Wave number-Explicit Analysis for Galerkin Discretizations of Lossy Helmholtz Problems

We present a stability and convergence theory for the lossy Helmholtz equation and its Galerkin discretization. The boundary conditions are of Robin type. All estimates are explicit with respect to the real and imaginary part of the complex wave number $\zeta\in\mathbb{C}$, $\operatorname{Re}\zeta\geq0$, $\left\vert \zeta\right\vert \geq1$. For the extreme cases $\zeta \in\operatorname*{i}\mathbb{R}$ and $\zeta\in\mathbb{R}_{\geq0}$, the estimates coincide with the existing estimates in the literature and exhibit a seamless transition between these cases in the right complex half plane.


Introduction
For many problems in time-harmonic acoustic scattering, the Helmholtz equation serves as a model problem, and its numerical discretization is a topic of vivid research.For homogeneous, isotropic material the differential operator is given by L The solution is highly oscillatory if |Im ζ| ≫ 1, which makes the discretization challenging with respect to both, stability and accuracy.To study this problem systematically the case of purely imaginary wave numbers ζ = − i k, k ∈ R, has often been used in the literature as a model problem for designing and analyzing numerical methods.However, in many applications waves are damped, e.g., by friction and viscoelastic effects in the material or loss via sound radiation or flow of vibration energy out of the physical scatterer (see, e.g., [18]).
Another important application is the approximation of the inverse Laplace transform by contour quadrature where the Helmholtz operator has to be discretized at many complex frequencies in the right complex half plane (see, e.g., [5]).
For the two extreme cases ζ = − i k and ζ = ν, k ∈ R, ν ∈ R ≥0 , a fairly complete theory for standard Galerkin hp-finite elements is available and the error estimates are explicit with respect to the wave number ζ, the mesh width h of the finite element mesh, and the polynomial degree p: a) For ζ = − i k and large |k| the problem is highly indefinite and a "resolution condition" of the form |k|h p ≤ C together with p ≥ C log |k| has to be imposed in order to ensure solvability of the Galerkin equations and quasi-optimality ( [9,10,8,2]); b) for ζ = ν > 0 and ν = O (1), the problem is properly elliptic and Céa's lemma ensures well-posedness and quasi-optimality without any resolution condition; c) for ζ = ν ≫ 1, the solution exhibits boundary layers.Although the Galerkin discretization is always well-posed in this last situation, special meshes should be used that are adapted to the boundary layers (see, e.g., [11,16,7] and references there).In this paper, we will develop a unified theory for Galerkin discretizations of L ζ with Robin boundary conditions that is applicable for all ζ ∈ C, Re ζ ≥ 0, and |ζ| ≥ 1.All estimates are explicit in terms of Re ζ and Im ζ and reproduce the limiting cases of purely real and imaginary ζ.It is shown that, for the sectorial case, i.e., the wave number lies in a sectorial neighborhood of the real axis in the right complex half plane, well-posedness and quasi-optimality is a consequence of coercivity while for Re ζ → 0 the estimates tend continuously to the purely imaginary case ζ = − i k.We follow the general theory developed in [9,10] and refine the estimates to be explicit with respect to the real and imaginary part of the wave number.
The paper is structured as follows.In Sect. 2 we introduce the Helmholtz model problem with Robin boundary conditions and formulate some geometric and algebraic assumptions on the data.Further, we define for the wave number the (well-behaved) sectorial and the (more critical) non-sectorial region.
The estimate of the continuity constant for the sesquilinear form is derived in Sect.3. Sect. 4 is devoted to the analysis of the inf-sup constant for the continuous sesquilinear form.If the real part of the wave number is positive the estimate follows simply from the coercivity of the sesquilinear form.However, this bound degenerates as Re ζ → 0. This can be remedied by a different proof: first one uses suitable test functions to derive stability estimates for an adjoint problem with L 2 right-hand sides and then by employing this result for the estimate of the inf-sup constant in a vicinity of the imaginary axis.
The key role for the analysis of the Galerkin discretization is played by a regular decomposition of the Helmholtz solution.In Sect.5, we introduce a splitting of the Helmholtz solution into a part with (low) H 2 -regularity and wave number-independent regularity constant and an analytic part with a more critical wave number dependence.First, this is derived for the full space solution by generalizing the results for purely imaginary frequencies in [9].In the case of bounded domains, we generalize the iteration argument in [10,Sect. 4] to general complex frequencies.In addition, this requires sharp estimates of frequencydepending lifting operators which we also present in this section.
Sect. 6 is devoted to the estimate of the discrete inf-sup constant for the standard Galerkin discretization of the Helmholtz equation.We will derive two type of estimates: one requires that the finite dimensional space for the Galerkin discretization satisfies a certain resolution condition and allows for robust (as Re ζ → 0) stability and quasi-optimal convergence estimates; the other one avoids a resolution condition while the constants in the estimates tend towards ∞ as Re ζ → 0 but stay robust for the sectorial case.Numerical examples in Sect.7 illustrate the application of our analysis in the context of hp-FEM.

Setting
We consider the Helmholtz problem for f ∈ L 2 (Ω) and g ∈ L 2 (Γ).We assume that the wave number (frequency where, for ρ ∈ R, Note that the choice ζ = − i k leads to the standard Helmholtz case.The frequency domain C • ≥0 is split into the sectorial and non-sectorial cases for some β > 0. Our focus is on the derivation of stability and error estimates that are explicit in the real and imaginary part of ζ but less on the development of a theory with minimal assumptions on the geometry of the domain.In this light we impose the following simplifying assumption.
Assumption 2.1 Ω ⊂ R 3 is a bounded domain with analytic boundary that is star-shaped with respect to a ball.
We note that our results can be extended to convex polygonal domains in a straightforward way following the arguments in [10].
Let L 2 (Ω) denote the usual Lebesgue space with scalar product denoted by (•, •) (complex conjugation is on the second argument) and norm denote the usual Sobolev space and let γ 0 : H 1 (Ω) → H 1/2 (Γ) be the standard trace operator.We introduce the sesquilinear forms The weak formulation of the Helmholtz problem with Robin boundary conditions (2.1) is given as follows: (2.3) In the following, we will omit explicitly writing the trace operator γ 0 when it is clear that it is implied.

The Continuity Constant
In this section, we will estimate the continuity constant of the sesquilinear form a ζ (•, •).We equip the Sobolev space V with the indexed norm • |ζ| , where, for ρ > 0, we set More generally, for measurable subsets T ⊂ Ω we write The L 2 -norm on Γ is denoted by • Γ .On H 1/2 (Γ) we introduce the weighted norm for ρ > 0.
Theorem 3.1 The sesquilinear form a ζ is continuous and Proof.The continuity estimate for the sesquilinear form b ζ (•, •) is a simple consequence of the multiplicative trace inequality (see [4, p.41, last formula]) Hence which implies the continuity of b Our goal in this section is to estimate the inf-sup constant which implies well-posedness of (2.3).This involves two different theoretical techniques: In Sect.4.1 we consider the case Re ζ > 0 and obtain estimates from the coercivity of the sesquilinear form.These estimates give stable bounds for the sectorial case but deteriorate as Re ζ → 0 in the non-sectorial case.In Sect.4.2 we employ the sesquilinear form with a suitably selected test function and obtain sharp estimates also for the non-sectorial case.

The Inf-Sup Constant for Re ζ > 0
The estimate of the inf-sup constant in the following Lemma 4.1 is a direct consequence of the technique used in [1].For every F ∈ V ′ , problem (2.3) has a unique solution.In particular if there Proof.We follow the idea of the proof in [1].We choose v = ζ |ζ| u.For the sesquilinear form with Robin boundary conditions we have The positivity of the inf-sup constant γ ζ implies unique solvability (see, e.g., [12,Thm. 2.1.44];the above argument can be used to show [12, (2.34b)]).We obtain A multiplicative trace inequality in the form of (3.5) leads to (4.3).
Lemma 4.2 Let Ω ⊂ R 3 be a smooth domain that is star-shaped with respect to a ball or let Ω be a convex polyhedron.Let the functional F ∈ V ′ be of the form 3) has a unique solution and satisfies for some , a stability estimate is proved that is related to (4.4) if Re ζ is sufficiently small.For ζ ∈ S c β , the estimate (4.4) is non-degenerate for Re ζ → 0 in contrast to (4.2) and the result in [3].
Proof.Without loss of generality, we assume that Ω is star-shaped with respect to the origin.We will fix a parameter β > 1 sufficiently large at the end of the proof.We distinguish between two cases.
Case a: ζ ∈ S β .The condition |ζ| ≥ 1 leads to and Lemma 4.1 becomes applicable: For Re ζ > 0, existence and uniqueness follows from Lemma 4.1 while the well-posedness in the case Re ζ = 0 is a consequence of [6,Prop. 8.1.3].We write |ζ| u and consider the real part of (2.3), which yields Young's inequality on the right-hand side leads to These two inequalities imply which is the desired (4.4) in view of ν ≥ 1.
The proof for ν < 1 is essentially a repetition of the arguments in the proof of [6,Prop. 8.1.4]using the inequalities for three different test functions in (2.3) and Young's inequality.For completeness, we show the relevant inequalities.The first test function is v = u yielding, after taking the real part, Next we choose v = − sign(k)u and consider the imaginary part to get As a last test function we use v (x) = x, ∇u (x) ; note that the assumptions on the domain imply via elliptic regularity theory that v ∈ V .Integration by parts yields with d = 3 (we write d to indicate the generalization to arbitrary spatial dimension d) Rearranging yields We remark that (4.8) and (4.7) give ) which allows for controlling u Γ and ∇u in terms of k u and the data f , g: ) Since Ω is assumed to be star-shaped, one has 0 < c 1 ≤ x, n(x) ≤ c 2 for all x ∈ Γ. Inserting this and (4.13) into (4.9)gives with The proof can be completed with suitable applications of Young's inequality, use of (4.12), (4.13), and selecting β sufficiently large to treat the term νk u x, ∇u ≤ c 3 νk u ∇u .

The Inf
In the following Theorem 4.4 we will prove an alternative estimate (compared to (4.2)) for the inf-sup constant that is robust as Re ζ → 0. To estimate this constant we employ the standard ansatz u ∈ V and v = u + z for some z ∈ V .Then The choice of z will be related to some adjoint problem.the next section.
We consider the adjoint problem: find z ∈ V such that which is well-posed according to Lemma 4.2 and satisfies For this choice of z, we consider the real part of (4.14) and obtain

Regular Decomposition of the Helmholtz Solution
In this section, we develop a regular decomposition of the solution of the Helmholtz problem (2.1) based on a frequency splitting of the right-hand side.The frequency splitting for functions defined on the full space R 3 is defined via their Fourier transform (Sect.5.1).For functions defined on finite domains, we derive the regular splitting using a lifting operator (Sect.5.3).This generalizes the theory developed in [9,10] to complex frequencies and the resulting estimates are explicit with respect to the real and imaginary part of the wave number.

The Full Space Adjoint Problem for ζ ∈ S c β
The first result concerns the adjoint problem for the full space Ω = R 3 .Let φ ∈ L 2 (Ω) be a function with compact support.We choose R > 0 sufficiently large so that the open ball B R with radius R centered at the origin contains supp φ.We consider the problem To analyze this equation we employ Fourier transformation and introduce a (5. 2) The fundamental solution to the Helmholtz operator The properties of µ ensure z µ | BR = z| BR .To analyze the stability and regularity of z µ we introduce a frequency splitting of the solution z µ = z H 2 + z A that depends on the complex frequency ζ ∈ C ≥0 and a parameter λ ≥ λ 0 > 1.
and the inversion formula Next, we introduce a frequency splitting of a function w ∈ L 2 (Ω) depending on ζ and a parameter λ > 1 by using the Fourier transformation.The low-and high-frequency part of w is given by where χ δ is the characteristic function of the open ball with radius δ > 0 centered at the origin.We construct a decomposition of z µ (5.9) as follows: We decompose the right-hand side φ in (5.1) via Accordingly, we define the decomposition of z µ by The Fourier transform of G ζ, • M is given by In the following we will analyze the symbol σ (ζ, •).We have: Applying integration by parts leads to This allows for the estimate with Hence, .
Since E 0 (t) ≤ 1 we end up with As a consequence, we have proved that so that we have This shows (5.3).In the following we estimate higher order derivatives.For the product s 2 σ (s), we get The estimates T II ≤ (2π) −3/2 4CE 0 (4Rν) (5.13) follow from the properties of µ (cf.(5.2)).As a simple consequence we obtain for m ≥ 2 and (5.15) Hence for α ∈ N 3 0 , |α| = 2, we have The bounds (5.16) expresses the desired estimate (5.6).A direct application of (5.14) does not lead to (5.5) as it introduces an undesired factor |ζ|.This is removed by noting that is suffices to consider s = ξ with s ≥ λ|k| and that only the estimates for T I need to be refined.This is achieved with an integration by parts: e −ζr sin (rs) µ ′ (r) dr . (5.17) Observe Also we have Hence, (5.18)This leads to and, in turn, From this, assertion (5.5) follows.

The Helmholtz Solution with Robin Boundary Conditions
In this section, we will derive a regularity result in the spirit of Lemma 5.1 for ζ ∈ S c β for the interior problem with Robin boundary conditions: Theorem 5.3 Let Assumption 2.1 be valid and fix β > 0. Then there exist constants C, γ > 0 such that for every f ∈ L 2 (Ω), g ∈ H 1/2 (Γ), and ζ ∈ S c β , the solution u = S ζ (f, g) of (5.19) can be written as u = u A + u H 2 , where, for all p ∈ N 0 , ) (5.23) The proof is the generalization of the proof in [10] for real wave numbers to more general ζ ∈ C • ≥0 with emphasis on the explicit dependence of the estimates on the real and imaginary part.It follows from Lemmata 5.11 and 5.12, which are presented in Sect.5.3 ahead.

and S ζ
For the analysis we introduce low-and high pass frequency filters for a bounded domain as well as for its boundary.Let E Ω : L 2 (Ω) → L 2 (R 3 ) be the extension operator of Stein, [17, Chap.VI].Then for f ∈ L 2 (Ω) we set for L R d and H R d defined in (5.8) for some λ > 1.By [10, Lemmas 4.2, 4.3], these operators have the following stability properties: where the constant C s depends on s and C s,s ′ depends on s, s ′ but is independent of λ and ζ.
To define frequency filters on the boundary we employ a lifting operator G N defined in Lemma 5.4 below with the mapping property G N : H s (Γ) → H 3/2+s (Ω) for every s > 0 and ∂ n G N g = g.We then define H N Γ and L N Γ by In particular, we have Then the following holds: Proof.The energy estimate (5.28) follows from the coercivity of the pertinent sesquilinear form.The H 2 -estimate follows from elliptic regularity theory.
Then there is λ > 1 in the definition of L N Γ and H N Γ such that the following holds (with implied constants independent of q):

.31)
Proof.Recall that L N Γ g := γ 0 g N , where and n * denotes an analytic extension of the normal n : Γ → S 2 on Ω to a tubular neighborhood T ⊂ Ω of Γ and γ 0 is the standard trace operator.Using (3.5) yields where ∇∇ ⊺ denotes the Hessian of a function.From (5.25) For s = 1/2, we note The proof of (5.31) is similar.We note where q is related to λ via (5.26) and can be made arbitrarily small by selecting λ appropriately.Hence, recalling that Next, we introduce the solution operators 1. We denote by u := N ζ f = G(ζ) * f the solution of the full space Helmholtz problem with Sommerfeld radiation condition (in the weak sense): ) with compact support.Here ∂/∂r denotes the derivative in radial direction x/ x .

S ∆ ζ (g) is the solution operator to the problem
for g ∈ L 2 (Γ).
3. We define ) as the solution operator to the problem (2.1) for analytic right-hand sides The proof of the next lemma is a direct consequence of Lemma 5.1.
For any λ > 1 (appearing in the definition of the operator H R 3 defined in (5.8)) there exist C > 0 depending only on R and µ such that (5.33a) Furthermore, for β > 0 the following is true: given q ∈ (0, 1) one can select λ > 1 such that for all (5.34b) Proof.(5.33) is a direct consequence of Lemma 5.1.The bounds (5.34) follow from (5.33).
The next two lemmata generalize the results in [10, Lemmas 4.5, 4.6] to complex wave numbers ζ.

Lemma 5.7 (properties of S ∆
ζ ) Let Ω be a bounded Lipschitz domain and ) Proof.The proof is essentially given in [10,Lemma 4.5].
A combination of Lemma 5.5 and Lemma 5.8 imply the following corollary.
Corollary 5.8 (properties of S ∆ ζ • H N Γ ) Let Assumption 2.1 be satisfied, β > 0, and let q ∈ (0, 1).There exists λ > 1 defining the high frequency filter H N Γ such that for every g ∈ H 1/2 (Γ) and every ζ ∈ S c β we have Lemma 5.9 (analyticity of S L ζ ) Let Assumption 2.1 be valid and let λ > 1 appearing in the definition of L Ω and L N Γ be fixed.Then there exist constants ) is analytic on Ω and satisfies for all p ∈ N 0 the estimates Proof.From Lemma 4.2, we have (5.39) The combination of (5.39), Lemma 5.4, Lemma 5.5 and (5.25) leads to To estimate higher derivatives, we employ [7,Prop. 5.4.5] in a similar way as in the proof of [10,Lemma 4.13].To apply [7,Prop. 5.4.5] an estimate of the constant is needed, where g N is defined in (5.32).We track the dependence of C G1 on |ζ| in a modified way (compared to [10, Proof.We proceed in the same way as in [10, Lemma 4.12] with k = Im ζ and estimate the constant C G1 in (5.40).Lemma 5.6 and (3.5) lead to This implies and In the same way as at the end of the proof of Lemma 5.9 we obtain 1 be valid, and ζ ∈ S c β .For every q ∈ (0, 1), there exist constants C, K > 0, depending on Here, the parameter λ defining the filter operators L Ω and H Ω is still at our disposal and will be selected at the end of the proof.Then, u I A satisfies the desired bounds by Lemma 5.9.Lemma 5.6 gives Also, the parameter q ′ ∈ (0, 1) depends on λ and is still at our disposal.In fact, in view of the statement of Lemma 5.6 it can be made sufficiently small by taking λ sufficiently large.The function u (5.43) Next, we define the functions u II A and u II H 2 by Then, the analytic part u II A satisfies again the desired analyticity bounds by Lemma 5.9 and Corollary 5.10 .For the function u II H 2 we obtain from Lemma 5.8 and inequalities (5.41) (set ũ = u I H 2 ) the estimates and conclude that the function u For f we obtain f ≤ C|ζ| u II H 2 |ζ| ≤ Cq ′ f .Hence, by taking λ sufficiently large so that q ′ is sufficiently small, we arrive at the desired bound.Lemma 5.12 (properties of S ζ (0, g)) Let β > 0 and Assumption 2.1 be valid.Let q ∈ (0, 1).Then there exist constants C, K > 0 independent of ζ ∈ S c β (but depending on β) such that for every g ∈ H 1/2 (Γ) the function u = S ζ (0, g) can be written as u = u A + u H 2 + u, where for all p ∈ N 0 For a g with g Γ,|ζ| ≤ g Γ,|ζ| the remainder u = S ζ (0, g) satisfies the equation Proof.The proof is very similar to that of Lemma 5.11.Define Then u I A is analytic and satisfies the desired analyticity estimates by Lemma 5.9.For u I H 2 we have by Corollary 5.8 Here, in order to apply the operator N ζ , we extend H Ω 2 k 2 + i νk u I H 2 by zero outside of Ω.By Lemma 5.9 and (5.46), we see that u II A satisfies the desired analyticity estimates.For the function u II H 2 , we obtain from Lemma 5.6 We set u A := u I A + u II A and u H 2 := u I H 2 + u II H 2 .Then u A and u H 2 satisfy the desired estimates and u := u − (u A + u H 2 ) satisfies The result follows by selecting λ sufficiently large so that q ′ is sufficiently small.

Discretization
We apply the regularity theory of the previous section to the of hp-finite element method.Let Sζ be the solution operator of the adjoint problem: Let S ⊂ V be a closed subspace and define the adjoint approximability η(S) := sup 6.1 Discrete Inf-Sup Constant γ disc and Quasi-Optimality for a constant c independent of ζ.Let z S ∈ V be the best approximation of z with respect to the and, in turn, we have proved A simple calculation shows that there exists a constant c > 0 independent of ζ ∈ C • ≥0 such that the right-hand side in (6.3) is bounded from below by the right-hand side in (6.2).Theorem 6.3 Assume that Re ζ > 0. Then the Galerkin method based on S is quasi-optimal, i.e., for every u ∈ V there exists a unique u S ∈ S with a(u − u S , v) − b(u − u S , v) = 0 for all v ∈ S, and Equation ( 6.4) is a direct consequence of the discrete inf-sup constant proved in Theorem 6.1.Estimate (6.5) follows from the proof of the next theorem (see (6.9)).We note here that for ζ ∈ S β , the ratio |ζ|/ Re ζ is bounded from above and no resolution assumption is required.In the next theorem, we find that under a resolution assumption, the estimate (6.4) can be improved, such that it is non-degenerate for Re ζ −→ 0.
then the Galerkin method based on S is quasi-optimal and

Impact on hp-FEM Approximation
We have shown in Sect.More specifically, one can prove that there exist constants C, σ > 0 that depend on the shape regularity of the triangulation such that for ever f ∈ L 2 (Ω) the function u = S|ζ| (f ) = S |ζ| (αf , 0) satisfies for the regular decomposition u = u A + u H 2 given by Theorem 5.3 (see [9,Sect. 5], in particular the proof of [9, Thm.5.5] for details).By choosing h and p as in (6.10) the right-hand sides in (6.11a) and (6.11b) imply the resolution assumption (6.6) and therefore the optimal convergence for the Galerkin solution.
If ζ ∈ S β no resolution condition is needed for the quasi-optimality of the problem (cf.Theorem 6.3).In that case, the solution is typically smooth in the domain and exhibits, for large Re ζ, a boundary layer.Such problems can be handled by suitable meshes capable to resolve the layers such as Shishkin meshes in the context of the h-version of the FEM [11,16,7] and "spectral boundary layer meshes" in the context of the hp-FEM, [15,7].
Using Bessel functions and polar coordinates, the solution is given as .
We The purely imaginary wave number corresponds to the choice α = π/2 and α = 0 to the real-valued case.We consider the h-FEM on quasi-uniform meshes for p ∈ {1, 2, 3, 4}.The results are presented Fig. 1, where the error is plotted versus the number of degrees of freedom per wavelength The calculations were carried out within the hp-FEM framework NgSolve, [13,14].The following features are visible in Fig. 1 c) The pollution effect decreases with increasing polynomial degree.In particular, the asymptotic behavior is reached for smaller values of N |ζ| as p is increased.
d) The pollution effect decreases with decreasing angle α.
The observation a) reflects a natural resolution condition for the problem class under consideration; that is, the best approximation error can only be expected to be small if N |ζ| ∼ |ζ|h/p is small.The pollution effect observed in b) is well-documented for the purely imaginary case Re ζ = 0. Fig. 1 shows that it is present also for Re ζ = 0 (and large Im ζ), albeit in a mitigated form.Theorem 6.4 quantifies how this pollution effect is weakened as the ratio Re ζ/ Im ζ increases.More specifically, the resolution condition (6.10), which results from applying Theorem 6.4 to high order methods, illustrates the helpful effect of Re ζ = 0.In the limiting case Im ζ = 0, the Galerkin method is an energy projection method and even monotone convergence can be expected in the energy norm on sequences of nested meshes.The observation c) is also well-documented for the purely imaginary case Re ζ = 0 and mathematically explained in [9,10].The regularity of the present work permits to extend the hp-FEM analysis of [9,10] to the case Re ζ = 0 as done in Sect.6.2.The observation that the asymptotic convergence regime is reached for smaller N |ζ| as p is increased can be understood qualitatively from Theorem 6.This shows that for larger p quasi-optimality of the hp-FEM may be expected for small N |ζ| .
Finally, observation d) can again be explained by Theorem 6.

Theorem 4 . 4 ≥ 1 1 + c |Im ζ| 1+Re ζ .
Let Ω ⊂ R 3 be a smooth domain that is star-shaped with respect to a ball or let Ω be a convex polyhedron.Then there exists a constant c > 0 such that for all ζ ∈ C • ≥0 the inf-sup constant γ f of (4.1) satisfies γ ζ Proof.Let ν = Re ζ and k = − Im ζ and set σ = 1/ √ 2. First, we consider the case ζ ∈ C • ≥0 with ν ≥ σ.From Lemma 4.1 we have for any ζ ∈ C • ≥σ the estimate 0) of radius R > 0 centered at the origin, and let µ be a cutoff function satisfying (5.2).Then there exists a constant C > 0 depending only on R and µ such that the solution z = G ζ * φ of (5.1) and z µ := G ζ M * φ satisfy z| BR = z µ | BR and

Remark 6 . 2
The resolution condition (6.2) is not an artifact of the theory: in[8, Ex. 3.7], a domain Ω, a finite element space S, and a purely imaginary wave number ζ = − i k are presented where the Galerkin discretization leads to a system matrix that is not invertible.Proof of Theorem 6.1.Let ζ = ν − i k.The first statement follows directly from the continuous inf-sup constant in Lemma 4.1.We prove the second statement.Let u ∈ S and choose v = u + z, where z = 2k 2 Sζ (u).Then it is simple to check that Re a(u, u + z) ≥ u 2 |ζ| .
: a) A plateau before convergence sets in.b) A pollution effect for ζ close to the imaginary axis (α = π/2).That is, asymptotic quasi-optimality sets in for larger N |ζ| as |ζ| becomes larger for Arg ζ close to π/2.
[10,9]hat the Galerkin solution u S ∈ S of the Helmholtz problem with Robin boundary conditions(5.19)withζ∈Sc β is quasi-optimal for any closed subspace S ⊂ V , if the adjoint approximability η(S) fulfills the Let S hp be the hp-FEM space described in[9, Sect.5].Similarly as in[10,9], one can show that the Galerkin method based on S hp is quasi-optimal if