after 13 adaptive iterations when ρ = 3R for Example 2.

Proof. First we have

∫ R

R

r

(∫ r

R

1

tdt

)

dr =

∫ R

R

r lnr

Rdr ≤1

2(R2 −R2) ln

R

R.

Next by (H1) and (5.1) we know that

∫ R

R

r

(∫ r

R

σ2

tdt

)

dr ≤ R

R

∫ R

R

(∫ r

R

σ2(t)dt

)

dr

=R

R

∫ R

R

σ20

2m+ 1

(r −R)2m+1

(ρ−R)2mdr

=R

R

σ20

(2m+ 1)(2m+ 2)

(R−R)2m+2

(ρ−R)2m

=R2(R −R)

(2m+ 1)R.

Thus∫ R

R

k2r

(∫ r

R

1 + σ2(t)

tdt

)

dr ≤ 1

2k2R2

[(

R2

R2− 1

)

lnR

R+

2R2(R −R)

(2m+ 1)R3

]

=1

2θ(R).

Now if σ20 ≥ ζ(Rmax), we know from the monotonicity of ζ that R = R(σ0) ≤ Rmax.

Thus θ(R) ≤ θ(Rmax) ≤ 1 by (H3). This completes the proof.Now we are ready to prove the main result of this appendix.Theorem 5.2. Under the assumptions (H1) and (H3) there exists a constant

C > 0 independent of k,R, ρ, and σ0 such that

Re [b(v, v)] ≥ C‖ v ‖2∗,ΩPML ∀v ∈ H1

0 (ΩPML).

25

Proof. For any v ∈ H10 (ΩPML), we have

Re [b(v, v)] =

∫ ρ

R

∫ 2π

0

[

1 + σσ

1 + σ2r

∣

∣

∣

∣

∂v

∂r

∣

∣

∣

∣

2

+1 + σσ

1 + σ2

1

r

∣

∣

∣

∣

∂v

∂θ

∣

∣

∣

∣

2

+ (σσ − 1)k2r|v|2]

.

By (5.2) we know that

∫ ρ

R

∫ 2π

0

(σσ − 1)k2r|v|2dr dθ

=

∫ R

R

∫ 2π

0

(σσ − 1)k2r|v|2dr dθ +

∫ ρ

R

∫ 2π

0

(σσ − 1)k2r|v|2dr dθ

≥ −3

2

∫ R

R

∫ 2π

0

k2r|v|2dr dθ +1

4

∫ ρ

R

∫ 2π

0

(1 + σσ)k2r|v|2dr dθ.

Notice that since v = 0 on ΓR,

|v(r)| =

∣

∣

∣

∣

∫ r

R

∂v

∂rdr

∣

∣

∣

∣

≤(

∫ r

R

1 + σσ

1 + σ2t

∣

∣

∣

∣

∂v

∂r

∣

∣

∣

∣

2

dt

)1/2(∫ r

R

1

t

1 + σ2

1 + σσdt

)1/2

,

which, by Lemma 5.1, yields

∫ R

R

∫ 2π

0

k2r|v|2dr ≤(

∫ R

R

∫ 2π

0

1 + σσ

1 + σ2r

∣

∣

∣

∣

∂v

∂r

∣

∣

∣

∣

2)

·∫ R

R

k2r

(∫ r

R

1

t

1 + σ2

1 + σσdt

)

≤ 1

2

∫ R

R

∫ 2π

0

1 + σσ

1 + σ2r

∣

∣

∣

∣

∂v

∂r

∣

∣

∣

∣

2

.

Thus

Re [b(v, v)] ≥ 1

4

∫ ρ

R

∫ 2π

0

[

1 + σσ

1 + σ2r

∣

∣

∣

∣

∂v

∂r

∣

∣

∣

∣

2

+1 + σσ

1 + σ2

1

r

∣

∣

∣

∣

∂v

∂θ

∣

∣

∣

∣

2

+ (1 + σσ)k2r|v|2]

.

This completes the proof.Acknowledgments. The authors wish to thank Peter Monk for several inspiring

discussions and the referees for constructive comments.

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27