Technical article APPLICATION OF DIFFERENTIAL FORMS IN THE FINITE ELEMENT FORMULATION OF ELECTROMAGNETIC PROBLEMS INTRODUCTION In many physical problems, we have to stuy the integral of a quantity over a p-imensional manifol in an n-imensional Eucliean space. In the stuy of these integrals, it is important to know in what manner the integral over the manifol epens on the position of the manifol in the Eucliean space. The purpose of ifferential forms is to stuy these integrals in a broa setting of geometry, topology, algebra an analysis. In the calculus of ifferential forms, the local fiel quantities are associate with the geometric an topological property of the manifol. One works with the integral quantities instea of local scalar or vector fiels. The integral quantities have usually physically measurable units an are more interesting in the engineering applications. In aition, the fact that ifferential forms carry the geometric information of the manifol makes clear the ifference between vectors relate to the line integral such as the electric or magnetic fiel intensity an vectors relate to the surface integral such as the flux or current ensity, which are ambiguously efine in vector algebra. Association of fiel quantities with the topological property of the manifol makes more unerstanable the moelling of multiply connecte omains. In one wor, the calculus of ifferential forms presents numerous avantages compare to the conventional vector or tensor algebra an turn out to be more efficient in the escription of physical problems [-4]. In electromagnetics, application of ifferential forms enables a simple an clear representation of Maxwell s equations. With the help of the exterior ifferential operator an Hoge operator foune in the calculus of ifferential forms, Maxwell equations can be presente in a formal flow iagram known as Tonti s iagram [5]. This iagram shows clearly the uality between the two systems of Maxwell equations. It is helpful for the erivation of ual formulations in the computation of electromagnetic fiels an allows a better unerstaning of ual approximation schemas. Application of ifferential forms in the numerical computation of electromagnetic fiels was notably marke by the appearance of Whitney elements [6]. Whitney elements consier the ifferential forms as egrees of freeom. Their avantages are principally their capacity of allowing natural iscretization of the systems with appropriate continuity of scalar an vector variables. With the help of ifferential forms, the high orer curl-conformal an iv-conformal elements propose in [7] can be put in a more clear circumstance. One of the important notions in the calculus of ifferential forms is De Rham s complex [-2]. De Rham s complex shows the relation between the spaces of ifferential forms of ifferent egrees. Application of De Rham s complex in electromagnetism shows the mathematical structure involve in electromagnetic theory. It allows a eep unerstaning in the stuy of ifferential operators gra, curl, iv with their kernels an their co-omains. De Rham s cohomology groups reveal the global topological property of the stuy omain. In three imensions, they are relate to the loops an the cavities in the stuy omain an have particular interest in the stuy of problems in multiply connecte regions. De Rham s complex can also be use to escribe the relation between the functional spaces of ifferential form base elements of ifferent egrees [8]. It illustrates not only the inclusion property of these elements, but also is helpful for the etermination of the rank of the matrix system of an electromagnetic fiel problem an enables an easy unerstaning of the gauge conition in the case of working with potential variables.
This paper gives a short presentation of the application of ifferential forms in the finite element computation of electromagnetic fiel. We give first a brief introuction of ifferential forms, in particular the notion of exact an close forms, an De Rham s complex. Maxwell equations in electromagnetics are then written in terms of ifferential forms an represente by Tonti s iagram. The very useful De Rham s complex is use to escribe the relation between the functional spaces (omains of ifferential operators) in electromagnetics. The iscrete spaces, i.e. elements base on ifferential forms of ifferent egrees, as well as the iscrete spaces of De Rham s cohomology groups for the moelling of cuts an links in the case of multiply connecte regions, are presente. Their relation is clarifie by using De Rham s complex. As an application, ual formulations of ey current problems in the case of non-trivial stuy omain are presente. The rank of the matrix system is etermine an the gauge conition in the case of working with vector potentials is iscusse with the help of De Rham s complex. We intentionally use alternately the vector notation an the ifferential form notation, whenever there is no confusion, in orer to have a comparison of two calculus. DIFFERENTIAL FORMS AND DE RHAM S COMPLEX Differential forms are expressions on which integration operates []. A ifferential form of egree p, or a p-form, is an expression where the integral is performe over a manifol of imension p in a space of imension n, i.e. the integran of a p-fol integral in an n-imensional space. The ifferential forms can be introuce accoring to the imension of the manifol on which the variable is integrate. In electromagnetics, a scalar potential is a 0-form; the circulation of a vector potential or a (electric or magnetic) fiel intensity along a small segment is a -form, a flux (or current) across a small area is a 2-form an charges containe in a small volume are a 3-form. For example, the electric fiel is ientifie with a -form, its expression is e l, representing the electromotive force along a short line. Another example is the current across a small surface, j s, which efines a 2-form. An example of 3-form is the charges containing in a small volume, given by ρv. We can note in particular that, the vectors relate to line integral such as electric an magnetic fiels, vector potentials, an the vectors relate to surface integral such as current an flux ensities correspon, respectively, to ifferential forms of egree an 2. These vectors, ifferent on their nature, cannot be istinguishe with the vector presentation. Differential forms operate in exterior algebra. Exterior (wege) prouct of a p-form ω an a q-form υ prouces a (p+q)-form with the skew symmetry property: ω υ = (-) pq υ ω, (where p+q<n, n enotes the imension of space). Two other operators permit transformation of a ifferential form of one egree to the other. One is the exterior erivation. Application of this operator to a ifferential form leas to a form of higher egree. In three imensions, it generalizes an unifies the familiar gra, curl an iv operators of vector algebra. The other operator is the star (Hoge) operator *. It transforms a p-form to an (n-p)-form. We will see later that this operator correspons to the constitutive laws in physic problems. Let D p (M) be the set of p-forms efine on an n-imensional ifferentiable manifol M. The inclusion property D p (M) D p+ (M) hols. This property is represente by a sequence calle De Rham s complex [2]. D 0 D p D p+ D n
A form ω is sai to be close if ω = 0. A form ω is sai to be exact if there exists a form υ (of one egree lower) such that ω = υ. Since (υ) 0, every exact form is close. Can we also say every close form is exact? The answer is positive for a manifol not too complex (topologically trivial omains). But in general, the answer is negative. Let Z p (M) be the set of close p-forms, B p (M) be the set of exact p-forms. We have in general B p (M) Z p (M). The complement of B p (M) in Z p (M), H p (M) = Z p (M) \ B p (M) is calle De Rham s p th cohomology group. The property of H p (M) epens on the topology of the manifol [2]. The imension of H p (M) is finite an calle p th Betti number of M. In particular, H 0 is equal to the number of connecte components of M. H p (for p>0) vanishes if M is topologically trivial. The spaces of exact forms an close forms are relate by the Hoge ecomposition: Z p (M) = B p (M) H p (M). Taking the Hoge ecomposition into account, De Rham s complex can be shown in the form of Fig.. D p (M) D p+ (M) B p (M) Z p (M) B p+ (M) Z p+ (M) H p (M) H p+ (M) Fig.. De Rham s complex showing relation of p th cohomology groups. De Rham s p th cohomology group has the particular interest because they are relate to the topology property of the manifol. In three imensions, H an H 2 are relate to the loops an cavities in the manifol M [9-0]. The first an secon Betti numbers, i.e. im(h ) an im(h 2 ), correspon respectively to the number of loops an the number of cavities in M. MAXWELL EQUATIONS IN TERMS OF DIFFRENTIAL FORMS The funamental equations of many physical problems can be put into a formal mathematical structure, they are classifie by efinition, balance an constitutive equations [5]. In electromagnetics, the Maxwell equations are put into two ual systems: Ampere s system an Faraay s system. In terms of ifferential forms, they are written as: Faraay s system: e = t b, b = 0, Ampere s system: h = j + t, = ρ (or j = t ρ), where e an h are -form electric an magnetic fiels, b, an j are 2-form magnetic, electric flux an current, ρ is a 3-form charge. The two ual systems are relate by the Hoge (star) transformation, they are constitutive laws of the meia: = ε*e, b = µ*h, j = σ*e, where ε, µ an σ are, respectively the permittivity, the permeability an the conuctivity. The -form electric fiel an magnetic fiel can also be expresse in terms of -form vector potentials a, t (or u) an the exterior erivative of 0-form scalar potentials v, φ :
e = t a v, h = t φ (h = t u φ). Different scalar an vector variables written in terms of ifferential forms in the two ual systems of electromagnetic problems are shown in Table. It can be note that ifferential forms are measurable quantities. Their units are also given in this table. Table. Differential forms an their units in electromagnetic systems ifferential forms Faraay system 0-form v = v φ = φ Ampere system -form e = e l a = a l h = h l t = t l u = u l 2-form b = b s j = j s = s 3-form ρ = ρ v Units Volt Weber Ampere Coulomb With the help of the exterior ifferential an the Hoge (star) operators, Maxwell s equations can be represente in a formal flow iagram known as Tonti iagram [5]. The case of the ey current problem ( t = 0, ρ = 0) is shown in Fig.2. The left han sie represents Faraay s equations an the unit of the forms is Weber or Volt. On the right han sie, we have the Ampere s equations an the forms take the unit Ampere. The two ual systems are relate by the constitutive laws (the Hoge transformation). This iagram illustrates clearly the sequence an the uality of Maxwell equations. It is very helpful for the erivation of ual finite element formulations. They are obtaine by performing a conformal approach of one system using appropriate elements (elements base on ifferential forms as escribe later) an by solving the other system using the weak variational principle (integration by parts). 0-form v 3-form (gra) (iv) -form a, e j= σ*e j 2-form (curl) (curl) 2-form b b =µ*h h, t -form (iv) (gra) 3-form 0 φ 0-form Fig.2. Tonti s ual flow iagram of ey current problems
FUNCTIONAL SPACES An electromagnetic fiel has a finite energy in a boune region. This implies that the fiel quantities, at least at the stationary state, are square integrable. It follows that the Hilbert spaces of square integrable forms are natural for the electromagnetism. Let be a connecte an open set (a boune omain) in a three-imensional Eucliean space, not necessarily topologically trivial. The fiel quantities (ifferential p-forms) belong to the following functional spaces (omains of ifferential operator ): H(, ) = {ω L p 2 (), ω L p+ 2 ()}, p = 0,, 2. Where L p 2 () is the space of square integrable p-forms over. In terms of more familiar vector presentation, they are written as H(gra, ) = {φ L 2 (), gra φ IL 2 ()} H(curl, ) = {u IL 2 (), curl u IL 2 ()} H(iv, ) = {u IL 2 (), iv u L 2 ()} where L 2 an IL 2 are the spaces of square integrable scalar an vector fiels over, respectively. These spaces have the following orthogonal ecompositions: H(gra, ) = ker(gra) co(gra) H(curl, ) = ker(curl) co(curl) H(iv, ) = ker(iv) co(iv) They are relate by the following sequence (De Rham s complex): H(gra, ) gra H(curl, ) curl H(iv, ) iv L 2 () We now consier the relation of De Rham s cohomology groups H () an H 2 () with the above functional spaces. In three imensions, H () an H 2 () are relate to the loops an cavities in the omain. They have the following properties []: H () = {u IL 2 () curl u = 0, iv u = 0, n u Γ = 0} H 2 () = {u IL 2 () curl u = 0, iv u = 0, n u Γ = 0} where Γ is the bounary of. The imensions of H an H 2 are finite an are equal to, respectively, the number of loops an the number of cavities in. Remin that H () ker(curl) but not in co(gra) an H 2 () ker(iv) but not in co(curl), as escribe by the Hoge ecomposition: ker(curl) = co(gra) H () ker(iv) = co(curl) H 2 () The functional spaces an De Rham s cohomology groups are relate by De Rham s complex as shown in Fig.3 [9]. It illustrates clearly the mathematical structure behin the electromagnetic theory an is very useful in the stuy of electromagnetic problems.
H(gra, ) (gra) ker(gra) co(gra) H(curl, ) (curl) ker(curl) H () co(curl) H(iv, ) ker(iv) H 2 () co(iv) (iv) L 2 () Fig.3. De Rham s complex showing mathematical structure of electromagnetic theory. DISCRETE SPACES - ELEMENTS BASED ON DIFFERRENTIAL FORMS In orer to approximate correctly an naturally the previous functional spaces, suitable elements must be aopte. These elements are erive with the help of the calculus of ifferential forms. Let the omain be pave with a tetraheral (3-simplex) mesh. The number of noes, eges, facets an tetrahera are note, respctively, by N, E, F an T. We consier the general case of high orer elements an note by W q p () the function space of q-orer p-form elements over. The case of the first orer (q = ) correspons to the well know Whitney elements [6]. The functional space of p-form element W q p can be ecompose into a null space of ifferential operator Z q p () (set of close forms) an a range space of ifferential operator Y q p (): W q p = Z q p Y q p. The element W q p fulfils the following requirements: moel correctly the null space Z q p of the ifferential operator an is complete to q- orer in the range space Y q p uner the ifferential operation. The basis functions of p-form elements take the p-form bases: λ i, λ i, λ i λ j, λ i λ j λ k, for p = 0,, 2, 3, respectively, an have polynomial coefficients, where λ i is the barycentric coorinates of a point relate the noe i. The egrees of freeom of a p-form element are assigne to r-simplexes accoring to the orer q (p r Min{p+q-, 3}) [8]. The p-form elements W q p () (p = 0,, 2 an 3) are iscrete spaces of the functional spaces H(gra, ), H(curl, ), H(iv, ) an L 2 (), respectively. The inclusion property W q p W q p+ hols an is shown with the help of De Rham s complex in Fig. 4, where W H an W H2, are, respectively, iscrete spaces of the cohomology groups H an H 2. The complex given in Fig.4 is the iscrete form of the complex shown in Fig.3.
W q 0 () (gra) Z q 0 () Y q 0 () W q () (curl) Z q () W H () Y q () W q 2 () W H2 () Z q 2 () Y q 2 () (iv) W q 3 () Fig.4.De Rham s complex showing the relation between p-form elements. Accoring to the analysis given in [8], the imension of the function spaces W q p () an the imension of the null spaces Z q p () are, respectively, im(w q 0 ) = N + E(q-) + F(q-2)(q-)/2 + T (q-3)(q-2)(q-)/6 im(w q ) = E q + F(q-)q + T (q-2)(q-)q/2 im(w q 2 ) = F q(q+)/2 + T (q-)q(q+)/2 im(w q 3 ) = T q(q+)(q+2) /6 im(z q 0 ) = im(z q ) = E (q-) + F(q-2)(q-)/2 + T (q-3)(q-2)(q-)/6 + N + N H im(z q 2 ) = F q(q+)/2 + T q(q+)(2q-5) /6 where N H = im(h ()) an N H2 = im(h 2 ()), are, respectively, number of loops an cavities in. An interest remark is that the famous Euler formula is embee in De Rham s complex. In fact, accoring to De Rham s complex shown in Fig.4, the equality im(y q ) = im(z q 2 ) N H2 hols. After the simplification, the Euler formula is straight forewor: N E + F T = N H + N H2 The basis functions of q-orer p-form elements have to fulfill the conformity an unisolvence requirement, i.e. a p-form element must match the continuity conition of p-form fiel on the interface of ajacent elements, an the basis functions must be inepenent to provie an unique solution of the fiel equation. Various high orer p-form elements have been evelope in recent years [2-7]. They are ivie into two main categories: the interpolatory basis [4-7] an the hierarchical basis [2-3]. Using the interpolatory bases, the egrees of freeom have usually a physical interpretation. However, the shape functions of ifferent orers are all ifferent. They are not aapte for mixing
elements of ifferent orer in the same mesh in the case of p-version aaptive mesh generation. In aition, the separation of null space an range spacein the case of interpolatory basis is not easy. That makes the gauging conition, whenever necessary, ifficult. Recent stuies showe the avantages of hierarchical bases [8]. The hierarchy means that the basis functions of the high orer elements inclue all basis functions of lower orer element spaces. This property allows mixing of ifferent orer of elements in the same mesh without the ifficulty of matching fiel continuities. It is helpful for mixe h- an p-version aaptive mesh generation or for the evelopment of aaptive multigri solvers. Another avantage of hierarchical bases, which is not much mentione before, is that the basis functions belonging to the null space an to the range space, respectively, can be easily ientifie. This is an important avantage for the application of a gauge conition. The basis functions belonging to the null space can be easily remove. We now consier W H an W H2, the iscrete spaces of De Rham s cohomology groups H an H 2. They are relate to the moelling of cuts an links in the case of a multiply connecte omain. Detaile escription about the moelling of H an H 2 can be foun in [0]. In this paper, the moelling of these spaces is presente in the context of De Rham s complex. We note by Σ i, i =, 2,, N H, the cutting surfaces which make the omain simply connecte, an by Λ i, i =, 2,, N H2 the links which connect all components of Γ. The cutting surface Σ i cuts a layer of tetrahera that we call the cutting omain an note by Σi. W H belongs to the null space of W q (curl free) an cannot be expresse by a graient. It is nonzero in the omain Σi an vanishes elsewhere. The imension of W H equals to the number of cuts. There is one egree of freeom in each cutting omain. The -form elements in Σi are correlate. Taking these conitions in to account, the space W H is spanne by the following constraint functions: w Σi = ± w e, i =,, N H. e E Σi here w e are shape functions of the Whitney -form element an E Σi is the set of eges across the cutting surface Σ i. The sign epens on the orientation of eges with respect to the normal of cutting surfaces. We note by Λi the linking omain constitute of one bunle of tetrahera crosse by the link Λ i. W H2 is a null space of 2-form elements W q 2 (having zero ivergence), which cannot be presente by a curl fiel. It is non-null only in Λi. The egree of freeom is one per linking omain. The 2-form elements in Λi are correlate. Shape functions of W H2 are hence given by w Λi = ± w f, i =,, N H2 f FΛ i here w f are basis functions of Whitney 2-form element, w f W 2, F Λj the set of facets passe by Λ j, an N H2 the number of links Λ j. The sign epens on the orientation of facets with respect to the irection of links. The spaces W H an W H2 are very useful in the computation of electromagnetic fiel in multiply connecte regions. W H is important in the moelling of a curl free fiel such as the magnetic fiel in a multiply connecte omain when there are non-zero currents flowing in the loops (conuctor containing holes). Similarly, W H2 is useful in the moelling of a ivergence free fiel such as the isplacement fiel when the cavities contain non-zero charges [9].
DUAL FINITE ELEMENT FORMULATIONS OF EDDY CURRENT PROBLEMS Γ e c j j 0 Σi Σ0 Γ h Fig.5. A typical ey current problem As an application, we consier the case of an ey current problem shown in Fig.5. The stuy omain contains a conucting region c an an excitation coil j with a given current ensity j 0. The bounary of is split into two parts = Γ h Γ e with Γ h Γ e = 0. On Γ h we have n h = 0 an on Γ e, n e = 0. Without loosing the generality, we consier the case where the conucting region c an the excitation coil j have holes (no trivial omains). Before the erivation of formulations, we express the excitation current j 0 in j by a source fiel t 0 (current vector potential) such that curl t 0 = j. It can be note that t 0 efine in this way is not unique, but any fiel fulfilling curl t 0 = j works. The most convenient way is to set t 0 in a simply connecte region t compose of the excitation coil j an the cutting omain Σ0, with the bounary conition n t 0 = 0 on t an to calculate it using a finite element approximation [20]. After oing that, the magnetic fiel h in t is split into a curl free fiel h r an the source fiel t 0 : h = h r + t 0. The excitation coil j is then remove from the stuy omain. In what follows, we will use a unifie notation for the magnetic fiel h an keep in min that h = h r in t. Application of ifferential form base elements to the two ual systems of Maxwell equations as shown in Tonti iagram (Fig.) leas to two ual formulations. The magnetic formulation is obtaine by a conformal approximation of the Ampere s theorem using ifferential form base elements an by solving the Faraay s law using the weak variational principle (integration by parts). Taking the magnetic fiel (the magneto-motive force) as the working variables, the variational formulation in terms of ifferential forms is written as: Fin h W h = {h W q h = 0 in \ c, h = 0 on Γ h } * h h' + t µ * ' + t µ * 0 ' = 0 σ h h t h, h' W h With the more familiar vector notation, the formulation is Fin h W h = {h W q curl h = 0 in \ c, n h = 0 on Γ h } j
curlh ' curlh + t µ ' + t µ ' 0 = 0 σ h h h t, h' W h The omain \ c is multiply connecte since the conuctor contains holes. Let Σi be cutting omains which make \ c simply connecte an note by = ( \ c ) \ Σi the simply connecte omain. We now can etermine the rank of the system with the help of De Ram s complex shown in Fig.5. Since n h = 0 on Γ h, the egrees of freeom relate to the simplexes on Γ h are remove. We note by h the omain excluing the bounary Γ h, an by c0 the conucting omain c excluing the bounary c. Accoring to De Rham s complex given in Fig.4, the conition curl h = 0 in the simply connecte omain h can be satisfie by using a scalar potential φ such that h = gra φ, φ W q 0 ( h ). The gauge conition for the scalar potential is assure by the fact that the egrees of freeom relate to φ on Γ h are omitte (Here we suppose that Γ h is connecte. Otherwise, φ cannot be put to zero on all components of Γ h. Constraints have to be introuce [2]). One can also work irectly with the variable h, by taking basis functions belonging to the null space of the -form element, i.e. h Z q ( h ). In the case of first orer (Whitney) element, it consists to set the egrees of freeom of h on a tree constitute by a set of eges. When the elements of higher orer are use, ientification of the null space is not that easy unless elements of hierarchical basis are use. In the cutting omains Σi, accoring to the analysis of the previous section, h W H ( Σi ). The rank of the whole system is im(w q 0 ( h )) + im(w q ( c0 )) + N H. This formulation ensures the tangential continuity of the magnetic fiel. The results give the circulation of magnetic fiels along eges, an hence the currents across facets. In the conucting omain, the magnetic fiel h can be written as the sum of a current vector potential t W q ( c ) an the graient of a scalar potential φ W q 0 ( c ). We get a formulation in terms of combine vector an scalar potentials. This constitutes an alternative of the previous fiel formulation. A gauge conition excluing the null space from the -form element is then necessary to ensure a unique solution of the vector potential. The imension of the null space to be exclue is the same as the number of unknown scalar variables introuce. The rank of the system is im(w q 0 ( h )) + im(w q ( c0 )) im(z q ( c0 )) + N H, same as the formulation in terms of h in c. The ual formulation, the electric one, is obtaine by using p-form elements to approximate the variables in Faraay s system an solving Ampere s theorem in weak sense. Working with the time integral of the electric fiel in the conucting region an the magnetic vector potential in nonconucting region, the variational formulation in terms of ifferential forms is Fin a W a = {a W q a = 0 on Γ e ) or using vector notation: * a a' + t σ* ' + 0 ' = 0 µ a a t a, a' W a j Fin a W a = {a W q n a = 0 on Γ e ) curla ' curla + t σ ' + curl ' 0 = 0 µ a a a t, a' W a j j
The variable (-form) a (such that curl a = b) must be seen as the time integral of electric fiel e in c. In the non-conucting region \ c, we solve a ivergence free fiel iv b = 0. Since the surface integral of normal component of b over the bounary of \ c is ientically zero, the space W H2 has no use in this application. We are not concerne by the trouble of non-simply connecte region. The ivergence free conition is simply assure by writing b = curl a. It can be seen from De Rham s complex that, the flux ensity b is in the null space of ivergence operator an the vector potential a is in the co-omain of the curl operator. The rank is hence the imension of the range space of -form element. Appropriate gauge conition removing the null space from the functional space of -form element shoul be introuce to ensure the uniqueness of a. Consiering n a = 0 on Γ e, the egrees of freeom relate to the simplexes on Γ e are omitte. We note by e the omain excluing the bounary Γ e. Supposing the intersection of c an Γ e is empty, the rank of the whole system is im(w q ( e )) - im(z q ( e \ c )). Where im(z q ( e \ c ) correspons to the number of unknowns remove in non-conucting region when a gauge conition is impose. This formulation ensures the tangential continuity of electric fiel an gives the circulation of vector potential along eges, an hence the fluxes across facets. The electric formulation has also an alternative in terms of combine vector an scalar potentials. The fiel a (the time integral of the electric fiel e) in the conucting region can be replace by the sum of a magnetic vector potential a W q ( c ) an the graient of a scalar potential ψ W q 0 ( c ). A gauge conition excluing the null space from the -form element is then necessary to ensure a unique solution of the vector potential. The scalar potential is gauge by imposing the value of ψ at one noe if the intersection of c an Γ e is empty. When this is one, the number of unknowns of the system is im(w q ( e )) + im(w q 0 ( c )) im(z q ( e )), same as the previous fiel formulation. It can be note that, in this formulation, the current ensity j 0 is replace by curl t 0 an the vector potential t 0 is projecte on the space of -form element with the help of integration by part. This projection ensures the compatibility of the equation an improves consierably the convergence behavior [20]. Above formulations show that, in the conucting region, we can work with either the fiel variable or the potential variables. In the case of potential formulations, gauge conition has to be introuce to ensure the uniqueness. The rank of the matrix system is the same in both situations. It has to be pointe out that the gauge conition for the vector potential is not inispensable when an iterative solver is use. It is foun that the convergence behavior is even better without the gauge conition [22]. Accoring to the analysis given in [23], removing the null space of the -form elements iminishes the minimal non-zero eigenvalue an leas to a worse conitioning of the matrix system. CONCLUSION Differential forms present consierable avantages over the classical vector or tensor calculus not only in the electromagnetic theory analysis but also in the numerical computation of electromagnetic fiel. The ual flow iagram (Tonti iagram) is helpful for the erivation of ual finite element formulations of electromagnetic problems an makes clear the uality an the complementarity of ual approximation schema. Elements base on ifferential forms of ifferent egrees constitute natural iscrete spaces of ifferent scalar an vector variables an ensure naturally the continuity requirement of ifferent fiel quantities. De Rham s complex reveals the mathematical framework behin the electromagnetic theory, incluing the geometric an topologic properties of the stuy omain. De Rham s cohomology groups (spaces of close but not exact forms) enables a better unerstaning of the moelling of curl-free or iv-free fiel in multiply connecte regions. With the
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Zhuoxiang Ren