An Adaptive Uniaxial Perfectly Matched Layer Method for Time ...

NUMERICAL MATHEMATICS: Theory, Methods and Applications 

Numer. Math. Theor. Meth. Appl., Vol. 1, No. 2, pp. 113-137 (2008) 

An Adaptive Uniaxial Perfectly Matched Layer Method 

for Time-Harmonic Scattering Problems 

Zhiming Chen ∗ and Xinming Wu 

LSEC, Institute of Computational Mathematics, Academy of Mathematics and 

System Science, Chinese Academy of Sciences, Beijing 100190, China. 

Received 1 July, 2007; Accepted (in revised version) 26 November, 2007 

Abstract. The uniaxial perfectly matched layer (PML) method uses rectangular domain 

to define the PML problem and thus provides greater flexibility and efficiency in dealing 

with problems involving anisotropic scatterers. In this paper an adaptive uniaxial 

PML technique for solving the time harmonic Helmholtz scattering problem is developed. 

The PML parameters such as the thickness of the layer and the fictitious medium 

property are determined through sharp a posteriori error estimates. The adaptive finite 

element method based on a posteriori error estimate is proposed to solve the PML equation 

which produces automatically a coarse mesh size away from the fixed domain and 

thus makes the total computational costs insensitive to the thickness of the PML absorbing 

layer. Numerical experiments are included to illustrate the competitive behavior of 

the proposed adaptive method. In particular, it is demonstrated that the PML layer can 

be chosen as close to one wave-length from the scatterer and still yields good accuracy 

and efficiency in approximating the far fields. 

AMS subject classifications: 65N30, 65N50 

Key words: Adaptivity, uniaxial perfectly matched layer, a posteriori error analysis, acoustic scattering 

problems. 

Dedicated to Professor Yucheng Su on the Occasion of His 80th Birthday 

1. Introduction 

We propose and study a uniaxial perfectly matched layer (PML) technique for solving 

Helmholtz-type scattering problems with perfectly conducting boundary: 

∆u+ k 2 u=0 in 2 \¯D, (1.1a) 

∂ u 

=−g onΓ D , (1.1b) 

∂ n D 

 

∂ u r 

∂ r − iku → 0 as r=|x|→∞. (1.1c) 

∗ Corresponding author. Email addresses:ÞÑÒÐ×ºººÒ(Z. 

M. Wu) 

Chen),ÜÑÛÙÐ×ºººÒ(X. 

http://www.global-sci.org/nmtma 113 c○2008 Global-Science Press

114 Z. Chen and X. M. Wu 

Here D⊂ 2 is a bounded domain with Lipschitz boundaryΓ D , g∈ H −1/2 (Γ D ) is determined 

by the incoming wave, and n D is the unit outer normal toΓ D . We assume the wave 

number k∈is a constant. We remark that the results in this paper can be extended to the 

case when k 2 (x) is a variable wave number inside some bounded domain, or to solve the 

scattering problems with other boundary conditions, such as Dirichlet or the impedance 

boundary condition onΓ D . 

Since the work of Bérénger[3] which proposed a PML technique for solving the time 

dependent Maxwell equations, various constructions of PML absorbing layers have been 

proposed and studied in the literature (cf., e.g., Turkel and Yefet[19], Teixeira and Chew 

[18] for the reviews). The basic idea of the PML technique is to surround the computational 

domain by a layer of finite thickness with specially designed model medium that 

would either slow down or attenuate all the waves that propagate from inside the computational 

domain. 

The convergence of the PML method is studied in Lassas and Somersalo[13], Hohage 

et al.[12] for the acoustic scattering problems for circular PML layers and in Lassas and 

Somersalo[14] for general smooth convex geometry. It is proved in[12–14] that the 

PML solution converges exponentially to the solution of the original scattering problem as 

the thickness of the PML layer tends to infinite. We remark that in practical applications 

involving PML techniques, one cannot afford to use a very thick PML layer if uniform 

meshes are used because it requires excessive grid points and hence more computer time 

and more storage. On the other hand, a thin PML layer requires a rapid variation of the 

artificial material property which deteriorates the accuracy if too coarse mesh is used in 

the PML layer. 

The adaptive PML technique was proposed in Chen and Wu[8] for a scattering problem 

by periodic structures (the grating problem), in Chen and Liu[6] for the acoustic scattering 

problem, and in Chen and Chen[5] for Maxwell scattering problems. The main idea of 

the adaptive PML technique is to use the a posteriori error estimate to determine the PML 

parameters and to use the adaptive finite element method to solve the PML equations. The 

adaptive PML technique provides a complete numerical strategy to solve the scattering 

problems in the framework of finite element which produces automatically a coarse mesh 

size away from the fixed domain and thus makes the total computational costs insensitive 

to the thickness of the PML absorbing layer. 

The purpose of this paper is to extend the adaptive PML technique developed for circular 

PML layer in[5, 6, 8] to deal with the uniaxial PML methods which are widely used 

in the engineering literature. The main advantage of the uniaxial PML method as opposing 

to the circular PML method is that it provides greater flexibility and efficiency to solve 

problems involving anisotropic scatterers. Our technique to prove the PML convergence 

is different from the techniques developed in[5, 6, 8] for circular PML layers. It is based 

on the integral representation of the exterior Dirichlet problem for the Helmholtz equation 

and the idea of the complex coordinate stretching. To the authors’ best knowledge, this is 

the first convergence proof of the uniaxial PML method in the literature. We remark that 

the boundary of the uniaxial PML layer is only Lipschitz and so the results in[14] cannot 

be applied.

An Adaptive Uniaxial PML Method for Time-Harmonic Scattering Problems 115 

The layout of the paper is as follows. In Section 2 we recall the uniaxial PML formulation 

for (1.1a)-(1.1c) by following the method of complex coordinate stretching in Chew 

and Weedon[4]. In Section 3 we prove the convergence of the uniaxial PML method. 

In Section 4 we introduce the finite element approximation. In Section 5 we derive the 

a posteriori error estimate which includes both the PML error and the finite element discretization 

error. Finally in Section 6 we describe our adaptive algorithm and present two 

examples to show the competitive behavior of the adaptive method. 

2. The PML equation 

Let D be contained in the interior of the rectangle B 1 ={x∈ 2 :|x 1 |< L 1 /2,|x 2 |< 

L 2 /2}. LetΓ 1 =∂ B 1 and n 1 the unit outer normal toΓ 1 . We start by introducing the 

Dirichlet-to-Neumann operator T : H 1/2 (Γ 1 )→ H −1/2 (Γ 1 ). Given f∈ H 1/2 (Γ 1 ), we define 

T f= ∂ξ 

∂ n 1 

on Γ 1 , 

whereξis the solution of the exterior Dirichlet problem of the Helmholtz equation 

∆ξ+ k 2 ξ=0 in 2 \¯B 1 , (2.1a) 

ξ= f onΓ 1 , (2.1b) 

 

∂ξ r 

∂ r − ikξ → 0 as r=|x|→∞. (2.1c) 

It is well-known that (2.1a)-(2.1c) has a unique solutionξ∈H 1 loc (2 \¯B 1 ) (cf., e.g., Colten- 

Kress[11]). Thus T : H 1/2 (Γ 1 )→H −1/2 (Γ 1 ) is well-defined and is a continuous linear 

operator. 

Let a : H 1 (Ω 1 )× H 1 (Ω 1 )→, whereΩ 1 = B 1 \¯D, be the sesquilinear form 

∫ 

a(ϕ,ψ)= 

Ω 1 

 

∇ϕ·∇ ¯ψ− k 2 ϕ ¯ψ d x−〈Tϕ,ψ〉 Γ1 

, (2.2) 

where〈·,·〉 Γ1 

stands for the inner product on L 2 (Γ 1 ) or the duality pairing between H −1/2 (Γ 1 ) 

and H 1/2 (Γ 1 ). The scattering problem (1.1a)-(1.1c) is equivalent to the following weak 

formulation (cf., e.g.,[11]): Given g∈ H −1/2 (Γ D ), find u∈ H 1 (Ω 1 ) such that 

a(u,ψ)=〈g,ψ〉 ΓD 

, ∀ψ∈ H 1 (Ω 1 ). (2.3) 

The existence of a unique solution of the scattering problem (2.3) is known (cf., e.g., 

[11], McLean[15]). Then the general theory in Babuška and Aziz[1, Chap. 5] implies 

that there exists a constantµ>0 such that the following inf-sup condition is satisfied 

sup 

0≠ψ∈H 1 (Ω 1 ) 

|a(ϕ,ψ)| 

≥ µ‖ϕ‖ 

‖ψ‖ H 1 (Ω 1 ) , ∀ϕ∈H1 (Ω 1 ). (2.4) 

H 1 (Ω 1 )


L 1 

d 1 

L 2 

D 

ÙÖ½ËØØÒÓØ×ØØÖÒÔÖÓÐÑÛØØÈÅÄÐÝÖº 

Γ D 

Γ 1 

d 2 

Γ 2 

Now we turn to the introduction of the absorbing PML layer. Let B 2 ={x∈ 2 : 

|x 1 |< L 1 /2+d 1 ,|x 2 |< L 2 /2+d 2 } be the rectangle which contains B 1 . Letα 1 (x 1 )= 

1+iσ 1 (x 1 ),α 2 (x 2 )=1+iσ 2 (x 2 ) be the model medium property which satisfy 

σ j ∈ C(), σ j ≥ 0, σ j (t)=σ j (−t), and σ j = 0 for|t|≤ L j /2, j= 1,2. 

Denote by ˜x j the complex coordinate defined by 

 

x j if|x j |< L j /2, 

˜x j = ∫ x j 

α 0 j(t)d t if|x j |≥ L j /2. 

(2.5) 

To derive the PML equation, we first notice that by the third Green formula, the solutionξ 

of the exterior Dirichlet problem (2.1a)-(2.1c) satisfies 

ξ=−Ψ k SL (λ)+Ψk DL (f) in2 \¯B 1 , (2.6) 

whereλ= T f∈ H −1/2 (Γ 1 ) is the Neumann trace ofξonΓ 1 , andΨ k SL ,Ψk DL 

are respectively 

the single and double layer potentials 

Ψ k SL (λ)(x)= ∫Γ 1 

G k (x, y)λ(y) ds(y), ∀λ∈H −1/2 (Γ 1 ), (2.7) 

Ψ k DL (f)(x)= ∫ 

∂ G k (x, y) 

∂ n 

Γ 1 1 (y) 

f(y) ds(y), ∀ f∈ H 1/2 (Γ 1 ). (2.8) 

Here G k is the fundamental solution of the Helmholtz equation satisfying the Sommerfeld 

radiation condition 

G k (x, y)= i 4 H(1) 0 

(k|x− y|), 

where H (1) 

0 

(z) is the first Hankel function of order zero.


We follow the method of complex coordinate stretching[4] to introduce the PML equation. 

For any z∈, denote by z 1/2 the analytic branch of z such that Im(z 1/2 )>0for 

any z∈\[0,+∞). Letρ(˜x, y)=[(˜x 1 − y 1 ) 2 +(˜x 2 − y 2 ) 2 ] 1/2 be the complex distance 

and define 

˜G k (x, y)= i 4 H(1) 0 

(kρ(˜x, y)). 

It is easy to see that ˜G k is smooth for x∈ 2 \¯B 1 and y∈¯B 1 . Now we can define the 

modified single and double layer potentials[14] 

˜Ψ k SL (λ)(x)= ∫Γ 1 

˜G k (x, y)λ(y) ds(y), ∀λ∈H −1/2 (Γ 1 ), (2.9) 

˜Ψ k DL (f)(x)= ∫ 

∂ ˜G k (x, y) 

∂ n 

Γ 1 1 (y) 

It is clear that ˜Ψ k SL (λ), ˜Ψ k DL (f) are smooth in2 \¯B 1 , and 

f(y) ds(y), ∀ f∈ H 1/2 (Γ 1 ). (2.10) 

γ + D ˜Ψ k SL (λ)=γ+ D Ψk SL (λ), ∀λ∈H−1/2 (Γ 1 ), 

γ + D ˜Ψ k DL (f)=γ+ D Ψk DL (f), ∀ f∈ H1/2 (Γ 1 ), 

(2.11) 

whereγ + D : H1 loc (2 \¯B 1 )→H 1/2 (Γ 1 ) is the trace operator. For any f ∈ H 1/2 (Γ 1 ), let 

(f)(x) be the PML extension given by 

By (2.11) and (2.6) we know that 

(f)(x)=− ˜Ψ k SL (T f)+ ˜Ψ k DL (f) for x∈2 \¯B 1 . (2.12) 

γ + D (f)=−γ+ D Ψk SL (T f)+γ+ D Ψk DL (f)=γ+ D ξ= f on Γ 1 

for any f∈ H 1/2 (Γ 1 ). For the solution u of the scattering problem (2.3), let ũ=(u| Γ1 

) 

be the PML extension of u| Γ1 which satisfiesγ + Dũ=u| Γ 1 

onΓ 1 . Since H (1) 

0 

(z) decays exponentially 

on the upper half complex plane[6], heuristically ũ(x) will decay exponentially 

when x is away fromΓ 1 . It is obvious that ũ satisfies 

∂ 2 ũ 

∂ ˜x 2 + ∂ 2 ũ 

1 

∂ ˜x 2 + k 2 ũ=0 in 2 \¯B 1 , 

2 

which yields the desired PML equation by the chain rule 

∇·(A∇ũ)+α 1 α 2 k 2 ũ=0 in 2 \¯B 1 , 

where A= diag(α 2 (x 2 )/α 1 (x 1 ),α 1 (x 1 )/α 2 (x 2 )) is a diagonal matrix. 

The PML solution û inΩ 2 = B 2 \¯D is defined as the solution of the following system 

∇·(A∇û)+α 1 α 2 k 2 û=0 inΩ 2 , (2.13a) 

∂ û 

∂ n D 

=−g onΓ D , û=0 onΓ 2 . (2.13b) 

The well-posedness of the PML problem (2.13a)-(2.13b) and the convergence of its solution 

to the solution of the original scattering problem will be studied in the next section.


3. Convergence analysis 

We start by considering the Dirichlet problem of the PML equation in the layer 

∇·(A∇w)+α 1 α 2 k 2 w= 0 inΩ PML = B 2 \¯B 1 , (3.1a) 

w= 0 onΓ 1 , w= q onΓ 2 , (3.1b) 

where q∈H 1/2 (Γ 2 ). Introduce the sesquilinear form c : H 1 (Ω PML )× H 1 (Ω PML )→ as 

∫ 

c(ϕ,ψ)= 

Ω PML (A∇ϕ·∇ ¯ψ−α 1 α 2 k 2 ϕ ¯ψ) dx. 

Then the weak formulation for (3.1a)-(3.1b) is: Given q∈H 1/2 (Γ 2 ), find w∈H 1 (Ω PML ) 

such that w= 0 onΓ 1 , w= q onΓ 2 , and 

Notice that, for anyϕ∈H 1 (Ω PML ), 

 

 

1+σ 1 σ 2 

∂ϕ 

2 

Re[c(ϕ,ϕ)]= 

∫Ω PML 1+σ 2 ∂ x 

1 1 

c(w,ψ)=0, ∀ψ∈ H 1 0 (ΩPML ). (3.2) 

+ 1+σ 1σ 2 

1+σ 2 2 

∂ϕ 

2 

+(σ 1 σ 2 − 1)k 2 |ϕ| 2 dx. 

∂ x 2 

Since 

1+σ 1 σ 2 

1+σ 2 1 

1 

≥ 

1+σ 2 , 

m 

1+σ 1 σ 2 

1+σ 2 2 

1 

≥ 

1+σ 2 , 

m 

whereσ m = max x∈¯B 2 

(σ 1 (x 1 ),σ 2 (x 2 ))>0, we know by using the spectral theory of compact 

operators that (3.2) has a unique solution for every real k except possibly for a discrete 

set of values of k (see Collino and Monk[10, Theorem 2] for a similar discussion on the 

PML equation in the polar coordinates). In this paper we will not elaborate on this issue 

and simply make the following assumption 

(H1) There exists a unique solution to the Dirichlet PML problem (3.2) in the layer. 

We remark that for the circular PML method, the unique existence of the PML equation 

in the layer can be proved under certain conditions on the PML medium property in[6,13]. 

The proof of the unique existence of the Dirichlet problem for the uniaxial PML equation 

is an interesting open problem. 

Throughout the paper we will use the weighted H 1 -norm 

 

1/2 

‖∇ϕ‖ 2 L 2 (Ω) +|Ω|−1 ‖ϕ‖ 2 L (Ω) 

, 

2 

‖ϕ‖ H 1 (Ω)= 

for any bounded domainΩ⊂ 2 , where|Ω| is the Lebesgue measure ofΩ. For anyϕ∈ 

H 1 (Ω PML ), we define 

⎡ 

 

1 

∂ϕ 

2 

‖ϕ‖ ∗,Ω PML= 

⎣ 

∫Ω PML 1+σ 2 + 1 

⎤ 

∂ϕ 

2 

1/2 

∂ x 

1 1 1+σ 2 +(1+σ 1 σ 2 )k 2 |ϕ| 2 dx⎦ 

. 

∂ x 

2 2


It is easy to see that‖·‖ ∗,Ω PML is an equivalent norm on H 1 (Ω PML ). Again by using the 

general theory in[1, Chap. 5] we know that there exists a constant Ĉ> 0 such that 

sup 

0≠ψ∈H 1 0 (ΩPML ) 

|c(ϕ,ψ)| 

‖ψ‖ ∗,Ω PML 

≥ Ĉ‖ϕ‖ ∗,Ω PML. (3.3) 

The constant Ĉ depends in general on the domainΩ PML and the wave number k. 

Before we state the main result of this section, we make the following assumptions 

on the fictitious medium property, which is rather mild in the practical applications of the 

uniaxial PML method 

(H2) 

∫ L 1 

2 +d 1 

0 

σ 1 (t) dt= 

∫ L 2 

2 +d 2 

0 

σ 2 (t) dt=σ, σ>0 is a constant; 

(H3) σ j (t)= ˜σ j 

|t|− L j /2 

d j 

m 

, m≥1 integer, ˜σ j > 0 is a constant, j= 1,2. 

The following theorem is the main result of this section. 

Theorem 3.1. Let(H1)-(H3) be satisfied. Then for sufficiently largeσ>0, the PML problem 

(2.13a)-(2.13b) has a unique solution û∈H 1 (Ω 2 ). Moreover, we have the following error 

estimate 

‖u−û‖ H 1 (Ω 1 )≤CĈ −1 |α m | 3 (1+ kL) 4 e −(γkσ−1) ‖û‖ H 1/2 (Γ 1 ) , 

(3.4a) 

where 

γ= 

min(d 1 , d 2 ) 

(L 1 + d 1 ) 2 +(L 2 + d 2 ) 2 , L= max(L 1, L 2 ). (3.4b) 

The proof of this theorem will be given in Section 3.3 which depends on the exponential 

decay estimates of the PML extension in Section 3.1 and the stability estimates of the 

Dirichelt problem of the PML equation in the layer in Section 3.2. 

3.1. Estimates for the PML extension 

We start with the following elementary lemma. 

Lemma 3.1. For any z 1 = a 1 + ib 1 ,z 2 = a 2 + ib 2 with a 1 , b 1 , a 2 , b 2 ∈ such that a 1 b 1 + 

a 2 b 2 ≥ 0 and a 2 1 + a2 2 

> 0, we have 

Im(z 2 1 + z2 2 )1/2 ≥ a 1b 1 + a 2 b 2 

. 

a 

2 

1 + a2 2


Proof. For any a, b∈ we know that 

 

Im(a+ib) 1/2 −a+ a 2 + b 2 

= 

. 

2 

Here we used the convention that z 1/2 is the analytic branch of z such that Im(z 1/2 )>0 

for any z∈\[0,+∞). It is easy to check that Im(a+ ib) 1/2 is a decreasing function in 

a∈. Let z 2 1 + z2 2 

= a+ ib. Then 

a+ib= 

⎛ 

⎝ a 2 1 + a2 2 + i a 1b 1 + a 2 b 2 

 

a 

2 

1 + a2 2 

⎞ 

⎠ 

2 

− (a 2b 1 − a 1 b 2 ) 2 

a 2 1 + . 

a2 2 

Let a ′ = a+(a 2 b 1 − a 1 b 2 ) 2 /(a 2 1 + a2 2 ). Since a 1b 1 + a 2 b 2 ≥ 0, we have 

Im(a ′ + ib) 1/2 = a 1b 1 + a 2 b 2 

. 

a 

2 

1 + a2 2 

On the other hand, since a ′ ≥ a, we know that Im(a+ib) 1/2 ≥ Im(a ′ + ib) 1/2 . This 

completes the proof. 

Now let 

z j = ˜x j − y j =(x j − y j )+i 

For any x∈Γ 2 , y∈¯Ω 1 , it is easy to see that 

(x j − y j ) 

∫ x j 

0 

∫ x j 

0 

σ j (t) dt≥ 0. 

Thus, by Lemma 3.1,ρ(˜x, y)=(z 2 1 + z2 2 )1/2 satisfies 

σ j (t) dt. 

Imρ(˜x, y)≥ |x 1− y 1 || ∫ x 1 

0 σ 1(t) dt|+|x 2 − y 2 || ∫ x 2 

0 σ 2(t) dt| 

. 

|x− y| 

Now by (H2) we have, for any x∈Γ 2 , y∈¯Ω 1 , 

Imρ(˜x, y)≥ 

min(d 1 , d 2 ) 

(3.5) (L 1 + d 1 ) 2 +(L 2 + d 2 ) 2σ=γσ, 

whereγis defined in (3.4b). 

We need the modified Bessel function K ν (z) of orderν,ν∈, which is the solution of 

the differential equation 

z 2 d2 w 

dz 2 + z dw 

dz −(z2 +ν 2 )w= 0


satisfying the asymptotic behavior 

K ν (z)∼i( π 2z )1/2 e −z−π 4 i 

as|z|→∞. 

K ν (z) is connected with H (1) 

ν 

(z) through the relation 

Thus we know that 

K ν (z)= 1 2 πie 1 2 νπi H (1) 

ν (iz). 

˜G k (x, y)= i 1 

4 H(1) 0 

(kρ(˜x, y))= 

2π K 0(−ikρ(˜x, y)). (3.6) 

We refer to the treatise Watson[20] for extensive studies on the special function K ν (z). 

Lemma 3.2. For anyν∈,θ 2 ≥θ 1 > 0, we have 

K ν (θ 2 )≤ e −(θ 2−θ 1 ) K ν (θ 1 ). 

Proof. We first recall the Schläfli integral representation formula[20, P. 181], for z∈ 

such that| arg z| 0 depending only onγbut independent of k,σ, L j , d j , 

j= 1,2, such that for any x∈Γ 2 , y∈ ¯Ω 1 , 

(i) | ˜G k (x, y)|≤ C e −(γkσ−1) ; 

(ii) ∂ ˜G 

 

k≤Ck|αm 

| e −(γkσ−1) , j= 1,2; 

∂ x j (iii) ∂ ˜G 

 

k≤ 

Ck e −(γkσ−1) , j= 1,2; 

∂ y j (iv) ∂ 2 ˜G 

 

k≤ 

Ck 2 |α m | e −(γkσ−1) , i, j= 1,2. 

∂ x i ∂ y j 

Hereα m = max(|α 1 (x 1 )|,|α 2 (x 2 )|). 

x∈Γ 2


Proof. By the Schläfli integral representation formula (3.7), it is easy to see that 

|K 0 (z)|0. Thus, by (3.6) and Lemma 

3.2, if k Imρ(˜x, y)≥1, 

| ˜G k |≤ 1 

2π |K 0(−ikρ(˜x, y))|≤ 1 

2π K 0(kImρ(˜x, y))≤ 1 

2π K 0(1)e −(kImρ(˜x,y)−1) . 

This proves (i) by (3.5) and (3.8). To show (ii) we first notice that 

∂ ˜G k 

=− ik 

y) 

∂ x j 2π K′ 0 

(−ikρ(˜x, y))∂ρ(˜x, 

= ik 

∂ x j 2π K 1(−ikρ(˜x, y)) (˜x j− y j )α j (x j ) 

, 

ρ(˜x, y) 

where we have used the identity K ′ 0 (z)=−K 1(z). Note that when|x− y|≥2σ, we have 

|ρ(˜x, y)|≥|Re(z 2 1 + z2 2 )|1/2 ≥(|x− y| 2 − 2σ 2 ) 1/2 ≥ 1 

 

2 

|x− y|, 

where z j =(˜x j − y j ). Thus for any x∈Γ 2 , y∈¯Ω 1 , 

|˜x j − y j | 

|ρ(˜x, y)| ≤ 2 (|x− y|2 +σ 2 ) 1/2 

≤ 1/2 

2 1+ σ2 

10 

|x− y| |x− y| 2 ≤ 

2 . (3.9) 

On the other hand, when|x− y|≤2σ, by (3.5) we know that Imρ(˜x, y)≥γσ. Thus 

for any x∈Γ 2 , y∈¯Ω 1 , 

|˜x j − y j | 

|ρ(˜x, y)| ≤(|x− y|2 +σ 2 ) 1/2 

≤γ −1(|x− y|2 +σ 2 ) 1/2 

≤ 5γ −1 . (3.10) 

Imρ(˜x, y) σ 

This proves (ii) again by using (3.8) and Lemma 3.2. Similarly, we can prove (iii). 

To prove (iv) we note that 

∂ 2 ˜G k 

∂ x i ∂ y j 

= k2 

2π K′ 1 (−ikρ(˜x, y))−(˜x i− y i )(˜x j − y j )α i (x i ) 

ρ(˜x, y) 2 

+ ik 

2π K 1(−ikρ(˜x, y)) −ρ2 δ i j α i (x i )+(˜x i − y i )(˜x j − y j )α i (x i ) 

ρ 3 . 

By using the identity K ′ 1 (z)=− 1 2 (K 0(z)+ K 2 (z)), we have 

|K ′ 1 (−ikρ(˜x, y))|≤ 1 2 (|K 0(−ikρ(˜x, y))|+|K 2 (−ikρ(˜x, y))|) 

≤ e −(γkσ−1) 1 2 (K 0(1)+K 2 (1))= e −(γkσ−1) |K ′ 1 (1)|. 

Therefore, by using (3.9)-(3.10) we obtain 

∂ 2 ˜G 

k 

≤ Ck 2 |α 

 

m | e −(γkσ−1) + Ck|α m |σ −1 e −(γkσ−1) 

∂ x i ∂ y j 

≤ Ck 2 |α m | e −(γkσ−1) ,


where in the last inequality we have used the assumption (3.8) to conclude thatσ −1 ≤ kγ. 

This completes the proof. 

Now we are in the position to estimate the modified single and double layer potentials 

˜Ψ k SL , ˜Ψ k DL . Throughout the paper we shall use the weighted H1/2 (Γ j ) norm, j= 1,2, 

‖ v‖ H 1/2 (Γ j ) = |Γ j | −1 ‖ v‖ 2 L 2 (Γ j ) +|v|2 1 

2 ,Γ j 

1/2 

, 

where 

|v| 2 1 = 

2 ,Γ j 

∫Γ j 

∫ 

|v(x)− v(x ′ )| 2 

Γ 

|x−x ′ | 2 ds(x) ds(x ′ ). 

j 

Lemma 3.4. For any f∈ H 1/2 (Γ 1 ), let 

v(x)= ˜Ψ k DL (f)= ∫ 

be the double layer potential. Then 

where L= max(L 1 , L 2 ). 

Hence 

∂ ˜G k (x, y) 

∂ n 

Γ 1 1 (y) 

f(y) ds(y) 

‖ v‖ H 1/2 (Γ 2 ) ≤ C|α m|(1+ kL) 2 e −(γkσ−1) ‖ f‖ H 1/2 (Γ 1 ) , 

Proof. For any x∈Γ 2 , by Lemma 3.3, we know that 

|v(x)|≤‖∂ n1 (y) ˜G k (x,·)‖ L ∞ (Γ 1 )‖f‖ L 1 (Γ 1 )≤Ck e −(γkσ−1) ‖f‖ L 1 (Γ 1 ). 

It is easy to see that, for any x, x ′ ∈Γ 2 , 

Thus 

|Γ 2 | −1/2 ‖ v‖ L 2 (Γ 2 ) ≤ Ck e−(γkσ−1) ‖ f‖ L 1 (Γ 1 ) . 

|v(x)− v(x ′ )|≤C‖∇ x v‖ L ∞ (Γ 2 )|x−x ′ |. 

|v| 1 

2 ,Γ 2 ≤ C|Γ 2|‖∇ x v‖ L ∞ (Γ 2 )≤C|Γ 2 | max 

x∈Γ 2 ,y∈Γ 1 

|∇ y ∇ x ˜G k (x, y)|‖f‖ L 1 (Γ 1 ) 

≤ Ck 2 |α m ||Γ 2 | e −(γkσ−1) ‖f‖ L 1 (Γ 1 ). 

Since‖f‖ L 1 (Γ 1 )≤C|Γ 1 |‖ f‖ H 1/2 (Γ 1 ) ≤ C L‖ f‖ H 1/2 (Γ 1 ), we conclude that 


‖ v‖ H 1/2 (Γ 2 ) ≤ C|α m|(1+ kL) 2 e −(γkσ−1) ‖ f‖ H 1/2 (Γ 1 ) .


Lemma 3.5. For anyλ∈H −1/2 (Γ 1 ), let 

v(x)= ˜Ψ k SL (λ)= ∫Γ 1 

˜G k (x, y)λ(y) ds(y) 

be the single layer potential. Then 


‖ v‖ H 1/2 (Γ 2 ) ≤ C|α m|(1+ kL) 2 e −(γkσ−1) ‖λ‖ H −1/2 (Γ 1 ) , 

Proof. For any x∈Γ 2 , we know that 

|v(x)|≤ C‖λ‖ H −1/2 (Γ 1 ) ‖ ˜G k (x,·)‖ H 1/2 (Γ 1 ) 

≤ C‖λ‖ H −1/2 (Γ 1 ) ‖ ˜G k (x,·)‖ H 1 (Ω 1 ) 

≤ C‖λ‖ H −1/2 (Γ 1 ) (‖ ˜G k (x,·)‖ L ∞ (Ω 1 )+|Ω 1 | 1/2 ‖∇ y ˜G k (x,·)‖ L ∞ (Ω 1 )) 

≤ C‖λ‖ H −1/2 (Γ 1 ) (1+ kL) e−(γkσ−1) , 

where we have used Lemma 3.3. Hence 

|Γ 2 | −1/2 ‖ v‖ L 2 (Γ 2 )≤C(1+ kL) e −(γkσ−1) ‖λ‖ H −1/2 (Γ 1 ) . 

On the other hand, similar argument as in Lemma 3.4 yields 

|v| 1 

2 ,Γ 2 ≤ C|Γ 2|‖∇ x v‖ L ∞ (Γ 2 ) 

≤ C L‖λ‖ H −1/2 (Γ 1 ) max 

x∈Γ 2 

‖∇ x ˜G k (x,·)‖ H 1/2 (Γ 1 ) 

≤ C L‖λ‖ H −1/2 (Γ 1 ) max 

x∈Γ 2 

‖∇ x ˜G k (x,·)‖ H 1 (Ω 1 ) 

≤ C L‖λ‖ H −1/2 (Γ 1 ) max 

x∈Γ 2 

 

‖∇x ˜G k (x,·)‖ L ∞ (Ω 1 )+|Ω 1 | 1/2 ‖∇ y ∇ x ˜G k (x,·)‖ L ∞ (Ω 1 ) 

Again by using Lemma 3.3 we obtain 


|v| 1 

2 ,Γ 2 ≤ C|α m|(1+ kL) 2 e −(γkσ−1) ‖λ‖ H −1/2 (Γ 1 ) . 

The following theorem is the main result of this subsection. 

Theorem 3.2. Let(H1)-(H3) and (3.8) be satisfied. For any f∈ H 1/2 (Γ 1 ), let(f) be the 

PML extension defined in (2.12). Then there exists a constant C depending only onγbut 

independent of k,σ, L j , d j , j= 1,2, such that 


‖(f)‖ H 1/2 (Γ 2 ) ≤ C|α m|(1+kL) 2 e −(γkσ−1) ‖ f‖ H 1/2 (Γ 1 ) , 

Proof. This theorem is a direct consequence of Lemmas 3.4-3.5 and the continuity of 

the Dirichlet-to-Neumann operator T : H 1/2 (Γ 1 )→H −1/2 (Γ 1 ). 

 

.


3.2. The PML equation in the layer 

In this subsection we derive the stability estimates for the Dirichlet problem of the PML 

equation in the layer. 

Theorem 3.3. Let(H1) be satisfied. For q∈H 1/2 (Γ 2 ), let w be the solution of the PML 

equation in the layer (3.1a)-(3.1b). Then there exists a constant C> 0 independent of k,σ 

such that 

‖∇w‖ L 2 (Ω PML ) ≤ CĈ−1 |α m | 2 (1+ kL)‖q‖ H 1/2 (Γ 2 ) , 

∂ w H ≤ CĈ −1 |α m | 2 (1+ kL) 2 ‖q‖ 

∂ n H 1/2 (Γ 2 ) . 

1 −1/2 (Γ 1 ) 

Proof. For anyζ∈H 1 (Ω PML ) such thatζ=q onΓ 2 andζ=0onΓ 1 . By the inf-sup 

condition in (3.3) and using (3.2), we know that 

|c(w−ζ,ψ)| 

Ĉ‖ w−ζ‖ ∗,Ω PML≤ sup = sup 

0≠ψ∈H 1 0 (ΩPML ) 


0≠ψ∈H 1 0 (ΩPML ) 

By Cauchy-Schwarz inequality 

|c(ζ,ψ)| 


 

|c(ζ,ψ)|≤C max |α 2 |,|α 1 |,|Ω PML | 1/2 k|α 1 ||α 2 | 

‖ζ‖ 

x∈Ω PML (1+σ 1 σ 2 ) 1/2 H 1 (Ω PML )‖ψ‖ ∗,Ω PML 

Notice that 

≤ C(1+ kL)|α m |‖ζ‖ H 1 (Ω PML )‖ψ‖ ∗,Ω PML. (3.11) 

‖ζ‖ ∗,Ω PML≤ C|α m |(1+ kL)‖ζ‖ H 1 (Ω PML ), 

by the triangle inequality and the trace inequality, we conclude that 

‖ w‖ ∗,Ω PML≤ CĈ −1 |α m |(1+ kL)‖q‖ H 1/2 (Γ 2 ) . (3.12) 

This shows the first estimate in the theorem by using the definition of‖·‖ ∗,Ω PML. 

Next, since A(x) reduces to the identity matrix onΓ 1 , we know that, for anyψ∈ 

H 1 (Ω PML ) such thatψ=0 onΓ 2 , 

∫ ∫ ∫ 

∂ w 

− ¯ψ= A∇w· n 

∂ n ¯ψ= (A∇w·∇ ¯ψ+∇·(A∇w)) dx 

Γ 1 1 ∂Ω PML Ω 

∫ 

PML 

= 

Ω PML (A∇w·∇ ¯ψ− k 2 α 1 α 2 w ¯ψ) dx, 

where we have used (3.1a) and the formula of integration by parts. Again by Cauchy- 

Schwarz inequality and the argument in (3.11) we get 

∫ 

∂ w 

¯ψ 

∂ n 1 ≤ C(1+ kL)|α m|‖ w‖ ∗,Ω PML‖ψ‖ H 1 (Ω PML ). 

Γ 1 

This completes the proof by using (3.12) and the trace inequality. 

.


3.3. Convergence of the PML problem 

The purpose of this section is to prove Theorem 3.1. We start by introducing the 

approximate Dirichlet-to-Neumann operator ˆT : H 1/2 (Γ 1 )→H −1/2 (Γ 1 ) associated with 

the PML problem. Given f∈ H 1/2 (Γ 1 ), letˆT f =∂ζ/∂ n 1 | Γ1 

, whereζ∈ H 1 (Ω PML ) be the 

solution of the PML problem in the layer 

By assumption (H1),ˆT is well-defined. 

∇·(A∇ζ)+α 1 α 2 k 2 ζ=0 inΩ PML , (3.13a) 

ζ= f onΓ 1 , ζ=0 onΓ 2 . (3.13b) 

Lemma 3.6. Let(H1)-(H3) and (3.8) be satisfied. We have, for any f∈ H 1/2 (Γ 1 ), 

‖T f−ˆT f‖ H −1/2 (Γ 1 ) ≤ CĈ−1 |α m | 3 (1+ kL) 4 e −(γkσ−1) ‖ f‖ H 1/2 (Γ 1 ) . 

Proof. For any f∈ H 1/2 (Γ 1 ), let(f) be the PML extension defined in (2.12). Denote 

byγ + N v=∂v/∂ n 1| Γ1 

be the Neumann trace onΓ 1 for any function v defined on 2 \¯B 1 . It 

is easy to see that 

γ + N ˜Ψ k SL (λ)=γ+ N Ψk SL (λ), 

γ+ N ˜Ψ k DL (f)=γ+ N Ψk DL (f) 

for anyλ∈H −1/2 (Γ 1 ) and f∈ H 1/2 (Γ 1 ). Thus, by (2.6), 

γ + N (f)=−γ+ N Ψk SL (T f)+γ+ N Ψk DL (f)=γ+ N ξ= T f . 

By (3.13a)-(3.13b), we know that T f−ˆT f=∂w/∂ n 1 | Γ1 , where w=(f)−ζ∈ H 1 (Ω PML ) 

satisfies 

∇·(A∇w)+α 1 α 2 k 2 w= 0 inΩ PML , 

w= 0 onΓ 1 , w=(f) onΓ 2 . 

By Theorem 3.3 and Theorem 3.2, 

∂ w H ≤ CĈ −1 |α m | 2 (1+ kL) 2 ‖(f)‖ 

∂ n H 1/2 (Γ 2 ) 

1 −1/2 (Γ 1 ) 


≤ CĈ −1 |α m | 3 (1+ kL) 4 e −(γkσ−1) ‖ f‖ H 1/2 (Γ 1 ) . 

Proof of Theorem 3.1: We first prove the estimate (3.4a). Let û be the solution of the 

PML problem (2.13a)-(2.13b). Simple integration by parts implies that 

Subtracting with (2.3) we get 

a(û,ψ)+〈Tû−ˆTû,ψ〉 Γ1 

=〈g,ψ〉 ΓD 

, ∀ψ∈ H 1 (Ω 1 ). 

a(u−û,ψ)=〈Tû−ˆTû,ψ〉 Γ1 , ∀ψ∈H 1 (Ω 1 ).


Thus, by the inf-sup condition (2.4) and Lemma 3.6, 

‖u−û‖ H 1 (Ω 1 )≤CĈ −1 |α m | 3 (1+ kL) 4 e −(γkσ−1) ‖û‖ H 1/2 (Γ 1 ) . (3.14) 

This is the desired estimate (3.4a). 

Now we turn to the well-posedness of the PML problem. By the Fredholm alternative 

theorem we only need to show the uniqueness of the PML problem (2.13a)-(2.13b). For 

that purpose we let g= 0 in (2.13a)-(2.13b). By the uniqueness of the scattering problem 

we know that the corresponding scattering solution u=0 inΩ 1 . Thus (3.14) implies 

‖û‖ H 1 (Ω 1 )≤CĈ −1 |α m | 3 (1+ kL) 4 e −(γkσ−1) ‖û‖ H 1/2 (Γ 1 ) 

≤ CĈ −1 |α m | 3 (1+ kL) 4 e −(γkσ−1) ‖û‖ H 1 (Ω 1 ). 

Thus for sufficiently largeσwe conclude that û=0onΩ 1 . That û also vanishes in 

Ω 2 is a direct consequence of the unique continuation theorem (cf., e.g.,[16, P. 92]). This 

completes the proof of Theorem 3.1. 

4. Finite element approximation 

In this section we introduce the finite element approximations of the PML problems 

(2.13a)-(2.13b). From now on we assume g∈L 2 (Γ D ). Let b : H 1 (Ω 2 )× H 1 (Ω 2 )→ be 

the sesquilinear form given by 

∫ 

b(ϕ,ψ)= 

Ω 2 

 

A∇ϕ·∇ ¯ψ−α1 α 2 k 2 ϕ ¯ψ d x. (4.1) 

Denote by H 1 (0) (Ω 2)={v∈H 1 (Ω 2 ) : v= 0 onΓ 2 }. Then the weak formulation of (2.13a)- 

(2.13b) is: Given g∈L 2 (Γ D ), find û∈H 1 (0) (Ω 2) such that 

∫ 

b(û,ψ)= 

Γ D 

g ¯ψ ds, ∀ψ∈ H 1 (0) (Ω 2). (4.2) 

Let h be a regular triangulation of the domainΩ 2 . We assume the elements K∈ h 

may have one curved edge align withΓ D so thatΩ 2 =∪ K∈h K. Let V h ⊂ H 1 (Ω 2 ) be the 

conforming linear finite element space overΩ 2 , and 

◦ 

V h ={v h ∈ V h : v h = 0 onΓ 2 }. 

The finite element approximation to the PML problem (2.13a)-(2.13b) reads as follows: 

Find u h ∈ ◦ V h such that 

b(u h ,ψ h )= 

∫ 

Γ D 

g ¯ψ h ds, ∀ψ h ∈ ◦ V h . (4.3)


Following the general theory in[1, Chap. 5], the existence of unique solution of the discrete 

problem (4.3) and the finite element convergence analysis depend on the following 

discrete inf-sup condition 

|b(ϕ h ,ψ h )| 

sup ≥ˆµ‖ϕ h ‖ 

0≠ψ h ∈V ◦ ‖ψ h ‖ H 1 (Ω 2 ), ∀ϕ h ∈ V ◦ h , (4.4) 

H 1 (Ω 2 ) 

h 

where the constant ˆµ>0is independent of the finite element mesh size. Since the 

continuous problem (4.2) has a unique solution by Theorem 3.1, the sesquilinear form 

b : H 1 (0) (Ω 2)× H 1 (0) (Ω 2)→ satisfies the continuous inf-sup condition. Then a general argument 

of Schatz[17] implies (4.4) is valid for sufficiently small mesh size h 0 

depending only onγand the minimum angle of the mesh h such that the following a 

posterior error estimate is valid 

⎛ 

‖u−u h ‖ H 1 (Ω 1 )≤CĈ −1 |α m | 2 (1+ kL) ⎝ ∑ 

K∈ h 

η 2 K 

⎞ 

⎠ 

1/2 

+CĈ −1 |α m | 3 (1+ kL) 4 e −(γkσ−1) ‖u h ‖ H 1/2 (Γ 1 ) .


5. A posteriori error analysis 

For anyϕ∈H 1 (Ω 1 ), let ˜ϕ be its extension inΩ PML such that 

∇·(Ā∇ ˜ϕ)+α 1 α 2 k 2 ˜ϕ= 0 inΩ PML , (5.1a) 

˜ϕ=ϕ onΓ 1 , ˜ϕ= 0 onΓ 2 . (5.1b) 

Lemma 5.1. Let (H1) be satisfied. For anyϕ,ψ∈H 1 (Ω PML ), we have 

〈ˆTϕ,ψ〉 Γ1 

=〈ˆT ¯ψ, ¯ϕ〉 Γ1 

. 

Proof. By definition,ˆTϕ=∂ξ/∂ n 1 onΓ 1 , whereξsatisfies 

∇·(A∇ξ)+α 1 α 2 k 2 ξ=0 inΩ PML , 

ξ=ϕ onΓ 1 , ξ=0 onΓ 2 . 

Similarly, ˆT ¯ψ=∂ζ/∂ n 1 onΓ 1 , whereζsatisfies the same equation butζ= ¯ψ onΓ 1 , 

ζ=0 onΓ 2 . Integrating by parts we know that 

∫ 

0=− (A∇ξ·∇ζ−α 1 α 2 k 2 ξ·ζ)− 

Ω 

∫ 

PML ∫ 

0=− 

Ω PML (A∇ζ·∇ξ−α 1 α 2 k 2 ζ·ξ)− 

∫ 

Γ 1 

Γ 1 

∂ξ 

∂ n 1·ζ, 

∂ζ 

∂ n 1·ξ. 

Since A is a diagonal matrix, the first integrals on the right-hand side of the above two 

equations are equal. Thus〈ˆTϕ, ¯ζ〉 Γ1 

=〈ˆT ¯ψ,ξ〉 Γ1 

. This completes the proof sinceξ= 

ϕ,ζ= ¯ψ onΓ 1 . 

Lemma 5.2 (Error representation formula). For anyϕ∈H 1 (Ω 1 ), which is extended to be a 

function ˜ϕ∈H 1 (Ω 2 ) according to (5.1a)-(5.1b), andϕ h ∈ V ◦ h , we have 

∫ 

a(u−u h ,ϕ)= g(ϕ−ϕ h )− b(u h , ˜ϕ− ˜ϕ h )+〈Tu h −ˆTu h ,ϕ〉 Γ1 . (5.2) 

Γ D 

Proof. By (2.3) and the definitions (2.2) and (4.1), 

= 

= 

a(u−u h ,ϕ) 

∫ ∫ 

g ¯ϕ− (A∇u h·∇ ¯ϕ−α 1 α 2 k 2 u h ¯ϕ)+〈Tu h ,ϕ〉 Γ1 

Γ D Ω 

∫ 

1 

∫ 

g ¯ϕ− b(u h , ˜ϕ)+ (A∇u h·∇ ¯˜ϕ−α 1 α 2 k 2 u h ¯˜ϕ)+〈Tu h ,ϕ〉 Γ1 

. (5.3) 

Γ D Ω PML


On the other hand, by multiplying (5.1a) by ū h , integrating by parts, and recalling that n 1 

is the unit outer normal toΓ 1 which points outsideΩ 1 , we deduce that 

− 

∫ 

which is equivalent to 

∫ 

Ω PML (Ā∇ ˜ϕ·∇ū h −α 1 α 2 k 2 ˜ϕū h )− 

Ω PML (A∇u h·∇ ¯˜ϕ−α 1 α 2 k 2 u h ¯˜ϕ)=− 

Since by the definition ofˆT : H 1/2 (Γ 1 )→ H −1/2 (Γ 1 ), 

∂ ¯˜ϕ 

∂ n 1 

Γ1 

=ˆT ¯ϕ, 

we obtain by substituting (5.4) into (5.3) that 

∫ 

a(u−u h ,ϕ)= 

∂ ˜ϕ 

∂ n 1 

,u h 

 

 

∂ ¯˜ϕ 

∂ n 1 

,ū h 

 

Γ 1 

= 0, 

Γ D 

g ¯ϕ− b(u h , ˜ϕ)+〈Tu h ,ϕ〉−〈ˆT ¯ϕ,ū h 〉. 

This completes the proof upon using Lemma 5.1 and (4.3). 

Γ 1 

. (5.4) 

Proof of Theorem 4.1: First, we construct a Clément type interpolation operatorΠ h : 

H 1 (0) (Ω 2)→ ◦ V h , where 

H 1 (0) (Ω 2)={v∈ H 1 (Ω 2 ) : v= 0 onΓ 2 }. 

Let h ={a i } N i=1 be the set of the nodes of h which is interior toΩ 2 or on the boundary 

Γ D , and{φ i } N i=1 be the corresponding nodal basis of V h. Define∆ i = suppφ i ∩Ω 2 . Then 

the interpolation operatorΠ h : H 1 (0) (Ω 2)→ V h is defined by 

Π h v(x)= 

 

N∑ 

i=1 

1 

|∆ i | 

∫ 

∆ i 

v(x)d x 

 

φ i (x). 

Since the nodes onΓ 2 are not included in the definition ofΠ h , we know thatΠ h v∈ ◦ V h . 

Moreover, by slightly modifying the argument in Chen and Nochetto[7, Lemmas 3.1-3.2], 

one can show that the operatorΠ h enjoys the following interpolation estimates, for any 

v∈H 1 (0) (Ω 2), 

‖ v−Π h v‖ L 2 (K) ≤ Ch K‖∇v‖ L 2 (˜K) , ‖ v−Π hv‖ L 2 (e) ≤ Ch1/2 e 

‖∇v‖ L 2 (ẽ) , (5.5) 

where ˜K and ẽ are the union of all elements in h having non-empty intersection with 

K∈ h and the side e, respectively.


Now we takeϕ h =Π h ˜ϕ∈ V ◦ h in the error representation formula (5.2) to get 

∫ 

a(u−u h ,ϕ) = g(ϕ−Π h ϕ)− b(u h , ˜ϕ−Π h ˜ϕ)+〈Tu h −ˆTu h ,ϕ〉 Γ1 

Γ D 

:= II 1 + II 2 + II 3 . (5.6) 

We observe that, by integration by parts and using (4.5)-(4.7), 

∫ 

∑ ∑ 1 

II 1 + II 2 = R h ( ˜ϕ−Π h ˜ϕ) dx+ J e ( ˜ϕ−Π h ˜ϕ) ds . 

2 

e 

K∈ h 

∫K 

e⊂∂ K 

By using standard argument in the a posteriori error analysis and (5.5) we get 

 

∑ 

|II 1 + II 2 |≤C ‖h K R h ‖ 2 L 2 (K) + 1 ∑ 

2 

K∈ h 

⎛ 

≤ C⎝ ∑ ⎞1/2 

⎠ ‖∇ ˜ϕ‖ L 2 (Ω 2 ). 

K∈ h 

η 2 K 

By the argument in Theorem 3.3, we deduce that 

e⊂∂ K 

‖h 1/2 

e 

J e ‖ 2 L 2 (e) 1/2 

‖∇ ˜ϕ‖ L 2 (˜K) 

‖∇ ˜ϕ‖ L 2 (Ω PML )≤CĈ −1 |α m | 2 (1+ kL)‖ϕ‖ H 1/2 (Γ 1 ) 

≤ CĈ −1 |α m | 2 (1+ kL)‖ϕ‖ H 1 (Ω 1 ) . 

Thus 

⎛ 

|II 1 + II 2 |≤CĈ −1 |α m | 2 (1+ kL) ⎝ ∑ 

K∈ h 

η 2 K 

⎞ 

⎠ 

1/2 

‖ϕ‖ H 1 (Ω 1 ). 

By Lemma 3.6, we obtain 

|II 3 |≤CĈ −1 |α m | 3 (1+ kL) 4 e −(γkσ−1) ‖u h ‖ H 1/2 (Γ 1 ) ‖ϕ‖ H 1/2 (Γ 1 ) . 

Therefore, by the inf-sup condition (2.4), we finally get 

‖u−u h ‖ H 1 (Ω 1 ) ≤ C sup |a(u−u h ,ϕ)| 

0≠ϕ∈H 1 (Ω 1 ) ‖ϕ‖ H 1 (Ω 1 ) 

⎛ 

≤ CĈ −1 |α m | 2 (1+ kL) ⎝ ∑ 

K∈ h 

η 2 K 

+CĈ −1 |α m | 3 (1+ kL) 4 e −(γkσ−1) ‖u h ‖ H 1/2 (Γ 1 ) . 


⎞ 

⎠ 

1/2


6. Implementation and numerical examples 

In this section, we present several numerical examples to illustrate the performance of 

the adaptive uniaxial PML method. The computations are carried out by using the PDE 

toolbox of MATLAB. The PML parameters are determined through the a posteriori error 

estimate in Theorem 4.1. Note that in Theorem 4.1 that the a posteriori error estimate 

consists of two parts: the PML error and the finite element discretization error. First we 

choose L 1 , L 2 such that D⊂B 1 and choose d 1 , d 2 such that 

d 1 

L 1 

= d 2 

L 2 

=χ, (6.1) 

whereχ is a constant. Then we chooseχ andσsuch that the exponentially decaying 

factor: 

χ 

 

min(L 

ω=e −(γkσ−1) − 

1 ,L 2 ) 

kσ−1 

χ+1 

L 

= e 

1 2+L2 2 ≤ 10 −8 , (6.2) 

which makes the PML error negligible compared with the finite element discretization 

errors. By (H3), 

σ j (t)= ˜σ j 

|t|− L j /2 

d j 

m 

, j= 1,2, (6.3) 

where ˜σ 1 , ˜σ 2 are determined fromσas follows 

˜σ j = (m+1)σ 

d j 

, j= 1,2. (6.4) 

Once the PML region and the medium property are fixed, we use the standard finite element 

adaptive strategy to modify the mesh according to the a posteriori error estimate. 

Now we describe the adaptive uniaxial PML method used in the paper. 

Algorithm 6.1. Given tolerance TOL> 0. Let m=2. 

• Choose L 1 , L 2 such that D⊂B 1 ; 

• Chooseχ andσsuch that the exponentially decaying factorω≤10 −8 ; 

• Set d 1 , d 2 and ˜σ 1 , ˜σ 2 according to (6.1),(6.4); 

• Set the computational domainΩ 2 = B 2 \¯D and generate an initial mesh h over 

Ω 2 ; 

• While F EM = ∑ 

K∈ h 

η 2 K 1/2>TOL 

do 

- refine the mesh h according to the strategy: 

ifη K > 1 2 max K∈ h 

η K , refine the element K∈ h 

- solve the discrete problem (4.3) on h 

- compute error estimators on h 

end while

ÌÐ½ÌÈÅÄÔÖÑØÖ×ÓÖÜÑÔÐ×º½Òº¾º ×Ø ×Ø 


Example 6.1 Example 6.2 

ÌÐ¾ÌÒÙÑÖÓÒÓ×ÖÕÙÖØÓÚØÚÒÖÐØÚÖÖÓÖÓØÖÐÓÖÖÒØ Ó×Ó×ØÓÖÜÑÔÐº½ºÌÜØÖÐ×¼º¾¾¼¼¹¼º¾¾¼¼iº 

χ σ χ σ 

0.1λ 1.0 9λ 1.0λ 0.5 20λ 

1.0λ 1.0 9λ 1.0λ 1.0 14λ 

5.0λ 1.0 9λ 1.0λ 2.0 10λ 

0.1λ 1.0 19λ 

Error×Ø=0.1λ×Ø=1.0λ×Ø=5.0λ 

10% 295 557 5111 

1% 997 5556 16845 

0.1% 2989 16006 >114848 

Now we report two numerical examples to demonstrate the efficiency of the proposed 

algorithm. We scale the error estimator for determining finite element meshes by a default 

factor 0.15 as in the PDE toolbox of MATLAB. 

Example 6.1. Let D=[−λ/2,λ/2]×[−λ/2,λ/2]. We consider the scattering problem 

whose exact solution is known: u=H (1) 

0 

(k|x|), where k=2π/λ and soλis the wavelength. 

Define 

×Ø= min{|x−x ′ |, x∈Γ D , x ′ ∈Γ 1 } 

as the minimum distance between the scatterer to the inner boundary of the PML layer. We 

want to test the influence of the different choices of the size of B 1 . Fixχ= 1.0 and take 

different×Ø, that is, in the first step we choose different L 1 and L 2 . Table 1 shows the 

different choices of the PML parameters×Ø,χ andσdetermined by the relation (6.2). 

In the following we simply takeλ=1.0 to fix the exact solution. 

Fig. 2 shows the log N k -log‖u−u k ‖ H 1 (Ω 1 ) curves, where N k is the number of nodes of 

the mesh k and u k is the finite element solution of (4.3) over the mesh k . It indicates 

that for different choices of×Ø, the meshes and the associated numerical complexity are 

quasi-optimal:‖u−u k ‖ H 1 (Ω 1 )≈CN −1/2 is valid asymptotically. 

k 

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

u ∞ (ˆx)= eiπ 4 

 

8πk 

∫ 

∂ D 

 

u(y) ∂ e−ikˆx·y 

∂ν(y) 

−∂ 

u(y) 

∂ν(y) e−ikˆx·y 

ds(y), 

ˆx= x 

|x| . 

We compute the far field u ∞ (ˆx),ˆx=(cos(θ),sin(θ)) T in the observation directionθ= 

π/4. Table 2 shows the number of nodes required to achieve the given relative error of 

the far field for different choices of PML parameter×Ø. It demonstrates clearly that for 

given relative error, a smaller×Øis preferred in terms of number of nodes used. Thus 

for the same accuracy of the far fields, we can choose small×Øwhich will largely save 

the computational costs.


10 3 Number of nodal points 

10 2 

10 1 

dist=0.1λ 

dist=1.0λ 

dist=5.0λ 

a line with slope −1/2 

2.5 

2 

1.5 

1 

ÙÖ¾ÌÕÙ×¹ÓÔØÑÐØÝÓØÔØÚ 

10 0 

)ÓÖÜÑÔÐº½º 

10 −1 

10 −2 

10 −3 

10 3 10 4 10 5 

−3 

Ñ×ÖÒÑÒØ×Ó‖u−u N ‖ H 1 (Ω 1 

||u−u N 

|| H 

1 (Ω1 ) 

A Posteriori Error Estimate 

10 2 Number of nodal points 

χ=0.5 

χ=1.0 

χ=2.0 

a line with slope −1/2 

10 1 

Ñ×ÖÒÑÒØ×ÓØÔÓ×ØÖÓÖÖÖÓÖ×ØÑ¹ ØÓÖÓÖÜÑÔÐº¾º ÙÖÌÕÙ×¹ÓÔØÑÐØÝÓØÔØÚ 

10 0 

10 −1 

10 2 10 3 10 4 10 5 

2 

0.5 

ÙÖ¿ÌÓÑØÖÝÓØ×ØØÖÖÓÖÜ¹ 

0 

ÑÔÐº¾º 

−0.5 

−1 

−1.5 

−2 

−2.5 

−2 −1 0 1 2 3 

The far−field pattern u ∞ 

−0.2 

−0.3 

−0.4 

−0.5 

χ=0.5 

χ=1.0 

χ=2.0 

−0.6 

−0.7 

ØÒÒØÖØÓÒÓÖÜÑÔÐº¾º ÙÖÌÖÐÔÖØÓØÖ¹ÐÔØØÖÒ×Ò 

−0.8 

−0.9 

−1 

−1.1 

4 6 8 10 12 

Number of nodal points 

x 10 4 

Example 6.2. This example is taken from[10] which concerns the scattering of the plane 

wave u I = e ikx 1 

from a perfectly conducting metal. The scatter D is contained in the box 

{x∈ 2 : −2< x 1 < 2.2,−0.7< x 2 < 0.7} as plotted in Fig. 3. We take k=2π, that 

is the wave lengthλ=1.0. Let×Ø=1.0 and thus the scatterer is contained in the 

box ¯B 1 =[−3.2,3.2]×[−1.7,1.7]. The different choices of PML parametersχ andσ 

determined by the relation (6.2) are shown in Table 1. 

Fig. 4 shows the log N k -log k curves, where N k is the number of nodes of the mesh k 

and the 

∑ 1/2 

k = η 2 K 

K∈ k 

is the associated a posteriori error estimate. It indicates that the meshes and the associated 

numerical complexity are quasi-optimal: k ≈ CN −1/2 is valid asymptotically. 

k 

Figs. 5 and 6 show the far fields in the incident directionθ = 0 and the reflective


1.5 

1.4 

1.3 

χ=0.5 

χ=1.0 

χ=2.0 

The far−field pattern u ∞ 

1.2 

1.1 

ÙÖÌÖÐÔÖØÓØÖ¹ÐÔØØÖÒ×ÒØÖØÚÖØÓÒÓÖÜÑÔÐº¾º 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

2 4 6 8 10 12 

Number of nodal points 

x 10 4 

5 

4 

3 

2 

º¾º 

1.0ÓÖÜÑÔÐ 

1 

0 

−1 

−2 

−3 

−4 

−5 

−8 −6 −4 −2 0 2 4 6 8 

ÙÖÌÑ×Ó¿ÒÓ×ØÖ½ÔØÚØÖØÓÒ×ÛÒ×Ø=1.0Òχ= 

ÙÖÌÖÐÔÖØÓØ×ÓÐÙØÓÒÓØÈÅÄÔÖÓÐÑÓÖÜÑÔÐº¾º 

directionθ=π. We observe that the far fields are insensitive to the thickness of the PML 

layers. 

Then we takeχ= 1.0 and test the influence of different choices of×Øas in Example 

6.1. Table 3 shows the number of nodes required to achieve the given relative error of the

ÌÐ¿ÌÒÙÑÖÓÒÓ×ÖÕÙÖØÓÚØÚÒÖÐØÚÖÖÓÖÓØÖÐ×ÒÓØ ÒÒØÒÖØÚÖØÓÒ×ÓÖÖÒØÓ×Ó×ØÓÖÜÑÔÐº¾ºÌÖÐ×ÒØÐ×Ø 

1.0ÖÓ×Ò×ØÜØÖÐ×º 


ÔØÚ×ØÔÛÒ×Ø=1.0λÒχ= 

Error_inc×Ø= 0.1λ×Ø=1.0λ Error_ref×Ø= 0.1λ×Ø= 1.0λ 

10% 568 1032 10% 549 492 

1% 7315 12614 1% 2871 5183 

0.1% 29882 62605 0.1% 9812 23156 

far fields in both incident and reflective directions for different choices of PML parameter 

×Ø. It again shows that for the same accuracy of the far fields, a smaller×Øis a better 

choice. 

In Fig. 7 we show the mesh after 16 adaptive iterations when×Ø= 1.0 andχ= 1.0. 

We observe that the mesh near the boundaryΓ 2 is rather coarse, because the solution is 

rather small there due to the exponential damping of the PML layer. Fig. 8 shows the real 

part of the PML solution when×Ø= 1.0 andχ= 1.0. 

Acknowledgments This work was supported in part by China NSF under the grant 

10428105 and by the National Basic Research Project under the grant 2005CB321701. 

The authors wish to thank Peter Monk for several inspiring discussions and the referees for 

constructive comments. 

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