review article differential forms in lattice field theories: an...

Hindawi Publishing CorporationISRNMathematical PhysicsVolume 2013 Article ID 487270 16 pageshttpdxdoiorg1011552013487270

Review ArticleDifferential Forms in Lattice Field Theories An Overview

F L Teixeira

ElectroScience Laboratory Department of Electrical and Computer Engineering The Ohio State UniversityColumbus OH 43212 USA

Correspondence should be addressed to F L Teixeira teixeiraeceosuedu

Received 13 November 2012 Accepted 11 December 2012

Academic Editors J Banasiak F Sugino and G F Torres del Castillo

Copyright copy 2013 F L Teixeira This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We provide an overview on the application of the exterior calculus of differential forms to the ab initio formulation of lattice fieldtheories with a focus on irregular or ldquorandomrdquo lattices

1 Introduction

The need to formulate field theories on a lattice (mesh grid)arises from two main reasons which may occur simultane-ously or not First the lattice provides a natural ldquoregulariza-tionrdquo of divergences in lieu of renormalization techniques [1]Such regularization does not need to be viewed as an ad hocstep but instead as a natural consequence of assuming thefield theory to be at some fundamental level an effective(ldquolowrdquo-energy) description [2] Second the lattice providesa direct route to compute in a nonperturbative fashionquantities of interest by numerical simulations Nontrivialdomains and complex boundary conditions can then beeasily treated as well [3ndash6] For these the use of irregular(ldquorandomrdquo) lattices are often of interest to gain geometricalflexibility Irregular lattices are also of interest as a means toprovide a potentially faster convergence to the continuumlimit near-isotropic lattice dispersion properties and betterldquoconservationrdquo of some (eg long-range translational androtational) symmetries [7 8] In some cases irregular latticesare useful for universality tests as well [9 10]

Lattice theories are typically developed by taking thecounterpart continuum theory as starting point and thenapplying discretization techniques whereby derivatives areapproximated by finite differences or some constraints areenforced on the functional space of admissible solutions tobe spanned by a finite set of ldquobasisrdquo functions (eg ldquoGalerkinmethodsrdquo such as spectral elements and finite elements)These discretization strategies have proved very useful inmany settings however they often produce difficulties in

the case of irregular (ldquorandomrdquo) lattices Among such dif-ficulties are (i) numerical instabilities in marching-on-timealgorithms (regardless of the time integration method used)(ii) convergence problems in algorithms relying on iterativelinear solvers and (iii) spurious (ldquoghostrdquo) modes andorextraneous degrees of freedomThese problems often (but notalways) appear associatedwith highly skewed or obtuse latticeelements or at the boundary between heterogeneous (hybrid)lattices subcomponent comprising overlapped domains orldquomesh-stitchingrdquo interfaces for example Clearly such dif-ficulties put a constraint on the geometric flexibility thatirregular lattices are intended for and may require stringent(and computationally demanding) mesh quality controlsThese difficulties also impact the ability to utilize ldquomeshrefinementrdquo strategies based on a priori error estimatesThe reasons behind these difficulties can be traced to aninconsistent rendering of the differential calculus and degreesof freedom on the lattice A rough classification of thoseinconsistencies is provided in Appendix B

Theobjective of this paper is to present an overviewon theapplication of the exterior calculus of differential forms to theab initio formulation of field theories on irregular lattices [11ndash29] In the exterior calculus framework the lattice is treatedas a cell complex (in the parlance of algebraic topology [30])instead of simply a collection of discrete points and dynamicfields are represented by means of discrete differential forms(cochains) of various degrees [28 31 32] This prescriptionprovides a basis for developing a consistent ldquodiscrete calculusrdquoon irregular lattices and discrete analogues to partial differ-ential equations that better adhere to the underlying physics

2 ISRNMathematical Physics

This topic intersects many disparate application areasFor concreteness we employ classical electrodynamics asthe standard example here however whenever particularlyrelevant to do so we provide brief pointers to other fieldtheories as well Although some familiarity with the exteriorcalculus of differential forms is assumed [18 19 33ndash39] thediscussion is mostly kept at a tutorial level Finally we stressthat this is a review paper and no claim of originality isintended

2 Premetric Lattice Equations

Let us denote the space of differential 119901-forms on a smoothconnected manifold Ω as Λ119901(Ω) From a geometric perspec-tive a differential 119901-form 120572

119901isin Λ

119901(Ω) can be viewed as

an oriented 119901-dimensional density or an object naturallyassociated with 119901-dimensional domains of integration 119880119901such that the lattice contraction (ldquopairingrdquo) below

⟨119880119901 120572119901⟩ ≐ int

119880119901

120572119901 (1)

gives a real number (in our context) for each choice of 119880119901[22] On a lattice K 119880119901 is restricted to be a union ofelements from the finite set of 119901-dimensional 119873119901-orientedlattice elements which we denote by Γ119901(K) = 120590119901119894 119894 =1 119873119901 These are collectively called ldquo119901-chainsrdquo In fourdimensions for example they correspond to the possibleunions of elements from the set of vertices (nodes) 1205900 edges(links) 1205901 facets (plaquettes) 1205902 volume cells (voxels) 1205903 andhypervolume cells 1205904 for 119901 = 1 4 respectively In thediscrete setting the degrees of freedom are reduced to the setof pairings (1) on each one of the lattice elements

On the lattice the pairing above can be understood as amapR119901

Λ119901(Ω) rarr Γ

119901(K) such that

R119901(120572119901) = ⟨120590119901119894 120572

119901⟩ ≐ int

120590119901119894

120572119901 (2)

defines its action on the basis of 119901-chains Note that we useΓ119901(K) to denote the space dual to Γ119901(K) that is the space

119901-cochains The latter can be viewed as the space of ldquodiscretedifferential formsrdquo Because of this and with some abuse oflanguage we use the terminology ldquodifferential formsrdquo andldquocochainsrdquo interchangeably to denote the sameobjects inwhatfollows The mapR119901 is called the de Rham map [22]

The basic differential operator of exterior calculus is theexterior derivative119889 applicable to anynumber of dimensionsThe discretization of 119889 on a general irregular lattice can beeffected by a straightforward application of the generalizedStokesrsquo theorem [22]

int120590119901+1

119889120572119901= int

120597120590119901+1

120572119901 (3)

with 119901 = 0 3 in 119899 = 4 In the above 120597 is the boundaryoperator which simply maps a 119901-dimensional lattice elementto the set of (119901minus1)-dimensional lattice elements that compriseits boundary preserving orientation This theorem sets 120597 asthe formal adjoint of119889 in terms of the pairing given in (1) that

is ⟨120590119901+1 119889120572119901⟩ = ⟨120597120590119901+1 120572

119901⟩ Computationally the boundary

operator can be implemented bymeans of incidencematrices[22 29 40] such that

120597120590119901+1119894 = sum

119895

119862119901

119894119895120590119901119895 (4)

where the indices 119894 and 119895 run over all (119901 + 1)- and 119901-dimensional lattice elements respectively The incidencematrix entries are such that 119862119901

119894119895isin minus1 0 1 for all 119901

with sign determined by the relative orientation of latticeelements 119894 and 119895 The restriction to this set of integer valuesreflects the ldquometric-freerdquo nature of the exterior derivativeonly information about element connectivity that is thecombinatorial aspects of the lattice is involved here It turnsout that the metric is fully encoded by Hodge star operatorsthe discretization of which will be discussed further downbelow

Using (3) and (4) one can write

int120590119901+1119894

119889120572119901= sum

119895

119862119901

119894119895int120590119901119895

120572119901

(5)

for all 119894 so that the derivative operation is replaced by a propersum over 119895 On the lattice the nilpotency of the operators 120597 ∘120597 = 119889 ∘ 119889 = 0 [41] is recovered by the constraint [22]

sum

119896

119862119901+1

119894119896119862119901

119896119895= 0 (6)

for all 119894 and 119895

3 Example Lattice Electrodynamics

We write Maxwellrsquos equations in a four-dimensionalLorentzian manifoldΩ as [34]

119889119865 = 0

119889119866 = lowastJ(7)

where 119889 is the four-dimensional exterior derivative 119865 and 119866are the so-called Faraday and Maxwell 2-forms respectivelyand lowastJ is the charge-current density 3-formThe Hodge staroperator lowast is an isomorphism that maps 119901-forms to (4 minus 119901)-forms and more generally 119901 forms to (119899 minus 119901) forms in a 119899-dimensional manifold and as mentioned before depends onthe metric of Ω [22 23 34 35 42ndash45] The above equationsare complemented by the relation 119866 = lowast119865 which indicatesthat 119865 and 119866 are ldquoHodge dualsrdquo of each other

31 Primal andDual Lattices Since119865 and119866 are 2-forms theyshould be discretized as 2-cochains residing on plaquettes (2-chains) of the 4-dimensional lattice however it is importantto recognize that these two forms are of different types 119865 isan ldquoordinaryrdquo (or ldquonontwistedrdquo) differential form whereas 119866(as well as lowastJ) is a ldquotwistedrdquo (or ldquooddrdquo) differential form [46]The basic difference here has to do with orientation ordinaryforms have internal orientation whereas twisted forms have

ISRNMathematical Physics 3

external orientation [20 22 46ndash48] These two types oforientations exhibit different symmetries under reflection adistinction akin to that between proper (or polar) tensorsand pseudo (or axial) tensors Only twisted forms admitintegration in nonorientable manifolds These two typesof forms are associated with two distinct ldquocell complexesrdquo(lattices) each one inheriting the corresponding orientationthe ordinary form 119865 is associated with the set of plaquettes Γ2on the ldquoordinary cell complexrdquo K thus belonging to Γ2(K)while the twisted forms 119866 and lowastJ are associated with the setof plaquettes Γ2 on the ldquotwisted cell complexrdquo K [22 27 4849] thus belonging to Γ2(K) Consequently we also have twosets of incidencematrices119862119901

119894119895and119862119901

119894119895 one for each lattice It is

convenient to denoteK as the ldquoprimal latticerdquo and K as theldquodual latticerdquo [22]

As detailed further below these two lattices becomeintertwined by the Hodge duality 119865 = lowast119866 The need for duallattices can also be motivated from a purely combinatorialstandpoint (as a means to preserve key topological propertiesfrom the continuum theory) [24] or from a strictly computa-tional standpoint (eg to provide higher-order convergenceto the continuum) [50ndash52]

32 3 + 1Theory At this point it is suitable to degeometrizetime and treat it simply as a parameter This corresponds tothe majority of low-energy applications involving Maxwellrsquosequations in which one is interested in predicting the fieldevolution along different spatial slices for a given set of initialand boundary conditions In this case we still use the symbolsK and K for the primal and dual lattices but they now referto three-dimensional spatial lattices Similarly Ω now refersto a three-dimensional Euclidean manifold In such a 3 + 1setting one can decompose 119865 and 119866 as

119865 = 119864 and 119889119905 + 119861

119866 = 119863 minus 119867 and 119889119905(8)

and the source density aslowastJ = minus119869 and 119889119905 + 120588 (9)

where and is the wedge product 119864 and119867 are the electric inten-sity and magnetic intensity 1-forms on Γ1 and Γ1 respectively119863 and 119861 are the electric flux and magnetic flux 2-forms onΓ2 and Γ2 respectively 119869 is the electric current density 2-form on Γ2 and 120588 is the electric charge density 3-form on Γ3(corresponding assignments for the 2 + 1 and 1 + 1 cases areprovided in [32]) As a result Maxwellrsquos equations reduce to

119889119864 = minus120597119905119861 (10)

119889119867 = 120597119905119863 + 119869 (11)

representing Faradayrsquos and Amperersquos laws respectively Here119889 stands for the 3-dimensional spatial exterior derivativeNote that both (10) and (11) are metric-free They are supple-mented by Hodge star relations given by

119863 = ⋆120598119864

119867 = ⋆120583minus1119861(12)

now involving two Hodge star maps in three-dimensionalspace ⋆120598 Λ

1(Ω) rarr Λ

2(Ω) and ⋆120583minus1 Λ

2(Ω) rarr Λ

1(Ω) On

the lattice we have the corresponding discrete counterparts[⋆120598] Γ

1(K) rarr Γ

2(K) and [⋆120583minus1] Γ

2(K) rarr Γ

1(K) The

subscripts 120598 and 120583 in ⋆120598 and ⋆120583minus1 serve to indicate that theseoperators also incorporate macroscopic constitutive materialproperties through the local permittivity and permeabilityvalues [53] (we assume dispersionless media for simplicity)InRiemannianmanifolds (and in particular Euclidean space)and reciprocal media these two Hodge star operators aresymmetric and positive-definite [54]

In what follows we employ the following short-handnotation for cochains ⟨1205901119894 119864⟩ = 119864119894 ⟨1119894 119867⟩ = 119867119894⟨2119894 119863⟩ = 119863119894 ⟨1205902119894 119861⟩ = 119861119894 ⟨2119894 119869⟩ = 119869119894 and ⟨3119894 120588⟩ = 120588119894where the indices run over the respective basis of 119901-chains ineitherK or K119901 = 1 2 3With the exception ofAppendix Awe restrict ourselves to the 3 + 1 setting throughout theremainder of this paper

4 Casting the Metric on a Lattice

41 Whitney Forms The Whitney map W Γ119901(K) rarr

Λ119901(Ω) is the right-inverse of the de Rham map (2) that

is R ∘ W = I where I is the identity operator Insimplicial lattices this morphism can be constructed usingthe so-called Whitney forms [15 22 36 43 55ndash61] whichare basic interpolants from cochains to differential forms [33](other interpolants are also possible [62 63]) By definitionall cell elements of a simplicial lattice are simplices that iscells whose boundaries are the union of a minimal numberof lower-dimensional cells In other words 0-simplices arenodes 1-simplices are links 2-simplices are triangles 3-simplices are tetrahedra and so on Note that if the primallattice is simplicial the dual lattice is not [31] For a 119901-simplex120590119901119894 the (lowest-order) Whitney form is given by

120596119901[120590119901119894]

≐ 119901

119901

sum

119895=0

(minus1)1198941205821198941198951198891205821198940 and 1198891205821198941 sdot sdot sdot 119889120582119894119895minus1 and 119889120582119894119895+1 sdot sdot sdot 119889120582119894119901

(13)

where 120582119894119895 119895 = 0 119901 are the barycentric coordinatesassociated with 120590119901119894 In the case of a 0-simplex (node) (13)reduces to 1205960[1205900119894] = 120582119894

From its definition it is clear that Whitney forms havecompact support Among its important structural propertiesare

⟨120590119901119894 120596119901[120590119901119895]⟩ = int

120590119901119894

120596119901[120590119901119895] = 120575119894119895 (14)

where 120575119894119895 is theKronecker delta which is simply a restatementofR ∘W = I and

120596119901[120597

119879120590119901minus1119894] = 119889 (120596

119901minus1[120590119901minus1119894]) (15)

where 120597119879 is the coboundary operator [56] consistent with thegeneralized Stokesrsquo theorem Further structural properties are


provided in [57 58] Higher-order version of Whitney formsalso exist [59 60] The key result W ∘R rarr I holds in thelimit of zero lattice spacing This is discussed together withother related convergence results in various contexts in [1533 64ndash68]

Using the short-hand 120596119901[120590119901119894] = 120596119901

119894 we can write the

following expansions for 119864 and 119861 in a irregular simpliciallattice in terms of its cochain representations

119864 = sum

119894

1198641198941205961

119894

119861 = sum

119894

1198611198941205962

119894

(16)

where the sums run over all primal lattice edges and facesrespectively

One could argue that Whitney forms are continuumobjects that should have no fundamental place on a trulydiscrete theory In our view this is only partially true Inmanyapplications (see eg the discussion on space-charge effectsbelow) it is less natural to consider the lattice as endowedwith some a priori discrete metric structure than it is toconsider it instead as embedded in an underlying continuum(say Euclidean) manifold with metric and hence inheritingall metric properties from it In the latter caseWhitney formsprovide the standard route to incorporatemetric informationinto the discrete Hodge star operators as described next

42 Discrete Hodge Star Operator In a source-free media wecan write the Hamiltonian as

H =1

2intΩ

(119864 and 119863 + 119867 and 119861) = intΩ

(119864 and ⋆120598119864 + ⋆120583minus1119861 and 119861)

(17)

Using (16) the lattice Hamiltonian assumes the expectedquadratic form

H = sum

119894

sum

119895

119864119894[⋆120598]119894119895119864119895 +sum

119894

sum

119895

119861119894[⋆120583minus1]119894119895119861119895 (18)

where we immediately identify the symmetric positive defi-nite matrices

[⋆120598]119894119895 = intΩ

1205961

119894and ⋆120598120596

1

119895

[⋆120583minus1]119894119895= int

Ω

(⋆120583minus11205962

119894) and 120596

2

119895

(19)

as the discrete realization of the Hodge star operator(s) on asimplicial lattice [23 69] so that

119863119894 = sum

119895

[⋆120598]119894119895119864119895

119867119894 = sum

119895

[⋆120583minus1]119894119895119861119895

(20)

From the above the Hamiltonian can be also expressed as

H = sum

119894

119864119894119863119894 +sum

119894

119867119894119861119894 (21)

43 Symplectic Structure and Dynamic Degrees of FreedomThe Hodge star matrices [⋆120598] and [⋆120583minus1] have different sizesThe number of elements in [⋆120598] is equal to1198731 times1198731 whereasthe number of elements in [⋆120583minus1] is equal to1198732 times1198732 In otherwords Θ(119864) = Θ(119863) =Θ(119861) = Θ(119867) where Θ denotes thenumber of (discrete) degrees of freedom in the correspondingfield

One important property of a Hamiltonian system is itssymplectic character associated with area preservation inphase space The symplectic character of the Hamiltonianin principle would require a canonical pair such as 119864 119861 tohave identical number of degrees of freedom This apparentcontradiction can be explained by the fact that Maxwellrsquosequations (10) and (11) can be thought as aconstraineddynamic system (by the divergence conditions) so that eventhough Θ(119864) =Θ(119861) we still have Θ119889

(119864) = Θ119889(119861) where Θ119889

denotes the number of dynamic degrees of freedom This isdiscussed further below in Section 6 in connection with thediscrete Hodge decomposition on a lattice

5 Semidiscrete Equations

51 Local and Ultralocal Lattice Coupling By using a contrac-tion in the form of (2) on both sides of (10) with every face1205902119895 ofK and using the fact that ⟨1205902119895 120596

2

119894⟩ = ⟨1205901119895 120596

1

119894⟩ = 120575119894119895

from (14) we get

⟨1205902119895 120597119905119861⟩ = 120597119905sum

119894

119861119894 ⟨1205902119895 1205962

119894⟩ = 120597119905119861119895

⟨1205902119895 119889119864⟩ = ⟨1205971205902119895 119864⟩ = sum

119894

119864119894sum

119896

1198621

119895119896⟨1205901119896 120596

1

119894⟩ = sum

119894

1198621

119895119894119864119894

(22)

so that

minus120597119905119861119894 = sum

119895

1198621

119894119895119864119895 (23)

where the index 119894 runs over all faces of the primal lattice Onthe dual lattice K we can similarly contract both sides of (11)with every dual face 2119895 to get

120597119905119863119894 = sum

119895

1198621

119894119895119867119895 (24)

where now the index 119894 runs over all faces of the dual latticeUsing (20) and the fact that in three-dimensions 1198621

119894119895= 119862

1

119895119894

[22] (up to possible boundary terms ignored here) we canwrite the last equation in terms of primal lattice quantities as

120597119905sum

119895

[⋆120598]119894119895119864119895 = sum

119895

1198621

119895119894sum

119896

[⋆120583minus1]119895119896119861119896 (25)

or by using the inverse Hodge star matrix [⋆120598]minus1

119894119895 as

120597119905119864119894 = sum

119895

Υ119894119895119861119895 (26)


with

Υ119894119895 ≐ sum

119896

sum

119897

[⋆120598]minus1

1198941198961198621

119897119896[⋆120583minus1]119897119895

(27)

The matrix [Υ] can be viewed as the discrete realization for119901 = 2 of the codifferential operator 120575 = (minus1)119901lowastminus1119889lowast thatmaps 119901-forms to (119899 minus 119901)-forms [35]

Since the continuum operators ⋆120598 and ⋆120583minus1 are local[46] and as seen Whitney forms (13) have local support itfollows that the matrices [⋆120598] and [⋆120583minus1] are sparse indicativeof an ultralocal coupling (in the terminology of [70]) Incontrast the numerical inverse [⋆120598]

minus1 used in (27) is ingeneral not sparse so that the field coupling between distantelements is nonzero The lack of sparsity is a potentialbottleneck in practical simulations However because thecoupling strength in this case decays exponentially [29 44]we can still say (using again the terminology of [70]) that theresulting discrete operator encoded by the matrix in (27) islocal In practical terms the exponential decay allows oneto set a cutoff on the nonzero elements of [⋆120598] based onelement magnitudes or on the sparsity pattern of the originalmatrix [⋆120598] to build a sparse approximate inverse for [⋆120598]and hence recover back an ultralocal representation for ⋆120598

minus1

[29 71] The sparsity pattern of [⋆120598] encodes the nearest-neighbor edge information of the mesh and consequentlythe sparsity pattern of [⋆120598]

119896 likewise encodes successive ldquo119896-levelrdquo neighbors The latter sparsity patterns can be usedto build quite efficiently sparse approximations for [⋆120598]

minus1as detailed in [29] Once such sparse representations areobtained (23) and (26) can be used in tandem to constructa marching-on-time algorithm (eg see Section 91 ahead)with a sparse structure and hence amenable for large-scaleproblems

52 Barycentric Dual and Barycentric Decomposition LatticesAn alternative approach aimed at constructing a sparsediscrete Hodge star for ⋆120598minus1 directly from the dual latticegeometry is described in [27] based on earlier ideas exposedin [24 72] This approach is based on the fact that bothprimal K and dual K lattices can be decomposed intoa third (underlying) lattice K by means of a barycentricdecomposition see [24] The dual lattice K in this case iscalled the barycentric dual lattice [27 72] and the underlyinglattice K is called the barycentric decomposition latticeImportantly K is simplicial andhence admitsWhitney formsbuilt on it using (13) Whitney forms on K can be used asbuilding blocks to construct (dual) Whitney forms on the(nonsimplicial) K and from that a sparse inverse discreteHodge star [⋆120598minus1] using integrals akin to (19) An explicitderivation of such dual lattice Whitney forms is provided in[73] Furthermore a recent comprehensive survey of this andother approaches based on dual lattices to construct discretesparse inverse Hodge stars is provided in [74]

The barycentric dual lattice has the important propertybelow associated with Whitney forms

⟨(119899minus119901)119894 ⋆120596119901[120590119901119895]⟩ = int

120590(119899minus119901)119894

⋆120596119901[120590119901119895] = 120575119894119895 (28)

where ⋆ stands for the spatial Hodge star operator (distilledfrom constitutive material properties) and (119899minus119901)119894 is the dualelement to 120590119901119894 on the barycentric dual latticeThe operator ⋆is such that

intΩ120596119901and ⋆120596

119901= int

Ω

|120596|2119889119907 (29)

where |120596|2 is the two-norm of 120596119901 and 119889119907 is the volumeelement

The identity (28) plays the role of structural property(14) on the dual lattice side We stress that identity (28) isa distinctively characteristic feature of the barycentric duallattice not shared by other geometrical constructions forthe dual lattice In other words compatibility with Whitneyforms via (28) naturally forces one to choose the dual latticeto be the barycentric dual

From the above one can also define a (Hodge) dualityoperator directly on the space of chains that is⋆119870 Γ119901(K) 997891rarrΓ119899minus119901(K) with ⋆119870(120590119901119894) = (119899minus119901)119894 and ⋆ Γ119901(K) 997891rarr Γ119899minus119901(K)with ⋆119870(119901119894) = (119899minus119901)119894 so that ⋆119870⋆ = ⋆⋆119870 = 1 Thisconstruction is detailed in [24]

53 Galerkin Duality Even though we have chosen to assign119864 and 119861 to the primal (simplicial) lattice and consequently119863119867 119869 and 120588 to the dual (nonsimplicial) lattice the reverseis equally possible In this case the fields 119863 119867 becomeassociated to a simplicial lattice and hence can be expressedin terms of Whitney forms the expressions dual to (16) arenow

119867 = sum

119894

1198671198941205961

119894

119863 = sum

119894

1198631198941205962

119894

(30)

with sums running over primal edges and primal facesrespectively and where

119864119894 = sum

119895

[⋆120598minus1]119894119895119863119895

119861119894 = sum

119895

[⋆120583]119894119895119867119895

(31)

with

[⋆120598minus1]119894119895 = intΩ

(⋆120598minus11205962

119894) and 120596

2

119895

[⋆120583]119894119895= int

Ω

1205961

119894and ⋆120583120596

1

119895

(32)

and the two Hodge star maps now used are such that in thecontinuum ⋆120598minus1 Λ

2(Ω) rarr Λ

1(Ω) and ⋆120583 Λ

1(Ω) rarr


Λ2(Ω) and on the lattice [⋆120598minus1] Γ

2(K) rarr Γ

1(K) and

[⋆120583] Γ1(K) rarr Γ

2(K) This alternate choice entails a

duality between these two formulations dubbed ldquoGalerkindualityrdquo This is explored in more detail in [44]

6 Discrete Hodge Decomposition andEulerrsquos Formula

For any 119901-form 120572119901 we can write

120572119901= 119889120577

119901minus1+ 120575120573

119901+1+ 120594

119901 (33)

where 120594119901 is a harmonic form [31]This Hodge decompositionis unique In the particular case of the 1-form 119864 we have

119864 = 119889120601 + 120575119860 + 120594 (34)

where 120601 is a 0-form and 119860 is a 2-form with 119889120601 representingthe static field 120575119860 the dynamic field and 120594 the harmonic fieldcomponent (if any) In a contractible domain 120594 is identicallyzero and the Hodge decomposition simplifies to

119864 = 119889120601 + 120575119860 (35)

more usually known as Helmholtz decomposition in threedimensions

In the discrete setting the degrees of freedom of 120601 areassociated to the nodes of the primal lattice Likewise thedegrees of freedom of 119860 are associated to the facets of theprimal lattice Consequently we have from (35) that

Θ119889(119864) = 119873

ℎ

119864minus 119873

ℎ

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

= 119873119864 minus 119873119881

(36)

where 119873119881 is the number of primal nodes 119873119864 the numberof primal edges and 119873119865 the number of primal facets withsuperscript 119887 standing for boundary (fixed) elements and ℎfor interior (free) elements

On the other hand once we identify the lattice as anetwork of (in general) polyhedra we can apply Eulerrsquospolyhedron formula on the primal lattice to obtain [44]

119873119881 minus 119873119864 = 1 minus 119873119865 + 119873119875 (37)

where119873119875 represents the number of volume cells comprisingthe primal lattice A similar Eulerrsquos polyhedron formulaapplies to the (closed two-dimensional) boundary of theprimal lattice

119873119887

119881minus 119873

119887

119864= 2 minus 119873

119887

119865 (38)

Combining (37) and (38) we have

(119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881) = (119873119865 minus 119873

119887

119865) minus (119873119875 minus 1) (39)

From the Hodge decomposition (35) we see that Θ119889(119864) is

Θ119889(119864) = 119873

119894119899

119864minus 119873

119894119899

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

(40)

Note that the divergence free condition 119889119861 = 0 producesone constraint on the 2-form 119861 for each volume elementThis constraint also spans the whole lattice boundary Thetotal number of the constrains for 119861 is therefore (119873119875 minus 1)Consequently we have

Θ119889(119861) = 119873

119894119899

119865minus (119873119875 minus 1)

= (119873119865 minus 119873119887

119865) minus (119873119875 minus 1)

(41)

so that

Θ119889(119861) = Θ

119889(119864) (42)

This discussion can be generalized to lattices on noncon-tractible domains with any number of holes (genus) wherethe identity Θ119889

(119861) = Θ119889(119864) is also satisfied [31] Moreover

from Hodge star isomorphism we have Θ119889(119863) = Θ

119889(119864) and

Θ119889(119867) = Θ

119889(119861)

In general we can trace a direct correspondence betweenquantities in the Euler polyhedron formula to the quantitiesin theHodge decomposition formula For example each termin the two-dimensional Eulerrsquos formula 119873119864 = 119873119881 + (119873119865 minus

1) + 119892 is associated to a corresponding term in 119864 = 119889120601 +120575119860 + 120594 that is the number of edges 119873119864 corresponds to thedimension of the space of lattice 1-forms 119864 which is thesum of the number of nodes 119873119881 (dimension of the space ofdiscrete 0-forms 120601) the number of faces (119873119865 minus1) (dimensionof the space of discrete 2-forms 119860) and the number ofholes 119892 (dimension of the space of harmonic forms 120594) Asimilar correspondence can be traced on a three-dimensionallattice [31]This correspondence provides a physical picture toEulerrsquos formula and a geometric interpretation to the Hodgedecomposition

7 Absorbing Boundary Conditions

In many wave scattering simulations the presence of long-range interactions with slow (algebraic) decay together withpractical limitations in computer memory resources impliesthat open-space problems necessitate the use of specialtechniques to suppress finite-volume effects and emulatefor example the Sommerfeld radiation condition at infinityPerfectly matched layers (PML) are absorbing boundaryconditions commonly used for this purpose [75ndash78] In thecontinuum limit the PML provides a reflectionless absorp-tion of outgoing waves in such a way that when the PMLis used to truncate a computational lattice finite-volumeeffects such as spurious reflections from the outer boundaryare exponentially suppressed When first introduced in theliterature [75] the PML relied upon the use of matchedartificial electric and magnetic conductivities in Maxwellrsquosequations and of a splitting of each vector field componentinto two subcomponents Because of this the resulting fieldsinside the PML layer are rendered ldquonon-Maxwellianrdquo ThePML concept was later shown to be equivalent in the Fourierdomain (120597119905 rarr minus119894120596) to a complex coordinate stretching of thecoordinate space (or an analytic continuation to a complex-valued coordinate space) [76ndash78] and as such applicable toany linear wave phenomena


Inside the PML the (local) spatial coordinate 120577 along theoutward normal direction to each lattice boundary point iscomplexified as

120577 997888rarr 120577 = int

120577

0

119904120577 (1205771015840) 119889120577

1015840 (43)

where 119904120577 is the so-called complex stretching variable writtenas 119904120577(120577 120596) = 119886120577(120577) + 119894Ω120577(120577)120596 with 119886120577 ge 1 andΩ120577 ge 0 (profilefunctions)The first inequality ensures that evanescent waveswill have a faster exponential decay in the PML region andthe second inequality ensures that propagating waves willdecay exponentially along 120577 inside the PML As opposed tosome other lattice truncation techniques the PML preservesthe locality of the underlying differential operators and henceretains the sparsity of the formulation

For Maxwellrsquos equations the PML can also be affectedby means of artificial material tensors (Maxwellian PML)[79] In three dimensions the Maxwellian PML can berepresented as a media with anisotropic permittivity andpermeability tensors exhibiting stratification along the nor-mal to the boundary 119878 that parametrizes the lattice trunca-tion boundary The PML tensors properties depend on thelocal geometry via the two principal curvatures of 119878 [80ndash82] The boundary surface 119878 is assumed (constructed) asdoubly differentiable with non negative radii of curvatureotherwise dynamic instabilities ensue during a marching-on-time evolution [83]

From (43) the PML also admits a straightforwardinterpretation as a complexification of the metric [38 84]As a result the use of differential forms readily unifiesthe Maxwellian and non-Maxwellian PML formulationsbecause the metric is explicitly factored out into the Hodgestar operatorsmdashany transformation the metric correspondsdually to a transformation on the Hodge star operators thatcan be mimicked by modified constitutive relations [37] Inthe differential forms framework the PML is obtained bya mapping on the Hodge star operators ⋆120598 rarr ⋆120598 and⋆120583minus1 rarr ⋆120583minus1 induced by the complexification of the metricThe resulting differential forms inside the PML 119864 119863 and 119861 therefore obey ldquomodifiedrdquo Hodge relations 119863 = ⋆120598119864and 119861 = ⋆120583minus1 but identical premetric equations (10) and(11) In other words (10) and (11) are invariant under thetransformation (43) [38 84]

8 Implementation of Space Charge Effects

In many applications related to plasma physics or electronicdevices it is necessary to include space charges (uncom-pensated charge effects) into lattice models of macroscopicMaxwellrsquos equations This is typically done by representingthe charged plasma media using particle-in-cell (PIC) meth-ods that track the individual particles on the lattice [85ndash87]The fieldcharge interaction is thenmodeled by (i) interpolat-ing lattice fields (cochains) to particle positions (gather step)(ii) advancing particle positions and velocities in time usingequations of motion and (iii) interpolating back charge den-sities and currents onto the lattice as cochains (scatter step)In general the ldquoparticlesrdquo do not need to be actual individual

particles but can be a collection thereof (macroparticles)To put it simply incorporation of space charges requirestwo extra steps during the field update in any marching-on-time algorithm which transfer information from the instan-taneous field distribution to the particle kinematic update andvice versa Conventionally this information transfer relies onspatial interpolations that often violates the charge continuityequation and as a result leads to spurious charge depositionon the lattice nodes On regular lattices this problem can becorrected for example using approaches that either subtracta static solution (charges) from the electric field solution(BorisDADI correction) or directly subtract the residualerror on the Gauss law (Langdon-Marder correction) ateach time step [88] On irregular lattices additional degreesof freedom can be added as coupled elliptical constraintsto produce an augmented Lagrange multiplier system [89]All these approaches necessitate changes on the originalequations while still allowing for small violations on chargeconservation In contrast Whitney forms provide a directroute to construct gather and scatter steps that satisfy chargeconservation exactly even on unstructured lattices [90 91]as explained next To conform to the vast majority of theplasma and electronic devices literature we once morerestrict ourselves here to the 3 + 1 setting even though afour-dimensional analysis in Minkowski space would haveprovided a more succinct discussion

For the gather stepWhitney forms can be used to directlycompute (interpolate) the fields at any location from theknowledge of its cochain values such as in (16) for exampleFor the scatter step charge movement can be modeled asthe Hodge-dual of the current 2-form 119869 that is as the 1-form ⋆119869which can be expanded in terms ofWhitney 1-formson the primal lattice Here ⋆ represents again the spatialHodge star in three dimensions distilled from macroscopicconstitutive properties The Hodge-dual current associatedto an individual point charge can be expressed as ⋆119869 =119902119907

where 119902 is the charge value 119907 is the associated velocityvector and is the ldquoflatrdquo operator or index-lowering canonicalisomorphism that maps a vector to a 1-form given by theEuclidean metric Similarly point charges can be encoded asthe Hodge-dual of the charge density 3-form 120588 that is asthe 0-form ⋆120588 which can be expanded in terms of Whitney0-forms on the primal lattice These two Whitney maps arelinked in such a way that the rate of change on the valueof the 0-cochain representing ⋆120588 at a node is associatedto the presence of a 1-cochain representing ⋆119869 along theedges that touch that particular node leading to exact chargeconservation at the discrete level To show this considerfor simplicity the two-dimensional case of a point charge 119902moving from point 119909(119904) to point 119909(119891) during a time interval 120591inside a triangular cell with nodes1205900012059001 and12059002 or simply0 1 and 2 At any point 119909 inside this cell the 0-form ⋆120588 canbe scattered to these three adjacent nodes via

⋆120588 = 119902

3

sum

119894=1

⟨119909 1205960

119894⟩120596

0

119894 (44)

where we are again using the short-hand 1205960[1205900119894] = 1205960

119894 and

the brackets represent the pairing expressed by (1) In this


case119901 = 0 and the pairing integral in (1) reduces to a functionevaluation at a point Since Whitney 0-forms are equal to thebarycentric coordinates associated of a given node that is⟨119909 120596

0

119894⟩ = 120582119894(119909) we have the scattered charge 119902120582119904

119894≐ 119902120582119894(119909

(119904))

on node 119894 for a charge 119902 at 119909(119904) and similarly the scatteredcharge 119902120582119891

119894on node 119894 for a charge 119902 at 119909(119891) The rate of

scattered charge variation on a givennode 119894 is therefore equalto 119902(120582

119891

119894minus 120582

119904

119894) where 119902 = 119902120591

During 120591 the particle travels through a path ℓ from 119909(119904)

to 119909(119891) and the corresponding ⋆119869 can be expanded as a sumof Whitney 1-forms 1205961

119894119895associated to the three adjacent edges

119894119895 = 01 12 20 that is

⋆119869 = 119902sum

119894119895

⟨ℓ 1205961

119894119895⟩120596

1

119894119895 (45)

The coefficients ⟨ℓ 1205961119894119895⟩ represent the (oriented) current flow

along the associated oriented edge that is the cochainrepresentation of ⋆119869 along edge 119894119895 Using (13) the sum of thetotal current magnitude scattered along edges 01 and 20 thatflows into node 0 is therefore

119902 (minus ⟨ℓ 1205961

01⟩ + ⟨ℓ 120596

1

20⟩) = 119902 int

ℓ

(minus1205961

01+ 120596

1

20) (46)

Using 1205961119894119895= 120582119894119889120582119895 minus 120582119895119889120582119894 and 1205821 + 1205822 + 1205823 = 1 the above

reduces to

119902 intℓ

1198891205820 = 119902 (120582119891

0minus 120582

119904

0) (47)

which exactly matches the rate of scattered charge variationon node 0 obtained before It is clear that similar equalitieshold for nodes 1 and 2 More fundamentally these equalitiesare a direct consequence of the structural property (15)

9 Outline of Related Discretization Methods

We outline below various discretization programs that relyone way or another on tenets exposed aboveThe delineationis informed mostly by applications related to electrodynam-ics As expected this delineation is not too sharp because theprograms share much in common

91 Finite-Difference Time-Domain Method In cubical lat-tices the (lowest-order) Whitney forms can be representedby means of a product of pulse and ldquorooftoprdquo functions onthe three Cartesian coordinates [92] This choice togetherwith the use of low-order quadrature rules to computethe Hodge star integrals in (19) leads to diagonal matrices[⋆120598] [⋆120583minus1] and consequently also diagonal [⋆120598]

minus1 [⋆120583minus1]minus1

and sparse [Υ] so that an ultralocal equation results for(26) In this fashion one obtains a ldquomatrix-freerdquo algorithmwhere no linear algebra is needed during a marching-on-time solution for the fieldsThis prescription exactly recoversthe Yeersquos scheme [50] that forms the basis for the celebratedfinite-difference time-domain (FDTD) method (see [51 93]

and references therein) FDTD adopts the simplest explicitenergy-conserving (symplectic) time-discretization for (23)and (26) which can be constructed by staggering the electricand magnetic fields in time and replacing time derivatives bycentral differences This results in the following ldquoleap-frogrdquomarching-on-time scheme

119861119899+12

119894= 119861

119899minus12

119894minus Δ119905(sum

119895

1198621

119894119895119864119899

119895)

119864119899+1

119894= 119864

119899

119894+ Δ119905(sum

119895

Υ119894119895119861119899+12

119895)

(48)

where the superscript 119899 denotes the time-step index andΔ119905 is the time increment (assumed uniform for simplicity)The staggering of the fields in both space and time isconsistent with the presence of two staggered hypercubicalspacetime lattices [48 94] that can be viewed as prismaticextrusions along the time coordinate from the two (dual)staggered spatial latticesThe staggering in time also providesa119874(Δ1199052) truncation error Higher-order FDTD schemes withfaster convergence to the continuum can be constructed byusing less local approximations for the spatial derivatives (orequivalently less sparse [⋆120598] and [Υ]) andor for the timederivatives [95ndash97]

92 Finite-Integration Technique Thefinite-integration tech-nique (FIT) [98ndash100] is closely related to FDTD with themain distinction being that in FIT the discretized equationsare derived from the integral form of Maxwellrsquos equationsapplied to every cell Assuming piecewise constant fields overeach cell the latter is equivalent to applying the (discreteversion) of the generalized Stokesrsquo theorem to the cochainsin (23) and (24) Another difference is that the incidencematrices and material (Hodge star) matrices are treatedseparately in FIT In other words metric-free and metric-dependent parts of the equations are factorized a priori in amanner akin to that exposed in Sections 3 and 4 Like FDTDFIT is based on dual staggered lattices and for cubical latticesit turns out that the lowest-order FIT is algorithmicallyequivalent to the lowest-order FDTDThe spatial operators inFIT can all be viewed as discrete incarnations of the exteriorderivative for the various 119901 and as such the exact sequenceproperty of the underlying de Rham complex is automaticallyenforced by construction [55] Because of this it couldperhaps be claimed that FIT provides amore systematic routefor generalizations to irregular lattices than Yeersquos FDTD His-torically FIT generalizations to irregular lattices have reliedon the use of either projection operators [101] or Whitneyforms [102] to construct discrete versions of the Hodge staroperators (or their procedural equivalents) however thesegeneralizations do not necessarily recover the specific formof the discrete Hodge matrix elements expressed in (19)

93 Cell Method Another related discretization methodbased on general principles originally put forth in [47ndash49]is the Cell method [103ndash108] Even though this method does


not rely on Whitney forms for constructing discrete Hodgestar operators (other geometrically based constructions areinstead used) it is nevertheless still based upon the use ofldquodomain-integratedrdquo discrete variables that conform to thenotion of discrete differential forms or cochains of variousdegrees and as such it is naturally suited for irregular latticesThe Cell method also employs metric-free discrete operatorsthat satisfy the exactness property of the de Rham complexand make explicit use of a dual lattice (but not necessarilybarycentric) motivated by the notion of inner and outerorientations The relationships between the various discreteoperators and ldquodomain-integratedrdquo field quantities (cochains)in the Cell method are built into general classification dia-grams referred to as ldquoTonti diagramsrdquo that reproduce correctcommuting diagram properties of the underlying operators[47 48]

94 Mimetic Finite Differences ldquoMimeticrdquo finite-differencemethods originally developed for nonorthogonal hexahe-dral structured lattices (ldquotensor-product gridsrdquo) and laterextended for irregular and polyhedral lattices [109ndash118] alsoshare many of the properties exposed above The thrusthere is towards the construction of discrete versions of thedifferential operators divergence gradient and curl of vectorcalculus having ldquocompatiblerdquo (in the sense of the exactnessproperty of the underlying de Rham complex) domains andranges and such that the resulting discrete equations exactlysatisfy discrete conservation laws In three dimensions thisnaturally leads to the definition of three ldquonaturalrdquo operatorsand three ldquoadjointrdquo operators that can be associated withexterior derivative 119889 and the codifferential 120575 respectively for119901 = 1 2 3 (although the exterior calculus terminology isoften not used explicitly in this context) Metric aspects arenot factored out into Hodge star operators because the latterdo not appear explicitly in the formulation instead theirprocedural analogues are embedded into the definition of thediscrete differential operators themselves through a properlydefined set of discrete inner products for discrete scalarand vector fields In mimetic finite differences the discreteanalogues of the codifferential operator 120575 are full matricesand the matrix-free character of FDTD is lacking even onorthogonal lattices In spite of that an obvious advantage ofmimetic finite differences versus conventional FDTD is thatthe formermethodology provides amore natural extension tononorthogonal and irregular lattices Note that higher-orderversions of mimetic finite differences also exist [119 120]

95 Compatible Discretizations and Finite-Element ExteriorCalculus In recent yearsmuch attention has been devoted tothe development of ldquocompatible discretizationsrdquo an umbrellaterm used to denote spatial discretizations of partial differ-ential equations seeking to provide finite-element spaces thatreproduce the exactness of the underlying de Rham com-plex (or the correct cohomology in topologically nontrivialdomains) [121ndash126] In this program Whitney forms playa role of providing ldquoconformingrdquo vector-valued functional(finite-element) spaces of Sobolev type Specifically Whitney

1-forms recover the space of ldquoNedelec edge-elementsrdquo or curl-conforming Sobolev space H(curl Ω) [127] and Whitney 2-forms recover the space of ldquoRaviart-Thomas elementsrdquo or div-conforming Sobolev space H(div Ω) [128] In this regard arelatively new advance here has been the development of newfinite-element spaces beyond those provided by Whitneyforms based on the Koszul complex [129] The latter iskey for the stable discretization of elastodynamics whichhad been an outstanding problem for many decades Anexcellent first-hand summary of these advances is providedin [130] Another recent comparable approach aimed at thestable discretization of elastodynamics using bundle-valueddiscrete differential forms is described in [131]

We should note that the link between stability conditionsof somemixed finite-elementmethods [127] and the complexof Whitney forms has a long history in the context ofelectrodynamics This link was first established in [55 132]and further explored for example in [18 19 21 23 32 36 61133ndash136]

96 Discrete Exterior Calculus The ldquodiscrete exterior cal-culusrdquo (DEC) is another discretization program aimed atdeveloping ab initio consistent discrete models to describefield theories [91 137ndash141] The main thrust of this pro-gram is not tied to any particular field theory but ratherseeks to develop fundamental discrete tools (field variablesoperators) amenable to tackle a whole gamut of theories(electrodynamics fluid dynamics elastodynamics etc) Thisdiscretization program recognizes the key role played bydiscrete differential forms as well as the need to defineprimal and dual cell complexes There is a perceived focuson the use of circumcentric dual lattices as opposed tobarycentric duals [138 139] (even though the former doesnot admit a metric-free construction) and the program doesnot emphasize the role of Whitney forms (at least on itsearlier stages) On the other hand it recognizes the needto address group-valued differential forms as well as themathematical objects that exist on the dual-bundle spacetogether with the associated operators (such as contractionsand Lie derivatives) in connection to discrete problems inmechanics optimal control and computer visiongraphics[137] A recent discussion on obstacles associated with someof the DEC underpinnings is provided in [142]

Appendices

A Differential Forms and Lattice Fermions

Differential 119901-forms can be viewed as antisymmetric covari-ant tensor fields on rank 119901 Therefore the ingredients dis-cussed above are applicable to any antisymmetric tensor fieldtheory including non-Abelian gauge field theories and eventopological field theories such as Chern-Simons theory [72]However for (Dirac) fermion fields the situation is differentand at first it would seem unclear how differential formscould be used to describe spinors Nevertheless a usefulconnection can indeed be established [1 16 143] To briefly


address this point we consider the lattice transcription of the(one-flavor) Dirac equation here

Needless to say the topic of lattice fermions is vast andwe cannot do much justice to it here we focus only onaspects that are more germane to main theme of this paperIn accordance to the related literature on lattice fermions wework on Euclidean spacetimewith ℏ = 119888 = 1 in this appendixand adopt the repeated index summation convention with120583 120584 as coordinate indices where 119909 is a point in four-dimensional space

It is well known that fermion fields defy a latticedescription with local coupling that gives the correct energyspectrum in the limit of zero lattice spacing and the correctchiral invariance [144] This is formally stated by the no-gotheorem of Nielsen-Ninomiya [145] and is associated to thewell-known ldquofermion-doublingrdquo problem [146] A perhapsless known fact is that it is possible to arrive at a ldquogeometricalrdquointerpretation of the source of this difficulty by consideringthe ldquogeneralizationrdquo of the Dirac equation (120574120583120597120583+119898)120595(119909) = 0given by the Dirac-Kahler equation

(119889 minus 120575)Ψ (119909) = minus119898Ψ (119909) (A1)

The square of the Dirac-Kahler operator can be viewed as thecounterpart of the Dirac operator in the sense that

(119889 minus 120575)2= minus (119889120575 + 120575119889) = minus◻ (A2)

recovers the Laplacian operator in the same fashion as theDirac operator squared does that is (120574120583120597120583)

2= minus120597120583120597

120583= minus◻

where 120574120583 represents Euclidean gamma matricesThe Dirac-Kahler equation admits a direct transcription

on the lattice because both the exterior derivative 119889 and thecodifferential 120575 can be simply replaced by its lattice analoguesas discussed before However for the Dirac equation theanalogy has to further involve the relationship between the 4-component spinor field 120595 and the object Ψ This relationshipwas first established in [16 17] for hypercubic lattices andlater extended to nonhypercubic lattices in [10 147] Theanalysis of [16 17] has shown that Ψ can be represented bya 16-component complex-valued inhomogeneous differentialform

Ψ (119909) =

4

sum

119901=0

120572119901(119909) (A3)

where 1205720(119909) is a (1-component) scalar function of positionor 0-form 1205721(119909) = 1205721

120583(119909)119889119909

120583 is a (4-component) 1-formand likewise for 119901 = 2 3 4 representing 2- 3- and 4-formswith 6- 4- and 1-components respectively By employing thefollowing Clifford algebra product

119889119909120583or 119889119909

120584= 119892

120583120584+ 119889119909

120583and 119889119909

120584 (A4)

as using the anticommutative property of the exterior productand we have

119889119909120583or 119889119909

120584+ 119889119909

120584or 119889119909

120583= 2119892

120583120584 (A5)

which exactly matches the anticommutator result of the 120574120583matrices 120574120583120574120584 + 120574120584120574120583 = 2119892120583120584 This suggests that 119889119909120583 plays

the role of the 120574120583 matrix in the space of inhomogeneousdifferential forms with Clifford product [148] that is

120574120583120597120583 997891997888rarr 119889119909

120583or 120597120583 (A6)

keeping in mind that while 120574120583120597120583 acts on spinors 119889119909120583 or120597120583 = (119889 minus 120575) acts on inhomogeneous differential formsThis analysis leads to a ldquogeometricalrdquo interpretation of thepopular Kogut-Susskind staggered lattice fermions [149 150]because the latter can be made identical to lattice Dirac-Kahler fermions after a simple relabeling of variables [17]

The 16-component object Ψ can be viewed as a 4 times 4matrix that produces a fourfold degeneracy with respect tothe Dirac equation for 120595 This degeneracy is actually not aproblem in the continuum because there is a well-definedprocedure to extract the 4-components of 120595 from those ofΨ [16 17] whereby the 16 scalar equations encoded by (A1)all reduce to the same copy of the four equations encodedby the standard Dirac equation This procedure is performedby a set of ldquoprojection operatorsrdquo that form a group [16151] On the lattice however the operators 119889 and 120597 as wellas lowast (which plays a role on the space of inhomogeneousdifferential forms Ψ analogous to that of 1205745 on the spaceof spinors 120595 [152]) behave in such a way that their actionleads to lattice translations This is because cochains withdifferent 119901 necessarily live on different lattice elements andalso because lowast is a map between different lattice elementsAs a consequence the product operation of such ldquogrouprdquo isnot closed anymoreThis nonclosure also stems from the factthat the lattice operators 119889 and 120575 do not satisfy Leibnitzrsquos rule[148] Because of this the degeneracy of the Dirac equationon the lattice is present at a more fundamental level and isharder to extricate using the Dirac-Kahler description thanthe analogous degeneracy in the continuum In this regard anew approach to identify the extraneous degrees of freedomaway from the continuum was recently described in [153] Inaddition a split-operator approach to solve Dirac equationbased on themethods of characteristics that purports to avoidfermion doubling while maintaining chiral symmetry on thelattice was very recently put forth in [154] This approachpreserves the linearity of the dispersion relation by a splittingof the original problem into a series of one-dimensionalproblems and the use of a upwind scheme with a Courant-Friedrichs-Lewy (CFL) number equal to one which providesan exact time evolution (ie with no numerical dispersioneffects) along each reduced one-dimensional problem Themain (practical) obstacle in this case is the need to use verysmall lattice elements

B Classification of Inconsistencies inNaıve Discretizations

We provide below a rough classification scheme of inconsis-tencies arising from naıve discretizations of the differentialcalculus on irregular lattices

(i) Premetric Inconsistencies of First KindWe call premetric inconsistencies of the first kind those thatare related to the primal or dual lattices taken as separate


objects and that occur when the discretization violates oneor more properties of the continuum theory that is invariantunder homeomorphismsmdashfor example conservations lawsthat relate a quantity on a region 119878 with an associatedquantity on the boundary of the region 120597119878 (a topologicalinvariant) Perhaps the most illustrative example is violationof ldquodivergence-freerdquo conditions caused by improper construc-tion of incidence matrices whereby the nilpotency of the(adjoint) boundary operator 120597 ∘ 120597 = 0 is not observed Thisimplies in a dual fashion that the identity 1198892 = 0 is violated[22] Stated in another way the exact sequence propertyof the underlying de Rham differential complex is violated[155] In practical terms this leads to the appearance spuriouscharges andor spurious (ldquoghostrdquo)modes As the classificationsuggests these properties are not related to metric aspectsof the lattice but only to its ldquotopological aspectsrdquo that ison how discrete calculus operators are defined vis-a-vis thelattice element connectivity Inmoremathematical terms onecan say that the structure of the (co)homology groups ofthe continuum manifold is not correctly captured by the cellcomplex (lattice) We stress again that given any dual latticeconstruction premetric inconsistencies of the first kind areassociated to the primal or dual lattice taken separately andnot necessarily on how they intertwine

(ii) Premetric Inconsistencies of Second KindThe second type of premetric inconsistency is associated tothe breaking of some discrete symmetry of the LagrangianIn mathematical terms this type of inconsistency can occurwhen the bijective correspondence between119901-cells of the pri-mal lattice and (119899 minus 119901)-cells of the dual lattice (an expressionof Poincare duality at the level of cellular homology [156]up to boundary terms) is violated This is typified by ldquonon-reciprocalrdquo constructions of derivative operators where theboundary operator effecting the spatial derivation on the pri-mal lattice 119870 is not the dual adjoint (or the incidence matrixtranspose) of the boundary operator on the dual latticeK forexample the identity 119862119901

119894119895= 119862

119899minus1minus119901

119895119894(up to boundary terms)

used to obtain (25) is violated One basic consequence of thisviolation is that the resulting discrete equations break time-reversal symmetry Consequently the numerical solutionswill violate energy conservation and produce either artificialdissipation or late-time instabilities [22] Many algorithmsdeveloped over the years for hyperbolic partial differentialequations do indeed violate these properties they are dissipa-tive and cannot be used for long integration times [157 158]

It should be noted at this point that lattice field theo-ries invariably break Lorentz covariance and many of thecontinuum Lagrangian symmetries and as a result violateconservation laws (currents) by virtue of Noetherrsquos theoremFor example angularmomentum conservation does not holdexactly on the lattice because of the lack of continuous rota-tional symmetry (note that discrete rotational symmetriescan still be present) However this latter type of symmetrybreaking is of a fundamentally different nature because it isldquocontrollablerdquo that is their effect on the computed solutionsis made arbitrarily small in the continuum limit Moreimportantly discrete transcriptions of the Noetherrsquos theorem

can be constructed for Lagrangian symmetries on a lattice [13159] to yield exact conservation laws of (properly defined)quantities such as discrete energy and discrete momentum[3]

(iii) Hodge Star InconsistenciesIn the third type of inconsistency we include those that arisein connection with metric properties of the lattice Becausethe metric is entirely encoded in the Hodge star operators[22 42 160] such inconsistencies can be simply understoodas inconsistencies on the construction of discrete Hodgestar operators (or their procedural analogues) For exampleit is not uncommon for naıve discretizations in irregularlattices to yield asymmetric discrete Hodge operators asnoted in [161 162] Even if symmetry is observed nonpositivedefinitenessmight ensue that is often associatedwith portionsof the lattice with highly skewed or obtuse cells [101] Lack ofeither of these properties leads to unconditional instabilitiesthat destroy marching-on-time solutions [22] When verylong integration times are needed asymmetry in the discreteHodgematrices can be a problem even if produced at the levelof machine rounding-off errors

Acknowledgments

The author thanks Weng Chew Burkay Donderici Bo Heand Joonshik Kim for discussions The author also thanksthe editorial board for the invitation to contribute with thispaper

References

[1] I Montvay and G Munster Quantum Fields on a LatticeCambridge Monographs on Mathematical Physics CambridgeUniversity Press Cambridge UK 1997

[2] A Zee Quantum Field Theory in a Nutshell Princeton Univer-sity Press Princeton NJ USA 2003

[3] W C Chew ldquoElectromagnetic field theory on a latticerdquo Journalof Applied Physics vol 75 no 10 pp 4843ndash4850 1994

[4] L S Martin and Y Oono ldquoPhysics-motivated numerical solversfor partial differential equationsrdquo Physical Review E vol 57 no4 pp 4795ndash4810 1998

[5] M A H Lopez S G Garcia A R Bretones and R G MartinldquoSimulation of the transient response of objects buried in dis-persive mediardquo in Ultrawideband Short-Pulse Electromagneticsvol 5 Kluwer Academic Press Dordrecht The Netherlands2000

[6] F L Teixeira ldquoTime-domain finite-difference and finite-element methods for Maxwell equations in complex mediardquoIEEE Transactions on Antennas and Propagation vol 56 no 8part 1 pp 2150ndash2166 2008

[7] N H Christ R Friedberg and T D Lee ldquoGauge theory on arandom latticerdquo Nuclear Physics B vol 210 no 3 pp 310ndash3361982

[8] J E Bolander and N Sukumar ldquoIrregular lattice model forquasistatic crack propagationrdquoPhysical Review B vol 71 ArticleID 094106 2005

[9] J M Drouffe and K J M Moriarty ldquoU(2) four-dimensionalsimplicial lattice gauge theoryrdquo Zeitschrift fur Physik C vol 24no 3 pp 395ndash403 1984


[10] M Gockeler ldquoDirac-Kahler fields and the lattice shape depen-dence of fermion flavourrdquo Zeitschrift fur Physik C vol 18 no 4pp 323ndash326 1983

[11] J Komorowski ldquoOn finite-dimensional approximations of theexterior differential codifferential and Laplacian on a Rieman-nian manifoldrdquo Bulletin de lrsquoAcademie Polonaise des Sciencesvol 23 no 9 pp 999ndash1005 1975

[12] J Dodziuk ldquoFinite-difference approach to the Hodge theory ofharmonic formsrdquo American Journal of Mathematics vol 98 no1 pp 79ndash104 1976

[13] R Sorkin ldquoThe electromagnetic field on a simplicial netrdquoJournal of Mathematical Physics vol 16 no 12 pp 2432ndash24401975

[14] DWeingarten ldquoGeometric formulation of electrodynamics andgeneral relativity in discrete space-timerdquo Journal of Mathemati-cal Physics vol 18 no 1 pp 165ndash170 1977

[15] W Muller ldquoAnalytic torsion and 119877-torsion of RiemannianmanifoldsrdquoAdvances inMathematics vol 28 no 3 pp 233ndash3051978

[16] P Becher and H Joos ldquoThe Dirac-Kahler equation andfermions on the latticerdquo Zeitschrift fur Physik C vol 15 no 4pp 343ndash365 1982

[17] J M Rabin ldquoHomology theory of lattice fermion doublingrdquoNuclear Physics B vol 201 no 2 pp 315ndash332 1982

[18] A Bossavit Computational Electromagnetism Variational For-mulations Complementarity Edge Elements ElectromagnetismAcademic Press San Diego Calif USA 1998

[19] A Bossavit ldquoDifferential forms and the computation of fieldsand forces in electromagnetismrdquo European Journal of Mechan-ics B vol 10 no 5 pp 474ndash488 1991

[20] C Mattiussi ldquoAn analysis of finite volume finite element andfinite difference methods using some concepts from algebraictopologyrdquo Journal of Computational Physics vol 133 no 2 pp289ndash309 1997

[21] L Kettunen K Forsman and A Bossavit ldquoDiscrete spaces fordiv and curl-free fieldsrdquo IEEE Transactions on Magnetics vol34 pp 2551ndash2554 1998

[22] F L Teixeira and W C Chew ldquoLattice electromagnetic theoryfrom a topological viewpointrdquo Journal of Mathematical Physicsvol 40 no 1 pp 169ndash187 1999

[23] T Tarhasaari L Kettunen and A Bossavit ldquoSome realizationsof a discreteHodge operator a reinterpretation of finite elementtechniquesrdquo IEEE Transactions on Magnetics vol 35 no 3 pp1494ndash1497 1999

[24] S Sen S Sen J C Sexton and D H Adams ldquoGeometricdiscretization scheme applied to the abelian Chern-Simonstheoryrdquo Physical Review E vol 61 no 3 pp 3174ndash3185 2000

[25] J A Chard and V Shapiro ldquoA multivector data structure fordifferential forms and equationsrdquo Mathematics and Computersin Simulation vol 54 no 1ndash3 pp 33ndash64 2000

[26] P W Gross and P R Kotiuga ldquoData structures for geomet-ric and topological aspects of finite element algorithmsrdquo inGeometric Methods in Computational Electromagnetics F LTeixeira Ed vol 32 of Progress in Electromagnetics Researchpp 151ndash169 EMW Publishing Cambridge Mass USA 2001

[27] F L Teixeira ldquoGeometrical aspects of the simplicial discretiza-tion of Maxwellrsquos equationsrdquo in Geometric Methods in Compu-tational Electromagnetics F L Teixeira Ed vol 32 of Progressin Electromagnetics Research pp 171ndash188 EMW PublishingCambridge Mass USA 2001

[28] T Tarhasaari and L Kettunen ldquoTopological approach to com-putational electromagnetismrdquo inGeometricMethods in Compu-tational Electromagnetics F L Teixeira Ed vol 32 of Progressin Electromagnetics Research pp 189ndash206 EMW PublishingCambridge Mass USA 2001

[29] J Kim and F L Teixeira ldquoParallel and explicit finite-elementtime-domain method for Maxwellrsquos equationsrdquo IEEE Transac-tions on Antennas and Propagation vol 59 no 6 part 2 pp2350ndash2356 2011

[30] A S Schwarz Topology for Physicists vol 308 of GrundlehrenderMathematischenWissenschaften Springer Berlin Germany1994

[31] B He and F L Teixeira ldquoOn the degrees of freedom of latticeelectrodynamicsrdquo Physics Letters A vol 336 no 1 pp 1ndash7 2005

[32] BHe and F L Teixeira ldquoMixed E-B finite elements for solving 1-D 2-D and 3-D time-harmonic Maxwell curl equationsrdquo IEEEMicrowave and Wireless Components Letters vol 17 no 5 pp313ndash315 2007

[33] HWhitneyGeometric IntegrationTheory PrincetonUniversityPress Princeton NJ USA 1957

[34] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman San Francisco Calif USA 1973

[35] G A Deschamps ldquoElectromagnetics and differential formsrdquoProceedings of the IEEE vol 69 pp 676ndash696 1982

[36] P R Kotiuga ldquoMetric dependent aspects of inverse problemsand functionals based on helicityrdquo Journal of Applied Physicsvol 73 no 10 pp 5437ndash5439 1993

[37] F L Teixeira and W C Chew ldquoUnified analysis of perfectlymatched layers using differential formsrdquoMicrowave and OpticalTechnology Letters vol 20 no 2 pp 124ndash126 1999

[38] F L Teixeira and W C Chew ldquoDifferential forms metrics andthe reflectionless absorption of electromagnetic wavesrdquo Journalof Electromagnetic Waves and Applications vol 13 no 5 pp665ndash686 1999

[39] F L Teixeira ldquoDifferential form approach to the analysis ofelectromagnetic cloaking andmaskingrdquoMicrowave and OpticalTechnology Letters vol 49 no 8 pp 2051ndash2053 2007

[40] A H Guth ldquoExistence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theoryrdquo Physical Review D vol21 no 8 pp 2291ndash2307 1980

[41] A Kheyfets and W A Miller ldquoThe boundary of a boundaryprinciple in field theories and the issue of austerity of the lawsof physicsrdquo Journal of Mathematical Physics vol 32 no 11 pp3168ndash3175 1991

[42] R Hiptmair ldquoDiscrete Hodge operatorsrdquo Numerische Mathe-matik vol 90 no 2 pp 265ndash289 2001

[43] BHe and F L Teixeira ldquoGeometric finite element discretizationofMaxwell equations in primal and dual spacesrdquo Physics LettersA vol 349 no 1ndash4 pp 1ndash14 2006

[44] B He and F L Teixeira ldquoDifferential forms Galerkin dualityand sparse inverse approximations in finite element solutionsof Maxwell equationsrdquo IEEE Transactions on Antennas andPropagation vol 55 no 5 pp 1359ndash1368 2007

[45] B Donderici and F L Teixeira ldquoConformal perfectly matchedlayer for the mixed finite element time-domain methodrdquo IEEETransactions on Antennas and Propagation vol 56 no 4 pp1017ndash1026 2008

[46] W L Burke Applied Differential Geometry Cambridge Univer-sity Press Cambridge UK 1985

[47] E Tonti ldquoThe reason for analogies between physical theoriesrdquoApplied Mathematical Modelling vol 1 no 1 pp 37ndash50 1976


[48] E Tonti ldquoFinite formulation of the electromagnetic fieldrdquo inGeometric Methods in Computational Electromagnetics F LTeixeira Ed vol 32 of Progress in Electromagnetics Researchpp 1ndash44 EMW Publishing Cambridge Mass USA 2001

[49] E Tonti ldquoOn the mathematical structure of a large class ofphysical theoriesrdquo Rendiconti della Reale Accademia Nazionaledei Lincei vol 52 pp 48ndash56 1972

[50] K S Yee ldquoNumerical solution of initial boundary value prob-lems involving Maxwellrsquosequation is isotropic mediardquo IEEETransactions on Antennas and Propagation vol 14 no 3 pp302ndash307 1969

[51] A Taflove Computational Electrodynamics Artech HouseBoston Mass USA 1995

[52] R A Nicolaides and X Wu ldquoCovolume solutions of three-dimensional div-curl equationsrdquo SIAM Journal on NumericalAnalysis vol 34 no 6 pp 2195ndash2203 1997

[53] L Codecasa R Specogna and F Trevisan ldquoSymmetric positive-definite constitutive matrices for discrete eddy-current prob-lemsrdquo IEEE Transactions on Magnetics vol 43 no 2 pp 510ndash515 2007

[54] B Auchmann and S Kurz ldquoA geometrically defined discretehodge operator on simplicial cellsrdquo IEEE Transactions onMagnetics vol 42 no 4 pp 643ndash646 2006

[55] A Bossavit ldquoWhitney forms a new class of finite elementsfor three-dimensional computations in electromagneticsrdquo IEEProceedings A vol 135 pp 493ndash500 1988

[56] P W Gross and P R Kotiuga Electromagnetic Theory andComputation A Topological Approach vol 48 of MathematicalSciences Research Institute Publications Cambridge UniversityPress Cambridge UK 2004

[57] A Bossavit ldquoDiscretization of electromagnetic problems theldquogeneralized finite differencesrdquo approachrdquo in Handbook ofNumerical Analysis vol 13 pp 105ndash197North-HollandPublish-ing Amsterdam The Netherlands 2005

[58] B He Compatible discretizations of Maxwell equations [PhDthesis] The Ohio State University Columbus Ohio USA 2006

[59] R Hiptmair ldquoHigher order Whitney formsrdquo in GeometricMethods in Computational Electromagnetics F L Teixeira Edvol 32 of Progress in Electromagnetics Research pp 271ndash299EMW Publishing Cambridge Mass USA 2001

[60] F Rapetti and A Bossavit ldquoWhitney forms of higher degreerdquoSIAM Journal on Numerical Analysis vol 47 no 3 pp 2369ndash2386 2009

[61] J Kangas T Tarhasaari and L Kettunen ldquoReading Whitneyand finite elements with hindsightrdquo IEEE Transactions onMagnetics vol 43 no 4 pp 1157ndash1160 2007

[62] A Buffa J Rivas G Sangalli and R Vazquez ldquoIsogeometricdiscrete differential forms in three dimensionsrdquo SIAM Journalon Numerical Analysis vol 49 no 2 pp 818ndash844 2011

[63] A Back and E Sonnendrucker ldquoSpline discrete differentialformsrdquo in Proceedings of ESAIM vol 35 pp 197ndash202 March2012

[64] S Albeverio and B Zegarlinski ldquoConstruction of convergentsimplicial approximations of quantum fields on Riemannianmanifoldsrdquo Communications in Mathematical Physics vol 132no 1 pp 39ndash71 1990

[65] S Albeverio and J Schafer ldquoAbelian Chern-Simons theory andlinking numbers via oscillatory integralsrdquo Journal of Mathemat-ical Physics vol 36 no 5 pp 2157ndash2169 1995

[66] S O Wilson ldquoCochain algebra on manifolds and convergenceunder refinementrdquo Topology and Its Applications vol 154 no 9pp 1898ndash1920 2007

[67] S O Wilson ldquoDifferential forms fluids and finite modelsrdquoProceedings of the American Mathematical Society vol 139 no7 pp 2597ndash2604 2011

[68] T G Halvorsen and T M Soslashrensen ldquoSimplicial gauge theoryand quantum gauge theory simulationrdquo Nuclear Physics B vol854 no 1 pp 166ndash183 2012

[69] A Bossavit ldquoComputational electromagnetism and geometry(5) the rdquo GalerkinHodgerdquo Journal of the Japan Society of AppliedElectromagnetics vol 8 pp 203ndash209 2000

[70] E Katz and U J Wiese ldquoLattice fluid dynamics from perfectdiscretizations of continuum flowsrdquo Physical Review E vol 58pp 5796ndash5807 1998

[71] B He and F L Teixeira ldquoSparse and explicit FETD viaapproximate inverse hodge (Mass) matrixrdquo IEEE Microwaveand Wireless Components Letters vol 16 no 6 pp 348ndash3502006

[72] D H Adams ldquoA doubled discretization of abelian Chern-Simons theoryrdquo Physical Review Letters vol 78 no 22 pp 4155ndash4158 1997

[73] A Buffa and S H Christiansen ldquoA dual finite element complexon the barycentric refinementrdquo Mathematics of Computationvol 76 no 260 pp 1743ndash1769 2007

[74] A Gillette and C Bajaj ldquoDual formulations of mixed finiteelement methods with applicationsrdquo Computer-Aided Designvol 43 pp 1213ndash1221 2011

[75] J-P Berenger ldquoA perfectly matched layer for the absorption ofelectromagnetic wavesrdquo Journal of Computational Physics vol114 no 2 pp 185ndash200 1994

[76] W C Chew andWHWeedon ldquo3D perfectlymatchedmediumfrommodifiedMaxwellrsquos equations with stretched coordinatesrdquoMicrowave andOptical Technology Letters vol 7 no 13 pp 599ndash604 1994

[77] F L Teixeira and W C Chew ldquoPML-FDTD in cylindrical andspherical gridsrdquo IEEE Microwave and Guided Wave Letters vol7 no 9 pp 285ndash287 1997

[78] F Collino and P Monk ldquoThe perfectly matched layer incurvilinear coordinatesrdquo SIAM Journal on Scientific Computingvol 19 no 6 pp 2061ndash2090 1998

[79] Z S Sacks D M Kingsland R Lee and J F Lee ldquoPerfectlymatched anisotropic absorber for use as an absorbing boundaryconditionrdquo IEEE Transactions on Antennas and Propagationvol 43 no 12 pp 1460ndash1463 1995

[80] F L Teixeira and W C Chew ldquoSystematic derivation ofanisotropic PML absorbing media in cylindrical and sphericalcoordinatesrdquo IEEE Microwave and Guided Wave Letters vol 7no 11 pp 371ndash373 1997

[81] F L Teixeira and W C Chew ldquoAnalytical derivation of a con-formal perfectly matched absorber for electromagnetic wavesrdquoMicrowave and Optical Technology Letters vol 17 no 4 pp 231ndash236 1998


[83] F L Teixeira and W C Chew ldquoOn Causality and dynamicstability of perfectly matched layers for FDTD simulationsrdquoIEEE Transactions onMicrowaveTheory and Techniques vol 47no 63 pp 775ndash785 1999

[84] F L Teixeira andW C Chew ldquoComplex space approach to per-fectly matched layers a review and some new developmentsrdquoInternational Journal of Numerical Modelling vol 13 no 5 pp441ndash455 2000


[85] R W Hockney and J W Eastwood Computer Simulation UsingParticles IOP Publishing Bristol UK 1988

[86] T Z Ezirkepov ldquoExact charge conservation scheme for particle-in-cell simularion with an arbitrary form-factorrdquo ComputerPhysics Communications vol 135 pp 144ndash153 2001

[87] Y A Omelchenko and H Karimabadi ldquoEvent-driven hybridparticle-in-cell simulation a new paradigm for multi-scaleplasma modelingrdquo Journal of Computational Physics vol 216no 1 pp 153ndash178 2006

[88] P J Mardahl and J P Verboncoeur ldquoCharge conservation inelectromagnetic PIC codes spectral comparison of BorisDADIand Langdon-Marder methodsrdquo Computer Physics Communi-cations vol 106 no 3 pp 219ndash229 1997

[89] F Assous ldquoA three-dimensional time domain electromagneticparticle-in-cell codeon unstructured gridsrdquo International Jour-nal of Modelling and Simulation vol 29 no 3 pp 279ndash2842009

[90] A Candel A Kabel L Q Lee et al ldquoState of the art inelectromagnetic modeling for the compact linear colliderrdquoJournal of Physics Conference Series vol 180 no 1 Article ID012004 2009

[91] J Squire H Qin and W M Tang ldquoGeometric integration ofthe Vlaslov-Maxwell system with a variational partcile-in-cellschemerdquo Physics of Plasmas vol 19 Article ID 084501 2012

[92] R A Chilton H- P- and T-refinement strategies for the finite-difference-time-domain (FDTD) method developed via finite-element (FE) principles [PhD thesis]TheOhio State UniversityColumbus Ohio USA 2008

[93] K S Yee and J S Chen ldquoThe finite-difference time-domain(FDTD) and the finite-volume time-domain (FVTD) methodsin solvingMaxwellrsquos equationsrdquo IEEE Transactions on Antennasand Propagation vol 45 no 3 pp 354ndash363 1997

[94] C Mattiussi ldquoThe geometry of time-steppingrdquo in GeometricMethods in Computational Electromagnetics F L Teixeira Edvol 32 of Progress in Electromagnetics Research pp 123ndash149EMW Publishing Cambridge Mass USA 2001

[95] J Fang Time domain finite difference computation for Maxwellequations [PhD thesis] University of California BerkeleyCalif USA 1989

[96] Z Xie C-H Chan and B Zhang ldquoAn explicit fourth-orderstaggered finite-difference time-domain method for Maxwellrsquosequationsrdquo Journal of Computational and Applied Mathematicsvol 147 no 1 pp 75ndash98 2002

[97] S Wang and F L Teixeira ldquoLattice models for large scalesimulations of coherent wave scatteringrdquo Physical Review E vol69 Article ID 016701 2004

[98] T Weiland ldquoOn the numerical solution of Maxwellrsquos equationsand applications in accelerator physicsrdquo Particle Acceleratorsvol 15 pp 245ndash291 1996

[99] R Schuhmann and T Weiland ldquoRigorous analysis of trappedmodes in accelerating cavitiesrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 3 no 12 pp 28ndash36 2000

[100] L Codecasa VMinerva andM Politi ldquoUse of barycentric dualgridsrdquo IEEE Transactions on Magnetics vol 40 pp 1414ndash14192004

[101] R Schuhmann and T Weiland ldquoStability of the FDTD algo-rithm on nonorthogonal grids related to the spatial interpola-tion schemerdquo IEEE Transactions onMagnetics vol 34 no 5 pp2751ndash2754 1998

[102] R Schuhmann P Schmidt and T Weiland ldquoA new Whitney-based material operator for the finite-integration technique on

triangular gridsrdquo IEEE Transactions onMagnetics vol 38 no 2pp 409ndash412 2002

[103] M Bullo F Dughiero M Guarnieri and E Tittonel ldquoIsotropicand anisotropic electrostatic field computation by means of thecell methodrdquo IEEE Transactions onMagnetics vol 40 no 2 pp1013ndash1016 2004

[104] P Alotto ADeCian andGMolinari ldquoA time-domain 3-D full-Maxwell solver based on the cell methodrdquo IEEE Transactions onMagnetics vol 42 no 4 pp 799ndash802 2006

[105] M Bullo F Dughiero M Guarnieri and E Tittonel ldquoNon-linear coupled thermo-electromagnetic problems with the cellmethodrdquo IEEETransactions onMagnetics vol 42 no 4 pp 991ndash994 2006

[106] P Alotto M Bullo M Guarnieri and F Moro ldquoA coupledthermo-electromagnetic formulation based on the cellmethodrdquoIEEE Transactions on Magnetics vol 44 no 6 pp 702ndash7052008

[107] P Alotto F Freschi andM Repetto ldquoMultiphysics problems viathe cell method the role of tonti diagramsrdquo IEEE Transactionson Magnetics vol 46 no 8 pp 2959ndash2962 2010

[108] L Codecasa R Specogna and F Trevisan ldquoDiscrete geometricformulation of admittance boundary conditions for frequencydomain problems over tetrahedral dual gridsrdquo IEEE Transac-tions onAntennas andPropagation vol 60 pp 3998ndash4002 2012

[109] M Shashkov and S Steinberg ldquoSupport-operator finite-difference algorithms for general elliptic problemsrdquo Journal ofComputational Physics vol 118 no 1 pp 131ndash151 1995

[110] J M Hyman and M Shashkov ldquoMimetic discretizations forMaxwellrsquos equationsrdquo Journal of Computational Physics vol 151no 2 pp 881ndash909 1999

[111] J M Hyman andM Shashkov ldquoThe orthogonal decompositiontheorems for mimetic finite difference methodsrdquo SIAM Journalon Numerical Analysis vol 36 no 3 pp 788ndash818 1999

[112] J M Hyman and M Shashkov ldquoAdjoint operators for thenatural discretizations of the divergence gradient and curl onlogically rectangular gridsrdquo Applied Numerical Mathematicsvol 25 no 4 pp 413ndash442 1997

[113] J M Hyman and M Shashkov ldquoMimetic finite differencemethods forMaxwellrsquos equations and the equations of magneticdiffusionrdquo in Geometric Methods in Computational Electromag-netics F L Teixeira Ed vol 32 of Progress in ElectromagneticsResearch pp 89ndash121 EMWPublishing CambridgeMass USA2001

[114] J Castillo and T McGuinness ldquoSteady state diffusion problemson non-trivial domains support operator method integratedwith direct optimized grid generationrdquo Applied NumericalMathematics vol 40 no 1-2 pp 207ndash218 2002

[115] K Lipnikov M Shashkov and D Svyatskiy ldquoThemimetic finitedifference discretization of diffusion problem on unstructuredpolyhedral meshesrdquo Journal of Computational Physics vol 211no 2 pp 473ndash491 2006

[116] F Brezzi and A Buffa ldquoInnovative mimetic discretizationsfor electromagnetic problemsrdquo Journal of Computational andApplied Mathematics vol 234 no 6 pp 1980ndash1987 2010

[117] N Robidoux and S Steinberg ldquoA discrete vector calculus intensor gridsrdquo Computational Methods in Applied Mathematicsvol 11 no 1 pp 23ndash66 2011

[118] K Lipnikov G Manzini F Brezzi and A Buffa ldquoThe mimeticfinite difference method for the 3D magnetostatic field prob-lems on polyhedral meshesrdquo Journal of Computational Physicsvol 230 no 2 pp 305ndash328 2011


[119] V Subramanian and J B Perot ldquoHigher-order mimetic meth-ods for unstructuredmeshesrdquo Journal of Computational Physicsvol 219 no 1 pp 68ndash85 2006

[120] L Beirao da Veiga andGManzini ldquoA higher-order formulationof the mimetic finite difference methodrdquo SIAM Journal onScientific Computing vol 31 no 1 pp 732ndash760 2008

[121] D N Arnold ldquoDifferential complexes and numerical stabilityrdquoin Proceedings of the International Congress of Mathematiciansvol 1 of Plenary Lectures pp 137ndash157 Beijing China 2002

[122] D N Arnold P B Bochev R B Lehoucq R A Nicolaides andM Shashkov EdsCompatible Spatial Discretizations vol 142 ofThe IMAVolumes inMathematics and Its Applications SpringerNew York NY USA 2006

[123] D A White J M Koning and R N Rieben ldquoDevelopmentand application of compatible discretizations of Maxwellrsquosequationsrdquo in Compatible Spatial Discretizations vol 142 ofTheIMAVolumes in Mathematics and Its Applications pp 209ndash234Springer New York NY USA 2006

[124] P Bochev and M Gunzburger ldquoCompatible discretizationsof second-order elliptic problemsrdquo Journal of MathematicalSciences vol 136 no 2 pp 3691ndash3705 2006

[125] D Boffi ldquoApproximation of eigenvalues in mixed form discretecompactness property and application to ℎ119901 mixed finiteelementsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 196 no 37ndash40 pp 3672ndash3681 2007

[126] P Bochev H C Edwards R C Kirby K Peterson and DRidzal ldquoSolving PDEs with Intrepidrdquo Scientific Programmingvol 20 pp 151ndash180 2012

[127] J-C Nedelec ldquoMixed finite elements in R3rdquo Numerische Math-ematik vol 35 no 3 pp 315ndash341 1980

[128] R Hiptmair ldquoCanonical construction of finite elementsrdquoMath-ematics of Computation vol 68 no 228 pp 1325ndash1346 1999

[129] VW Guillemin and S Sternberg Supersymmetry and Equivari-ant de Rham Theory Mathematics Past and Present SpringerBerlin Germany 1999

[130] D N Arnold R S Falk and R Winther ldquoFinite elementexterior calculus homological techniques and applicationsrdquoActa Numerica vol 15 pp 1ndash155 2006

[131] A Yavari ldquoOn geometric discretization of elasticityrdquo Journal ofMathematical Physics vol 49 no 2 article 022901 2008

[132] A Bossavit ldquoMixed finite elements and the complex ofWhitneyformsrdquo inTheMathematics of Finite Elements and ApplicationsVI (Uxbridge 1987) J RWhiteman Ed pp 137ndash144 AcademicPress London UK 1988

[133] M F Wong O Picon and V F Hanna ldquoFinite element methodbased onWhitney forms to solveMaxwell equations in the timedomainrdquo IEEE Transactions on Magnetics vol 31 no 3 pp1618ndash1621 1995

[134] M Feliziani and F Maradei ldquoMixed finite-differenceWhitney-elements time-domain (FDWE-TD) methodrdquo IEEE Transac-tions on Magnetics vol 34 no 5 pp 3222ndash3227 1998

[135] P Castillo J Koning R Rieben andDWhite ldquoA discrete differ-ential forms framework for computational electromagnetismrdquoComputer Modeling in Engineering amp Sciences vol 5 no 4 pp331ndash345 2004

[136] R N Rieben G H Rodrigue and D A White ldquoA highorder mixed vector finite element method for solving the timedependent Maxwell equations on unstructured gridsrdquo Journalof Computational Physics vol 204 no 2 pp 490ndash519 2005

[137] M Dsebrun A N Hirani and J E Mardsen ldquoDiscreteexterior calculus for variational problem in computer vision

and graphicsrdquo in Proceedings of the 42nd IEEE Conference onDecision and Control pp 4902ndash4907 Maui Hawaii USA 2003

[138] A N HiraniDiscrete exterior calculus [PhD thesis] CaliforniaInstitute of Technology Pasadena Calif USA 2003

[139] MDesbrun A NHiraniM Leok and J EMardsen ldquoDiscreteexterior calculusrdquo 2005 httparxivorgabsmath0508341

[140] A Gillette ldquoNotes on discrete exterior calculusrdquo Tech RepUniversity of Texas at Austin Austin Texas USA 2009

[141] J B Perot ldquoDiscrete conservation properties of unstructuresmesh schemesrdquo Annual Review of Fluid Mechanics vol 43 pp299ndash318 2011

[142] P R Kotiuga ldquoTheoretical limitation of discrete exterior cal-culus in the context of computational electromagneticsrdquo IEEETransactions on Magnetics vol 44 pp 1162ndash1165 2008

[143] W Graf ldquoDifferential forms as spinorsrdquo Annales de lrsquoInstitutHenri Poincare A Physique Theorique vol 29 no 1 pp 85ndash1091978

[144] DHAdams ldquoFourth root prescription for dynamical staggeredfermionsrdquo Physical Review D vol 72 Article ID 114512 2005

[145] D Friedan ldquoA proof of the Nielsen-Ninomiya theoremrdquo Com-munications inMathematical Physics vol 85 no 4 pp 481ndash4901982

[146] I F Herbut ldquoTime reversal fermion doubling and the massesof lattice Dirac fermions in three dimensionsrdquo Physical ReviewB vol 83 no 24 Article ID 245445 2011

[147] H Raszillier ldquoLattice degeneracies of fermionsrdquo Journal ofMathematical Physics vol 25 no 6 pp 1682ndash1693 1984

[148] I Kanamori and N Kawamoto ldquoDirac-Kahler femion withnoncommutative differential forms on a latticerdquoNuclear PhysicsBmdashProceedings Supplements vol 129 pp 877ndash879 2004

[149] L Susskind ldquoLattice fermionsrdquo Physical Review D vol 16 no10 pp 3031ndash3039 1977

[150] MG doAmaralMKischinhevsky CAA deCarvalho andFL Teixeira ldquoAn efficient method to calculate field theories withdynamical fermionsrdquo International Journal ofModern Physics Cvol 2 no 2 pp 561ndash600 1991

[151] I M Benn and R W Tucker ldquoThe Dirac equation in exteriorformrdquo Communications in Mathematical Physics vol 98 no 1pp 53ndash63 1985

[152] V de Beauce S Sen and J C Sexton ldquoChiral dirac fermions onthe latice using geometric discretizationrdquo Nuclear Physics BmdashProceedings Supplements vol 129 pp 468ndash470 2004

[153] D H Adams ldquoTheoretical foundation for the index theorem onthe lattice with staggered fermionsrdquo Physical Review Letters vol104 Article ID 141602 2010

[154] F Fillion-Gourdeau E Lorin and A D Bandrauk ldquoNumericalsolution of the time-dependent Dirac equation in coordinatespace without fermion-doublingrdquo Computer Physics Communi-cations vol 183 no 7 pp 1403ndash1415 2012

[155] D N Arnold ldquoDifferential complexes and numerical stabilityrdquoin Proceedings of the International Congress of Mathematiciansvol 1 of Plenary Lecture pp 137ndash157 Higher Ed Press BeijingChina 2002

[156] J R Munkres Topology Pearson 2nd edition 2000[157] M W Chevalier R J Luebbers and V P Cable ldquoFDTD local

grid withmaterial traverserdquo IEEE Transactions on Antennas andPropagation vol 45 pp 411ndash421 1997

[158] M J White Z Yun and M F Iskander ldquoA new 3-D FDTDmultigrid technique with dielectric traverse capabilitiesrdquo IEEETransactions on Microwave Theory and Techniques vol 49 no3 pp 422ndash430 2001


[159] S H Christiansen and T G Halvorsen ldquoA simplicial gaugetheoryrdquo Journal of Mathematical Physics vol 53 no 3 ArticleID 033501 17 pages 2012

[160] A Bossavit ldquoGeneralized finite differencesrsquoin computationalelectromagneticsrdquo in Geometric Methods for ComputationalElectromagnetics F L Teixeira Ed vol 32 of Progress in Electro-magnetics Research pp 45ndash64 EMW Publishing CambridgeMass USA 2001

[161] PThoma and TWeiland ldquoA consistent subgridding scheme forthe finite difference time domainmethodrdquo International Journalof Numerical Modelling vol 9 no 5 pp 359ndash374 1996

[162] K M Krishnaiah and C J Railton ldquoPassive equivalent circuitof FDTD an application to subgriddingrdquoElectronics Letters vol33 no 15 pp 1277ndash1278 1997

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


This topic intersects many disparate application areasFor concreteness we employ classical electrodynamics asthe standard example here however whenever particularlyrelevant to do so we provide brief pointers to other fieldtheories as well Although some familiarity with the exteriorcalculus of differential forms is assumed [18 19 33ndash39] thediscussion is mostly kept at a tutorial level Finally we stressthat this is a review paper and no claim of originality isintended

2 Premetric Lattice Equations

Let us denote the space of differential 119901-forms on a smoothconnected manifold Ω as Λ119901(Ω) From a geometric perspec-tive a differential 119901-form 120572

119901isin Λ

119901(Ω) can be viewed as

an oriented 119901-dimensional density or an object naturallyassociated with 119901-dimensional domains of integration 119880119901such that the lattice contraction (ldquopairingrdquo) below

⟨119880119901 120572119901⟩ ≐ int

119880119901

120572119901 (1)

gives a real number (in our context) for each choice of 119880119901[22] On a lattice K 119880119901 is restricted to be a union ofelements from the finite set of 119901-dimensional 119873119901-orientedlattice elements which we denote by Γ119901(K) = 120590119901119894 119894 =1 119873119901 These are collectively called ldquo119901-chainsrdquo In fourdimensions for example they correspond to the possibleunions of elements from the set of vertices (nodes) 1205900 edges(links) 1205901 facets (plaquettes) 1205902 volume cells (voxels) 1205903 andhypervolume cells 1205904 for 119901 = 1 4 respectively In thediscrete setting the degrees of freedom are reduced to the setof pairings (1) on each one of the lattice elements

On the lattice the pairing above can be understood as amapR119901

Λ119901(Ω) rarr Γ

119901(K) such that

R119901(120572119901) = ⟨120590119901119894 120572

119901⟩ ≐ int

120590119901119894

120572119901 (2)

defines its action on the basis of 119901-chains Note that we useΓ119901(K) to denote the space dual to Γ119901(K) that is the space

119901-cochains The latter can be viewed as the space of ldquodiscretedifferential formsrdquo Because of this and with some abuse oflanguage we use the terminology ldquodifferential formsrdquo andldquocochainsrdquo interchangeably to denote the sameobjects inwhatfollows The mapR119901 is called the de Rham map [22]

The basic differential operator of exterior calculus is theexterior derivative119889 applicable to anynumber of dimensionsThe discretization of 119889 on a general irregular lattice can beeffected by a straightforward application of the generalizedStokesrsquo theorem [22]

int120590119901+1

119889120572119901= int

120597120590119901+1

120572119901 (3)

with 119901 = 0 3 in 119899 = 4 In the above 120597 is the boundaryoperator which simply maps a 119901-dimensional lattice elementto the set of (119901minus1)-dimensional lattice elements that compriseits boundary preserving orientation This theorem sets 120597 asthe formal adjoint of119889 in terms of the pairing given in (1) that

is ⟨120590119901+1 119889120572119901⟩ = ⟨120597120590119901+1 120572

119901⟩ Computationally the boundary

operator can be implemented bymeans of incidencematrices[22 29 40] such that

120597120590119901+1119894 = sum

119895

119862119901

119894119895120590119901119895 (4)

where the indices 119894 and 119895 run over all (119901 + 1)- and 119901-dimensional lattice elements respectively The incidencematrix entries are such that 119862119901

119894119895isin minus1 0 1 for all 119901

with sign determined by the relative orientation of latticeelements 119894 and 119895 The restriction to this set of integer valuesreflects the ldquometric-freerdquo nature of the exterior derivativeonly information about element connectivity that is thecombinatorial aspects of the lattice is involved here It turnsout that the metric is fully encoded by Hodge star operatorsthe discretization of which will be discussed further downbelow

Using (3) and (4) one can write

int120590119901+1119894

119889120572119901= sum

119895

119862119901

119894119895int120590119901119895

120572119901

(5)

for all 119894 so that the derivative operation is replaced by a propersum over 119895 On the lattice the nilpotency of the operators 120597 ∘120597 = 119889 ∘ 119889 = 0 [41] is recovered by the constraint [22]

sum

119896

119862119901+1

119894119896119862119901

119896119895= 0 (6)

for all 119894 and 119895

3 Example Lattice Electrodynamics

We write Maxwellrsquos equations in a four-dimensionalLorentzian manifoldΩ as [34]

119889119865 = 0

119889119866 = lowastJ(7)

where 119889 is the four-dimensional exterior derivative 119865 and 119866are the so-called Faraday and Maxwell 2-forms respectivelyand lowastJ is the charge-current density 3-formThe Hodge staroperator lowast is an isomorphism that maps 119901-forms to (4 minus 119901)-forms and more generally 119901 forms to (119899 minus 119901) forms in a 119899-dimensional manifold and as mentioned before depends onthe metric of Ω [22 23 34 35 42ndash45] The above equationsare complemented by the relation 119866 = lowast119865 which indicatesthat 119865 and 119866 are ldquoHodge dualsrdquo of each other

31 Primal andDual Lattices Since119865 and119866 are 2-forms theyshould be discretized as 2-cochains residing on plaquettes (2-chains) of the 4-dimensional lattice however it is importantto recognize that these two forms are of different types 119865 isan ldquoordinaryrdquo (or ldquonontwistedrdquo) differential form whereas 119866(as well as lowastJ) is a ldquotwistedrdquo (or ldquooddrdquo) differential form [46]The basic difference here has to do with orientation ordinaryforms have internal orientation whereas twisted forms have



119894119895and119862119901





119865 = 119864 and 119889119905 + 119861

119866 = 119863 minus 119867 and 119889119905(8)



119889119864 = minus120597119905119861 (10)

119889119867 = 120597119905119863 + 119869 (11)


119863 = ⋆120598119864

119867 = ⋆120583minus1119861(12)


1(Ω) rarr Λ


2(Ω) rarr Λ

1(Ω) On


1(K) rarr Γ

2(K) and [⋆120583minus1] Γ

2(K) rarr Γ

1(K) The







120596119901[120590119901119894]

≐ 119901

119901

sum

119895=0


(13)



⟨120590119901119894 120596119901[120590119901119895]⟩ = int

120590119901119894

120596119901[120590119901119895] = 120575119894119895 (14)


120596119901[120597

119879120590119901minus1119894] = 119889 (120596

119901minus1[120590119901minus1119894]) (15)







119864 = sum

119894

1198641198941205961

119894

119861 = sum

119894

1198611198941205962

119894

(16)




H =1

2intΩ

(119864 and 119863 + 119867 and 119861) = intΩ

(119864 and ⋆120598119864 + ⋆120583minus1119861 and 119861)

(17)


H = sum

119894

sum

119895

119864119894[⋆120598]119894119895119864119895 +sum

119894

sum

119895

119861119894[⋆120583minus1]119894119895119861119895 (18)


[⋆120598]119894119895 = intΩ

1205961

119894and ⋆120598120596

1

119895

[⋆120583minus1]119894119895= int

Ω

(⋆120583minus11205962

119894) and 120596

2

119895

(19)


119863119894 = sum

119895

[⋆120598]119894119895119864119895

119867119894 = sum

119895

[⋆120583minus1]119894119895119861119895

(20)


H = sum

119894

119864119894119863119894 +sum

119894

119867119894119861119894 (21)



(119864) = Θ119889(119861) where Θ119889




2

119894⟩ = ⟨1205901119895 120596

1

119894⟩ = 120575119894119895

from (14) we get

⟨1205902119895 120597119905119861⟩ = 120597119905sum

119894

119861119894 ⟨1205902119895 1205962

119894⟩ = 120597119905119861119895

⟨1205902119895 119889119864⟩ = ⟨1205971205902119895 119864⟩ = sum

119894

119864119894sum

119896

1198621

119895119896⟨1205901119896 120596

1

119894⟩ = sum

119894

1198621

119895119894119864119894

(22)

so that

minus120597119905119861119894 = sum

119895

1198621

119894119895119864119895 (23)


120597119905119863119894 = sum

119895

1198621

119894119895119867119895 (24)


119894119895= 119862

1

119895119894


120597119905sum

119895

[⋆120598]119894119895119864119895 = sum

119895

1198621

119895119894sum

119896

[⋆120583minus1]119895119896119861119896 (25)


119894119895 as

120597119905119864119894 = sum

119895

Υ119894119895119861119895 (26)


with

Υ119894119895 ≐ sum

119896

sum

119897

[⋆120598]minus1

1198941198961198621

119897119896[⋆120583minus1]119897119895

(27)




minus1






⟨(119899minus119901)119894 ⋆120596119901[120590119901119895]⟩ = int

120590(119899minus119901)119894

⋆120596119901[120590119901119895] = 120575119894119895 (28)


intΩ120596119901and ⋆120596

119901= int

Ω

|120596|2119889119907 (29)





119867 = sum

119894

1198671198941205961

119894

119863 = sum

119894

1198631198941205962

119894

(30)


119864119894 = sum

119895

[⋆120598minus1]119894119895119863119895

119861119894 = sum

119895

[⋆120583]119894119895119867119895

(31)

with

[⋆120598minus1]119894119895 = intΩ

(⋆120598minus11205962

119894) and 120596

2

119895

[⋆120583]119894119895= int

Ω

1205961

119894and ⋆120583120596

1

119895

(32)


2(Ω) rarr Λ

1(Ω) and ⋆120583 Λ

1(Ω) rarr



2(K) rarr Γ

1(K) and

[⋆120583] Γ1(K) rarr Γ





120572119901= 119889120577

119901minus1+ 120575120573

119901+1+ 120594

119901 (33)


119864 = 119889120601 + 120575119860 + 120594 (34)


119864 = 119889120601 + 120575119860 (35)



Θ119889(119864) = 119873

ℎ

119864minus 119873

ℎ

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

= 119873119864 minus 119873119881

(36)



119873119881 minus 119873119864 = 1 minus 119873119865 + 119873119875 (37)


119873119887

119881minus 119873

119887

119864= 2 minus 119873

119887

119865 (38)


(119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881) = (119873119865 minus 119873

119887

119865) minus (119873119875 minus 1) (39)


Θ119889(119864) = 119873

119894119899

119864minus 119873

119894119899

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

(40)


Θ119889(119861) = 119873

119894119899

119865minus (119873119875 minus 1)

= (119873119865 minus 119873119887

119865) minus (119873119875 minus 1)

(41)

so that

Θ119889(119861) = Θ

119889(119864) (42)




119889(119864) and

Θ119889(119867) = Θ

119889(119861)







120577 997888rarr 120577 = int

120577

0

119904120577 (1205771015840) 119889120577

1015840 (43)









⋆120588 = 119902

3

sum

119894=1

⟨119909 1205960

119894⟩120596

0

119894 (44)


119894 and




0


119894≐ 119902120582119894(119909

(119904))




119891

119894minus 120582

119904

119894) where 119902 = 119902120591




119894119895 = 01 12 20 that is

⋆119869 = 119902sum

119894119895

⟨ℓ 1205961

119894119895⟩120596

1

119894119895 (45)



119902 (minus ⟨ℓ 1205961

01⟩ + ⟨ℓ 120596

1

20⟩) = 119902 int

ℓ

(minus1205961

01+ 120596

1

20) (46)


reduces to

119902 intℓ

1198891205820 = 119902 (120582119891

0minus 120582

119904

0) (47)








119861119899+12

119894= 119861

119899minus12

119894minus Δ119905(sum

119895

1198621

119894119895119864119899

119895)

119864119899+1

119894= 119864

119899

119894+ Δ119905(sum

119895

Υ119894119895119861119899+12

119895)

(48)











Appendices







(119889 minus 120575)Ψ (119909) = minus119898Ψ (119909) (A1)




2= minus120597120583120597

120583= minus◻



Ψ (119909) =

4

sum

119901=0

120572119901(119909) (A3)


120583(119909)119889119909


119889119909120583or 119889119909

120584= 119892

120583120584+ 119889119909

120583and 119889119909

120584 (A4)


119889119909120583or 119889119909

120584+ 119889119909

120584or 119889119909

120583= 2119892

120583120584 (A5)



120574120583120597120583 997891997888rarr 119889119909

120583or 120597120583 (A6)









119894119895= 119862







Acknowledgments


References

















































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894119895and119862119901





119865 = 119864 and 119889119905 + 119861

119866 = 119863 minus 119867 and 119889119905(8)



119889119864 = minus120597119905119861 (10)

119889119867 = 120597119905119863 + 119869 (11)


119863 = ⋆120598119864

119867 = ⋆120583minus1119861(12)


1(Ω) rarr Λ


2(Ω) rarr Λ

1(Ω) On


1(K) rarr Γ

2(K) and [⋆120583minus1] Γ

2(K) rarr Γ

1(K) The







120596119901[120590119901119894]

≐ 119901

119901

sum

119895=0


(13)



⟨120590119901119894 120596119901[120590119901119895]⟩ = int

120590119901119894

120596119901[120590119901119895] = 120575119894119895 (14)


120596119901[120597

119879120590119901minus1119894] = 119889 (120596

119901minus1[120590119901minus1119894]) (15)







119864 = sum

119894

1198641198941205961

119894

119861 = sum

119894

1198611198941205962

119894

(16)




H =1

2intΩ

(119864 and 119863 + 119867 and 119861) = intΩ

(119864 and ⋆120598119864 + ⋆120583minus1119861 and 119861)

(17)


H = sum

119894

sum

119895

119864119894[⋆120598]119894119895119864119895 +sum

119894

sum

119895

119861119894[⋆120583minus1]119894119895119861119895 (18)


[⋆120598]119894119895 = intΩ

1205961

119894and ⋆120598120596

1

119895

[⋆120583minus1]119894119895= int

Ω

(⋆120583minus11205962

119894) and 120596

2

119895

(19)


119863119894 = sum

119895

[⋆120598]119894119895119864119895

119867119894 = sum

119895

[⋆120583minus1]119894119895119861119895

(20)


H = sum

119894

119864119894119863119894 +sum

119894

119867119894119861119894 (21)



(119864) = Θ119889(119861) where Θ119889




2

119894⟩ = ⟨1205901119895 120596

1

119894⟩ = 120575119894119895

from (14) we get

⟨1205902119895 120597119905119861⟩ = 120597119905sum

119894

119861119894 ⟨1205902119895 1205962

119894⟩ = 120597119905119861119895

⟨1205902119895 119889119864⟩ = ⟨1205971205902119895 119864⟩ = sum

119894

119864119894sum

119896

1198621

119895119896⟨1205901119896 120596

1

119894⟩ = sum

119894

1198621

119895119894119864119894

(22)

so that

minus120597119905119861119894 = sum

119895

1198621

119894119895119864119895 (23)


120597119905119863119894 = sum

119895

1198621

119894119895119867119895 (24)


119894119895= 119862

1

119895119894


120597119905sum

119895

[⋆120598]119894119895119864119895 = sum

119895

1198621

119895119894sum

119896

[⋆120583minus1]119895119896119861119896 (25)


119894119895 as

120597119905119864119894 = sum

119895

Υ119894119895119861119895 (26)


with

Υ119894119895 ≐ sum

119896

sum

119897

[⋆120598]minus1

1198941198961198621

119897119896[⋆120583minus1]119897119895

(27)




minus1






⟨(119899minus119901)119894 ⋆120596119901[120590119901119895]⟩ = int

120590(119899minus119901)119894

⋆120596119901[120590119901119895] = 120575119894119895 (28)


intΩ120596119901and ⋆120596

119901= int

Ω

|120596|2119889119907 (29)





119867 = sum

119894

1198671198941205961

119894

119863 = sum

119894

1198631198941205962

119894

(30)


119864119894 = sum

119895

[⋆120598minus1]119894119895119863119895

119861119894 = sum

119895

[⋆120583]119894119895119867119895

(31)

with

[⋆120598minus1]119894119895 = intΩ

(⋆120598minus11205962

119894) and 120596

2

119895

[⋆120583]119894119895= int

Ω

1205961

119894and ⋆120583120596

1

119895

(32)


2(Ω) rarr Λ

1(Ω) and ⋆120583 Λ

1(Ω) rarr



2(K) rarr Γ

1(K) and

[⋆120583] Γ1(K) rarr Γ





120572119901= 119889120577

119901minus1+ 120575120573

119901+1+ 120594

119901 (33)


119864 = 119889120601 + 120575119860 + 120594 (34)


119864 = 119889120601 + 120575119860 (35)



Θ119889(119864) = 119873

ℎ

119864minus 119873

ℎ

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

= 119873119864 minus 119873119881

(36)



119873119881 minus 119873119864 = 1 minus 119873119865 + 119873119875 (37)


119873119887

119881minus 119873

119887

119864= 2 minus 119873

119887

119865 (38)


(119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881) = (119873119865 minus 119873

119887

119865) minus (119873119875 minus 1) (39)


Θ119889(119864) = 119873

119894119899

119864minus 119873

119894119899

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

(40)


Θ119889(119861) = 119873

119894119899

119865minus (119873119875 minus 1)

= (119873119865 minus 119873119887

119865) minus (119873119875 minus 1)

(41)

so that

Θ119889(119861) = Θ

119889(119864) (42)




119889(119864) and

Θ119889(119867) = Θ

119889(119861)







120577 997888rarr 120577 = int

120577

0

119904120577 (1205771015840) 119889120577

1015840 (43)









⋆120588 = 119902

3

sum

119894=1

⟨119909 1205960

119894⟩120596

0

119894 (44)


119894 and




0


119894≐ 119902120582119894(119909

(119904))




119891

119894minus 120582

119904

119894) where 119902 = 119902120591




119894119895 = 01 12 20 that is

⋆119869 = 119902sum

119894119895

⟨ℓ 1205961

119894119895⟩120596

1

119894119895 (45)



119902 (minus ⟨ℓ 1205961

01⟩ + ⟨ℓ 120596

1

20⟩) = 119902 int

ℓ

(minus1205961

01+ 120596

1

20) (46)


reduces to

119902 intℓ

1198891205820 = 119902 (120582119891

0minus 120582

119904

0) (47)








119861119899+12

119894= 119861

119899minus12

119894minus Δ119905(sum

119895

1198621

119894119895119864119899

119895)

119864119899+1

119894= 119864

119899

119894+ Δ119905(sum

119895

Υ119894119895119861119899+12

119895)

(48)











Appendices







(119889 minus 120575)Ψ (119909) = minus119898Ψ (119909) (A1)




2= minus120597120583120597

120583= minus◻



Ψ (119909) =

4

sum

119901=0

120572119901(119909) (A3)


120583(119909)119889119909


119889119909120583or 119889119909

120584= 119892

120583120584+ 119889119909

120583and 119889119909

120584 (A4)


119889119909120583or 119889119909

120584+ 119889119909

120584or 119889119909

120583= 2119892

120583120584 (A5)



120574120583120597120583 997891997888rarr 119889119909

120583or 120597120583 (A6)









119894119895= 119862







Acknowledgments


References

















































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119864 = sum

119894

1198641198941205961

119894

119861 = sum

119894

1198611198941205962

119894

(16)




H =1

2intΩ

(119864 and 119863 + 119867 and 119861) = intΩ

(119864 and ⋆120598119864 + ⋆120583minus1119861 and 119861)

(17)


H = sum

119894

sum

119895

119864119894[⋆120598]119894119895119864119895 +sum

119894

sum

119895

119861119894[⋆120583minus1]119894119895119861119895 (18)


[⋆120598]119894119895 = intΩ

1205961

119894and ⋆120598120596

1

119895

[⋆120583minus1]119894119895= int

Ω

(⋆120583minus11205962

119894) and 120596

2

119895

(19)


119863119894 = sum

119895

[⋆120598]119894119895119864119895

119867119894 = sum

119895

[⋆120583minus1]119894119895119861119895

(20)


H = sum

119894

119864119894119863119894 +sum

119894

119867119894119861119894 (21)



(119864) = Θ119889(119861) where Θ119889




2

119894⟩ = ⟨1205901119895 120596

1

119894⟩ = 120575119894119895

from (14) we get

⟨1205902119895 120597119905119861⟩ = 120597119905sum

119894

119861119894 ⟨1205902119895 1205962

119894⟩ = 120597119905119861119895

⟨1205902119895 119889119864⟩ = ⟨1205971205902119895 119864⟩ = sum

119894

119864119894sum

119896

1198621

119895119896⟨1205901119896 120596

1

119894⟩ = sum

119894

1198621

119895119894119864119894

(22)

so that

minus120597119905119861119894 = sum

119895

1198621

119894119895119864119895 (23)


120597119905119863119894 = sum

119895

1198621

119894119895119867119895 (24)


119894119895= 119862

1

119895119894


120597119905sum

119895

[⋆120598]119894119895119864119895 = sum

119895

1198621

119895119894sum

119896

[⋆120583minus1]119895119896119861119896 (25)


119894119895 as

120597119905119864119894 = sum

119895

Υ119894119895119861119895 (26)


with

Υ119894119895 ≐ sum

119896

sum

119897

[⋆120598]minus1

1198941198961198621

119897119896[⋆120583minus1]119897119895

(27)




minus1






⟨(119899minus119901)119894 ⋆120596119901[120590119901119895]⟩ = int

120590(119899minus119901)119894

⋆120596119901[120590119901119895] = 120575119894119895 (28)


intΩ120596119901and ⋆120596

119901= int

Ω

|120596|2119889119907 (29)





119867 = sum

119894

1198671198941205961

119894

119863 = sum

119894

1198631198941205962

119894

(30)


119864119894 = sum

119895

[⋆120598minus1]119894119895119863119895

119861119894 = sum

119895

[⋆120583]119894119895119867119895

(31)

with

[⋆120598minus1]119894119895 = intΩ

(⋆120598minus11205962

119894) and 120596

2

119895

[⋆120583]119894119895= int

Ω

1205961

119894and ⋆120583120596

1

119895

(32)


2(Ω) rarr Λ

1(Ω) and ⋆120583 Λ

1(Ω) rarr



2(K) rarr Γ

1(K) and

[⋆120583] Γ1(K) rarr Γ





120572119901= 119889120577

119901minus1+ 120575120573

119901+1+ 120594

119901 (33)


119864 = 119889120601 + 120575119860 + 120594 (34)


119864 = 119889120601 + 120575119860 (35)



Θ119889(119864) = 119873

ℎ

119864minus 119873

ℎ

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

= 119873119864 minus 119873119881

(36)



119873119881 minus 119873119864 = 1 minus 119873119865 + 119873119875 (37)


119873119887

119881minus 119873

119887

119864= 2 minus 119873

119887

119865 (38)


(119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881) = (119873119865 minus 119873

119887

119865) minus (119873119875 minus 1) (39)


Θ119889(119864) = 119873

119894119899

119864minus 119873

119894119899

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

(40)


Θ119889(119861) = 119873

119894119899

119865minus (119873119875 minus 1)

= (119873119865 minus 119873119887

119865) minus (119873119875 minus 1)

(41)

so that

Θ119889(119861) = Θ

119889(119864) (42)




119889(119864) and

Θ119889(119867) = Θ

119889(119861)







120577 997888rarr 120577 = int

120577

0

119904120577 (1205771015840) 119889120577

1015840 (43)









⋆120588 = 119902

3

sum

119894=1

⟨119909 1205960

119894⟩120596

0

119894 (44)


119894 and




0


119894≐ 119902120582119894(119909

(119904))




119891

119894minus 120582

119904

119894) where 119902 = 119902120591




119894119895 = 01 12 20 that is

⋆119869 = 119902sum

119894119895

⟨ℓ 1205961

119894119895⟩120596

1

119894119895 (45)



119902 (minus ⟨ℓ 1205961

01⟩ + ⟨ℓ 120596

1

20⟩) = 119902 int

ℓ

(minus1205961

01+ 120596

1

20) (46)


reduces to

119902 intℓ

1198891205820 = 119902 (120582119891

0minus 120582

119904

0) (47)








119861119899+12

119894= 119861

119899minus12

119894minus Δ119905(sum

119895

1198621

119894119895119864119899

119895)

119864119899+1

119894= 119864

119899

119894+ Δ119905(sum

119895

Υ119894119895119861119899+12

119895)

(48)











Appendices







(119889 minus 120575)Ψ (119909) = minus119898Ψ (119909) (A1)




2= minus120597120583120597

120583= minus◻



Ψ (119909) =

4

sum

119901=0

120572119901(119909) (A3)


120583(119909)119889119909


119889119909120583or 119889119909

120584= 119892

120583120584+ 119889119909

120583and 119889119909

120584 (A4)


119889119909120583or 119889119909

120584+ 119889119909

120584or 119889119909

120583= 2119892

120583120584 (A5)



120574120583120597120583 997891997888rarr 119889119909

120583or 120597120583 (A6)









119894119895= 119862







Acknowledgments


References

















































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










with

Υ119894119895 ≐ sum

119896

sum

119897

[⋆120598]minus1

1198941198961198621

119897119896[⋆120583minus1]119897119895

(27)




minus1






⟨(119899minus119901)119894 ⋆120596119901[120590119901119895]⟩ = int

120590(119899minus119901)119894

⋆120596119901[120590119901119895] = 120575119894119895 (28)


intΩ120596119901and ⋆120596

119901= int

Ω

|120596|2119889119907 (29)





119867 = sum

119894

1198671198941205961

119894

119863 = sum

119894

1198631198941205962

119894

(30)


119864119894 = sum

119895

[⋆120598minus1]119894119895119863119895

119861119894 = sum

119895

[⋆120583]119894119895119867119895

(31)

with

[⋆120598minus1]119894119895 = intΩ

(⋆120598minus11205962

119894) and 120596

2

119895

[⋆120583]119894119895= int

Ω

1205961

119894and ⋆120583120596

1

119895

(32)


2(Ω) rarr Λ

1(Ω) and ⋆120583 Λ

1(Ω) rarr



2(K) rarr Γ

1(K) and

[⋆120583] Γ1(K) rarr Γ





120572119901= 119889120577

119901minus1+ 120575120573

119901+1+ 120594

119901 (33)


119864 = 119889120601 + 120575119860 + 120594 (34)


119864 = 119889120601 + 120575119860 (35)



Θ119889(119864) = 119873

ℎ

119864minus 119873

ℎ

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

= 119873119864 minus 119873119881

(36)



119873119881 minus 119873119864 = 1 minus 119873119865 + 119873119875 (37)


119873119887

119881minus 119873

119887

119864= 2 minus 119873

119887

119865 (38)


(119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881) = (119873119865 minus 119873

119887

119865) minus (119873119875 minus 1) (39)


Θ119889(119864) = 119873

119894119899

119864minus 119873

119894119899

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

(40)


Θ119889(119861) = 119873

119894119899

119865minus (119873119875 minus 1)

= (119873119865 minus 119873119887

119865) minus (119873119875 minus 1)

(41)

so that

Θ119889(119861) = Θ

119889(119864) (42)




119889(119864) and

Θ119889(119867) = Θ

119889(119861)







120577 997888rarr 120577 = int

120577

0

119904120577 (1205771015840) 119889120577

1015840 (43)









⋆120588 = 119902

3

sum

119894=1

⟨119909 1205960

119894⟩120596

0

119894 (44)


119894 and




0


119894≐ 119902120582119894(119909

(119904))




119891

119894minus 120582

119904

119894) where 119902 = 119902120591




119894119895 = 01 12 20 that is

⋆119869 = 119902sum

119894119895

⟨ℓ 1205961

119894119895⟩120596

1

119894119895 (45)



119902 (minus ⟨ℓ 1205961

01⟩ + ⟨ℓ 120596

1

20⟩) = 119902 int

ℓ

(minus1205961

01+ 120596

1

20) (46)


reduces to

119902 intℓ

1198891205820 = 119902 (120582119891

0minus 120582

119904

0) (47)








119861119899+12

119894= 119861

119899minus12

119894minus Δ119905(sum

119895

1198621

119894119895119864119899

119895)

119864119899+1

119894= 119864

119899

119894+ Δ119905(sum

119895

Υ119894119895119861119899+12

119895)

(48)











Appendices







(119889 minus 120575)Ψ (119909) = minus119898Ψ (119909) (A1)




2= minus120597120583120597

120583= minus◻



Ψ (119909) =

4

sum

119901=0

120572119901(119909) (A3)


120583(119909)119889119909


119889119909120583or 119889119909

120584= 119892

120583120584+ 119889119909

120583and 119889119909

120584 (A4)


119889119909120583or 119889119909

120584+ 119889119909

120584or 119889119909

120583= 2119892

120583120584 (A5)



120574120583120597120583 997891997888rarr 119889119909

120583or 120597120583 (A6)









119894119895= 119862







Acknowledgments


References

















































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2(K) rarr Γ

1(K) and

[⋆120583] Γ1(K) rarr Γ





120572119901= 119889120577

119901minus1+ 120575120573

119901+1+ 120594

119901 (33)


119864 = 119889120601 + 120575119860 + 120594 (34)


119864 = 119889120601 + 120575119860 (35)



Θ119889(119864) = 119873

ℎ

119864minus 119873

ℎ

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

= 119873119864 minus 119873119881

(36)



119873119881 minus 119873119864 = 1 minus 119873119865 + 119873119875 (37)


119873119887

119881minus 119873

119887

119864= 2 minus 119873

119887

119865 (38)


(119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881) = (119873119865 minus 119873

119887

119865) minus (119873119875 minus 1) (39)


Θ119889(119864) = 119873

119894119899

119864minus 119873

119894119899

119881

= (119873119864 minus 119873119887

119864) minus (119873119881 minus 119873

119887

119881)

(40)


Θ119889(119861) = 119873

119894119899

119865minus (119873119875 minus 1)

= (119873119865 minus 119873119887

119865) minus (119873119875 minus 1)

(41)

so that

Θ119889(119861) = Θ

119889(119864) (42)




119889(119864) and

Θ119889(119867) = Θ

119889(119861)







120577 997888rarr 120577 = int

120577

0

119904120577 (1205771015840) 119889120577

1015840 (43)









⋆120588 = 119902

3

sum

119894=1

⟨119909 1205960

119894⟩120596

0

119894 (44)


119894 and




0


119894≐ 119902120582119894(119909

(119904))




119891

119894minus 120582

119904

119894) where 119902 = 119902120591




119894119895 = 01 12 20 that is

⋆119869 = 119902sum

119894119895

⟨ℓ 1205961

119894119895⟩120596

1

119894119895 (45)



119902 (minus ⟨ℓ 1205961

01⟩ + ⟨ℓ 120596

1

20⟩) = 119902 int

ℓ

(minus1205961

01+ 120596

1

20) (46)


reduces to

119902 intℓ

1198891205820 = 119902 (120582119891

0minus 120582

119904

0) (47)








119861119899+12

119894= 119861

119899minus12

119894minus Δ119905(sum

119895

1198621

119894119895119864119899

119895)

119864119899+1

119894= 119864

119899

119894+ Δ119905(sum

119895

Υ119894119895119861119899+12

119895)

(48)











Appendices







(119889 minus 120575)Ψ (119909) = minus119898Ψ (119909) (A1)




2= minus120597120583120597

120583= minus◻



Ψ (119909) =

4

sum

119901=0

120572119901(119909) (A3)


120583(119909)119889119909


119889119909120583or 119889119909

120584= 119892

120583120584+ 119889119909

120583and 119889119909

120584 (A4)


119889119909120583or 119889119909

120584+ 119889119909

120584or 119889119909

120583= 2119892

120583120584 (A5)



120574120583120597120583 997891997888rarr 119889119909

120583or 120597120583 (A6)









119894119895= 119862







Acknowledgments


References

















































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120577 997888rarr 120577 = int

120577

0

119904120577 (1205771015840) 119889120577

1015840 (43)









⋆120588 = 119902

3

sum

119894=1

⟨119909 1205960

119894⟩120596

0

119894 (44)


119894 and




0


119894≐ 119902120582119894(119909

(119904))




119891

119894minus 120582

119904

119894) where 119902 = 119902120591




119894119895 = 01 12 20 that is

⋆119869 = 119902sum

119894119895

⟨ℓ 1205961

119894119895⟩120596

1

119894119895 (45)



119902 (minus ⟨ℓ 1205961

01⟩ + ⟨ℓ 120596

1

20⟩) = 119902 int

ℓ

(minus1205961

01+ 120596

1

20) (46)


reduces to

119902 intℓ

1198891205820 = 119902 (120582119891

0minus 120582

119904

0) (47)








119861119899+12

119894= 119861

119899minus12

119894minus Δ119905(sum

119895

1198621

119894119895119864119899

119895)

119864119899+1

119894= 119864

119899

119894+ Δ119905(sum

119895

Υ119894119895119861119899+12

119895)

(48)











Appendices







(119889 minus 120575)Ψ (119909) = minus119898Ψ (119909) (A1)




2= minus120597120583120597

120583= minus◻



Ψ (119909) =

4

sum

119901=0

120572119901(119909) (A3)


120583(119909)119889119909


119889119909120583or 119889119909

120584= 119892

120583120584+ 119889119909

120583and 119889119909

120584 (A4)


119889119909120583or 119889119909

120584+ 119889119909

120584or 119889119909

120583= 2119892

120583120584 (A5)



120574120583120597120583 997891997888rarr 119889119909

120583or 120597120583 (A6)









119894119895= 119862







Acknowledgments


References

















































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0


119894≐ 119902120582119894(119909

(119904))




119891

119894minus 120582

119904

119894) where 119902 = 119902120591




119894119895 = 01 12 20 that is

⋆119869 = 119902sum

119894119895

⟨ℓ 1205961

119894119895⟩120596

1

119894119895 (45)



119902 (minus ⟨ℓ 1205961

01⟩ + ⟨ℓ 120596

1

20⟩) = 119902 int

ℓ

(minus1205961

01+ 120596

1

20) (46)


reduces to

119902 intℓ

1198891205820 = 119902 (120582119891

0minus 120582

119904

0) (47)








119861119899+12

119894= 119861

119899minus12

119894minus Δ119905(sum

119895

1198621

119894119895119864119899

119895)

119864119899+1

119894= 119864

119899

119894+ Δ119905(sum

119895

Υ119894119895119861119899+12

119895)

(48)











Appendices







(119889 minus 120575)Ψ (119909) = minus119898Ψ (119909) (A1)




2= minus120597120583120597

120583= minus◻



Ψ (119909) =

4

sum

119901=0

120572119901(119909) (A3)


120583(119909)119889119909


119889119909120583or 119889119909

120584= 119892

120583120584+ 119889119909

120583and 119889119909

120584 (A4)


119889119909120583or 119889119909

120584+ 119889119909

120584or 119889119909

120583= 2119892

120583120584 (A5)



120574120583120597120583 997891997888rarr 119889119909

120583or 120597120583 (A6)









119894119895= 119862







Acknowledgments


References

















































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Appendices







(119889 minus 120575)Ψ (119909) = minus119898Ψ (119909) (A1)




2= minus120597120583120597

120583= minus◻



Ψ (119909) =

4

sum

119901=0

120572119901(119909) (A3)


120583(119909)119889119909


119889119909120583or 119889119909

120584= 119892

120583120584+ 119889119909

120583and 119889119909

120584 (A4)


119889119909120583or 119889119909

120584+ 119889119909

120584or 119889119909

120583= 2119892

120583120584 (A5)



120574120583120597120583 997891997888rarr 119889119909

120583or 120597120583 (A6)









119894119895= 119862







Acknowledgments


References

















































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(119889 minus 120575)Ψ (119909) = minus119898Ψ (119909) (A1)




2= minus120597120583120597

120583= minus◻



Ψ (119909) =

4

sum

119901=0

120572119901(119909) (A3)


120583(119909)119889119909


119889119909120583or 119889119909

120584= 119892

120583120584+ 119889119909

120583and 119889119909

120584 (A4)


119889119909120583or 119889119909

120584+ 119889119909

120584or 119889119909

120583= 2119892

120583120584 (A5)



120574120583120597120583 997891997888rarr 119889119909

120583or 120597120583 (A6)









119894119895= 119862







Acknowledgments


References

















































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894119895= 119862







Acknowledgments


References

















































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









review article differential forms in lattice field theories: an...

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