Regularized Combined Field Integral Equations

Digital Object Identifier (DOI) 10.1007/s00211-004-0579-9Numer. Math. (2005) 100: 1–19 Numerische

Mathematik

Regularized Combined Field Integral Equations

A. Buffa1, R. Hiptmair2

1 Istituto di Matematica applicata e tecnologie informatiche del CNR, Pavia, Italye-mail: [email protected]

2 SAM, ETH Zurich, 8092 Zurich, e-mail: [email protected]

Received September 22, 2003 / Revised version received Septemebr 27, 2004Published online: February 16, 2005 – c© Springer-Verlag 2005

Summary. Many boundary integral equations for exterior boundary valueproblems for the Helmholtz equation suffer from a notorious instability forwave numbers related to interior resonances. The so-called combined fieldintegral equations are not affected. However, if the boundary � is not smooth,the traditional combined field integral equations for the exterior Dirichletproblem do not give rise to an L2(�)-coercive variational formulation. Thisfoils attempts to establish asymptotic quasi-optimality of discrete solutionsobtained through conforming Galerkin boundary element schemes.

This article presents new combined field integral equations on two-dimen-sional closed surfaces that possess coercivity in canonical trace spaces. Themain idea is to use suitable regularizing operators in the framework of bothdirect and indirect methods. This permits us to apply the classical conver-gence theory of conforming Galerkin methods.

Mathematics Subject Classification(2000): 35J05, 65N38, 65N12

1 Introduction

Given a bounded domain �− ⊂ R3 we consider the exterior Dirichlet prob-

lem for the Helmholtz equation, cf. [30, Ch. 9]

�U + κ2U = 0 on �+ := R \ �− ,(1.1)

U = g on ∂� for some g ∈ H12 (�) ,(1.2)

∂U

∂r(x) − iκU(x) = o(r−1) uniformly as r := |x| → ∞.(1.3)

Correspondence to: Ralf Hiptmair

2 A. Buffa, R. Hiptmair

Here κ > 0 stands for a fixed wave number. The above equations are amodel for acoustic scattering at a sound-hard obstacle [14, Sect. 2.1].

We take for granted that the boundary � := ∂� is Lipschitz continu-ous. Thus, it will possess an exterior unit normal vectorfield n ∈ L∞(�)

pointing from �− into �+ . Numerical approximation in mind, we will evenassume that � is a curvilinear Lipschitz polyhedron in the parlance of [16].This will cover most geometric arrangements that occur in practical simula-tions. We emphasize that non-smooth geometries are the main focus of thispaper.

It is well known that solutions of the above exterior boundary value areunique, see [30, Thm. 9.10]:

Theorem 1.1 The exterior Dirichlet problem (1.1), (1.2), and (1.3) for theHelmholtz equation has at most one weak solution U ∈ H 1

loc(�+).

Many different strategies are available to tackle (1.1)–(1.3) numerically:one could truncate �+ and use standard finite elements in conjunction withabsorbing boundary conditions [24]. An alternative is provided by infinitefinite elements in �+ [5,4] or the method of fundamental solutions [20].

However, in this article we focus on integral equations methods, whichreduce the problem (1.1)–(1.3) to equations on the bounded surface �. Thisclass of methods comprises a variety of approaches, among them direct andindirect methods. Unfortunately, the formulations that can be derived froman integral representation formula for Helmholtz solutions in a straightfor-ward fashion display a worrisome instability: if κ2 agrees with a Dirichlet orNeumann eigenvalue (resonant frequency) of the Laplacian in �−, then theintegral equations may fail to possess a unique solution. In light of Thm. 1.1this has been dubbed a spurious resonance phenomenon.

Spurious resonances are particularly distressing for numerical proceduresbased on the integral equations, because whenever κ2 is close to an interiorresonant frequency the resulting linear systems of equations may not be use-less, but will be extremely ill-conditioned: see the profound analysis of theimpact of spurious resonances in the case of electromagnetic scattering givenin [13].

One way to deal with spurious resonances is the use of integral oper-ators with modified kernels [35,26]. Here we will restrict our attention toanother remedy, namely the widely used combined field integral equations(CFIE). They owe their name to the typical complex linear combinationof different boundary integral operators on the left hand side of the fi-nal boundary integral equation. In the case of indirect schemes this trickhas independently been discovered by Brakhage and Werner [6], Leis [29],and Panich [31] in 1965. In 1971 Burton and Miller used the same ideato obtain direct boundary integral equations without spurious resonances

Regularized Combined Field Integral Equations 3

[11]. Meanwhile, CFIEs have become the foundation for numerous numer-ical methods in direct and inverse acoustic and electromagnetic scattering[14, Ch. 3 & 6].

In terms of mathematical analysis many combined field integral equationsare challenging. This is particularly true for non-smooth surfaces, for whichthe double layer integral operator is no longer a compact perturbation of theidentity in L2(�). Thus, in the case of the exterior Dirichlet problem for non-smooth scatterers, Fredholm theory in L2(�) can no longer be used to settlethe issue of existence and uniqueness of weak solutions of the traditionalCFIE. Of course, this also thwarts any attempt to establish convergence forGalerkin boundary element discretizations.

Hence, modified CFIE involving a regularizing operator have been sug-gested, mainly for theoretical purposes [14,31], though. An exception knownto us is [3] where regularization in the spirit of this paper was used to get ridof a hypersingular operator in the boundary integral equation for the Neu-mann problem. Yet, little analysis was provided and the authors did not aimat a mixed Galerkin formulation. A profound investigation of regularizationstrategies for indirect CFIEs with numerical applications in mind is given inthe recent article [10]. There, in contrast to the current paper, regularizationis based on smoothing integral operators, and no Galerkin error analysis isprovided.

In this paper we take the cue from the idea to introduce regularizing opera-tors. We derive new variational formulations that are coercive in natural tracespaces and cast them in mixed form, in order to avoid products of boundaryintegral operators. Galerkin boundary element discretizations are shown toyield asymptotically quasi-optimal approximate solutions. A similar analysishas successfully been applied to electromagnetic boundary integral equations[9].

The focus of the present article is the rigorous theoretical examinationof the new combined field integral equations and of their Galerkin dis-cretization. The optimal choices of parameters and actual numerical per-formance are not addressed yet and will be treated in future work. The sameis true for the key issue of how the estimates for the Galerkin discretiza-tion error depend on the wave number κ . For certain (regularized) com-bined field integral equations on a sphere, this dependence is investigated in[10,21].

2 Boundary integral operators

For the sake of completeness, in this section we review important proper-ties of boundary integral operators related to Helmholtz’ equation. The main


reference is the pioneering work by M. Costabel [15] and the textbooks [30,33]. Throughout, for known results we will mainly refer to [30]. Referencesto original work can be found in the bibliography of [30].

Without further explanation we will use Sobolev spaces Hs , s ∈ R, ondomains and boundaries, in particular H 1(�), H

12 (�), and H− 1

2 (�), cf. [1],[30, Ch. 2]. The corresponding Frechet spaces on unbounded domains willbe tagged by a subscript loc, e.g.

H 1loc(�) := {u : � �→ C : u|�∩K ∈ H 1(� ∩ K) for any compact K ⊂ R

3}.

Further, we adopt the notation Hscomp(�) for compactly supported distribu-

tions in Hs(�). For s < 0 these space are to be read as dual spaces ofHs

loc(�).Writing

Hloc(�, �) := {U ∈ H 1loc(�), �U ∈ L2

loc(�)} .

for the domain of the Laplacian, we have continuous and surjective traceoperators, cf. [15, Lemma 3.2],

Dirichlet trace γD : H 1loc(�) �→ H

12 (�) ,

Neumann trace γN : Hloc(�, �) �→ H− 12 (�)

that generalize the following pointwise traces of smooth U ∈ C∞(�),

(γDU)(x) := U(x) and (γNU)(x) := grad U(x) · n(x) , x ∈ � ,

respectively.So far � ⊂ R

3 has been a generic Lipschitz domain. Returning to ourparticular setting, superscripts + and − will tag traces from �+ and �−,respectively. Jumps are defined as

[γDU

]�

= γ +D U − γ −

D U ,[γNU

]�

= γ +N U − γ −

N U .

Averages are denoted by

{γDU}� = 12 (γ +

D U + γ −D U) , {γNU}� = 1

2 (γ +N U + γ −

N U) .

We recall that the bi-linear symmetric pairing

〈ϕ, v〉� :=∫

�

ϕv dS , ϕ, v ∈ L2(�) ,


can be extended to a duality pairing on H− 12 (�)×H

12 (�). We also recall the

integration by parts formulas∫

�−

grad U · grad V + �U V dx = ⟨γ −

N U, γ −D V

⟩�

,(2.1)

−∫

�+

grad U · grad V + �U V dx = ⟨γ +

N U, γ +D V

⟩�

,(2.2)

for U ∈ Hloc(�, �±), V ∈ H 1comp(�

±).For fixed wavenumber κ > 0 a distribution U in R

3 is called a radiatingHelmholtz solution, if

�U + κ2U = 0 in �− ∪ �+ ,

∂U

∂r(x) − iκU(x) = o(r−1) uniformly as r := |x| → ∞ .(2.3)

Based on the Helmholtz kernel

�κ(x, y) := exp(iκ|x − y|)4π |x − y| .

we can state the transmission formula for radiating Helmholtz solution U

[30, Thm. 6.10]

U = −κSL(

[γNU

]�) + κ

DL([γDU

]�)(2.4)

with potentials

single layer potential: κSL(λ)(x) = ∫

�

�κ(x, y)λ(y) dS(y) ,

double layer potential: κDL(u)(x) = ∫

�

∂�κ(x, y)

∂n(y)u(y) dS(y) .

The potentials themselves provide radiating Helmholtz solutions, that is

(� + κ2)κSL = 0 , (� + κ2)κ

DL = 0 in �− ∪ �+ .(2.5)

Moreover, they give rise to continuous mappings, see [30, Thm. 6.12],

κSL : H− 1

2 (�) �→ H 1loc(R

3) ∩ Hloc(�, �− ∪ �+) ,(2.6)

κDL : H

12 (�) �→ Hloc(�, �− ∪ �+) .(2.7)

This means that we can apply the trace operators to the potentials. Thiswill yield the following four continuous boundary integral operators, cf. [30,


Thm. 7.1], but also [15] and [27].

Vκ : Hs(�) �→ Hs+1(�), −1 ≤ s ≤ 0 , Vκ := {γDκ

SL

}�

,

Kκ : Hs(�) �→ Hs(�), 0 ≤ s ≤ 1 , Kκ := {γDκ

DL

}�

,

K∗κ : Hs(�) �→ Hs(�), −1 ≤ s ≤ 0 , K∗

κ := {γNκ

SL

}�

,

Dκ : Hs(�) �→ Hs−1(�), 0 ≤ s ≤ 1 , Dκ := − {γNκ

DL

}�

.

By the jump relations [30, Thm. 6.11]

[γDκ

SL(λ)]�

= 0 ,[γNκ

SL(λ)]�

= −λ , ∀λ ∈ H− 12 (�) ,[

γDκDL(u)

]�

= u ,[γNκ

DL(u)]�

= 0 , ∀u ∈ H12 (�) .

we find

γ −D κ

DL = Kκ − 12Id , γ +

D κDL = Kκ + 1

2Id ,

γ −N κ

SL = K∗κ + 1

2Id , γ +N κ

SL = K∗κ − 1

2Id .(2.8)

Crucial will be the ellipticity of the single layer boundary integral operatorin the natural trace norms [30, Cor. 8.13]

∃cV > 0 : 〈ϕ, V0ϕ〉� ≥ cV ‖ϕ‖2

H− 1

2 (�)∀ϕ ∈ H− 1

2 (�) .(2.9)

Lemma 2.1 The operator Vκ − V0 : H− 12 (�) �→ H

12 (�), is compact.

Proof Note that

exp(iκ|z|) − 1

4π |z| = |z|G1(|z|2) + iG2(|z|2) , z ∈ R3 ,

with bounded analytic functions G1, G2 : R �→ R, This shows that all secondderivatives

∂2

∂zi∂zj

exp(iκ|z|) − 1

4π |z| , i, j = 1, 2, 3 ,(2.10)

are weakly singular, that is, grow like O(|z|−1) as |z| → 0. As a conse-quence, the kernels (2.10) will each give rise to an integral operator of order−2. Therefore, the Newton potential

(Nκf )(x) :=∫

R3

exp(iκ|x − y|) − 1

4π |x − y| f (y) dy(2.11)


will provide a pseudo-differential operator of order −4, since, for any i, j =1, 2, 3,

∂2Nκf

∂xi∂xj

(x) =∫

R3

{∂2

∂zi∂zj

exp(iκ|z|) − 1

4π |z|}

z=x−y

f (y) dy

belongs to H 1loc(R

3), if f ∈ H−1comp(R

3). In other words, the Newton potential(2.11) provides a continuous mapping from H−1

comp(R3) to H 3

loc(R3).

From this we conclude the continuity of

Vκ − V0 = γD ◦ Nκ ◦ γ ∗D : H− 1

2 (�) �→ H 1(�) ,

where we have used a representation of the single layer boundary integraloperators given, for instance, in [33, Sect. 3.1]. The compact embeddingH 1(�) ↪→ H

12 (�) finishes the proof. ��

3 Indirect boundary integral equations

We recall that indirect methods are based on potential representations for(exterior) radiating Helmholtz solutions in �+. By virtue of (2.5) we may set

U = κSL(φ), φ ∈ H− 1

2 (�) or U = κDL(u), u ∈ H

12 (�) .(3.1)

Applying γ +D to (3.1) and using (2.8) we obtain the following integral equa-

tions for the exterior Dirichlet problem:

Vκ(φ) = g or (Kκ + 12Id)u = g .(3.2)

However, these boundary integral equations are haunted by the problem of“resonant frequencies” [12, Sect. 7.7]: if κ2 is a Dirichlet eigenvalue of −�

in �−, then the Neumann traces of the corresponding eigenfunctions willbelong to the kernel of Vκ . Conversely, the kernel of Kκ + 1

2Id consists of theDirichlet traces of interior Neumann eigenfunctions of −�. This destroysinjectivity of the operators in the boundary integral equations and bars usfrom applying the powerful Fredholm theory.

3.1 Classical CFIE

As pointed out in the introduction, the awkward lack of uniqueness of solu-tions of (3.2) at resonant frequencies led to the development of the classicalcombined field integral equation [14, Sect. 3.2]. It can be obtained by anindirect approach starting from the trial expression

U = κDL(u) + iηκ

SL(u) ,(3.3)


with real η �= 0. Applying the exterior Dirichlet trace results in the boundaryintegral equation

g = ( 12Id + Kκ)u + iηVκu .(3.4)

Actually, this equation is set in H12 (�) and the density u should be sought in

H− 12 (�). Since we plug it into the double layer potential, this is not possible,

unless we use a pairing in H− 12 (�) to convert the equation into weak form.

Yet, this will introduce products of non-local operators, which have to bediscretized with great care. The fundamental difficulty is that we cannot usematching trial and test spaces, because in light of (2.6) and (2.7) the poten-tials involved in (3.1) should be applied to functions with different regularity.Hence, we have to abandon the framework of natural trace spaces, shift theequation (3.4) into L2(�) and seek the unknown density u in L2(�), too.

A key argument in the theoretical treatment of (3.4) in L2(�) is the com-pactness of the double layer potential operator Kκ : L2(�) �→ L2(�) onsmooth surfaces, which relegates the boundary integral operator associatedwith (3.4) to a compact perturbation of the identity. On non-smooth surfacesthis argument is not available.

3.2 Regularized formulation

In order to remedy the mismatch of the regularity of the arguments in (3.3),regularization is a natural idea. It amounts to introducing another operatorthat lifts either the argument of the double layer potential or the argumentof the single layer potential into the “right” trace space. Here, following[31], regularization will target the double layer potential. The objective isto force the boundary integral operator arising from the regularized doublelayer potential to become a compact perturbation of the single layer boundaryintegral operator.

First we adopt an abstract perspective and call an operator M : H− 12 (�) �→

H12 (�) a regularizing operator, if it satisfies

(i) M is compact and(3.5)

(ii) Re{〈ϕ, Mϕ〉�} > 0 for all ϕ ∈ H− 12 (�) \ {0} .(3.6)

The new regularized indirect method is based on the trial expression

U = κDL(Mϕ) + iηκ

SL(ϕ) , ϕ ∈ H− 12 (�) ,(3.7)

for a fixed η ∈ R \ {0}. As above, U is a radiating Helmholtz solution in�− ∪ �+. If we apply the Dirichlet trace, we arrive at the boundary integralequation

g = (( 12Id + Kκ) ◦ M)(ϕ) + iηVκϕ in H

12 (�) .(3.8)


This prompts us to introduce the boundary integral operator

Sκ := ( 12Id + Kκ) ◦ M + iηVκ .

Recalling (2.9) and the compactness of M, it is clear that Sκ will spawn aH− 1

2 (�)-coercive bi-linear form in the sense that there is a compact bi-linearform k : H− 1

2 (�) × H− 12 (�) �→ C such that

∃C > 0 : | 〈ϕ, Sκϕ〉� + k(ϕ, ϕ)| ≥ C ‖ϕ‖2

H− 1

2 (�)∀ϕ ∈ H− 1

2 (�) .

Lemma 3.1 The boundary integral operator Sκ : H− 12 (�) �→ H

12 (�) is

injective.

Proof The same idea as in the proof of injectivity of the boundary inte-gral operators of the classical CFIE (3.4) can be applied: we assume thatϕ ∈ H− 1

2 (�) solves

Sκϕ = (( 12Id + Kκ) ◦ M)(ϕ) + iηVκϕ = 0 .

It is immediate from the jump relations that U given by (3.7) is a Helmholtzsolution with γ +

D U = 0, which, by Thm. 1.1, implies U = 0 in �+. Hence,the jump relations confirm that

γ −D U = −Mϕ , γ −

N U = iηϕ .

Next, we use the integration by parts formula (2.1) and get

−iη 〈ϕ, Mϕ〉� = ⟨γ −

N U, γ −D U

⟩�

=∫

�−| grad U |2 − κ2|U |2 dx ∈ R

Necessarily, Re{〈Mϕ, ϕ〉�} = 0, which, by requirement (3.6), can only besatisfied, if ϕ = 0. ��

Eventually, Lemma 3.1 allows to deduce existence of solutions of (3.8)by means of a Fredholm argument, cf. [30, Thm. 2.34].

In the construction of our first concrete regularizing operator, we makeexplicit use of �− being a (curvilinear) Lipschitz polyhedron: denote by�1, . . . , �p, p ∈ N, its smooth (curved) polygonal faces and introduce thespace

H 1pw,0(�) := H 1

0 (�1) × · · · × H 10 (�p) ⊂ H 1(�) .(3.9)

Then define the regularizing operator M : H−1(�) �→ H 1pw,0(�) by

⟨grad� Mϕ, grad� v

⟩�

= 〈ϕ, v〉� ∀v ∈ H 1pw,0(�), ϕ ∈ H−1(�) .(3.10)

In words, M is a combination of inverse Laplace–Beltrami operators on theindividual faces �i , i = 1, . . . , p. Continuity of M is straightforward. Thenext lemma shows that M is injective, when restricted to H− 1

2 (�).


Lemma 3.2 The space H 1pw,0(�) is dense in H

12 (�).

Proof Denote by� the union of closed edges of�−.We can rely on Lemma 2.6in [17] that claims that the embedding

C∞� := {u ∈ C∞(�−), supp u ∩ � = ∅} ⊂ H 1(�−)

is dense. Obviously, γ −D (C∞

� ) ⊂ H 1pw,0(�) and the continuity of

γ −D : H 1(�−) �→ H

12 (�) finishes the proof. ��

We conclude that for ϕ ∈ H− 12 (�)

Mϕ=0 ⇒ 〈ϕ, v〉� =0 ∀v ∈ H 1pw,0(�) by (3.10)

⇒ 〈ϕ, v〉� =0 ∀v ∈ H12 (�) by Lemma 3.2

⇒ ϕ = 0 by duality of H12 (�) and H− 1

2 (�).

In particular, this involves

〈ϕ, Mϕ〉� = |Mϕ|2H 1(�) > 0 ∀ϕ ∈ H− 1

2 (�) \ {0} ,(3.11)

which amounts to requirement (3.6).Now, regard M as an operator M : H− 1

2 (�) �→ H 1pw,0(�).As such it inher-

its compactness from the embeddings H 1(�) ↪→ H12 (�) and, consequently,

meets requirement (3.5).As the equation (3.8) is set in the space H

12 (�), a natural weak formula-

tion can be obtained by testing with functions in H− 12 (�). However, in the

context of a Galerkin discretization it is not entirely clear how to deal withthe products of boundary integral operators occurring in the definition of Sκ .The usual trick to avoid operator products is to switch to a mixed formulation.Here, this is done by introducing the new unknown u := Mϕ ∈ H 1

pw,0(�).The definition of M is used as second variational equation, which leads to thefollowing saddle point problem: seek ϕ ∈ H− 1

2 (�), u ∈ H 1pw,0(�), such that

iη 〈ξ, Vκϕ〉� + ⟨( 1

2Id + Kκ)u, ξ⟩�

= 〈g, ξ〉� ∀ξ ∈ H− 12 (�) ,

− 〈ϕ, v〉� + ⟨grad� u, grad� v

⟩�

= 0 ∀v ∈ H 1pw,0(�) .

(3.12)

It goes without saying that the first component of a solution (ϕ, u) of (3.12)will give us a solution of (3.8). Thus, Lemma 3.1 and the injectivity of Mconfirm the uniqueness of solutions of (3.12).

Next, we aim to identify compact perturbations of the bi-linear forma : (H− 1

2 (�) × H 1pw,0(�)) × (H− 1

2 (�) × H 1pw,0(�)) �→ C associated with

(3.12). First of all the term 〈ϕ, v〉� : H 1pw,0(�) × H− 1

2 (�) �→ C is compact


thanks to the compactness of the embedding H 1pw,0(�) ↪→ H

12 (�). More-

over, recall the continuity Kκ : H12 (�) �→ H

12 (�) and take into account

Lemma 2.1. Hence, up to compact perturbations we need only to examinethe modified bi-linear form that comprises the principal parts of the diagonalterms of (3.12)

a

((λ

u

),

(µ

v

)):= iη 〈µ, V0λ〉� + ⟨

grad� u, grad� v⟩�

,(3.13)

λ, µ ∈ H− 12 (�), u, v ∈ H 1

pw,0(�). By (2.9) and the Poincare-Friedrichsinequalities on the faces �i , i = 1, . . . , p, it is obvious that a is ellipticon H− 1

2 (�) × H 1pw,0(�). This permits us to conclude that the bilinear form

belonging to (3.12) is coercive on H− 12 (�) × H 1

pw,0(�).

Remark 3.1 It is also possible to use M := (−�� + Id)−1, where �� :H 1(�) �→ H−1(�) is the Laplace-Beltrami operator on all of �. The ratio-nale why we opted for a localized operator M is explained in Sect. 5.

4 Direct boundary integral equations

The direct approach to the derivation of boundary integral equations uses therepresentation formula

U = κDL(γ +

D U) − κSL(γ +

N U) ,(4.1)

valid for any exterior Helmholtz solution. Applying both the Dirichlet andNeumann trace operator to (4.1), we obtain the formulas of the Calderonprojector

γ +D U = (Kκ + 1

2Id)(γ +D U) − Vκ(γ

+N U) ,(4.2)

γ +N U = −Dκ(γ

+D U) − (K∗

κ − 12Id)(γ +

N U) .(4.3)

From these equations we can extract two boundary integral equations re-lated to the exterior Dirichlet problem for the Helmholtz equation. Since, theboundary integral operators applied to the unknown Cauchy datum γ +

N U arethe same as in (3.2), these boundary integral equations will also be affectedby spurious resonances.

4.1 Classical CFIE

It was the idea of Burton and Miller in [11] to consider the following complexlinear combination of the two equations (4.2) and (4.3)

(iη(Kκ − 12Id) − Dκ)(γ

+D U) − (iηVκ + 1

2Id + K∗κ)(γ

+N U) = 0 ,(4.4)


where η �= 0 is a real parameter. Then, the boundary integral equation for theexterior Dirichlet problem integral equation reads

ϕ ∈ H− 12 (�) : (iηVκ + 1

2Id + K∗κ)(ϕ) = (iη(Kκ − 1

2Id) − Dκ)(γ+D U) .

As before this equation eludes a simple variational analysis in natural tracespaces, because it cannot be tested against functions in H− 1

2 (�). Lifting itto L2(�) is a remedy only on smooth surfaces, cf. Sect. 3.1. Again, it takesregularization to get a coercive variational formulation.

4.2 Regularized formulation

The strategy for regularization closely follows the “double layer regulariza-tion” of the indirect CFIE elaborated in Sect. 3.2. Again, we assume that aregularizing operator M : H− 1

2 (�) �→ H12 (�) that satisfies (3.5) and (3.6) is

at our disposal.Now, the trick is to apply M to (4.3) before adding it to iη·(4.2). Doing

so is strongly suggested by the fact that Dirichlet traces and Neumann tracesbelong to different spaces so that γ +

N U should be lifted into H12 (�) before

adding it to iη · γ +D U . This yields the following boundary integral equation

for the exterior Dirichlet problem

Sκ(ϕ) = (iη(Kκ − 12Id) − M ◦ Dκ)g ,(4.5)

where ϕ ∈ H− 12 (�) is the unknown Neumann datum and

Sκ := M ◦ (K∗κ + 1

2Id) + iηVκ .

The first result corresponds to Lemma 3.1.

Lemma 4.1 The boundary integral operator Sκ : H− 12 (�) �→ H

12 (�) is

injective.

Proof We consider ϕ ∈ H− 12 (�) with Sκϕ = 0 and set U = κ

SL(ϕ). Thanksto (2.5) and the jump relations U|�− is a solution of

�U + κ2U = 0 in �− ,

M(γ −N U) + iη γ −

D U = 0 on � .(4.6)

This is clear from the jump relations for the single layer potential. The Helm-holtz solution U ∈ H 1(�−) satisfies∫

�−grad U · grad V − κ2U V dx − ⟨

γ −N U, γ −

D V⟩�

= 0 ∀V ∈ H 1(�−) .


Now, test with V = U and use the boundary conditions from (4.6) to expressγ −

D U

∫

�−| grad U |2 − κ2|U |2 dx + i

η

⟨γ −

N U, M(γ −N U)

⟩�

= 0 .

Equating imaginary parts and using the assumption (3.6) on M we findγ −

N U = 0, which implies γ −D U = 0. Hence, U|�− ≡ 0. The jump relations

for the single layer potential involve γ +D U = 0, which, by Thm. 1.1, means

U|�+ = 0.As another consequence of the jump relations,ϕ = − [γNU

]�

= 0.��

This paves the way for applying the Fredholm alternative to (4.5): since

Sκ = iηV0 + iη(Vκ − V0) + M ◦ (K∗κ + 1

2Id) ,

we can invoke Lemma 2.1, the continuity of K∗κ , and the compactness of M

to conclude that Sκ : H− 12 (�) �→ H

12 (�) is bijective.

As a concrete incarnation of M we could use the same operator as inSect. 3.2, namely the one given by (3.10). Yet, in the case of the regularizeddirect CFIE there is no reason to eschew the choice M = (−�� + Id)−1,that is, this time we introduce M : H−1(�) �→ H 1(�) by

⟨grad� Mϕ, grad� v

⟩�

+ 〈Mϕ, v〉� = 〈ϕ, v〉� ∀v ∈ H 1(�) .(4.7)

It goes without saying that this M restricted to H− 12 (�) satisfies both (3.5)

and (3.6). This choice of M makes it possible to switch to a simple mixedformulation by introducing the new unknown

u := M(( 12Id + K∗

κ)ϕ + Dκg) ∈ H12 (�) .(4.8)

Actually, u is mislabelled, because it is by no means an unknown: recalling(4.3) we quickly realize that u = 0, if ϕ is the exact Neumann trace. Whatis the point of introducing u, nevertheless? The reason is that we aim to geta variational formulation suitable for Galerkin discretization and in the dis-crete setting the approximation of u does not necessarily vanish. The concretevariational problem reads: seek ϕ ∈ H− 1

2 (�), u ∈ H 1(�) such that for allξ ∈ H− 1

2 (�), v ∈ H 1(�)

(4.9)

iη 〈ξ, Vκϕ〉� + 〈ξ, u〉� = iη⟨ξ, (Kκ − 1

2Id)g⟩�

,

− ⟨( 1

2Id + K∗κ)ϕ, v

⟩�

+ ⟨grad� u, grad� v

⟩�

= 〈Dκg, v〉� .

It is clear that ϕ ∈ H− 12 (�) solves (4.5) if and only if (ϕ, u), u given by (4.8),

solves (4.9). Hence, Lemma 4.1 along with (3.6) also implies uniqueness ofsolutions of (4.9).


As in Sect. 3.2, it is immediate to see that the off-diagonal terms in (4.9)are compact. Eventually, up to compact perturbations, it turns out that thebi-linear form associated with (4.9) equals the H− 1

2 (�) × H 1(�)-ellipticbi-linear form a defined in (3.13). Hence, it is a coercive bi-linear form onH− 1

2 (�) × H 1(�) that underlies the variational problem (4.9).

5 Galerkin discretization

Both variational problems (3.12) and (4.9) have been shown to be based onbi-linear forms that are coercive in the respective function spaces. Moreoever,we have established that they possess unique solutions. It is well known thatthese properties ensure asymptotic quasi-optimality of approximate Galer-kin solutions in the sense that on sufficiently fine meshes the discretizationerror measured in the norm of H− 1

2 (�) × H 1pw,0(�)/H 1(�), is bounded by a

constant times the best approximation error of the trial spaces, see [36], butalso [19,34].

Conforming boundary element spaces for the approximation of functionsin H 1(�), H 1

pw,0(�), and H− 12 (�), respectively, are standard. First, we equip

� with a family {Th}h of triangulations comprising (curved) triangles and/orquadrilaterals. The meshes Th have to resolve the shape of the curvilinearpolyhedron �− in the sense that none of their elements may reach across anedge of �−. Then, we introduce boundary element spaces Sh ⊂ H 1(�) andQh ⊂ H− 1

2 (�), which contain piecewise polynomials of fixed total/maximaldegree k + 1 and k, k ∈ N0, respectively. We will restrict ourselves to thesestraditional boundary element spaces, but we remark that for smooth scatterersspectral discretization schemes are an attractive alternative [22].

For the Galerkin discretization of (3.12) we will need boundary elementsubspaces Spw

h of H 1pw,0(�). They can be constructed as subspaces of Sh by

setting all degrees of freedom located on edges of � to zero.Let h denote the meshwidth of Th and assume uniform shape-regularity,

which, sloppily speaking, imposes a uniform bound on the distortion of theelements. Then we can find constants Cs, Cq > 0 such that [7, Sect. 4.4]

infφh∈Qh

‖φ − φh‖H

− 12 (�)

≤ Cqht+ 1

2 ‖φ‖Ht (�)(5.1)

∀ϕ ∈ Ht(�), ∀h, 0 ≤ t ≤ k + 1 ,

infvh∈Sh

‖v − vh‖H 1(�) ≤ Csht−1 ‖v‖Ht (�)(5.2)

∀v ∈ Ht(�), ∀, 1 ≤ t ≤ k + 2 .

Thus, the quantitative investigation of convergence entails establishing theSobolev regularity of the continuous solutions. We start the investigation ofit for the variational boundary integral equations (3.12) and (4.9).


It is useful to characterize the lifting properties of Neumann-to-Dirich-let maps for the interior/exterior Helmholtz problem by means of two realnumbers α+/α−. In particular, let α−/α+ be a real number such that for an inte-rior/exterior Helmholtz solution γ ±

N U ∈ Hs− 12 (�) implies γ ±

D U ∈ Hs+ 12 (�)

for all s ≤ α± and vice-versa. It is known that for mere Lipschitz domainsα−, α+ ≥ 1

2 , see [25] or [30, Thm. 4.24].We first examine the indirect regularized formulation (3.12) introduced in

Sect. 3.2. If ϕ ∈ H− 12 (�) is the solution of (3.8) and the Helmholtz solution

U is given by (3.7), the jump relations give us[γDU

]�

= Mϕ ,[γNU

]�

= −iηϕ .(5.3)

It is clear that the regularizing properties of M will come into play. To measurethem define for s ≥ 1

Hspw,0(�) := {v ∈ H 1

pw,0(�), v|�i∈ Hs(�i), i = 1, . . . , p} .

We will write β for a real number such that Mv ∈ Hs−1(�) implies v ∈Hs+1

pw,0(�) for all s ≤ β. From [23] we know that β > 12 , and that β ≥ 1 can

be choosen, if all �i are diffeomorphic images of convex polygons.Assume that the Dirichlet boundary values g belong to Hσ+ 1

2 (�). Thismeans that γ +

N U ∈ H− 12 +min{σ,α+}(�). In addition, U|�− satisfies the inho-

mogeneous boundary conditions

iηM(γ −N U) − γ −

D U = iηM(γ +N U) − g .(5.4)

Since γ +N U ∈ H− 1

2 +min{σ,α+}(�), using the mapping property of M, we de-duce that the right hand side of (5.4) belongs to Hr(�), with r = min{ 3

2 +α+, 1 + β, 1

2 + σ }.We first have thatγ −

D U ∈ H min{1,r}(�), thusγ −N U ∈ H

min{0,r−1,− 12 +α−}

(�).Now, a bootstrap argument can be used: by the shift theorem for M,we obtain an improved regularity for M(γ −

N U), namely, M(γ −N U) ∈

Hmin{2,r+1,

32 +α−,1+β}

(�).

Using again (5.4), we then have γ −D U ∈ H

min{2,r,32 +α−,1+β}

(�). Thus, fi-

nally, recalling the definition of r , we have γ −N U ∈ H

min{1,− 12 +α−,β,− 1

2 +σ }(�).

By (5.3), this involves

ϕ ∈ H min{1,β,− 12 +σ,− 1

2 +α+,− 12 +α−}(�) .(5.5)

Note that, since � is a polyhedron, either α+ or α− is smaller than 1. Withoutloss of generality, we can then reduce (5.5) to:

ϕ ∈ H− 12 +min{σ,α+,α−}(�) ,(5.6)


which means that the regularity of ϕ depends only on the regularity of theDirichlet datum and of the interior and exterior Dirichlet-to-Neumann maps.

For the mixed variational problem (3.12) convergence will also hinge onthe regularity of the auxiliary variable u := Mϕ. The regularity (5.6) of ϕ

will directly translate into the regularity

u ∈ Hmin{1+β, 3

2 +σ, 32 +α+, 3

2 +α−}pw,0 (�) .(5.7)

We point out that the approximation estimate (5.2) remains true when wereplace Sh with Spw

h and Ht(�) with Htpw,0(�).

Summing up, in light of asymptotic quasi-optimality, by combining (5.6)and (5.7) with (5.1) and (5.1), we arrive at the following asymptotic a priorierror estimate for the Galerkin boundary element solutions ϕh ∈ Qh anduh ∈ Sh of (4.9)

‖ϕ − ϕh‖H

− 12 (�)

+ ‖u − uh‖H 1(�) ≤ Chmin{β,σ,α+,α−,k+1} ,(5.8)

where C > 0 does not depend on the meshwidth h.

Remark 5.1 Estimate (5.8) highlights the need for good regularity of u, thatis, the possibility to choose large β. This motivates the concrete choice of Min Sect. 3.2. If we had opted for M = (−�� + Id)−1, cf. Rem. 3.1, Thm. 5.3of [8] tells us that u may be only slightly more regular than merely belongingto H 1(�). This could make u severely limit the overall rate of convergence.

A much simpler argument suffices for the direct formulation (4.9) forthe exterior Dirichlet problem. The variational problem features the Cauchydatum ϕ := γ +

N U as the principal unknown.Assuming g ∈ H12 +σ (�), σ ≥ 0,

we conclude ϕ ∈ H min{− 12 +σ,− 1

2 +α+}(�). Moreover, the exact solution for theauxiliary unknown u will be u = 0, which means that it does not affect theasymptotic convergence of the Galerkin scheme. Summing up, we find an apriori estimate O(hmin{σ,α+,k+ 3

2 }) for the rate of convergence.

Remark 5.2 What is the point of trying to “approximate” u = 0 at all?If the variational problem (4.9) was truly elliptic, then a piecewise constant“approximation” of u on the connected components of � would really suffice.However, the considerations in [36] show that for merely coercive variationalproblems some minimal approximation power of the discrete trial spaces isrequired to allow a statement about existence and quality of Galerkin solu-tions. It is an interesting question what this barely sufficient approximationof u may be in the case of (4.9).

Remark 5.3 The behavior of an exterior Helmholtz solution at edges andcorners of � is well known [18]. This gives a lot of information about thelocal behavior of the Neumann trace γ +

N U . This knowledge can be exploited


to construct more efficient locally adapted approximations by means of localanisotropic refinement in conjunction with hp-adaptivity [32].

Remark 5.4 An issue is still looming, namely the “optimal” choice of theparameter η. For the classical CFIE this has been addressed in [2] for a directmethod and in [21,28] for the indirect BIE (3.4). Investigations for a reg-ularized formulations are carried out in [10]. In each case spectral analysison a sphere was the main tool. It should be straightforward to apply it to theformulations proposed in this article.

6 Conclusion

We found that, in the case of non-smooth scatterers, the rigorous variationalanalysis of CFIEs for the exterior Dirichlet problem for the Helmholtz equa-tion has to rely on special regularizing operators. For the indirect methodregularization is aimed at the double layer potential. In the case of the directmethod regularization amounts to a lifting of Neumann traces. In both cases,after introducing an auxiliary unknown, we end up with a coercive mixed var-iational formulation in natural trace spaces, for which conforming Galerkinboundary element schemes enjoy asymptotic quasi-optimality. On top of thatthe regularized direct CFIE instantly yields a priori error estimates that onlydepend on the regularity of the Neumann data: the auxiliary unknown hasno influence. In this respect the direct method seems superior to the indirectapproach. Still, comprehensive numerical experiments and comparisons willhave to be carried out to bear out the practical viability of the proposed newCFIEs.

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Regularized Combined Field Integral Equations

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