Wavenumber-explicit analysis for the Helmholtz $h$-BEM: error estimates and iteration counts for the Dirichlet problem

We consider solving the exterior Dirichlet problem for the Helmholtz equation with the $h$-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber $k$ to have the error in the iterative solution bounded independently of $k$ as $k\rightarrow \infty$ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how $h$ must decrease with $k$ to maintain $k$-independent quasi-optimality of the Galerkin solutions as $k \rightarrow \infty$ when the obstacle is nontrapping.


Introduction
This paper is concerned with the wavenumber-explicit numerical analysis of boundary integral equations (BIEs) for the Helmholtz equation ∆u + k 2 u = 0, (1.1) where k > 0 is the wavenumber, posed in the exterior of a 2-or 3-dimensional bounded obstacle Ω with Dirichlet boundary conditions on Γ := ∂Ω. We consider the standard second-kind combined-field integral equation formulations of this problem: the so-called "direct" formulation (arising from Green's integral representation) (1.2) and the so-called "indirect" formulation (arising from an ansatz of layer potentials not related to Green's integral representation) where A k,η := from the Dirichlet boundary conditions is contained in the right-hand side f k,η ; our results are independent of the particular form of f k,η , and so we can simplify the presentation by restricting attention to the particular exterior Dirichlet problem corresponding to scattering by a point source or plane wave, i.e. the sound-soft scattering problem (Definition 1.7 below). We consider solving the equation (1.2) in L 2 (∂Ω) using the Galerkin method; this method seeks an approximation v N to the solution v from a finite-dimensional approximation space V N (where N is the dimension, i.e. the total number of degrees of freedom). In the majority of the paper Γ is C 2 , in which case V N will be the space of piecewise polynomials of degree p, for some fixed p ≥ 0, on shape-regular meshes of diameter h, with h decreasing to zero; this is the so-called h-version of the Galerkin method, and we denote V N and v N by V h and v h , respectively, and note that N ∼ h −(d−1) , where d is the dimension. To find the Galerkin solution v h , one must solve a linear system of dimension N ; in practice this is usually done using Krylov-subspace iterative methods such as the generalized minimal residual method (GMRES).
For the numerical analysis of this situation when k is large, there are now, roughly speaking, two main questions: Q1. How must h decrease with k in order to maintain accuracy of the Galerkin solution as k → ∞?
Q2. How does the number of GMRES iterations required to achieve a prescribed accuracy grow with k?
The goal of this paper is to prove rigorous results about these two questions, and then compare them with the results of numerical experiments. We now give short summaries of the main results. These results depend on the choice of the coupling parameter η; for the results on Q1 we need |η| ∼ k and for the results on Q2 we need η ∼ k, where we use the notation a ∼ b to mean that there exists C 1 , C 2 > 0, independent of h and k, such that C 1 b ≤ a ≤ C 2 b. We also use the notation a b to mean that there exists C > 0, independent of h and k, such that a ≤ Cb.
Summary of main results regarding Q1 and their context. Numerical experiments indicate that, in many cases, the condition hk 1 is sufficient for the Galerkin method to be quasi-optimal (with the constant of quasi-optimality independent of k; i.e., (1.20) below holds); see [42, §5]. This feature can be described by saying that the h-BEM does not suffer from the pollution effect (in constrast to the h-FEM; see, e.g., [7], [51,Chapter 4]). The best existing result in the literature is that k-independent quasi-optimality of the Galerkin method applied to the integral equation (1.2) holds when hk (d+1)/2 1 for 2-and 3-d C 2,α obstacles that are star-shaped with respect to a ball [42,Theorem 1.4]. In this paper we improve this result by showing that the k-independent quasioptimality holds for 2-d nontrapping obstacles when hk 3/2 1, for 3-d nontrapping obstacles when hk 3/2 log k 1, and for 2-and 3-d smooth (i.e. C ∞ ) convex obstacles with strictly positive curvature when hk 4/3 1 (see Theorem 1.15 below).
The ideas behind the proofs of these results are summarised in Remark 1.18 below, but we highlight here that all the integral-operator bounds used in these arguments are sharp up to a factor of log k. Therefore, to lower these thresholds on h for which quasi-optimality is proved, one would need to use different arguments than in the present paper. We also highlight that recent experiments by Marburg [59], [10], [60] give examples where pollution occurs, and therefore determining the sharp threshold on h for k-independent quasi-optimality to hold in general is an exciting open question.
Summary of main results regarding Q2 and their context. There has been a large amount of research effort expended on understanding empirically how iteration counts for integral-equation formulations of scattering problems involving the Helmholtz or Maxwell equations depend on k; see, e.g, [1], [4], [14], [15], [84], and the references therein.
To our knowledge, however, there are no sharp k-explicit bounds in the literature, for any integral-equation formulation of a Helmholtz or Maxwell scattering problem, on the number of iterations GMRES requires to achieve a prescribed accuracy. The main reason, in this current setting of the Helmholtz sound-soft scattering problem, is that the operator A k,η is non-normal for all obstacles other than the circle and sphere [13], [12], and so one cannot use the well-known every χ ∈ C ∞ comp (Ω + ) := {χ| Ω+ : χ ∈ C ∞ (R d ) is compactly supported}. Let γ + denote the trace operator from Ω + to ∂Ω. Let n be the outward-pointing unit normal vector to Ω (i.e. n points out of Ω and in to Ω + ), and let ∂ + n denote the normal derivative trace operator from Ω + to ∂Ω that satisfies ∂ + n u = n · γ + (∇u) when u ∈ H 2 loc (Ω + ). (We also call γ + u the Dirichlet trace of u and ∂ + n u the Neumann trace.) Definition 1.1 (Star-shaped, and star-shaped with respect to a ball) (i) Ω is star-shaped with respect to the point x 0 ∈ Ω if, whenever x ∈ Ω, the segment [x 0 , x] ⊂ D.
(ii) Ω is star-shaped with respect to the ball B a (x 0 ) if it is star-shaped with respect to every point in B a (x 0 ).
(iii) Ω is star-shaped with respect to a ball if there exists a > 0 and x 0 ∈ Ω such that Ω is star-shaped with respect to the ball B a (x 0 ). Definition 1.2 (Nontrapping) We say that Ω ⊂ R d , d = 2, 3 is nontrapping if ∂Ω is smooth (C ∞ ) and, given R such that Ω ⊂ B R (0), there exists a T (R) < ∞ such that all the billiard trajectories (in the sense of Melrose-Sjöstrand [63,Definition 7.20]) that start in Ω + ∩ B R (0) at time zero leave Ω + ∩ B R (0) by time T (R).

Definition 1.3 (Smooth hypersurface)
We say that Γ ⊂ R d is a smooth hypersurface if there exists Γ a compact embedded smooth d − 1 dimensional submanifold of R d , possibly with boundary, such that Γ is an open subset of Γ and the boundary of Γ can be written as a disjoint union where each Y is an open, relatively compact, smooth embedded manifold of dimension d − 2 in Γ, Γ lies locally on one side of Y , and Σ is closed set with d − 2 measure 0 and Σ ⊂ n l=1 Y l . We then refer to the manifold Γ as an extension of Γ.
For example, when d = 3, the interior of a 2-d polygon is a smooth hypersurface, with Y i the edges and Σ the set of corner points.

Definition 1.4 (Curved)
We say a smooth hypersurface is curved if there is a choice of normal so that the second fundamental form of the hypersurface is everywhere positive definite.
Recall that the principal curvatures are the eigenvalues of the matrix of the second fundamental form in an orthonormal basis of the tangent space, and thus "curved" is equivalent to the principal curvatures being everywhere strictly positive (or everywhere strictly negative, depending on the choice of the normal). Definition 1.5 (Piecewise smooth) We say that a hypersurface Γ is piecewise smooth if Γ = ∪ N i=1 Γ i where Γ i are smooth hypersurfaces and Γ i ∩ Γ j = ∅. Definition 1.6 (Piecewise curved) We say that a piecewise-smooth hypersurface Γ is piecewise curved if Γ is as in Definition 1.5 and each Γ j is curved.

The boundary value problem and integral equation formulation
Definition 1.7 (Sound-soft scattering problem) Given k > 0 and an incident plane wave u I (x) = exp(ikx · a) for some a ∈ R d with | a| = 1, find u S ∈ C 2 (Ω + ) ∩ H 1 loc (Ω + ) such that the total field u := u I + u S satisfies the Helmholtz equation (1.1) in Ω + , γ + u = 0 on ∂Ω, and u S satisfies the Sommerfeld radiation condition as r := |x| → ∞, uniformly in x := x/r.
The incident field in the sound-soft scattering problem of Definition 1.7 is a plane wave, but this could be replaced by a point source or, more generally, a solution of the Helmholtz equation in a neighbourhood of Ω; see [18,Definition 2.11].
Obtaining the direct integral equation (1.2). If u satisfies the sound-soft scattering problem of Definition 1.7 then Green's integral representation implies that ). Taking the exterior Dirichlet and Neumann traces of (1.5) on ∂Ω and using the jump relations for the singleand double-layer potentials (see, e.g., [18,Equation 2.41]) we obtain the integral equations S k ∂ + n u = γ + u I and where S k and D k are the single-and adjoint-double-layer operators defined by for φ ∈ L 2 (∂Ω) and x ∈ ∂Ω. Later we will also need the definition of the double-layer potential, The first equation in (1.7) is not uniquely solvable when −k 2 is a Dirichlet eigenvalue of the Laplacian in Ω, and the second equation in (1.7) is not uniquely solvable when −k 2 is a Neumann eigenvalue of the Laplacian in Ω (see, e.g., [18,Theorem 2.25]). The standard way to resolve this difficulty is to take a linear combination of the two equations, which yields the integral equation (1.2) where A k,η is defined by (1.4), f k,η := ∂ + n u I − iη γ + u I , (1.10) and we use the notation that v = ∂ + n u (this makes denoting the Galerkin solution below easier since we then have v h instead of (∂ + n u) h ). The space L 2 (∂Ω) is a natural space for the practical solution of second-kind integral equations since it is self-dual, and, for η ∈ R \ {0}, A k,η is a bounded invertible operator from L 2 (∂Ω) to itself [18,Theorem 2.27]. Furthermore the right-hand side f k,η is in L 2 (∂Ω) (since u I ∈ C ∞ (Ω + )) and thus we consider the equation (1.2) as an equation in L 2 (∂Ω).
The Galerkin method. Given a finite-dimensional approximation space V N ⊂ L 2 (∂Ω), the Galerkin method for the integral equation (1.2) is (1.11) , the Galerkin method (1.11) is equivalent to solving the linear system Av = f .
We consider the h-version of the Galerkin method, and we then denote V N and v N by V h and v h respectively. The main results for Q1 and Q2 will be stated under the following assumption on V h . Assumption 1.8 (Assumptions on V h ) V h is a space of piecewise polynomials of degree p for some fixed p ≥ 0 on shape-regular meshes of diameter h, with h decreasing to zero (see, e.g., [72,Chapter 4] for specific realisations). Furthermore where · 2 denotes the l 2 (i.e. euclidean) vector norm.

Statement of the main results and discussion
We split the statement of the main results into three sections • k-explicit bounds on S k , D k , and D k as mappings from L 2 (∂Ω) to H 1 (∂Ω) ( §1.2.1).
If ∂Ω is a piecewise-smooth hypersurface (in the sense of Definition 1.5), then, given k 0 > 1, (1.14) for all k ≥ k 0 . Moreover, if ∂Ω is piecewise curved (in the sense of Definition 1.6), then, given k 0 > 1, the following stronger estimate holds for all k ≥ k 0 If ∂Ω is a piecewise smooth, C 2,α hypersurface, for some α > 0, then, given k 0 > 1, for all k ≥ k 0 . Moreover, if ∂Ω is piecewise curved, then, given k 0 > 1, the following stronger estimates hold for all k ≥ k 0 (c) If Ω is convex and ∂Ω is C ∞ and curved (in the sense of Definition 1.4) then, given k 0 > 0,

Remark 1.12 (Comparison to previous results)
The only previously-existing bounds on the L 2 (∂Ω) → H 1 (∂Ω)-norms of S k , D k , and D k were the following: when ∂Ω is Lipschitz, and We see that (1.17) is a factor of log k sharper than the bound (1.14) when d = 2, but otherwise all the bounds in Theorem 1.10 are sharper than (1.17) and (1.18). Remark 1.13 (Bounds for general dimension and k ∈ R) We have restricted attention to 2and 3-dimensions because these are the most practically interesting ones. From a semiclassical point of view, it is natural work in d ≥ 1, and the results of Theorem 1.10 apply for any d ≥ 1 (although when d = 1 it is straightforward to get sharper bounds). We have also restricted attention to the case when k is positive and bounded away from 0. Nevertheless, the methods used to prove the bounds in Theorem 1.10 show that if one replaces log k by log k (where · = (2 + | · | 2 ) 1/2 ) and includes an extra factor of log k −1 when d = 2, then the resulting bounds hold for all k ∈ R This paper is concerned with second-kind Helmholtz BIEs posed in L 2 (∂Ω), but there is also a large interest in both first-and second-kind Helmholtz BIEs posed in the trace spaces H −1/2 (∂Ω) and H 1/2 (∂Ω) (see, e.g., [72, §3.9], [78, §7.6]). The k-explicit theory of Helmholtz BIEs in the trace spaces is much less developed than the theory in L 2 (∂Ω), so we therefore highlight that the L 2 (∂Ω) → H 1 (∂Ω) bounds in Theorem 1.10 can be converted to H s−1/2 (∂Ω) → H s+1/2 (∂Ω) bounds for |s| ≤ 1/2. Corollary 1.14 (Bounds from H s−1/2 (∂Ω) → H s+1/2 (∂Ω) for |s| ≤ 1/2) Theorem 1.10 is valid with all the norms from L 2 (∂Ω) → H 1 (∂Ω) replaced by norms from H s−1/2 (∂Ω) → H s+1/2 (∂Ω) for |s| ≤ 1/2.

Results concerning Q1
Theorem 1.15 (Sufficient conditions for the Galerkin method to be quasi-optimal) Let u be the solution of the sound-soft scattering problem of Definition 1.7, let |η| ∼ k, and let V h satisfy Part (a) of Assumption 1.8.
(a) If either (i) Ω is nontrapping, or (ii) Ω is star-shaped with respect to a ball and ∂Ω is C 2,α and piecewise smooth, then given k 0 > 0, there exists a C > 0 (independent of k and h) such that if for all k ≥ k 0 . (b) In case (ii) above, if ∂Ω is piecewise curved, then given k 0 > 0, there exists a C > 0 (independent of k and h) such that if Having established quasi-optimality, it is then natural to ask how the best approximation error inf w h ∈V h v − w h L 2 (∂Ω) depends on k, h, and v L 2 (∂Ω) . Theorem 1.16 (Bounds on the best approximation error) Let u be the solution of the sound-soft scattering problem of Definition 1.7 and let V h satisfy Assumption 1.8.
Combining Theorems 1.15 and 1. 16 we can obtain bounds on the relative error of the Galerkin method. For brevity, we only state the ones corresponding to cases (a) and (c) in Theorems 1.15 and 1.16. (a) If either (i) Ω is nontrapping, or (ii) Ω is star-shaped with respect to a ball and ∂Ω is C 2,α and piecewise smooth, then given k 0 > 0, there exists a C > 0 (independent of k and h) such that if h and k satisfy (1.19) then the Galerkin equations (1.11) have a unique solution which satisfies If Ω is convex and ∂Ω is C ∞ and curved, then given k 0 > 0, there exists a C > 0 (independent of k and h) such that if hk 4/3 ≤ C the Galerkin equations (1.11) have a unique solution which satisfies Remark 1.18 (The main ideas behind the proofs of Theorems 1.15 and 1.16) The proof of Theorem 1.15 uses the classic projection-method analysis of second-kind integral equations (see, e.g., [6]), with A k,η be treated as a compact perturbation of the identity. In [42], this argument was used to reduce the question of finding k-explicit bounds on the mesh threshold h for k-independent quasi-optimality to hold to finding k-explicit bounds on We use the sharp bounds on the first two of these norms from Theorem 1.10, and the sharp bounds on the third of these norms from [ [42]: the mesh thresholds for quasi-optimality in Theorem 1.15 are sharper than the corresponding ones in [42], and the results are valid for a wider class of obstacles. This sharpening is due to the new, sharp bounds on L 2 (∂Ω) → H 1 (∂Ω)-norms of S k , D k , and D k from Theorem 1.10, and the widening of the class of obstacles is due to the bound on (A k,η ) −1 L 2 (∂Ω)→L 2 (∂Ω) for nontrapping obstacles from [9,Theorem 1.13]. In more detail: Theorem 1.4 of [42] is the analogue of our Theorem 1.15 except that the former is only valid when Ω is star-shaped with respect to a ball and C 2,α and the mesh threshold is hk (d+1)/2 ≤ C. Comparing this result to Theorem 1.15 we see that we've sharpened the threshold in the d = 3 case, expanded the class of obstacles to nontrapping ones, and added the additional results (b) and (c). Theorem 1.16 on the best approximation error is again proved using the L 2 (∂Ω) → H 1 (∂Ω)-bounds in Theorem 1.10 and thus we see similar improvements over the corresponding theorem in [42] As discussed in Remark 1.18, both the present paper and [42] use the classic projection-method argument to obtain k-explicit results about quasi-optimality of the h-BEM. There are two other sets of results about quasi-optimality of the h-BEM in the literature: (a) results that use coercivity [30], [76], [77], and (b) results that give sufficient conditions for quasi-optimality to hold in terms of how well the spaces V h approximate the solution of certain adjoint problems [8], [56], [62].
These two sets of results are discussed in detail in [42, pages 181-182] and [42, §4.2] respectively, and neither give results as strong as those in Theorem 1.15. Finally, in this paper we have only considered the h-BEM; a thorough k-explicit analysis of the hp-BEM for the exterior Dirichlet problem was conducted in [56] and [62]. In particular, this analysis, combined with the bound on (A k,η ) −1 L 2 (∂Ω)→L 2 (∂Ω) for nontrapping obstacles from [9, Theorem 1.13], proves that k-independent quasi-optimality can be obtained for nontrapping obstacles through a choice of h and p that keeps the total number of degrees of freedom proportional to k d−1 [ [42, §5] show that for a wide variety of obstacles (including certain trapping obstacles) the h-BEM is quasi-optimal with constant independent of k (i.e. (1.20) holds), when hk ∼ 1. The closest we can get to proving this is the result for strictly convex obstacles in Theorem 1.15 part (c), with the threshold being hk 4/3 ≤ C. The recent results of Marburg [59], [10], [60], however, give examples of cases where hk ∼ 1 is not sufficient to keep the error bounded as k → ∞.

Result concerning Q2
We now consider solving the linear system Av = f with the generalised minimum residual method (GMRES) introduced by Saad and Schultz in [71]; for details of the implementation of this algorithm, see, e.g., [70], [43].

Theorem 1.21 (A bound on the number of GMRES iterations)
Let Ω be a 2-or 3-d convex obstacle whose boundary ∂Ω is analytic and curved. Let V h satisfy Part (b) of Assumption 1.8, let the Galerkin matrix corresponding to (1.11) be denoted by A, and consider GMRES applied to Av = f There exist constants η 0 > 0 and k 0 > 0 (with η 0 = 1 if Ω is a ball) such that if k ≥ k 0 and η 0 k ≤ η k, then, given 0 < ε < 1, there exists a C (independent of k, η, and ε) such that if then the mth GMRES residual r m := Av m − f satisfies where · 2 denotes the l 2 (i.e. euclidean) vector norm. Furthermore, when Ω is a ball (i.e. ∂Ω is a circle or sphere), then the constant η 0 = 1.
In other words, Theorem 1.21 states that, for convex, analytic, curved Ω, the number of iterations growing like k 1/3 is a sufficient condition for GMRES to maintain accuracy as k → ∞.

Remark 1.22 (How sharp is the result of Theorem 1.21?) Numerical experiments in §6
show that for the sphere the number of GMRES iterations grows like k 0.29 , and for an ellipsoid they grow like k 0.28 . The bound in Theorem 1.21 is therefore effectively sharp (at least for the range of k considered in the experiments).
Remark 1.23 (The main ideas behind the proof of Theorem 1.21) The two ideas behind Theorem 1.21 are that: (a) A sufficient (but not necessary) condition for iterative methods to be well behaved is that the numerical range (also known as the field of values) of the matrix is bounded away from zero, and in this case the Elman estimate [37,36] and its refinement due to Beckermann, Goreinov, and Tyrtyshnikov [11] can be used to bound the number of GMRES iterations in terms of (i) the distance of the numerical range to the origin, and (ii) the norm of the matrix.
(b) When Ω is convex, C 3 , piecewise analytic, and ∂Ω is curved, [77] proved that A k,η is coercive for sufficiently large k (with η ∼ k) . The k-dependence of the coercivity constant, along with the k-dependence of A k,η L 2 (∂Ω)→L 2 (∂Ω) then give the information needed about the numerical range of the Galerkin matrix A required in (a).

Remark 1.24 (Comparison to previous results)
The bound m k 2/3 when ∂Ω is a sphere was stated in [77, §1.3]; this bound was obtained using the original Elman estimate (see Remark 5.4 below), and the fact that the sharp bound A k,η L 2 (∂Ω)→L 2 (∂Ω) k 1/3 was known for the circle and sphere; see [18, §5.4]. To our knowledge, there are no other k-explicit bounds in the literature, for any Helmholtz BIE, on the number of GMRES iterations required to achieve a prescribed accuracy. The closest related work is [24], which uses a second-kind integral equation to solve the Helmholtz equation in a half-plane with an impedance boundary condition. The special structure of this integral equation allows a two-grid iterative method to be used, and [24] prove that there exists C > 0 such that if kh ≤ C, then, after seven iterations, the difference between the solution and the Galerkin solution computed via the iterative method is bounded independently of k and h.
for x ∈ Ω + , φ ∈ L 2 (∂Ω), and η ∈ R \ {0}. Imposing the boundary condition γ + u S = −γ + u I on ∂Ω and using the jump relations for the single-and double-layer potentials leads to the integral equation (1.3) where A k,η is defined by (1.4) and g = −γ + u I . One can use (2.53) below to show that A k,η and A k,η are adjoint with respect to the real-valued L 2 (∂Ω) inner product (see, e.g., [ The bounds on the best approximation error in Theorem 1.16 hold for the indirect equation These powers of k are all slightly higher than those for the direct equation; the reason for this is essentially that we have more information about the unknown in the direct equation (since it is ∂ + n u) than about the unknown in the indirect equation (one can express φ in terms of the difference of solutions to interior and exterior boundary value problems -see [18, Page 132]but it is harder to make use of this fact than for the direct equation). 2), and therefore hold for the general Dirichlet problem with Dirichlet data in H 1 (∂Ω) (this assumption is needed so that A k,η can still be considered as an operator on L 2 (∂Ω); see, e.g., [18, §2.6]). The results of Theorem 1.16 and Corollary 1.17, however, do not immediately hold for the general Dirichlet problem, since they use the particular form of the right-hand side in (1.10).
Outline of the paper In §2 we prove Theorem 1.10 (the L 2 (∂Ω) → H 1 (∂Ω) bounds) and Corollary 1.14, and in §3 we show that these bounds are sharp in their k-dependence. In §4 we prove Theorems 1.15 and 1.16 (the results concerning Q1). In §5 we prove Theorem 1.21 (the result concerning Q2), and then in §6 we give numerical experiments showing that Theorem 1.21 is sharp in its k-dependence.
2 Proof of Theorem 1.10 (the L 2 (∂Ω) → H 1 (∂Ω) bounds) and Corollary 1.14 In this section we prove Theorem 1.10 and Corollary 1.14. The vast majority of the work will be in proving Parts (a) and (b) of Theorem 1.10, with Part (c) of Theorem 1.10 following from the results in [38,Chapter 4], and Corollary 1.14 following from the results of [42]. The outline of this section is as follows: In §2.1 we discuss some preliminaries from the theory of semiclassical pseudodifferential operators, with our default references the texts [85] and [31]. In §2.2 we recap facts about function spaces on piecewise-smooth hypersurfaces. In §2.3 we recap restriction bounds on quasimodes -these results are central to our proof of Theorem 1.10. In §2.4 we prove of Parts (a) and (b) of Theorem 1.10, in §2.5 we prove Part (c) of Theorem 1.10 §2.5, and in §2.6 we prove Corollary §2.6. We drop the notation in this section and state every bound with a constant C; we do this because later in the proof it will be useful to be able to indicate whether or not the constant in our estimates depends on the order s of the Sobolev space, or on a particular hypersurface Γ (we do this via the subscript s and Γ -see, e.g., (2.17) below).

Symbols and quantization
We define the symbol class S m (R 2d ) by For an element a ∈ S m , we define its quantization to be the operator for u ∈ S(R d ). These operators can be defined by duality on u ∈ S (R d ). We denote the set of pseudodifferential operators of order m by for some a ∈ S comp . Here, we say that an operator

Action on semiclassical Sobolev spaces
We define the Semiclassical Sobolev spaces H s where ξ := (1 + |ξ| 2 ) 1/2 ∈ S 1 and D := −i∂. Note that for s an integer, this norm is equivalent to The definition of the semiclassical Sobolev spaces on a smooth compact manifold of dimension With these definitions in hand, we have the following lemma on boundedness of pseudodifferential operators.

Ellipticity
For A ∈ Ψ m (R d ), we say that (x, ξ) ∈ R 2d is in the elliptic set of A, denoted ell(A), if there exists U a neighborhood of (x, ξ) such that for some δ > 0, We then have the following lemma then the same conclusions hold with R i ∈ Ψ m2−m1 (R d ).

Pseudodifferential operators on manifolds
Since we only use the notion of a pseudodifferential operator on a manifold in passing (in Lemma 2.15 and §2.5 below), we simply note that it is possible to define pseudodifferential operators on manifolds (see, e.g., [85,Chapter 14] We make two remarks: 1. The definition of the norm H s (Γ) depends on Γ, χ, and the choice of charts (U j , ψ j ) and partition of unity (χ j ). One can however prove that two different choices of charts (U j , ψ j ) and partition of unity (χ j ) lead to equivalent norms H s (Γ). In what follows, (U j , ψ j , χ j ) shall be traces on Γ of charts and partition of unity on R d .
2. When Γ is a compact embedded submanifold without boundary, the norm on H s (Γ) coincides with usual H s (Γ) norm.
We similarly define the normsH s . The following lemma shows that, when S k , D k , and D k map L 2 (∂Ω) to H 1 (∂Ω), to bound the H 1 (∂Ω) norms of S k φ, D k φ, and D k φ, it is sufficient to bound their H 1 (∂Ω) norms.

Lemma 2.7
Let Ω be a bounded Lipschitz open set such that its open complement is connected and ∂Ω is a piecewise smooth hypersurface (in the sense of Definition 1.5). If u ∈ H 1 (∂Ω) then Proof. Then, and the proof is complete.

6)
and where ∂ ν is a choice of normal derivative to Γ.
In the context of the wave equation on smooth Riemannian manifolds with restriction to a submanifold, the estimates (2.6) along with their L p generalizations appear in the work of Tataru [81] who also notes that the L 2 bounds are a corollary of an estimate of Greenleaf and Seeger [44]. The semiclassical version was studied by Burq, Gérard and Tzvetkov in [16], Tacy [79] and Hassell-Tacy [47].
Estimates like (2.7) first appeared in the work of Tataru [81] in the form of regularity estimates for restrictions of solutions to hyperbolic equations. Semiclassical analogs of this estimate were proved in Christianson-Hassell-Toth [27] and Tacy [80]. Remark 2.9 (Smoothness of Γ required for the quasimode estimates) The k 1/4 -bound in (2.6) is valid when Γ is only C 1,1 , and the k 1/6 -bound is valid when Γ is C 2,1 and curved. Therefore, with some extra work it should be possible to prove that the bounds on S k in Theorem 1.10 hold with the assumption "piecewise smooth" replaced by "piecewise C 1,1 " and "piecewise C 2,1 and curved" respectively. On the other hand, the bound (2.29) is not known in the literature for lower regularity Γ.

Proof of Parts (a) and (b) of Theorem 1.10
When proving these results, it is more convenient to work in semiclassical Sobolev spaces, i.e. to prove the bounds from L 2 (∂Ω) to H 1 k (∂Ω). We therefore now restate Theorem 1.10 as Theorem 2.10 below, working in these spaces. Theorem 2.10 (Restatement of Theorem 1.10 as bounds from L 2 (∂Ω) → H 1 k (∂Ω)) (a) If ∂Ω is a piecewise-smooth hypersurface (in the sense of Definition 1.5), then, given k 0 > 1, there exists C > 0 (independent of k) such that for all k ≥ k 0 . Moreover, if ∂Ω is piecewise curved (in the sense of Definition 1.6), then, given k 0 > 1, the following stronger estimate holds for all k ≥ k 0 (2.9) (b) If ∂Ω is a piecewise smooth, C 2,α hypersurface, for some α > 0, then, given k 0 > 1, there exists C > 0 (independent of k) such that (2.10) Moreover, if ∂Ω is piecewise curved, then, given k 0 > 1, there exists C > 0 (independent of k) such that the following stronger estimates hold for all k ≥ k 0 (c) If Ω is convex and ∂Ω is C ∞ and curved (in the sense of Definition 1.4) then, given k 0 > 1, there exists C such that, for k ≥ k 0 , This theorem is actually stronger than Theorem 1.10 in that it now contains the L 2 (∂Ω) → L 2 (∂Ω) estimates originally proved in [ In §2.4.2 below, we give an outline of the proof of Parts (a) and (b). This outline, however, requires the definitions of S k , D k , and D k in terms of the free resolvent, given in the next subsection.
2.4.1 S k , D k , and D k written in terms of the free resolvent R 0 (k) We now recall the definitions of S k , D k , and D k in terms of the free resolvent R 0 (k), these expressions are well-known in the theory of BIEs on Lipschitz domains [29], [61,Chapters 6 and 7]. We then specialise these to the case when ∂Ω is a piecewise-smooth hypersurface (in the sense of Definition 1.5) Let R 0 (k) be the free (outgoing) resolvent at k; i.e. for ψ ∈ C ∞ comp (R d ) we have Similarly, the result about the normal-derivative traces of the single-layer potential S k implies that We now consider the case when ∂Ω is a piecewise-smooth hypersurface (in the sense of Definition 1.5) and use the notation that Γ i are the compact embedded smooth manifolds of R d such that, for each i, Γ i is an open subset of Γ i . Let L i be a vector field whose restriction to Γ i is equal to ∂ νi , the normal to Γ i that is outward pointing with respect to ∂Ω.
Hausdorff measure on Γ. Finally, we let γ ± i denote restrictions from the interior and exterior respectively, where "interior" and "exterior" are defined via considering Γ i as a subset of ∂Ω. With these notations, we have that and (2.15) the advantage of these last two expressions over (2.12) and (2.13) is that they involve γ i and L i instead of ∂ * n and ∂ ± n . In the rest of this section, we use the formulae (2.11), (2.14), and (2.15) as the definitions of S k , D k , and D k . Note that we slightly abuse notation by omitting the sums in (2.14) and (2.15) and instead writing

Outline of the proof of Parts (a) and (b) of Theorem 2.10
The proof of Parts (a) and (b) of Theorem 2.10 will follow in two steps. In Lemma 2.11, we obtain estimates on frequencies ≤ M k and in Lemma 2.19 we complete the proof by estimating the high frequencies (≥ M k).
To estimate the low frequency components, we spectrally decompose the resolvent using the Fourier transform. We are then able to reduce the proof of the low-frequency estimates to the estimates on the restriction of eigenfunctions (or more precisely quasimodes) to ∂Ω that we recalled in §2.3. To understand this reduction, we proceed schematically; from the description of S k in terms of the free resolvent, (2.11), the spectral decomposition of S k via the Fourier transform is schematically where u(r) is a generalized eigenfunction of −∆ with eigenvalue r 2 . Using this decomposition we see that estimating S k amounts to estimating the restriction of the generalized eigenfunction u(r) to ∂Ω.
At very high frequency, we compare the operators S k , D k , and D k with the corresponding operators when k = 1 (recall that the mapping properties of boundary integral operators with k = 1 have been extensively studied on rough domains; see, e.g. [65,Chapter 15], [61], [66]). By using a description of the resolvent at very high frequency as a pseudodifferential operator, we are able to see that these differences gain additional regularity and hence to obtain estimates on them easily.
The new ingredients in our proof compared to [39] and [46] are that we have H s norms in Lemma 2.11 and Lemma 2.19 rather than the L 2 norms appearing in the previous work.

Proof of Parts (a) and (b) of Theorem 2.10
Low-frequency estimates. Following the outline in §2.4.2, our first task is to estimate frequencies ≤ kM . We start by proving a conditional result that assumes a certain estimate on restriction of the Fourier transform of surface measures to the sphere of radius r (Lemma 2.11). In Lemma 2.13 we then show that the hypotheses in Lemma 2.11 are a consequence of restriction estimates for quasimodes. In Lemma 2.16 we show how the low-frequency estimates on S k , D k , and D k follow from Lemma 2.11.
In this section we denote the sphere of radius r by S d−1 r and we denote the surface measure on S d−1 r by dσ.

Lemma 2.11
Suppose that for Γ ⊂ R d any precompact smooth hypersurface, s ≥ 0, f ∈Ḣ −s (Γ), and some α , β > 0, Let Γ 1 , Γ 2 ⊂ R d be compact embedded smooth hypersurfaces. Recall that L i is a vector field with L i = ∂ ν on Γ i for some choice of normal ν on Γ i and ψ ∈ C ∞ c (R) with ψ ≡ 1 in neighborhood of 0. With the frequency cutoff ψ(k −1 D) defined as in (2.1), we then define for f ∈Ḣ −s1 (Γ 1 ), g ∈Ḣ −s2 (Γ 2 ), s i ≥ 0, Then there exists C Γ1,Γ2 so that for k > 1, The key point is that, modulo the frequency cutoff ψ(k −1 D), Q S (f, g), Q D (f, g), and Q D (f, g) are given respectively by S k f, g Γ , D k f, g Γ , and D k f, g Γ ,where f is supported on Γ 1 and g on Γ 2 . Proof of Lemma 2.11. We follow [39] [46] to prove the lemma. First, observe that due to the compact support of f δ Γi , (2.17) and (2.18) imply that for Γ ⊂ R d precompact,

Remark 2.12
The estimate (2.24) is the only term where the log |k| appears, which leads to the log k factors in the bounds of Theorem 1.10 (without which these bounds would be sharp). To prove this lemma, we need to understand certain properties of the operator T r defined by with T r defined by (2.25). Before proving Lemma 2.13 we prove two lemmas (Lemma 2.14 and 2.15) collecting properties of T r . Lemma 2.14 Let T r be defined by (2.25) and χ ∈ C ∞ c (R d ). Then, Proof of Lemma 2.14. We estimate B := (χT r ) * χT r : . This operator has kernel B(ξ, η) = R d χ 2 (y) exp (i y, ξ − η ) dy = χ 2 (η − ξ). Now, for η ∈ S d−1 r , and any N > 0, Thus, by Schur's inequality, B is bounded on L 2 S d−1 r uniformly in r. Therefore,
Proof of Lemma 2.13. The key observation for the proof of Lemma 2.13 is that for χ ∈ C ∞ c (R d ), χT r φ is a quasimode of the Laplacian with k = r in the sense of (2.5) in Theorem 2.8. To see this, observe first that −∆T r φ = r 2 T r φ by the definition (2.25). Therefore, Therefore, taking such aχ withχ ≡ 1 in a neighborhood, U of Γ shows that χT r φ is a quasimode.
To prove (2.18), we let A = I. Then, by the bounds (2.6) in Theorem 2.8 together with Lemmas 2.14 and 2.15, for s ≥ 0, and if Γ is curved then To prove (2.17), we take A = L. Observe that Hence, using the fact that L = ∂ ν on Γ together with the bound (2.7) in Theorem 2.8, we can estimate LT r φ. χLT r φ L 2 ( Γ) = LχT r φ L 2 ( Γ) ≤ C r χT r φ L 2 (R d ) . (2.29) In particular, for s ≥ 0, and if Γ is curved, Lemma 2.16 (Low-frequency estimates) Let s 2 be either 0 or 1. If ∂Ω is piecewise smooth and Lipschitz, then is a smoothing operator on S , by elliptic regularity R 0 (k)ψ(k −1 D) is smoothing and hence its restriction to ∂Ω maps compactly supported distributions into H 1 (∂Ω). Applying (2.33)-(2.35) with s 1 = 0, Γ = Γ i , and then summing over i, we find that, for 0 ≤ s 2 ≤ 1, High frequency estimates. Next, we obtain an estimate on the high frequency (≥ kM ) components of S k , D k , and D k . We start by analyzing the high frequency components of the free resolvent, proving two lemmata on the structure of the free resolvent there.

Now, by Lemma 2.3 there exists
and A 0 has (Indeed, since we are working on R d , with no remainder.) Composing (2.39) on the right with A 0 , we have Now, applying the same arguments, there exists A n ∈ k −2 Ψ −2 (R d ) such that Hence, by induction as desired. The proof of the statement for B 2 is identical.
Next, we prove an estimate on the difference between the resolvent at high energy and that at fixed energy.
Proof of Lemma 2.18. We proceed as in the proof of Lemma 2.17. Let (2.44) In particular, iterating using the same argument to write we see that the right hand side of (2.44) is in k −2 Ψ −4 (R d ).

Proof of Part (c) of Theorem 2.10
Proof of Part (c) of Theorem 2.10.

Sharpness of the bounds in Theorem 1.10
We now prove that the powers of k in the S k L 2 (∂Ω)→H 1 (∂Ω) bounds in Theorem 2.10 are optimal. The analysis in [46, §A.3] proves that the powers of k in the D k L 2 (∂Ω)→L 2 (∂Ω) bounds are optimal, but can be adapted in a similar way to below to prove the sharpness of the D k L 2 (∂Ω)→H 1 (∂Ω) bounds.
In this section we write x ∈ R d as x = (x , x d ) for x ∈ R d−1 , and x = (x 1 , x ) (in the case d = 2, the x variable is superfluous). for some δ > 0 and is C 2 in a neighborhood thereof (i.e. ∂Ω contains a line segment), then there exists k 0 > 0 and C > 0 (independent of k), such that, for all k ≥ k 0 , This result shows that the bound (1.14), when ∂Ω is piecewise smooth, is sharp up to a factor of log k.
Lemma 3.2 (General lower bound on S k L 2 (∂Ω)→H 1 (∂Ω) ) If ∂Ω is C 2 in a neighborhood of a point then there exists k 0 > 0 and C > 0 (independent of k), such that, for all k ≥ k 0 , This result shows that the bound (1.15), when ∂Ω is piecewise curved, is sharp up to a factor of log k and that the bound (1.16), when when ∂Ω is smooth and curved, is sharp. By the definition of the operator norm, it is sufficient to prove that there exists u k ∈ L 2 (∂Ω) with supp u k ⊂ Γ, k 0 > 0, and C > 0 (independent of k), such that, for all k ≥ k 0 , We begin by observing that the definition of Φ k (x, y) (1.6) and the asymptotics of Hankel functions for large argument and fixed order (see, e.g., [69, §10.17]) imply that In what follows, we suppress the dependence of u on k for convenience. Let u(x , γ(x )) := e ikx1 χ ,0,1/2 (x ). The definition of χ implies that supp u = (x , γ(x )) : |x 1 | ≤ 2 , |x | ≤ 2 k −1/2 , and thus supp u ⊂ Γ for sufficiently small and k sufficiently large (say < (2 √ 2) −1 δ and k > 1); for the rest of the proof we assume that and k are such that this is the case. Observe also that the motivation for this choice comes from the analysis in Remark 4.5 below. Indeed, we know that S k is largest microlocally near points that are glancing in both the incoming and outgoing variables. Since u concentrates microlocally at x = 0, ξ = (1, 0) up to scale k −1/2 , the billiard trajectory emanating from this point is {t (1, 0) : t > 0}. This ray is always glancing since Γ is flat. Therefore, we choose U to contain this ray up to scale k −1/2 . Then for x ∈ U , y ∈ supp u, Then, observe that by Taylor's formula Since γ(x 1 , 0) = 0 for |x 1 | < δ, In particular, We have from the Hankel-function asymptotics (3.2) and the definition of u that, for x ∈ U , and then using the asymptotics (3.6) in the exponent of the integrand and the asymptotics (3.7) in the rest of the integrand, we have, for x ∈ U , Therefore, with M large enough, small enough, and then k 0 large enough, the contribution from the integral over Γ is determined by the cutoff χ ,0,1/2 , yielding k −(d−2)/2 , and thus In the step of taking sufficiently small, we can also take small enough to ensure that U ⊂ Γ for all k ≥ 1. Using (3.8), along with the fact that the measure of U ∼ k −(d−2)/2 , we have that Since we have ensured that U ⊂ Γ, (3.9) and (3.5) imply that the first bound in (3.1) holds. It easy to see that if we repeat the argument above but with (3.3) instead of (3.2), then we obtain the second bound in (3.1).
Similar to the proof of Lemma 3.1, it is sufficient to prove that there exists u k ∈ L 2 (∂Ω) with supp u k ⊂ Γ, k 0 > 0, and C > 0 (independent of k), such that for all k ≥ k 0 . The idea in the curved case is the same as in the flat case: choose u concentrating as close as possible to a glancing point and measure near the point given by the billiard map. More practically, this amounts to ensuring that |x − y| looks like x 1 − y 1 modulo terms that are much smaller than k −1 . The fact that Γ may be curved will force us to choose u differently and cause our estimates to be worse than in the flat case (leading to the weaker -but still sharp -lower bound).
With χ ,γ1,γ2 defined by (3.4), let u(x , γ(x )) := e ikx1 χ ,1/3,2/3 (x ) where, as in the proof of Lemma 3.1, we have x = (x 1 , x ) and as in Lemma 3.1, supp u ⊂ Γ for sufficiently small and k sufficiently large, and for the rest of the proof we assume that this is the case. Then Then, for y ∈ supp u and x ∈ U , From (3.2) and the definition of u, we have for x ∈ U , and then, using (3.12) in the exponent of the integrand and (3.13) in the rest, we have, for x ∈ U , Thus, fixing M large enough, then small enough, then k 0 large enough, we have In the step of taking sufficiently small, we can also take small enough so that when x ∈ U , |x | < δ, and thus x ∈ Γ. Using the lower bound (3.14), and the fact that the measure of U ∼ k −1/3 k −2(d−2)/3 , we have that and so using (3.11) we obtain the first bound in (3.10). Similar to before, if we repeat this argument with (3.3) instead of (3.2), we find the second bound in (3.10). The heart of the proof of Theorem 1.15 is the following lemma.
Lemma 4.1 There exists aC > 0 such that under the condition the Galerkin equations (1.11) have a unique solution satisfying (1.20).
Proof of Theorem 1.15 using Lemma 4.1. Using the triangle inequality, a sufficient condition for (4.1) to hold is we show in Remark 4.5 below that we do not lose anything by doing this, i.e., (4.2) is no less sharp than (4.1) in terms of k-dependence. 1 from Theorem 4.2 and the different bounds on D k L 2 (∂Ω)→H 1 (∂Ω) and S k L 2 (∂Ω)→H 1 (∂Ω) in Theorem 1.10 (apart from when d = 2 when we use the bound on S k (1.17) instead of (1.14)).
To prove Theorem 1.15 we therefore only need to prove Lemma 4.1. This was proved in [42,Corollary 4.1], but since the proof is short we repeat it here for completeness. We first introduce some notation: let P h denote the orthogonal projection from L 2 (∂Ω) onto V h (see, e.g, [6, §3.1.2]); then the Galerkin equations (1.11) are equivalent to the operator equation The proof requires us to treat A k,η as a (compact) perturbation of the identity, and thus we let L k,η := D k − iηS k . Furthermore, to make the notation more concise, we let λ = 1/2. Therefore, the left-hand side of (4.3) becomes (λI + P h L k,η )v h , and the question of existence of a solution to (4.3) boils down to the invertibility of (λI + P h L k,η ). Note also that, with the P h -notation, the best approximation error on the right-hand side of (1.20) The heart of the proof of Lemma 4.1 is the following lemma.

Lemma 4.3 If
for some δ > 0, then the Galerkin equations have a unique solution, v h , which satisfies the quasi- Therefore, choosing, say, δ = 1, we find that there exists aC > 0 such that (4.1) implies that (4.4) holds.
Thus, to prove Theorem 1.15, we only need to prove Lemma 4.3.
Proof of Lemma 4.3. Since Writing the direct equation as (λI + L k,η )v = f and the Galerkin equation as (λI and the result (4.5) follows from using the bound (4.6) in (4.7).
Remark 4.4 (Is there a better choice of η than |η| ∼ k?) Theorem 1.15 is proved under the assumption that |η| ∼ k. This choice of η is widely recommended from studies of the condition number of A k,η ; see [18,Chapter 5] for an overview of these. From (4.2) we see that the best choice of η, from the point of view of obtaining the least-restrictive threshold for k-independent quasi-optimality, will minimise the k-dependence of There does not yet exist a rigorous proof that |η| ∼ k minimises this quantity, but [9, §7.1] outlines exactly the necessary results still to prove.
Remark 4.5 (Using the triangle inequality on D k − iηS k L 2 (∂Ω)→H 1 (∂Ω) ) We now show that we do not lose anything, from the point of view of k-dependence, by using the triangle in- . First, recall that D k and S k have wavefront set relation given by the billiard ball relation (see for example [38,Chapter 4]). Denote the relation by C β ⊂ B * ∂Ω × B * ∂Ω i.e.
where β is the billiard ball map (see Figure 1). To see that the optimal bound in terms of powers of k for D k − iηS k L 2 (∂Ω)→H 1 (∂Ω) is equal to that for D k L 2 (∂Ω)→H 1 (∂Ω) + |η| S k L 2 (∂Ω)→H 1 (∂Ω) , observe that the largest norm for S k corresponds microlocally to points (q 1 , q 2 ) ∈ C β ∩(S * ∂Ω×S * ∂Ω) (i.e. "glancing" to "glancing"). On the other hand, these points are damped (relative to the worst bounds) for D k . In particular, microlocally near such points, one expects that where f q2 L 2 (∂Ω) = 1 and f q2 is microlocalized near q 2 . Figure 1: A recap of the billiard ball map. Let q = (x, ξ) ∈ B * ∂Ω (the unit ball in the cotangent bundle of ∂Ω). The solid black arrow on the left denotes the covector ξ ∈ B * x ∂Ω, with the dashed arrow denoting the unique inward-pointing unit vector whose tangential component is ξ. The dashed arrow on the right is the continuation of the dashed arrow on the left, and the solid black arrow on the right is ξ(β(q)) ∈ B * πx(β(q)) ∂Ω. The center of the left circle is x and that of the right is π x (β(q)). If this process is repeated, then the dashed arrow on the right is reflected in the tangent plane at π x (β(q)): the standard "angle of incidence equals angle of reflection" rule.

Proof of Theorem 1.16
Proof of Theorem 1.16. By the polynomial-approximation result (1.12), we only need to prove that the bound (1.24) hold with the different functions A(k). The idea is to take the H 1 norm of the integral equation (1.2) and then use the L 2 (∂Ω) → L 2 (∂Ω) and L 2 (∂Ω) → H 1 (∂Ω) bounds from Theorems 2.10 and 1.10 respectively.
Taking the H 1 norm of (1.2) and using the notation that A k,η = 1 2 I + L k,η and v := ∂ + n u as in the proof of Theorem 1.15 above, we have In this inequality, η is just a parameter that appears in L k,η and f k,η , with the equation holding for all values of η; in other words, the unknown v(= ∂ + n u) does not depend on the value of η. We now seek to minimise the k-dependence of L k,η L 2 (∂Ω)→H 1 (∂Ω) . Looking at the k-dependence of the L 2 (∂Ω) → H 1 (∂Ω)-bounds on S k and D k in Theorem 1.10, we see that, under each of the different geometric set-ups, the best choice is η = 0, and thus where we have explicitly worked out the k-dependence of f k,η H 1 (∂Ω) using the definition (1.10).
Taking the L 2 norm of (1.2) (with η = 0), and noting that f k,η L 2 (∂Ω) ∼ k, we have that Using (4.9) in (4.8), we have Since the bounds on the L 2 (∂Ω) → H 1 (∂Ω)-norm of D k in Theorem 1.10 are one power of k higher that the L 2 (∂Ω) → L 2 (∂Ω)-bounds in Theorem 2.10, using these norm bounds in (4.10) results in the bound v H 1 (∂Ω) A(k) v L 2 (∂Ω) with the functions of A(k) as in the statement of theorem (and equal to the right-hand sides of the bounds on D k L 2 (∂Ω)→H 1 (∂Ω) in Theorem 1.10).
5 Proofs of Theorem 1.21 (the result concerning Q2) To prove Theorem 1.21 we need to recall (i) the result about coercivity of A k,η when Ω is convex, C 3 , piecewise analytic, and curved from [77], and (ii) the refinement of the Elman estimate in [11].
Theorem 5.1 (Coercivity of A k,η for Ω convex, C 3 , piecewise analytic, and curved [77]) Let Ω be a convex domain in either 2-or 3-d whose boundary, ∂Ω, is curved and is both C 3 and piecewise analytic. Then there exist constants η 0 > 0, k 0 > 0 (with η 0 = 1 when Ω is a ball) and a function of k, α k > 0, such that for k ≥ k 0 and η ≥ η 0 k, In stating this result we have used the bound (2.9) on S k in [77, Remark 3.3] to get the asymptotics (5.2). The fact that η 0 = 1 when Ω is a ball follows from [76,Corollary 4.8]. When we apply the estimate (5.4) to A, we find that β = π/2 − δ, where δ = δ(k) is such that δ → 0 as k → ∞. We therefore specialise the result (5.4) to this particular situation in the following corollary.
That is, choosing m δ −1 is sufficient for GMRES to converge in an δ-independent way as δ → 0.
Note that the assumption in the theorem that ∂Ω is analytic comes from the fact that if ∂Ω is both piecewise analytic and C ∞ , then ∂Ω must be analytic, where the notion of piecewise analyticity in Theorem 5.1 is inherited from [26, Definition 4.1].
Remark 5.5 (The star-combined operator) The bound on the number of iterations in Theorem 1.21 crucially depended on the coercivity result of Theorem 5.1. Although numerical experiments in [13] indicate that A k,η is coercive, uniformly in k, for a wider class of obstacles that those in Theorem 5.1, this has yet to be proved. Nevertheless, there does exist an integral operator that (i) can be used to solve the sound-soft scattering problem, and (ii) is provable coercive for a wide class of obstacles. Indeed, the star-combined operator A k , introduced in [76] and defined by The refinement of the Elman estimate in Theorem 5.2 can therefore be used to prove results about the number of iterations required when GMRES is applied to the Galerkin discretisation of (5.10). Since the coercivity constant of the star-combined operator is independent of k, the kdependence of the analogue of the bound (1.25) for A k rests on the bounds on A k L 2 (∂Ω)→L 2 (∂Ω) .
For convex Ω with smooth and curved ∂Ω, Theorems 2.10 and Theorem 1.10 imply that A k L 2 (∂Ω)→L 2 (∂Ω) k 1/3 , and we therefore obtain the same bound on m as for A k,η (i.e. (1.25)). For general piecewise-smooth Lipschitz obstacles that are star-shaped with respect to a ball, Theorems 2.10 and 1.10, along with the bound  .17), show that A k L 2 (∂Ω)→L 2 (∂Ω) k 1/2 when d = 2 and k 1/2 log k when d = 3. Corollary 5.3 then implies that m k 1/2 for d = 2 and m k 1/2 log k for d = 3. Recall that GMRES always converges in at most N steps (in exact arithmetic), and when h ∼ 1/k we have that N ∼ k d−1 ; these bounds on m are therefore nontrivial.

Numerical experiments concerning Q2
The main purpose of this section is to show that the k 1/3 growth in the number of iterations given by Theorem 1.21 is effectively sharp.
Details of the scattering problems considered We solve the sound-soft scattering problem of Definition 1.7 with a = (1, 0, 0) (i.e the incident plane wave propagates in the x 1 -direction), using the direct integral equation (1.2) and the Galerkin method (1.11). The subspace V h is taken to be piecewise constants on a shape regular mesh, and the meshwidth h is taken to be 2π/(10k), i.e. we are choosing ten points per wavelength. We solve the resulting linear system with GMRES, with tolerance 1 × 10 −5 . We consider two obstacles: 1. Ω the unit sphere, and 2. Ω the ellipsoid with semi-principal axes of lengths 3, 1, and 1 (in the x 1 -, x 2 -, and x 3 -directions respectively. The computations were carried out using version 3.0.3 of the BEM++ library [74] on one node of the "Balena" cluster at the University of Bath. The cluster consists of Intel Xeon E5-2650 v2 (Ivybridge, 2.60 GHz) CPUs and the used node had 512GB of main memory. BEM++ was compiled with version 5.2 of the GNU C compiler and the Python code was run under Anaconda 2.3.0.
Numerical results Tables 1 and 2 displays the number of degrees of freedom, number of iterations required for GMRES to converge, and time taken to converge, with η = k, and with Ω the sphere or ellipsoid. The difference between Tables 1 and 2 is that, in the first, k starts as 2 and then doubles until it equals 128, and in the second, k starts as 3 and then doubles until it equals 96; we performed the second set of experiments when the k = 128 run for the ellipsoid failed to complete. Figure 2 plots the iteration counts from both tables and compares them to the k 1/3 rate   from Theorem 1.21 (the graph is plotted on a log-log scale so that a dependence # iterations ∼ k α appears as a straight line with gradient α).
We see from Figure 2 that the k 1/3 growth predicted by Theorem 1.21 appears to be sharp. Indeed, the plot of the iterations for the ellipsoid becomes roughly linear from k = 12 onwards, and estimating the slope of this line using the numbers of iterations at k = 12 and k = 96 we have that the # iterations ∼ k 0.28 . Using the numbers of iterations at k = 12 and k = 96 to estimate the rate of growth for the sphere we have that # iterations ∼ k 0.29 .
Finally, Table 3 compares the iteration counts and times for the sphere when η = k and when η = −k. We see that, for every value of k considered, the number of iterations when η = −k is much greater than when η = k. Table 3 only goes up to k = 32, since the k = 64 run for the sphere with η = −k did not complete.
We performed the experiment in Table 3 because, in the engineering acoustics literature, Marburg recently considered collocation discretisations of the direct integral equation for the Neumann problem (i.e. the Neumann-analogue of equation (1.2)) and showed that the analogue of the choice η = k leads to much slower growth than the analogue of the choice η = −k [57], [58].
A heuristic explanation for this dependence of the number of iterations on the sign of η is essentially contained in the work of Levadoux and Michielsen [54], [55], and Antoine and Darbas [3]; the understanding is that iη should, in some sense, approximate the Dirichlet-to-Neumann map in Ω + , and (at least for smooth convex obstacles) ik is a better approximation to the Dirichlet-to-Neumann map than −ik.  Table 3: With Ω the sphere and η = k or η = −k, the number of iterations required for GMRES to converge (with tolerance 1 × 10 −5 ) and time taken to converge, when GMRES is applied to the Galerin matrix corresponding to the direct integral equation (1.2).