Numerical Multiscale Methods for Waves in High-Contrast Media

Waves are around us in all parts of everyday live — be it water waves when throwing a stone into the lake, be it sound waves when we speak, be it electromagnetic waves in our mobile phones, microwave ovens or even simply the (sun)light. All these phenomena are described by various types of wave equations, whose solutions share a common property: they propagate and oscillate in time and space. The materials through which the waves propagate have specific material properties that influence the wave speed and that enter into the wave equations as coefficients.

Humans are ever trying to manipulate and control the propagation by building specific materials. One large class, on which we focus in this article, are so-called metamaterials. They are artificially constructed assemblies of small components. The exciting feature is that the metamaterial may behave effectively in a way unattainable for its (ordinary) material building blocks. One striking example is a negative refractive index: While common natural materials have a positive index, metamaterials have been constructed with an (effective) negative refractive index. This has lead to a plethora of astonishing applications such as perfect lenses [1] or optical cloaking [2]; see [3] for further examples.

From the mathematical perspective, we therefore have to study wave propagation in multiscale media, i.e., media with the above described small components of different materials. Assuming the length scale of the material bulk to be of order one, the small components are characterized by another length scale \(\varepsilon \ll 1\). The effective behavior of the metamaterial is described by “zooming out” of the fine structures and this process is called homogenization. It can, for instance, be achieved analytically by considering the limit \(\varepsilon \to 0\) in the models.

Let us finally motivate the advent of a high contrast in the material properties. From the application side, in the metamaterials produced for, e.g., a negative refractive index, the electric permittivities of the two components are of very different magnitude. This implies that waves propagate at very different speeds in those materials. Although the structures are smaller than the wavelength, the different wave speeds can nevertheless lead to (sub-wavelength) resonant effects. Roughly speaking, the wave amplitude changes dramatically inside the small structures, leading to global effects such as exponential damping. As an illustrative example, Fig. 1 shows the solution of Helmholtz equations in one dimension with constant, periodic moderate-contrast, and periodic high-contrast coefficient. We observe a significant influence of the contrast on the wave pattern, in particular, the high contrast (right) leads to an exponential damping in this example. Further details on the model are given in Sect. 2 below. In the field of homogenization, coefficients taking two values 1 and \(\varepsilon ^{2}\) on the fine scales lead to very different limit equations than if the two values are both of the order one. Note that the scaling \(\epsilon ^{2}\) is decisive in the sense that other scalings like \(\varepsilon ^{\alpha}\) for \(\alpha \neq 2\) do not lead to unusual limit equations. This coupling between the (relative) length scale of the fine structures and the contrast of the material properties is at the heart of this article. We will highlight in different examples that it leads to astonishing mathematical as well as application-immanent phenomena. We refer to [4, 5] and references therein for further examples of interesting wave phenomena in high-contrast media.

Numerical simulations play an important role to explore such phenomena because of the time and costs required for real experiments. The key challenge is the oscillatory behavior of solutions, which has three reasons. First, waves themselves are oscillatory as mentioned above and the higher the frequency, the more oscillations we expect. Second, the fine material structures or the multiscale nature of the problem, respectively, introduce additional oscillations, which we expect to happen on the scale \(\varepsilon \). Third, depending on the interplay or coupling of the frequency and \(\varepsilon \), the high contrast can lead to the above mentioned resonance effects, which typically express themselves in very different amplitudes of the solution over the entire domain. See Fig. 1 for illustrations.

All these effects, summarized as oscillations, require a suitable numerical resolution: we need a sufficiently large number of data points – called degrees of freedom – to faithfully represent the solution. This is illustrated for the example of the periodic, medium-contrast Helmholtz problem in Fig. 2. Only if the mesh size fully resolves the jumps of the coefficients, i.e., if \(H<\varepsilon \), the finite element solution gives reasonable approximations. Note that also an appropriate resolution of the wavelength by the mesh size is required, which in this example is already fulfilled through \(H<\varepsilon \). We emphasize that such resolutions, especially of the finescale structures, often require an impractical computational effort. However, in many cases it is sufficient to have a faithful representation of the effective or macroscopic behavior of the solution, which is expected to be less oscillatory. For this effective behavior, we can hope to achieve good approximations with far less degrees of freedom — roughly speaking, this is the goal of numerical homogenization or numerical multiscale methods. We emphasize that the effective behavior can not be simulated and represented by just using a standard discretization scheme with a small number of degrees of freedom. In fact, the numerical methods have to take the fine material structures into account and have to “average them out” in a suitable manner.

The above described general recipe has been turned into several successful methods. Discussing and classifying the huge list of computational multiscale methods is beyond the scope of this article, but we refer, for instance, to [6, 7] for surveys. The additional challenges of wave propagation (at high-frequency) and of the high-contrast, however, need particular attention. Recalling that, for instance, a high contrast can lead to astonishing, non-standard limit equations, we expect that the numerical methods also have to be tuned and adapted. The article, therefore, aims to highlight some of the developed strategies at the example of two classes of multiscale methods, namely the Heterogeneous Multiscale Method and the Localized Orthogonal Decomposition. In the end, this allows to perform simulations of some interesting phenomena.

The article is structured as follows: In Sect. 2, we describe wave propagation in high-contrast media through a mathematical model serving as example. We discuss the non-standard limit equations occurring if we let \(\varepsilon \to 0\), where \(\varepsilon \) characterizes the fine scale as well as the contrast, see above. Section 3 is devoted to the numerical treatment of the model problem. We introduce the above mentioned two classes of numerical multiscale methods and discuss how they cope with the challenges of high-contrast wave propagation. In Sect. 4, simulations showcase the astonishing effects of wave propagation through high-contrast media. We close with a brief, most possible non-exhaustive, outlook to further topics in wave propagation in high-contrast media in Sect. 5.

Throughout the article, we assume the reader to be familiar with standard Lebesgue and Sobolev spaces. In addition, Sect. 3 assumes some basic knowledge about finite element methods. Moreover, Sect. 3.2 contains some more technical paragraphs, in particular concerning error estimates for certain multiscale methods. Readers more interested in wave propagation phenomena may skip these parts and directly proceed to the simulation results in Sect. 4.

As an example model for this article, we consider the following Helmholtz problem where \(\Omega \subset \mathbb{R}^{d}\), \(d\geq 2\), is a given bounded Lipschitz domain, \(\omega >0\) is the fixed frequency, \(a_{\varepsilon}\) models the material (see below), and \(u_{\varepsilon}\) is the sought solution. Additionally, suitable boundary conditions have to be posed, which we however do not specify further in this overview. We only stress that these boundary conditions should be independent of \(\varepsilon \). Many of the results reviewed have been obtained with Robin boundary conditions, but the application of perfectly matched layers should yield similar results. We understand (1) in a weak or variational sense and denote by the associated sesquilinear form.

The Helmholtz model problem can describe some of the waves from the introduction in certain simplified settings. For instance, it can be derived from the classical wave equation \(\partial _{tt}u_{\varepsilon}-\nabla \cdot (a_{\varepsilon}\nabla u_{ \epsilon})=0\) under the time-harmonic ansatz \(u_{\epsilon}(x,t)=u_{\varepsilon}(x)e^{-i\omega t}\), which is valid under the assumption that the wave oscillates with a given (imposed) frequency \(\omega \). The classical wave equation, in turn, is a prototypical model for the propagation of sound waves, where \(u_{\varepsilon}\) corresponds to the pressure and \(a_{\varepsilon}\) to the inverse speed of sound. The Helmholtz equation also occurs as special case of the time-harmonic Maxwell equations describing the propagation of electromagnetic waves under a superimposed frequency. If the full three-dimensional geometry is in fact invariant in one coordinate direction, one can reduce the Maxwell system to the Helmholtz problem by assuming a transversal mode for the electric or the magnetic field. Transversal mode means that from the general vector-valued field only one component is non-zero. Consequently, \(u_{\varepsilon}\) corresponds to this component of the electric or magnetic field – depending on which of the two fields is transversal — and \(a_{\varepsilon}\) describes the (inverse of) the magnetic permeability or the electric permittivity.

In the very general setting, our material coefficient \(a_{\varepsilon}\) takes two different values where \(0<\varepsilon \ll 1\) is a small parameter and \(\Omega _{1}\) and \(\Omega _{\varepsilon}\) are disjoint subsets, i.e., \(\Omega _{\varepsilon}\cap \Omega _{1}=\emptyset \), that form a partition of \(\Omega \), i.e., \(\overline{\Omega}=\overline{\Omega}_{1}\cup \overline{\Omega}_{ \varepsilon}\). Note that \(\Omega _{1}\) and \(\Omega _{\varepsilon}\) do not need to be connected. We denote (3) as a high-contrast medium in contrast to a standard or low-contrast medium, where the value in \(\Omega _{\varepsilon}\) would be of order 1. Note that (3) implies that \(a_{\varepsilon}\) is almost degenerate for small \(\varepsilon \). In the subdomain \(\Omega _{\varepsilon}\), waves will oscillate with a frequency \(\omega \varepsilon ^{-1}\) much faster than in \(\Omega _{1}\).

In the general setting, we do not specify the geometry of \(\Omega _{\varepsilon}\), in particular it could fill the whole domain. However, the type of setting we have in mind is best described in the periodic setting. Let \(Y:= [0,1]^{d}\) be the \(d\)-dimensional unit cell and \(\Sigma \subset Y\) a bounded, connected Lipschitz subdomain which is compactly embedded in \(Y\) in the sense that \(\overline{\Sigma}\) does not touch the boundary of \(Y\). \(\Omega _{\varepsilon}\) is now formed out of \(\varepsilon \)-scaled and shifted copies of \(\Sigma \) see Fig. 3 (left) for an illustration. \(\Omega _{1}\) is the complement of \(\Omega _{\varepsilon}\) and, thus, connected. Note that the volume of \(\Omega _{\varepsilon}\) is of order one independent of \(\varepsilon \). In particular, \(\Omega _{\varepsilon}\) does not “vanish” in the limit \(\varepsilon \to 0\). The general setting should be imagined as some generalization of this periodic setting: \(\Omega _{\varepsilon}\) still (roughly) consists of \(O(\varepsilon ^{d})\) copies of diameter \(O(\varepsilon )\), but not all small inclusions have to be of the same shape, be arranged in a periodic manner, or be separated. We again refer to Fig. 3 (right) for an illustrative example.

Throughout the article, we assume that our model problem (1)–(3) is well-posed in the sense that a solution \(u_{\varepsilon}\) exists in the weak/variational sense, is unique and its energy norm is bounded by the data. Note that the so-called stability constant in this bound in general depends on \(\omega \) and \(\varepsilon \).

As mentioned in the introduction, homogenization is concerned with the questions to what function \(u_{\varepsilon}\) converges in the limit \(\varepsilon \to 0\) and whether this limit function is the solution to, again, a Helmholtz-type equation. We emphasize that homogenization always considers a fixed frequency \(\omega \). As we pass to the limit \(\varepsilon \to 0\), we are eventually in the regime \(\varepsilon \ll \omega ^{-1}\), which we denote as the homogenization regime. The typical size of the connected components in \(\Omega _{\varepsilon}\) is thus smaller than the wavelength. In the following, we restrict ourselves to the periodic setting, cf. (4). Consequently, we can write \(a_{\varepsilon}(x)=a(x/\varepsilon )\), where \(a\) is a \(Y\)-periodic function.

In the low-contrast setting, \(u_{\varepsilon}\) converges weakly in \(H^{1}(\Omega )\) — and hence strongly in \(L^{2}(\Omega )\) — to \(u_{0}\in H^{1}(\Omega )\), which solves the Helmholtz equation (1) with \(a_{\varepsilon}\) replaced by the effective or homogenized matrix \(a_{0}\), where Here, the functions \(w_{j}\) for \(j=1,\ldots ,d\) are periodic \(H^{1}\)-functions restricted to \(Y\) and solve up to an additive constant in the weak sense, where \(e_{i}\) are the canonical basis vectors of \(\mathbb{R}^{d}\). (7) are called cell problems and \(w_{j}\) the cell problem solutions. We stress that the formula for \(a_{0}\) is the same as in the elliptic setting [8]. Intuitively, this means that \(u_{\varepsilon}\) converges to a Helmholtz solution with effective/homogenized material properties. For instance, in the one-dimensional setting \(a_{0}\) is simply the harmonic mean of \(a_{\varepsilon}\).

In the high-contrast setting, however, the convergence of \(u_{\varepsilon}\) as well as the limit problem change, cf. [9]. Precisely, \(u_{\varepsilon}\) converges weakly in \(L^{2}(\Omega )\) to \(\mu _{0}u_{0}\), where \(u_{0}\in H^{1}(\Omega )\) solves a homogenized Helmholtz equation of the form Before addressing the two homogenized coefficients \(a_{0}\), \(\mu _{0}\), let us emphasize that the convergence of \(u_{\varepsilon}\) in fact is only weak in \(L^{2}(\Omega )\) and that the weak limit is not the homogenized solution \(u_{0}\). Typically, the domain \(\Omega \) plays the role of a penetrable scatterer, i.e., we consider the Helmholtz equation in fact on a lager domain \(B\supset \Omega \) with \(a_{\varepsilon}\equiv 1\) outside \(\Omega \). In this case, \(\mu _{0}\) and \(a_{0}\) are the identity outside \(\Omega \) and in this exterior region, \(u_{\varepsilon}\) converges strongly to \(u_{0}\) in \(C^{\infty}\). Summarizing, the high contrast drastically influences the homogenization limit. The main reason is that \(\|\sqrt{a_{\varepsilon}}\nabla u_{\varepsilon}\|_{L^{2}(\Omega )}\) is bounded, which implies that \(\nabla u_{\varepsilon}\) is bounded in \(\Omega _{1}\) and \(\varepsilon \nabla u_{\varepsilon}\) is bounded in \(\Omega _{\varepsilon}\). Therefore, the limiting behavior differs in these two subdomains.

This is also reflected in the homogenized coefficients \(a_{0}\) and \(\mu _{0}\). In detail, \(a_{0}\) is very similar to the low-contrast case (6), except that the cell problem and the integration are over the complement \(\Sigma ^{*}:=Y\setminus \overline{\Sigma}\) of \(\Sigma \). Precisely, where \(Y\)-periodic \(H^{1}\)-functions \(w_{i}\) solve up to an additive constant in the weak sense. Note that we omitted \(a\) in (9) and (10) because it is equal to 1 in \(\Sigma ^{*}\), but of course the formulas can be extended in a straightforward manner, which we do not further discuss in detail.

The new homogenized coefficient \(\mu _{0}\) is a scalar, defined as where \(w\in H^{1}_{0}(\Sigma )\) solves Note that \(\omega \) must not be an eigenvalue of the Dirichlet Laplacian on \(\Sigma \) in order for \(w\) to be well-defined. This issue is often circumvented by adding a small imaginary part to \(a_{\varepsilon}\) in \(\Omega _{\varepsilon}\) in (3). The homogenized coefficient \(\mu _{0}\) occurs in the \(L^{2}\)-term of the Helmholtz equation although no multiscale coefficient was present there in the original problem (1). Further, we highlight that \(\mu _{0}\) depends on \(\omega \). If \(\omega \) is close to an eigenvalue of the Dirichlet Laplacian on \(\Sigma \), \(\mu _{0}\) in fact may not only be very large in value, but can even change sign and become negative. An example behavior of \(\mu _{0}\) is plotted in Fig. 4 for the setting described in Sect. 4.1. \(w\) and \(\mu _{0}\) can equivalently be expressed in terms of the Laplace eigenvalues and eigenfunctions on \(\Sigma \), see [9], where this “resonant” behavior becomes clearly visible.

As numerical examples in Sect. 4.1 and [10] underline, \(u_{0}(x)+\chi _{\Omega _{\varepsilon}}\omega ^{2}u_{0}(x)w(x/ \varepsilon )\) is an even better approximation to \(u_{\varepsilon}\) than \(u_{0}\) alone. We even conjecture that \(u_{\varepsilon}\) converges strongly to \(u_{0}(x)+\chi _{\Omega _{\varepsilon}}\omega ^{2}u_{0}(x)w(x/ \varepsilon )\) in \(L^{2}(\Omega )\). This also implies the strong convergence of \(u_{\varepsilon}\) to \(u_{0}\) outside of \(\Omega _{\varepsilon}\). These strong convergences are shown in the elliptic high-contrast setting in [8]. However, the proof is not applicable in the Helmholtz case, so that the convergence remains a conjecture.

In the physical interpretation of (8), \(\mu _{0}\) is linked to the effect of artificial magnetism. In the transversal mode for the electric field, \(\mu _{0}\) represents the magnetic permeability. The homogenization result thus mathematically describes the effect that suitable composites with non-magnetic constituents can effectively behave as magnets. As seen, the effective magnetic permeability is frequency-dependent and can be negative for certain values of \(\omega \). The change of sign in \(\mu _{0}\) obviously changes the mathematical nature of the problem, as for negative \(\mu _{0}\) the PDE is not indefinite anymore. Physically, a negative \(\mu _{0}\) — or with negative real part — implies a damping of the incoming wave. In other words, for those \(\omega \) where \(\mu _{0}<0\), waves cannot propagate in the multiscale materials but will be (exponentially) damped to zero. Such frequencies lie in so-called band gaps of the material. We emphasize that this effect is independent of the angle of the incoming wave. The discussed effect of a new effective coefficient in the setting of high-contrast media is not limited to the Helmholtz equation. It already occurs for elliptic reaction-diffusion equations, even if the reaction-coefficient in the multiscale problem is constant, see [8]. Related to artificial magnetism, the above discussed effect occurs in similar form also in time-harmonic Maxwell equations with high contrast, which we discuss in Sect. 5.

Finally, we emphasize once more two important (geometric) assumptions in our model. First, the dimension \(d\) has to be at least two. The one-dimensional case is special: there is no homogenized problem to be solved, but only a cell problem of similar form as (12), cf. [11]. Second, it is crucial that the high-contrast region \(\Sigma \) in the unit cell does not touch the boundaries of \(Y\), i.e., it is an inclusion. If \(\Sigma \) is such that \(\Omega _{1}\) is no longer connected, i.e., a layered high-contrast medium in the two-dimensional setting, the homogenized coefficient \(a_{0}\) becomes degenerate. For instance, this implies the occurrence of so-called surface plasmons on the surface of the layers, see [12].

As announced, we assume the periodic setting (4) and that \(\varepsilon \ll \omega ^{-1}\). When \(\varepsilon \) is small, \(u_{\varepsilon}\) is well approximated by \(u_{0}\) or, as discussed above, by \(u_{0}(x)+\chi _{\Omega _{\varepsilon}}\omega^{2}u_{0}(x)w(x/\varepsilon )\). Therefore, we will explore the idea to use the homogenization result (8)–(12) for our numerical approach.

An approximation of \(u_{0}\) in (8) can be found by any standard finite element method. Let \(\mathcal {T}_{H}\) be an admissible, i.e., no hanging nodes, and shape regular triangulation of \(\Omega \). Denote by \(V_{H}\) the associated linear Lagrange finite element space, possibly including boundary conditions. Functions in \(V_{H}\) are continuous over \(\Omega \) and affine functions on each mesh element \(K\in \mathcal {T}_{H}\). Since the homogenized coefficients \(a_{0}\) and \(\mu _{0}\) are constant, the mesh size \(H\) of \(\mathcal {T}_{H}\) can be chosen independent of \(\varepsilon \). In particular, this allows us to use coarse meshes with \(H>\varepsilon \) and a moderate number of degrees of freedom.

However, the values of \(a_{0}\) and \(\mu _{0}\) are not directly available by the homogenization result. Instead, we have to compute approximations for them in a second step. Putting it simple, we introduce another triangulation \(\mathcal {T}_{h}\), namely of \(Y\), and the corresponding linear finite element space with periodic boundary conditions \(V_{h}\). \(\mathcal {T}_{h}\) can be chosen independent of \(\varepsilon \) as well because we are working on the unit cell where the re-scaled material variations occur on length scales of order one. Note that the mesh for \(Y\) has to resolve the interface of \(\Sigma \). Then, we can also define the linear Lagrange finite element space with Dirichlet boundary conditions on \(\Sigma \) and we denote it by \(V_{h,0}(\Sigma )\). We then solve the cell problems (10) and the additional cell problem (12) with this so-called microscopic discretization. This means we compute \(w_{h,i}\in V_{h}\) and \(w\in V_{h,0}(\Sigma )\) by solving Note that both problems lead to small linear systems as the number of degrees of freedom is moderate. We calculate approximations \(a_{0}^{h}\) and \(\mu _{0}^{h}\) via (9) and (11) with \(w_{i}\) and \(w\) replaced by \(w_{h,i}\) and \(w_{i}\), respectively.

The HMM approximation \(u_{H,h}\in V_{H}\) is then found as the finite element solution to the homogenized equation (8) with \(a_{0}\) and \(\mu _{0}\) replaced by \(a_{0}^{h}\) and \(\mu _{0}^{h}\), respectively. Precisely, we find \(u_{H,h}\in V_{H}\) via where we stress that terms from the boundary conditions may have to be added appropriately. Again, the resulting linear system is of moderate dimension.

We emphasize that the traditional HMM approach does not directly proceed in that way. Instead, one exploits that the integral in (13) is approximated with a quadrature rule on \(\mathcal {T}_{H}\). Then, one needs to calculate the term \(a_{0}^{h}\nabla u_{H,h}\cdot \nabla v_{H}\) at any quadrature point. The latter is re-formulated in such a way that it can be computed on the fly from \(a_{\varepsilon}\), \(u_{H,h}\) and \(v_{H}\) at the quadrature point. For this, the cell problems are transferred to small cells of length \(\varepsilon \) around the quadrature points of \(\mathcal {T}_{H}\). This procedure is advantageous in the locally periodic case, where \(a_{\varepsilon}\) includes an additional, non-multiscale \(x\)-dependence, and also allows to apply the methodology beyond purely periodic settings. For ease of presentation and to convey the main ideas, we do not go into further details and refer to [19].

We want to focus on three peculiarities of the high-contrast case in comparison to the standard Helmholtz or elliptic setting. All of them have their roots in special features of the homogenization result.

As announced, we now drop the assumption of periodicity and allow rather general geometries \(\Omega _{\varepsilon}\). Still, some (implicit) assumptions will be required as we indicate below. Moreover, the following methodology directly approximates \(u_{\varepsilon}\) and does not assume \(\varepsilon \) to be small. Further, no scale separation in the form of \(\varepsilon \ll \omega ^{-1}\) is needed. Our goal, hence, is to considerably enlarge the set of treatable settings. We will review the (classical) Localized Orthogonal Decomposition (LOD) method and in particular highlight aspects that come with time-harmonic wave propagation and high-contrast media. Finally, we will comment on recent developments of the methodology.

Recall that in the periodic setting, homogenization theory predicts the occurrence of an effective magnetic permeability which is frequency dependent. We further discussed in Sect. 2.2 that the behavior of the wave crucially depends on the sign of the effective magnetic permeability \(\mu _{0}\). Precisely, if \(\mu _{0}<0\), waves cannot propagate inside the medium — a so-called band gap occurs. This phenomenon is independent from the incident angle of the incoming wave.

Using the HMM from Sect. 3.1, we simulate the phenomenon in the following set-up. Our computational domain is \((0.25, 0.75)^{2}\), with the periodic high-contrast material included in \((0.375,0.625)^{2}\). In the definition of \(\Omega _{\varepsilon}\), we choose the inclusions \(\Sigma \) in the unit cell as \(\Sigma =(0.25,0.75)^{2}\). We refer to Fig. 5, left and middle, for the geometric set-up. We take \(a_{\varepsilon}=10\) outside the inclusions and \(a_{\varepsilon}=\varepsilon ^{2}(10-0.01i)\) inside. This corresponds to the high-contrast coefficient (3) scaled with a factor 10 and adding a small imaginary part inside the inclusions to prevent ill-posedness of the cell problem (12). We equip the Helmholtz equation with inhomogeneous Robin boundary conditions, where the boundary data are computed from the incoming wave \(u_{\mathrm{in}}(x)=\exp (-i\omega x\cdot \nu )\) with \(\nu =(0.6,0.8)\). This means that the wave incidents from the top right in the following figures.

From the behavior of \(\mu _{0}\) versus \(\omega \) depicted in Fig. 4, we select \(\omega =29\) and \(\omega =38\) as illustrative frequencies: For \(\omega =38\), we expect standard transmission of the wave through the medium since \(\mu _{0}\) is positive. On the other hand, \(\omega =29\) corresponds to a negative \(\mu _{0}\) and we expect a band gap here, i.e., no wave propagation. Note that because of the scaling of \(a_{\varepsilon}\), the finescale structure is sub-wavelength for both choices of \(\omega \). Figure 6 compares the approximation of the homogenized solution \(u_{0}\), approximated with the HMM, for these two frequencies. Our physical expectations are met: On the left (\(\omega =38\)), the wave propagates as usual, the bended wavefronts are caused by the different wavenumber inside the medium. In contrast, for \(\omega =29\), the wave amplitude quickly decays to zero inside the medium so that there is large region where the wave amplitude is close to zero. More precisely, the wave is exponentially damped inside the medium as we expect for a frequency in the band gap. The same behavior is also observed for plane incidence, see [19].

Including the discrete correctors in our simulations, as explained in Sect. 3.1, allows to visualize the behavior inside the inclusions. Figure 7 shows that much higher wave amplitudes are observed in the inclusions for \(\omega =29\) in comparison to \(\omega =38\) (note the different scaling of the colorbar). Hence, we observe the expected resonance effects inside the inclusions for \(\omega =29\), which cause the exponential damping. We emphasize once more that this effect is only possible because of the high contrast.

Recall that homogenization theory considers the limit \(\varepsilon \to 0\) and we kept the frequency fixed in our results from Sect. 2.2. This means that, eventually, the high-contrast inclusions from the previous experiment will be sub-wavelength in the sense that \(\varepsilon \ll \omega ^{-1}\). Physically, we also expect resonance effects if the wavelength and the fine length scale are comparable, i.e., if \(\omega ^{-1}\sim \varepsilon \). However, these resonance effects are of a different nature. We cannot hope to simulate the macroscopic behavior in this scenario by the homogenization results from Sect. 2.2. In fact, numerical experiments for periodic waveguides in [31] underline the necessity of the sub-wavelength assumption for (classical) homogenization results.

Therefore, our simulation will rely on the classical LOD from Sect. 3.2. We consider the computational domain \((0,1)^{2}\) with the periodic high-contrast material included in the slab \((0.25,0.75)\times (0,1)\). We choose the inclusions \(\Sigma \) as in the previous example, i.e., \(\Sigma = (0.25, 0.75)^{2}\), see Fig. 5, middle and right, for an illustration. We take \(a_{\varepsilon}\) as defined in (3) with the rather moderate choice \(\epsilon =2^{-3}\). We consider homogeneous Robin boundary conditions, but an inhomogeneous right-hand side for the Helmholtz equation. The latter is chosen as an approximation of a very localized source centered at (0.25,0.5) and we refer to [25] for the precise formula.

Due to the rather moderate contrast, we can still use the same interpolation operator for the LOD as in the low-contrast case. Further, we set \(H=2^{-6}\), \(m=2\) and \(h=2^{-9}\) for the discretization of the correctors. The latter clearly resolves the finescale structure of \(a_{\varepsilon}\). We compare the frequencies \(\omega =20\) and \(\omega =38\). Because of the different choice of \(a_{\varepsilon}\) in comparison to the previous section, these frequencies are no longer in the homogenization regime.

For both frequencies, we observe larger amplitudes in the inclusions. In order to focus on the overall wave behavior, we truncate the color bar to the interval \([-2,2]\) in the figures for better visibility. While we observe usual transmission for \(\omega =38\) (Fig. 8, right), the periodic high-contrast structure causes a sort of lensing or mirroring effect for \(\omega =20\) (Fig. 8, left). By lensing or mirroring effect, we mean that the wave pattern on the right — behind the scatterer — looks qualitatively similar to the original wave pattern on the left of the scatterer. In particular, the wave pattern on the right part behaves as if it originated from an (image) source located close to \((0.75, 0.5)\). Since the image source on the right seems to be located slightly inside the metamaterial, there is no perfect symmetry between the wave patterns on the left and the right of the scatterer. In other words, a point source is re-focused by the metamaterial to a point, which is commonly observed in lenses. These, however, have classically curved interfaces, whereas here lensing effects are observed with flat interfaces. To summarize, while the set-up of our numerical experiments seems very close to Sect. 4.1, the different relation between fine geometric scale and frequency leads to considerably different effects, which the LOD is able to capture.

Imagine that a periodic high-contrast metamaterial with some very favorable effective properties – for instance, the lensing effect of the previous section — has been theoretically designed. It should now be fabricated, but there is a certain low risk that some mistakes happen in this process. Consequently, the actually produced material will slightly deviate from the “optimal” theoretical design. We will call any such deviation a defect in the following. The overall question is how robust the material properties are under such defects. Since the previously simulated effects – all-angle band gaps and flat lenses – exploit resonances, it is by no ways clear that a small material defect may not destroy these properties.

For our simulations, we consider the same set-up as in Sect. 4.3, but change the coefficient \(a_{\varepsilon}\). We consider two different defects in Fig. 9 – without discussing in how far they are “small” in sense of the above philosophical discussion. For the first defect, we “erase” one of the high-contrast inclusions by setting its value to 1 instead of \(\varepsilon ^{2}\). Although this rather local defect is far away from the source location, it has some impact on the wave pattern in comparison to the periodic setting. Overall, the wave patterns still look very similar, but on close inspection one can observe that the symmetry around the central axis \(y=0.5\) is broken in the defect case. It is now the question of the concrete application whether this change is already significant or still tolerable. For instance, one might argue that the lost symmetry is an instantiation of a destroyed lensing/mirroring effect.

Not unexpectedly, the wave pattern changes more dramatically if the introduced defect is more drastic. As example, a sort of line defect is depicted on the very right, where the area \(a_{\varepsilon}=1\) around the central axis \(y=0.5\) is enlarged in comparison to the periodic setting. The wave now mainly travels through this channel, which may be seen as a wave guide. Besides the influence of defects, this examples furthermore indicates another possible application of high-contrast materials, namely as coating for wave guides.

We emphasize that our numerical studies focus in some sense on the question how robust the properties of the periodic metamaterial are with respect to defects in the periodic structure. In passing, we mention that deviations from the periodic set-up may also be introduced on purpose in order to obtain new interesting phenomena. One important example is the so-called localization of waves by defects in high-contrast media. A detailed review of the related literature is beyond the scope of this article and we refer to [4] and the references therein.

While our previous numerical examples focused on the simulation of physically interesting phenomena, we now want to study the behavior of the different LOD-type methods (cf. Sect. 3.2) for high-contrast media. The goal is not to identify a best-performing method, but to illustrate how the error depends on the different discretization parameters and to relate this to the mentioned theoretical error bounds. As challenging set-up, we choose a setting very similar to Sect. 4.2: We consider the computational domain \((0,1)^{2}\) with the periodic high-contrast material included in the slab \((0.25,0.75)\times (0,1)\). We choose the inclusions \(\Sigma \) as throughout this section, i.e., \(\Sigma = (0.25, 0.75)^{2}\), see again Fig. 5, middle and right, for an illustration. We take \(a_{\varepsilon}\) as defined in (3) with the rather moderate choice \(\epsilon =2^{-3}\). We consider homogeneous Robin boundary conditions, but an inhomogeneous right-hand side for the Helmholtz equation. The latter is chosen as an approximation of a very localized source centered at (0,0.5) and we refer to [25] for the precise formula. As underlying fine mesh, where we also computed the reference solution, we use \(h=2^{-9}\). We fix the frequency \(\omega =9\) in our studies, which is related to resonant effects and therefore considered to be a challenging example. In the following, all errors are measured in the energy norm \(\|\cdot \|_{1,a,\omega}\) defined in (5).

Since the super-localized versions of the LOD use piecewise constant approximations of the right-hand side, we slightly change the classical LOD method, which in Sect. 3.2 operates on piece-wise linear polynomials, for a fairer comparison. We instead use a classical LOD using \(L^{2}\)-projections onto piece-wise constants as interpolation operator and we refer to [32, 33] for the technical details. Figure 10 shows the convergence of the relative LOD error depending on the coarse mesh size \(H\) and the patch size \(\ell \). Due to the higher regularity of the right-hand side, we asymptotically expect a quadratic convergence in \(H\). This is illustrated in Fig. 10 if the resolution condition between \(H\), \(\omega \) and \(\varepsilon \) is met and if the patch size \(\ell \) is chosen large enough. Particularly, we need \(\ell \in \{5,6\}\) here to obtain small errors at finer meshes. We emphasize that we did not adapt the interpolation operator to the high-contrast setting as advocated in Sect. 3.2. This might reduce the required patch size, see [25] for related experiments.

For the SLOD, the same asymptotic convergence rate is observed in Fig. 11 (left). Similar as for the classical LOD, the coarse mesh size \(H\) has to meet a resolution condition, we refer to [34] for an analysis in the low-contrast case. Due to the super-exponential decay, smaller patches are sufficient to achieve the asymptotic convergence rate in contrast to the classical LOD. Consequently, the SLOD improves the computational efficiency. Overall, Fig. 11 (left) underlines the good performance of the SLOD also for the high-contrast Helmholtz equation, which was already briefly addressed – without a detailed study of the convergence rates – in a numerical experiment in [34].

For the SLgFEM, an additional discretization parameter needs to be chosen in practice, namely the number of local eigenfunctions \(n\). Figure 11 (right) illustrates the convergences for the SLgFEM. As for the SLOD, we observe a rather small patch size \(\ell \in \{2,3\}\) to be sufficient for convergence. However, choosing a sufficiently large number of local eigenfunctions \(n\) is crucial as well. In particular, for \(\ell \in \{2,3\}\), \(n=10\) leads to higher errors in the SLgFEM than in the SLOD. Moreover, for \(n=10\), the errors of SLgFEM tend to stagnate with decreasing mesh size. In contrast, with more local eigenfunctions \(n=20\), the SLgFEM errors converge and are even smaller than the errors for the SLOD. In conclusion, the super-exponential decay makes the SLOD and SLgFEM very attractive from the viewpoint of computational efficiency also in the high-contrast case. To obtain good accuracy, a sufficient number of local eigenfunctions is crucial in the SLgFEM. Possibly, with suitable adaptions to the high contrast as mentioned at the end of Sect. 3.2, the super-localized methods could perform even better, which is a topic for future research.