Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains S. N. Chandler-Wilde E. A. Spence

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Coercive second-kind boundary integral equations for the

Laplace Dirichlet problem on Lipschitz domains

S. N. Chandler-Wilde∗, E. A. Spence†

May 28, 2024

Abstract

We present new second-kind integral-equation formulations of the interior and exterior

Dirichlet problems for Laplace’s equation. The operators in these formulations are both con-

tinuous and coercive on general Lipschitz domains in Rd,d≥2, in the space L2(Γ), where Γ

denotes the boundary of the domain. These properties of continuity and coercivity immedi-

ately imply that (i) the Galerkin method converges when applied to these formulations; and

(ii) the Galerkin matrices are well-conditioned as the discretisation is reﬁned, without the need

for operator preconditioning (and we prove a corresponding result about the convergence of

GMRES). The main signiﬁcance of these results is that it was recently proved (see Chandler-

Wilde and Spence, Numer. Math., 150(2):299-271, 2022) that there exist 2- and 3-d Lipschitz

domains and 3-d star-shaped Lipschitz polyhedra for which the operators in the standard

second-kind integral-equation formulations for Laplace’s equation (involving the double-layer

potential and its adjoint) cannot be written as the sum of a coercive operator and a compact

operator in the space L2(Γ). Therefore there exist 2- and 3-d Lipschitz domains and 3-d

star-shaped Lipschitz polyhedra for which Galerkin methods in L2(Γ) do not converge when

applied to the standard second-kind formulations, but do converge for the new formulations.

1 Introduction

1.1 Boundary integral equations for Laplace’s equation

If an explicit expression for the fundamental solution of a linear PDE is known, then boundary

value problems (BVPs) for that PDE can be converted to integral equations on the boundary

of the domain. The main advantage of this procedure is that the dimension of the problem is

reduced; indeed, the problem is converted from one on a d-dimensional domain to one on a (d−1)-

dimensional domain. Futhermore, if the original domain is the exterior of a bounded obstacle, then

the problem is reduced from one on a d-dimensional inﬁnite domain, to one on a (d−1)-dimensional

ﬁnite domain.

This reduction to the boundary has both theoretical and practical beneﬁts: on the theoret-

ical side, C. Neumann famously used boundary integral equations (BIEs) to prove existence of

the solution of the Dirichlet problem for Laplace’s equation in convex domains in [91] (see, e.g.,

the account in [76, Chapter 1]), and BIEs have a long history of use in the harmonic analysis

literature to prove wellposedness of BVPs on rough domains (see, e.g., [25], [114], [16], [59,§2.1],

[83], [79, Chapter 15], [109, Chapter 4], [81]). On the more practical side, numerical methods

based on Galerkin, collocation, or numerical quadrature discretisation of BIEs, coupled with fast

matrix-vector multiply and compression algorithms, and iterative solvers such as GMRES, provide

spectacularly eﬀective computational tools for solving a range of linear boundary value problems,

for example in potential theory, elasticity, and acoustic and electromagnetic wave scattering (see,

e.g., [96,4,65,11,27,24,8,119,48,98,18]).

∗Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading, RG6

6AX, UK, S.N.Chandler-Wilde@reading.ac.uk

†Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK, E.A.Spence@bath.ac.uk

arXiv:2210.02432v3 [math.NA] 27 May 2024

Interior Dirichlet Interior Neumann Exterior Dirichlet Exterior Neumann

problem problem problem problem

Direct S H S H

2I−D′1

2I+D1

2I+D′1

2I−D

Indirect S H S H

2I−D1

2I+D′1

2I+D1

2I−D′

Table 1.1: The integral operators involved in the standard boundary-integral-equation formula-

tions of the interior and exterior Dirichlet and Neumann problems for Laplace’s equation.

Let Φ(x,y) be the fundamental solution for Laplace’s equation:

Φ(x,y) := 1

2πlog a

|x−y|, d = 2,:= 1

(d−2)Cd|x−y|d−2, d ≥3,(1.1)

where Cdis the surface area of the unit sphere Sd−1⊂Rdand a > 0. With Γ the boundary of a

bounded Lipschitz domain, the boundary integral operators (BIOs) S,D,D′, and H, the single-

layer,double-layer,adjoint double-layer, and hypersingular operators, respectively, are deﬁned for

ϕ∈L2(Γ), ψ∈H1(Γ), and x∈Γ by

Skϕ(x) = ZΓ

Φk(x,y)ϕ(y) ds(y), Dϕ(x) = ZΓ

∂Φ(x,y)

∂n(y)ϕ(y) ds(y),(1.2)

and

D′ϕ(x) = ZΓ

∂Φ(x,y)

∂n(x)ϕ(y) ds(y), Hψ(x) = ∂

∂n(x)ZΓ

∂Φk(x,y)

∂n(y)ψ(y) ds(y).(1.3)

When Γ is Lipschitz, the integrals in Dand D′are deﬁned as Cauchy principal values, in general

only for almost all x∈Γ with respect to the surface measure ds. The deﬁnition of Hon spaces

larger than H1(Γ) is complicated (it must be understood either as a ﬁnite-part integral, or as

the non-tangential limit of a potential; see [76, Chapter 7], [20, Page 113] respectively), but these

details are not essential to the present paper. The standard mapping properties of S, D, D′, and

Hon Sobolev spaces on Γ are recalled in Appendix A(see (A.3)).

The BIE operators involved in the standard ﬁrst- and second-kind BIEs for the Dirichlet and

Neumann problems for Laplace’s equation are shown in Table 1.1; although we do not explicitly

consider the Neumann problem in this paper, we use the information in this table in what follows.

For the details of the right-hand sides and unknowns for the integral equations corresponding to

the operators in Table 1.1, see, e.g., [98,§3.4], [76, Chapter 7], [105, Chapter 7], [20,§2.5]. Recall

that the adjective “direct” in the table refers to equations where the unknown is either the Dirichlet

or Neumann trace of the solution to the corresponding BVP, and the adjective “indirect” refers to

equations where the unknown does not have immediate physical relevance.

Following [98, Pages 9 and 10], we call BIEs ﬁrst kind where the unknown function only appears

under the integral, and second kind where the unknown function appears outside the integrand as

well as inside; by this deﬁnition, the BIEs in the ﬁrst and third row of Table 1.1 are ﬁrst kind, and

in the second and fourth row second kind. An alternative deﬁnition of second kind BIEs is that,

in addition to the unknown function appearing outside the integrand as well as inside, the BIO

is Fredholm of index zero (i.e., the Fredholm alternative applies to the BIE); see, e.g., [4,§1.1.4].

Every BIE that we describe in the paper as second-kind is second-kind in both senses above.

1.2 The Galerkin method

We focus on solving Laplace BIEs with the Galerkin method. The Galerkin method applied to the

equation Aϕ =f, where ϕ, f ∈ H,A:H → H is a continuous (i.e. bounded) linear operator, and

His a complex1Hilbert space, is: given a sequence (HN)∞

N=1 of ﬁnite-dimensional subspaces of H

with dim(HN)→ ∞ as N→ ∞,

ﬁnd ϕN∈ HNsuch that AϕN, ψN)H=f, ψNHfor all ψN∈ HN.(1.4)

We say that the Galerkin method converges for the sequence (HN)∞

N=1 if, for every f∈ H, the

Galerkin equations (1.4) have a unique solution for all suﬃciently large Nand ϕN→A−1fas

N→ ∞. We say that (HN)∞

N=1 is asymptotically dense in Hif, for every ϕ∈ H,

min

ψN∈HN∥ϕ−ψN∥H→0 as N→ ∞.(1.5)

A necessary condition for the convergence of the Galerkin method is that (HN)∞

N=1 is asymp-

totically dense in H. Indeed, a standard necessary and suﬃcient condition (e.g., [46, Chapter II,

Theorem 2.1]) for convergence of the Galerkin method is that (HN)∞

N=1 is asymptotically dense

and that, for some N0∈Nand Cdis >0,

∥PNAψN∥H

∥ψN∥H≥Cdis for all non-zero ψN∈ HNand N≥N0,(1.6)

where PNis orthogonal projection of Honto HN. Importantly, if (1.6) holds, then ([46, Chapter

II, Equation (2.5)] or see [98, Theorem 4.2.1 and Remark 4.2.5])

∥ϕ−ϕN∥H≤1 + ∥A∥H→H

Cdis min

ψN∈HN∥ϕ−ψN∥H,for N≥N0,(1.7)

where ϕ=A−1fand ϕNis the unique solution of the Galerkin equations (1.4). We note that (1.7)

is known as a quasioptimal error estimate.

We now recap the main abstract theorem on convergence of the Galerkin method; this theorem

uses the deﬁnition that an operator A:H → H is coercive2if there exists Ccoer >0 such that

(Aψ, ψ)H≥Ccoer ∥ψ∥2

Hfor all ψ∈ H.(1.8)

Theorem 1.1 (The main abstract theorem on convergence of the Galerkin method.)

(a) If Ais invertible then there exists a sequence (HN)∞

N=1 for which the Galerkin method con-

verges.

(b) If Ais invertible then the following are equivalent:

(i) The Galerkin method converges for every asymptotically-dense sequence (HN)∞

N=1 in H.

(ii) A=A0+Kwhere A0is coercive and Kis compact.

N=1 and every N∈N, the

Galerkin equations (1.4)have a unique solution ϕNand, where ϕ=A−1f,



ϕ−ϕN

H≤∥A∥H→H

Ccoer

min

ψ∈HN

ϕ−ψ

H,(1.9)

(so that ϕN→ϕas N→ ∞ if (HN)∞

N=1 is asymptotically dense in H).

References for the proof. Part (a) was ﬁrst proved in [75, Theorem 1]; see also [46, Chapter II,

Theorem 4.1]. Part (b) was ﬁrst proved in [75, Theorem 2], with this result building on results in

[112]; see also [46, Chapter II, Lemma 5.1 and Theorem 5.1]. Part (c) is C´ea’s Lemma, ﬁrst proved

in [17].

1It is convenient, since we deal with non-self-adjoint operators and talk at some points about spectra and

numerical ranges, to assume throughout that all Hilbert spaces and function spaces are complex. Of course results

for the corresponding real case are easily deduced, if needed, from the complex function space case.

2In the literature, the property (1.8) (and its analogue for operators A:H → H′, where H′is the dual of H) is

sometimes called “H-ellipticity” (as in, e.g., [98, Page 39], [105,§3.2], and [57, Deﬁnition 5.2.2]) or “strict coercivity”

(e.g., [62, Deﬁnition 13.22]), with “coercivity” then used to mean either that Ais the sum of a coercive operator

and a compact operator (as in, e.g., [105,§3.6] and [57,§5.2]) or that Asatisﬁes a G˚arding inequality (as in [98,

Deﬁnition 2.1.54]).

1.3 The rationale for using second-kind BIEs posed in L2(Γ)

The BIOs in Table 1.1 are coercive in the trace spaces H±1/2(Γ) (or certain subspaces of these) for

Lipschitz Γ, thus insuring convergence of the associated Galerkin methods by Part (c) of Theorem

1.1; this coercivity theory was established for ﬁrst-kind equations by N´ed´elec and Planchard [90],

Le Roux [66], [67], and Hsiao and Wendland [56], and for second-kind equations by Steinbach and

Wendland [107]. These arguments involve transferring boundedness/coercivity properties of the

PDE solution operator to the associated boundary integral operators via the trace map and layer

potentials; the generality of these arguments is why coercivity holds with Γ only assumed to be

Lipschitz, and Costabel [29] highlighted how these ideas can be traced back to the work of Gauss

and Poincar´e.

Despite convergence of the associated Galerkin methods, using the ﬁrst-kind formulations in

the trace spaces has the disadvantage that the condition numbers of the Galerkin matrices grow as

the discretisation is reﬁned; e.g., for the h-version of the Galerkin method (where convergence is

obtained by decreasing the mesh-width hand keeping the polynomial degree ﬁxed), the condition

numbers grow like h−1; see, e.g. [98,§4.5]. The design of appropriate preconditioning strategies for

these Galerkin matrices has therefore been a classic topic of study in the BIE community for over 20

years, with proposed solutions including (i) preconditioning with an opposite-order operator [106]

(see also the survey [55]), (ii) using wavelets, either as an approximation space (e.g., [116,52,53])

or in preconditioning (e.g., [111,99]); using domain decomposition methods; see, e.g., [54] and the

recent book [108]. Furthermore, using the second-kind formulations in the trace spaces has the

disadvantage that the inner products on H±1/2(Γ) are non-local and non-trivial to evaluate; even

if the basis functions ϕNand ψNin (1.4) have supports only on a subset of Γ, (AϕN, ψN)His an

integral over all of Γ, and the calculation of the Galerkin matrix in this case is impractical.

For the second-kind BIEs, an attractive alternative to working in the trace spaces is to work in

L2(Γ). When Γ is C1,Dand D′are compact in L2(Γ) by the results of Fabes, Jodeit, and Rivi`ere

[41, Theorems 1.2 and 1.9] and thus each of the second-kind BIOs 1

2I±Dand 1

2I±D′is the sum of

a coercive operator and a compact operator, and convergence of the associated Galerkin methods

in L2(Γ) is ensured by Part (b) of Theorem 1.1. Since the L2(Γ) norm is local, (AϕN, ψN)His

an integral over the support of ψN, and the Galerkin matrix is much more easily computable.

Furthermore, when Dand D′are compact, the condition numbers of the Galerkin matrices of

2I±Dand 1

2I±D′are independent of the discretisation (without preconditioning); see [4,§3.6.3],

[51,§4.5.5].

1.4 Convergence of the Galerkin method in L2(Γ) for the standard

second-kind integral equations on polyhedral and Lipschitz domains.

The disadvantage of using second-kind BIEs in L2(Γ) is that convergence of the Galerkin method is

harder to establish when Γ is only Lipschitz, or Lipschitz polyhedral. Indeed, in these cases Dand

D′are not compact; e.g., when Γ has a corner or edge their spectra are not discrete; see, e.g. [4,

§8.1.3]. When Γ is only Lipschitz, Dand D′are bounded on L2(Γ) by the results on boundedness

of the Cauchy integral on Lipschitz Γ of Coifman, McIntosh, and Meyer [25] (following earlier work

by Calder´on [15] on boundedness for Γ with small Lipschitz character). Verchota [114] showed that

the operators 1

2I±Dand 1

2I±D′are Fredholm of index zero on L2(Γ); when Γ is connected,

2I−Dand 1

2I−D′are invertible on L2(Γ) and 1

2I+Dand 1

2I+D′invertible on L2

0(Γ), the set

of ϕ∈L2(Γ) with mean value zero; see [114, Theorems 3.1 and 3.3(i)]. 3

A long-standing open question has been

Can 1

2I±Dand 1

2I±D′be written as the sum of a coercive operator and a compact

operator in the space L2(Γ) when Γ is only assumed to be Lipschitz?

3The invertibility of 1

2I−D′on L2(Γ) implies that the bilinear form of the associated least-squares formulation

a(ϕ, ψ) = 1

2I−D′ϕ, 1

2I−D′ψL2(Γ)

is coercive. This formulation, however, suﬀers from the same disadvantages as the variational formulation of 1

2I−D′

in H−1/2(Γ), including that computing the entries of the Galerkin matrix requires computing integrals over all of

Γ, even when the basis functions have support on (small) subsets of Γ.

Figure 1.1: Views from above and below of the open-book polyhedron Ωθ,n of [22, Deﬁnition 5.7],

with n= 4 pages and opening angle θ=π/2

By Part (b) of Theorem 1.1, this question is equivalent to the question: does the Galerkin method

applied to 1

2I±Dand 1

2I±D′in L2(Γ) converge for every asymptotically-dense sequence of

subspaces when Γ is only assumed to be Lipschitz?

Until recently, this question was answered only in the following two cases, both in the aﬃr-

mative: (i) Γ is a 2d curvilinear polygon with each side C1,α for some 0 < α < 1 and with each

corner angle in the range (0,2π). (ii) Γ is Lipschitz, with suﬃciently small Lipschitz character.

Regarding (i): this result was announced by Shelepov in [100], with details of the proof given in

[101], and with the analogous result for polygons following from the result of Chandler [19,§3]; see,

e.g. [9, Lemma 1.5]. Regarding (ii): Wendland [118,§4.2] recognised that the results of I. Mitrea

[82, Lemma 1, Page 392] about the essential spectral radius could be adapted to prove this result,

with this result proved in full in [22, Corollary 3.5]; for more discussion on both (i) and (ii), see

[22,§1].

The recent paper [22] ﬁnally settled the question above negatively by giving examples of 2-d

Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which 1

2I±Dand 1

2I±D′cannot

be written as the sum of a coercive operator and a compact operator in the space L2(Γ). The

3-d star-shaped Lipschitz polyhedra are deﬁned in [22, Deﬁnition 5.7], and called the open-book

polyhedra; see Figure 1.1 for an example, where we use the notation that Ωθ,n is the open-book

polyhedron with npages and opening angle θ. Given ϵ > 0 there exists θ0∈(0, π] such that the

essential numerical range of Din L2(Γ) contains the interval [−√n/2 + ϵ, √n/2−ϵ] [22, Theorem

1.3]. By the deﬁnition of the essential numerical range (see, e.g., [22, Equation 2.3]), this result

implies that if θis suﬃciently small and n≥2, then 1

2I±Dand 1

2I±D′cannot be written as the

sum of a coercive operator and a compact operator in the space L2(Γ) when Γ = ∂Ωθ,n.

Nevertheless, Part (b) of Theorem 1.1 only shows that the Galerkin method applied to these

domains does not converge for every asymptotically dense sequence (HN)∞

N=1 ⊂L2(Γ), leaving

opening the possibility that all Galerkin methods used in practice (based on boundary element

method discretisation [105,98]) are in fact convergent. However, the following result from [22]

clariﬁes that this is not the case.

Theorem 1.2 ([22, Theorem 1.4]) Suppose that Ais invertible but Acannot be written in the

form A=A0+K, where A0is coercive and Kis compact, and that (H∗

N)∞

N=1 is a sequence of

ﬁnite-dimensional subspaces of H, with H∗

1⊂ H∗

2⊂..., for which the Galerkin method converges.

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Coercivesecond-kindboundaryintegralequationsfortheLaplaceDirichletproblemonLipschitzdomainsS.N.Chandler-Wilde∗,E.A.Spence†May28,2024AbstractWepresentnewsecond-kindintegral-equationformulationsoftheinteriorandexteriorDirichletproblemsforLaplace’sequation.Theoperatorsintheseformulationsarebothcon-tinu...

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Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains S. N. Chandler-Wilde E. A. Spence

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