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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 5, MAY 2007 1359

Differential Forms, Galerkin Duality, and SparseInverse Approximations in Finite Element

Solutions of Maxwell EquationsBo He, Member, IEEE, and Fernando L. Teixeira, Senior Member, IEEE

Abstract—We identify primal and dual formulations in the finiteelement method (FEM) solution of the vector wave equation usinga geometric discretization based on differential forms. These twoformulations entail a mathematical duality denoted as Galerkinduality. Galerkin-dual FEM formulations yield identical nonzero(dynamical) eigenvalues (up to machine precision), but have static(zero eigenvalue) solution spaces of different dimensions. Alge-braic relationships among the degrees of freedom of primal anddual formulations are explained using a deep-rooted connectionbetween the Hodge–Helmholtz decomposition of differential formsand Descartes–Euler polyhedral formula, and verified numeri-cally. In order to tackle the fullness of dual formulation, algebraicand topological thresholdings are proposed to approximate inversemass matrices by sparse matrices.

Index Terms—Differential forms, finite element methods(FEMs), Maxwell equations, sparse matrices.

I. INTRODUCTION

THE finite element method (FEM) has been widely used tosolve Maxwell equations in complex geometries [1]–[5].

FEM is traditionally based upon seeking solutions by properlyweighting the residual of the second-order vector wave equa-tion, with stability and convergence issues being addressedby variational principles. Another route to derive stable FEMdiscretizations, first suggested by Bossavit and Kotiuga [6]–[8],and increasingly adopted in recent years [9]–[23], is basedon a representation of the electromagnetic field in terms ofdifferential forms [24]–[34]. In this geometric discretiza-tion approach, Whitney forms (elements) [6]–[9], [14], [35],[36] become interpolants for cochains (discrete differentialforms) [35] representing the discretized electromagnetic field.When the discrete field is associated with a 1-form and thesecond-order vector wave equation is used, this approach

Manuscript received June 20, 2006; revised October 3, 2006. This work wassupported in part by the AFOSR under MURI Grant FA 9550-04-1-0359, theNSF under CAREER Grant ECS-0347502, and in part by the OSC under GrantsPAS-0061 and PAS-0110.

B. He was with the ElectroScience Laboratory and the Department of Elec-trical and Computer Engineering, Ohio State University, Columbus, OH, 43210,USA. He is now with the Department of Electrical and Computer Engineering,University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail:[email protected]).

F. L. Teixeira is with the ElectroScience Laboratory and Department of Elec-trical and Computer Engineering, Ohio State University, Columbus, OH 43210USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAP.2007.895619

recovers the conventional FEM based upon edge elements.In other contexts, this approach recovers certain mixed FEMsdeveloped over the years to fulfill inf-sup stability conditions[6]. Differential forms bring some advantages by providingsystematic and unified discretization rules to obtain stable FEMschemes. Differential forms also facilitate the factorizationof FEM matrices into 1) topological components (dependingonly on the mesh connectivity and, hence, invariant underhomeomorphisms) that involve only integer arithmetic and 2)metric components (depending on element shapes) [14], [23].This factorization fits the general prescription for field theoriesexpressed by Tonti diagrams [37]–[39]. Discretizations basedon differential forms automatically fulfill a discrete version deRham diagram in an exact fashion [40], a necessary conditionto avoid problems such as spectral pollution by spurious modes.A recent comprehensive discussion on the relative merits ofdiscretizations based on differential forms versus traditionalapproaches can be found in [41].

This work is divided into two main parts. In the first part, weutilize a geometric discretization to identify two distinct FEMformulations based on the second-order vector wave equation.These dual FEM formulations involve either the electric fieldintensity (primal formulation) or the magnetic field inten-sity (dual formulation), and generalize the prior two-dimen-sional (2-D) analysis [23] to 3-D. The primal formulation re-covers the conventional FEM based on edge elements, while thedual formulation suggests a new type of FEM. The connectionbetween the two is established through a mathematical transfor-mation denoted as Galerkin duality [23]. The primal system ma-trix can be decomposed into a pair , that recoversthe conventional (primal) stiffness and mass matrices. The dualsystem matrix can be decomposed into another pair ,

, denoted as dual stiffness matrix and dual mass matrix,respectively. We verify that primal and dual FEM formulationsyield identical nonzero (dynamical) eigenvalues (up to numer-ical roundoff error), while the dimensions of their null spaces(zero eigenvalues or static solutions) are different. Both and

are sparse, while and are, in general, not.In the second part of this work, we investigate the dual FEM

system in more detail to show that and are quasi-sparse (in a sense to be made precise in Section IV), and in-troduce strategies to approximate them by sparse matrices. Fi-nally, we discuss how the combination of primal and approxi-mate sparse dual systems can be used to construct (conditionallystable) time-domain FEM updates which are both sparse and ex-plicit.

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1360 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 5, MAY 2007

II. PRIMAL AND DUAL FEM

A. Discrete Maxwell Equations in 3-D Simplicial Meshes

The (source-free) discrete Maxwell equations in a simplicialgrid can be cast as [14], [21]

(1)

with convention . , , , and are column vectors (ar-rays) for the degrees of freedom (DoFs) of the electric field in-tensity, magnetic flux density, magnetic field intensity, and elec-tric flux density, respectively [14], [21], [23]. In 3-D, each ele-ment of is associated with a primal grid edge, and similarlyfor , , on primal grid faces, dual grid edges, and dual gridfaces, respectively [14], [21]. Here, the primal grid refers to theFEM mesh itself. These DoF arrays represent the discrete coun-terparts of differential forms of various degrees representing theelectromagnetic field, viz., the (ordinary) 1-form , the (or-dinary) 2-form , the (twisted) 1-form , and the (twisted)2-form , [14] (this classification refers to the 3-D case; forother dimensions, see Table I) [14]. The matrices , ,

, are (sparse, non-square) incidence matrices thatdiscretize the exterior derivative , as detailed, e.g., in [14]. Thematrices and are discrete counterparts of the curland divergence operators on the primal grid, respectively, dis-tilled from their metric structure. Similarly, , arediscrete counterparts of the curl and divergence on the dualgrid. Since the incidence matrices are metric-free, their ele-ments assume only values. From the nilpotency ofthe exterior derivative, , we have the identities (exactsequence property) and full-filled by construction, which is important for energy and chargeconservation at the discrete level [14]. Moreover, from the du-ality pairing between geometric elements of the primal and dualgrids [14], and, in particular, between primal edges with dualfaces and vice-versa, we have (up to do-main boundary terms), where the superscript denotes transpose.Combining the above identities yields and

, where the transpose incidence matrices acton nodal DoFs (associated with discrete 0-forms) and can beidentified with (metric-free) gradient-like operators. At the dis-crete level, (1) has close analogy with the finite integration tech-nique (FIT) developed over the years by Weiland and colleagues[42].

The discrete version of constitutive equations is written as

(2)

where and are (discrete) Hodge star operators [10],[14], [15], [43], [44] represented by square invertible matrices.In 3-D, the Hodge operators in (2) establish an isomorphism

TABLE IDIFFERENTIAL FORMS DEGREES

between 1-forms and 2-forms. More generally, Hodge isomor-phisms map -forms to -forms, where is the number ofspatial dimensions. Explicitely manifest in the above is the fac-torization of Maxwell equations into a purely topological part(i.e., invariant under homeomorphisms)1 represented by (1), anda metric part represented by (2).2

B. Discrete Hodge Star Operator: 3-D Galerkin Hodges

In 3-D, we expand a differential -form , , interms of Whitney -forms , [14] as follows:

(3)

where are the degrees of freedom (complex numbers in theFourier domain), and the sum run over internal (free) nodes for

, internal edges for , internal faces for ,and volume cells for . A discrete (matrix) representationfor the Hodge star operator in an Euclidean 3-D manifold ,mapping a -form in the space spanned by Whitney -forms to a

-form, is obtained by the following (Galerkin) projection[23]:

(4)

where is the exterior product of differential forms. The aboverepresent volume integrals for the various . In terms of vectorproxies, we have for and

(5)

respectively. In the above, , , areWhitney edge elements and , , areWhitney face elements [9], [23], where and are thenumber of internal edges and faces, respectively, of the FEMmesh. The domain is the union of the (compact) sup-ports of and . Note that Whitney elements are defined interms of the primal grid only. Primal grid arrays such as and

1In the continuum, the invariance of the topological part is under diffeomor-phisms.

2This factorization is also explored, albeit in a less explicit fashion, bymimetic finite-difference methods [45].

HE AND TEIXEIRA: DIFFERENTIAL FORMS, GALERKIN DUALITY, AND SPARSE INVERSE APPROXIMATIONS 1361

(ordinary forms) are in the domain of the and ,whereas dual grid arrays such as and (twisted forms) are inthe image of and . From the above, the matrices in(2) write as [9], [23], [43], [44]

(6)

where and are the (isotropic) permittivity and permeability,respectively (both assumed uniform and strictly positive overeach element). The matrices in (6) are referred to as Galerkindiscrete Hodges [15], [23], [44] and are symmetric positive-def-inite (SPD) by construction.

C. Galerkin-Dual FEM Formulations in 3-D

From (1) and (2), two discrete vector wave equations can beobtained

(7)

(8)

corresponding to primal and dual formulation, respectively.These are discrete analogues of the vector wave equations

(9)

(10)

This analogy does not mean, however, that (7) and (8) are bothedge element discretizations of (9) and (10), respectively. In-deed, (7) is the edge element discretization of (9) on the FEMmesh (primal grid) so that can be expanded in terms of edgeelements with coefficients in . On the other hand, (8) cor-responds to the discretization of (10) on the dual grid and noton the FEM mesh [14] (cf. Table I), and cannot be expandedin terms of . Since there is a one-to-one correspondence be-tween and via (2) (Hodge isomorphism), where is asso-ciated with FEM mesh faces (DoFs of a 2-form), the DoFs in(8) can be (indirectly) associated only with faces of the FEMmesh. In this sense, can be expanded in terms of face ele-ments from coefficients in by solving the linear system

in terms of . Note that this expansion carries anadditional source of error stemming from the Galerkin projec-tion (discretization of the Hodge operator) in (4). Note also thatby combining (2) and (8), one arrives at

(11)

where the DoFs of associate with primal mesh faces andis expanded in terms of with coefficients in . This latterexpansion carries no (additional) Galerkin projection error.

The matrix is identical to the conven-tional stiffness matrix by FEM using edge elements [23],given by

(12)

Moreover, is identical to the conventional mass matrix .Hence, the primal formulation recovers the conventional edge-element FEM with and ,and suggests a geometric foundation for it. For the dual formula-tion, we can introduce dual stiffness and mass ma-

trices as and .These matrices have no direct counterpart in conventional FEM.We can further define system matrices and

for the primal and dual eigenvalue prob-lems in (7) and (8), respectively.

D. Low-dimensional Cases

The above discussion has been focused the 3-D case only.For completeness, Table I lists the degrees of the differentialforms representing , , , , as well as the electric po-tential , the electric charge density , and the electric cur-rent density , for low dimensional cases and various polar-izations. The association between DoFs of discrete differentialforms of various degrees and grid elements (nodes, edges, faces,and volumes) is also provided for reference.3 Such classificationtable provides general design rules for representing the variousfields and sources in terms of the appropriate type of finite el-ements (nodal, edge, face, or volume elements [14]), on eitherthe primal or dual grid. An example of use of this classificationtable to construct consistent mixed FEM in 1-D, 2-D, and 3-Dis provided in [46].

III. GALERKIN DUALITY: 3-D CASE

The primal and dual formulations above are connectedthrough Galerkin duality [23]. The nonzero eigenvalues (as-sociated with the dynamical solutions or the range space of

and ) is invariant under Galerkin duality. However,the same is not true for the null space. This was verifiedpreviously in 2-D [23]. In this Section, we present examplesto verify this in 3-D. Note that Galerkin duality is not to beconfused with conventional electromagnetic duality [47]. Theformer establishes two distinct mathematical formulationsfor the same physical problem, whereas the latter providesthe same mathematical formulation for two distinct physicalproblems. Galerkin duality is also distinct from other kinds ofduality [7], [48], [49] that arise in variational FEM formula-tions. In terms of the boundary conditions, Galerkin dualitytransforms Dirichlet boundary conditions into Neumann, andNeumann into Dirichlet. Note that a pure Dirichlet boundayvalue problem may be easier to solve numerically than a pureNeumann problem [50].

3In concordance with the choice made earlier, we assign ordinary forms to theprimal grid and twisted forms to the dual grid. More generally, the reciprocalchoice could have been made as well.


TABLE IIEIGENFREQUENCIES OF SPHERICAL CAVITY

TABLE IIIEIGENFREQUENCIES OF INHOMOGENEOUS CYLINDRICAL CAVITY

Fig. 1. Cylindrical cavity with dimensions a = 1, b = 0:1, and d = 0:3.

In the following, the eigenfrequencies of resonant modesare computed by solving eigenvalue (7) and (8).

A. Numerical Examples: Eigenfrequencies

Table II presents numerical results for eigenfrequencies ofa 3-D spherical cavity using both primal and dual FEM for-mulations. For simplicity, we set radius , and materialparameters . The tetrahedral 3-D FEM mesh iscomposed of 94 nodes, 122 boundary faces, and 326 tetrahedra.The analytical solutions for and modes have a

-fold degeneracy for fixed and , but the numericalsolutions break the degeneracy due to the mesh asymmetry.

Table III presents the eigenfrequencies of the dominantmodes for a 3-D inhomogeneous cylindrical cavity, as illus-trated in Fig. 1, using both primal and dual FEM formulations.Also listed are the total number of zero and nonzero eigenvaluesolutions. In what follows, we denote these as zero and nonzeromodes, respectively. We again set , and usedifferent values for and in the material region as indicatedin Table III. The 3-D FEM mesh for this cylindrical cavity has69 nodes, 118 boundary faces, and 174 tetrahedra.

TABLE IVNUMBER OF MODES FROM NUMERICAL DATA

TABLE VNULL SPACE AND RANGE SPACE OF [X ] AND [X ]

B. Discrete DoF and Hodge–Helmholtz Decomposition

From Tables II and III, we find that the number of zero modesof the primal formulation is equal to the number of internalnodes, , while the number of zero modes of dual formulationis equal to the number of tetrahedral cells minus one, .Moreover, the last rows of Tables II and III show that both primaland dual formulations yield identical number of nonzero modes.These identities can be verified true for any tetrahedral mesh,and are summarized in Table IV. They are a consequence of thediscrete Hodge–Helmholtz decomposition [21], [28], which in3-D for the electric field intensity 1-form write as

(13)

where is a 0-form, is a 2-form, is a harmonic 1-form,and is the codifferential operator (pre-Hilbert adjoint of ) [28]. In a contractible domain, is iden-tically zero. For Maxwell equations, in (13) represents thestatic electric field and represents the dynamic electric field.In the vector calculus framework, the above recovers

.By identifying the FEM mesh as a network of polyhedra, the

3-D Descartes-Euler polyhedral formula [51] states thatfor any 3-D FEM mesh, where is

the total number of vertices (nodes), the total number ofedges, the total number of faces, and the total numberof volume cells (tetrahedra) in the mesh. Applying a 2-D variantof the same theorem for the closed 2-D boundary of the mesh,we have , where the subscript refersto boundary elements. Using the fact that ,

, and , we arrive at. The latter relation can be paired with

the discrete DoFs in the Hodge–Helmholtz decomposition in thefollowing fashion [21]

(14)

The l.h.s. of (14) corresponds to the (dimension of the) rangespace of , while the r.h.s. corresponds to the range space of

. Furthermore, corresponds to the null space ofand corresponds to the null space of . These re-sults are summarized in Table V, which exactly matches the nu-merical results in Table IV. The number of rows of and

equal the total (static plus dynamic) number of DoF of


primal and dual formulations, respectively. The DoF in the nullspace of and represent the zero (static) modes ofprimal formulation and dual formulations, respectively. Further-more, the DoF in the range space of and representthe nonzero (dynamic) modes of primal and dual formulations,respectively. For the primal formulation, it is a well known factthat the number of zero modes (null space dimension) ofequals the number of internal nodes of the FEM mesh [21], [52].Recall also that zero modes are solutions of the eigenvalue prob-lems in (5) and (6) for , but are not (divergence-free) so-lutions of (1).

IV. SPARSE APPROXIMATE INVERSE MASS MATRICES

Although the dual formulation yields the same nonzero eigen-values as the primal formulation, and do not retainthe sparse nature of and . This is rooted on the lack ofsparsity of the inverse mass matrices and . We

next discuss that and are quasi-sparse, i.e., theycan be well approximated by sparse matrices due to their stronglocalization properties.

A. Strong Localization Property

To illustrate the localization properties of , we define,for each edge , a vector field given by

(15)

where the subscripts , represent edge indexing. By construc-tion, the function is such that the integral

(16)

is equal to one for and zero otherwise. Sinceis full, is non-zero over the entire domain , in general.However, does not exhibit inherent long-range interac-tions, cf. Section IV-E below. The lack of (exact) sparsity of

is simply a consequence of the lack of orthogonality be-tween edges in a simplicial FEM mesh. As a result, the elements

are relatively very small unless edges and are inclose proximity. In other words, is strongly localized aroundedge . Strong localization can be illustrated by plotting, in a logscale, the magnitude of the for different edges of a 2-D FEMmesh, as in Figs. 2 and 3. An identical analysis can be done for

in terms of the face elements (Whitney 2-forms) onthe grid.

B. Algebraic Thresholding

Since most of its elements are relatively very small, the ma-trix can be well approximated by a sparse matrix .This can be done, for example, by algebraic thresholding. Inthis case, a parameter is chosen such that if the ratio of theabsolute value of an element of to the maximum abso-lute value of its diagonal entries is below , then the element

Fig. 2. Plot of log (j~� j) for an edge � (indicated by the bold segment ofthe mesh in the inset) near the center of a 2-D TE circular cavity, showing thestrong localization property of ~� .

Fig. 3. Plot of log (j~� j) for edge � (indicated by the bold segment of themesh in the inset) near the boundary of a 2-D TE circular cavity, again showingthe strong localization property of ~� .

of is set to zero. Otherwise, the element of is setequal to the corresponding element of . The threshold isin the range , whereand are the minimum and maximum absolute valuesof diagonal entries of . A similar procedure can be ap-plied for . Algebraic thresholding helps in ver-ifying the quasi-sparse nature of , but relies on explicitknowledge of . A more practical strategy to obtainthat does not require explicit knowledge of is to use atopological thresholding, as discussed ahead. But we first ex-amine the tradeoff between sparsity and sparsification error inSection IV-C.

C. Tradeoff Between Sparsity and Sparsification Error

1) Eigenvalues of Inverse Mass Matrices: We compare theeigenvalues of and with those of and


Fig. 4. Sparsity pattern of [? ] , with r = 0:005.

Fig. 5. Sparsity pattern of ? , with r = 0:005.

to quantify the sparsification error. Consider the samespherical cavity and FEM mesh of Section III-A. For amatrix , the density is defined as ,with the number of nonzero elements. The sparsity pat-terns of and for are depicted in

Figs. 4 and 5, respectively, with and

. Let be an eigenvalue of a

matrix . The relative error of an eigenvalue is defined as

where the superscript stands for relative. We plotin Fig. 6 and

Fig. 6. Relative sparsification errors on the eigenvalues of [? ] .

Fig. 7. Relative sparsification errors on the eigenvalues of ? .

in Fig. 7 for all eigenvalues. In this case, the relative errorsare consistently below 1.8% for all eigenvalues of andbelow 1.3% for all eigenvalues of . with .

2) Eigenvalues of Dual System Matrices: Letbe the eigenvalue of original dual system,be the eigenvalue of dual system after sparse approximationmatrix, and be the exact eigenvalue. We define the trun-cation (discretization) error , thesparsification error ,and the total error . It is easyto show that . If is chosen such that ,then and the total error is bounded by trun-cation error . Because the eigenvalues can vary much inmagnitude, a normalization is convenient. In this case, rel-ative errors are defined as ,

, and.

Fig. 8 shows the relative errors for the spherical cavityproblem with . For visualization purposes, only thelowest 23 modes are shown. We observe that the sparsificationerror is less than the truncation error for all modes with thischoice of . The total error may be smaller than truncation error


Fig. 8. Relative truncation, sparsification, and total errors on FEM eigenvalues.

Fig. 9. Two-dimensional illustration of the neighbor edge classification by dif-ferent k-levels of topological thresholding. The reference edge is 12 (indicatedin bold). Level-0 edge: 12. Level-1 edges: level-0 edge plus edges 13, 14, 23,24. Level-2 edges: level-0 edge plus level-1 edges plus edges 16, 17, 25, 28, 35,36, 47, 48.

because the differences andmay have opposite signs.

D. Topological Thresholding

The strong localization property illustrated in Figs. 2 and 3suggests that only closeby edges have significant coupling, andthe coupling decays very quickly with the distance betweenedges. For each edge, one can define various neighbor levelsusing, for example, mesh connectivity (topological) criteria[53], [54]. We define a level- neighbor ina 2-D triangular mesh as follows (similar definitions applyfor 3-D, and for DoF defined on nodes, faces, or tetrahedra):For each edge , level-0 neighbor include only edge itself.Level-1 neighbors include edge and the four (nearest neigh-bors) edges belonging to the two triangles that share edge .Level-2 neighbors include all level-1 edges plus the edges in theneighboring triangles, and so forth for . This is illustratedin Fig. 9.

By retaining only interactions among level- neighbors foreach edge, one obtains a sparse approximate inverse mass ma-trix . From (3), is a sparse matrix that includes onlylevel-1 coupling. As a result, the sparsity pattern of is

equal to that of the th power of , i.e., . For example,is diagonal, whereas the sparsity pattern of is the

Fig. 10. Two FEM meshes for a circular cavity. Mesh (a) has 41 nodes and 64triangles. Mesh (b) has 178 nodes and 312 triangles.

Fig. 11. Sparsity pattern of matrix [? ] for the coarse mesh in Fig. 9 bytopological thresholding. (a) k = 0, (b) k = 1, (c) k = 2, (d) and k = 3.

same as the sparsity pattern of . This is important in prac-tice because can be obtained without the need to obtain

. In particular, can be obtained by minimizing theEuclidean (Frobenius) norm of the difference ,where has a prescribed sparsity pattern. This minimiza-tion problem decouples into local and independent least-squareprocedures that are naturally parallelizable [55].

To illustrate topological thresholding, we present resultsfrom two FEM meshes for a 2-D TE circular cavity as depictedin Fig. 10. The resulting sparsity patterns from various level-topological thresholdings for the coarse mesh are shown inFig. 11. Fig. 12 shows the relative sparsification errors for theTE eigenvalues (together with the relative truncation errors).Figs. 13 and 14 repeat the same for the finer mesh. In bothcases, level-2 topological thresholdings already work very well,with the sparsification error consistently below the truncationerror. The numerical results illustrate the increase of the trun-cation error with frequency, as expected. The sparsificationerror shows no such visible trend. Note that in order to fullyexamine the sparsification error, we plot the entire discreteeigenvalue spectra in these figures. Of course, only the lowerend of the discrete spectra provides a good approximation ofthe continuum spectra.


Fig. 12. Relative sparsification error for each eigenvalue, using level-k topo-logical thresholding (star) and relative truncation error (diamond). (a) k = 0,(b) k = 1, (c) k = 2, and (d) k = 3.

Fig. 13. Sparsity pattern of [? ] for the fine mesh in Fig. 9 from level-ktopological thresholding. (a) k = 0, (b) k = 1, (c) k = 2, and (d) k = 3.

E. Relation With SPAI Preconditioners

The sparsification described above mirrors the strategy usedby sparse approximate inverse (SPAI) preconditioners [55].However, a fundamental difference here is that SPAI precon-ditioners are most often used to approximate the inverse of(discrete) differential operators (or, sometimes, integral opera-tors), i.e., Green’s functions with long range interactions. By

Fig. 14. Relative sparsification error using level-k topological thresholding(stars) versus relative truncation error (diamonds). (a) k = 0, (b) k = 1, (c)k = 2, and (d) k = 3.

contrast, sparse approximations, as applied here, approximateinverse Hodge star operators, which are local operators in thecontinuum limit. In other words, and do nothave inherent long range interactions that would remain presentin the continuum limit.

F. Application to Time-Domain FEM

A major factor limiting the efficiency of the FEM in thetime-domain is that mass matrices are non-diagonal. As aresult, a sparse linear system solution is required at each timestep. This becomes a major bottleneck of the computationaleffort and is in contrast to the finite-difference time-domain(FDTD) method, for example, which is a matrix-free explicitscheme.4 One strategy used to overcome this problem is masslumping [56], whereby the mass matrix is reduced to a diagonalor block-diagonal matrix. However, mass lumping often resultsin non positive definite matrices, leading to (unconditional)instabilities that destroy the solutions [57]. Another strategy isbased upon the use of orthogonal basis functions [58], however,it requires three times more DoF.

To overcome this limitations, sparse and explicit, time-do-main FEMs based on the sparse approximation approaches ex-plained here have been described in [54] (topological) and [59](algebraic). Combining (1) and (2), we arrive at

(17)

4Of course, implicit time-domain schemes that rely on sparse matrix solverscan be, in cases such as highly refined grids, more advantageous than matrix-freeexplicit ones because the former can provide unconditionally stability.


where matrices , , , and are all sparse.By inverting and inverse Fourier transforming the aboveequation, one has

(18)

A leap-frog time discretization of the above leads to an explicittime-domain update that unfortunately is full becauseis full. However, by approximating by , an ex-plicit and sparse update scheme (FDTD-like) is obtained. Usingtopological thresholding, the resulting algorithm is highly par-allelizable, both for obtaining from and for thetime-domain update itself. Since the eigenvalues of areonly slightly perturbed, remains SPD and the scheme isconditionally stable [14], [57]. 5

From a finite-difference standpoint, the above provides astrategy to derive stable and consistent—hence convergent[1]—finite-difference stencils in irregular simplicial meshes.This is contrast to some previous explicit time-domain schemesbased on Whitney elements [61]–[63] that have (proceduralequivalents of) discrete Hodge operators which are not neces-sarily SPD by construction. The above framework recognizesthe need for two distinct finite-difference representations ofthe curl operators appearing in Maxwell curl equations, viz.,

and , for a discretization based upona single mesh (as opposed to obe based upon dual meshessuch as Yee’s scheme), a property also shared by mimeticfinite-difference methods [45].

V. SUMMARY AND CONCLUSION

Galerkin duality has been discussed in connection withdiscrete FEM solutions of Maxwell equations in 3-D, andverified in examples involving both homogeneous and inho-mogeneous media. Basic algebraic features of the primal anddual discrete solutions were explained by means of the discreteHodge–Helmholtz decomposition and Descartes–Euler polyhe-dral formula. It was verified that primal and dual formulationsyield identical nonzero eigenvalues while having null spaces (ofstatic eigenfunctions) of different dimensions. The quasi-sparsenature of the inverse mass matrices and arisingin the dual formulation was discussed and explained in termsof the strong localization property. Two different approaches toapproximate the inverse mass matrices by sparse matrices werediscussed: algebraic thresholding and topological thresholding.The former is of mainly theoretical interest only, while thelatter can be used to obtain sparse approximate inverses withoutthe need for direct inversion. We have discussed how to controlthe (sparsification) error of sparse dual FEM solutions. Finally,we discussed how these results can be applied to construct(conditionally stable) sparse and explicit FEM in time-domain.

5Other solvers can also be quite efficient in obtaining sparse apporximationsfor [? ] . For example, since [? ] is SPD, an incomplete Cholesky factoriza-tion of is particularly attractive for implementation in serial computers [60].

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Bo He (S’03–M’06) was born in Changzhi, China, in 1969. He received the B.S.degree from Shanxi University, China, in 1991, the M.S. degree from East ChinaNormal University, in 1994, the Ph.D. degree in physics from the Universityof Missouri, Rolla/St. Louis, in 2002, and the Ph.D. degree in electrical andcomputer engineering from Ohio State University, Columbus, in 2006.

From 1994 to 1999, he was a faculty member with the Shanghai Jiao TongUniversity. Since 2006, he has been a Postdoctoral Research Associate at theCenter for Computational Electromagnetics, University of Illinois at Urbana-Champaign. His research interests include theoretical particle physics and com-putational electromagnetics.

Fernando L. Teixeira (S’89–M’93–SM’05) received the B.S. and M.S. de-grees in electrical engineering from the Pontifical Catholic University of Rio deJaneiro (PUC-Rio), Brazil, in 1991 and 1995, respectively, and the Ph.D. degreein electrical engineering from the University of Illinois at Urbana-Champaign,in 1999.

From 1999 to 2000, he was a Postdoctoral Research Associate with the Re-search Laboratory of Electronics, Massachusetts Institute of Technology (MIT),Cambridge. Since 2000, he has been with the Department of Electrical and Com-puter Engineering and the ElectroScience Laboratory, The Ohio State University(OSU), where he is now an Associate Professor. His current research interestsinclude modeling of wave propagation, scattering, and transport phenomena forcommunications, sensing, and device applications. He has edited one book Geo-metric Methods for Computational Electromagnetics (Cambridge, MA: PIER32 EMW, 2001) and has authored over 70 journals articles and book chapters.

Dr. Teixeira is a Member of Phi Kappa Phi, Sigma Xi, and an Elected Memberof the International Scientific Radio Union (URSI) Commission B. He wasawarded many prizes for his research including the including the Lumley Re-search Award from the OSU in 2003, the CAREER Award from the NSF in 2004and the triennial Henry Booker Fellowship from USNC/URSI in 2005.

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