THE finite element method (FEM) has been widely used to

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1 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 5, MAY Differential Forms, Galerkin Duality, Sparse Inverse Approximations in Finite Element Solutions of Maxwell Equations Bo He, Member, IEEE, Ferno L. Teixeira, Senior Member, IEEE Abstract We identify primal dual formulations in the finite element method (FEM) solution of the vector wave equation using a geometric discretization based on differential forms. These two formulations entail a mathematical duality denoted as Galerkin duality. Galerkin-dual FEM formulations yield identical nonzero (dynamical) eigenvalues (up to machine precision), but have static (zero eigenvalue) solution spaces of different dimensions. Algebraic relationships among the degrees of freedom of primal dual formulations are explained using a deep-rooted connection between the Hodge Helmholtz decomposition of differential forms Descartes Euler polyhedral formula, verified numerically. In order to tackle the fullness of dual formulation, algebraic topological thresholdings are proposed to approximate inverse mass 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 to solve Maxwell equations in complex geometries [1] [5]. FEM is traditionally based upon seeking solutions by properly weighting the residual of the second-order vector wave equation, with stability convergence issues being addressed by variational principles. Another route to derive stable FEM discretizations, first suggested by Bossavit Kotiuga [6] [8], increasingly adopted in recent years [9] [23], is based on a representation of the electromagnetic field in terms of differential forms [24] [34]. In this geometric discretization approach, Whitney forms (elements) [6] [9], [14], [35], [36] become interpolants for cochains (discrete differential forms) [35] representing the discretized electromagnetic field. When the discrete field is associated with a 1-form the second-order vector wave equation is used, this approach Manuscript received June 20, 2006; revised October 3, This work was supported in part by the AFOSR under MURI Grant FA , the NSF under CAREER Grant ECS , in part by the OSC under Grants PAS-0061 PAS B. He was with the ElectroScience Laboratory the Department of Electrical Computer Engineering, Ohio State University, Columbus, OH, 43210, USA. He is now with the Department of Electrical Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL USA ( bohe@uiuc.edu). F. L. Teixeira is with the ElectroScience Laboratory Department of Electrical Computer Engineering, Ohio State University, Columbus, OH USA ( teixeira.5@osu.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TAP recovers the conventional FEM based upon edge elements. In other contexts, this approach recovers certain mixed FEMs developed over the years to fulfill inf-sup stability conditions [6]. Differential forms bring some advantages by providing systematic unified discretization rules to obtain stable FEM schemes. Differential forms also facilitate the factorization of FEM matrices into 1) topological components (depending only on the mesh connectivity, hence, invariant under homeomorphisms) that involve only integer arithmetic 2) metric components (depending on element shapes) [14], [23]. This factorization fits the general prescription for field theories expressed by Tonti diagrams [37] [39]. Discretizations based on differential forms automatically fulfill a discrete version de Rham diagram in an exact fashion [40], a necessary condition to avoid problems such as spectral pollution by spurious modes. A recent comprehensive discussion on the relative merits of discretizations based on differential forms versus traditional approaches can be found in [41]. This work is divided into two main parts. In the first part, we utilize a geometric discretization to identify two distinct FEM formulations based on the second-order vector wave equation. These dual FEM formulations involve either the electric field intensity (primal formulation) or the magnetic field intensity (dual formulation), generalize the prior two-dimensional (2-D) analysis [23] to 3-D. The primal formulation recovers the conventional FEM based on edge elements, while the dual formulation suggests a new type of FEM. The connection between the two is established through a mathematical transformation denoted as Galerkin duality [23]. The primal system matrix can be decomposed into a pair, that recovers the conventional (primal) stiffness mass matrices. The dual system matrix can be decomposed into another pair,, denoted as dual stiffness matrix dual mass matrix, respectively. We verify that primal dual FEM formulations yield identical nonzero (dynamical) eigenvalues (up to numerical roundoff error), while the dimensions of their null spaces (zero eigenvalues or static solutions) are different. Both are sparse, while are, in general, not. In the second part of this work, we investigate the dual FEM system in more detail to show that are quasisparse (in a sense to be made precise in Section IV), introduce strategies to approximate them by sparse matrices. Finally, we discuss how the combination of primal approximate sparse dual systems can be used to construct (conditionally stable) time-domain FEM updates which are both sparse explicit X/$ IEEE

2 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 simplicial grid can be cast as [14], [21] TABLE I DIFFERENTIAL FORMS DEGREES with convention.,,, are column vectors (arrays) for the degrees of freedom (DoFs) of the electric field intensity, magnetic flux density, magnetic field intensity, electric flux density, respectively [14], [21], [23]. In 3-D, each element of is associated with a primal grid edge, similarly for,, on primal grid faces, dual grid edges, dual grid faces, respectively [14], [21]. Here, the primal grid refers to the FEM mesh itself. These DoF arrays represent the discrete counterparts of differential forms of various degrees representing the electromagnetic field, viz., the (ordinary) 1-form, the (ordinary) 2-form, the (twisted) 1-form, the (twisted) 2-form, [14] (this classification refers to the 3-D case; for other dimensions, see Table I) [14]. The matrices,,, are (sparse, non-square) incidence matrices that discretize the exterior derivative, as detailed, e.g., in [14]. The matrices are discrete counterparts of the curl divergence operators on the primal grid, respectively, distilled from their metric structure. Similarly,, are discrete counterparts of the curl divergence on the dual grid. Since the incidence matrices are metric-free, their elements assume only values. From the nilpotency of the exterior derivative,, we have the identities (exact sequence property) fullfilled by construction, which is important for energy charge conservation at the discrete level [14]. Moreover, from the duality pairing between geometric elements of the primal dual grids [14],, in particular, between primal edges with dual faces vice-versa, we have (up to domain boundary terms), where the superscript denotes transpose. Combining the above identities yields, where the transpose incidence matrices act on nodal DoFs (associated with discrete 0-forms) can be identified with (metric-free) gradient-like operators. At the discrete level, (1) has close analogy with the finite integration technique (FIT) developed over the years by Weil colleagues [42]. The discrete version of constitutive equations is written as where 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 (1) (2) between 1-forms 2-forms. More generally, Hodge isomorphisms map -forms to -forms, where is the number of spatial dimensions. Explicitely manifest in the above is the factorization of Maxwell equations into a purely topological part (i.e., invariant under homeomorphisms) 1 represented by (1), a metric part represented by (2). 2 B. Discrete Hodge Star Operator: 3-D Galerkin Hodges In 3-D, we exp a differential -form,, in terms of Whitney -forms, [14] as follows: where are the degrees of freedom (complex numbers in the Fourier domain), the sum run over internal (free) nodes for, internal edges for, internal faces for, volume cells for. A discrete (matrix) representation for 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]: where is the exterior product of differential forms. The above represent volume integrals for the various. In terms of vector proxies, we have for respectively. In the above,,, are Whitney edge elements,, are Whitney face elements [9], [23], where are the number of internal edges faces, respectively, of the FEM mesh. The domain is the union of the (compact) supports of. Note that Whitney elements are defined in terms of the primal grid only. Primal grid arrays such as 1 In the continuum, the invariance of the topological part is under diffeomorphisms. 2 This factorization is also explored, albeit in a less explicit fashion, by mimetic finite-difference methods [45]. (3) (4) (5)

3 HE AND TEIXEIRA: DIFFERENTIAL FORMS, GALERKIN DUALITY, AND SPARSE INVERSE APPROXIMATIONS 1361 (ordinary forms) are in the domain of the, whereas dual grid arrays such as (twisted forms) are in the image of. From the above, the matrices in (2) write as [9], [23], [43], [44] The matrix is identical to the conventional stiffness matrix by FEM using edge elements [23], given by (12) where are the (isotropic) permittivity permeability, respectively (both assumed uniform strictly positive over each element). The matrices in (6) are referred to as Galerkin discrete Hodges [15], [23], [44] are symmetric positive-definite (SPD) by construction. C. Galerkin-Dual FEM Formulations in 3-D From (1) (2), two discrete vector wave equations can be obtained corresponding to primal dual formulation, respectively. These are discrete analogues of the vector wave equations (6) (7) (8) (9) (10) This analogy does not mean, however, that (7) (8) are both edge element discretizations of (9) (10), respectively. Indeed, (7) is the edge element discretization of (9) on the FEM mesh (primal grid) so that can be exped in terms of edge elements with coefficients in. On the other h, (8) corresponds to the discretization of (10) on the dual grid not on the FEM mesh [14] (cf. Table I), cannot be exped in terms of. Since there is a one-to-one correspondence between via (2) (Hodge isomorphism), where is associated with FEM mesh faces (DoFs of a 2-form), the DoFs in (8) can be (indirectly) associated only with faces of the FEM mesh. In this sense, can be exped in terms of face elements from coefficients in by solving the linear system in terms of. Note that this expansion carries an additional source of error stemming from the Galerkin projection (discretization of the Hodge operator) in (4). Note also that by combining (2) (8), one arrives at (11) where the DoFs of associate with primal mesh faces is exped in terms of with coefficients in. This latter expansion carries no (additional) Galerkin projection error. Moreover, is identical to the conventional mass matrix. Hence, the primal formulation recovers the conventional edgeelement FEM with, suggests a geometric foundation for it. For the dual formulation, we can introduce dual stiffness mass matrices as. These matrices have no direct counterpart in conventional FEM. We can further define system matrices for the primal dual eigenvalue problems in (7) (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 differential forms representing,,,, as well as the electric potential, the electric charge density, the electric current density, for low dimensional cases various polarizations. The association between DoFs of discrete differential forms of various degrees grid elements (nodes, edges, faces, volumes) is also provided for reference. 3 Such classification table provides general design rules for representing the various fields sources in terms of the appropriate type of finite elements (nodal, edge, face, or volume elements [14]), on either the primal or dual grid. An example of use of this classification table to construct consistent mixed FEM in 1-D, 2-D, 3-D is provided in [46]. III. GALERKIN DUALITY: 3-D CASE The primal dual formulations above are connected through Galerkin duality [23]. The nonzero eigenvalues (associated with the dynamical solutions or the range space of ) is invariant under Galerkin duality. However, the same is not true for the null space. This was verified previously in 2-D [23]. In this Section, we present examples to verify this in 3-D. Note that Galerkin duality is not to be confused with conventional electromagnetic duality [47]. The former establishes two distinct mathematical formulations for the same physical problem, whereas the latter provides the same mathematical formulation for two distinct physical problems. Galerkin duality is also distinct from other kinds of duality [7], [48], [49] that arise in variational FEM formulations. In terms of the boundary conditions, Galerkin duality transforms Dirichlet boundary conditions into Neumann, Neumann into Dirichlet. Note that a pure Dirichlet bounday value problem may be easier to solve numerically than a pure Neumann problem [50]. 3 In concordance with the choice made earlier, we assign ordinary forms to the primal grid twisted forms to the dual grid. More generally, the reciprocal choice could have been made as well.

4 1362 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 5, MAY 2007 TABLE II EIGENFREQUENCIES OF SPHERICAL CAVITY TABLE IV NUMBER OF MODES FROM NUMERICAL DATA TABLE V NULL SPACE AND RANGE SPACE OF [X ] AND [X ] TABLE III EIGENFREQUENCIES OF INHOMOGENEOUS CYLINDRICAL CAVITY B. Discrete DoF Hodge Helmholtz Decomposition From Tables II III, we find that the number of zero modes of the primal formulation is equal to the number of internal nodes,, while the number of zero modes of dual formulation is equal to the number of tetrahedral cells minus one,. Moreover, the last rows of Tables II III show that both primal dual formulations yield identical number of nonzero modes. These identities can be verified true for any tetrahedral mesh, are summarized in Table IV. They are a consequence of the discrete Hodge Helmholtz decomposition [21], [28], which in 3-D for the electric field intensity 1-form write as (13) Fig. 1. Cylindrical cavity with dimensions a =1, b =0:1, d =0:3. In the following, the eigenfrequencies of resonant modes are computed by solving eigenvalue (7) (8). A. Numerical Examples: Eigenfrequencies Table II presents numerical results for eigenfrequencies of a 3-D spherical cavity using both primal dual FEM formulations. For simplicity, we set radius, material parameters. The tetrahedral 3-D FEM mesh is composed of 94 nodes, 122 boundary faces, 326 tetrahedra. The analytical solutions for modes have a -fold degeneracy for fixed, but the numerical solutions break the degeneracy due to the mesh asymmetry. Table III presents the eigenfrequencies of the dominant modes for a 3-D inhomogeneous cylindrical cavity, as illustrated in Fig. 1, using both primal dual FEM formulations. Also listed are the total number of zero nonzero eigenvalue solutions. In what follows, we denote these as zero nonzero modes, respectively. We again set, use different values for in the material region as indicated in Table III. The 3-D FEM mesh for this cylindrical cavity has 69 nodes, 118 boundary faces, 174 tetrahedra. where is a 0-form, is a 2-form, is a harmonic 1-form, is the codifferential operator (pre- Hilbert adjoint of ) [28]. In a contractible domain, is identically zero. For Maxwell equations, in (13) represents the static electric field 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 that for any 3-D FEM mesh, where is the total number of vertices (nodes), the total number of edges, the total number of faces, the total number of volume cells (tetrahedra) in the mesh. Applying a 2-D variant of the same theorem for the closed 2-D boundary of the mesh, we have, where the subscript refers to boundary elements. Using the fact that,,, we arrive at. The latter relation can be paired with the discrete DoFs in the Hodge Helmholtz decomposition in the following fashion [21] (14) The l.h.s. of (14) corresponds to the (dimension of the) range space of, while the r.h.s. corresponds to the range space of. Furthermore, corresponds to the null space of corresponds to the null space of. These results are summarized in Table V, which exactly matches the numerical results in Table IV. The number of rows of equal the total (static plus dynamic) number of DoF of

5 HE AND TEIXEIRA: DIFFERENTIAL FORMS, GALERKIN DUALITY, AND SPARSE INVERSE APPROXIMATIONS 1363 primal dual formulations, respectively. The DoF in the null space of represent the zero (static) modes of primal formulation dual formulations, respectively. Furthermore, the DoF in the range space of represent the nonzero (dynamic) modes of primal dual formulations, respectively. For the primal formulation, it is a well known fact that the number of zero modes (null space dimension) of equals the number of internal nodes of the FEM mesh [21], [52]. Recall also that zero modes are solutions of the eigenvalue problems in (5) (6) for, but are not (divergence-free) solutions of (1). IV. SPARSE APPROXIMATE INVERSE MASS MATRICES Although the dual formulation yields the same nonzero eigenvalues as the primal formulation, do not retain the sparse nature of. This is rooted on the lack of sparsity of the inverse mass matrices.we next discuss that are quasi-sparse, i.e., they can be well approximated by sparse matrices due to their strong localization properties. Fig. 2. Plot of log (j~ j) for an edge (indicated by the bold segment of the mesh in the inset) near the center of a 2-D TE circular cavity, showing the strong localization property of ~. A. Strong Localization Property To illustrate the localization properties of for each edge, a vector field given by,we define, (15) where the subscripts, represent edge indexing. By construction, the function is such that the integral (16) is equal to one for zero otherwise. Since is full, is non-zero over the entire domain, in general. However, does not exhibit inherent long-range interactions, cf. Section IV-E below. The lack of (exact) sparsity of is simply a consequence of the lack of orthogonality between edges in a simplicial FEM mesh. As a result, the elements are relatively very small unless edges are in close proximity. In other words, is strongly localized around edge. Strong localization can be illustrated by plotting, in a log scale, the magnitude of the for different edges of a 2-D FEM mesh, as in Figs An identical analysis can be done for in terms of the face elements (Whitney 2-forms) on the grid. B. Algebraic Thresholding Since most of its elements are relatively very small, the matrix can be well approximated by a sparse matrix. This can be done, for example, by algebraic thresholding. In this case, a parameter is chosen such that if the ratio of the absolute value of an element of to the maximum absolute value of its diagonal entries is below, then the element Fig. 3. Plot of log (j~ j) for edge (indicated by the bold segment of the mesh in the inset) near the boundary of a 2-D TE circular cavity, again showing the strong localization property of ~. of is set to zero. Otherwise, the element of is set equal to the corresponding element of. The threshold is in the range, where are the minimum maximum absolute values of diagonal entries of. A similar procedure can be applied for. Algebraic thresholding helps in verifying the quasi-sparse nature of, but relies on explicit knowledge of. A more practical strategy to obtain that does not require explicit knowledge of is to use a topological thresholding, as discussed ahead. But we first examine the tradeoff between sparsity sparsification error in Section IV-C. C. Tradeoff Between Sparsity Sparsification Error 1) Eigenvalues of Inverse Mass Matrices: We compare the eigenvalues of with those of

6 1364 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 5, MAY 2007 Fig. 6. Relative sparsification errors on the eigenvalues of [? ]. Fig. 4. Sparsity pattern of [? ], with r =0:005. Fig. 7. Relative sparsification errors on the eigenvalues of?. Fig. 5. Sparsity pattern of?, with r =0:005. to quantify the sparsification error. Consider the same spherical cavity FEM mesh of Section III-A. For a matrix, the density is defined as, with the number of nonzero elements. The sparsity patterns of for are depicted in Figs. 4 5, respectively, with matrix. Let be an eigenvalue of a. The relative error of an eigenvalue is defined as where the superscript sts for relative. We plot in Fig. 6 in Fig. 7 for all eigenvalues. In this case, the relative errors are consistently below 1.8% for all eigenvalues of below 1.3% for all eigenvalues of. with. 2) Eigenvalues of Dual System Matrices: Let be the eigenvalue of original dual system, be the eigenvalue of dual system after sparse approximation matrix, be the exact eigenvalue. We define the truncation (discretization) error, the sparsification error, the total error. It is easy to show that.if is chosen such that, then the total error is bounded by truncation error. Because the eigenvalues can vary much in magnitude, a normalization is convenient. In this case, relative errors are defined as,,. Fig. 8 shows the relative errors for the spherical cavity problem with. For visualization purposes, only the lowest 23 modes are shown. We observe that the sparsification error is less than the truncation error for all modes with this choice of. The total error may be smaller than truncation error

7 HE AND TEIXEIRA: DIFFERENTIAL FORMS, GALERKIN DUALITY, AND SPARSE INVERSE APPROXIMATIONS 1365 Fig. 10. Two FEM meshes for a circular cavity. Mesh (a) has 41 nodes 64 triangles. Mesh (b) has 178 nodes 312 triangles. Fig. 8. Relative truncation, sparsification, total errors on FEM eigenvalues. Fig. 9. Two-dimensional illustration of the neighbor edge classification by different k-levels of topological thresholding. The reference edge is 12 (indicated in 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 may have opposite signs. D. Topological Thresholding The strong localization property illustrated in Figs. 2 3 suggests that only closeby edges have significant coupling, the coupling decays very quickly with the distance between edges. For each edge, one can define various neighbor levels using, for example, mesh connectivity (topological) criteria [53], [54]. We define a level- neighbor in a 2-D triangular mesh as follows (similar definitions apply for 3-D, for DoF defined on nodes, faces, or tetrahedra): For each edge, level-0 neighbor include only edge itself. Level-1 neighbors include edge the four (nearest neighbors) edges belonging to the two triangles that share edge. Level-2 neighbors include all level-1 edges plus the edges in the neighboring triangles, so forth for. This is illustrated in Fig. 9. By retaining only interactions among level- neighbors for each edge, one obtains a sparse approximate inverse mass matrix. From (3), is a sparse matrix that includes only level-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. 11. Sparsity pattern of matrix [? ] for the coarse mesh in Fig. 9 by topological thresholding. (a) k =0, (b) k =1, (c) k =2, (d) k =3. same as the sparsity pattern of. This is important in practice because can be obtained without the need to obtain. In particular, can be obtained by minimizing the Euclidean (Frobenius) norm of the difference, where has a prescribed sparsity pattern. This minimization problem decouples into local independent least-square procedures that are naturally parallelizable [55]. To illustrate topological thresholding, we present results from two FEM meshes for a 2-D TE circular cavity as depicted in Fig. 10. The resulting sparsity patterns from various leveltopological thresholdings for the coarse mesh are shown in Fig. 11. Fig. 12 shows the relative sparsification errors for the TE eigenvalues (together with the relative truncation errors). Figs repeat the same for the finer mesh. In both cases, level-2 topological thresholdings already work very well, with the sparsification error consistently below the truncation error. The numerical results illustrate the increase of the truncation error with frequency, as expected. The sparsification error shows no such visible trend. Note that in order to fully examine the sparsification error, we plot the entire discrete eigenvalue spectra in these figures. Of course, only the lower end of the discrete spectra provides a good approximation of the continuum spectra.

8 1366 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 5, MAY 2007 Fig. 12. Relative sparsification error for each eigenvalue, using level-k topological thresholding (star) relative truncation error (diamond). (a) k =0, (b) k =1, (c) k =2, (d) k =3. 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, (d) k =3. contrast, sparse approximations, as applied here, approximate inverse Hodge star operators, which are local operators in the continuum limit. In other words, do not have inherent long range interactions that would remain present in the continuum limit. F. Application to Time-Domain FEM Fig. 13. Sparsity pattern of [? ] for the fine mesh in Fig. 9 from level-k topological thresholding. (a) k =0, (b) k =1, (c) k =2, (d) k =3. E. Relation With SPAI Preconditioners The sparsification described above mirrors the strategy used by sparse approximate inverse (SPAI) preconditioners [55]. However, a fundamental difference here is that SPAI preconditioners are most often used to approximate the inverse of (discrete) differential operators (or, sometimes, integral operators), i.e., Green s functions with long range interactions. By A major factor limiting the efficiency of the FEM in the time-domain is that mass matrices are non-diagonal. As a result, a sparse linear system solution is required at each time step. This becomes a major bottleneck of the computational effort is in contrast to the finite-difference time-domain (FDTD) method, for example, which is a matrix-free explicit scheme. 4 One strategy used to overcome this problem is mass lumping [56], whereby the mass matrix is reduced to a diagonal or block-diagonal matrix. However, mass lumping often results in non positive definite matrices, leading to (unconditional) instabilities that destroy the solutions [57]. Another strategy is based upon the use of orthogonal basis functions [58], however, it requires three times more DoF. To overcome this limitations, sparse explicit, time-domain FEMs based on the sparse approximation approaches explained here have been described in [54] (topological) [59] (algebraic). Combining (1) (2), we arrive at (17) 4 Of course, implicit time-domain schemes that rely on sparse matrix solvers can be, in cases such as highly refined grids, more advantageous than matrix-free explicit ones because the former can provide unconditionally stability.

9 HE AND TEIXEIRA: DIFFERENTIAL FORMS, GALERKIN DUALITY, AND SPARSE INVERSE APPROXIMATIONS 1367 where matrices,,, are all sparse. By inverting inverse Fourier transforming the above equation, one has (18) A leap-frog time discretization of the above leads to an explicit time-domain update that unfortunately is full because is full. However, by approximating by, anex- plicit sparse update scheme (FDTD-like) is obtained. Using topological thresholding, the resulting algorithm is highly parallelizable, both for obtaining from for the time-domain update itself. Since the eigenvalues of are only slightly perturbed, remains SPD the scheme is conditionally stable [14], [57]. 5 From a finite-difference stpoint, the above provides a strategy to derive stable consistent hence convergent [1] finite-difference stencils in irregular simplicial meshes. This is contrast to some previous explicit time-domain schemes based on Whitney elements [61] [63] that have (procedural equivalents of) discrete Hodge operators which are not necessarily SPD by construction. The above framework recognizes the need for two distinct finite-difference representations of the curl operators appearing in Maxwell curl equations, viz.,, for a discretization based upon a single mesh (as opposed to obe based upon dual meshes such as Yee s scheme), a property also shared by mimetic finite-difference methods [45]. V. SUMMARY AND CONCLUSION Galerkin duality has been discussed in connection with discrete FEM solutions of Maxwell equations in 3-D, verified in examples involving both homogeneous inhomogeneous media. Basic algebraic features of the primal dual discrete solutions were explained by means of the discrete Hodge Helmholtz decomposition Descartes Euler polyhedral formula. It was verified that primal dual formulations yield identical nonzero eigenvalues while having null spaces (of static eigenfunctions) of different dimensions. The quasi-sparse nature of the inverse mass matrices arising in the dual formulation was discussed explained in terms of the strong localization property. Two different approaches to approximate the inverse mass matrices by sparse matrices were discussed: algebraic thresholding topological thresholding. The former is of mainly theoretical interest only, while the latter can be used to obtain sparse approximate inverses without the need for direct inversion. We have discussed how to control the (sparsification) error of sparse dual FEM solutions. Finally, we discussed how these results can be applied to construct (conditionally stable) sparse explicit FEM in time-domain. 5 Other solvers can also be quite efficient in obtaining sparse apporximations for [? ]. 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Lett., vol. 7, no. 14, pp , Bo He (S 03 M 06) was born in Changzhi, China, in He received the B.S. degree from Shanxi University, China, in 1991, the M.S. degree from East China Normal University, in 1994, the Ph.D. degree in physics from the University of Missouri, Rolla/St. Louis, in 2002, the Ph.D. degree in electrical computer engineering from Ohio State University, Columbus, in From 1994 to 1999, he was a faculty member with the Shanghai Jiao Tong University. Since 2006, he has been a Postdoctoral Research Associate at the Center for Computational Electromagnetics, University of Illinois at Urbana- Champaign. His research interests include theoretical particle physics computational electromagnetics. Ferno L. Teixeira (S 89 M 93 SM 05) received the B.S. M.S. degrees in electrical engineering from the Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Brazil, in , respectively, the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in From 1999 to 2000, he was a Postdoctoral Research Associate with the Research Laboratory of Electronics, Massachusetts Institute of Technology (MIT), Cambridge. Since 2000, he has been with the Department of Electrical Computer Engineering the ElectroScience Laboratory, The Ohio State University (OSU), where he is now an Associate Professor. His current research interests include modeling of wave propagation, scattering, transport phenomena for communications, sensing, device applications. He has edited one book Geometric Methods for Computational Electromagnetics (Cambridge, MA: PIER 32 EMW, 2001) has authored over 70 journals articles book chapters. Dr. Teixeira is a Member of Phi Kappa Phi, Sigma Xi, an Elected Member of the International Scientific Radio Union (URSI) Commission B. He was awarded many prizes for his research including the including the Lumley Research Award from the OSU in 2003, the CAREER Award from the NSF in 2004 the triennial Henry Booker Fellowship from USNC/URSI in 2005.