Complex-scaled infinite elements for resonance problems in heterogeneous open systems

The technique of complex scaling for time harmonic wave-type equations relies on a complex coordinate stretching to generate exponentially decaying solutions. In this work, we use a Galerkin method with ansatz functions with infinite support to discretize complex-scaled scalar Helmholtz-type resonance problems with inhomogeneous exterior domains. We show super-algebraic convergence of the method with respect to the number of unknowns in radial direction. Numerical examples underline the theoretical findings.

standard finite element method. Complex scaling (i.e., the choice of the exterior domain and the complex coordinate stretching) can be done in various ways: parallel to the coordinate axes, resulting in so-called cartesian scalings (cf. [5]), in radial direction (cf. [6]), or in normal direction with respect to a convex interface (cf. [23]). In this work we will focus on radial scalings, although the method can be extended to cartesian or normal scalings in a straightforward way (cf. Remark 4.1).
PMLs are rather easy to implement in standard finite element codes but have the downside that there are many method parameters to choose: the scaling function, the thickness of the layer, and the finite element discretization of the layer. All these method parameters have to be balanced to ensure the efficiency of the method. The parameter choice becomes even more delicate if there are inhomogeneities such as potentials in the exterior domain.
In this work we present and analyze a method which is also based on complex scaling but omits the truncation of the exterior domain. In this way we simplify the choice of method parameters. Another idea to omit truncation was introduced in [4], where singular scaling profiles in combination with standard finite elements are used. In contrast, we use a linear scaling profile combined with non-standard basis functions. Convergence of our method is established based on the abstract results of [18,19] for holomorphic Fredholm operator functions.
As radial basis functions we choose generalized Laguerre functions leading to the complex-scaled infinite elements. They converge super-algebraically with respect to the number of radial unknowns , i.e., the error decays faster than any polynomial in 1 . Moreover, they lead to sparse and well-conditioned discretization matrices, and are simple to couple to interior problems. Note that a standard PML would lead to an exponentially decreasing truncation error. Nevertheless, if a finite element discretization with piecewise polynomials is used in the perfectly matched layer, the total error will decrease merely polynomially with respect to 1 due to the finite element error. It turns out that, for homogeneous exterior problems, the complex-scaled infinite elements are equivalent to the Hardy space infinite elements introduced in [14].
The remainder of the paper is organized as follows: In Section 2 we define the complex-scaled problems in question and give a brief explanation of the method of complex scaling. In Section 3 we explain the used tensor product exterior discretizations. The complex-scaled infinite elements are defined in Section 4. Section 5 contains the convergence analysis. The main results are summarized in Section 5.4. The numerical experiments of Section 6 underline our theoretical findings. The numerical tests also include a performance study of the infinite elements compared to a conventional radial PML. We end with a short conclusion.

Complex coordinate stretching
Let 3 be an unbounded open domain such that can be decomposed into a bounded interior part int with Lipschitz boundary int , an unbounded exterior part ext , and an interface . The open sets int ext int ext and the interface should fulfill the following assumptions: (i) int ext , where denotes the union of two disjoint sets, (ii) there exists 0, such that int 0 , ext int and x , and (iii) ext 1 x x 0 .
Note that these conditions imply that for each x ext there exists a unique pair x 0 , such that For the mapping defined by (2.1) we also write x x and x x x for the inverse mapping. Figure 1 illustrates the cross section of two examples of the setting described above.
In this work we consider linear complex scalings of the form for a given parameter 0 . Note that although we define the scaling for general non-zero parameters we will assume positive real and imaginary part for later on. We denote the Jacobian of the scaling by Note that is smooth on int and ext respectively.

The complex-scaled problem for homogeneous exterior domains
To motivate the use of the complex-scaled problem for inhomogeneous exterior domains, we use this subsection to sketch the derivation of this technique in the case of homogeneous exterior domains (cf. [6,10,21]). A version of the classical Helmholtz eigenvalue problem is to find an eigenpair 2 loc 0 , such that with boundary conditions on , a suitable potential function with x 1 for x ext , and a radiation condition for . For resonance problems this radiation condition can be specified by imposing that in ext the eigenfunctions are linear combinations of functions which are spherical Hankel functions of the first kind (with respect to ) and spherical harmonics (with respect to x).
Since the spherical Hankel functions are analytic we may consider the analytic continuation of an eigenfunction to the complex-scaled domain . Due to the asymptotic behavior of the spherical Hankel functions of the first kind the function is exponentially decreasing in ext with respect to and therefore an element of 1 ext as long as 0. Thus (assuming homogeneous Neumann boundary conditions) the pair satisfies the complex-scaled weak formulation for every 1 . A radiation condition is implicitly incorporated in (2.4) by imposing that 1 .
The equivalence of this complex scaling radiation condition to an expansion of the eigenfunctions in (spherical) Hankel functions of the first kind is shown in [6,10,21] for resonances with 0 and scaling parameters with positive real and imaginary part. The equivalence to a radiation condition based on boundary integrals (see [7]) is addressed in [23,32]. Note that a resonance , as well as the interior part of the corresponding complex-scaled resonance function, is independent of as long as 0. An alternative approach to derive (2.4) which is also useful for dealing with inhomogeneous exterior domains is inspired by [ x for and any x , we may replace the original integration over ext by the integration over the set ext . An application of the transformation rule then leads to the weak formulation.

The complex-scaled problem for inhomogeneous exterior domains
Contrary to above we also consider potential functions which are not constant in ext . In this case we state the problem under investigation in this paper as follows. Clearly, int and are independent of . Note that the assumptions int 1, 1, and to 1 for in Problem 2.1, which are necessary for the convergence analysis, can be relaxed using positive constants instead of 1. Similar to the case of a homogeneous exterior the radiation condition is implicitly included in Problem 2.1 in the assumption that the resonance functions of the complex-scaled problem are elements of 1 (i.e., they decay fast enough for x to be square integrable).

A brief discussion of the complex scaling radiation condition for inhomogeneous exterior
In the case where 1 in ext , justifying the use of the complex-scaled weak formulation and therefore the complex scaling radiation condition is more delicate than as it is sketched in Section 2.2 for the homogeneous case. In the following, we discuss the difficulties occurring in the described specific configurations.
In (2.5) three types of integrals over appear: The integral on the left-hand side splits into a standard 2 scalar product and the sesquilinear form induced by the negative Laplace-Beltrami operator. Additionally, we have a weighted 2 -inner product with weight function on the right-hand side. Pure radial inhomogeneity: For 1 0 and x x , i.e., 1, as in the homogeneous case, the spherical harmonics diagonalize all these three integrals. Along the lines of [16,17], and in particular using a separation into Bessel-like equations, for rational potentials as used in Section 6.2.2 for some of the numerical experiments, a solution representation by functions similar to spherical Hankel functions of the first kind is possible. This allows the definition of a radiation condition and the application of similar arguments as in the homogeneous case. However, one has to take into account that the holomorphic extension of such rational might have singularities. Thus, the application of Cauchy's integral theorem in (2.5) is limited to domains given by scaling parameters with sufficiently small argument. Surface inhomogeneity: If the potential is not constant with respect to x , in general the eigenfunctions of the Laplace-Beltrami operator on are not orthogonal with respect to the weighted 2 scalar product with the weight function . Hence, they cannot diagonalize all three terms. Therefore, to the best of our knowledge, even the definition of a radiation condition is not perfectly clear for such problems. Nevertheless, under reasonable conditions the convergence theory for Problem 2.1 holds true for surface inhomogeneities as well. Moreover using Cauchy's integral theorem as above one can argue that for suitable scaling parameters , the complex scaling radiation condition is in fact independent of .

The discrete problem
In this section we exploit the inherent structure of the exterior domain to derive a simple way of discretizing it without having to mesh it explicitly.

Abstract form of the discrete problem
Our goal is to discretize Problem 2.1. To this end we pick and a family of functions 0 1 and define the discrete space by span 1 .
Defining the mass and stiffness matrices by respectively, we may formulate the discrete problem as follows. The discrete Problem 3.1 can be solved using standard eigenvalue solvers (see, e.g., [27]). In the following our task will be to find a suitable basis . In the subsections up to Section 4.2 we focus on the exterior problem. The full discrete space will be defined in Section 4.3.

Exterior variational formulation
For the remainder of this section we assume that there exists a finite set of diffeomorphisms covering and let 2 be one of these mappings. For the case x 3 x , it is well-known that such a set exists.

Lemma 3.2
We can calculate the Jacobian of the coordinate transformation 0 ext 1 its inverse, and its determinant by is the pseudo inverse of a matrix 3 2 with full rank.
Proof The Jacobian can be obtained by straightforward differentiation. Its inverse can be easily verified using the facts that 0, † , and By taking the square root we obtain the desired result. It can be shown that the surface gradient defined above is independent of the specific embedding . After applying the transformation rule we immediately obtain the formula for ext . For the formula for ext we calculate Plugging this into the integral and applying the transformation rule leads to the desired result. To discretize the exterior problem, we use a tensor product space of the form

Tensor product discretization of the exterior problem
To obtain the entries of the mass and stiffness matrices defined in (3.1) and (3.2), we need to evaluate the exterior sesquilinear forms for all pairs of basis functions. Since our basis functions are composed of a radial and a tangential part, we can decompose the sesquilinear forms accordingly and obtain for where is a finite element space. Differing from this approach, we will choose basis functions with infinite support to omit truncation and ensure faster convergence. Our requirements to the basis functions and the discrete space are: (R 1) The basis functions can be evaluated in a numerically stable manner, (R 2) the radial part of the solution can be well approximated by functions from , (R 3) it is easy to couple the interior to the exterior problem, (R 4) the integrals 0 and 0 can be computed or approximated numerically for suitable coefficients , (R 5) the discretization matrices are sparse, and (R 6) the condition numbers of the discretization matrices behave well for large values of .

Infinite elements based on complex scaling
Complex scaling leads to solutions with anisotropic behavior. In the interior domain, as well as in the tangential direction of the exterior domain, the oscillating behavior of the function dominates. In radial direction of the exterior domain the exponential decay is crucial. Therefore, in order to reduce computational costs it is natural to choose suitable basis functions for the different parts of the solution.
Remark 4.1 Although we confined ourselves to an exterior domain which is (part of) the complement of a sphere, our method can be also applied to more general exterior domains ext , if a parametrization of ext with respect to a surface and a radial coordinate and a decomposition of the sesquilinear forms into surface and radial parts similar to (3.3) exists (for details see [32,Section 7.1]).
In the case of cartesian scalings (see [22]) such a decomposition is not possible due to the existence of edge and corner regions. However, using tensor products of complex-scaled infinite elements in these edge and corner regions (as in [25, Section 2.3.1] for Hardy space infinite elements) makes our method applicable to cartesian scalings as well.

Interior and interface discretization
For discretizing the interior problem, basically any discrete space int span 0 1 int such that int int 1 can be used. The trace space of this interior discrete space is then used for the interface discretization (cf. Section 3), i.e., int span 0 1 .
In our examples we will choose int as a standard high-order conforming finite element space. Since in this case all of the basis functions corresponding to inner nodes in int are zero on the interface , we expect the dimension of to be much smaller than the dimension of int in usual configurations.

Radial discretization
For the radial discretization we use the space of generalized Laguerre functions. These functions are used as basis functions of spectral methods for equations on unbounded domains with exponentially decreasing solutions (cf. [30,Section 7.4]). We will see in the following that they are a suitable choice considering our requirements (R 1)-(R 6). Following [30, Section 7.1], we define the generalized Laguerre polynomials and functions as follows. (4.1) We shorten the notation by writing 0 and 1 . Moreover, we define the radial discrete space by span 0 .  [32,Section 6.4]). Hence, up to some minor modifications the discretization matrices of the complex-scaled infinite elements are the same as those of the Hardy space infinite element method in [14]. Note however that the functional framework of the complex-scaled infinite elements is more natural than the one of Hardy space infinite elements. The convergence analysis for Hardy space infinite elements in [9,14] is very challenging and for resonance problems not complete. In the context of complex scalings on the other hand we are able to use standard approximation results of generalized Laguerre functions to derive a complete convergence analysis.
We proceed to study whether the basis functions defined in Definition 4.2 satisfy our requirements (R 1)-(R 6). To this end we state a few properties of the generalized Laguerre functions. Proof All of the statements are easily checked by the reader and/or can be found, e.g., in [1,Chapter 22] Item (v) of Lemma 4.4 shows that only the first radial basis function has to be coupled to an interior basis function, i.e., (R 3) is fulfilled. Moreover, items (i) and (ii) tell us that the resulting matrices will be sparse (cf. (R 5)) as long as the radial coefficients are polynomials with low degree.
Since the functions form an orthogonal basis with respect to the 2 0 -inner product, and our basis functions are linear combinations of them (cf. 4.4(iii)) it is plausible that our discretization matrices are well-conditioned. We refer to [32, Section 9.1] for numerical experiments underlining this fact.

Coupling the interior and the exterior problem
Since we want to create a conforming discrete space for the whole problem, we need to couple our interior and exterior discrete spaces in a manner such that the resulting space is equivalent to a subspace of 1 . We achieve this by using To obtain a basis of we have to couple the basis functions such that the resulting functions are continuous. This can be done by identifying an interior basis function with non-vanishing trace on with the exterior basis function 0 . Note that, due to the tensor product structure of the exterior space, the parts of S and M that correspond to the exterior domain can be assembled by computing the radial and interface part separately and tensorizing them appropriately.

Stable evaluation and numerical integration
The generalized Laguerre functions can be evaluated numerically by using the stable three-term recursion given in Lemma 4.4(vi). We use Gauss rules for 0 with weighting function exp to obtain exact quadrature rules for the Laguerre functions (see [30,Chapter 7.1.2]). This enables us to also deal with inhomogeneous potentials in the exterior domain which is not possible in a straightforward way using classical Hardy space infinite elements.

Analysis
The analysis of our method is based on the results of Karma [18,19]. This work deals with (linear) Fredholm operators, which depend holomorphically on a complex parameter. The parameters for which the Fredholm operator is not uniquely invertible are the sought eigenvalues. For a certain class of approximations of these operators spectral convergence of the approximated eigenvalues is shown. In [10,Theorem 3.17] or [11] it is shown that these results are applicable for compact perturbations of coercive bilinear forms and their Galerkin discretizations.
Note that the standard eigenvalue convergence theory [2] of the Laplace operator cannot be used for the complex-scaled resonance problem (2.6) since for unbounded domains the embedding 1 2 is not compact. Hence, the solution operator to (2.6) is not compact, which would be the starting point of the theory in [2]. Therefore, rigorous convergence studies for complex-scaled resonance problems are very rare. In [21,22] the error induced by the truncation to a perfectly matched layer is studied for specific scattering profiles. Subsequently, on the bounded, truncated domain standard arguments are used. [15] introduces the interpretation of a perfectly matched layer as a projection method for a waveguide structure. Based on this some basic convergence results are obtained. Finally, [10] or [12] give a full convergence analysis for a class of perfectly matched layer methods based on the results of [18,19].
In this section we show that the complex-scaled infinite elements fit into this framework as well. Moreover, we derive convergence rates. Note that parts of the analysis are already contained in [32] although the analysis therein is not complete. However, in [32, Section 6.5.2] the approximation of spherical Hankel functions by Laguerre functions is studied. These studies can be helpful for choosing suitable scaling parameters.

Weak coercivity
We define the weak coercivity of an operator as follows.
Definition 5.1 For a Hilbert space , let (i.e., is a bounded, linear operator that maps to ). Then we call weakly coercive if the induced bilinear form is weakly coercive, i.e., there exists a compact operator and a constant 0 such that 2 for all .
Next we show that the sesquilinear form induced by the eigenvalue problem in question is bounded for every fixed frequency .  Next, we show that is coercive. Since we assumed that arg 0 2 , 0, and 0, we obtain arg 0 2 and therefore also 2 2 0. Because of this and 2 0 we can find 0 2 such that exp 2 0 and exp 2 2 0.
We use this to bound for . This gives that , and therefore also , is coercive.
Lemma 5.4 shows that we may indeed apply the theory of Karma to the complex-scaled problem to obtain spectral convergence of our method. The approximation quality of an eigenvalue in a discrete space is governed by the best-approximation errors inf 1 inf 1 where are eigenfunctions and are eigenfunctions of the adjoint problem corresponding to the eigenvalue (cf. [10,Theorem 3.17]). Therefore, to prove convergence of our method we have to show that these errors tend to zero with respect to the discretization parameters.
Since the approximation error in the interior domain int can be treated using standard approximation results for 1 -conforming finite elements we focus on the approximation error in the exterior domain ext in the following sections. (5.2b)

Approximation of eigenfunctions by infinite elements
In the following we analyze the dependence of these errors on the discretization parameters . where denotes the 2 0 -orthogonal projection onto the space .

Super-algebraic decay of coefficients
Proof [30,Theorem 7.9] with 0 and the function 2 . Note that the arguments of the basis functions in [30] are scaled by the factor 1 2 compared to the basis functions .
Proposition 5.5 shows that the interpolation error of smooth and sufficiently fast decaying functions on 0 by (generalized) Laguerre functions decays superalgebraically. More precisely, if the right-hand side of (5.3) is bounded for each , then the error on the left-hand side is bounded for each by the algebraic factor whereas depend on and but not on . To obtain superalgebraic convergence of our method we need to assume such a behavior with respect to the radial direction of the complex-scaled resonance functions. For homogeneous exterior domains, this can, e.g., be proven by using boundary integral representations (cf. [31] is continuous. Thus, we may integrate the right-and left-hand side of (5.4) over and obtain the claim.

The tangential error
The tangential error is given by Remark 5.7 For the surface discretization, high-order 1 -conforming surface finite elements can be used. For the construction of these elements we refer to [8]. Such elements lead to error bounds 0 , 1 1 , where is the element order.
The following Lemma shows that the entries of the (infinite) matrices defined above grow at most polynomially. Proof We exemplarily only prove the assertion for S , we obtain the claim.
Using the previous Lemma and the fact that the 2 -norm of the functions decays super-algebraically in we can prove the following theorem: Theorem 5.9 Let be an eigenfunction, and a surface discretization parameter such that (5.5) holds. Then the tangential error can be bounded by max 0 1 for some constant 0 depending on and independent of and .
Proof Since is smooth we may apply Lemma 5.6 to obtain that the quantities for a constant independent of .
The same arguments can be repeated for the term containing S 1 2 . Using (5.5) we obtain the claim.

The radial error
The radial error is given by x x x.
Using the same notation as above we obtain Using again the decay of the coefficients we can prove the following theorem.

Summary of the convergence results
In this subsection we sum up the main convergence results. First, we collect the assumptions made in the previous sections: We assume a linear complex scaling with parameter 0 and an interface with 0. Moreover, we assume that is an eigenvalue of Problem 2.1 such that 0, is a sequence of discretization parameters with lim 0 , (C 3) is an eigenfunction corresponding to and is an eigenfunction of the adjoint problem corresponding to such that (C 3a) and all their derivatives with respect to the radial variable decay super-algebraically, uniformly in the surface variable, (C 3b) the surface discretizations are chosen such that (5.5) holds, and (C 4) , 0, and 0.
Remark 5.11 (C 1) is mainly needed to ensure that the complex-scaled resonance functions decay exponentially in radial direction and to ensure that the resonances are independent of (see for homogeneous exterior domains, e.g., [32]). (C 2) implies, together with the other assumptions, pointwise 1 -convergence of the projection operators onto the discrete spaces. The assumption (C 3) is needed to derive convergence rates. In particular, (C 3a) is used in Prop. 5.5 for the radial error and (C 3b) for the surface error. Finally, (C 4) is used to prove weak coercivity in Lem. 5.4.
The following Theorem 5.12 is basically a restatement of [10, Thm. 3.17.(i-iv)] for our specific case. It ensures that for each resonance of the continuous problem there exist sequences of discrete eigenvalues converging with the correct multiplicity towards this resonance. Moreover, there is no fail convergence, i.e., if a sequence of discrete eigenvalues converges, then the limit is a resonance of the continuous problem.

Numerical experiments
In the following we illustrate our theoretical findings from the previous sections by numerical examples. All numerical examples were computed using the high-order finite element software NGSolve [29] and the mesh generator Netgen [28].

Comparison to perfectly matched layers
In this subsection we compare the computational costs of our infinite elements and a conventional PML by approximating the resonances of Problem 2.1 on All computations in this section were done on a desktop computer with an Intel i3 CPU with 2x3.5GHz and 16GiB memory. The eigenvalues were calculated using a shift-and-invert Arnoldi algorithm (cf. [27]) and a direct inverse via a Cholesky factorization for complex symmetric matrices. All given times are for the factorization of the given system matrix only, since this is the main contribution to the overall computational costs. The largest problems in the examples had 99425 degrees of freedom (24 in radial direction). Factorizing the inverse took up about 7GiB of memory. Figure 2 shows the error plotted against factorization times for infinite elements and a PML using the same tensor product method described in Section 3 but with one-dimensional high-order finite element basis functions in radial direction on an interval 0 . We applied uniform -refinement to obtain a succession of discretizations. In Fig. 2a and c the error generated by the truncation of the exterior domain can be observed at approximately 10 3 . In Fig. 2c and d the infinite elements already reach the error generated by the surface discretization which is approximately 10 7 . All experiments show that the infinite elements are clearly superior to the used PML discretizations with respect to computational efficiency.
Note that due to the fact that we used the tensor product ansatz also for the PML discretizations, this version of PML is already more efficient than a typical PML based on an unstructured exterior mesh. On the other hand, there are many ways to optimize the PML even more, e.g., using non-uniform meshes on 0 or nonlinear complex scalings. We are not claiming the complex-scaled infinite elements to be superior to all possible versions of perfectly matched layers. Nevertheless, for complex-scaled infinite elements only the scaling parameter and the number of degrees of freedom in radial direction have to be chosen. Typically, for an efficient PML more parameters have to be optimized.

Application of infinite elements to problems with inhomogeneous exterior domains
As described in Section 2.4 the theoretical justification of complex scaling and the induced radiation condition for problems with an inhomogeneous exterior is more and different exterior discretizations. We use PMLs with truncation at and radial elements of order delicate than in the homogeneous case. Therefore, in the following examples we start from well-justified reference configurations. We monitor the changes in the numerical results when using continuous perturbations of this configuration. In this way we make sure to obtain reasonable results.

An example with a surface inhomogeneity
We use an example with 3 and a potential function , which is piecewise constant. It takes three different values in one bounded (a sphere) and two unbounded (part of a cone and the remainder of 3 ) subdomains (cf. Fig. 3). In the cases 0 2 the exterior domain is homogeneous. Moreover, in these cases the problem For all of our experiments we use fixed parameters 0 1, 1 1, 2 5, and 3 10. For the complex scaling we use 1.1 (i.e., int , , , and varying parameters . For the discretization we use 20 degrees of freedom in radial direction, a tetrahedral mesh of int with mesh size 0.3 and polynomials of degree 3. Figure 4 shows resonances for 0.6 and two different complex scaling parameters . We observe spurious resonances in the vicinity of the sets 0 . This has to be expected from the results with homogeneous exterior domains in [6,21,26]. These resonances are a discretization of an essential spectrum. Moreover, it can be seen that the spurious resonances with in absolute values larger imaginary part move away from this axis. Probably, this effect is generated by discretization errors of the interior domain (see, e.g., [26]). More important is the fact that the resonances with 0 are independent of and in the vicinity of the reference resonances for the homogeneous exterior domain.
In Fig. 5 cross-sections of selected resonance functions are given. They are in the vicinity of a resonance to the homogeneous exterior domain, i.e., 0. The multiplicity of the resonance in the homogeneous case is three, since the restrictions of the resonance functions on are spherical harmonics with index one. Due to the inhomogeneous exterior domain, this resonance is split into two perturbed resonances, one with multiplicity one (Fig. 5a) and the other (Fig. 5b) with multiplicity two. This Fig. 4 Example with inhomogeneous exterior (cf. Fig. 3) with 0.6 , 1 1, 2 5, 3 10 and different parameters . The dashed lines mark the regions in the complex plane where 0 and therefore where we expect the essential spectra. Note that the reference resonances are for the separable problem with 0 and give merely an indication where we expect correct resonances is due to the fact that the rotational symmetry of the domain with respect to the -axis is preserved (cf. Fig. 3). In Figs. 6 and 7 we studied the transformation of resonances from the reference configuration 0 to the other reference configuration 2 . The resonances in Figs. 6b and 7 correspond to spherical harmonics of index one.

Radial and surface inhomogeneities
In this subsection we approximate the resonances of Problem 2.1 on  examples we used 200. In this example we refrain from a presentation of results with different scaling constants . We merely note that the following results are in principle independent of the choice of as long as 0. Note that the poles of the complex continuation of have negative real parts. Thus any choice of scaling parameter with positive real and imaginary part leads to the correct complex scaling radiation condition, see Section 2.4. Of course, the approximation quality depends on .
In Fig. 8 we fixed 0 and use 0 1.5 . The analytically given resonances for 0 are marked by squares. Moreover, we solved the separated problems and the full three-dimensional problems. For larger values of the eigenvalues move  closer to the real axis. The approximated eigenvalues of the full three-dimensional simulation show a good agreement with the ones of the separated problem. The resonances located close to the negative imaginary axis in Fig. 8 are again part of the discretization of the essential spectrum. Figure 9 shows resonances of the same problem but with 0. This problem is not separable any more. Thus only a three-dimensional simulation is possible. Due to the disturbed symmetry, the multiple eigenvalues fan out. Figures 10 and 11 show selected resonance functions corresponding to the previously approximated resonances. To visualize the resonance functions, int 1.8 0 1 0 was chosen here. In Fig. 10 is zero, i.e., we have a rotationally invariant problem with radial inhomogeneity. In Fig. 11 the problem is not rotational invariant leading to resonance functions with perturbed symmetry.

Conclusion
In summary we can say that complex-scaled infinite elements are a very effective method for scalar resonance problems. They are more efficient compared to a standard PML due to their super-algebraic convergence.
Using continuous perturbations of well-established reference configurations, the numerical results indicate that complex-scaled infinite elements are well-suited for problems with inhomogeneous exterior domains. The convergence analysis covers these problems. Nevertheless, a rigorous derivation of radiation conditions in these situations, as well as the general applicability of complex scaling methods to these problems, remains an open question.
Funding Open access funding provided by TU Wien (TUW). This study was funded by the Austrian Science Fund (FWF): P26252.

Conflict of interest The authors declare no competing interests.
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