Journal of Computational Physics 230 (2011) 7300–7310

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Journal of Computational Physics

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Spurious solutions in mixed finite element method for Maxwell’sequations: Dispersion analysis and new basis functions

Luis Tobón a,b, Jiefu Chen a, Qing Huo Liu a,⇑a Department of Electrical and Computer Engineering at Duke University, 130 Hudson Hall, Box 90291, Durham, NC 27708, United Statesb Departamento de Ciencias e Ingeniería de la Computación, Pontificia Universidad Javeriana, Cali, Colombia

a r t i c l e i n f o

Article history:Received 29 November 2010Received in revised form 18 May 2011Accepted 28 May 2011Available online 16 June 2011

Keywords:Mixed finite element methodSpurious solutionsDispersion analysis

0021-9991/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.jcp.2011.05.035

⇑ Corresponding author.E-mail addresses: luis.tobonllano@duke.edu (L. T

a b s t r a c t

The finite element method is a well known computational technique used to obtain numer-ical solutions to boundary-value problems including Maxwell’s equations. This paper firstpresents a brief description of the mathematical structure, based on the De Rham diagram,to discretize Maxwell’s equations. Then it uses a numerical dispersion analysis of themixed finite element method with both electric and magnetic fields as unknowns to eval-uate the presence of spurious solutions for different basis functions. These unwanted spu-rious solutions appear when the same order of element is used for electric and magneticfields, while the system is free of spurious modes when different orders of elements areemployed for electric and magnetic fields. In this work, finite elements in both frequencyand time domain are studied, and the effects of these spurious solutions in both domainsare analyzed in one- and three-dimensional cases.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

One of the most active areas of research in the finite element method is the appropriate choices of basis functions that cansolve correctly the topological and constitutive laws in electromagnetics by Maxwell’s equations [1–5]. A problematic situ-ation is the presence of extraneous or spurious solutions when inappropriate basis functions are used [5–13]. These spurioussolutions are non-physical. We can differentiate two categories of spurious solutions [6]: the first set has modes with verylow (theoretically zero) frequencies, and are several orders of magnitude away from the rest in the spectrum. The origin ofthis kind of solutions is that the divergence condition for electric and magnetic fields is not satisfied. A typical situation forsuch modes is when the curl–curl operator is used, without taking into account of the divergence operator [5,9,8], thereforesolutions in the form of gradient of a potential are valid mathematical solutions but are not physical. A correct formulation ofthe problem based on the four Maxwell’s equations, and the inclusion of compatibility relations applied to boundaries be-tween elements [9,14] solve this problem.

The second category of spurious solutions intermingle with the correct ones [15–17,7]. These non-physical solutionsappear when the mixed finite element method is used to solve the first order Maxwell’s equations with the same order ele-ments for electric and magnetic fields. Previously the cause of this subset of spurious solutions has not been completelyclear, and it is supposed to be related to a compatibility problem between the electric and magnetic field formulation[2,3,18,13]. Some characteristics of these solutions are their rapid spatial variations (high numerical wavenumbers), and highand negative group velocities [19]. Under these considerations a dispersion analysis [10] is presented in this work as a simpleand practical procedure to detect the second kind of spurious solutions in finite element formulations. The mathematical

. All rights reserved.

obón), jiefu.chen@duke.edu (J. Chen), qhliu@ee.duke.edu (Q.H. Liu).

L. Tobón et al. / Journal of Computational Physics 230 (2011) 7300–7310 7301

structure to obtain correct solutions in the finite element method applied to Maxwell’s Equations using both electric andmagnetic fields is described, based on functional analysis [20,21,1,22].

The objective of this work is to perform numerical dispersion analysis of the mixed finite element method so that thepresence of spurious modes can be predicted. A continuous Galerkin (CG) finite element formulation is described usingthe first-order Maxwell’s equations in both frequency and time domains. A Fourier analysis is used to evaluate the accuracyof the solutions in frequency domain, and the presence of spurious solutions in time domain. Furthermore, we solve theeigenvalue problem and the Rayleigh quotient to describe the numerical dispersion curve [23,24], and find good agreementwith the theoretical analysis of the semi-discrete continuous Galerkin procedure.

2. Formulations

2.1. Maxwell’s equations

Maxwell’s equations for electric field and magnetic field due to imposed current densities in an anisotropic and inhomo-geneous medium are considered:

@D@t¼ r�H� Jc � Js ð1Þ

� @B@t¼ r� EþMc þMs ð2Þ

r � D ¼ qe ð3Þ

r � B ¼ qm ð4Þ

D ¼ �E ð5Þ

B ¼ lH ð6Þ

Jc ¼ reE ð7Þ

Mc ¼ rmH ð8Þ

where E and H are the electric and magnetic fields respectively; D and B are the electric and magnetic fluxes; Jc and Mc arethe electric and magnetic current densities due to conductive properties of the medium, and Js and Ms are the electric andmagnetic current densities due to imposed sources; and �, l, re, and rm are the electrical and magnetic properties of themedium corresponding respectively to permittivity, permeability, electric conductivity, and magnetic conductivity. Theseequations define, based on the Tonti approach [1], the topological laws (1)–(4), and constitutive laws (5)–(8) of electromag-netic physics, and they can be summarized (with an abstraction of the time derivatives) in the following diagram:

/E !r E !r� M;B !r� qm

re; � # rm;l "

qe r� J;D r� H r /H

ð9Þ

where /E and /H are scalar electric and magnetic potentials.

2.2. Finite element method

A finite element is defined, using the Ciarlet approach [25,21,22], as a triad K ¼ ðK;Q ;RÞ, where

� K is a domain in Rd: Intervals (d = 1), triangles and quadrilaterals (d = 2), and tetrahedrons, hexahedrons and prisms(d = 3). The set form by all Ka is called a Mesh T ¼ fK1;K2; . . . ;KMg, and the union of the all closures of Ka representsthe domain X ¼

Sall aKa.

� Q is a space of polynomials on K of dimension dim(Q) = NQ.� R ¼ fL1; L2; . . . ; LNPg is a set of linear forms. The elements of R are called the degrees of freedom (DoF).

2.3. Hilbert space and the De Rham diagram

The family of functions used in finite elements belongs to the Hilbert space of functions (H1, H(curl), H(div), and L2), in abounded domain X � Rd; d being the spatial dimensionality:

7302 L. Tobón et al. / Journal of Computational Physics 230 (2011) 7300–7310

H1ðXÞ ¼ u 2 L2ðXÞ; @u@xi2 L2ðXÞ;1 6 i 6 d

� �ð10Þ

HðX; curlÞ ¼ u 2 ½L2ðXÞ�d;r� u 2 ½L2ðXÞ�dn o

ð11Þ

HðX; divÞ ¼ u 2 ½L2ðXÞ�d;r � u 2 L2ðXÞn o

ð12Þ

L2ðXÞ ¼ u : X! R

ZXjuj2dx <1

���� ��ð13Þ

Global conformity requirements are defined for these functions.

Lemma 1. Assume two neighboring elements K1, K2 2 T, with a face in common f ¼ K1T

K2, and n a normal vector to the face f

� ujKa2 H1ðKaÞ imposes the continuity of u between neighboring elements: ujK1

¼ ujK2on f.

� ujKa2 HðKa; curlÞ imposes the continuity of the tangential components: n� ujK1

¼ n� ujK2on f.

� ujKa2 HðKa; divÞ imposes the continuity of the normal components: n � ujK1

¼ n � ujK2on f.

The relations between these function spaces can be analyzed by the De Rham diagram, by means of differentialoperators.:

H1!r HðcurlÞ!r�HðdivÞ!r� L2 ð14Þ

In this diagram, the range of each operator is the null-space of the next one. Hence, finite elements are understood as asequence of scalar and vector-value elements satisfying the De Rham diagram on the discrete level [22]. Therefore, this dia-gram has two main roles: the design of vector-valued finite elements, and the mathematical foundation for stability and con-vergence analysis. From the evident connection between (9) and (14), we can deduce that the choice of the family of basisfunctions depends directly on the field which will be solved. In our case, because E and H are employed, curl-conformingvector basis functions are used. Furthermore, based on numerical experiments and the dispersion analysis, we found thatdifferent orders of interpolation in E and H are required to guarantee a non-spurious vector element. More rigorous studyof this topic will be presented in a future paper.

2.4. Galerkin’s method

Galerkin’s method seeks the solution by weighting the residual of the partial differential equation over the entire region[8,26] . We used curl-conforming vector-basis functions, U(M)and W(N), to discretize the electric and magnetic fields respec-tively in each element:

E ¼Xne

i¼1

eiUðMÞi ð15Þ

H ¼Xnh

j¼1

hjWðNÞj ð16Þ

where M and N determine the orders of interpolation used in each polynomial associated with each degree of freedom ei andhj, and ne and nh represent the number of degrees of freedom associated with electric and magnetic fields, respectively. Byassembling all elements the following discrete system of equations is obtained:

Mdudt¼ Suþ s ð17Þ

where the vector of unknowns, the imposed source vector, and the mass and the stiffness matrices are defined by

u ¼e

h

� �; s ¼

jm

� �; M ¼

Mee 00 Mhh

� �; S ¼

Kee Keh

Khe Khh

� �

In addition, the elemental terms of these matrices are defined by the following integrals

ðMeeÞepq ¼Z

XeUT

p�UqdV ð18Þ

L. Tobón et al. / Journal of Computational Physics 230 (2011) 7300–7310 7303

ðMhhÞepq ¼Z

XeWT

plWqdV ð19Þ

ðKehÞepq ¼Z

XeUT

pðr �WqÞdV ð20Þ

ðKeeÞepq ¼ �Z

XeUT

preUqdV ð21Þ

ðKheÞepq ¼ �Z

XeWT

qðr �UpÞdV ð22Þ

ðKhhÞepq ¼ �Z

XeWT

prmWqdV ð23Þ

2.5. Eigenproblem

The system of Eq. (17) is a set of ordinary differential equations in time domain, which can be easily transformed intofrequency domain by changing d

dt to jx (a time dependence exp(jxt) is implied here). This transformation allows us to detectspurious solutions in the form of non-physical eigensolutions, and to analyze the convergence rate of the method. In thisway, in the one-dimensional case with a periodic boundary condition with a source-free and non-conductive medium, wecan rewrite (17) in the form:

jxMeee� Kehh ¼ 0 ð24Þ

jxMhhh� Khee ¼ 0 ð25Þ

e1 ¼ eP�Mþ1; h1 ¼ hP�Nþ1 ð26Þ

Here the subindex P defines the number of elements in a physical region with length L. Therefore, with Eqs. (24) and (25) theeigenproblem can be formulated as:

Ye ¼ vXe ð27Þ

where

Y ¼ �Keh½Mhh��1Khe; X ¼ c20Mee

vector e represents eigenvectors, and v ¼ xc0

� �2¼ k2 represents eigenvalues. For reasonable numerical results, we expect the

eigenvectors to be close to the true electric field patterns, and the wavenumbers (eigenvalues) allowed in the physical sys-tem modeled and solved by the numerical method. Consequently, eigensolutions without a physical meaning are spuriousones, and must be removed. In frequency domain, checking the eigenvectors is enough to discard spurious solutions, but intime domain, this method of removing spurious solution is not feasible. In Section 2.6, a method based on the Rayleigh quo-tient to verify the presence of spurious modes is presented.

2.5.1. Field accuracy based on Fourier AnalysisInformation related with field accuracy can be extracted from the set of eigenvectors obtained in the procedure described

in (27). Consider a domain 0 6 x 6 L with the periodic boundary condition. Based on this condition, it is known that the exactsolutions are pure sinusoidal functions

epðxÞ ¼ a cosðkpxþuÞ ¼ ap cos kpxþ bp sin kpx ð28Þ

where kp = 2pp/L is the pth numerical wavenumber with p ¼ 0;1;2; . . . ;Np ¼ PM2 , and arbitrary values of ap and bp.

In general, the numerical solution is not a pure sinusoidal function, i.e. it is a sum of multiple harmonics including theexact solution (28) for ki = kp, and error terms for ki – kp:

~epðxÞ ¼ a0 þXNp

i¼1

ai cos kixþXNp

i¼1

bi sin kix ð29Þ

where ai and bi represent the intensities of all harmonics associated with cosine and sine functions, respectively. Thus, sub-tracting (28) from (29) the error signal in the numerical solution associated with the wavenumber equal to kp can beobtained:

7304 L. Tobón et al. / Journal of Computational Physics 230 (2011) 7300–7310

~epðxÞ � epðxÞ ¼ a0 þXNp

i¼1i – p

ai cos kixþXNp

i¼1i – p

bi sin kix ð30Þ

and the error in the numerical solution is

error ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVarð~epðxÞ � epðxÞÞ

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2N2p

XNp

i¼1i – p

jeEij2vuuut ð31Þ

where jeEij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijaij2 þ jbij2

qis the magnitude of the ith component of the Fourier transform of the pth eigenvector.

2.6. Modal analysis

Let us define an approximate solution e to the 1D periodic problem presented in (27) in the form of a cosine function,similar to (28):

e ¼ cosðkxÞ ð32Þ

where e represents an approximate eigenvector with wavenumber given by k. Thus, an approximate solution v to the prob-lem (27) is given when the residual (X � vY)e is orthogonal to e [27,23]. Hence, the best approximation ~v exists and is givenby the Rayleigh quotient :

~v ¼~xc0

�2

¼ ~k2 ¼ eYeeXe

ð33Þ

In order to simplify the discussions of the results, we define the normalized theoretical and numerical wavenumbers:

XR ¼kL

P �minðM;NÞ ð34Þ

XN ¼~kL

P �minðM;NÞ ð35Þ

Thus, the curve XN as a function of XR represents the numerical dispersion curve of the scheme defined by Eq. (17). Sev-eral important features can be extracted from this curve: phase and group velocity; accuracy of the system; and, the mostimportant in this work, spurious modes. The desirable dispersion curves must have a monotonic behavior, i.e. one numericalwavenumber is associated with one theoretical wavenumber, and vice versa.

2.7. Semi-discretization

For clarity we will first present the analysis in detail for the one-dimensional TEMx (both electric and magnetic fields aretransverse to the x direction) case for a source-free homogeneous medium, and subsequently consider the extension to high-er dimensions. Under these considerations Eqs. (1) and (2) take the following forms:

�@Ey

@tþ @Hz

@x¼ 0 ð36Þ

l @Hz

@tþ @Ey

@x¼ 0 ð37Þ

For this analysis, we seek periodic solutions of (36) and (37) of the form:

Eyðx; tÞHzðx; tÞ

� �¼

bEyðxÞbHzðxÞ

( )ejxt ð38Þ

where bEz and bHy are amplitudes, and x is the angular frequency. Eqs. (36) and (37) can be rewritten as

j�xbEy þdbHz

dx¼ 0 ð39Þ

jlxbHz þdbEy

dx¼ 0 ð40Þ

L. Tobón et al. / Journal of Computational Physics 230 (2011) 7300–7310 7305

The functions bEy and bHz belong to H1. In the Galerkin finite element method, an approximate solution bEy and bHz fromfinite dimensional subspaces He and Hh, of the infinite dimensional space H1, is sought. Hence, the approximate solutionsbEy and bHz are typically continuous piecewise-polynomials (with the same or different degrees) satisfying the essentialboundary conditions.

Let /p � /p(x), p = 1, . . . ,ne, and wq � wq(x), q = 1, . . . ,nh, be piecewise-polynomial finite element basis functions spanningthe approximation spaces He and Hh, respectively. Then, for any solution we have the expansions:

bEyðxÞ ¼Xne

p¼1

Ep/ðMÞp ðxÞ; bHzðxÞ ¼

Xnh

q¼1

HqwðNÞq ðxÞ ð41Þ

where Lagrange-type bases are assumed, and Ep and Hq are nodal degrees-of-freedom for the problem with Ep ¼ bEyðxpÞ, andHq ¼ bHzðxqÞ for specified nodes p and q, at points xp and xq, respectively, in a uniform mesh of P elements in the physicaldomain with length L, and the element size is defined as d ¼ L

P. It is important to note that the number of degrees of freedomassociated with electric and magnetic field, ne and nh, respectively, is function of the number of elements, P, and the order ofthe polynomials, M and N. Using the appropriate basis functions, /q and wq, as test functions and substituting the expansions(41) into (39) and (40) to determine the semi-discrete weak system:

jxXne

p¼1

Ep

Z L

0�/ðMÞq /ðMÞp dxþ

Xnh

p¼1

Hp

Z L

0/ðMÞq

dwðNÞp

dxdx ¼ 0 ð42Þ

Xne

p¼1

Ep

Z L

0wðNÞq

d/ðMÞp

dxdxþ jx

Xnh

p¼1

Hp

Z L

0lwðNÞq wðNÞp dx ¼ 0 ð43Þ

Now, for the dispersion analysis, the dependent variables in wave vector space are decomposed as follows

Ep

Hp

� �¼

E

H

� �ejkxp ð44Þ

As an example, considering a linear common interpolation in a uniform mesh, and first order functions in both E and H, wehave

XM

p¼1

Ep

Z L

0/ð1Þq /ð1Þp dx ¼ Eejkxq

d3ðcosðkdÞ þ 2Þ ð45Þ

XN

p¼1

Hp

Z L

0/ð1Þq

dwð1Þp

dxdx ¼ Hejkxq j sinðkdÞ ð46Þ

Similarly, we can obtain results for all terms. Then, based on those formulations we can obtain the matrix equation

x�d3 cos kdð Þ þ 2ð Þ sinðkdÞ

sinðkdÞ xld3 ðcosðkdÞ þ 2Þ

" #E

H

� �¼

00

� �ð47Þ

The normalized numerical wavenumber can be obtained from the dispersion relation

XN ¼ ~kd ¼~xdc¼ 3 sinðXRÞ

cosðXRÞ þ 2ð48Þ

for the linear interpolation basis functions (M = N = 1), where XR and XN are defined by (34) and (35).The same procedure can be used to obtain the numerical dispersion for the mixed-order basis functions. In particular, if

the second-order basis function is used for electric field, and first-order basis function is used for magnetic field, the expres-sion for the dispersion relation is:

XN ¼ ~kd ¼~xdc¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5ð3� cosðXRÞÞð1� cosðXRÞÞ þ sin2ðXRÞ

ð3� cosðXRÞÞð2þ cosðXRÞÞ

sð49Þ

where M = 2 and N = 1 have been used. In following numerical results, a very good agreement between the numericaldispersion from semi-discretization method, modal analysis, and eigensolutions will be presented. The method will beextended to higher order schemes, and to three-dimensional cases.

7306 L. Tobón et al. / Journal of Computational Physics 230 (2011) 7300–7310

3. Numerical results

Below we first show the one-dimensional case to compare the numerical dispersion curves for different schemes, andshow that the new basis functions are immune of spurious modes. Then numerical results are further demonstrated for athree-dimensional problem.

3.1. One-dimensional case: periodic boundary condition

The first case is a one-dimensional domain with length L = 1 m, and a uniform mesh with P = 100 elements is used to dis-cretize the periodic problem. To facilitate the discussion, we use a notation EmHn to denote the orders of interpolation in theelectric and magnetic fields in a numerical scheme, where mth order basis functions are used for E, while nth order basisfunctions are used for H. Table 1 shows a comparison between the exact wavenumber and numerical wavenumber in(27) using E1H1 and E2H1 schemes. There are low frequency spurious solutions (such as Mode 2 in E1H1 scheme and Mode1 in E2H1 scheme) in both numerical schemes due to the non-divergence-conforming basis functions used in both schemes;such spurious modes are in general of little concern in practice and they are well understood. However, at high frequencies,interspersed spurious eigenvalues are presented in the E1H1 scheme and are shown in parentheses, while the E2H1 schemeis free of high-frequency spurious modes. For the correct eigenvalues in both schemes, similar accuracy in the numericalwavenumber is obtained; thus, it is difficult to know whether a particular solution is a spurious mode in a complex casewhen no analytical solutions are available. Therefore, it is an important goal for this study to identify which scheme will givenon-spurious solutions. Fig. 1 presents the eight eigenvectors corresponding to the eigenvalues in Table 1 from the E1H1scheme. We observe the fast spatially varying patterns in both the zero-frequency (1st panel) and high-frequency (4thand 8th panels) spurious modes.

Next, we show the modal analysis and semi-discrete analysis applied to the E1H1 and E2H1 schemes; in addition, a ref-erence dispersion curve from Walters and Carey is presented [10]. First, we note an excellent agreement between modalanalysis results and the semi-discrete analysis results for both schemes E1H1 and E2H1 in Fig. 2(a), and the Walters–Careyexpression [10]. Secondly, double solutions are present in the E1H1 scheme, i.e., with one given numerical wavenumber,there exist two different real values of wavenumbers; these solutions correspond to one correct and one spurious solutions,respectively. Furthermore, we can note a monotonic behavior of the E2H1 scheme, i.e., it is a scheme free of spurious solu-tions. Similarly, results for higher order schemes are presented in Fig. 2(b); the results present the same conclusion: The spu-rious modes are present when the same order of interpolation is used in E and H (i.e., E1H1, E2H2, and E3H3 schemes), whilethe schemes with different orders of interpolation for E and H (i.e., E2H1, E3H1, E3H2 schemes) are free of spurious solutions.

Fig. 3(a) and (b) show the numerical errors for wavenumber and for the electric field versus the sampling density in termsof the number of points per wavelength (PPW) for different schemes. We have the following observations:

� From the wavenumber accuracy analysis in Fig. 3(a), we note that the scheme E3H3 presents the best performance, fol-lowed by E3H2, E2H2, E2H1, and E3H1. Thus, the first conclusion is that such schemes with the same order of interpo-lation in electric and magnetic fields have very good convergence, with order of convergence two times the order of theelement, but they are not free of spurious solutions.� The schemes with one order difference in interpolation orders for E and H (i.e., EmHn where n = m � 1) have good per-

formance in wavenumber convergence, with a convergence rate equal to 2m, and are free of spurious solutions.� If the difference between m and n in the EmHn scheme is greater than one (for example E3H1), the scheme has a poorer

performance than that with n = m � 1, but are also free of spurious solutions.� Similar results are shown in the electric field accuracy plots, but the convergence order is lower than the wavenumber

plots; the best performance is for the E3H3 scheme and the worst for E3H1 scheme.� All these analyses show that the EmHm schemes (i.e., m = n) have excellent performance in wavenumber and field accu-

racy for the correct eigenvectors, but they produce spurious solutions. Therefore, such schemes are not appropriate forMaxwell’s equations in the finite-element methods.

Table 1Eigenvalues for the 1D-PBC case calculated by the analytical method, the E1H1 scheme, and the E2H1 scheme. High-frequency spurious modes are denoted with parentheses.

Analytical Mode E1H1 Numerical E1H1 Mode E2H1 Numerical E2H1

� � � 2 0.00000125 1 0.000004416.28318531 4 6.28318476 3 6.2831864612.56637061 6 12.56635317 5 12.56640743� � � 8 (18.80005824) � � � � � �18.84955592 10 18.84942316 7 18.8498338725.13274123 12 25.13217993 9 25.1339029331.41592654 14 31.41420634 11 31.41943484� � � 16 (37.30580267) � � � � � �

0 0.5 1−1

−0.5

0

0.5

1

x

E y (mod

e 2

)

0 0.5 1−1

−0.5

0

0.5

1

x

E y (mod

e 4

)

0 0.5 1−1

−0.5

0

0.5

1

x

E y (mod

e 6

)

0 0.5 1−1

−0.5

0

0.5

1

x

E y (mod

e 8

)

0 0.5 1−1

−0.5

0

0.5

1

x

E y (m

ode

10 )

0 0.5 1−1

−0.5

0

0.5

1

x

E y (m

ode

12 )

0 0.5 1−1

−0.5

0

0.5

1

x

E y (m

ode

14 )

0 0.5 1−1

−0.5

0

0.5

1

x

E y (m

ode

16 )

Fig. 1. Eigenvectors corresponding to the eight eigenvalues in Table 1 in the 1D periodic case using the E1H1 scheme. The first panel shows the zero-frequency spurious mode, while the 4th and 8th panels show the high-frequency spurious modes.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Ωx

Ωn

E1H1 NumE2H1 NumIdeal behaviorE1H1 TheoE2H1 TheoWalter and Carey

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

ΩR

ΩN

E1H1E2H1E3H1E2H2E3H2E3H3Ideal

(b)(a)

Fig. 2. Dispersion curves for various schemes for the 1D periodic problem. (a) Comparison of the analytical dispersion solution and numerical dispersionsolution for E1H1 and E2H1, together with the exact dispersion relation and the result from Walters and Carey [10] and (b) dispersion curves for higherorder schemes.

101 102

10−10

10−5

100

ppw

rela

tive

erro

r Ω N

E2H1 − order 4E3H1 − order 4E2H2 − order 4E3H2 − order 6E3H3 − order 6

101 10210−10

10−8

10−6

10−4

10−2

100

PPW

Abso

lute

erro

r E F

ield

Accuracy E field

E2H1 − order 2E3H1 − order 1E2H2 − order 2E3H2 − order 2E3H3 − order 4

(a) (b)Fig. 3. Normalized wavenumber error (a), and electric field error (b) using different schemes in the 1D periodic problem.

L. Tobón et al. / Journal of Computational Physics 230 (2011) 7300–7310 7307

The presence of spurious waves is notorious in time domain as they introduce rapid spatial variations in the field solu-tions. To demonstrate this, we study a pulse of the first-order Blackman–Harris window function [28] with a characteristicfrequency fc = 3 GHz. The first three plots in Fig. 4(a)–(c) show how the correct wave (a first-derivative of the Blackman–Harris Window pulse) and the spurious waves move along the axis of propagation in this 1D problem. These figures are snap-shots at three different times (0.25, 0.50, and 0.75 ns), and show a spurious wave faster than the correct one. This spuriouswave is characterized by a rapid variation in space, i.e. a high wavenumber, in agreement with the dispersion analysis.

0 0.2 0.4 0.6 0.8 1−300

−200

−100

0

100

200

300

x (m)

Elec

tric

Fiel

d (V

/m)

0 0.2 0.4 0.6 0.8 1−300

−200

−100

0

100

200

300

x (m)

Elec

tric

Fiel

d (V

/m)

0 0.2 0.4 0.6 0.8 1−300

−200

−100

0

100

200

300

x (m)

Elec

tric

Fiel

d (V

/m)

0 0.5 1 1.5 2 2.5−200

−100

0

100

200

time (ns)

Elec

tric

field

(V/m

)

(a) (b)

(c) (d)

Fig. 4. (a)–(c) Snapshots of real and spurious solutions using second order in both E and H (E2H2 scheme) at 0.25, 0.50, and 0.75 ns, respectively, and (d)time response at x = 0.8 m, for the 1D periodic problem.

7308 L. Tobón et al. / Journal of Computational Physics 230 (2011) 7300–7310

However, it is evident that this spurious wave has a slow variation in the time response, as shown in Fig. 4(d), which hindersthe detection of such non-physical solutions from correct signals taken by receivers located at a single point in space. There-fore, in practice, it is important to use basis functions that are free of spurious solutions, such as the EmHn scheme withm = n ± 1 or in general (but less favorable) with m – n with jm � njP 2.

3.2. Three-dimensional case: PEC cavity problem

To our knowledge, the above analysis is the first analysis of the presence of spurious modes in the EmHm scheme andtheir absence in the EmHn scheme with m – n, even though previously we have shown numerical evidence of these phe-nomena in [29,18]. Here, a simple 3D case is further presented to show the presence of spurious solutions when the sameorder of interpolation in E and H is used; these basis functions are the edge-based vector basis functions, i.e., the Nedlec edgeelements. The case under study is a perfect electric conductor cavity (0.23 m � 0.11 m � 0.03 m) with a dipole source locatedat (0.01,0.01,0.003) m, with polarization in the direction ðxþ yþ zÞ=

ffiffiffi3p

, and a receiver located at (0.20,0.07,0.013) m. Afirst-derivative of the Blackman–Harris Window with characteristic frequency 2.0 GHz was used as the source time function.The mesh (Fig. 5) is formed by brick elements with size (0.01,0.01,0.01) m.

Fig. 6(a) shows the correct eigenvalues using E1H2, and while in Fig. 6(b) a large number of spurious eigenvalues are pres-ent with the E1H1 scheme. The eigenvectors corresponding to the first four correct eigenvalues are on the top of Fig. 7, and

(a) (b)Fig. 5. A PEC cavity of dimensions 0.23 m � 0.11 m � 0.03 m centered at the origin with a source and receiver. (a) Geometry and (b) mesh.

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

order of eigenvalue (E1H2)

Freq

uenc

y (G

Hz)

0 500 10000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

order of eigenvalue (E1H1)(a) (b)

Fig. 6. Eigenvalues of the 3D problem in Fig. 5. (a) Correct values using the E1H2 scheme and (b) non-physical, spurious values using the E1H1 scheme. Notethat the correct analytical eigenvalues are represented by the horizontal lines.

L. Tobón et al. / Journal of Computational Physics 230 (2011) 7300–7310 7309

eigenvectors with closest eigenvalue but from E1H1 solutions on the bottom. We can notice similar behaviors as thoseobserved in one dimension, with fast spatial variations in the spurious solutions. The time domain solution is presented

Fig. 7. Eigenvectors of the 3D problem in Fig. 5 for first four modes on the plane z = 0.01. Top: correct solutions obtained using the E1H2 scheme. Bottom:spurious solutions with the closest frequencies to those above using the E1H1 scheme.

0 0.5 1 1.5 2 2. 5 3−8

−6

−4

−2

0

2

4

6

8x 104

time (ns)

Ez (V

/m)

SETD/FETD E1H2SETD/FETD E1H1Wavenology

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

x 108

Frequency (GHz)

FFT

Ez

SETD/FETD E1H2SETD/FETD E1H1WavenologyTheory

(a) (b)

Fig. 8. Ez signals at the receiver in the 3D problem in Fig. 5 obtained using the E1H1 and E1H2 schemes and their comparison with results from commercialsoftware Wavenology. (a) Time domain signals up to 3 ns (later signals are truncated for clarity) and (b) spectrum magnitude of the signals, where the E1H1and E1H2 results are shifted up by 2 � 10�8 and 4 � 10�8, respectively for clarity. Theoretical values of the resonance frequencies are also marked.

7310 L. Tobón et al. / Journal of Computational Physics 230 (2011) 7300–7310

in Fig. 8, with a spurious solution in dashed line produced by the E1H1 scheme, but a correct one (verified by commercialsoftware) when the E1H2 scheme is used. The presence of spurious solutions is also evident in the spectrum (dotted line) inFig. 8. A good agreement in both time and frequency domains is obtained in the E1H2 scheme.

4. Conclusions

From the 1D periodic boundary case and the 3D PEC cavity case, we observed spurious solutions when the same order ofinterpolation is used in E and H. These solutions can be noted in eigenvalues and eigenvectors, as the non-physical modeshave high values of wavenumber and thus rapid spatial variations. This behavior is presented in the time domain solution asa wave with rapid spatial variations with a high group velocity; however, it is difficult to detect such spurious solutions at asingle receiver because of their smooth variations with time. In the three-dimensional case, the spectra obtained using thesame-order basis functions show a large number of non-physical peaks from low to high frequencies. All these issues aresolved if non-spurious elements, i.e., different order basis functions, are used in E and H. Numerical solutions in 1D and3D verified the accuracy of these new basis functions in the finite-element time-domain solution of Maxwell’s equations.

Future research will be carried out to analyze the mathematical fundamentals of these non-spurious elements, and amixed continuous-discontinuous Galerkin method will be analyzed to allow different orders of elements in different subdo-mains within a multi-domain multiscale problem.

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