A projection penalization approach for the high-order DG-FEM in the time domain



[1] We investigate the Discontinuous Galerkin-Finite Element Method (DG-FEM) for the solution of Maxwell's equations in the time domain. The charge conservation laws are known to be violated by this method which consequently gives rise to spurious numerical solutions. It is, however, possible to introduce additional constrains in the formulation which impose charge conservation either exactly or approximately. In this work, a new approach based on a topological orthogonal projector into a high-order H(curl)-conforming approximation space is proposed. Using this approach we derive a constrained DG-FEM formulation which is strictly free of spurious modes. Furthermore, we construct a time domain penalization scheme which allows to separate the spurious modes related to unphysical charges from the physical solutions while maintaining accuracy and energy conservation.

1. Introduction

[2] Various Galerkin-type methods using high-order conforming approximation spaces for the solution of Maxwell's equations in the time domain have been proposed [cf. Webb, 1999; Rieben et al., 2005; Jin and Riley, 2008]. These methods provide more flexibility in the use of unstructured meshes as well as a higher numerical accuracy compared to, e.g., the conventional FDTD method [Yee, 1966]. Since tangentially (or normally) continuous approximation spaces are employed, these methods automatically fulfill discrete charge conservation and thus are free of spurious solutions [Hipmair, 2002; Rieben et al., 2005]. We will refer to them as CG-FEM standing for Continuous Galerkin-FEM as opposed to DG-FEM to be discussed later. An important disadvantage of CG-FEM is that for time domain simulations a system of linear equations involving the mass matrix of the formulation needs to be solved at each time step. This procedure is typically associated with a large computational effort. Indeed, for many practical time domain problems it is more appropriate to apply the older (but numerically extremely efficient) FDTD method than the accurate (but slow) CG-FEM. In order to deal with this issue, mass lumping techniques have been proposed [Cohen et al., 2001; Fisher et al., 2005]. These techniques work well for low-order approximations and for simple (hexahedral element) meshes. However, the application of mass lumping techniques is not obvious at all in the general case of high-order CG-FEM.

[3] DG-FEM appeared as an alternative to CG-FEM for the solution of Maxwell's equations in the time domain with high-order accuracy [see, e.g., Hesthaven and Warburton, 2002; Canouet et al., 2005]. This formulation leads to block-diagonal mass matrices so that the time stepping procedure can be efficiently performed. A major issue related with DG-FEM, however, is the unphysical charge distribution induced by discretization which gives rise to spurious numerical solutions. In the work by Gjonaj et al. [2007] it was shown on topological grounds that, except for tensor product formulations on Cartesian grids, any other DG-FEM will violate charge conservation independently from the type of the approximation spaces used. The appearance of unphysical solutions related to the violation of charge conservation has consequences on both the numerical accuracy and on the stability of time domain simulations. In particular, for particle-in-cell (PIC) simulations involving moving charges, it has been early recognized that only a discrete electromagnetic field solution which is strictly charge conserving is guaranteed to be numerically stable [Villasenor and Buneman, 1992; Mardahl and Verboncoeur, 1997]. An interesting study in the context of ADI-FDTD methods showing how numerical instability arises in PIC simulations when charge conservation is violated is given, e.g., in the work by Smithe et al. [2009].

[4] A very thorough investigation on the spurious modes appearing in the case of nodal DG-FEM using central as well as upwind fluxes can be found in the work by Hesthaven and Warburton [2004]. In this work, the authors introduce also an effective stabilization technique which allows to separate those spurious modes from the physical ones in eigenvalue computations. The present paper is very much inspired by this work following the same path in an attempt to reduce or possibly eliminate the influence of spurious modes in DG-FEM simulations. However, the approach proposed in this paper is a distinct one. First, we introduce a specific class of DG-FEM where the approximation spaces are chosen such that the definition of the basis functions within every element of the mesh is identical with the one used in CG-FEM. Then, having the charge conservation issue in mind, we establish a direct relationship between the CG-FEM and DG-FEM in form of an orthogonal subspace projection. This relationship allows us to derive a constrained DG-FEM formulation which is free of spurious solutions in the full dynamic range. This formulation is, to our knowledge, the only known in the context of DG-FEM fulfilling the charge conservation property in the strict sense. The idea of the orthogonal subspace projection is, furthermore, used to construct a time domain scheme based on the standard Lagrange multiplier penalty method which imposes the charge conservation constraint in an approximate manner. The latter scheme does not fully eliminate spurious solutions. Rather, is allows to systematically clean up the DG-FEM spectrum at low frequencies from those mode solutions.

2. Basic Theory

[5] Maxwell's equations in the time domain read

equation image
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where E and H are the electric and magnetic field intensities, respectively, and D and B denote the corresponding flux densities. The charge and current densities are given by ρ and J, respectively. In the following, heterogeneous but linear, isotropic, nondispersive and nonconducting media will be considered. Under these conditions, the constitutive material laws are written as

equation image

where ε and μ are the dielectric permittivity and magnetic permeability of the medium, respectively. Furthermore, we will restrict the discussion on solutions of (1) and (2) in closed domains Ω; that is, the region of interest is completely enclosed by a perfectly conducting boundary. Then, charge and current densities fulfill the continuity equation,

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[6] Equation (4) states charge conservation. It implies that in absence of currents and if the initial electric flux density is divergence-free, it should remain so at all later times. The purpose is to establish a numerical field approximation for (1) and (2) fulfilling at the same time a discrete equivalent of (4).

2.1. CG-FEM

[7] In order to introduce the formulation we assume that the domain Ω is discretized by a finite element mesh into a number of nonoverlapping cells (elements) Ωi; each of being them bounded by ∂Ωi. The electric field intensity E is approximated by a linear combination of vector basis functions. The appropriate approximation space to be used for the CG-FEM is the tangentially continuous H(curl) space defined as, H(curl) = {ϕ ∈ L2:curlϕ ∈ L2}. Among the many possible choices available, a set of high-order and hierarchical H(curl) basis functions for different types of polygonal mesh elements is given, e.g., by Schöberl and Zaglmayr [2005].

[8] Specifying the approximation space is completely sufficient to derive the classical CG-FEM for the second-order wave equation as previously described by many authors [cf. Jin and Riley, 2008, and references therein]. The matrix equation resulting from the standard Galerkin procedure reads

equation image

where eFE and jFE are the numerical degrees of freedom of the CG-FEM formulation associated with the electric field intensity and current density, respectively. The matrices appearing in (5) are given elementwise by the following expressions:

equation image
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The curl-curl operator, Kμ−1FE, can be further decomposed as

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where CFE is a (topological) curl operator and Mμ−1FE is the mass matrix in the H(div) vector space of the normally continuous fields; H(div) = {ψL2:divψL2}. This decomposition becomes obvious in the mixed CG-FEM formulation for Maxwell's equations in the time domain introduced by Rieben et al. [2005], where the magnetic flux density B is explicitly discretized in a H(div)-conforming space. The latter approach is, however, computationally equivalent with (5) and will, therefore, not be further considered in the following.

[9] Equation (5) can be integrated in time by employing either an implicit Newmark scheme or an explicit symplectic integrator [Rieben et al., 2004]. Both procedures require the solution of at least one linear system of equations involving the mass matrix MεFE at each time step. Note that the bandwidth of MεFE increases with approximation order, so that the iterative solution of this system of equations in the high-order case becomes a numerically very costly procedure. This certainly represents the major disadvantage of CG-FEM for time domain applications. On the other hand, this formulation has been shown to be free of spurious solutions [cf. Rieben et al., 2005]; that is, CG-FEM is consistent with the charge conservation law (4). This property is solely due to the global tangential continuity the electric field approximation as implied by the H(curl)-conforming approximation spaces used.

2.2. DG-FEM

[10] The semidiscrete DG-FEM equations using a central flux approach [cf. Canouet et al., 2005] read

equation image

where Mε, Mμ are the mass operators, C is the curl operator and e, h and j are the numerical degrees of freedom, respectively. DG-FEM solutions (either for E or H) are sought for in an arbitrary broken approximation space; that is, the basis functions are defined separately for every cell Ωe of the mesh such that they are continuous within the cell and vanish identically outside of it. The form of the matrix equations (9) is independent from the choice of these basis functions (as long as the central flux approach is employed). For the purpose of establishing charge conservation, however, one may be tempted to introduce an approximation space following within every cell, Ωe, the same definition for the basis functions as in the CG-FEM case. Such an approach has been proposed, e.g., by Gedney et al. [2007]. Then, the mass and curl operators are given by

equation image
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respectively, where α: = {ε, μ}. The central flux at the element boundary ∂Ωe is defined as

equation image

where n is the surface normal on ∂Ωe. Note the additional index, e, in these equations indicating the cellwise definition of the basis functions. The global approximation of the field intensities remains discontinuous although within every cell of the mesh the same set of basis functions as in (5) is used. Contrary to what is stated, e.g., by Gedney et al. [2007], these approximations do not satisfy the divergence-free nature of the fields. Unphysical charges will emerge at the interfaces between neighboring cells where the field approximation becomes discontinuous. Thus, spurious numerical solutions exist which are not compatible with (4). This statement is, in fact, part of a general result obtained by Gjonaj et al. [2007] showing that any DG-FEM will violate charge conservation independently from the type of the approximation spaces used in the formulation.

[11] Equations (9) can be brought into the same form as (5) by eliminating the magnetic field intensity:

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Also (13) can be integrated in time using explicit time stepping. Due to the block-diagonal structure of Mε, however, this procedure is numerically much more efficient than for the CG-FEM, although here a larger number of degrees of freedom is involved for the same mesh and approximation order.

3. Relationship to CG-FEM

[12] A basic observation is that when the basis functions used in the CG-FEM and in the DG-FEM are cellwise identical, CG-FEM solutions belong to a particular solution subspace of the DG-FEM. Indeed, any globally continuous field represented by the discrete degrees of freedom eFE can be written as a discontinuous, DG-FEM solution by the projection,

equation image

Referring explicitly to the high order, hierarchical basis functions introduced by Schöberl and Zaglmayr [2005], it is sufficient to assign the CG-FEM degrees of freedom eFE associated with an arbitrary edge of the mesh to the incident DG-FEM edge degrees of freedom belonging to all the cells sharing this edge. Similarly, the CG-FEM degrees of freedom associated with mesh faces must be duplicated to the two sets of DG-FEM face degrees of freedom residing in each of the two neighboring cells sharing this face. Thus, L is a topological operator with entries 0 or 1 corresponding to the incidence between edges and faces on the one side and cells of the mesh on the other.

[13] Vice versa, if a DG-FEM solution represented by the degrees of freedom e happens to be continuous it can be projected without loss of accuracy into the space of CG-FEM solutions as

equation image

The operator W is not uniquely defined. A possible choice is to take the average of all DG-FEM edge degrees of freedom coming from the different cells sharing this particular edge. Similarly, the average of two incident DG-FEM face degrees of freedom equals the unique CG-FEM degree of freedom associated with this face. Thus, W is again a topological operator with entries 0 or 1/Ni, where Ni is the number of cells sharing the ith edge or face, respectively. Since the projection (14) is always exact (not so (15)), W is a pseudoinverse of L, i.e.,

equation image

[14] Now, by right multiplying (13) with W and using (14), we obtain

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so that (13) reduces to (5). Equations (17) and (18) can be verified in a straightforward manner. Note, e.g., that the DG-FEM matrix blocks (10) are nothing but the local element mass matrices of the CG-FEM. Then, (17) corresponds to the usual assembly procedure employed in numerical implementations for constructing global matrices from local element matrices. It is, thus, possible to start with a pure DG-FEM discretization framework and then reduce the formulation algebraically exactly to the CG-FEM by projecting degrees of freedom and discrete operators as in (15)–(18).

[15] The topological projectors W and L provide the link between the two formulations showing that the only difference between the two consists in the DG-FEM subspace of tangentially discontinuous fields representing spurious solutions. Note that this equivalence at the discrete level can only be established for the particular choice of the DG-FEM basis functions described in section 2.2. Note also that this procedure provides a practical way for hybridizing the two methods. Applying the projections (17)–(18) blockwise only within some preselected portions of the mesh, it is possible to obtain a mixed formulation from an existing DG-FEM code without need for a concrete implementation of the CG-FEM. Such a hybrid approach is particularly useful in simulations with highly inhomogeneous meshes. In these cases, in order to be able to operate with a larger time step, it can be advantageous to apply implicit time integration in the fine regions of the mesh. Clearly, using a CG-FEM approach in these regions is the appropriate choice because of the smaller number of unknowns involved in this formulation.

3.1. Constrained DG-FEM

[16] In order to impose tangential continuity within the DG-FEM framework, we introduce the operator,

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P is an orthogonal projector into the tangentially continuous DG-FEM solution subspace since

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The first property follows directly from the fact that W is pseudoinverse to L(16). The second can be shown by introducing the decomposition,

equation image

where N is a positive diagonal matrix containing the inverses, 1/Ni, of the incidence counts between edges (faces) and cells as described above.

[17] The DG-FEM constrained to tangentially continuous solutions can now be written as

equation image
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[18] Equations (22) and (23) can be explicitly solved for the time derivative term by employing the Bott-Duffin inverse [Rao and Mitra, 1973]:

equation image

where P = 1 − P is the complementary subspace projector of P. Equation (24) represents the semidiscrete form of a DG-FEM which admits only tangentially continuous solutions and thus is free of spurious modes. The derivation of (24) relies on the fact that the orthogonal projector P can be constructed explicitly. This is here possible because of the special choice of the basis functions in section 2.2 and it is, again, not possible in the general DG-FEM case. Contrary to the stabilization approach in the work by Hesthaven and Warburton [2004], where tangential continuity is imposed weakly (see also below), the constrained DG-FEM (24) results in a modification of the inverse mass matrix rather than of the curl-curl operator.

[19] Equation (24) is, to our knowledge, the only known DG-FEM which is strictly free of spurious solutions in the full dynamic range. Unfortunately, this form is not adequate for time domain simulations since computing the Bott-Duffin inverse requires the solution of a linear system of equations at each time step. The tangentially continuous DG-FEM subspace can only be accessed at the cost of losing the simple structure of Mε. However, the orthogonal projectors derived above can be used to construct an alternative DG-FEM time domain scheme, where tangential continuity is weakly imposed.

4. Penalization Approach

[20] We propose the following scheme:

equation image
equation image
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where w is a new vector field and γ a scheme parameter. In (25)–(27) the standard Lagrange multiplier technique for constrained solutions is applied. The orthogonal projector P provides, thereby, a natural way to implement the penalty term in (25). The latter penalizes the displacement current by an amount which is exactly proportional to the tangentially discontinuous “part” of the numerical solution. The Lagrange multiplier w can be interpreted as an electric current flowing at the interfaces between neighboring cells (edges and faces) to compensate for the unphysical DG-FEM interface charges.

[21] In order to demonstrate the effect of penalization, we rewrite (25)–(27) as a second-order wave equation by eliminating w and h as usual. The equation reads

equation image

Obviously, since P is a singular operator with eigenvalues 0 and 1, the effective eigenvalues of the curl-curl operator are shifted to higher frequencies. Note, however, that only solutions with discontinuous tangential components are affected, whereas physical solutions with Pe = 0 remain unchanged. Thus, the effect of penalization is to clean up the solution spectrum at low frequencies by pushing spurious solutions to the less relevant high frequencies. This frequency spectrum cleaning is expected to improve by selecting higher values for γ.

[22] The crucial question in this context is, how to select the penalization parameter γ such that field solutions close to a given target frequency f0 are not polluted by spurious modes. Equation (28) suggests that an upper frequency bound for the clean range of the spectrum can be written as

equation image

where the smallest eigenvalue of the inverse mass matrix appearing in (29) can be computed in an element-by-element fashion. According to (29), the optimum penalization parameter for any given frequency f0 becomes smaller with increasing mesh density. Thus, less penalization is needed for fine meshes. The behavior with respect to approximation order is not quite obvious. Numerical tests show, however, that also with increasing order a smaller penalization parameter can be used for the same target frequency f0. In all cases, however, the criterion (29) appears to be too restrictive compared to what is observed in simulations.

[23] The scheme (25)–(27) is clearly skew symmetric and thus energy conserving. Compared to the original DG-FEM with central fluxes (9), the operation count in (25)– is increased by two matrix vector multiplications involving the complementary projector P. The fully explicit scheme (25)–(27) is still by far more efficient than the conventional CG-FEM (5) which requires the application of a linear solver at each time step.

[24] Finally, note that the idea of applying Lagrange multiplier techniques to impose given constrains is very common in numerical simulations. Different penalization methods differ only on the choice of the penalty terms which has an impact on simulation performance and accuracy. In the context of DG-FEM, e.g., a similar approach has been earlier proposed by Hesthaven and Warburton [2004] and further discussed by Warburton and Embree [2006]. There, however, the penalty term emerges from the continuity condition for the interface fluxes along the mesh faces rather than from the topological-type conditions employed here. In contrast to Hesthaven and Warburton [2004], scheme (25)–(27) imposes tangential continuity not only along the faces but also along the edges of each cell in the mesh. This results in a different behavior of the two approaches with respect to accuracy for large values of the penalization parameter (see section 4.1). In addition, the orthogonal projector approach provides a natural way to combine the penalization method with the constrained DG-FEM (24) or even hybridize with CG-FEM according to (17) and (18); all within one and the same simulation framework.

4.1. Box Resonator Tests

[25] The actual performance of the scheme is illustrated in Figures 1, 2, and 3. The test structure is a box resonator with side lengths 1 × 1 × 1 m. An impulse current initially excites electromagnetic fields within the box which then oscillate freely at the resonator frequencies. The solution spectrum is extracted by discrete Fourier analysis of the time domain data obtained by the application of (25)–(27) with leapfrog time stepping.

Figure 1.

Frequency spectra in the box resonator computed in the time domain according to (25)–(27) for different values of the penalization parameter γ using incomplete first-order basis functions (edge elements).

Figure 2.

Frequency spectra in the box resonator computed in the time domain according to (25)–(27) for different values of the penalization parameter γ using full-order quadratic basis functions. The top left graph shows the resulting spectrum for the unpenalized DG-FEM.

Figure 3.

Frequency spectra in the box resonator computed in the time domain according to (25)–(27) for different values of the penalization parameter γ using full-order quadratic basis functions and a regular Cartesian grid. The top left graph shows the resulting spectrum for the unpenalized DG-FEM.

[26] As shown in Figure 1 the spectrum of the standard DG-FEM (top left graph) is completely polluted by spurious modes which overlap with the physical frequencies of the resonator. The shifting of these modes toward higher frequencies is already observed for small values of the penalization parameter. For γ = 1 the field spectrum appears completely clean from spurious modes up to at least twice the ground frequency of the resonator. In this example, incomplete first-order vector basis functions (edge elements) are used for defining the discontinuous approximation space of DG-FEM. The mesh consists of tetrahedral elements with a fixed resolution of approximately 11 steps/wavelength for the ground mode of the resonator at ≈212 MHz.

[27] Figure 2 shows the resulting spectra when full-order quadratic basis functions on a tetrahedral mesh with the same resolution are used. The effect of the penalization on the spurious modes is here the same as in the low-order case, whereas, the physical resonance peaks are obviously much closer to the analytical value.

[28] The same experiment is repeated in Figure 3 using, however, a regular Cartesian grid and full-order quadratic basis functions. The DG-FEM in this tensor product case is known to be charge conserving [cf. Gjonaj et al., 2007]. Thus, no spurious solutions appear in the dynamic range of the spectrum (see top left graph in Figure 3). These modes, however, exist as static solutions at zero frequency. When the penalization is applied, the spurious modes are pushed into the dynamic range, thus, initially polluting the physical spectrum. For larger values of the penalization parameter the same behavior of frequency spectrum cleaning as in the tetrahedral case is observed.

[29] A closer inspection of the numerical results indicates that the value of the penalization parameter does have an impact on numerical solution for the resonance frequencies. As it can be easier recognized in Figure 1 using low-order simulations the resonance peak corresponding to the ground mode is shifted to the right toward the analytical value with increasing γ. This raises the question on the accuracy of this approach. Figure 4 shows the convergence behavior of the ground mode frequency for different mesh resolutions, approximation orders and penalization parameters. For reference, the convergence curve for the CG-FEM with lowest-order edge elements is also shown. In the later case, however, it is hardly possible to obtain comparable convergence results for higher-order elements due to the enormous numerical effort required in the computation of the frequency spectra by time domain simulations.

Figure 4.

Numerical convergence of the lowest resonance frequency in the box resonator versus mesh size for different values of the penalization parameter γ and approximation orders.

[30] As seen in Figure 4 the convergence rate of the method for any fixed value of γ cases is optimal. On the other hand, the smaller the penalization parameter the less accurate is the resonance frequency. This can be explained by the mixing of the physical and spurious solution subspaces, respectively, which are not sufficiently separated for small values of γ. As the penalization parameter is increased the solution tends to the one obtained by the CG-FEM. Indeed, for very large penalizations the resonance frequencies obtained by DG-FEM and CG-FEM coincide to machine accuracy. This is to be expected by the way the DG-FEM approach has been constructed. In the work by Hesthaven and Warburton [2004], however, the opposite is reported. In this investigation, the authors state that choosing the penalization parameter too large leads to loss of accuracy. The different behavior of the two methods may be due to the different scaling used for the penalization parameter with respect to mesh size. This discrepancy, however, may also be the result of the different nature of the penalty term applied in (25)–(27).

[31] A critical issue for all penalization methods is the additional stiffness introduced into the system. In the present case, shifting spurious modes to higher frequencies implies that the highest eigenvalue of the modified curl-curl operator also increases. This leads to a reduction of the maximum stable time step for explicit time integration below its original limit. Figure 5 shows the relative decrease in the stability limit versus γ for different approximation orders. The maximum stable time step for low-order simulations is almost unaffected by the penalization. This effect, however, becomes critical at high orders. The impact of the maximum stable time step reduction in practical simulations depends on the relevant portion of the physical spectrum one is interested at in the investigation. Electrical devices are typically operated close to their lowest resonance frequency. Thus, in many applications it may be sufficient to apply the frequency spectrum cleaning only to a moderate portion of the spectrum at low frequencies. In this case, the stability limit of the method is only slightly decreased. Nevertheless, a more detailed investigation on this issue is actually needed as this effect represents the most important disadvantage of this approach.

Figure 5.

Maximum stable time step reduction versus penalization parameter γ for different approximation orders.

4.2. TEAM Workshop Problem 19

[32] The following example is taken from the TEAM Workshop problems for testing electromagnetic analysis methods. It represents a standard benchmark for eigenvalue computations. The structure consists of a cylindrical cavity which is fed by a rectangular waveguide. The resonator is loaded by inserting a dielectric rod of diameter 9 mm and relative permittivity εr = 2.7 (Plexiglas). The detailed description of the problem is given by Bossavit et al. [1992]. Figure 6 shows the geometry of the resonator as well as the electric field distribution of the ground mode.

Figure 6.

(a) Geometry of the dielectric loaded resonator from the TEAM Workshop problem 19. (b) Electric field strength distribution of the ground mode.

[33] The time domain simulations are performed using a very sparse grid of only 2906 curved tetrahedral elements. In order to resolve the resonance frequencies at the required resolution of 1 MHz, however, a huge number of time steps is needed. In the most accurate simulation (with third-order elements) this number amounts to ∼106. The simulation results for incomplete first-order, full-order linear, full-order quadratic and full-order cubic basis functions are shown in Figure 7. In all cases, a fixed penalization parameter, γ = 2, is used to ensure a clean spectrum in the relevant frequency range. The resulting ground frequencies for each approximation order are 2.358 GHz, 2.480 GHz, 2.468 GHz and 2.464 GHz corresponding to e relative error of 4%, 0.7%, 0.2% and 0.08% for the incomplete first-order, full-order linear, full-order quadratic and full-order cubic cases, respectively. Note that there is a second resonance at 2.518 GHz which is located very close to the ground frequency of the resonator. The method is able to resolve the basic mode pattern of the spectrum even when (because of the sparse mesh and/or low approximation order) the accuracy of the single resonance frequencies is low. This is not necessarily the case when a dissipative method is used. In this case, the numerical spectrum features broad resonances which may overlap if the frequencies are sufficiently close to each other. Finally, note that in terms of numerical performance, time domain simulations are generally not adequate for the computation of resonance frequencies in electromagnetic cavities. The fact that we are able to do so for this example demonstrates the efficiency of the proposed method.

Figure 7.

Frequency spectra of the dielectric loaded cavity computed by time domain simulations using different approximation orders.

5. Conclusion

[34] A new orthogonal projector method for the high-order DG-FEM in the time domain is introduced. The method uses the equivalence between the CG-FEM and the subspace of tangentially continuous DG-FEM solutions when a specific set of approximation functions is used in the formulation. For this particular case, a constrained formulation for DG-FEM which is strictly free of spurious solutions is presented. Furthermore, for time domain simulations using an explicit time integrator, a conservative penalization approach is proposed which allows to systematically clean up the DG-FEM spectrum from spurious modes at any desired level of accuracy. The method converges at optimal rate for physical solutions in the clean portion of the spectrum independently from the choice of the penalization parameter. The accuracy of the method was demonstrated in simulation examples involving a simple box resonator and the dielectric loaded cavity benchmark of the TEAM Workshop problem 19.