AN ADAPTIVE FINITE ELEMENT DTN METHOD FOR THE ELASTIC

WAVE SCATTERING PROBLEM IN THREE DIMENSIONS

GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

Abstract. Consider the elastic scattering of an incident wave by a rigid obstacle in three dimensions,which is formulated as an exterior problem for the Navier equation. By constructing a Dirichlet-to-Neumann (DtN) operator and introducing a transparent boundary condition, the scattering problemis reduced equivalently to a boundary value problem in a bounded domain. The discrete problemwith the truncated DtN operator is solved by using the a posteriori error estimate based adaptivefinite element method. The estimate takes account of both the finite element approximation errorand the truncation error of the DtN operator, where the latter is shown to converge exponentiallywith respect to the truncation parameter. Moreover, the generalized Woodbury matrix identityis utilized to solve the resulting linear system efficiently. Numerical experiments are presented todemonstrate the superior performance of the proposed method.

1. Introduction

As a basic problem in classical scattering theory, the obstacle scattering problem refers to asthe scattering of a time-harmonic wave by an impenetrable medium of compact support. It playsa fundamental role in diverse scientific areas such as radar and sonar, geophysical exploration,medical imaging, and nondestructive testing. The obstacle scattering problems have been extensivelyinvestigated in both the engineering and mathematical communities. A great number of numericaland mathematical results are available, especially for acoustic and electromagnetic waves [9,26,27].Recently, the scattering problems for elastic waves have received ever-increasing attention due tothe significant applications in geophysics, seismology, and elastography [2, 6, 7, 21]. For example, inmedical diagnostics, by mapping the elastic properties and stiffness of soft tissues, they are able togive diagnostic information about the presence or status of disease. Compared with acoustic andelectromagnetic waves, the scattering problems for elastic waves remain many issues on theoreticalanalysis and numerical computation because of the complexity of the model equation [8, 20].

The elastic obstacle scattering problem is imposed in an unbounded domain, which needs tobe truncated into a bounded domain in practice. Therefore, an appropriate boundary conditionis required on the boundary of the truncated domain so that no artificial wave reflection occurs.Such a boundary condition is called a non-reflecting boundary condition or transparent boundarycondition (TBC). Despite the large amount of work done so far, it is still one of the important andactive research subjects on developing effective non-reflecting boundary conditions in the area ofcomputational wave propagation [5,11–14,25,30]. In this work, we construct a Dirichlet-to-Neumann(DtN) operator and develop a TBC for solving the elastic obstacle scattering problem in threedimensions. Based on the Helmholtz decomposition, the scattered field of the elastic displacement issplit into the compressional and shear wave components which satisfy the Helmholtz equation andthe Maxwell equation, respectively. Therefore, the DtN operator for the elastic wave equation canbe obtained from the well-studied DtN operators for the Helmholtz and Maxwell equations. Since

2010 Mathematics Subject Classification. 65N30, 65N12, 78A45.Key words and phrases. Elastic wave equation, obstacle scattering problem, Dirichlet-to-Neumann operator, trans-

parent boundary condition, adaptive finite element method, a posteriori error estimate.The work of GB is supported in part by an NSFC Innovative Group Fund (No.11621101). The research of PL is

supported in part by the NSF grant DMS-1912704.

1

arX

iv:2

101.

0583

5v1

[m

ath.

NA

] 1

4 Ja

n 20

21

2 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

the TBC is exact, the artificial boundary could be put as close as possible to the obstacle in orderto reduce the computational complexity [19,24].

To design an efficient numerical method, there are two more issues which need to be considered.The first issue concerns the truncation of the DtN operator. The nonlocal DtN operator is given asan infinite series, which has to be truncated into a sum of finitely many terms in actual computation.However, it is known that the convergence of the truncated DtN operator could be arbitrarily slow inthe operator norm [16]. From the computational viewpoint, it is important to answer the questionhow many terms are required in the summation in order to maintain a certain level of accuracy.Second, the solution may have local singularity when the obstacle has edges. The mesh should befine around the nonsmooth part of the obstacle in order to capture the singularity of the solution;while the mesh could be coarse in other part of the domain where the solution is smooth. Hence, it iscrucial to design an algorithm for mesh modification which can distribute equally the computationaleffort and optimize the computation.

In this paper, we propose an adaptive finite element method with the truncated DtN operator toovercome the two difficulties mentioned above. Specifically, we consider the scattering of a planewave by an elastically rigid obstacle in three dimensions. The exterior domain is assumed to be filledwith a homogeneous and isotropic elastic medium. The elastic wave propagation is governed by theNavier equation. Based on the TBC, the exterior scattering problem is formulated equivalently intoa boundary value problem in a bounded domain. The discrete problem is solved by using the finiteelement method with the truncated DtN operator. Based on the Helmholtz decomposition, a newduality argument is developed to obtain an a posteriori error estimate between the solution of theoriginal scattering problem and the discrete problem. The a posteriori error estimate takes accountof both the finite element approximation error and the truncation error of the DtN operator, wherethe latter is shown to decay exponentially with respect to the truncation parameter. The estimateis used to design the adaptive finite element algorithm to choose elements for refinements and todetermine the truncation parameter. The stiffness matrix is made of a sparse, real and symmetricmatrix, which comes from the discretization of the variational formulation in the interior of thedomain, and a dense low rank matrix given by vector products, which arises from the nonlocalTBC. The generalized Woodbury matrix identity is utilized to solve the resulting linear systemefficiently. Numerical experiments are presented to demonstrate the superior performance of theproposed method.

Recently, the adaptive finite element DtN method has been developed to solve many acoustic andelectromagnetic scattering problems, such as the obstacle scattering problems [3,17], the diffractiongrating problems [18, 31], and the open cavity scattering problem [33]. This paper is a non-trivialextension of our previous work on the two-dimensional elastic obstacle scattering problem [23].Apparently, the analysis is more sophisticated and the computation is more intensive for the three-dimensional problem. This work adds a significant contribution to designing efficient computationalmethods for solving the elastic wave scattering problems.

The paper is organized as follows. In Section 2, the elastic wave equation is introduced; theboundary value problem is formulated by using the TBC; the corresponding weak formulation isdiscussed. In Section 3, the discrete problem is considered by using the finite element approximationwith the truncated DtN operator. Section 4 is devoted to the a posteriori error analysis and servesas the basis of the adaptive algorithm. In Section 5, we discuss the numerical implementation ofthe adaptive algorithm, the construction of the stiffness matrix, and an efficient solver for the linearsystem; two numerical examples are presented to illustrate the performance of the proposed method.The paper concludes with some general remarks in Section 6.

2. Problem formulation

Let D Ă R3 be an elastically rigid obstacle with Lipschitz continuous boundary BD. Denote byν the unit outward normal vector on BD. The exterior domain R3zD is assumed to be filled with a

AN ADAPTIVE FINITE ELEMENT DTN METHOD 3

homogeneous and isotropic elastic medium with a unit mass density. Let BR “

x P R3 : |x| ă R(

and BR1 “

x P R3 : |x| ă R1(

be balls with radii R and R1, where 0 ă R1 ă R. Denote by ΓR and

ΓR1 the surfaces of BR and BR1 , respectively. Let Ω “ BRzD be the bounded domain enclosed bythe surfaces BD and ΓR.

Let the obstacle be illuminated by an incident wave uinc, which can be a point source or a planewave. Due to the interaction between the incident wave and the obstacle, the displacement of thescattered field u satisfies the elastic wave equation

µ∆u` pλ` µq∇∇ ¨ u` ω2u “ 0 in R3zD, (2.1)

where ω ą 0 is the angular frequency and λ, µ are the Lame constants satisfying µ ą 0, λ ` µ ą 0.Since the obstacle is assumed to be elastically rigid, the displacement of the total field vanishes onthe surface of the obstacle, i.e., we have

u “ g “ ´uinc on BD.

Introduce the Helmholtz decomposition

u “ ∇φ`∇ˆψ, ∇ ¨ψ “ 0, (2.2)

where φ and ψ are called the potential functions. Substituting (2.2) into (2.1), we may verify that thescalar potential φ satisfies the Helmholtz equation and the vector potential ψ satisfies the Maxwellequation in R3zD, i.e.,

∆φ` κ2pφ “ 0, ∇ˆ p∇ˆψq ´ κ2

sψ “ 0,

where

κp “ ωpλ` 2µq12, κs “ ωµ12

are known as the the compressional wavenumber and the shear wavenumber, respectively. In addi-tion, the potential functions φ and ψ are required to satisfy the Sommerfeld radiation condition andthe Silver–Muller radiation condition, respectively:

limρÑ8

ρ pBρφ´ iκpφq “ 0, limρÑ8

pp∇ˆψq ˆ x´ iκsρψq “ 0, ρ “ |x|,

which is known as the Kupradze–Sommerfeld radiation condition for the elastic wave equation.The obstacle scattering problem is defined in the unbounded domain R3zD. It needs to be reduced

equivalently into the bounded domain Ω. Next we introduce a transparent boundary condition onΓR. Due to the page limit, the details are given as supplementary materials. Define a boundaryoperator for the displacement of the scattered wave

Du :“ µBρu` pλ` µqp∇ ¨ uqeρ on ΓR.

In the spherical coordinates, the scattered field admits the following expansion in R3zBR:

upR, θ, ϕq “8ÿ

n“0

nÿ

m“´n

um1nUmn pθ, ϕq ` u

m2nV

mn pθ, ϕq ` u

m3nX

mn pθ, ϕqeρ.

The transparent boundary condition is

Du “ T u :“8ÿ

n“0

nÿ

m“´n

bm1nUmn ` b

m2nV

mn ` b

m3nX

mn eρ on ΓR, (2.3)

where the Fourier coefficients bmn “ pbm1n, bm2n, b

m3nq

J and umn “ pum1n, um2n, u

m3nq

J are connected bybmn “ Mnu

mn with the 3 ˆ 3 matrix Mn being given in Appendix D, and T is called the DtN

operator.Based on the transparent boundary condition (2.3), the scattering problem can be reformulated

as the variational problem: find u PH1pΩq with u “ g on BD such that

bpu,vq “ 0 @v PH1BDpΩq, (2.4)

4 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

where the sesquilinear form b : H1pΩq ˆH1pΩq Ñ C is defined as

bpu,vq “ µ

ż

Ω∇u : ∇vdx` pλ` µq

ż

Ωp∇ ¨ uq p∇ ¨ vq dx

´ω2

ż

Ωu ¨ vdx´

ż

ΓR

T u ¨ vds.

Here A : B “ tr`

ABJ˘

is the Frobenius inner product of square matrices A and B.

It is shown in [22] that the variational problem admits a unique weak solution u PH1pΩq, whichsatisfies the estimate

uH1pΩq À gH12pBDq À uincH1pΩq. (2.5)

Hereafter, the notation a À b stands for a ď Cb, where C ą 0 is a generic constant whose value isnot required and may change step by step in the proofs.

Since the variational problem is well-posed, it follows from the general theory in [1] that thereexists a constant γ ą 0 such that the following inf-sup condition holds:

sup0‰vPH1pΩq

|bpu,vq|

vH1pΩqě γuH1pΩq @u PH1pΩq.

3. The discrete problem

The DtN operator T in (2.3) is given as an infinite series, which needs to be truncated into asum of finitely many terms in computation. Given a sufficiently large N , define the truncated DtNoperator

TNu “Nÿ

n“0

nÿ

m“´n

bm1nUmn ` b

m2nV

mn ` b

m3nX

mn eρ. (3.1)

The truncated variational problem is to find uN PH1pΩq with uN “ g on BD such that

bN puN ,vq “ 0 @v PH1BDpΩq, (3.2)

where the sesquilinear form bN : H1pΩq ˆH1pΩq Ñ C is given by

bN pu,vq “ µ

ż

Ω∇u : ∇vdx` pλ` µq

ż

Ωp∇ ¨ uq p∇ ¨ vq dx

´ω2

ż

Ωu ¨ vdx´

ż

ΓR

TNu ¨ vds.

Let Mh be a regular tetrahedral mesh of Ω, where h denotes the maximum diameter of all theelements in Mh. For simplicity, we assume that the surfaces BD and ΓR are polyhedral and ignorethe approximation error on the surfaces, which allows us to focus on deducing the a posteriori errorestimate. Thus any face e PMh is a subset of BΩ if it has three boundary vertices.

Let V h ĂH1pΩq be a conforming finite element space, i.e.,

V h :“

v P CpΩq3 : v|T P PmpT q3 @T PMh

(

,

where m is a positive integer and PmpT q denotes the set of all polynomials of degree no more than m.The finite element approximation to the variational problem (3.2) is to find uhN P V h with uhN “ g

h

on BD such that

bN puhN ,v

hN q “ 0 @vh P V h,BD, (3.3)

where gh is the finite element approximation of g and V h,BD “ tv P V h : v “ 0 on BDu.Following the idea in [16] and the discussion of [22], we may show that for sufficiently large N

the variational problem (3.2) is well-posed. Meanwhile, for sufficiently small h, the discrete inf-supcondition of the sesquilinear form bN may also be established by following the approach in [29].Based on the general theory in [1], the truncated variational problem (3.3) can be shown to have a

AN ADAPTIVE FINITE ELEMENT DTN METHOD 5

unique solution uhN P V h. The details are omitted since our focus is the a posteriori error estimateand the convergence analysis for the truncated DtN operator.

4. The a posteriori error analysis

For any tetrahedral element T P Mh, denoted by hT its diameter. Let BF denote the set of allthe face of T and hF be the size of the face F . For any interior face F which is the common face oftetrahedral element T1, T2 PMh, define the jump residual across F as

JF “ ´”

µ∇uhN ¨ ν1 ` pλ` µqp∇ ¨ uhN qν1 ` µ∇uhN ¨ ν2 ` pλ` µqp∇ ¨ uhN qν2

ı

,

where νj is the unit outward normal vector on the boundary of Tj , j “ 1, 2. For any boundary edgeF Ă ΓR, the jump residual is

JF “ 2´

TNuhN ´ µ∇uhN ¨ eρ ´ pλ` µqp∇ ¨ uhN qeρ

¯

,

For any tetrahedral element T PMh, define the local error estimator as

ηT “ hT RuhNL2pT q `

˜

1

2

ÿ

FPBT

hF JF 2L2pF q

¸12

,

where R is the residual operator given by

RuhN “ µ∆uhN ` pλ` µq∇p∇ ¨ uhN q ` ω2uhN .

Introduce the following weighted norm ~ ¨ ~H1pΩq:

~u~2H1pΩq

“ µ

ż

Ω|∇u|2dx` pλ` µq

ż

Ω|∇ ¨ u|2dx` ω2

ż

Ω|u|2dx. (4.1)

It is easy to check that for any u PH1pΩq we have

minpµ, ω2qu2H1pΩq

ď ~u~2H1pΩq

ď maxp2λ` 3µ, ω2qu2H1pΩq

,

which implies that the norms ¨ H1pΩq and ~ ¨ ~H1pΩq are equivalent.

Now, let us state the main result of this paper.

Theorem 4.1. Let u and uhN be the solutions of the variational problems (2.4) and (3.3), respec-

tively. Let ξ “ u ´ uhN . Then for sufficiently large N , the following a posteriori error estimateholds:

~ξ~H1pΩq À

˜

ÿ

TPMh

η2T

¸12

` g ´ ghH12pBDq `N

ˆ

R1

R

˙N

uincH1pΩq.

It can be seen from the theorem that the a posteriori error estimate consists of three parts: thefirst two parts arise from the finite element discretization error; the third part accounts for thetruncation error of the DtN operator. Apparently, the DtN truncation error decreases exponentiallywith respect to N since R1 ă R.

Using (4.1) and the integration by parts, we obtain

~ξ~2H1pΩq

“ µ

ż

Ω∇ξ : ∇ξdx` pλ` µq

ż

Ωp∇ ¨ ξqp∇ ¨ ξqdx` ω2

ż

Ωξ ¨ ξdx

“ <bpξ, ξq ` 2ω2

ż

Ωξ ¨ ξdx` <

ż

ΓR

T ξ ¨ ξds

“ <bpξ, ξq ` <ż

ΓR

pT ´TN q ξ ¨ ξds` 2ω2

ż

Ωξ ¨ ξdx` <

ż

ΓR

TNξ ¨ ξds. (4.2)

To prove Theorem 4.1, it suffices to estimate the four terms given on the right-hand side of (4.2).The following lemma concerns the trace theorem. The proof is standard and omitted for brevity.

6 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

Lemma 4.2. For any u P H1pΩq, the following estimates hold:

uH12pΓRqÀ uH1pΩq, uH12pΓR1 q

À uH1pΩq.

Lemma 4.3. Let u P H1pΩq be the solution of the variational problem (2.4). For any v P H1pΩq,the following estimate holds:

ˇ

ˇ

ˇ

ˇ

ż

ΓR

pT ´TN qu ¨ vds

ˇ

ˇ

ˇ

ˇ

ď CN

ˆ

R1

R

˙N

uincH1pΩqvH1pΩq,

where C is a positive constant independent of N .

Proof. Let φ and ψ be the potentials of the Helmholtz decomposition for the solution u. It can beverified from (D.1)–(D.2) that

φmn pRq “ zpnφmn pR

1q, ψm2npRq “ zsnψm2npR

1q, ψm3npRq “

ˆ

R1

R

˙

zsnψm3npR

1q, (4.3)

where

zpn “hp1qn pκpRq

hp1qn pκpR1q

, zsn “hp1qn pκsRq

hp1qn pκsR1q

.

Substituting (4.3) into (D.3) and using (D.4) with R being replaced with R1, we obtain

umn pRq “ Qnumn pR

1q, (4.4)

where the entries of the matrix Qn are

Qn,11 “R1

RΛnpR1q

”

´npn` 1qzpn ` zp1qn pκpR

1qp1` zp1qn pκsRqqzsn

ı

,

Qn,13 “R1

RΛnpR1q

a

npn` 1q”

p1` zp1qn pκsR1qqzpn ´ p1` z

p1qn pκsRqqz

sn

ı

,

Qn,31 “R1

RΛnpR1q

a

npn` 1q”

´zp1qn pκpRqzpn ` z

p1qn pκpR

1qzsn

ı

,

Qn,33 “R1

RΛnpR1q

”

p1` zp1qn pκsR1qqzp1qn pκpRqz

pn ´ npn` 1qzsn

ı

,

and

Qn,22 “ zsn, Qn,12 “ Qn,21 “ Qn,23 “ Qn,32 “ 0.

We use the element Qn,11 as an example to show the estimate of the matrix Qn since all the otherelements can be similarly estimated. A simple calculation yields

Qn,11 “R1

RΛnpR1qnpn` 1qpzsn ´ z

pnq

`R1

RΛnpR1q

”

zp1qn pκpR1qp1` zp1qn pκsRqq ´ npn` 1q

ı

zsn.

It follows from Lemma E.5 and (E.9)–(E.10) that

|Qn,11| À n

ˆ

R1

R

˙n

.

Similarly, we may show that all the other entries of Kn satisfy

|Qn,ij | À n

ˆ

R1

R

˙n

, i, j “ 1, 2, 3.

AN ADAPTIVE FINITE ELEMENT DTN METHOD 7

Substituting the estimate of Qn into (4.4), we have

|umn pRq| À n

ˆ

R1

R

˙nˇ

ˇumn pR1qˇ

ˇ .

Combining the above estimate with (D.6) and using Theorem 4.2 and (2.5), we haveˇ

ˇ

ˇ

ˇ

ż

ΓR

pT ´TN qu ¨ v ds

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

nąN

nÿ

m“´n

Mnumn pRq ¨ v

mn pRq

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

nąN

nÿ

m“´n

MnQnumn pR

1q ¨ vmn pRq

ˇ

ˇ

ˇ

ˇ

ˇ

À

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

nąN

nÿ

m“´n

n2

ˆ

R1

R

˙n

umn pR1q ¨ vmn pRq

ˇ

ˇ

ˇ

ˇ

ˇ

À maxnąN

ˆ

n

ˆ

R1

R

˙n˙

uincH1pΩqvH1pΩq.

The proof is completed by noting that n pR1Rqn decreases for sufficiently large n.

Based on Lemma 4.3, the estimate of the first two terms in (4.2) is given in the following lemma.The proof follows directly from the discussion in [23], where the basic idea is to use the integrationby parts and the interpolation theory. The details are omitted.

Lemma 4.4. Let v be any function in H1BDpΩq, the following estimate holds:

ˇ

ˇ

ˇ

ˇ

bpξ,vq `

ż

ΓR

pT ´TN q ξ ¨ vds

ˇ

ˇ

ˇ

ˇ

À

¨

˝

˜

ÿ

TPMh

η2T

¸12

`N

ˆ

R1

R

˙N

uincH1pΩq

˛

‚vH1pΩq,

which gives by taking v “ ξ that

ˇ

ˇ

ˇ

ˇ

bpξ, ξq `

ż

ΓR

pT ´TN q ξ ¨ ξ ds

ˇ

ˇ

ˇ

ˇ

À

˜˜

ÿ

TPMh

η2T

¸12

`N

ˆ

R1

R

˙N

uincH1pΩq

` g ´ ghH12pBDq

¸

ξH1pΩq.

It is proved in [22] that the matrix Mn “ ´12 pMn `M

˚n q is positive definite for sufficiently large

n, where the star denotes the complex transpose. The following result can be obtained easily byfollowing the proof of [23, Lemma 4.6].

Lemma 4.5. For any δ ą 0, there exists a positive constant Cpδq independent of N such thatż

ΓR

TNξ ¨ ξds ď Cpδqξ2L2pBRzBR1 q

`

ˆ

R

R1

˙

δξ2H1pBRzBR1 q

.

To estimate the third term of (4.2), we introduce the dual problem

bpv,pq “

ż

Ωv ¨ ξdx @v PH1

BDpΩq. (4.5)

It is easy to verify that p satisfies the boundary value problem$

’

&

’

%

µ∆p` pλ` µq∇∇ ¨ p` ω2p “ ´ξ in Ω,

p “ 0 on BD,

Dp “ T ˚p on ΓR,

(4.6)

where T ˚ is the adjoint operator to the DtN operator T .

8 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

Letting v “ ξ in (4.5) leads to

ξ2L2pΩq

“ bpξ,pq `

ż

ΓR

pT ´TN q ξ ¨ pds´

ż

ΓR

pT ´TN q ξ ¨ pds. (4.7)

The first two terms of (4.7) can be estimated in the same way as that of Lemma 4.4. Thus it sufficesto estimate the third term of (4.7). Next we deduce the solution of the dual problem (4.5) which iscrucial for the estimate.

Consider the system

#

∇ζ `∇ˆZ “ ξ, ∇ ¨Z “ 0 in BRzBR1 ,

ζpRq “ 0, ZpRq “ 0 on ΓR.(4.8)

By a straightforward calculation, the Fourier coefficients for the solution of (4.8) are given by

ζmn pρq “1

2n` 1

ż R

ρr´nc2 ´ pn` 1qc4s ξ

m3npτq ´

a

npn` 1qpc2 ´ c4qξm1npτqdτ, (4.9)

Zm1npρq “1

2n` 1

ż R

ρr´nc1 ´ pn` 1qc3s ξ

m2npτqdτ, (4.10)

Zm2npρq “1

2n` 1

ż R

ρ

a

npn` 1qpc2 ´ c4qξm3npτq ` rpn` 1qc2 ` nc4s ξ

m1npτqdτ, (4.11)

Zm3npρq “1

2n` 1

ż R

ρ

a

npn` 1qpc1 ´ c3qξm2npτqdτ, (4.12)

where

c1 “

´ρ

τ

¯´n´2, c2 “

´ρ

τ

¯´n´1, c3 “

´ρ

τ

¯n´1, c4 “

´ρ

τ

¯n.

Consider the following boundary value problem:

$

’

’

’

&

’

’

’

%

∆g ` κ2pg “ ´

1λ`2µζ, ´∇ˆ p∇ˆ qq ` κ2

sq “ ´1µZ in BRzBR1 ,

∇ ¨ q “ ∇ ¨Z “ 0 in BRzBR1 ,

Bρg “ T ˚1 g, p∇ˆ qq ˆ eρ “ ´iκsT ˚

2 qΓR on ΓR,

g “ g, q “ q on ΓR1 ,

(4.13)

where T ˚1 and T ˚

2 are the adjoint operators to the DtN operators T1 (cf. (D.7)) and T2 (cf. (D.8)),respectively, and qΓR “ ´eρ ˆ peρ ˆ qq is the tangential component of q on ΓR.

Lemma 4.6. If pζ,Zq and pg, qq are the solutions of the systems (4.8) and (4.13), respectively, thenp “ ∇g `∇ˆ q is the solution of the dual problem (4.6) in BRzBR1.

Proof. Letting p “ ∇g `∇ˆ q and substituting it into the elastic wave equation, we get

µ∆ p∇g `∇ˆ qq ` pλ` µq∇∇ ¨ p∇g `∇ˆ qq ` ω2 p∇g `∇ˆ qq“ ∇

`

pλ` 2µq∆g ` ω2g˘

`∇ˆ`

´µ∇ˆ∇ˆ q ` µ∇∇ ¨ q ` ω2q˘

“ ´∇ζ ´∇ˆZ “ ´ξ,

which shows that p satisfies the elastic wave equation in (4.6). The rest of the proof is to show thatp satisfies the boundary condition Dp “ T ˚p on ΓR.

AN ADAPTIVE FINITE ELEMENT DTN METHOD 9

It follows from (B.1)–(B.2) that

p “ ∇g `∇ˆ q

“ÿ

nPN

ÿ

|m|ďn

«

a

npn` 1q

ρgmn pρq ´

1

ρqm2npρq ´ q

m1

2n pρq

ff

Umn (4.14)

`

«

1

ρqm1npρq ` q

m1

1n pρq ´

a

npn` 1q

ρqm3npρq

ff

V mn `

«

gm1

n pρq ´

a

npn` 1q

ρqm2npρq

ff

Xmn eρ.

Let vmn pρq “ ρqm3npρq. Since ∇ ¨ q “ 0, we have from (B.3) that

qm1npρq “1

a

npn` 1q

1

ρ

´

vmn pρq ` ρvm1

n pρq¯

,

qm1

1n pρq “1

a

npn` 1q

ˆ

´1

ρ2vmn pρq `

1

ρvm

1

n pρq ` vm2

n pρq

˙

.

It follows from Lemma F.3 that

1

ρqm1npρq ` q

m1

1n pρq ´

a

npn` 1q

ρqm3npρq “

1a

npn` 1q

ˆ

vm2

n pρq `2

ρvm

1

n pρq ´npn` 1q

ρ2vmn pρq

˙

“ ´1

a

npn` 1q

`

βmn pρq ` κ2svmn pρq

˘

. (4.15)

Substituting (4.15) into (4.14) and taking derivative of p, we obtain

p1pρq “ÿ

nPN

ÿ

|m|ďn

„

a

npn` 1qB

Bρ

ˆ

1

ρgmn pρq

˙

´B

Bρ

ˆ

1

ρqm2npρq

˙

´ qm2

2n pρq

Umn (4.16)

´1

a

npn` 1q

”

βm1

n pρq ` κ2svm1

n pρqı

V mn `

„

gm2

n pρq ´a

npn` 1qB

Bρ

ˆ

1

ρqm2npρq

˙

Xmn eρ.

A simple calculation yields

∇ ¨ p “ ∆g “ ´1

λ` 2µζ ´ κ2

pg. (4.17)

Substituting (4.16)–(4.17) into the boundary operator D , we have

Dp “ µBρp` pλ` µqp∇ ¨ pqeρ

“ÿ

nPN

ÿ

|m|ďn

"

”

µgm2

n pρq ´ pλ` µqκ2pgmn pρq

ı

Xmn eρ ` µ

a

npn` 1qB

Bρ

ˆ

1

ρgmn pρq

˙

Umn

*

` µÿ

nPN

ÿ

|m|ďn

#

´

„

B

Bρ

ˆ

1

ρqm2npρq

˙

` qm2

2n pρq

Umn ´

a

npn` 1qB

Bρ

ˆ

1

ρqm2npρq

˙

Xmn eρ

´1

a

npn` 1q

”

βm1

n pρq ` κ2svm1

n pρqı

V mn

+

´

ˆ

λ` µ

λ` 2µ

˙

ζeρ

“: I1 ` I2 ` I3, (4.18)

where I1, I2 and I3 accounts for the summation of gmn , qm2n and qm3n, respectively.It follows from (B.4) that

∆g ` κ2p g “ ´

1

λ` 2µζ,

10 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

which gives

ÿ

nPN

ÿ

|m|ďn

„

gm2

n pρq `2

ρgm

1

n pρq ´npn` 1q

ρ2gmn pρq ` κ

2p g

mn

Xmn `

1

λ` 2µζ “ 0.

Substituting the above equation into I1 and replacing the second order derivative, we obtain

I1 “ÿ

nPN

ÿ

|m|ďn

"

”

µgm2

n pρq ´ pλ` µqκ2pgmn pρq

ı

Xmn eρ ` µ

a

npn` 1qB

Bρ

ˆ

1

ρgmn pρq

˙

Umn

*

“ÿ

nPN

ÿ

|m|ďn

#

„

´µ2

ρgm

1

n pρq ` µnpn` 1q

ρ2gmn pρq ´ ω

2 gmn pρq

Xmn eρ

` µa

npn` 1q

„

1

ρgm

1

n pρq ´1

ρ2gmn pρq

Umn

+

Replacing the first order derivative by the transparent boundary condition

gm1

n pRq “zp2qn pκpRq

Rgmn pRq,

we have from a straightforward computation that

I1pRq “1

R2

ÿ

nPN

ÿ

|m|ďn

#

”

´2µzp2qn pκpRq ` npn` 1qµ´ ω2R2ı

gmn pRqXmn eρ

` µa

npn` 1q”

zp2qn pκpRq ´ 1ı

gmn pRqUmn

+

.

Using (4.13) and (B.5), we get

´1

ρ

B2

Bρ2pρqm2npρqq `

npn` 1q

ρ2qm2npρq ´ κ

2sqm2npρq “

1

µZm2npρq,

which implies that

qm2

2n pρq “ ´2

ρqm

1

2n pρq `npn` 1q

ρ2qm2npρq ´ κ

2sqm2npρq ´

1

µZm2npρq.

AN ADAPTIVE FINITE ELEMENT DTN METHOD 11

Substituting the above equation into I2 and replacing the second derivative lead to

I2 “ µÿ

nPN

ÿ

|m|ďn

#

´

„

qm2

2n pρq `

ˆ

´1

ρ2qm2npρq `

1

ρqm

1

2n pρq

˙

Umn

´a

npn` 1q

ˆ

´1

ρ2qm2npρq `

1

ρqm

1

2n pρq

˙

Xmn eρ

+

“ µÿ

nPN

ÿ

|m|ďn

#

„

2

ρqm

1

2n pρq ´npn` 1q

ρ2qm2npρq ` κ

2sqm2npρq `

1

µZm2npρq `

1

ρ2qm2npρq ´

1

ρqm

1

2n pρq

Umn

´a

npn` 1q

ˆ

´1

ρ2qm2npρq `

1

ρqm

1

2n pρq

˙

Xmn eρ

+

“ µÿ

nPN

ÿ

|m|ďn

#

„

1

ρqm

1

2n pρq `

ˆ

κ2s `

1

ρ2´npn` 1q

ρ2

˙

qm2npρq

Umn

´a

npn` 1q

„

1

ρqm

1

2n pρq ´1

ρ2qm2npρq

Xmn eρ

+

.

Note that the terms Zm1n and ζ have been dropped since they vanish on ΓR. Substituting thetransparent boundary condition back to I2 and eliminating the first order terms, we deduce

I2pRq “µ

R2

ÿ

nPN

ÿ

|m|ďn

"

”

zp2qn pκsRq ` κ2sR

2 ` 1´ npn` 1qı

qm2npRqUmn

´a

npn` 1q”

zp2qn pκsRq ´ 1ı

qm2npRqXmn eρ

*

.

By (4.12), we have βm1

n pRq “ 0. It follows from Lemma F.3 that

I3pRq “ ´µÿ

nPN

ÿ

|m|ďn

1a

npn` 1q

”

βm1

n pRq ` κ2svm1

n pRqı

V mn

“ ´ÿ

nPN

ÿ

|m|ďn

µκ2s

a

npn` 1qzp2qn pκsRqq

m3npRqV

mn .

Substituting I1, I2 and I3 into (4.18), we get

Dp “ I1pRq ` I2pRq ` I3pRq (4.19)

“1

R2

ÿ

nPN

ÿ

|m|ďn

#

”

´2µ zp2qn pκpRq ` npn` 1qµ´ ω2R2ı

gmn pRqXmn eρ

` µa

npn` 1q”

zp2qn pκpRq ´ 1ı

gmn pRqUmn

+

`µ

R2

ÿ

nPN

ÿ

|m|ďn

#

”

zp2qn pκsRq ` κ2sR

2 ` 1´ npn` 1qı

qm2npRqUmn

´a

npn` 1q”

zp2qn pκsRq ´ 1ı

qm2npRqXmn eρ ´

zp2qn pκsRqκ

2sR

2

a

npn` 1qqm3npRqV

mn

+

12 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

Note that (4.19) is the complex conjugate of (16) in [22]. Using the fact that the transparentboundary conditions of g and q are the complex conjugate of the original transparent boundaryconditions, we deduce Dp “ T ˚p on ΓR, which completes the proof.

Lemma 4.7. Let

ppρ, θ, ϕq “8ÿ

n“0

nÿ

m“´n

pm1npρqUmn pθ, ϕq ` p

m2npρqV

mn pθ, ϕq ` p

m3npρqX

mn pθ, ϕqeρ

be the solution of (4.6) in BRzBR1. Then the following estimate holds:

|pmjnpRq| À n

ˆ

R1

R

˙n 3ÿ

i“1

|pminpR1q| `

1

nξL8prR1,Rsq, j “ 1, 2, 3.

Proof. It follows from (4.14), (F.2), and (F.5) that

pm1npRq “

a

npn` 1q

Rgmn pRq ´

1

Rqm2npRq ´

1

Rzp2qn pκsRqq

m2npRq

“

a

npn` 1q

R

„

SpnpRqgmn pR

1q `iκp2

ż R

R1t2SpnpRqW

pnpR

1, tqζmn ptqdt

´1

R

”

1` zp2qn pκsRqı

„

SsnpRqqm2npR

1q `iκs2

ż R

R1t2SsnpRqW

snpR

1, tqZm2nptqdt

.

Using (4.14) and (F.7) yields

pm2npRq “ ´1

a

npn` 1qκ2sRq

m3npRq

“ ´1

a

npn` 1qκ2sR

„

R1

RSsnpRqq

m3npR

1q `iκs2R

ż R

R1t3SsnpRqW

snpR

1, tqZm3nptqdt

“ ´1

a

npn` 1qκ2s

„

R1SsnpRqqm3npR

1q `iκs2

ż R

R1t3SsnpRqW

snpR

1, tqZm3nptqdt

.

We have from (4.14), (F.2), and (F.5) that

pm3npRq “1

Rzp2qn pκpRqg

mn pRq ´

a

npn` 1q

Rqm2npRq

“1

Rzp2qn pκpRq

„

SpnpRqgmn pR

1q `iκp2

ż R

R1t2SpnpRqW

pnpR

1, tqζmn ptqdt

´

a

npn` 1q

R

„

SsnpRqqm2npR

1q `iκs2

ż R

R1t2SsnpRqW

snpR

1, tqZm2nptqdt

.

Denote a diagonal matrix by

Mdiag “ diag

˜

hp1qn pκpRq

hp1q1n pκpR1q

,hp1qn pκsRq

hp1q1n pκsR1q

,hp1qn pκsRq

hp1q1n pκsR1q

¸

.

It is easy to check that

ppm1npRq, pm2npRq, p

m3npRqq

J“ KnpRqMdiag

`

gmn pR1q, qm2npR

1q, qm3npR1q˘J

`KnpRqMdiag pbm1n, b

m2n, b

m3nq

J , (4.20)

AN ADAPTIVE FINITE ELEMENT DTN METHOD 13

where

bmn “ pbm1n, b

m2n, b

m3nq

J“

ˆ

iκp2

ż R

R1t2W p

npR1, tqζmn ptqdt,

iκs2

ż R

R1t2W s

npR1, tqZm2nptqdt,

iκs2RR1

ż R

R1t3W s

npR1, tqZm3nptqdt

˙J

.

Next is to estimate p at ρ “ R1. By (4.14) and (F.8), we have

pm1npR1q “

a

npn` 1q

R1gmn pR

1q ´1

R1qm2npR

1q ´ qm1

2n pR1q

“

a

npn` 1q

R1gmn pR

1q ´1

R1qm2npR

1q ´1

R1zp2qn pκsR

1qqm2npR1q ´

2κsπR1

ż R

R1t2SsnptqZ

m2nptqdt

“

a

npn` 1q

R1gmn pR

1q ´

”

1` zp2qn pκsR1q

ı 1

R1qm2npR

1q ´2κsπR1

ż R

R1t2SsnptqZ

m2nptqdt. (4.21)

A simple calculation from (4.14)–(4.15) yields

pm2npR1q “ ´

1a

npn` 1q

”

κ2sR1qm3npR

1q `R1Zm3npR1q

ı

. (4.22)

It follows from (4.14) and (F.3) that

pm3npR1q “ gm

1

n pR1q ´

a

npn` 1q

R1qm2npR

1q

“1

R1zp2qn pκpR

1qgmn pR1q `

2κpπR1

ż R

R1t2Spnptqζ

mn ptqdt´

a

npn` 1q

R1qm2npR

1q. (4.23)

We get from (4.21)–(4.23) that

`

pm1npR1q, pm2npR

1q, pm3npR1q˘J“ KnpR1q

`

gmn pR1q, qm2npR

1q, qm3npR1q˘J` pdm1n, d

m2n, d

m3nq

J , (4.24)

where

dmn “ pdm1n, d

m2n, d

m3nq

J“

ˆ

´2κsπR1

ż R

R1t2SsnptqZ

m2nptqdt,

´1

a

npn` 1qR1Zm3npR

1q,2κpπR1

ż R

R1t2Spnptqζ

mn ptqdt

˙J

. (4.25)

Substituting (4.24) into (4.20) and using the definition (4.4), we obtain

pmn pRq “ Qn pmn pR

1q ´Qn dmn `KnpRqMdiag b

mn . (4.26)

By (4.9), it can be verified that

bm1n “1

λ` 2µ

iκp2

ż R

R1t2W p

npR1, tqζmn ptqdt

“1

λ` 2µ

iκp2

1

2n` 1

ż R

R1t2W p

npR1, tq

«

ż R

t

«

´n

ˆ

t

τ

˙´n´1

´ pn` 1q

ˆ

t

τ

˙nff

ξm3npτq

´a

npn` 1q

«

ˆ

t

τ

˙´n´1

´

ˆ

t

τ

˙nff

ξm1npτqdτ

ff

dt.

14 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

We have from (4.10) that

bm2n “iκs2µ

ż R

R1t2W s

npR1, tqZm2nptqdt

“1

2n` 1

iκs2µ

ż R

R1t2W s

npR1, tq

„ż R

t

a

npn` 1qpc2 ´ c4qξm3npτq ` rpn` 1qc2 ` nc4s ξ

m1npτqdτ

dt

“1

2n` 1

iκs2µ

ż R

R1t2W s

npR1, tq

«

ż R

t

a

npn` 1q

«

ˆ

t

τ

˙´n´1

´

ˆ

t

τ

˙nff

ξm3npτq

`

«

pn` 1q

ˆ

t

τ

˙´n´1

` n

ˆ

t

τ

˙nff

ξm1npτqdτ

ff

dt.

It follows from (4.12) that

bm3n “iκs

2µRR1

ż R

R1t3W s

npR1, tqZm3nptqdt

“iκs

2µRR11

2n` 1

ż R

R1t3W s

npR1, tq

„ż R

t

a

npn` 1qpc1 ´ c3qξm2npτqdτ

dt

“iκs

2µRR11

2n` 1

ż R

R1t3W s

npR1, tq

«

ż R

t

a

npn` 1q

«

ˆ

t

τ

˙´n´2

´

ˆ

t

τ

˙n´1ff

ξm2npτqdτ

ff

dt.

Substituting (4.9) – (4.12) into (4.25), we obtain

dm1n “ ´2κsπµR1

ż R

R1t2SsnptqZ

m2nptqdt

“ ´2κsπµR1

1

2n` 1

ż R

R1t2Ssnptq

«

ż R

t

a

npn` 1q

«

ˆ

t

τ

˙´n´1

´

ˆ

t

τ

˙nff

ξm3npτq

`

«

pn` 1q

ˆ

t

τ

˙´n´1

` n

ˆ

t

τ

˙nff

ξm1npτqdτ

ff

dt,

dm2n “ ´1

a

npn` 1q

R1

µZm3npR

1q

“ ´1

a

npn` 1q

R1

µ

1

2n` 1

ż R

R1

a

npn` 1q

«

ˆ

R1

τ

˙´n´2

´

ˆ

R1

τ

˙n´1ff

ξm2npτqdτ,

dm3n “1

λ` 2µ

2κpπR1

ż R

R1t2Spnptqζ

mn ptqdt

“1

λ` 2µ

2κpπR1

1

2n` 1

ż R

R1t2Spnptq

«

ż R

t

«

´n

ˆ

t

τ

˙´n´1

´ pn` 1q

ˆ

t

τ

˙nff

ξm3npτq

´a

npn` 1q

«

ˆ

t

τ

˙´n´1

´

ˆ

t

τ

˙nff

ξm1npτqdτ

ff

dt.

For sufficiently large n, it is shown in Lemma 4.3 and [3] that

Qn À n

ˆ

R1

R

˙n

, KnpRqMdiag À n

ˆ

R1

R

˙n

,ˇ

ˇWnpR1, tq

ˇ

ˇ À1

n

ˆ

t

R1

˙n

.

AN ADAPTIVE FINITE ELEMENT DTN METHOD 15

Substituting the above estimates into bmn and dmn leads to

ˇ

ˇbmjnˇ

ˇ À1

n2

ˆ

R

R1

˙n

ξL8prR1,Rsq,ˇ

ˇdmjnˇ

ˇ À1

n2

ˆ

R

R1

˙n

ξL8prR1,Rsq.

Hence we have from (4.26) that

|pmn pRq| ďˇ

ˇQnpmn pR

1qˇ

ˇ`ˇ

ˇQndmn

ˇ

ˇ`

ˇ

ˇ

ˇKnpRqMdiagb

mn

ˇ

ˇ

ˇ

À n

ˆ

R1

R

˙n 3ÿ

i“1

ˇ

ˇpminpR1qˇ

ˇ`1

nξL8prR1,Rsq `

1

nξL8prR1,Rsq,

which completes the proof.

Lemma 4.8. Let p be the solution of the dual problem (4.6). For sufficiently large N , the followingestimate holds:

ˇ

ˇ

ˇ

ˇ

ż

ΓR

pT ´TN q ξ ¨ p

ˇ

ˇ

ˇ

ˇ

ds À1

Nξ2

H1pΩq.

Proof. It is shown in [22] that all the elements of the DtN matrix Mn have an order n. Hence,

ˇ

ˇ

ˇ

ˇ

ż

ΓR

pT ´TN q ξ ¨ p ds

ˇ

ˇ

ˇ

ˇ

ď

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

n“N`1

ÿ

|m|ďn

MnξpRq ¨ ppRq

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ďÿ

n“N`1

ÿ

|m|ďn

n p|ξm1npRq| ` |ξm2npRq| ` |ξ

m3npRq|q p|p

m1npRq| ` |p

m2npRq| ` |p

m3npRq|q

Àÿ

n“N`1

”

n p1` npn` 1qq12ı´12

»

–

ÿ

|m|ďn

p1` npn` 1qq12´

|ξm1npRq|2` |ξm2npRq|

2` |ξm3npRq|

2¯

fi

fl

12

ˆ

»

–

ÿ

|m|ďn

n3´

|pm1npRq|2` |pm2npRq|

2` |pm3npRq|

2¯

fi

fl

12

À1

NξH12pΓRq

»

–

ÿ

n“N`1

ÿ

|m|ďn

n3´

|pm1npRq|2` |pm2npRq|

2` |pm3npRq|

2¯

fi

fl

12

À1

NξH1pΩq

»

–

ÿ

n“N`1

ÿ

|m|ďn

n3´

|pm1npRq|2` |pm2npRq|

2` |pm3npRq|

2¯

fi

fl

12

. (4.27)

It follows from Lemma 4.7 thatÿ

n“N`1

ÿ

|m|ďn

n3´

|pm1npRq|2` |pm2npRq|

2` |pm3npRq|

2¯

Àÿ

n“N`1

ÿ

|m|ďn

n3

«

n2

ˆ

R1

R

˙2n 3ÿ

i“1

|pminpR1q|2 `

1

n2ξ2L8prR1,Rsq

ff

Àÿ

n“N`1

ÿ

|m|ďn

n5

ˆ

R1

R

˙2n 3ÿ

i“1

|pminpR1q|2 `

ÿ

n“N`1

ÿ

|m|ďn

p1` np1` nqq12ξ2L8prR1,Rsq

:“ J1 ` J2.

16 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

Noting that the function t4e´2t is bounded on p0,`8q, we have

J1 À maxn“N`1

˜

n4

ˆ

R1

R

˙2n¸

ÿ

n“N`1

ÿ

|m|ďn

n3ÿ

i“1

|pminpR1q|2 À p2H12pΓR1 q

À ξH1pΩq.

It is shown in [17] that

ξmjnptq2L8prR1,Rsq ď

ˆ

2

R´R1` n

˙

ξmjnptq2L2prR1,Rsq `

1

nξm

1

jn ptq2L2prR1,Rsq,

ξmjn2H1pBRzBR1 q

ě

ż R

R1

„ˆ

R1 `n2

R

˙

|ξmjnpρq|2 `R1

ˇ

ˇ

ˇξm

1

jn pρqˇ

ˇ

ˇ

2

dρ.

Substituting them into J2 gives

J2 Àÿ

n“N`1

ÿ

|m|ďn

p1` np1` nqq123ÿ

j“1

„ˆ

2

R´R1` n

˙

ξmjnptq2L2prR1,Rsq `

1

nξm

1

jn ptq2L2prR1,Rsq

À ξ2H1pBRzBR1 q

ď ξ2H1pΩq

,

which yieldsÿ

n“N`1

ÿ

|m|ďn

n3´

|pm1npRq|2` |pm2npRq|

2` |pm3npRq|

2¯

À ξ2H1pΩq

.

Plugging the above estimate into (4.27), we obtainˇ

ˇ

ˇ

ˇ

ż

ΓR

pT ´TN q ξ ¨ pds

ˇ

ˇ

ˇ

ˇ

À1

Nξ2

H1pΩq,

which completes the proof.

We are now in position to show the proof of Theorem 4.1.

Proof. It follows from (4.2) that

~ξ~2H1pΩq

“ <bpξ, ξq ` <ż

ΓR

pT ´TN q ξ ¨ ξds` 2ω2

ż

Ωξ ¨ ξdx` <

ż

ΓR

TNξ ¨ ξds

ď C1

»

–

˜

ÿ

TPMh

η2T

¸12

`N

ˆ

R1

R

˙N

uincH1pΩq ` g ´ ghH12pBDq

fi

fl ξH1pΩq

` pC2 ` Cpδqq ξ2L2pΩq

`

ˆ

R

R1

˙

δξ2H1pΩq

.

Choosing δ such that RR

δminpµ,ω2q

ă 12 , we get

~ξ~2H1pΩq

ď 2C1

»

–

˜

ÿ

TPMh

η2T

¸12

`N

ˆ

R1

R

˙N

uincH1pΩq ` g ´ ghH12pBDq

fi

fl ξH1pΩq

` 2 pC2 ` Cpδqq ξ2L2pΩq

. (4.28)

Using Lemmas 4.2, 4.4, and 4.8 yields

ξ2L2pΩq

“ bpξ,pq `

ż

ΓR

pT ´TN q ξ ¨ pds´

ż

ΓR

pT ´TN q ξ ¨ pds

À

»

–

˜

ÿ

TPMh

η2T

¸12

`N

ˆ

R1

R

˙N

uincH1pΩq ` g ´ ghH12pBDq

fi

fl ξH1pΩq `1

Nξ2H1pΩq.

AN ADAPTIVE FINITE ELEMENT DTN METHOD 17

Table 1. The adaptive finite element DtN method.

(1) Given the tolerance ε ą 0, θ P p0, 1q;(2) Fix the computational domain Ω “ BRzD by choosing the radius R;(3) Choose R1 and N such that εN ď 10´8;(4) Construct an initial mesh Mh over Ω and compute error estimators;(5) While εh ą ε do(6) Refine the mesh Mh according to the strategy:

if ηT ą θ maxTPMh

ηT , then refine the element T PMh;

(7) Solve the discrete problem (3.3) on the new mesh which is still denoted as Mh;(8) Compute the corresponding error estimators;(9) End while.

Substituting the above estimate into (4.28) and taking sufficiently large N such that

2 pC2 ` Cpδqq

N

1

minpµ, ω2qă 1,

we obtain

~u´ uhN~H1pΩq À

˜

ÿ

TPMh

η2T

¸12

` g ´ ghH12pBDq `N

ˆ

R1

R

˙N

uincH1pΩq,

which completes the proof.

5. Implementation and numerical experiments

In this section, we discuss the algorithmic implementation of the adaptive finite element DtNmethod and present two numerical examples to demonstrate the effectiveness of the proposedmethod.

5.1. Adaptive algorithm. Based on the a posteriori error estimate in Theorem 4.1, we adopt theFreeFem [15] to implement the adaptive algorithm of the linear finite elements. By Theorem 4.1,the a posteriori error estimator consists of three parts: the first two parts are related to the finiteelement discretization error εh and the third one is the DtN truncation error εN which depends onthe truncation number N . Explicitly,

εh “

˜

ÿ

TPMh

η2T

¸12

` g ´ ghH12pBDq, εN “ N

ˆ

R1

R

˙N

uincH1pΩq.

In the implementation, the parameters R1, R, and N are chosen such that the finite elementdiscretization error is not polluted by the DtN truncation error, i.e., εN is required to be very smallcompared to εh, for example, εN ď 10´8. For simplicity, in the following numerical experiments,R1 is chosen such that the obstacle is exactly contained in the ball BR1 , and N is taken to be thesmallest positive integer such that εN ď 10´8. Table 1 shows the algorithm of the adaptive finiteelement DtN method for solving the elastic obstacle scattering problem.

5.2. TBC matrix construction. Denote by tΨjuLj“1 the basis of the finite element space V h.

Then the finite element approximation of the solution is

uN “Lÿ

j“1

ujΨj . (5.1)

18 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

Recall that the truncated DtN operator (3.1) is

TNuN “Nÿ

n“0

nÿ

m“´n

#

”

Mpn,mq11 um1npRq `M

pn,mq13 um3npRq

ı

Umn `M

pn,mq22 um2npRqV

mn

`

”

Mpn,mq31 um1npRq `M

pn,mq33 um3npRq

ı

Xmn eρ

+

, (5.2)

where

pum1npRq, um2npRq, u

m3npRqq

J“

ż

ΓR

uN ¨`

Umn ,V

mn ,

`

Xmn eρ

˘˘Jds. (5.3)

Substituting (5.1) into (5.3) yields

pum1npRq, um2npRq, u

m3npRqq

J“

Lÿ

j“1

uj

ż

ΓR

Ψj ¨`

Umn ,V

mn ,

`

Xmn eρ

˘˘Jds. (5.4)

Define the components of the vectors Φpn,mqU ,Φ

pn,mqV , and Φ

pn,mqX as follows

Φpn,mqU,j “

ż

ΓR

Ψj ¨Umn ds, Φ

pn,mqV,j “

ż

ΓR

Ψj ¨ Vmn ds, Φ

pn,mqX,j “

ż

ΓR

Ψj ¨ pXmn eρq ds.

Denote by B the TBC matrix. It follows from (5.2) and (5.4) that

Lÿ

j“1

Bijuj “

ż

ΓR

TNuN ¨Ψids

“

Nÿ

n“0

nÿ

m“´n

#«

Mpn,mq11

ÿ

j

ujΦpn,mqU,j `M

pn,mq13

ÿ

j

ujΦpn,mqX,j

ff

Φpn,mqU,i

`

«

Mpn,mq31

ÿ

j

ujΦpn,mqU,j `M

pn,mq33

ÿ

j

ujΦpn,mqX,j

ff

Φpn,mqX,i

`Mpn,mq22

ÿ

j

ujΦpn,mqV,j Φ

pn,mqV,i

+

,

which indicates that the TBC matrix is a low rank matrix constructed by vector products. Specifi-cally, we have

B “Nÿ

n“0

nÿ

m“´n

´

Mpn,mq11 Φ

pn,mqU Φ

pn,mq˚

U `Mpn,mq13 Φ

pn,mqU Φ

pn,mq˚

X `Mpn,mq22 Φ

pn,mqV Φ

pn,mq˚

V

`Mpn,mq31 Φ

pn,mqX Φ

pn,mq˚

U `Mpn,mq33 Φ

pn,mqX Φ

pn,mq˚

X

¯

. (5.5)

To simplify the notation, define matrices U and V as

U “´

Φpn,mqU ,Φ

pn,mqU ,Φ

pn,mqV ,Φ

pn,mqX ,Φ

pn,mqX

¯

pn,mq,

V “´

Mpn,mq11 Φ

pn,mq˚

U ,Mpn,mq13 Φ

pn,mq˚

X ,Mpn,mq22 Φ

pn,mq˚

V ,Mpn,mq31 Φ

pn,mq˚

U ,Mpn,mq33 Φ

pn,mq˚

X

¯

pn,mq.

Then we have from (5.5) that

B “ UV. (5.6)

Denote by A the stiffness matrix corresponding to the variational problem with the Neumanncondition Bνu “ 0 on ΓR and the Dirichlet boundary condition on BD, which has the sesquilinear

AN ADAPTIVE FINITE ELEMENT DTN METHOD 19

0 1000 2000 3000 4000 5000

nz = 186209

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Figure 1. Sparsity pattern of the coefficient matrix. (left) Matrix A; (right) Matrix M .

form

bpu,vq “ µ

ż

Ω∇u : ∇v dx` pλ` µq

ż

Ωp∇ ¨ uq p∇ ¨ vq dx´ ω2

ż

Ωu ¨ v dx

Then the stiffness matrix W for the variational problem (3.3) takes the form

W “ A´B. (5.7)

Since the DtN operator is nonlocal, it is clear to note from (5.6) that the nonzeros of B and Aare Oph´4q and Oph´3q, respectively. Figure 1 plots the sparsity patterns of matrices A and Wwith 1805 nodal points in the mesh. Hence, the bandwidth of matrix W is much larger than thebandwidth of matrix A. For the above reason, we do not assemble matrix W and solve the resultinglinear system directly.

The matrix A is sparse, real and symmetric. It can be handled effectively by many parallel directsolvers. Since B is a low rank matrix given in (5.6), the linear system Wz “ b can be solvedeffectively via the generalized Woodbury matrix identity by using the following steps: (1) constructmatrices U and V ; (2) solve Az1 “ b with a parallel direct solver; (3) perform matrix-vector productz2 “ V z1; (4) construct matrix C “ A´1U and H “ I ` V C; (5) solve a dense but small size linearsystem Hz3 “ z2; (6) perform matrix-vector product z4 “ Cz3; (7) construct the solution of thesystem Wz “ b by z “ z1 ´ z4. The above approach has several advantages: (1) it is only neededto store matrices A, U and V . The nonzeros of U, V are Oph´2q; (2) the bandwidth of matrix A ismuch smaller, so it is faster to solve Az1 “ b. Once the linear solver is set up, the construction ofmatrix C “ A´1U is very fast.

5.3. Numerical examples. We present two examples to show the performance of the proposedmethod. In the first example, the obstacle is a ball so that the explicit solution is available. Theincident wave is chosen as the third column of the Green Tensor. By comparing the numericalsolution with the explicit solution, we are able to report the accuracy of the proposed algorithm. Inthe second example, we consider a more complex geometry: a rectangular U-shaped obstacle. Theobstacle is assumed to be illuminated by a compressional plane wave. We pay particular attentionon the mesh refinement around the corners of the obstacle, where the solution has singularity. Inboth examples, the a posteriori error is plotted against the number of unknowns in order to showthe convergence rate.

Example 1. In this example, we intend to test the accuracy of the proposed algorithm. Theobstacle is taken as a ball with radius 0.5. The transparent boundary condition is set on the sphereΓR with radius R “ 1. Denote by Gpx,y;ωq the Green Tensor of the three-dimensional elastic waveequation. More explicitly, we have

Gpx,y;ωq “1

µgpx,y;κsqI `

1

ω2∇x∇Jx pgpx,y;κsq ´ gpx,y;κpqq ,

20 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

104 105 106

Number of unknowns

100

H1 e

rro

r

Slope -1/3

A posteriori error

A priori error

Figure 2. Example 1. Quasi-optimality of the a priori and a posteriori error estimates.

where I is the 3ˆ 3 identity matrix and

gpx,y;κq “1

4π

eiκ|x´y|

|x´ y|

is the fundamental solution of the three-dimensional Helmholtz equation. The incident wave ischosen as a multiple of the third column of Green tensor, i.e.,

uincpx;yq “ 10Gpx,y;ωqp:, 3q,

where the source point y “ p0, 0, 0qJ is taken as the origin, the angular frequency ω “ π, and theLame parameters µ “ 1, λ “ 2. Then it is easy to check that the scattered field is u “ ´uinc.

Denote the a priori error by eh “ u´uhNH1pΩq. Figure 2 displays the curves of log eh and log εh

against log DoFh for the adaptive mesh refinements, where DoFh stands for the degrees of freedomor the number of knowns for the mesh Mh. The red circle line represents the a posteriori errorestimate and the yellow star line stands for the a priori error estimate; the slope of the straight blueline is ´13. It indicates that the meshes and the associated numerical complexity are quasi-optimal,

i.e., u´ uhNH1pΩq “ O`

DoF´13h

˘

holds asymptotically.

Example 2. This example demonstrates the effectiveness of the proposed method to handle theproblem where the solution has singularity. We consider a more complex geometry: a rectangularU-shaped obstacle, which is shown in the left of Figure 3. In this example, the parameters of therectangular U-shaped geometry are L “ 1, D “ 0.6, H “ W “ 0.2. The obstacle is assumed to beilluminated by a compressional plane wave

uincpx;dq “ deiκpx¨d,

where d “ p0,´1, 0qJ is the incident direction, the angular frequency ω “ π, and the lame parametersµ “ 1, λ “ 2. Thus the compressional wavenumber κp “ π2. The transparent boundary conditionis set on the sphere ΓR with radius R “ 1. Figure 3 plots the curves of log εh versus log DoFh for theadaptive mesh refinements. The blue circle line represents for the a posteriori error estimate and thered line is a straight line with slope ´13. Again, it indicates that the meshes and the associatednumerical complexity are quasi-optimal. Figure 4 shows the initial mesh (1805 nodal points) andthe refined mesh after 2 iterative steps with 13352 nodal points. It is clear to note that the mesh isrefined mainly around the corners and the interior part of the U-shaped obstacle, where the solutionhas singularity, and keeps relatively coarse near the TBC surface, where the solution is smooth.

6. Conclusion

In this paper, we have presented an adaptive finite element DtN method for the elastic obstaclescattering problem in three dimensions. Based on the Helmholtz decomposition, a new duality

AN ADAPTIVE FINITE ELEMENT DTN METHOD 21

104 105 106

Number of unknowns

100

H1 e

rro

r

A posteriori error

Slope -1/3

Figure 3. Example 2: Quasi-optimality of the a posteriori error estimates.

Figure 4. Example 2: Cross sections of the initial mesh (left) and refined mesh (right).

argument is developed to obtain the a posteriori error estimate. It not only takes into account of thefinite element discretization error but also includes the truncation error of the DtN operator. Weshow that the truncation error decays exponentially with respect to the truncation parameter. Thea posteriori error estimate for the solution of the discrete problem serves as a basis for the adaptivefinite element approximation. Numerical results show that the proposed method is accurate andeffective.

Appendix A. Basis functions

The spherical coordinates pρ, θ, ϕq are related to the Cartesian coordinates x “ px1, x2, x3q byx1 “ ρ sin θ cosϕ, x2 “ ρ sin θ sinϕ, x3 “ ρ cos θ, with the local orthonormal basis teρ, eθ, eϕu:

eρ “ psin θ cosϕ, sin θ sinϕ, cos θqJ,

eθ “ pcos θ cosϕ, cos θ sinϕ,´ sin θqJ,

eϕ “ p´ sinϕ, cosϕ, 0qJ,

where θ P r0, πs and ϕ P r0, 2πs are the Euler angles.

22 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

Denote by tY mn pθ, ϕq : |m| ď n, n “ 0, 1, 2, . . . u the orthonormal sequence of spherical harmonics

of order n on the unit sphere S2 “ tx P R3 : |x| “ 1u. Explicitly, they are given by

Y mn pθ, ϕq “

d

p2n` 1qpn´ |m|q!

4πpn` |m|q!P |m|n pcos θqeimϕ,

where

Pmn ptq “ p1´ t2q

m2

dm

dtmPnptq, 1 ď t ď 1, 0 ď m ď n,

are the associated Legendre functions and Pn is the Legendre polynomial of degree n. Define asequence of rescaled spherical harmonics of order n:

Xmn pθ, ϕq “

1

RY mn pθ, ϕq.

It is clear to note that tXmn pθ, ϕq : |m| ď n, n “ 0, 1, 2, ...u forms a complete orthonormal system in

L2pΓRq.For a smooth scalar function w defined on ΓR, let

∇Γw “Bw

Bθeθ `

1

sin θ

Bw

Bϕeϕ

be the tangential gradient on ΓR. It follows from [10, Theorem 6.25] that

Umn pθ, ϕq “

1a

npn` 1q∇ΓX

mn pθ, ϕq, V m

n pθ, ϕq “ eρ ˆUmn

for m “ ´n, ..., n, n “ 1, 2, ... form a complete orthonormal system in L2t pΓRq “ tw P L2pΓRq

3 :w ¨ eρ “ 0u. For the convenience of notation, define

U00 “ 0, V 0

0 “ 0.

Appendix B. Some basic identities

In this section, we list some basic identities in the spherical coordinates which are frequently usedin the paper. The proofs are based on straightforward calculations.

Let fpρq be a smooth function. It can be verified that the gradient operator satisfies

∇ pfpρqXmn q “ f 1pρqXm

n eρ ` fpρq

a

npn` 1q

ρUmn , (B.1)

the curl operator satisfies$

’

’

’

’

’

’

’

&

’

’

’

’

’

’

’

%

∇ˆ pfpρqUmn q “

1

ρ

B

BρpρfpρqqV m

n ,

∇ˆ pfpρqV mn q “ ´

1

ρ

B

BρpρfpρqqUm

n ´

a

npn` 1q

ρfpρqXm

n eρ,

∇ˆ pfpρqXmn eρq “ ´

a

npn` 1q

ρfpρqV m

n ,

(B.2)

the divergence operator satisfies$

’

’

’

’

&

’

’

’

’

%

∇ ¨ pfpρqUmn q “ ´fpρq

a

pn` 1qn1

ρXmn ,

∇ ¨ pfpρqV mn q “ 0,

∇ ¨ pfpρqXmn eρq “

1

ρ2

B

Bρ

`

ρ2fpρq˘

Xmn ,

(B.3)

AN ADAPTIVE FINITE ELEMENT DTN METHOD 23

the Laplacian operator satisfies

∆fpρq “ÿ

nPN

ÿ

|m|ďn

1

ρ2

B

Bρ

ˆ

ρ2 B

Bρfmn pρq

˙

Xmn `

1

ρ2fmn pρq∆ΓX

mn

“ÿ

nPN

ÿ

|m|ďn

„

2

ρfm

1

n pρq ` fm2

n pρq ´npn` 1q

ρ2fmn pρq

Xmn , (B.4)

and the double curl operator satisfies$

’

’

’

’

’

’

’

&

’

’

’

’

’

’

’

%

∇ˆ p∇ˆ pfpρqUmn qq “ ´

1

ρ

B2

Bρ2pρfpρqqUm

n ´

a

pnpn` 1qq

ρ2

B

BρpρfpρqqXm

n eρ,

∇ˆ p∇ˆ pfpρqV mn qq “

„

´1

ρ

B2

Bρ2pρfpρqq `

npn` 1q

ρ2fpρq

V mn ,

∇ˆ p∇ˆ pfpρqXmn eρqq “

a

npn` 1q

ρ

B

BρfpρqUm

n `npn` 1q

ρ2fpρqXm

n eρ,

(B.5)

where ∆Γ is the Laplace–Beltrami operator on ΓR and is defined by

∆Γ “1

sin θ

B

Bθ

ˆ

sin θB

Bθ

˙

`1

sin2 θ

B2

Bϕ.

Appendix C. Function spaces

Denote by L2pΩq the square integrable functions on Ω. Let L2pΩq “ L2pΩq3 be equipped withthe inner product and norm given as follows:

pu,vq “

ż

Ωu ¨ v dx, uL2pΩq “ pu,uq

12.

Let H1pΩq the standard Sobolev space with the norm given by

uH1pΩq “

ˆż

Ω|upxq|2 ` |∇upxq|2 dx

˙12

.

Define H1BDpΩq “ H1

BDpΩq3, where H1

BDpΩq :“ tu P H1pΩq : u “ 0 on BDu.Denote by HspΓRq the standard trace functional space which is equipped with the norm

uHspΓRq “

˜

8ÿ

n“0

nÿ

m“´n

p1` npn` 1qqs|umn |2

¸12

, upR, θ, ϕq “8ÿ

n“0

nÿ

m“´n

umn Xmn pθ, ϕq.

Let HspΓRq “ HspΓRq3 which is equipped with the norm

uHspΓRq “

˜

8ÿ

n“0

nÿ

m“´n

p1` npn` 1qqs|umn |2

¸12

,

where umn “ pum1n, u

m2n, u

m3nq and

upR, θ, ϕq “8ÿ

n“0

nÿ

m“´n

um1nUmn pθ, ϕq ` u

m2nV

mn pθ, ϕq ` u

m3nX

mn pθ, ϕqeρ.

It can be verified that H´spΓRq is the dual space of HspΓRq with respect to the inner product

xu,vyΓR “

ż

ΓR

u ¨ vds “8ÿ

n“0

nÿ

m“´n

um1nvm1n ` u

m2nv

m2n ` u

m3nv

m3n,

24 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

where

vpR, θ, ϕq “8ÿ

n“0

nÿ

m“´n

vm1nUmn pθ, ϕq ` v

m2nV

mn pθ, ϕq ` v

m3nX

mn pθ, ϕqeρ.

Appendix D. The DtN operator

The DtN operator T defined in (2.3) was introduced in [22]. For the self-contained purpose, wesummary the results here.

Let u admit the Fourier expansion

upρ, θ, ϕq “8ÿ

n“0

nÿ

m“´n

um1npρqUmn pθ, ϕq ` u

m2npρqV

mn pθ, ϕq ` u

m3npρqX

mn pθ, ϕqeρ.

Consider the Helmholtz decomposition

u “ ∇φ`∇ˆψ, ∇ ¨ψ “ 0,

where the potential functions φ and ψ have the Fourier expansions

φpρ, θ, ϕq “8ÿ

n“0

nÿ

m“´n

hp1qn pκpρq

hp1qn pκpRq

φmn pRqXmn pθ, ϕq, (D.1)

ψpρ, θ, ϕq “8ÿ

n“0

nÿ

m“´n

1a

npn` 1q

R

ρ

hp1qn pκsρq ` κsρh

p1q1

n pκsρq

hp1qn pκsRq

ψm3npRqUmn pθ, ϕq

´hp1qn pκsρq

hp1qn pκsRq

ψm2npRqVmn pθ, ϕq `

˜

R

ρ

hp1qn pκsρq

hp1qn pκsRq

¸

ψm3npRqXmn pθ, ϕqeρ. (D.2)

Here hp1qn is the spherical Hankel function of the first kind with order n.

Substituting (D.1) and (D.2) into the Helmholtz decomposition yields

»

–

um1npRqum2npRqum3npRq

fi

fl “1

R

»

—

—

–

a

npn` 1q ´1´ zp1qn pκsRq 0

0 0 ´κ2sR

2?npn`1q

zp1qn pκpRq ´

a

npn` 1q 0

fi

ffi

ffi

fl

»

–

φmn pRqψm2npRqψm3npRq

fi

fl

“ KnpRq

»

–

φmn pRqψm2npRqψm3npRq

fi

fl , (D.3)

where zp1qn ptq “ th

p1q1

n ptqhp1qn ptq. It follows from a simple calculation that the inverse of KnpRq is

K´1n pRq “

R

ΛnpRq

»

—

–

´a

npn` 1q 0 1` zp1qn pκsRq

´zp1qn pκpRq 0

a

npn` 1q

0 ´a

npn` 1qΛnpRqpκ2sR

2q 0

fi

ffi

fl

, (D.4)

where

ΛnpRq “ zp1qn pκpRq´

1` zp1qn pκsRq¯

´ npn` 1q. (D.5)

It is shown in [22] that

=ΛnpRq “ <zp1qn pκpRq=zp1qn pκsRq `´

1` <zp1qn pκsRq¯

=zp1qn pκpRq ă 0,

which guarantees the existence of the inverse for KnpRq.By the Helmholtz decomposition and (D.1)–(D.2), the boundary operator D can be expressed in

terms of the potential functions φ and ψ on ΓR; it can be further written in terms of u on ΓR by

AN ADAPTIVE FINITE ELEMENT DTN METHOD 25

(D.3)–(D.4), which deduce the DtN operator T . Explicitly, it is shown in [22] that the entries ofthe matrix MnpRq defined in (2.3) are

Mn,11pRq “ ´µ

R

˜

1`zp1qn pκpRq

ΛnpRqκ2sR

2

¸

, Mn,13pRq “a

npn` 1qµ

R

ˆ

1`κ2sR

2

ΛnpRq

˙

,

Mn,22pRq “µ

Rzp1qn pκsRq, Mn,31pRq “

a

npn` 1qµ

R

„

1`ω2R2

µΛnpRq

, Mn,32pRq “ 0,

Mn,33pRq “ ´µ

R

„

2`ω2R2

µΛnpRq

´

1` zp1qn pκsRq¯

, Mn,12pRq “Mn,21pRq “Mn,23pRq “ 0.

Let Mn “ ´12 pMn `M

˚n q, where M˚

n is the adjoint of Mn. It is also shown in [22] that for a fixed

R ą 0 and a sufficiently large n, Mn is positive definite and its entries satisfy the estimate

|Mn,ijpRq| À n, i, j “ 1, 2, 3. (D.6)

Introduce the DtN operators T1 and T2 for the potentials φ and ψ on ΓR, respectively. In [3], itis shown that

Bρφ “ T1φ, T1φ “8ÿ

n“0

nÿ

m“´n

zp1qn pκpRqφmn pRqX

mn pθ, ϕq. (D.7)

In [4], it is shown that

p∇ˆψq ˆ eρ “ ´iκsT2ψΓR , ψΓR “ ´eρ ˆ peρ ˆψq ,

where

T2ψ “8ÿ

n“0

nÿ

m“´n

iκsR

1` zp1qn pκsRq

ψm1npRqUmn pθ, ϕq `

1` zp1qn pκsRq

iκsRψm2npRqV

mn pθ, ϕq. (D.8)

Appendix E. Bessel functions

This section is concerned with some properties of the Bessel functions, which are needed in theproof of Theorem 4.1.

The proof of Lemma E.1 may be found in [28].

Lemma E.1. Let ν “ n´ 12, where n is an arbitrary integer. For 0 ă z ă ν, it holds that

Yνpzq “ ´

ˆ

2

π

˙12 1

pν2 ´ z2q14eηpz,νq r1` rpz, νqs ,

where

ηpz, νq “ ν log

˜

ν

z`

„

´ν

z

¯2´ 1

12¸

´`

ν2 ´ z2˘12

and

|rpz, νq| ď eg2 g2, g2 “ν ´ z

ν13, g2 “

2

3 g322

.

As a consequence, the results in Lemma E.1 can be used to estimate Yν´1pzqYν .

Lemma E.2. For a fixed z P R` and sufficiently large ν, it holds that

z

2ν´

1

6

z

ν2`O

ˆ

1

n3

˙

ďYν´1pzq

Yνpzqď

z

2ν`

7

6

z

ν2`O

ˆ

1

ν3

˙

.

26 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

Proof. It follows from Lemma E.1 that

Yν´1pzq

Yνpzq“

`

ν2 ´ z2˘14

rpν ´ 1q2 ´ z2s14

1`Rpz, ν ´ 1q

1`Rpz, νqeηpz,ν´1q´ηpz,νq :“ I1I2I3. (E.1)

A direct calculation shows that

I1 “

`

ν2 ´ z2˘14

rpν ´ 1q2 ´ z2s14“ 1`

1

2ν`O

ˆ

1

ν2

˙

(E.2)

and

g2pz, νq “2

3

˜

ν13

ν ´ z

¸32

“2

3ν`O

ˆ

1

ν2

˙

,

eg2pz,νqg2pz, νq “2

3ν`O

ˆ

1

ν2

˙

,

eg2pz,ν´1qg2pz, ν ´ 1q “2

3ν`O

ˆ

1

ν3

˙

.

Substituting the above estimates into I2, we have

I2 “1`Rpz, ν ´ 1q

1`Rpz, νqď

1` g2pz, ν ´ 1qeg2pz,ν´1q

1´ g2pz, νqeg2pz,νq“ 1`

4

3

1

ν`O

ˆ

1

ν2

˙

,

I2 “1`Rpz, ν ´ 1q

1`Rpz, νqě

1´ g2pz, ν ´ 1qeg2pz,ν´1q

1` g2pz, νqeg2pz,νq“ 1´

4

3ν`O

ˆ

1

ν2

˙

,

which give that

1´4

3ν`O

ˆ

1

ν2

˙

ď I2 ď 1`4

3ν`O

ˆ

1

ν2

˙

. (E.3)

It follows from a simple calculation that

ηpz, ν ´ 1q ´ ηpz, νq “ pν ´ 1q log

¨

˝

ν ´ 1

z`

«

ˆ

ν ´ 1

z

˙2

´ 1

ff12˛

‚

´ ν log

˜

ν

z`

„

´ν

z

¯2´ 1

12¸

``

ν2 ´ z2˘12

´`

pν ´ 1q2 ´ z2˘12

:“ A`B

It is easy to check that

B “`

ν2 ´ z2˘12

´`

pν ´ 1q2 ´ z2˘12

“ 1`z2

2ν2`O

ˆ

1

ν3

˙

(E.4)

and

log

˜

ν

z`

„

´ν

z

¯2´ 1

12¸

“ log

ˆ

2ν

z

˙

´1

4

´z

ν

¯2`O

ˆ

1

ν3

˙

,

log

¨

˝

ν ´ 1

z`

«

ˆ

ν ´ 1

z

˙2

´ 1

ff12˛

‚“ log

ˆ

2ν

z

˙

´1

ν´

ˆ

1

2`z2

4

˙

1

ν2`O

ˆ

1

ν3

˙

,

AN ADAPTIVE FINITE ELEMENT DTN METHOD 27

which lead to

A “ pν ´ 1q log

¨

˝

ν ´ 1

z`

«

ˆ

ν ´ 1

z

˙2

´ 1

ff12˛

‚´ ν log

˜

ν

z`

„

´ν

z

¯2´ 1

12¸

“ ´ log

ˆ

2ν

z

˙

´ 1`1

2ν`O

ˆ

1

ν2

˙

. (E.5)

Combining (E.5) and (E.4), we obtain

I3 “ exp pηpz, ν ´ 1q ´ ηpz, νqq “ exp pA`Bq “z

2ν`z

4

1

ν2`O

ˆ

1

ν3

˙

. (E.6)

Substituting (E.2) and (E.6) into (E.1) gives

Yν´1pzq

Yνpzq“ I1I2I3 “

„

z

2ν`z

2

1

ν2``O

ˆ

1

ν3

˙

I2.

We have from (E.3) that

Yν´1pzq

Yνpzqď

„

z

2ν`z

2

1

ν2``O

ˆ

1

ν3

˙„

1`4

3ν`O

ˆ

1

ν2

˙

“z

2ν`

7z

6

1

ν2`O

ˆ

1

ν3

˙

and

Yν´1pzq

Yνpzqě

„

z

2ν`z

2

1

ν2``O

ˆ

1

ν3

˙„

1´4

3ν`O

ˆ

1

ν2

˙

“z

2ν´z

6

1

ν2`O

ˆ

1

ν3

˙

,

which completes the proof.

Lemma E.3. Let Gνpzq “zY 1νpzqYνpzq

. The following estimate holds for sufficiently large ν:

´ν `z2

2ν´

1

6

z2

ν2`O

ˆ

1

ν3

˙

ď Gνpzq ď ´ν `z2

2ν`

7

6

z2

ν2`O

ˆ

1

ν3

˙

ă 0.

Proof. Recall the following recurrence relation for the Bessel function (cf. [32]):

Y 1νpzq “ Yν´1pzq ´ν

zYνpzq,

which gives

Gνpzq “zY 1νpzq

Yνpzq“ z

„

Yν´1pzq

Yνpzq´ν

z

.

By Theorem E.2, we obtain for sufficiently large ν that

Gνpzq ď z

„

z

2ν`

7

6

z

ν2`O

ˆ

1

ν3

˙

´ν

z

“ ´ν `z2

2ν`

7

6

z2

ν2`O

ˆ

1

ν3

˙

and

Gνpzq ě z

„

z

2ν´z

6

1

ν2`O

ˆ

1

ν3

˙

´ν

z

“ ´ν `z2

2ν´

1

6

z2

ν2`O

ˆ

1

ν3

˙

,

which completes the proof.

Lemma E.4. Assume R1 ă R and κp ă κs. Then for sufficiently large positive ν, it holds thatˇ

ˇ

ˇ

ˇ

YνpκpRq

YνpκpR1q´YνpκsRq

YνpκsR1q

ˇ

ˇ

ˇ

ˇ

ď7

3

κspκs ´ κpq

νR`

R´R1˘

ˆ

R1

R

˙ν

.

28 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

Proof. Let Fνpzq “ YνpzRqYνpzR1q. By the mean value theorem, we have

Fνpκpq ´ Fνpκsq “ F 1νpξqpκp ´ κsq

“RY 1νpξRqYνpξR

1q ´R1YνpξRqY1νpξR

1q

YνpξR1q2pκp ´ κsq

““

GνpξRq ´GνpξR1q‰ YνpξRq

YνpξR1q

κp ´ κsξ

“ G1νpηqpR´R1qpκp ´ κsq

YνpξRq

YνpξR1q, (E.7)

where η satisfies κpR1 ă η ă κsR. A simple calculation yields

G1νpzq “

ˆ

zY 1νpzq

Yνpzq

˙1

“1

z

zY 1νpzqYνpzq ` z2Y 2ν pzqYνpzq ´ z

2Y 1νpzq2

Y 2ν pzq

.

Using the identityz2Y 2ν pzq “ ´zY

1νpzq ´ pz

2 ´ ν2qYνpzq,

we obtain

G1νpzq “1

z

zY 1νpzqYνpzq `“

´zY 1νpzq ´ pz2 ´ ν2qYνpzq

‰

Yνpzq ´ z2Y 1νpzq

2

Y 2ν pzq

“ ´z2 ´ ν2

z´

1

zG2νpzq. (E.8)

It is shown in Lemma E.3 that Gνpzq is always negative for sufficiently large ν. Hence we have

G2νpzq ď

„

´ν `z2

2ν´

1

6

z2

ν2`O

ˆ

1

ν3

˙2

“ ν2 ´ z2 `1

3

z2

ν`O

ˆ

1

ν2

˙

and

G2νpzq ě

„

´ν `z2

2ν`

7

6

z2

ν2`O

ˆ

1

n3

˙2

“ ν2 ´ z2 ´7

3

z2

ν`O

ˆ

1

ν2

˙

.

Substituting the above estimates into (E.8) leads to

G1νpzq ď ´z `ν2

z´

1

z

„

ν2 ´ z2 ´7

3

z2

ν`O

ˆ

1

ν2

˙

“7

3

z

ν`O

ˆ

1

ν2

˙

and

G1νpzq ě ´z `ν2

z´

1

z

„

ν2 ´ z2 `1

3

z2

ν`O

ˆ

1

ν2

˙

“ ´1

3

z

ν`O

ˆ

1

ν2

˙

.

Combining the above estimates, we get from (E.7) that

G1νpηqpR´R1qpκp ´ κsq ď ´

1

3

η

νpR´R1qpκp ´ κsq ď

1

3

κsR

νpR´R1qpκs ´ κpq

and

G1νpηqpR´R1qpκp ´ κsq ě

7

3

η

νpR´R1qpκp ´ κsq ě ´

7

3

κsR

νpR´R1qpκs ´ κpq,

which shows

|G1νpηqpR´R1qpκp ´ κsq| ď

7

3

κsR

νpR´R1qpκs ´ κpq.

The proof is completed by noting that

YνpzRq

YνpzR1q„

ˆ

R1

R

˙ν

.

AN ADAPTIVE FINITE ELEMENT DTN METHOD 29

Lemma E.5. Let 0 ă κp ă κs and 0 ă R ă R. For sufficiently large n, the following estimate holdsfor j “ 1, 2:

ˇ

ˇ

ˇ

ˇ

ˇ

hpjqn pκpRq

hpjqn pκpR1q

´hpjqn pκsRq

hpjqn pκsR1q

ˇ

ˇ

ˇ

ˇ

ˇ

ď14

3

κspκs ´ κpq

nR`

R´R1˘

ˆ

R1

R

˙n`1

,

where hp1qn and h

p2qn are the spherical Hankel functions of the first and second kind with order n,

respectively.

Proof. Since the spherical Hankel functions of the first and second kind are complex conjugate toeach other, it is only needed to prove the result for the first kind of spherical Hankel functions.

Recalling the relationship between the spherical Bessel functions and the Bessel functions

hpjqn pzq “

c

π

2zHpjq

n` 12

, ynpzq “

c

π

2zYn` 1

2, jnpzq “

c

π

2zJn` 1

2,

we have from [32] that

jnpzq „1

z

c

1

2e

ˆ

ez

2n` 1

˙n`1

, ynpzq „ ´1

z

c

2

e

ˆ

ez

2n` 1

˙´n

.

Let Snpzq “jnpzqynpzq

. Then it is easy to check that

Snpzq „ ´1

2

ˆ

ez

2n` 1

˙2n`1

.

It is shown in [23] thatˇ

ˇ

ˇ

ˇ

ˇ

hp1qn pκpRq

hp1qn pκpR1q

´hp1qn pκsRq

hp1qn pκsR1q

ˇ

ˇ

ˇ

ˇ

ˇ

ď

ˇ

ˇ

ˇ

ˇ

ynpκpRq

ynpκpR1q´ynpκsRq

ynpκsR1q

ˇ

ˇ

ˇ

ˇ

`

ˇ

ˇ

ˇ

ˇ

ynpκpRq

ynpκpR1q

SnpκpR1q

1´ iSnpκpR1q

ˇ

ˇ

ˇ

ˇ

`

ˇ

ˇ

ˇ

ˇ

ynpκpRq

ynpκpR1q

SnpκpRq

1´ iSnpκpR1q

ˇ

ˇ

ˇ

ˇ

`

ˇ

ˇ

ˇ

ˇ

ynpκsRq

ynpκsR1q

SnpκsR1q

1´ iSnpκsR1q

ˇ

ˇ

ˇ

ˇ

`

ˇ

ˇ

ˇ

ˇ

ynpκsRq

ynpκsR1q

SnpκsRq

1´ iSnpκsR1q

ˇ

ˇ

ˇ

ˇ

.

A simple computation yields that

ˇ

ˇ

ˇ

ˇ

SnpzRq

1´ iSnpzR1q

ˇ

ˇ

ˇ

ˇ

ď

ˆ

ezR

2n` 1

˙2n`1

,

ˇ

ˇ

ˇ

ˇ

SnpzR1q

1´ iSnpzR1q

ˇ

ˇ

ˇ

ˇ

ď

ˆ

ezR1

2n` 1

˙2n`1

.

Combining the above estimates, we have for R ą R and κs ą κp that

ˇ

ˇ

ˇ

ˇ

ynpκpRq

ynpκpR1q

SnpκpR1q

1´ iSnpκpR1q

ˇ

ˇ

ˇ

ˇ

`

ˇ

ˇ

ˇ

ˇ

ynpκpRq

ynpκpR1q

SnpκpRq

1´ iSnpκpR1q

ˇ

ˇ

ˇ

ˇ

`

ˇ

ˇ

ˇ

ˇ

ynpκsRq

ynpκsR1q

SnpκsR1q

1´ iSnpκsR1q

ˇ

ˇ

ˇ

ˇ

`

ˇ

ˇ

ˇ

ˇ

ynpκsRq

ynpκsR1q

SnpκsRq

1´ iSnpκsR1q

ˇ

ˇ

ˇ

ˇ

ď 2

ˆ

eκsR

2n` 1

˙2n`1 ˆˇˇ

ˇ

ˇ

ynpκpRq

ynpκpR1q

ˇ

ˇ

ˇ

ˇ

`

ˇ

ˇ

ˇ

ˇ

ynpκsRq

ynpκsR1q

ˇ

ˇ

ˇ

ˇ

˙

.

By Lemma E.4, we have

ˇ

ˇ

ˇ

ˇ

ynpκpRq

ynpκpR1q´ynpκsRq

ynpκsR1q

ˇ

ˇ

ˇ

ˇ

“

c

R1

R

ˇ

ˇ

ˇ

ˇ

Yν`1pκpRq

Yν`1pκpR1q´Yν`1pκsRq

Yν`1pκsR1q

ˇ

ˇ

ˇ

ˇ

ď7

3

κspκs ´ κpq

n` 1R`

R´R1˘

ˆ

R1

R

˙n`1

.

30 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

Therefore,ˇ

ˇ

ˇ

ˇ

ˇ

hp1qn pκpRq

hp1qn pκpR1q

´hp1qn pκsRq

hp1qn pκsR1q

ˇ

ˇ

ˇ

ˇ

ˇ

ď7

3

κspκs ´ κpq

n` 1R`

R´R1˘

ˆ

R1

R

˙n`1

` 2

ˆ

eκsR

2n` 1

˙2n`1 ˆˇˇ

ˇ

ˇ

ynpκpRq

ynpκpR1q

ˇ

ˇ

ˇ

ˇ

`

ˇ

ˇ

ˇ

ˇ

ynpκsRq

ynpκsR1q

ˇ

ˇ

ˇ

ˇ

˙

ď14

3

κspκs ´ κpq

nR`

R´R1˘

ˆ

R1

R

˙n`1

,

which completes the proof.

Lemma E.6. Given κp, κs, and R, it holds for sufficiently large n that

ΛnpRq “ ´`

κ2p ` κ

2s

˘ R2

2`O

ˆ

1

n

˙

. (E.9)

Proof. It is clear to note for sufficiently large n that

zp1qn ptq “thp1q1

n ptq

hp1qn ptq

„typ1q1

n ptq

yp1qn ptq

.

We have from the recursive equation of the spherical Bessel functions that

typ1q1

n ptq

yp1qn ptq

“ t

„

yn´1ptq

ynptq´n` 1

t

“ t

„

Yνptq

Yν`1ptq´ν ` 1

t

´1

2“ Gν`1ptq ´

1

2.

By Lemma E.3, we get

zp1qn ptq “ Gν`1ptq ´1

2ď ´n´ 1`

t2

2n`

11

12

t2

n2`O

ˆ

1

n3

˙

,

zp1qn ptq “ Gν`1ptq ´1

2ě ´n´ 1`

t2

2n´

5

12

t2

n2`O

ˆ

1

n3

˙

,

which give

´ n´ 1`t2

2n´

5

12

t2

n2`O

ˆ

1

n3

˙

ď zp1qn ptq ď ´n´ 1`t2

2n`

11

12

t2

n2`O

ˆ

1

n3

˙

. (E.10)

The proof is completed by substituting (E.10) into (D.5).

Appendix F. The dual problems

The dual problems in (4.13) for the Helmholtz equation and the Maxwell equation are consideredin [3] and [4], respectively. For the self-contained purpose, we summarize the related results here.

Lemma F.1. Let ζ “ 1λ`2µζ. The boundary value problem

$

’

&

’

%

∆g ` κ2pg “ ´ζ in BRzBR1 ,

Bρg “ T ˚1 g on BBR,

g “ g on BBR1

has a unique solution given by

gmn pρq “ Spnpρqgmn pR

1q `iκp2

ż ρ

R1t2W p

npρ, tqζmn ptqdt`

iκp2

ż R

R1t2SpnptqW

pnpR

1, ρqζmn ptqdt, (F.1)

AN ADAPTIVE FINITE ELEMENT DTN METHOD 31

where gmn and ζmn are the Fourier coefficients of g and ζ with respect to the basis functions Xmn , and

Spnpρq “hp2qn pκpρq

hp2qn pκpR1q

, W pnpρ, tq “ det

«

hp1qn pκpρq h

p2qn pκpρq

hp1qn pκptq h

p2qn pκptq

ff

.

Taking ρ “ R in (F.1) yields

gmn pRq “ SpnpRqgmn pR

1q `iκp2

ż R

R1t2SpnpRqW

pnpR

1, tqζmn ptqdt. (F.2)

Taking the derivative of (F.1) and estimating it at ρ “ R1, we get

gm1

n pR1q “

1

R1zp2qn pκpR

1qgmn pR1q `

2κpπR1

ż R

R1t2Spnptqζ

mn ptqdt. (F.3)

Lemma F.2. Let Zm2n “1µZ

m2n, where Zm2n are the Fourier coefficients of Z under the basis V m

n . The

two-point boundary value problem$

’

’

&

’

’

%

qm2

2n pρq `2ρqm12n pρq `

´

κ2s ´

npn`1qρ2

¯

qm2npρq “ ´Zm2npρq, ρ P pR1, Rq,

qm1

2n pRq ´zp2qn pκsRqR qm2npRq “ 0, ρ “ R,

qm2npR1q “ qm2npR

1q, ρ “ R1

has a unique solution given by

qm2npρq “ Ssnpρqqm2npR

1q `iκs2

ż ρ

R1t2W s

npρ, tqZm2nptqdt`

iκs2

ż R

R1t2SsnptqW

snpR

1, ρqZm2nptqdt, (F.4)

where

Ssnpρq “hp2qn pκsρq

hp2qn pκsR1q

, W snpρ, tq “ det

«

hp1qn pκsρq h

p2qn pκsρq

hp1qn pκstq h

p2qn pκstq

ff

.

Taking ρ “ R in (F.4) gives

qm2npRq “ SsnpRqqm2npR

1q `iκs2

ż R

R1t2SsnpRqW

snpR

1, tqZm2nptqdt. (F.5)

Taking the derivative of (F.4) and estimating it at ρ “ R1, we obtain

qm1

2n pR1q “

1

R1zp2qn pκsR

1qqm2npR1q `

2κsπR1

ż R

R1t2SsnptqZ

m2nptqdt. (F.6)

Lemma F.3. Let vmn pρq “ ρqm3npρq. Then vmn satisfies the two-point boundary value problem$

’

’

&

’

’

%

vm2

n pρq ` 2ρv

m1n pρq `

´

κ2s ´

npn`1qρ2

¯

vmn pρq “ ´βmn pρq, ρ P pR1, Rq,

vm1

n pRq ´zp2qn pκsRqR vmn pRq “ 0, ρ “ R,

vmn pR1q “ vmn pR

1q, ρ “ R1,

where βmn pρq “1µρZ

m3npρq. Moreover, vmn pRq and vm

1

n pR1q are given by

vmn pRq “ SsnpRqvmn pR

1q `iκs2

ż R

R1t2SsnpRqW

snpR

1, tqβmn ptqdt,

vm1

n pR1q “

1

R1zp2qn pκsR

1qvmn pR1q `

2κsπR1

ż R

R1t2Ssnptqβ

mn ptqdt.

32 GANG BAO, PEIJUN LI, AND XIAOKAI YUAN

Let Zm3n “1µZ

m3n. We have from Lemma F.3 that

qm3npRq “R1

RSsnpRqq

m3npR

1q `iκs2R

ż R

R1t3SsnpRqW

snpR

1, tqZm3nptqdt, (F.7)

qm1

3n pR1q “

1

R1

”

zp2qn pκsR1q ´ 1

ı

qm3npR1q `

2κsπpR1q2

ż R

R1t3SsnptqZ

m3nptqdt. (F.8)

References

[1] I. Babuska and A. Aziz, Survey Lectures on Mathematical Foundations of the Finite Element Method, in theMathematical Foundation of the Finite Element Method with Application to the Partial Differential Equations,ed. by A.Aziz, Academic Press, New York, 1973, 5–359.

[2] G. Bao, G. Hu, J. Sun, and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. PuresAppl., 117 (2018), 263–301.

[3] G. Bao, M. Zhang, B. Hu, and P. Li, An adaptive finite element DtN method fro the three-dimensional acousticscattering problem, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 61–79.

[4] G. Bao, M. Zhang, X. Jiang, P. Li, and X. Yuan, An adaptive finite element DtN method for the electromagneticscattering problem, preprint.

[5] A. Bayliss and E. Turkel, Radiation boundary conditions for numerical simulation of waves, Comm. Pure Appl.Math., 33 (1980), 707–725.

[6] J. H. Bramble, J. E. Pasciak, and D. Trenev, Analysis of a finite PML approximation to the three dimensionalelastic wave scattering problem, Math. Comp., 79 (2010), 2079–2101.

[7] Z. Chen, X. Xiang, and X. Zhang, Convergence of the PML method for elastic wave scattering problems, Math.Comp., 85 (2016), 2687–2714.

[8] P. G. Ciarlet, Mathematical Elasticity, vol. I : Three-Dimensional Elasticity, Studies in Mathematics and itsApplications, North-Holland, Amsterdam, 1988.

[9] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.[10] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998.[11] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp.,

31 (1977), 629–651.[12] G. K. Gachter and M. J. Grote, Dirichlet-to-Neumann map for three-dimensional elastic waves, Wave Motion, 37

(2003), 293–311.[13] D. Givoli and J. B. Keller, Non-reflecting boundary conditions for elastic waves, Wave Motion, 12 (1990), 261–279.[14] M. Grote and J. B. Keller, On nonreflecting boundary conditions, J. Comput. Phys., 122 (1995), 231–243.[15] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251–265.[16] G. C. Hsiao, N. Nigam, J. E. Pasiak, and L. Xu, Error analysis of the DtN-FEM for the scattering problem in

acoustic via Fourier analysis, J. Comput. Appl. Math., 235 (2011), 4949–4965.[17] X. Jiang, P. Li, J. Lv, and W. Zheng, An adaptive finite element method for the wave scattering with transparent

boundary condition, J. Sci. Comput., 72 (2017), 936–956.[18] X. Jiang, P. Li, J. Lv, Z. Wang, H. Wu, and W. Zheng, An adaptive finite element DtN method for Maxwell’s

equations in biperiodic structures, arXiv:1811.12449.[19] X. Jiang, P. Li, and W. Zheng, Numerical solution of acoustic scattering by an adaptive DtN finite element

method, Commun. Comput. Phys., 13 (2013), 1277–1244.[20] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Theory of Elasticity, Oxford: Pergamon Press, 1986.[21] P. Li, Y. Wang, Z. Wang, and Y. Zhao, Inverse obstacle scattering for elastic waves, Inverse Problems, 32 (2016),

115018.[22] P. Li and X. Yuan, Inverse obstacle scattering for elastic waves in three dimensions, Inverse Problems and Imaging,

13 (2019), 545–573.[23] P. Li and X. Yuan, An adaptive finite element DtN method for the elastic wave scattering problem,

arXiv:1903.03606.[24] Y. Li, W. Zheng, and X. Zhu, A CIP-FEM for high-frequency scattering problem with the truncated DtN boundary

condition, CSIAM Trans. Appl. Math., 1 (2020), 530–560.[25] M. Grote and C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comput.

Phys., 201 (2004), 630–650.[26] P. Monk, Finite Element Methods for Maxwell’s Equations, Oxford University Press, 2003.[27] J.-C. Nedelec, Acoustic and Electromagnetic Equations Integral Representations for Harmonic Problems, Springer-

Verlag, New York, 2001.[28] L. Rondi and M. Sini, Stable determination of a scattered wave from its farfield pattern: the high frequency

asymptotics, Arch. Ration. Mech. Anal., 218 (2015), 1–54.

AN ADAPTIVE FINITE ELEMENT DTN METHOD 33

[29] A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comp., 28(1974), 959–962.

[30] H. Thomas, Radiation boundary conditions for the numerical simulation of waves, Acta Numer., 8 (1999), 47–106.[31] Z. Wang, G. Bao, J. Li, P. Li, and H. Wu, An adaptive finite element method for the diffraction grating problem

with transparent boundary condition, SIAM J. Numer. Anal., 53 (2015), 1585–1607.[32] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1995.[33] X. Yuan, G. Bao, and P. Li, An adaptive finite element DtN method for the open cavity scattering problems,

CSIAM Trans. Appl. Math., 1 (2020), 316-345.

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China.Email address: baog@zju.edu.cn

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA.Email address: lipeijun@math.purdue.edu

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China.Email address: yuan170@zju.edu.cn