static.uni-graz.at · XXXX, 1–59 © De Gruyter YYYY High-order topological expansions for...

XXXX, 1–59 © De Gruyter YYYY

High-order topological expansions forHelmholtz problems in 2d

Victor A. Kovtunenko

Abstract. Methods of topological analysis are inherently related to singular perturbations. Fortopology variation, a trial geometric object put in a test domain is examined by reducing theobject size from a finite to an infinitesimal one. Based on the singular perturbation of the for-ward Helmholtz problem, a topology optimization approach, which is a direct one, is describedfor the inverse problem of object identification from boundary measurements.

Relying on the 2d setting in a bounded domain, the high-order asymptotic result is provedrigorously for the Neumann, Dirichlet, and Robin type conditions stated at the object boundary.In particular, this implies the first-order asymptotic term called a topological derivative. Foridentifying arbitrary test objects, a variable parameter of the surface impedance is successful.The necessary optimality condition of minimum of the objective function with respect to trialgeometric variables is discussed and realized for finding the center of the test object.

Keywords. Inverse problem, object identification, shape and topology optimization, topologi-cal derivative, derivative-free optimality condition, singular perturbation, asymptotic analysis,variational method, Helmholtz problem.

AMS classification. 35R30, 35Q93, 49Q10, 65K10.

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Background Helmholtz problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Helmholtz problems for geometric objects under Neumann (sound hard) bound-ary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Helmholtz problems for geometric objects under Dirichlet (sound soft) boundarycondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Helmholtz problems for geometric objects under Robin (impedance) boundarycondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

1 Introduction

We consider both the forward and inverse Helmholtz problems with respect to a vari-able geometric object put in a bounded test domain. It has numerous applications for

The results were obtained with the support of the Austrian Science Fund (FWF) in the framework of theproject P26147-N26: "Object identification problems: numerical analysis" (PION) and NAWI Graz.

2 V. A. Kovtunenko

testing methodologies by scattering with acoustic, elastic, and electromagnetic waves.The literature on this subject is numerous, so we give selected references only.

The classic methods of analysis available for the Helmholtz problem, see [16, 20,22, 39, 58], are based mostly on the potential operator theory which is well estab-lished in case of unbounded domains. Possible approaches to inverse problems canbe distinguished to iterative, see [23, 35], as well as non-iterative ones called direct.Within direct approaches to the inverse scattering, there are well known sampling,probe, factorization, singular source, MUSIC-type, and other relevant techniques, e.g.[2, 4, 27, 32, 50].

The inverse Helmholtz problem of identification of an unknown geometric objectbelongs to the field of shape and topology optimization [1, 24, 25, 34, 42, 44, 55],as well as parameter estimation, see [10, 15, 40, 48] for the common methods here.Recently, the concept of topological derivatives was adapted to this field, e.g. in [6,11, 56]. The reason is that a trial object of finite size in comparison to infinitesimalone implies variation of topology of the test domain. For the analysis of topologicalchanges we refer to the methods of singular perturbations in [28, 33, 49].

The high-order topological expansions were considered e.g. in [5, 13, 17, 18, 30] forthe Poisson equation, and in [47] for a screened Poisson equation. They were the sub-ject of discussion in [12, 54]. Regarding the inverse problems, in [17, 18] the Kohn–Vogelius cost functional over a domain was suggested as alternative to the boundarymisfit functional. Numerically, the high-order terms usually have a high computa-tional cost because they require to solve a PDE at every point where the topologicalderivative is computed.

To derive even formally asymptotic representations under singular perturbations isitself a hard task requiring a huge number of fine calculations. Its rigorous mathemat-ical justification is the challenging problem. In this chapter, we obtain the high-orderasymptotic formula for solutions of the singularly perturbed state problems and thecorresponding optimal value functions. The first-order term implying a topologicalderivative is the particular case here. All subsequent asymptotic expansions are provedby rigorous estimates of the residual error in appropriate function spaces.

Our consideration addresses the topology optimization problem for identificationand reconstruction of arbitrary geometries. By “arbitrary geometry” we mean the im-plicitly defined set of geometric variables which is parameterized by admissible triplesof shape - center - size. Such general geometric assumptions can be treated within aproper variational formulation.

We provide the underlying singularly perturbed, forward and inverse, Helmholtzproblem with suitable primal and dual variational principles according to the Fenchel–Legendre duality. The asymptotic terms are expressed by auxiliary Helmholtz prob-lems given in bounded domains as well as boundary layers described by the Laplaceproblems in exterior domains. They all are stated in the weak form in proper Sobolevspaces. In the exterior domains, weighted Sobolev spaces are useful.

A crucial issue of the topological analysis concerns the fact that boundary conditions

High-order topological expansions for Helmholtz problems in 2d 3

of the test object should be prescribed a-priori. In the present chapter, relying on a 2dspatial setting, in a unified way we consider the Helmholtz problem under Dirichlet(sound soft), Neumann (sound hard), and Robin (impedance) conditions stated at theobject boundary, respectively, in Sections 3, 4, and 5. The background problem in thereference domain is described in Section 2.

From the perspective of topology optimization, the geometric variables enter theobjective in a fully implicit way through the geometry-dependent state problem. Thisfact does not allow to find optimality conditions based on directional derivatives ofthe objective. Nevertheless, in Section 5.4 we show that an unknown parameter of theboundary impedance is well suitable for the purpose of variation of trial geometries.In fact, passing its imaginary part to the limit in the asymptotic expansion of the opti-mal value function, this gets a necessary optimality condition determining the optimalcenter, and this condition is derivative-free.

We start with the description of the reference configuration.

2 Background Helmholtz problemLet Ω ⊂ R2 be a reference domain with the Lipschitz boundary ∂Ω. With ν =(ν1, ν2)

> the unit normal vector at the boundary ∂Ω and outward to Ω is denoted.Let ∂Ω = ΓN ∪ ΓD consist of two nonempty, mutually disjoint, parts ΓN and ΓDassociated to the Neumann and Dirichlet boundary conditions, respectively.

For the fixed Neumann and Dirichlet data g ∈ L2(ΓN ;C) and h ∈ H1/2(ΓD;C),and for a wave number k ∈ R+, the reference (called background) Helmholtz problemfor the wave potential u0(x), x = (x1, x2)> ∈ Ω, is given by:

−[∆ + k2]u0 = 0 in Ω, (2.1a)

∂u0

∂ν = g on ΓN , (2.1b)

u0 = h on ΓD. (2.1c)

In (2.1) and in what follows, ∆ is the Laplace operator, the usual notation ∂∂ν := ν ·∇ =ν>∇ stands for the normal derivative, " · " means for the inner product of vectors, ∇is the gradient, and the upper > denotes transposition swapping columns and rows.

In the weak form, (2.1) is described by the following variational problem: Findu0 ∈ H1(Ω;C) such that

u0 = h on ΓD, (2.2a)∫Ω(∇u0 · ∇u− k2u0u) dx =

∫ΓNgu dSx

for all u ∈ H1(Ω;C) : u = 0 on ΓD.(2.2b)

Here u = Re(u) − ıIm(u) is the complex conjugate of u = Re(u) + ıIm(u), and ı isthe imaginary unit.

4 V. A. Kovtunenko

Proposition 2.1. For the strong solution to (2.1), the boundary value problem (2.1) andthe variational equation (2.2) are equivalent. There exists the unique weak solutionu0 to (2.2). Moreover, it implies the first-order necessary optimality condition for theCherkaev–Gibiansky variational principle:

P(u0) = minRe(v)

maxIm(v)P(v) over v ∈ H1(Ω;C) : v = h on ΓD (2.3)

with the Lagrangian P : H1(Ω;C) 7→ R of the form

P(v) = Re{

12

∫Ω(∇v · ∇v − k2v2) dx−

∫ΓNgv dSx

}. (2.4)

Proof. The weak formulation can be verified by standard variational arguments. In-deed, (2.2) is derived by multiplying the equation (2.1a) with a test function u ∈H1(Ω;C) and subsequent integration by parts over Ω due to boundary conditions(2.1b) and u = 0 on ΓD.

In return, the following Green’s formula holds for every function u ∈ H1(Ω;C)such that ∆u ∈ L2(Ω;C):∫

Ω(∇u · ∇u+ u∆u) dx = 〈∂u∂ν , u〉ΓN

for all u ∈ H1(Ω;C) : u = 0 on ΓD.(2.5)

Here 〈∂u∂ν , u〉ΓN stands for the duality pairing between u ∈ H1/200 (ΓN ;C) and

∂u∂ν ∈

H1/200 (ΓN ;C)

? in the Lions–Magenes dual space, see e.g. [36, Section 1.4] for detail.With the help of (2.5), we get from (2.2b)

−∫

Ω([∆ + k2]u0)u dx = 〈g − ∂u0∂ν , u〉ΓN , 〈g, u〉ΓN =

∫ΓNgu dSx

and derive (2.1a) and (2.1b) by fundamental lemma of the calculus of variations whenvarying the test function u such that first u = 0 on ΓN and then u 6= 0 on ΓN .

The unique solvability of the variational equation (2.2) can be argued as usuallywith a Garding inequality and injectivity by using the Fredholm alternative, see e.g.[46, Theorem 5.2, Chapter 3].

Now rewriting (2.4) component-wisely for v = Re(v) + ıIm(v) as

P(v) = 12∫

Ω

{|∇(Re(v))|2 − |∇(Im(v))|2

− k2(Re(v)2 − Im(v)2

)}dx−

∫ΓN

(Re(g)Re(v)− Im(g)Im(v)

)dSx

and differentiating P(v) with respect to Re(v) and Im(v), the necessary optimalitycondition for (2.3) implies two variational inequalities〈

∂∂Re(v)P(u

0),Re(v − u0)〉≥ 0,

〈∂

∂Im(v)P(u0), Im(v − u0)

〉≤ 0


holding for all v ∈ H1(Ω;C) such that v = h on ΓD. Inserting here v = u0 ± u withu ∈ H1(Ω;C) such that u = 0 on ΓD we get the following two variational equations:∫

Ω

(∇Re(u0) · ∇Re(u)− k2Re(u0)Re(u)

)dx =

∫ΓN

Re(g)Re(u) dSx,∫Ω

(∇Im(u0) · ∇Im(u)− k2Im(u0)Im(u)

)dx =

∫ΓN

Im(g)Im(u) dSx.

The summation of these equations for u = u and for u = ıu constitutes respectivelythe real and the imaginary parts of (2.2b), see (5.5a). This completes the proof.

In the next section we give a local representation of the solution to (2.2) in thenear-field of a trial point, which is called "inner" asymptotic expansion.

2.1 Inner asymptotic expansion by Fourier series in near-field

For a fixed center x0 ∈ Ω, a local polar coordinate system associated to x0 can beintroduced with the help of the polar radius ρ ∈ R+ and the polar angle θ ∈ (−π, π]such that x− x0 = ρx̂, that is

ρ := |x− x0|, x̂ := x−x0|x−x0| = (cos θ, sin θ)>, x̂′ = (− sin θ, cos θ)>. (2.6)

We set R > 0 such that BR(x0) ⊂ Ω, where BR(x0) denotes the ball of radius R andcenter x0.

In what follows, Bessel functions of the first kind Jn(kρ) and the second kind (calledthe Neumann functions) Yn(kρ), n ∈ N0, will be employed for the argument kρ. Thesefunctions are two linearly independent solutions of the Bessel equation:

(un)′′ρ +

1ρ(un)

′ρ +

(k2 − n2

ρ2

)un = 0 for kρ 7→ un : R+ 7→ R. (2.7)

In particular, J0, J1, and Y0 yield the expansions for kρ↘ +0:

J0(kρ) =∞∑m=0

(−1)m(m!)2 (

kρ2 )

2m = 1 + a0(kρ), a0(kρ) = − (kρ)2

4 + O((kρ)4), (2.8a)

J1(kρ) = −J ′0(kρ) = 12(kρ+ a1(kρ)), a1(kρ) = −(kρ)3

8 + O((kρ)5), (2.8b)

Y0(kρ) =2π (ln

kρ2 + γ)J0(kρ) + a2(kρ), a2(kρ) = O((kρ)

2) (2.8c)

with the Euler constant γ > 0. Using (2.6)–(2.8) we prove the following truncatedFourier series.

Lemma 2.2. The solution u0 of (2.2) admits the local asymptotic representation

u0(x) = u0(x0)J0(kρ) + U00 (x) for x ∈ BR(x0) ⊂ Ω (2.9)

6 V. A. Kovtunenko

holding in the near-field, with the residual U00 ∈ H1(BR(x0);C) such that∫ π−πU00 dθ = 0, (2.10a)

U00 (x0 + ρx̂) = O(ρ) for ρ ∈ [0, R), θ ∈ (−π, π]. (2.10b)

Proof. In the ball Bδ(x0) with δ ∈ [0, R) we set

u00(ρ) :=1

2π

∫ π−πu0 dθ and U00 := u

0 − u00, (2.11)

thus decomposing u0 into the radial u00 and residual U00 functions:

u0(x) = u00(ρ) + U00 (x) for x ∈ Bδ(x0). (2.12)

According to (2.11), the residual U00 ∈ H1(BR(x0);C) and it fulfills (2.10a).Using (2.12) and (2.10a), the substitution of a smooth cut-off function η(ρ) sup-

ported in Bδ(x0) as the test function u = η into (2.2b) and integration by parts gives

0 =∫Bδ(x0)

(∇u0 · ∇η − k2u0η) dx

=

∫ δ0

{∂∂ρ

(∫ π−π

(u00 + U00 ) dθ

)η′ − k2

∫ π−π

(u00 + U00 ) dθ η

}ρdρ

= 2π∫ δ

0

((u00)

′ρη′ − k2u00η

)ρdρ = −2π

∫ δ0

((ρ(u00)

′ρ

)′ρ+ k2ρu00

)η dρ

for all η. This results in the Bessel equation (2.7) for n = 0:

(u00)′′ρ +

1ρ(u

00)′ρ + k

2u00 = 0 for ρ ∈ (0, δ). (2.13)

Its general solution has the form

u00(ρ) = K00J0(kρ) + S

00Y0(kρ), K

00 , S

00 ∈ C.

But the Neumann function Y0(kρ) = O(| ln ρ|) in (2.8c) disagrees the inclusion u00 ∈H1((0, δ);C) in (2.11), hence S00 = 0 and

u00(ρ) = K00J0(kρ), K

00 ∈ C. (2.14)

To justify (2.10b) we apply the Saint–Venant estimate to the residual U00 in (2.12).Equation (2.13) implies −[∆ + k2]u00 = 0 which together with (2.1a) yields the

Helmholtz equation for the residual

−[∆ + k2]U00 = 0 in Bδ(x0). (2.15)


With the help of (2.15) after integration by parts we have

I(δ) :=∫Bδ(x0)

|∇U00 |2 dx =∫Bδ(x0)

k2|U00 |2 dx+∫∂Bδ(x0)

∂U00∂ρ U

00 dSx. (2.16)

Thanks to (2.10a), the Poincare and the Wirtinger (cf. (2.29)) inequalities hold:∫Bδ(x0)

|U00 |2 dx ≤(2δ)2

π2

∫Bδ(x0)

|∇U00 |2 dx, (2.17a)

∫ π−π|U00 |2 dθ ≤

∫ π−π

∣∣∂U00∂θ

∣∣2 dθ. (2.17b)From (2.17b) we can estimate the boundary integral in (2.16) as∫

∂Bδ(x0)

∂U00∂ρ U

00 dSx =

∫ π−π

∂U00∂ρ U

00 δdθ ≤

∫ π−π

(δ2

∣∣∂U00∂ρ

∣∣2 + 12δ |U00 |2) δdθ≤∫ π−π

(δ2

∣∣∂U00∂ρ

∣∣2 + 12δ ∣∣∂U00∂θ ∣∣2) δdθ = δ2 ∫∂Bδ(x0)

|∇U00 |2 dSx.

Therefore, together with (2.17a) and the co-area formula

ddδ

∫Bδ(x0)

|∇U00 |2 dx =∫∂Bδ(x0)

|∇U00 |2 dSx, (2.18)

from (2.16) we get the differential inequality for I(δ):(1− k2 4δ2

π2

)I(δ) ≤ δ2

ddδ I(δ) (2.19)

Integrating (2.19) with respect to δ ∈ (r,R) as

ln( I(R)I(r)

)=

∫ Rr

dII ≥

∫ Rr

( 2δ −

8k2π2δ)dδ = ln

(Rr

)2 − 4k2π2

(R2 − r2)

we derive the resulting estimate∫Br(x0)

|∇U00 |2 dx = I(r) ≤(rR

)2I(R)e

4k2

π2(R2−r2)

= O(r2). (2.20)

Due to the fundamental theorem of calculus and using homogeneity argument, thefunction oscillation in R2 can be estimated from above (see e.g. [38]) as

supx,y∈Br(x0)

|U00 (x)− U00 (y)|2 ≤ C∫Br(x0)

(|∇U00 |2 + r2|∆U00 |2) dx

= C

∫Br(x0)

(|∇U00 |2 + r2k2|U00 |2

)dx, (C > 0)

(2.21)

8 V. A. Kovtunenko

where we have used (2.15). Combining estimates (2.17a) for δ = ρ with (2.20) and(2.21) for r = ρ, where ρ ∈ [0, R), we derive that

U00 (x0 + ρx̂) = U00 (x0) + O(ρ) in BR(x0).

But U00 (x0) = 0 due to (2.10a), thus following (2.10b). Now passing ρ ↘ +0 in(2.12), in view of (2.8a) and (2.10b) from (2.14) we find the factor K00 = u

0(x0),hence (2.9) holds. This completes the proof.

We generalize Lemma 2.2 to the high-order inner asymptotic expansion below, seethe related result in [49].

Proposition 2.3. For every N ∈ N, the solution u0 of (2.2) admits the following localasymptotic representation in the near-field BR(x0) ⊂ Ω:

u0(x) = K00J0(kρ) +N∑n=1

Jn(kρ)K0n · x̂n + U0N (x), (2.22)

where the notation x̂n := (cos(nθ), sin(nθ))> with convention x̂1 = x̂ for n = 1, andK0n =

((K0n)1, (K

0n)2)∈ C2. The residual U0N ∈ H1(BR(x0);C) is such that∫ π

−πU0N dθ = 0,

∫ π−πU0N x̂

n dθ = 0 for n = 1, . . . , N, (2.23a)

U0N (x0 + ρx̂) = O(ρN+1) for ρ ∈ [0, R), θ ∈ (−π, π]. (2.23b)

Proof. Starting with u00 and U00 given in (2.9) and (2.10), for every n = 1, . . . , N we

set recursively the radial and residual functions:

u0n(ρ) :=1π

∫ π−πu0x̂n dθ and U0n := U

0n−1 − u0n · x̂n, (2.24)

which constitute the local asymptotic representation with δ ∈ [0, R):

u0(x) = u00(ρ) +

N∑n=1

u0n(ρ) · x̂n + U0N (x) for x ∈ Bδ(x0). (2.25)

According to (2.24), all the residuals U0n, n = 1, . . . , N , fulfill∫ π−πU0n dθ = 0,

∫ π−πU0nx̂

m dθ = 0 for m = 1, . . . , n, (2.26)

and (2.26) for n = N implies (2.23a).We show that every radial function u0n satisfies the respective Bessel equation (2.7).


Indeed, for every n = 1, . . . , N , plugging into (2.2b) the test vector-function u =x̂nη(ρ) supported in Bδ(x0) and the representation u0 = u00 +

∑nm=1 u

0m · x̂m + U0n

according to (2.25), recalling trigonometric calculus for x̂n = (x̂n1 , x̂n2 )>:

∂x̂n1∂θ = −nx̂

n2 ,

∂x̂n2∂θ = nx̂

n1 ,

∫ π−πx̂n dθ = 0, n,m ∈ N,

∫ π−πx̂mi x̂

nj dθ =

{π for m = n and i = j0 otherwise

, i, j = 1, 2(2.27)

and using orthogonality conditions in (2.23a), it succeeds in

0 =∫Bδ(x0)

(∇u0 · ∇(x̂nη)− k2u0x̂nη

)dx =

∫ δ0

{∂∂ρ

(∫ π−π

(u00 +n∑

m=1

u0m · x̂m

+ U0n)x̂n dθ

)η′ + η

ρ2

∫ π−π

( n∑m=1

u0m · ∂x̂m

∂θ +∂U0n∂θ

)∂x̂n

∂θ dθ − k2η

∫ π−π

(u00

+

n∑m=1

u0m · x̂m + U0n) dθ}ρdρ = π

∫ δ0

((u0n)

′ρη′ + n

2

ρ2u0nη − k2u0nη

)ρdρ.

After integration by parts we obtain the Bessel equation (2.7) possessing the generalsolution

u0n(ρ) = K0nJn(kρ), K

0n ∈ C2 (2.28)

since the Neumann function Yn(kρ) is singular for ρ ↘ +0 and disagrees the H1-regularity of the solution u0.

It remains to prove (2.23b). For this task, since (2.23a) holds, we refine the Wirtingerinequality (compare with (2.17b)):∫ π

−π|U0N |2 dθ ≤ 1(N+1)2

∫ π−π

∣∣∂U0N∂θ

∣∣2 dθ. (2.29)Indeed, employment of the Fourier series

U0N = cN0 +

∞∑n=1

cNn · x̂n, cN0 := 12π∫ π−πU0N dθ, c

Nn :=

1π

∫ π−πU0N x̂

n dθ

with scalar cN0 ∈ C and vectors cNn ∈ C2, together with the derivative

∂U0N∂θ =

∞∑n=1

ncNn · (x̂n)′, where (x̂n)′ = (− sin(nθ), cos(nθ))>,

this leads to the following Parseval identities

12π

∫ π−π|U0N |2 dθ = |cN0 |2 + 12

∞∑n=1

|cNn |2, 12π∫ π−π

∣∣∂U0N∂θ

∣∣2 dθ = 12 ∞∑n=1

n2|cNn |2.

10 V. A. Kovtunenko

Conditions (2.23a) imply cN0 = · · · = cNN = 0 that allows us to estimate∫ π−π|U0N |2 dθ = π

∞∑n=N+1

|cNn |2 ≤ π(N+1)2∞∑

n=N+1

n2|cNn |2 = 1(N+1)2∫ π−π

∣∣∂U0N∂θ

∣∣2 dθ,thus concluding with (2.29).

Due to (2.29) the boundary integral in IN (δ) introduced below can be estimated as∫∂Bδ(x0)

∂U0N∂ρ U

0N dSx ≤

∫ π−π

(δ

2(N+1)

∣∣∂U0N∂ρ

∣∣2 + N+12δ |U0N |2)δdθ≤∫ π−π

(δ

2(N+1)

∣∣∂U0N∂ρ

∣∣2 + 12δ(N+1) ∣∣∂U0N∂θ ∣∣2)δdθ = δ2(N+1) ∫∂Bδ(x0)

|∇U0N |2 dSx.

Therefore, considering similar to (2.16) the residual in the energy norm

IN (δ) :=∫Bδ(x0)

|∇U0N |2 dx =∫Bδ(x0)

k2|U0N |2 dx+∫∂Bδ(x0)

∂U0N∂ρ U

0N dSx,

applying here the Poincare inequality from (2.17a) and the co-area formula from (2.18),we derive the corresponding differential inequality(

1− k2 4δ2π2

)IN (δ) ≤ δ2(N+1)

ddδ IN (δ). (2.30)

Integration of (2.30) over δ ∈ (r,R) leads to the estimate

0 ≤ IN (r) ≤(rR

)2(N+1)I(R)e

4(N+1)k2

π2(R2−r2)

= O(r2(N+1)). (2.31)

Applying to U0N (x0 + ρx̂) the point-wise estimate in the manner of (2.21), hence

|U0N (x0 + ρx̂)|2 ≤ CIN (ρ) with C > 0, (2.32)

from (2.31) and (2.32) it follows (2.23b). The proof is completed.

As the consequence, below we specify Proposition 2.3 for N = 1, where K00 =u0(x0) and K01 =

2k∇u

0(x0) are calculated due to (2.8a) and (2.8b).

Corollary 2.4. The solution u0 of (2.2) admits the first-order local asymptotic repre-sentation in the near-field BR(x0) ⊂ Ω as

u0(x) = u0(x0)J0(kρ) +2kJ1(kρ)∇u

0(x0) · x̂+ U01 (x) (2.33)

with the residual U01 ∈ H1(BR(x0);C) such that∫ π−πU01 dθ =

∫ π−πU01 x̂ dθ = 0, (2.34a)


U01 (x0 + ρx̂) = O(ρ2) for ρ ∈ [0, R), θ ∈ (−π, π]. (2.34b)

Moreover, the differentiation in (2.33) according to (2.6) and using the rule

∂∂x1

= cos θ ∂∂ρ −sin θρ

∂∂θ ,

∂∂x2

= sin θ ∂∂ρ +cos θρ

∂∂θ

yields the following formula for the gradient in BR(x0):

∇u0(x) = ∇u0(x0) + b0u(x) +∇U01 (x), ∇U01 = O(ρ) (2.35a)

with the term b0u(x) :=(u0(x0)ka

′0(kρ) + a

′1(kρ)∇u0(x0) · x̂

)x̂

+ a1(kρ)kρ (∇u0(x0) · x̂′)x̂′, b0u = O(ρ).

(2.35b)

In the next sections we proceed with the "outer" asymptotic expansion which willbe given in the far-field with respect to a test geometric object put in the referencedomain. We will see that it depends crucially on conditions imposed at the boundaryof the test object. The Neumann, Dirichlet, and Robin boundary conditions will beconsidered separately in Sections 3, 4, and 5, respectively.

3 Helmholtz problems for geometric objects underNeumann (sound hard) boundary condition

We start with the geometric description of a test object (inclusion, obstacle, scatterer).Let ω stand for a generic geometric shape implying the compact set in R2 with the

piecewise Lipschitz boundary ∂ω and the normal vector ν = (ν1, ν2)> outward to ω.We require that 0 ∈ ω and the unit ball B1(0) separating the near and far fields is theminimum enclosing ball centered at origin 0 such that ω ⊂ B1(0). Consequently, theshapes are invariant to translations and isotropic scaling. We call by Gω the set of suchshapes ω.

Definition 3.1. Rescaling a shape ω ∈ Gω by a size parameter ε > 0, it produces afamily of admissible geometric objects

ωε(x0) ={x ∈ R2 : x−x0ε ∈ ω

}⊂ Bε(x0) (3.1)

posed at a center x0 ∈ R2.

Such admissible geometries in (3.1) admit parametrization by the triple of geometricvariables (ω, ε, x0) ∈ Gω × R+ × R2. In particular, the shape ω itself is equal to theparametrized object ω1(0).

Definition 3.2. We call by admissible geometries G = Gω×Gε×Gx in the referencedomain Ω those triples: the shape ω ∈ Gω, the size ε ∈ Gε ⊂ R+, and the centerx0 ∈ Gx ⊂ Ω of objects ωε(x0) in (3.1) which satisfy the consistency condition:

ωε(x0) ⊂ Bε(x0) ⊂ Ω. (3.2)

12 V. A. Kovtunenko

The set G of admissible geometries in (3.2) will be used further for the sake ofshape variation of test objects in the reference domain Ω. Moreover, passing ε ↘ +0diminishes the object and represents the topology change.

For forward problems, we fix the object ωε(x0) in Ω, or, equivalently, (ω, ε, x0) ∈G, and we mark the dependence of functions on the size ε for the subsequent asymp-totic analysis as ε↘ +0.

Given the boundary data g ∈ L2(ΓN ;C) and h ∈ H1/2(ΓD;C), the Neumann(sound hard) problem for the Helmholtz equation (compare with the background prob-lem (2.1)) is considered for the wave potential uε(x), x ∈ Ω \ ωε(x0), satisfying:

−[∆ + k2]uε = 0 in Ω \ ωε(x0), (3.3a)

∂uε

∂ν = 0 on ∂ωε(x0), (3.3b)∂uε

∂ν = g on ΓN , (3.3c)

uε = h on ΓD. (3.3d)

The weak solution to (3.3) is described by the variational problem: Find uε ∈ H1(Ω \ωε(x0);C) such that

uε = h on ΓD, (3.4a)∫Ω\ωε(x0)

(∇uε · ∇u− k2uεu) dx =∫

ΓNgu dSx

for all u ∈ H1(Ω \ ωε(x0);C) : u = 0 on ΓD.(3.4b)

For every wave number k ∈ R+, which can be also large, the well-posedness of prob-lem (3.4) can be argued by using the Fredholm alternative similarly to Proposition 2.1.Moreover, (3.4) follows necessarily from the corresponding variational principle (com-pare with (2.3) and (2.4)):

Pε(uε) = minRe(v)

maxIm(v)Pε(v) over v ∈ H1(Ω \ ωε(x0);C) : v = h on ΓD (3.5)

with the Lagrangian Pε : H1(Ω \ ωε(x0);C) 7→ R given by

Pε(v) = Re{

12

∫Ω\ωε(x0)

(∇v · ∇v − k2v2) dx−∫

ΓNgv dSx

}. (3.6)

From the variational problem (3.4) we can determine the boundary traction ∂uε

∂ν in(3.3b) and (3.3c) in the weak sense using the following Green’s formula. By recallingthat ν denotes both the outer normal on ∂Ω and ∂ωε(x0), we have for every functionu ∈ H1(Ω \ ωε(x0);C) such that ∆u ∈ L2(Ω \ ωε(x0);C) (cf. (2.5)):∫

Ω\ωε(x0)(∇u · ∇u+ u∆u) dx = 〈∂u∂ν , u〉ΓN − 〈

∂u∂ν , u〉∂ωε(x0)

for all u ∈ H1(Ω \ ωε(x0);C) : u = 0 on ΓD.(3.7)


Here 〈∂u∂ν , u〉∂ωε(x0) denotes the duality pairing between u ∈ H1/2(∂ωε(x0);C) and

∂u∂ν ∈ H

−1/2(∂ωε(x0);C).In order to derive residual error estimates following further we require the uniform

inf-sup condition (see e.g. [41]): there exists β0 > 0 such that

0 < β0 ≤ infu

supv

∣∣∣∫Ω\ωε(x0)

(∇u · ∇v − k2uv) dx∣∣∣

‖u‖H1(Ω\ωε(x0);C)

‖v‖H1(Ω\ωε(x0);C)

(3.8)

for all u, v ∈ H1(Ω \ ωε(x0);C) : u = v = 0 on ΓD, admissible geometries(ω, ε, x0) ∈ G, and moderate wave numbers k ∈ [0, k0], k0 > 0. By (3.8) the well-posedness of (3.4) follows directly from the Babuska–Lions–Necas–Lax–Milgram the-orem, see [31, Section 4.4].

Proposition 3.3. The solutions u0 of (2.2) and uε of (3.4) satisfy the residual estimate

‖uε − u0‖H1(Ω\ωε(x0);C)

= O(ε). (3.9)

Proof. With the help of strong formulation (2.1) we derive from Green’s formula (3.7)the variational equation for u0 over the perturbed domain Ω \ ωε(x0) in the form∫

Ω\ωε(x0)(∇u0 · ∇u− k2u0u) dx =

∫ΓNgu dSx −

∫∂ωε(x0)

∂u0

∂ν u dSx

for all u ∈ H1(Ω \ ωε(x0);C) : u = 0 on ΓD.(3.10)

Here we have used the fact that ∂u0

∂ν ∈ L2(∂ωε(x0);C) since the solution u0 is locally

H2-smooth, see e.g. [46, Theorem 10.1, Chapter 3].Subtracting (3.10) from (3.4) and inserting the test function u = uε − u0 we have∫

Ω\ωε(x0)

(|∇(uε − u0)|2 − k2|uε − u0|2

)dx =

∫∂ωε(x0)

∂u0

∂ν (uε − u0) dSx.

Applying here the inf-sup condition (3.8) and the Cauchy–Schwarz inequality, it re-sults in the following estimate

‖uε − u0‖2H1(Ω\ωε(x0);C)

≤ 1β0 ‖∂u0

∂ν ‖L2(∂ωε(x0);C)‖uε − u0‖

L2(∂ωε(x0);C). (3.11)

For u ∈ H1(Ω \ ωε(x0);C), the boundary trace theorem provides with 0 < c < c:

c‖u‖H1(Ω\ωε(x0);C)

≤ ‖u‖H1/2(∂Ω;C)

+ ‖u‖H1/2(∂ωε(x0);C)

≤ c‖u‖H1(Ω\ωε(x0);C)

, (3.12)

where, by homogeneity argument, the H1/2-norm at ∂ωε(x0) implies

‖u‖2H1/2(∂ωε(x0);C)

= 1ε‖u‖2L2(∂ωε(x0);C)

+

∫∫∂ωε(x0)

|u(x)−u(y)|2|x−y|2 dSxdSy. (3.13)

14 V. A. Kovtunenko

Therefore, applying (3.12) and (3.13) with u = uε − u0 to (3.11) we get

‖uε − u0‖H1(Ω\ωε(x0);C)

≤ cβ0√ε‖∂u0∂ν ‖L2(∂ωε(x0);C) .

From (2.35) in Corollary 2.4 it follows ∂u0

∂ν = O(1) for ρ = O(ε) at ∂ωε(x0), hence

‖∂u0∂ν ‖L2(∂ωε(x0);C) =(∫

∂ωε(x0)

∣∣∂u0∂ν

∣∣2 dSx)1/2 = O(√ε) (3.14)and we conclude with (3.9). The proof is completed.

We observe that the last term on the right hand side of (3.10) expresses the residualerror near ∂ωε(x0). This boundary integral constitutes the leading order O(ε) in theresidual error estimate (3.9). To refine this estimate in Proposition 3.3 to the ordero(ε), we construct a corrector for ∂u

0

∂ν = ∇u0(x0) · ν + O(ε) (see (3.44)) in form of

the boundary layer, see e.g. [59]. In will be expressed via outer asymptotic expansionin the far-field with respect to the reference geometric object ω.

3.1 Outer asymptotic expansion by Fourier series in far-field

In this section we state an auxiliary problem in the exterior domain for y ∈ R2 \ ωwith respect to the stretched variable y = x−x0ε according to Definition 3.1.

For this reason we introduce the weighted Sobolev spaces (see [7, 51, 52]):

W 1,pµ (R2 \ ω;C) = {v : vµ ,∇v ∈ Lp(R2 \ ω;C)}, p ∈ (1,∞),

µ(y) = O(|y| ln |y|) in R2 \B2(0), µ(y) = O(1) in B2(0) \ ω,

with the weight µ suggested by the weighted Poincare inequality in exterior domains∫R2\B2(0)

(v

|y| ln |y|)2dy ≤ 4

∫R2\B2(0)

|∇v|2dy if∫∂B2(0)

v dSx = 0. (3.15)

We note that constant function is allowed for p ≥ 2 and logarithm for p > 2 in W 1,pµ .For p = 2 we consider the real-valued exterior Neumann problem: Find vector-

function wν(y) = ((wν)1, (wν)2)> ∈(W 1,2µ (R2 \ ω;R) \ P0

)2 such that∫R2\ω

Dwν∇v dy =∫∂ωνv dSy for all v ∈W 1,2µ (R2 \ ω;R), (3.16)

where the derivative matrix (Dwν)ij = (wν)i,j for i, j = 1, 2, the normal vectorν = (ν1, ν2)

>, and P0 stands for polynomials of degree zero, i.e. constant. Excludingconstant solutions, this implies the boundary value problem:

−∆wν = 0 in R2 \ ω, (3.17a)


Dwνν = −ν on ∂ω, (3.17b)

wν = O( 1|y|)

as |y| ↗ ∞. (3.17c)

The existence of a solution to (3.16) follows from the result of [7]. After rescalingy = x−x0ε we reduce the problem to the bounded domain Ω \ ωε(x0) as follows.

Lemma 3.4. The rescaled solution wεν(x) := wν(x−x0ε ) to (3.16) implies the function

wεν ∈ H1(Ω \ ωε(x0);R)2 which fulfills the following relation:∫Ω\ωε(x0)

Dwεν∇u dx =∫

ΓN(Dwενν)u dSx +

1ε

∫∂ωε(x0)

νu dSx

for all u ∈ H1(Ω \ ωε(x0);R) : u = 0 on ΓD.(3.18)

It admits the far-field representation in the Fourier series

wεν(x) =ερ

12πMω x̂+W

εν (x) for x ∈ R2 \Bε(x0), (3.19)

with the 2-by-2 real matrix Mω and the residual function W εν = ((Wεν )1, (W

εν )2)

> ∈H1(Ω \Bε(x0);R)2 such that∫ π

−πW εν dθ =

∫ π−πW εν x̂ dθ = 0, (3.20a)

W εν (x0 + ρx̂) = O(( ερ)

2) for ρ > ε, θ ∈ (−π, π]. (3.20b)Moreover, the uniform estimates hold

‖Dwεν‖L2(Ω\ωε(x0);R)2×2 = O(1), ‖wεν‖L2(Ω\ωε(x0);R)2 = O(ε

√| ln ε|), (3.21a)

‖wεν‖L2(∂ωε(x0);R)2 = O(√ε), ‖Dwενν‖L2(∂ωε(x0);R)2 = O(

1√ε). (3.21b)

Proof. The local coordinate system (2.6) after stretching implies the polar radius |y| ∈R+ and the polar angle θ ∈ (−π, π] such that

y = x−x0ε = |y|x̂, |y| =ρε , x̂ = (cos θ, sin θ)

>. (3.22)

By using the radial vector-functions x̂n from Proposition 2.3, the harmonic vector-valued function in (3.17) admits the Fourier series in the far-field as

wν(y) =∞∑n=1

1|y|nC

νnx̂

n, Cνn ∈ R2×2, for y ∈ R2 \B1(0) (3.23)

with unknown coefficient matrices Cνn , n ∈ N. Formula (3.23) implies

wν(y) =1|y|

12πMωx̂+Wν(y),

12πMω := C

ν1 , y ∈ R2 \B1(0) (3.24)

16 V. A. Kovtunenko

with the square matrix Mω ∈ R2×2 and the vector-valued residual function Wν =((Wν)1, (Wν)2)

> ∈W 1,2µ (R2 \ ω;R)2 such that∫ π−πWν dθ =

∫ π−πWν x̂ dθ = 0, (3.25a)

Wν(y) = O(|y|2) for |y| > 1, θ ∈ (−π, π], (3.25b)

where (3.25a) is obtained with the help of trigonometric calculus in (2.27). Applyingthe coordinate transformation y = x−x0ε to (3.24) and (3.25), due to (3.22) it followsstraightforwardly (3.19) and (3.20) forwεν(x) := wν(

x−x0ε ) andW

εν (x) :=Wν(

x−x0ε ).

Next we apply the coordinate transformation y = x−x0ε to the problem (3.17) andemploy the differential calculus according to (3.22)

∂∂y = ε

∂∂x , dy =

1ε2dx, dSy =

1εdSx (3.26)

to derive the following relations

−∆wεν = 0 in R2 \ ωε(x0), (3.27a)

Dwενν = − 1εν on ∂ωε(x0). (3.27b)

Since wεν ∈ W1,2µ (R2 \ ωε(x0);R)2 follows wεν ∈ H1(Ω \ ωε(x0);R)2 and in view of

(3.27a), the Green formula (3.7) can be applied to the vector-function u = wεν :∫Ω\ωε(x0)

(Dwεν∇u+ u∆wεν) dx = 〈Dwενν, u〉ΓN − 〈Dwενν, u〉∂ωε(x0)

for all u ∈ H1(Ω \ ωε(x0);R) : u = 0 on ΓD.

As the result, using (3.27) and the fact that the solution wεν is locally smooth near ΓN ,hence Dwενν ∈ L2(ΓN ;R)2, we arrive at formulation (3.18) in the bounded domain.

It remains to justify estimates (3.21). The first inequality in (3.21a) follows from∫Ω\ωε(x0)

|Dwεν |2 dx ≤∫R2\ωε(x0)

|Dwεν |2 dx =∫R2\ω|Dwν |2 dy = O(1)

which is calculated according to (3.26). To get the second inequality in (3.21a) weinscribe Ω in a ball BR(x0) of radius R > 0 sufficiently large and decompose it inBR(x0) \Bε(x0) and Bε(x0) \ ωε(x0) such that∫

Ω\ωε(x0)|wεν |2 dx ≤

∫BR(x0)\Bε(x0)

|wεν |2 dx+∫Bε(x0)\ωε(x0)

|wεν |2 dx

=

∫ π−π

∫ Rε

(( ερ)

2| 12πMω x̂|2 + |W εν |2

)ρdρdθ + ε2

∫B1(0)\ω

|wν |2 dy = O(ε2| ln ε|)

due to (3.19), (3.20), and (3.26).


Finally, by homogeneity and the trace theorem from (3.12) and (3.13) it follows

1√ε‖wεν‖L2(∂ωε(x0);R)2 ≤ c‖w

εν‖H1(Ω\ωε(x0);R)2 = O(1)

and the former inequality in (3.21b), while the latter one is confirmed from (3.27b) by

‖Dwενν‖2L2(∂ωε(x0);R)2

=

∫∂ωε(x0)

∣∣νε

∣∣2 dSx = 1ε2 meas2(∂ωε(x0)) = 1εmeas2(∂ω)where meas2(∂ωε(x0)) and meas2(∂ω) mean the Hausdorff measure of the sets in R2.This completes the proof.

We remark that the matrix Mω in Lemma 3.4 is called added or virtual mass tensorin [53, Note G]. Its properties are given below in Lemma 3.5 following [6, 14, 26, 45].

Lemma 3.5. The entries of Mω have the implicit expression:

(Mω)ij = δij meas2(ω) +∫∂ω

(wν)iνj dSy, i, j = 1, 2. (3.28)

The matrix Mω ∈ Spsd(R2×2), i.e. symmetric positive semi-definite, and positivedefinite if meas2(ω) > 0. For ellipsoidal shapes ω it has the explicit expression

Mω = Θ(α)Mω′Θ(α)>, Θ(α) :=

(cosα − sinαsinα cosα

), (3.29a)

Mω′ = π(a+ b)

(b 00 a

)(3.29b)

with the ellipse major a = 1 and minor b ∈ (0, 1] semi-axes, where the major axis hasan angle of α ∈ (−π2 ,

π2 ) with the y1-axis counted in the anti-clockwise direction.

Proof. We split the exterior domain in the far-field R2 \ B1(0) and the near-fieldB1(0) \ ω. In the far-field, the truncated Fourier series (3.24) holds. In the near-field,for i, j = 1, 2 we have the second Green formula

0 =∫B1(0)\ω

{∆(wν)iyj − (wν)i∆yj} dy

=

∫∂B1(0)

{∂(wν)i∂|y| yj − (wν)i

∂yj∂|y|}dSy −

∫∂ω

{∂(wν)i∂ν yj − (wν)i

∂yj∂ν

}dSy.

Using here the Neumann condition (3.17b), ∂yj∂|y| = x̂j according to (3.22), yj = x̂j

since |y| = 1 at ∂B1(0), and ∂yj∂ν = νj it follows

−∫∂B1(0)

{∂(wν)i∂|y| − (wν)i

}x̂j dSy =

∫∂ω

{νiyj + (wν)iνj

}dSy. (3.30)

18 V. A. Kovtunenko

We employ (3.24), (3.25a), and the trigonometric calculus (2.27) to calculate the inte-gral over ∂B1(0) on the left-hand side of (3.30) as∫

∂B1(0)

{−∂(wν)i∂|y| + (wν)i

}x̂j dSy =

∫ π−π

{ 12π

2∑l=1

(Mω)ilx̂l − ∂(Wν)i∂|y|

+ 12π

2∑l=1

(Mω)ilx̂l + (Wν)i}x̂j dθ =

1π

∫ π−π

2∑l=1

(Mω)ilx̂lx̂j dθ = (Mω)ij .

Applying to the right-hand side of (3.30) the divergence theorem∫∂ωνiyj dSy =

∫ωyj,i dy = δij meas2(ω), δij =

{1 , if i = j,0 , if i 6= j,

(3.31)

this together results in expression (3.28).To prove the symmetry of Mω, we insert v = (wν)j as the test-function in (3.16):∫

R2\ω∇(wν)i · ∇(wν)j dy =

∫∂ωνi(wν)j dSy =

∫∂ωνj(wν)i dSy,

written component-wisely for i, j = 1, 2, which follows (Mω)ij = (Mω)ji in (3.28).For arbitrary ξ ∈ R2, from (3.16) we have the non-negative linear combinations

0 ≤∫R2\ω|∇(ξ1(wν)1 + ξ2(wν)2

)|2 dy =

2∑i,j=1

∫∂ω

(wν)iξiνjξj dSy.

Therefore, multiplying (3.28) with ξiξj and summing the result over i, j = 1, 2, thepositive semi-definiteness, which is strict if meas2(ω) > 0, follows:

2∑i,j=1

(Mω)ijξiξj = |ξ|2 meas2(ω) +2∑

i,j=1

∫ω(wν)iξinjξj dSy ≥ |ξ|2 meas2(ω).

Finally, we derive the explicit representation of Mω for ellipsoidal shapes ω.Let the canonical ellipse ω′ enclosed in the ball B1(0) have the major a = 1 and the

minor b ∈ (0, 1] semi-axes with respect to y′-coordinates, i.e.

ω′ = {y′ ∈ R2 : (y′1a )

2 + (y′2b )

2 < 1}, a = 1.

Let the major axis of the reference ellipse ω written in y-coordinates have an angle ofα ∈ (−π2 ,

π2 ) with the y1-axis counted in the anti-clockwise direction, i.e.

ω = {y ∈ R2 : (y1 cosα+y2 sinαa )2 + (−y1 sinα+y2 cosαb )

2 < 1}, a = 1.


The y-coordinates are after rotation of y′-coordinates with the angle of−α, and y′ ∈ ω′when Θ(−α)y = Θ>(α)y ∈ ω with the orthogonal matrix Θ(α) given in (3.29a).Therefore, we will prove formula (3.29b) for ω′ and then transform y′ = Θ>y.

We introduce the elliptic coordinates r ∈ R+ and ψ ∈ (−π, π] such that

y′1 = c cosh(r) cosψ, y′2 = c sinh(r) sinψ, c =

√a2 − b2, a = 1, (3.32)

where c is called the linear eccentricity. By setting the distance r0 ∈ R+ such that

a = c cosh(r0) b = c sinh(r0), (3.33a)

the geometry ω′ can be restated as

∂ω′ = {r = r0, ψ ∈ (−π, π]}, R2 \ ω′ = {r > r0, ψ ∈ (−π, π]}. (3.33b)

From (3.32) it follows the differential calculus in elliptic coordinates:{∂∂y′1

= 1κ2(r,ψ)(c cosh(r) cosψ ∂∂r − c cosh(r) sinψ

∂∂ψ

),

∂∂y′2

= 1κ2(r,ψ)(c cosh(r) sinψ ∂∂r + c sinh(r) cosψ

∂∂ψ

),

dy′ = κ2(r, ψ)drdψ, κ(r, ψ) = c√

sinh2(r) + sin2 ψ

(3.34)

with the scale factor κ(r, ψ). In particular, at the ellipse boundary as r = r0 in (3.33b)with the normal vector ν ′, using (3.32) for constant ψ and (3.33a) we have

ν ′ = 1κ(r0,ψ)(b cosψ, a sinψ)>, ∂∂ν′ =

1κ(r0,ψ)

∂∂r ,

dSy′ = κ(r0, ψ)dψ, κ(r0, ψ) =√a2 cos2 ψ + b2 sin2 ψ.

(3.35)

Applying the coordinate transformation (3.32) and calculus in (3.34) and (3.35),the exterior problem (3.17), formulated for wν′ in R2 \ ω′ in y′-coordinates, can berewritten in elliptic coordinates. In fact, similarly to Proposition 2.3, the harmonicvector-function wν′ admits the following Fourier series in R2 \ ω′ (cf. (3.23)):

wν′ =

∞∑n=1

e−nrCν′n (cos(nψ), sin(nψ))

> for r > r0 (3.36)

according to (3.17c) for r ↗ ∞. Coefficient matrices Cν′n ∈ R2×2 in (3.36) can befound from the boundary condition (3.17b) written due to (3.35) and (3.36) as

1κ(r0,ψ)

∂wν′∂r

∣∣r=r0

= −∞∑n=1

ne−nr0κ(r0,ψ)C

ν′n (cos(nψ), sin(nψ))

> = − (b cosψ,a sinψ)>

κ(r0,ψ) .

20 V. A. Kovtunenko

Henceforth, e−r0Cν′

1 (cosψ, sinψ)> = (b cosψ, a sinψ)>, Cν

′n = 0 for all n ≥ 2, and

we obtain the following analytic expression for the solution

wν′ = er0−r(b cosψ, a sinψ)> for r ≥ r0. (3.37)

Now the matrix Mω′ can be calculated analytically. After substitution of (3.37) inthe representation formula (3.28) it implies the following two vectors for j = 1, 2:

(Mω′)( · ,j) = δ( · ,j)meas2(ω′) +

∫∂ω′

(b cos(ψ)ν ′j , a sin(ψ)ν′j)> dSy′ . (3.38)

We extend (b cosψ, a sinψ)> from the boundary inside ω′ with the smooth function( bay′1,ab y′2)> and use the divergence theorem to calculate the elliptic integral in (3.38)∫

∂ω′( bay′1ν′j ,ab y′2ν′j)> dSy′ =

∫ω′( bay′1,j

ab y′2,j)> dy′ = meas2(ω′)

{( ba , 0)

>, j = 1,(0, ab )

>, j = 2.

Together with meas2(ω′) = πab from (3.38) we arrive at (3.29b).The transformation formula in (3.29a) can be justified by rotation y′ = Θ>y applied

to the variational equation in the manner of (3.16):∫R2\ω′

Dwν(Θy′)∇v dy′ =∫∂ω′

νv dSy′ for all v ∈W 1,2µ (R2 \ ω′;R),

which, after the left multiplication with Θ>, results in∫R2\ω′

D(Θ>wν(Θy′))∇v dy′ =∫∂ω′

Θ>νv dSy′ . (3.39)

Since Θ>ν = ν ′ in (3.39) and using the representation (3.24) this proves the identity

wν′(y′) = Θ>wν(Θy′) = 1|Θy′|

12πΘ

>MωΘy′|Θy′| + Θ

>Wν(Θy′)

= 1|y′|1

2πΘ>MωΘ y

′

|y′| + Θ>Wν(Θy′) = 1|y′|

12πMω′

y′

|y′| +Wν′(y′),

which implies Θ>MωΘ =Mω′ , thus (3.29a), and completes the proof.

We remark that, in the limit case when b ↘ +0, formulas in Lemma 3.5 describethe singular matrix of virtual mass for the straight crack, see [9, 45].

3.2 Uniform asymptotic expansion of solution of the Neumann problem

With the help of the boundary layer wεν described in Lemma 3.4 we can improve theresidual error estimate (3.9) according to (3.43) in Theorem 3.6:

‖uε − u0 − ε∇u0(x0) · wεν‖H1(Ω\ωε(x0);C) = O(ε2√| ln ε|). (3.40)


The leading order here is contributed by εwεν over the domain Ω \ ωε(x0) due to thesecond inequality in (3.21a). Further we construct a refined asymptotic term of orderε2 using the auxiliary Helmholtz problem: Find u1 ∈ H1(Ω;C)2 such that

u1 = 0 on ΓD, (3.41a)∫Ω(Du1∇u− k2u1u) dx = k2

ε√| ln ε|

∫Ω\ωε(x0)

wενu dx

for all u ∈ H1(Ω;C) : u = 0 on ΓD.(3.41b)

We stress that the solution u1 to problem (3.41) is estimated uniformly with respect toε since the right-hand side of (3.41b) is bounded by (3.21a)∣∣∣ 1ε√| ln ε|

∫Ω\ωε(x0)

wενu dSx

∣∣∣ ≤ ‖wεν‖L2(Ω\ωε(x0);R)2ε√| ln ε|

‖u‖L2(Ω\ωε(x0);C)

≤ C‖u‖L2(Ω\ωε(x0);C)

with C > 0. Using Green’s formula (3.7) and the local H2-regularity of the solutionu1 in Bε(x0) ⊃ ωε(x0) we restate (3.41b) over the perturbed domain∫

Ω\ωε(x0)(Du1∇u− k2u1u) dx = k2

ε√| ln ε|

∫Ω\ωε(x0)

wενu dx

−∫∂ωε(x0)

(Du1ν)u dSx for all u ∈ H1(Ω \ ωε(x0);C) : u = 0 on ΓD.(3.42)

Theorem 3.6. The solutions u0 of (2.2), uε of (3.4), wν of (3.16), and u1 of (3.41)satisfy the following residual error estimate

‖qε1‖H1(Ω\ωε(x0);C) = O(ε2), (3.43a)

qε1 := uε − u0 − ε∇u0(x0) · (wεν + ε

√| ln ε|u1). (3.43b)

Proof. Subtracting from (3.4b) equations (3.10), (3.18) multiplied with ε∇u0(x0) and(3.42) multiplied with ε2

√| ln ε|∇u0(x0), using

∂u0

∂ν −∇u0(x0) · ν = b0u · ν +

∂U01∂ν = O(ε) on ∂ωε(x0) (3.44)

according to (2.35), the differential identity

∇(∇u0(x0) · (wεν + ε

√| ln ε|u1)

)= ∇u0(x0) · (Dwεν + ε

√| ln ε|Du1),

and the notation of qε1 introduced in (3.43b), we obtain the variational equation∫Ω\ωε(x0)

(∇qε1 · ∇u− k2qε1u) dx = −ε∫

ΓN(∇u0(x0) ·Dwενν)u dSx

+

∫∂ωε(x0)

(b0u · ν +

∂U01∂ν + ε

2√| ln ε|∇u0(x0) ·Du1ν

)u dSx

for all u ∈ H1(Ω \ ωε(x0);C) : u = 0 on ΓD.

(3.45)

22 V. A. Kovtunenko

One difficulty is that qε1 is inhomogeneous, namely qε1 = −ε∇u0(x0) · wεν = O(ε2)

at ΓD due to (3.19). For its lifting, we take a smooth cut-off function ηΓD supported ina neighborhood of ΓD such that ηΓD = 1 at ΓD. Henceforth, for

Qε1 := qε1 +R

ε1, R

ε1 := ε(∇u0(x0) · wεν)ηΓD = O(ε

2), Qε1 = 0 on ΓD,

applying to (3.45) with qε1 = Qε1−Rε1 the Cauchy–Schwarz inequality, the asymptotic

estimates in (3.19), (3.44), and the trace theorems (3.12) and (3.13), we get∣∣∣∫Ω\ωε(x0)

(∇Qε1 · ∇u− k2Qε1u) dx∣∣∣ ≤ ∣∣∣∫

Ω\ωε(x0)(∇Rε1 · ∇u− k2Rε1u) dx

∣∣∣+ C1ε

2‖u‖L2(ΓN ;C)

+ C2ε

∫∂ωε(x0)

|u| dSx ≤ Cε2‖u‖H1(Ω\ωε(x0);C)

, C, C1, C2 > 0.

This upper bound together with the inf-sup condition (3.8) proves ‖Qε1‖H1(Ω\ωε(x0);C) =O(ε2), hence (3.43a) and the assertion of the theorem.

As the consequence, from (3.45) we infer the boundary value problem for qε1:

−[∆ + k2]qε1 = 0 in Ω \ ωε(x0), (3.46a)∂qε1∂ν = −ν ·

(b0u +∇U01 + ε2

√| ln ε|(Du1)∇u0(x0)

)on ∂ωε(x0), (3.46b)

∂qε1∂ν = −ε∇u

0(x0) ·Dwενν on ΓN , (3.46c)

qε1 = −ε∇u0(x0) · wεν on ΓD. (3.46d)Theorem 3.6 is useful for the asymptotic expansion with respect to ε ↘ +0 of the

state-constrained objective function as suggested in the following section.

3.3 Inverse Helmholtz problem under Neumann boundary condition

In the inverse setting of the problem, the shape ω? ∈ Gω, the size ε? ∈ Gε, andthe center x? ∈ Gx for an unknown geometric object ω?ε?(x?) being tested are to beidentified and reconstructed from the known boundary measurement u? ∈ L2(ΓN ;C).The admissible set G = Gω ×Gε ×Gx is introduced in Definitions 3.1 and 3.2.

For this purpose, a trial geometric object ωε(x0) with admissible (ω, ε, x0) ∈ G isput in Ω. For such trial variables we find a family of solutions uε to problem (3.4) anddetermine the square function of the misfit at the boundary

J : G 7→ R+, J(ω, ε, x0) := 12∫

ΓN|uε − u?|2 dSx. (3.47)

The objective function (3.47) serves for the state-constrained, topology optimizationproblem: Find (ω?, ε?, x?) ∈ G which is the argument of the trivial minimum

0 = J(ω?, ε?, x?) = min(ω,ε,x0)∈G

J(ω, ε, x0) subject to (3.4). (3.48)


Since the test geometry (ω?, ε?, x?) ∈ G is feasible, the trivial minimum in (3.48) isattained at the solution uε

?of (3.4) for the test object ω?ε?(x

?) when uε?= u? at ΓN

in (3.47). Uniqueness of the minimum is open.To bring (3.48) in a form suitable for analysis, we give primal-dual arguments.While uε in (3.47) implies the primal state variable, a dual state variable vε can be

obtained by a Fenchel–Legendre duality corresponding to the variational principle:

Lε(uε, vε) = minRe(u),Re(v)

maxIm(u),Im(v)

Lε(u, v)

over u, v ∈ H1(Ω \ ωε(x0);C) : u = h, v = 0 on ΓD,(3.49)

where the Lagrangian Lε has the form (compare with (3.5) and (3.6)):

Lε(u, v) :=Re{∫

Ω\ωε(x0)(∇u · ∇v − k2uv) dx−

∫ΓNgv dSx

+ 12

∫ΓN

(u− u?)2 dSx}.

(3.50)

Lemma 3.7. The first-order necessary optimality conditions for (3.49) imply the pri-mal problem (3.4) together with the dual variational problem: Find vε ∈ H1(Ω \ωε(x0);C) such that

vε = 0 on ΓD, (3.51a)∫Ω\ωε(x0)

(∇vε · ∇u− k2vεu) dx = −∫

ΓN(uε − u?)u dSx

for all u ∈ H1(Ω \ ωε(x0);C) : u = 0 on ΓD.(3.51b)

Proof. Indeed, using variational calculus from Proposition 2.1, the first order optimal-ity condition for (3.49) necessitates four variational inequalities:〈

∂∂Re(v)Lε(u

ε, vε),Re(v − vε)〉≥ 0,

〈∂

∂Im(v)Lε(uε, vε), Im(v − vε)

〉≤ 0,〈

∂∂Re(u)Lε(u

ε, vε),Re(u− uε)〉≥ 0,

〈∂

∂Im(u)Lε(uε, vε), Im(u− uε)

〉≤ 0

holding for all u, v ∈ H1(Ω \ ωε(x0);C) such that u = h, v = 0 on ΓD. Insertinghere v = vε±v and u = uε±u with v,u ∈ H1(Ω \ωε(x0);C) such that v = u = 0on ΓD we get the following four variational equations:∫

Ω\ωε(x0)

(∇Re(uε) · ∇Re(v)− k2Re(uε)Re(v)

)dx =

∫ΓN

Re(g)Re(v) dSx,∫Ω\ωε(x0)

(∇Im(uε) · ∇Im(v)− k2Im(uε)Im(v)

)dx =

∫ΓN

Im(g)Im(v) dSx,∫Ω\ωε(x0)

(∇Re(vε) · ∇Re(u)− k2Re(vε)Re(u)

)dx =

∫ΓN

Re(u? − uε)Re(u) dSx,∫Ω\ωε(x0)

(∇Im(vε) · ∇Im(u)− k2Im(vε)Im(u)

)dx =

∫ΓN

Im(u? − uε)Im(u) dSx.

24 V. A. Kovtunenko

The summation of the first and the second equations for v = u and v = ıu constitutesthe real and imaginary parts of (3.4b), while the third and the fourth equations foru = u and u = ıu contribute to (3.51b), respectively. This completes the proof.

We emphasize that the Helmholtz problem (3.51) is analogous to (3.4) and differsfrom it by the boundary data at ∂Ω. Therefore, the well-posedness result stated for uεremains true also for vε. It implies the weak solution to (cf. (3.3))

−[∆ + k2]vε = 0 in Ω \ ωε(x0), (3.52a)

∂vε

∂ν = 0 on ∂ωε(x0), (3.52b)

∂vε

∂ν = −(uε − u?) on ΓN , (3.52c)

vε = 0 on ΓD. (3.52d)

Plugging the representation (3.52c) in (3.47) we establish the following equivalence.

Proposition 3.8. The topology optimization problem (3.48) can by equivalently statedwith respect to the primal and dual state variables: Find (ω?, ε?, x?) ∈ G such that

0 = min(ω,ε,x0)∈G

Re{−12∫

ΓN(uε − u?)∂vε∂ν dSx

}subject to (3.4) and (3.51). (3.53)

Further we provide asymptotic analysis of the objective as ε↘ +0.As ε = 0, problem (3.51) turns into the dual background problem stated in the

reference domain Ω: Find v0 ∈ H1(Ω;C) such that

v0 = 0 on ΓD, (3.54a)∫Ω(∇v0 · ∇u− k2v0u) dx = −

∫ΓN

(u0 − u?)u dSx

for all u ∈ H1(Ω;C) : u = 0 on ΓD,(3.54b)

which implies the weak solution to (cf. (2.1))

−[∆ + k2]v0 = 0 in Ω, (3.55a)

∂v0

∂ν = −(u0 − u?) on ΓN , (3.55b)

v0 = 0 on ΓD. (3.55c)

Problem (3.54) is similar to the primal background problem (2.2), henceforth, all theresults of Section 2 hold true for (3.54), too. In particular, the inner asymptotic expan-sion holds in the near-field BR(x0) ⊂ Ω in the form

v0(x) = v0(x0)J0(kρ) + V0

0 (x), V0

0 (x) =2kJ1(kρ)∇v

0(x0) · x̂+ V 01 (x) (3.56)


with the residuals V 00 , V0

1 ∈ H1(BR(x0);C) such that∫ π−πV 00 dθ =

∫ π−πV 01 dθ =

∫ π−πV 01 x̂ dθ = 0, (3.57a)

V 00 (x0 + ρx̂) = O(ρ), V0

1 (x0 + ρx̂) = O(ρ2) for θ ∈ (−π, π], (3.57b)

and it implies similar to (2.35) representation of the gradient

∇v0(x) = ∇v0(x0) + b0v(x) +∇V 01 (x), ∇V 01 = O(ρ), (3.58a)

b0v(x) :=(v0(x0)ka

′0(kρ) + a

′1(kρ)∇v0(x0) · x̂

)x̂

+ a1(kρ)kρ (∇v0(x0) · x̂′)x̂′, b0v = O(ρ).

(3.58b)

We note that, expansion of vε − v0 for ε ↘ +0 in the manner of Theorem 3.6would be a hard task since the right-hand side of problem (3.51) itself depends on uε.In this respect, Proposition 3.8 will be not helpful. Instead, decomposing uε − u? =uε − u0 + u0 − u? and using (3.55b) we express the objective in (3.47) equivalently

J(ω, ε, x0) = J0 − Re{∫

ΓN(uε − u0)∂v0∂ν dSx

}+ 12

∫ΓN|uε − u0|2 dSx,

J0 := 12

∫ΓN|u0 − u?|2 dSx = O(1), 12

∫ΓN|uε − u0|2 dSx = O(ε4| ln ε|)

(3.59)

for ε↘ +0 due to Theorem 3.6. From (3.59) we infer the asymptotic result below.

Theorem 3.9. The objective in (3.47) admits the high-order asymptotic expansion

J(ω, ε, x0) = J0 + Re{ε2JN1 (ω, x0) + J

ε2 + J

ε3 + J

ε4}+ O(ε4| ln ε|), (3.60)

where the asymptotic terms are expressed by formulas:

JN1 (ω, x0) := −∇u0(x0)>Mω∇v0(x0) + k2meas2(ω)u0(x0)v0(x0), (3.61a)

Jε2 := ε3√| ln ε|

∫ π−π∇u0(x0) ·

(∂u1

∂ρ v0(x0)− u1∇v0(x0) · x̂

)dθ

= O(ε3√| ln ε|),

(3.61b)

Jε3 := ε2∫ π−π∇u0(x0) ·

(∂W εν∂ρ V

01 −W

εν∂V 01∂ρ

)dθ

−∫∂ωε(x0)

ν ·(qε1∇v0(x0) + (b0u +∇U01 )V 00

)dSx = O(ε3),

(3.61c)

Jε4 := ε2√| ln ε|

(ε

∫ π−π∇u0(x0) ·

{∂u1

∂ρ ε∇v0(x0) · x̂− u1(v0(x0)ka′0

+∂V 01∂ρ

)}dθ −

∫∂ωε(x0)

ν · (Du1)∇u0(x0)V 00 dSx)= O(ε4

√| ln ε|).

(3.61d)

26 V. A. Kovtunenko

Proof. To verify (3.60) it needs to expand up to O(ε4| ln ε|)-order asymptotic termsthe boundary integral in (3.59)

I(uε − u0, v0) := −∫

ΓN(uε − u0)∂v0∂ν dSx.

For this task we employ the second Green formula in Ω \Bε(x0) and rewrite I as

I(uε − u0, v0) =∫

Ω\Bε(x0)

(∆(uε − u0)v0 − (uε − u0)∆v0

)dx

+

∫∂Bε(x0)

(∂(uε−u0)∂ρ v

0 − (uε − u0)∂v0∂ρ)dSx,

(3.62)

where the domain integral disappears due to the Helmholtz equations (2.1a), (3.3a),and (3.55a). On the one hand, we note that I is an invariant integral which canbe written over arbitrary Lipschitz-smooth boundary ∂O of a domain O such thatωε(x0) ⊆ O ⊂ Ω. On the other hand, the integral over the circle ∂Bε(x0) is advan-tageous while it can be calculated by substitution of the uniform expansion (3.43) foruε − u0 and the inner expansion (3.56) and (3.58) for v0.

By doing so, we decompose I(uε−u0, v0) = I(qε1, v0)+ I(uε−u0− qε1, v0) withthe residual qε1 from Theorem 3.6 and calculate these two integrals separately.

First, applying to I(qε1, v0) the second Green formula in Bε(x0) \ ωε(x0) we get

I(qε1, v0) :=∫∂Bε(x0)

(∂qε1∂ρ v

0 − qε1 ∂v0

∂ρ

)dSx =

∫∂ωε(x0)

(∂qε1∂ν v

0 − qε1 ∂v0

∂ν

)dSx

= −∫∂ωε(x0)

ν ·{qε1(∇v0(x0) + b0v +∇V 01 )

+(b0u +∇U01 + ε2

√| ln ε|(Du1)∇u0(x0)

)(v0(x0)(1 + a0) + V 00

)}dSx

after substitution of (3.46b), (3.56), and (3.58a). The divergence theorem provides

−∫∂ωε(x0)

ν ·(b0u +∇U01 + ε2

√| ln ε|(Du1)∇u0(x0)

)v0(x0) dSx

= −∫ωε(x0)

div(b0u +∇U01 + ε2

√| ln ε|(Du1)∇u0(x0)

)v0(x0) dx.

(3.63)

Here −div(∇U01 ) = −∆U01 = k2U01 due to (2.1a) and (2.33), b0u = −u0(x0)k2ρ

2 x̂ +O(ρ2) according to (2.8) and (2.35b), and since ρx̂ = x− x0 then

Iq1 :=∫ωε(x0)

div(u0(x0)

k2(x−x0)2 v

0(x0))dx = k2u0(x0)v0(x0)

∫ωε(x0)

dx. (3.64)

With the help of (3.63) and (3.64) we collect the asymptotic terms of the same order

I(qε1, v0) = Iq1 + I

q3 + I

q4 + I

q5 (the term I

q2 of order O(ε

3√| ln ε|) is zero),


Iq3 := −∫∂ωε(x0)

ν ·(qε1∇v0(x0) + (b0u +∇U01 )V 00

)dSx = O(ε3),

Iq4 := −ε2√| ln ε|

∫∂ωε(x0)

ν · (Du1)∇u0(x0)V 00 dSx = O(ε4√| ln ε|),

Iq5 := −∫ωε(x0)

div(b0u + u

0(x0)k2(x−x0)

2 + ε2√| ln ε|(Du1)∇u0(x0)

)v0(x0) dx

−∫∂ωε(x0)

{ν ·(b0u +∇U01 + ε2

√| ln ε|(Du1)∇u0(x0)

)v0(x0)a0

+ qε1(b0v +∇V 01 )

}dSx +

∫ωε(x0)

k2U01 v0(x0) dx = O(ε4)

in view of the following asymptotic relations:

qε1 = U01 = V

01 = O(ε

2),∂qε1∂ρ =

∂U01∂ρ =

∂V 01∂ρ = O(ε), u

1 = ∂u1

∂ρ = O(1),

V 00 = b0u = b

0v = O(ε),

∫ωε(x0)

dx = ε2meas2(ω),∫∂ωε(x0)

dSx = εmeas1(∂ω),(3.65)

which hold in Bε(x0) due to (2.34b), (2.35b), (3.43), and (3.57b).Second, inserting in I the representations (3.19), (3.43), (3.56), and (3.58a) we have

I(uε − u0 − qε1, v0) :=∫∂Bε(x0)

(∂(uε−u0−qε1 )∂ρ v

0 − (uε − u0 − qε1)∂v0

∂ρ

)dSx

= ε

∫ π−π∇u0(x0) ·

{(− 12πεMω x̂+

∂W εν∂ρ + ε

√| ln ε|∂u1∂ρ

)(v0(x0)(1 + a0)

+ (ε+ a1k )∇v0(x0) · x̂+ V0

1

)−( 1

2πMω x̂+Wεν + ε

√| ln ε|u1

)×(∇v0(x0) · x̂+ b0v · x̂+

∂V 01∂ρ

)}εdθ = Iw1 + I

w2 + I

w3 + I

w4 + I

w5 .

(3.66)

We calculate the asymptotic terms Iw1 , Iw2 , I

w3 , I

w4 , I

w5 in (3.66) by using the orthogo-

nality (3.20a) and (3.57a), calculus (2.27) providing∫ π−π∇u0(x0) · 12π (Mω x̂)(∇v0(x0) · x̂) dθ =

12∇u

0(x0)>Mω∇v0(x0), (3.67)

and the following relations holding at ∂Bε(x0) due to (2.8), (3.20b), and (3.58b):

b0v · x̂ = v0(x0)ka′0 + a′1∇v0(x0) · x̂, W εν = O(1),∂W εν∂ρ = O(

1ε),

a0 = O(ε2), a′0 = O(ε), a1 = O(ε3), a′1 = O(ε

2).(3.68)

With the help of (3.65), (3.67), and (3.68) the calculation results in

Iw1 = −ε2∇u0(x0)>Mω∇v0(x0),

28 V. A. Kovtunenko

Iw2 := ε3√| ln ε|

∫ π−π∇u0(x0) ·

(∂u1

∂ρ v0(x0)− u1∇v0(x0) · x̂

)dθ = O(ε3

√| ln ε|),

Iw3 := ε2∫ π−π∇u0(x0) ·

(∂W εν∂ρ V

01 −W

εν∂V 01∂ρ

)dθ = O(ε3),

Iw4 := ε3√| ln ε|

∫ π−π∇u0(x0) ·

{∂u1

∂ρ ε∇v0(x0) · x̂− u1(v0(x0)ka′0 + ∂V 01∂ρ )} dθ

= O(ε4√| ln ε|),

Iw5 := ε3√| ln ε|

∫ π−π∇u0(x0) ·

{∂u1

∂ρ

(v0(x0)a0 +

a1kε∇v0(x0) · x̂+ V

01

)− u1(a′1∇v0(x0) · x̂)

}dθ − ε2(

a1k + εa

′1)∇u0(x0)>Mω∇v0(x0) = O(ε4).

Finally, the summation Iq1 + Iw1 = ε

2JN1 , Iw2 = J

ε2 , I

q3 + I

w3 = J

ε3 , and I

q4 + I

w4 = J

ε4

gathers the asymptotic terms in (3.61) and finishes the proof.

We make a few remarks on corollaries following from Theorem 3.9.The first-order asymptotic term Re(JN1 ) given in formula (3.61a) is called the topo-

logical derivative of the objective J following the terminology of [57] since

Re(JN1 (ω, x0)) = limε↘+0

1ε2

(J(ω, ε, x0)− J0

). (3.69)

It was found e.g. in [3, 8].After approximation by (3.60), the optimization problem (3.48) forces the shape-

topological control problem: For fixed ε ∈ Gε, find (ω?, x?) ∈ Gω ×Gx such that

Re(JN1 (ω?, x?)) = min

(ω,x0)∈Gω×GxRe(JN1 (ω, x0)). (3.70)

The approximated objective in (3.70) does not depend on the perturbed state. It isexpressed by the reference solutions u0 and v0 of the primal (2.2) and the dual (3.54)background Helmholtz problems, as well as the solution wν of the exterior Neumannproblem for the Laplace operator (3.16).

The boundary layer wν enters formula (3.61a) via the virtual mass tensor Mω.Therefore, employing the explicit description of Mω given for ellipsoidal shapes inLemma 3.5, the control problem (3.70) can be relaxed by reducing the set of admissi-ble shapes Gω to a family of ellipses depending on rotation and compression.

The problem of identification of the center x? of the test object will be discussedfurther in Section 5.4.

In the next Section 4 we modify our methods to treat forward and inverse problemsfor the Helmholtz equation under Dirichlet boundary conditions at ∂ωε(x0).


4 Helmholtz problems for geometric objects under Dirichlet(sound soft) boundary condition

Given g ∈ L2(ΓN ;C) and h ∈ H1/2(ΓD;C), the (forward) Dirichlet problem for theHelmholtz equation consists in finding the wave potential uε(x) fulfilling:

−[∆ + k2]uε = 0 in Ω \ ωε(x0), (4.1a)

uε = 0 on ∂ωε(x0), (4.1b)

∂uε

∂ν = g on ΓN , (4.1c)

uε = h on ΓD, (4.1d)

which is described by the variational problem: Find uε ∈ H1(Ω\ωε(x0);C) such that

uε = h on ΓD, uε = 0 on ∂ωε(x0), (4.2a)∫Ω\ωε(x0)

(∇uε · ∇u− k2uεu) dx =∫

ΓNgu dSx

for all u ∈ H1(Ω \ ωε(x0);C) : u = 0 on ΓD ∪ ∂ωε(x0).(4.2b)

Well-posedness of the Dirichlet problem (4.2) is argued similarly to the Neumannproblem (3.4). It implies necessary condition for the variational principle with Pεfrom (3.6):

Pε(uε) = minRe(v)

maxIm(v)Pε(v)

over v ∈ H1(Ω \ ωε(x0);C) : v = h on ΓD, v = 0 on ∂ωε(x0).(4.3)

To evaluate the difference of uε from the background solution u0, the outer asymp-totic expansion in the far-field is needed, see [33, Section 3.3].

4.1 Outer and inner asymptotic expansions by Fourier series

We construct three auxiliary problems: two boundary layers in the exterior domainR2 \ ω and a regularized Helmholtz problem in Ω for the logarithm.

First, we define the kernel of the Laplace operator in R2 by means of the logarithmiccapacity, see e.g. [21]. We consider the following homogeneous Dirichlet problem:

−∆w00 = 0 in R2 \ ω, (4.4a)

w00 = 0 on ∂ω, (4.4b)

w00 = O(ln |y|) as |y| ↗ ∞. (4.4c)

30 V. A. Kovtunenko

The weak variational formulation to (4.4) can be given in the weighted Sobolev spacesintroduced in Section 3.1: For p > 2 and p′ < 2 such that 1p +

1p′ = 1, find w00 ∈

W 1,pµ (R2 \ ω;R) such thatw00 = 0 on ∂ω, (4.5a)∫

R2\ω∇w00 · ∇v dy = 0 for all v ∈W 1,p

′µ (R2 \ ω;R) : v = 0 on ∂ω. (4.5b)

The existence of a nontrivial solution to (4.5) is argued as follows. Following [7],there exists a unique solution of the inhomogeneous Dirichlet problem: Find w̃ ∈W 1,2µ (R2 \ ω;R) such that

w̃ = ln |y| on ∂ω,∫R2\ω∇w̃ · ∇v dy = 0 for all v ∈W 1,2µ (R2 \ ω;R) : v = 0 on ∂ω.

Since ln |y| ∈W 1,pµ (R2 \ω;R) for p > 2, then w00 = ln |y|− w̃ solves (4.5). It admitsthe Fourier series (cf. (3.23)) in R2 \B1(0):

w00(y) = − ln(cap(ω)) + ln |y|+W00, W00 =∞∑n=1

1|y|nC

00n x̂

n, (4.6)

with C00n ∈ R2×2, n ∈ N, and cap(ω) ∈ R+ called logarithmic capacity of the set ω.After rescaling y = x−x0ε , the exterior problem (4.5) can be reduced to the bounded

domain Ω \ ωε(x0) with the help of the following Green formula holding for everyfunction u ∈ H1(Ω \ ωε(x0);C) such that ∆u ∈ L2(Ω \ ωε(x0);C) (cf. (3.7)):∫

Ω\ωε(x0)(∇u · ∇u+ u∆u) dx = 〈∂u∂ν , u〉ΓN

for all u ∈ H1(Ω \ ωε(x0);C) : u = 0 on ΓD ∪ ∂ωε(x0).(4.7)

Then relations (4.5)–(4.7) prove the assertion of the Lemma 4.1 below.

Lemma 4.1. The rescaled solution wε00(x) := w00(x−x0ε ) to (4.5) fulfills

wε00 = 0 on ∂ωε(x0), (4.8a)∫Ω\ωε(x0)

∇wε00 · ∇u dx =∫

ΓN

∂wε00∂ν u dSx

for all u ∈ H1(Ω \ ωε(x0);R) : u = 0 on ΓD ∪ ∂ωε(x0).(4.8b)


wε00(x) = − ln(εcap(ω)) + ln ρ+W ε00(x) for x ∈ R2 \Bε(x0), (4.9)

with the residual function W ε00 ∈ H1(Ω \ ωε(x0);R) such that∫ π−πW ε00 dθ = 0, W

ε00 = O(

ερ) for ρ > ε, θ ∈ (−π, π]. (4.10)


Second, to compensate the logarithm in (4.9) which is unbounded as ρ ↘ +0, weconstruct the regularized Helmholtz problem in Ω: Find uln ∈ H1(Ω;R) such that

uln = ln ρ on ΓD, (4.11a)∫Ω(∇uln · ∇u− k2ulnu) dx =

∫ΓN

∂(ln ρ)∂ν u dSx −

∫Ωk2u ln ρ dx

for all u ∈ H1(Ω;R) : u = 0 on ΓD.(4.11b)

Since ∆[ln ρ] = 0, the solution of (4.11) describes the boundary value problem:

−[∆ + k2]uln = −[∆ + k2] ln ρ in Ω, (4.12a)

∂uln

∂ν =∂(ln ρ)∂ν on ΓN , (4.12b)

uln = ln ρ on ΓD. (4.12c)

Using Green’s formula (4.7) we restate (4.11b) over the perturbed domain as∫Ω\ωε(x0)

(∇uln · ∇u− k2ulnu) dx =∫

ΓN

∂(ln ρ)∂ν u dSx −

∫Ω\ωε(x0)

k2u ln ρ dx

for all u ∈ H1(Ω \ ωε(x0);R) : u = 0 on ΓD ∪ ∂ωε(x0).(4.13)

Similarly to Lemma 2.2, below we establish the inner asymptotic expansion for uln.

Lemma 4.2. The solution uln of (4.11) admits the representation in the near-field

uln(x) = uln(x0) + (uln(x0)− ln ρ)a0 − π2 a2 + U

ln0 (x) in BR(x0) ⊂ Ω, (4.14)

with a0 and a2 given in (2.8) and the residual U ln0 ∈ H1(BR(x0);R) such that∫ π−πU ln0 dθ = 0, U

ln0 = O(ρ) for ρ ∈ [0, R), θ ∈ (−π, π]. (4.15)

Proof. For BR(x0) ⊂ Ω, we decompose uln into the radial and residual functions:

uln(x) = uln0 (ρ) + Uln0 (x) in Bδ(x0), δ ∈ [0, R), where

uln0 (ρ) :=1

2π

∫ π−πuln dθ, U ln0 := u

ln − uln0 , hence∫ π−πU ln0 dθ = 0.

(4.16)

Using (4.16) we substitute a smooth cut-off function η(ρ) supported in Bδ(x0) as thetest function u = η into (4.11b) and integrate it by parts to derive that

0 =∫Bδ(x0)

(∇uln · ∇η − k2(uln − ln ρ)η

)dx = 2π

∫ δ0

((uln0 )

′ρη′

− k2(uln0 − ln ρ)η)ρdρ = −2π

∫ δ0

((ρ(uln0 )

′ρ

)′ρ+ k2ρ(uln0 − ln ρ)

)η dρ

32 V. A. Kovtunenko

which implies the inhomogeneous Bessel equation

(uln0 )′′ρ +

1ρ(u

ln0 )′ρ + k

2uln0 = k2 ln ρ for ρ ∈ (0, δ). (4.17)

Together with the particular integral ln ρ, its general solution has the form

uln0 (ρ) = Kln0 (1 + a0) + S

ln0( 2π (ln

k2 + ln ρ+ γ)(1 + a0) + a2

)+ ln ρ (4.18)

with the Bessel and Neumann functions written according to (2.8a) and (2.8c), andtwo unknown coefficients K ln0 , S

ln0 ∈ R. The factor Sln0 = −

π2 avoids the leading

logarithmic term, thus providing the function regularity uln0 ∈ H1((0, δ);R) in (4.16).The inhomogeneous Helmholtz equation (4.12a) and (4.17) imply−[∆+k2]U ln0 = 0

which argues a Fourier series representation of U ln0 for ρ↘ +0. From∫ π−π U

ln0 dθ = 0

in (4.16) we get the Wirtinger inequality (2.17b) for the residual U ln0 . This followsthe asymptotic order U ln0 = O(ρ) in (4.15), see for detail the proof of Lemma 2.2.Now passing ρ ↘ +0 in (4.16) and (4.18) we find K ln0 = uln(x0) + (ln

k2 + γ) and,

consequently, we arrive at formula (4.14). This completes the proof.

With the help of Lemma 4.1 and Lemma 4.2 we construct the first-order correctionfunction wεln to u

ε − u0 which will be used further in Theorem 4.5.

Lemma 4.3. Combining the rescaled solutionwε00 to (4.5) and the solution uln to (4.11)

it forms the first-order correction function in H1(Ω \ ωε(x0);R)

wεln := wε00 + ln(εcap(ω))− uln (4.19)

which fulfills the following relations:

wεln =Wε00 on ΓD, (4.20a)

wεln = ln(εcap(ω))−uln(x0)+ (ln ρ−uln(x0))a0 + π2 a2−Uln0 on ∂ωε(x0), (4.20b)∫

Ω\ωε(x0)(∇wεln · ∇u− k2wεlnu) dx =

∫ΓN

∂W ε00∂ν u dSx −

∫Ω\ωε(x0)

k2W ε00u dx

for all u ∈ H1(Ω \ ωε(x0);R) : u = 0 on ΓD ∪ ∂ωε(x0)(4.20c)

and admits the representation in the Fourier series

wεln =(ln ρ− uln(x0)

)(1 + a0) +W ε00 +

π2 a2 − U

ln0 in BR(x0) \Bε(x0). (4.21)

Proof. Indeed, formulas (4.20) and (4.21) are obtained by substitution of the represen-tations (4.8) and (4.9) holding for wε00 as well as the representations (4.11a), (4.13),and (4.14) for uln in the combination of functions wε00 and u

ln defined in (4.19).


Third, we construct a boundary layer which realizes the second-order correction touε − u0 as it will be proved further in Theorem 4.5.

For this task, we consider the vector-valued exterior Dirichlet problem:

−∆wy = 0 in R2 \ ω, (4.22a)

wy = −y on ∂ω, (4.22b)

wy = O( 1|y|)

as |y| ↗ ∞. (4.22c)

The boundary value problem (4.22) admits the following weak formulation: Findwy = ((wy)1, (wy)2)

> ∈W 1,2µ (R2 \ ω;R)2 such that

wy = −y on ∂ω, (4.23a)∫R2\ω

Dwy∇v dy = 0 for all v ∈W 1,2µ (R2 \ ω;R) : v = 0 on ∂ω. (4.23b)

The unique solution to (4.23) is guaranteed by existence theorems in [7].Moreover, Dwyν ∈ H−1/2(∂ω;R)2 is well defined at the boundary ∂ω by Green’s

formula holding for harmonic functions u ∈W 1,2µ (R2 \ ω;C) with ∆u = 0 (see [7]):∫R2\ω∇u · ∇u dy = −〈∂u∂ν , u〉∂ω for all u ∈W

1,2µ (R2 \ ω;C) (4.24)

with the paring 〈∂u∂ν , u〉∂ω between u ∈ H1/2(∂ω;C) and ∂u∂ν ∈ H

−1/2(∂ω;C).After rescaling y = x−x0ε we reduce the exterior Dirichlet problem to the bounded

domain Ω \ ωε(x0) which is described in the following lemma.

Lemma 4.4. The rescaled solution wεy(x) := wy(x−x0ε ) to (4.23) implies the vector-

function wεy ∈ H1(Ω \ ωε(x0);R)2 which fulfills the following relations:

wεy = −x−x0ε on ∂ωε(x0), (4.25a)∫

Ω\ωε(x0)Dwεy∇u dx =

∫ΓN

(Dwεyν)u dSx

for all u ∈ H1(Ω \ ωε(x0);R) : u = 0 on ΓD ∪ ∂ωε(x0).(4.25b)


wεy(x) =ερ

12πPω x̂+W

εy (x) for x ∈ R2 \Bε(x0), (4.26)

with Pω called polarization matrix in [53, Note G] and the residual function W εy =((W εy )1, (W

εy )2)

> ∈ H1(Ω \ ωε(x0);R)2 such that for ρ > ε, θ ∈ (−π, π] it holds∫ π−πW εy dθ =

∫ π−πW εy x̂ dθ = 0, W

εy = O

(( ερ)

2). (4.27)

34 V. A. Kovtunenko

The entries of the 2-by-2 matrix Pω have the implicit expression (cf. (3.28)):

(Pω)ij = −δij meas2(ω)− 〈∂(wy)i∂ν , yj〉∂ω, i, j = 1, 2. (4.28)

−Pω is symmetric positive semi-definite (Spsd), and symmetric positive definite (Spd)if meas2(ω) > 0. For ellipsoidal shapes ω it has the explicit expression (cf. (3.29))

Pω = Θ(α)Pω′Θ(α)>, Pω′ = −π(a+ b)

(a 00 b

)(4.29)

with the ellipse major a = 1 and minor b ∈ (0, 1] semi-axes, where the major axis hasan angle of α ∈ (−π2 ,

π2 ) with the y1-axis counted in the anti-clockwise direction.

Proof. Following the proof of Lemma 3.4 and employing the radial functions x̂n fromProposition 2.3, the harmonic function wy in (4.22) obeys the Fourier series

wy(y) =∞∑n=1

1|y|nC

ynx̂

n for y ∈ R2 \B1(0)

with unknown coefficient matrices Cyn ∈ R2×2, n ∈ N. This implies

wy(y) =1|y|

12πPωx̂+Wy(y),

12πPω := C

y1 , y ∈ R

2 \B1(0) (4.30)

with the residual Wy = ((Wy)1, (Wy)2)> ∈W 1,2µ (R2 \ ω;R)2 such that∫ π−πWy dθ =

∫ π−πWy x̂ dθ = 0, Wy = O(|y|2) for |y| > 1, θ ∈ (−π, π]. (4.31)

After the transformation y = x−x0ε using the calculus (3.22), from (4.30) and (4.31)we get (4.26) and (4.27) for wεy(x) := wy(

x−x0ε ) and W

εy (x) :=Wy(

x−x0ε ).

The coordinate transformation y = x−x0ε applied to the boundary value problem(4.22) and supported by the differential calculus in (3.26) leads to the relations :

−∆wεy = 0 in R2 \ ωε(x0), (4.32a)

wεy = −x−x0ε on ∂ωε(x0). (4.32b)

The solution to (4.32a) is locally H2-smooth, hence Dwεyν ∈ L2(ΓN ;R)2, and theinclusion wεy ∈W

1,2µ (R2 \ ωε(x0);R)2 implies wεy ∈ H1(Ω \ ωε(x0);R)2. Therefore,

applying Green’s formula (4.7) to wεy, from (4.32) we derive the weak formulation(4.25) of the transformed problem in the bounded domain.

To get the expressions of matrix Pω, we follow the lines in the proof of Lemma 3.5.In the near-field B1(0) \ ω, due to (4.22a) from the second Green formula we have∫∂B1(0)

{∂(wy)i∂|y| yj − (wy)i

∂yj∂|y|}dSy = 〈∂(wy)i∂ν , yj〉∂ω −

∫∂ω(wy)i

∂yj∂ν dSy, i, j = 1, 2,


with the duality pairing defined in (4.24), and the Dirichlet condition (4.22b) follows

−∫∂B1(0)

{∂(wy)i∂|y| − (wy)i

}x̂j dSy = −〈∂(wy)i∂ν , yj〉∂ω −

∫∂ωyiνj dSy. (4.33)

Using (4.30) and (4.31) we calculate the integral on the left-hand side of (4.33) as

∫∂B1(0)

{−∂(wy)i∂|y| + (wy)i

}x̂j dSy =

1π

∫ π−π

2∑l=1

(Pω)ilx̂lx̂j dθ = (Pω)ij

due to (2.27). Applying to the right-hand side of (4.33) the divergence theorem (3.31)this results in expression (4.28).

With the help of the Green formula (4.24) and relations (4.22) we derive that∫R2\ω∇(wy)i · ∇(wy)j dy = 〈∂(wy)i∂ν , yj〉∂ω = 〈

∂(wy)j∂ν , yi〉∂ω, (4.34)

which proves the symmetry (Pω)ij = (Pω)ji in (4.28) as well as the non-negativeness

0 ≤∫R2\ω|∇(ξ1(wy)1 + ξ2(wy)2

)|2 dy =

2∑i,j=1

〈∂(wy)i∂ν ξi, yjξj〉∂ω

for arbitrary ξ = (ξ1, ξ2)> ∈ R2. Therefore, multiplying (4.28) with −ξiξj we get

−2∑

i,j=1

(Pω)ijξiξj = |ξ|2 meas2(ω) +2∑

i,j=1

〈∂(wy)i∂ν ξi, yjξj〉∂ω ≥ |ξ|2 meas2(ω).

This implies that −Pω ∈ Spsd(R2×2), and −Pω ∈ Spd(R2×2) if meas2(ω) > 0.Finally, let ω′ be the canonical ellipsoidal shape with the major a = 1 and the minor

b ∈ (0, 1] semi-axes in respect to Cartesian coordinates (y′1, y′2)>. The ellipse can bewritten in the elliptic coordinates (3.32) in the form (3.33).

Composing the Fourier series for the solution wy′ of (4.22) in R2 \ ω′ (cf. (3.36)):

wy′ =

∞∑n=1

e−nrCy′n (cos(nψ), sin(nψ))

> for r > r0, (4.35)

the unknown coefficient matrices Cy′n ∈ R2×2 in (4.35) are determined from the

boundary condition (4.22b)

wy′∣∣r=r0

=∞∑n=1

e−nr0Cy′n (cos(nψ), sin(nψ))

> = −(a cosψ, b sinψ)> on ∂ω′

36 V. A. Kovtunenko

as e−r0Cy′

1 (cosψ, sinψ)> = −(a cosψ, b sinψ)> and Cy

′n = 0 for all n ≥ 2. Hence-

forth, we obtain the solution analytically

wy′ = −er0−r(a cosψ, b sinψ)> for r ≥ r0. (4.36)

Now we calculate the matrix Pω′ explicitly by substituting (4.36) in the representa-tion formula (4.28). Indeed, using (3.35) the normal derivative of wy′ at ∂ω′ is found

∂wy′∂ν′ =

1κ(r0,ψ)

∂wy′∂r

∣∣r=r0

= 1κ(r0,ψ)(a cosψ, b sinψ)> = (ab ν

′1,baν′2)>,

and it leads to the following two vectors in (4.28) for j = 1, 2:

(Pω′)( · ,j) = −δ( · ,j)meas2(ω′)− I( · ,j), I( · ,j) :=∫∂ω′

(ab ν′1y′j ,baν′2y′j

)>dSy. (4.37)

Using the divergence theorem we calculate two elliptic integrals I( · ,j) in (4.37)

I( · ,j) =

∫ω′(ab y′j,1

bay′j,2)> dy′ = meas2(ω′)

{(ab , 0)

> for j = 1,(0, ba)

> for j = 2,

and arrive at the second formula in (4.29).The first formula in (4.29) is justified by rotation of y′ ∈ ω′ to Θ>(α)y ∈ ω using

the orthogonal matrix Θ(α) given in (3.29a) with the angle of α ∈ (−π2 ,π2 ). Indeed,

the coordinate transformation y = Θy′ applied to the Dirichlet problem (4.23) reads

wy(Θy′) = −Θy′ on ∂ω′,∫R2\ω′

Dwy(Θy′)∇v dy′ = 0 for all v ∈W 1,2µ (R2 \ ω′;R) : v = 0 on ∂ω′.

Henceforth, wy′(y′) = Θ>wy(Θy′) and the representation (4.30) provides Θ>PωΘ =Pω′ , for detail see the proof of Lemma 3.5. Our proof is finished.

We remark that, in the limit case when b↘ +0, explicit formulas (4.29) in Lemma 4.4describe the singular polarization matrix Pω for the straight crack, see [45].

4.2 High-order uniform asymptotic expansion of the Dirichlet problem

With the help of the first and second order correction terms wεln and wεy given in Lem-

mas 4.3 and 4.4 we decompose the residual uε − u0 for the Dirichlet problem (4.2).

Theorem 4.5. The solutions u0 of (2.2), uε of (4.2), w00 of (4.5) and uln of (4.11)composed together in the function wεln given in (4.19), and the solution wy of (4.23)satisfy the following residual error estimate

‖qε2‖H1(Ω\ωε(x0);C) = O(

ε√| ln ε|

), (4.38a)

qε2 := uε − u0 − u

0(x0)wεln

uln(x0)−ln(εcap(ω))− ε∇u0(x0) · wεy. (4.38b)


Proof. For qε2 introduced in (4.38b), we use the boundary conditions (2.2a), (4.2a),and (4.20a) at ΓD to get (4.41a). We apply the local representations (4.9), (4.21), and

u0(x) = u0(x0)(1 + a0) + (ρ+ a1k )∇u0(x0) · x̂+ U01 in Bε(x0) (4.39)

holding due to (2.8) and (2.33), the boundary conditions (4.32b) and (4.20b) implying

wεlnuln(x0)−ln(εcap(ω))

= −1 + (ln ρ−uln(x0))a0+

π2 a2−U

ln0

uln(x0)−ln(εcap(ω))on ∂ωε(x0), (4.40)

and ρx̂ = x− x0 to calculate the residual at ∂ωε(x0):

qε2 = −u0(x0)a0 − a1k ∇u0(x0) · x̂− U01 − u0(x0)

(ln ρ−uln(x0))a0+π2 a2−U

ln0

uln(x0)−ln(εcap(ω)),

hence (4.41b). We subtract from (4.2b) the equations (3.10), (4.20c) multiplied withu0(x0)

uln(x0)−ln(εcap(ω)), and (4.25b) multiplied with ε∇u0(x0), thus obtaining the relations:

qε2 = −u0(x0)W

ε00

uln(x0)−ln(εcap(ω))− ε∇u0(x0) · wεy on ΓD, (4.41a)

qε2 = −u0(x0)(ln ρ−ln(εcap(ω)))a0+

π2 a2−U

ln0

uln(x0)−ln(εcap(ω))− a1k ∇u

0(x0) · x̂− U01

on ∂ωε(x0),(4.41b)

∫Ω\ωε(x0)

(∇qε2 · ∇u− k2qε2u) dx =∫

Ω\ωε(x0)k2( u0(x0)W ε00uln(x0)−ln(εcap(ω))

+ ε∇u0(x0) · wεy)u dx−

∫ΓNu ∂∂ν

( u0(x0)W ε00uln(x0)−ln(εcap(ω))

+ ε∇u0(x0) · wεy)dSx

for all u ∈ H1(Ω \ ωε(x0);C) : u = 0 on ΓD ∪ ∂ωε(x0).

(4.41c)

For lifting in the boundary conditions (4.41a) and (4.41b), respectively, the cut-offfunctions ηΓD supported in a neighborhood of ΓD such that ηΓD = 1 at ΓD and ηεx0with a local support in B2ε(x0) such that ηεx0 = 1 in Bε(x0) are taken. We define

Rε2 :=(u0(x0)

(ln ρ−ln(εcap(ω)))a0+π2 a2−U

ln0

uln(x0)−ln(εcap(ω))+ a1k ∇u

0(x0) · x̂+ U01)ηεx0

+( u0(x0)W ε00uln(x0)−ln(εcap(ω))

+ ε∇u0(x0) · wεy)ηΓD = O(

ε| ln ε|), Q

ε2 := q

ε2 +R

ε2.

(4.42)

Since 1uln(x0)−ln(εcap(ω))

= O( 1| ln ε|), the asymptotic order in (4.42) is provided by a0 =

a2 = U01 = O(ε

2), a1 = O(ε3), U ln0 = O(ε) holding due to the representations (2.8),(2.34b), and (4.15) in B2ε(x0). At ΓD it is argued by wεy = W ε00 = O(ε) due to(4.10), (4.26), and (4.27). Applying the Cauchy–Schwarz inequality to (4.41c) andusing (4.42), where ‖ηεx0‖H1(Ω\ωε(x0);R) = O(1), we estimate with C,C3, C4 > 0:∣∣∣∫

Ω\ωε(x0)(∇Qε2 · ∇u− k2Qε2u) dx

∣∣∣ ≤ ∣∣∣∫Ω\ωε(x0)

(∇Rε2 · ∇u− k2Rε2u) dx∣∣∣

+ C3ε√| ln ε|‖u‖

L2(Ω\ωε(x0);C)+ C4ε| ln ε|‖u‖L2(ΓN ;C) ≤

Cε√| ln ε|‖u‖

H1(Ω\ωε(x0);C).

(4.43)

38 V. A. Kovtunenko

Here we have utilized wεy =Wε00 = O(ε) at ΓN due to (4.10), (4.26), and (4.27).

The asymptotic order ε√| ln ε|

in Ω \ ωε(x0) was calculated as follows.Inscribing Ω in a ball BR(x0) of radius R > 0 sufficiently large, we decompose it

in the far-field BR(x0) \Bε(x0) and the near-field Bε(x0) \ ωε(x0). In the far-field,∫Ω\Bε(x0)

|W ε00|2 dx ≤∫BR(x0)\Bε(x0)

|W ε00|2 dx ≤ C∫ Rε( ερ)

2 ρdρ = O(ε2| ln ε|) (4.44)

due to (4.10), hence ‖ Wε00

uln(x0)−ln(εcap(ω))‖L2(BR(x0)\ωε(x0);C)

= O(

ε√| ln ε|

), and analog

∫Ω\Bε(x0)

|wεy|2 dx ≤∫BR(x0)\Bε(x0)

|wεy|2 dx ≤ C∫ Rε

( ερ)2 ρdρ = O(ε2| ln ε|) (4.45)

due to (4.26) and (4.27), with C > 0, hence ‖εwεy‖L2(BR(x0)\ωε(x0);C) = O(ε2√| ln ε|).

In the near-field, the transformation y = x−x0ε with the calculus (3.26) provides∫Bε(x0)\ωε(x0)

|W ε00|2 dx = ε2∫B1(0)\ω

|W00|2 dy = O(ε2) (4.46)

since W00 in (4.6) does not depend on ε. Analogously for wy from (4.23) we get∫Bε(x0)\ωε(x0)

|wεy|2 dx = ε2∫B1(0)\ω

|wy|2 dy = O(ε2). (4.47)

The estimates (4.42) and (4.43) together with the inf-sup condition (3.8) holding forall u, v ∈ H1(Ω \ ωε(x0);C) : u = v = 0 on ΓD ∪ ∂ωε(x0) prove the asymptoticorder in (4.38a) and the assertion of the theorem.

As a corollary of Theorem 4.5 we state two asymptotic expansions of the low orderin two propositions below.

Proposition 4.6. The solutions u0 of (2.2), uε of (4.2), w00 of (4.5) and uln of (4.11)composed together in the function wεln given in (4.19), satisfy the error estimate

‖qε1‖H1(Ω\ωε(x0);C) = O(ε), qε1 := u

ε − u0 − u0(x0)w

εln

uln(x0)−ln(εcap(ω)). (4.48)

Indeed, using the coordinate transformation y = x−x0ε and calculus (3.26) we have∫Ω\ωε(x0)

|Dwεy|2 dx ≤∫R2\ωε(x0)

|Dwεy|2 dx =∫R2\ω|Dwy|2 dy = O(1),

hence ‖wεy‖H1(Ω\ωε(x0);C)2 = O(1) due to (4.45) and (4.47), and from (4.38) it follows

‖qε1‖H1(Ω\ωε(x0);C) = ‖q2ε + ε∇u0(x0) · wεy‖H1(Ω\ωε(x0);C) = O(ε).


Proposition 4.7. The solutions u0 of (2.2) and uε of (4.2) satisfy the error estimate

‖uε − u0‖H1(Ω\ωε(x0);C)

= O( 1| ln ε|

). (4.49)

Moreover, at ΓN the estimate (4.49) can be improved as

‖uε − u0‖L2(ΓN ;C)

= O(

ε√| ln ε|

). (4.50)

Proof. The homogeneity argument and Lemma 4.3 supported by estimates (4.44) and(4.46) provides similarly ‖wεln‖H1(Ω\ωε(x0);C) = O(1) and leads to (4.49) for u

ε − u0 =

qε1 +u0(x0)w

εln

uln(x0)−ln(εcap(ω))in view of Proposition 4.6.

The estimate (4.50) is argued by the representation (4.38) and wεy =Wε00 = O(ε) at

ΓN holding due to (4.10), (4.26), and (4.27). The proof is completed.

We note that the residual estimate (4.50) will be needed for (4.53) in the next section.

4.3 Inverse Helmholtz problem under Dirichlet boundary condition

For the objective function J defined in (3.47) and characterizing the misfit

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