IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002 341

Generating Whitney Forms of Polynomial DegreeOne and Higher

Alain Bossavit

Abstract—A rationale for Whitney forms is proposed: they areseen as a device to approximatemanifolds, with approximation ofdifferential forms as a byproduct. A recursive generating formulais derived. A natural way to build higher-degree forms then follows.

Index Terms—Convergence, edge elements, Whitney forms.

I. INTRODUCTION: NOTATIONS

WE STUDY Whitney forms on a simplicial mesh .Labels , , , are used for nodes, edges, etc., each

with its own inner orientation [1], [2], and , , etc., referto the corresponding Whitney forms.Boldface connotes “dis-crete” objects, especially degrees of freedom (DoF). The setsof nodes, edges, faces, volumes (i.e., tetrahedra), are denotedby . They are linked, as in [2], by the incidencematrices , , (for which the generic notation willoccasionally be used, suffixed byif needed). For instance,

expresses the boundary of faceas a formallinear combination of edges (such a thing is called a-chain,with here). Symbol will serve for , i.e., as a genericnotation for the transposes , , . Recall that is theboundary map for chains: e.g., ,with in this case. Our results hold for any spatialdimension and all simplicial dimensions , but are statedas if was 3 and or 2 (edge and face elements). So weshall assume a specific in proofs (see, e.g., Proposition 2),and prefer , , or , , to or , but it should be cleareach time that the proof has general validity.

We use for the barycentric weight of point with re-spect to node (so : this dual notation will make someformulas more readable). Moreover, we define ,when edge , as well as for face

, etc. When and , wedenote nodeby . Thus refers to (in that case) , andequals . Liberal use is made of the wedge productandexterior derivative of differential geometry, but this should notdeter anyone: Just read as and [resp. , ] as grad

[resp. , ], according to the nature of the operand.

II. WHY WHITNEY FORMS?

In electromagnetism, a natural way to assign DoFs is to havea pair of them at each face and a pairat each edge . Hence arrays , etc.Constitutive laws are then expressed as (or )

Manuscript received July 2, 2001.The author is with Électricité de France, 92141 Clamart, France, and is also

with CNRS, 91192 Palaiseau, France (e-mail: alain.bossavit@der.edfgdf.fr).Publisher Item Identifier S 0018-9464(02)01246-3.

and (variant for Ohm’s law: ), where , , etc.,are square positive definite matrices. These can be built to bediagonal, as in the MAFIA codes [3] or Tonti’s cellular method[1], when one is able to form a dual mesh with cells orthogonalto their primal mates [2]. Otherwise, the Galerkin approach isalways an option: Denoting by ( ), with bold parentheses,the sum , we want ( ) to equal (themagnetic energy, up to a factor 2), when an induction fieldis reconstructed from the DoF array. This is easy withfaceelements : Then is set equal to (which weshall denote by ) and entries of are taken as

. The underlying notion of dual mesh, this time, isthe “barycentric” one [2]. Edge elements play a similar role withrespect to the electric field: from the arrayof edge emfs, onebuilds (note the generic use of ),hence if .

This motivates the search for such interpolants. (Even if oneshuns the Galerkin approach, they are needed for error anal-ysis.) So we shall pretend wedon’t know about , , etc.,and “custom-design” them. One notes that fields shouldhave normal continuity across faces, while fieldsshould betangentially continuous. In view of Faraday’s law, (aliasthe curl of ) should be a combination of face elements.More, the desire to preserve at the discrete level the validity of“exact sequence” properties) like ,which plays the leading role in spurious-modes elimination[4]), puts severe constraints on the structure of thecomplexof Whitney forms (all simplicial dimensions consideredtogether). One may marvel at the fact that these forms,

, (when ),

(when ), and vol , do satisfy theserequirements. Are these expressionsforced on us, in someway, by them? We shall answer (affirmatively) by derivinga generatingrecursiveformula for Whitney forms. Next, ifmore than one DoF per simplex is allowed, what would thecorresponding Whitney forms (of higher polynomial degree,presumably) look like, and are there analogous generatingformulas? As an answer, a heuristic principle (Whitney formsas “partition of unity”) is advanced, which sheds some light onrecent proposals about such higher-degree interpolants [5]–[7].

III. A R ATIONALE FOR WHITNEY FORMS

The question is addressed by taking a detour, changing theviewpoint. Fields, in electromagnetism, are observed via num-bers, such as emf’s, intensities, etc., which correspond to lineintegrals, surface integrals, etc. A field (say, for definiteness,)

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342 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002

then maps a-manifold ( for points, 1 for lines, etc., and2 in our example, where is a surface) to a number, here .Suppose we replaceby a -chain , and letus interpret the DoFs as the elementary values (fluxes,here). Then a natural approximation to is what obtains bysubstituting for , i.e., the sum (bold-face brackets). Hence an approximate knowledge of the field,i.e., of all its measurable attributes, from the DoF array. Thequestion has thus become “how best to representby a chain?”Fig. 1 (where , so a curve replaces ) gives the idea.A definition of Whitney forms then ensues: (for instance),which is, like the field itself, a map from surfaces to num-bers, is defined as the map , whose value we denoteby or by . (Again, such notational redundancywill be useful.) Note that, with this convention,

, which justifies the “ ”notation. A transposition is indeed involved.

We now construct the weights in the case of a “small mani-fold,” such as a point, a segment, etc. (the rigorous underlyingconcept, not used here, is that ofmultivectorat point ). Whatto use for nonsmall ones, i.e., the maps, , etc., from lines,surfaces, etc., to reals, will follow by linearity. The principle ofthis construction is to enforce the followingcommutative dia-gramproperty

(1)

which implies, by transposition, , the re-quired structural property of the Whitney complex. (Ifmoreover , i.e., trivial topology, then

, by transposition, hence, the Whitneyspaces form an exact sequence.) We shall not prove (1) im-mediately, however, rather let it inspire the heuristic approachthat leads to Prop. 1 below. Then, we shall directly prove

[which in turn implies (1)].

IV. FORMULAS

Proposition 1: Starting from , one has

(2)

(and so on, recursively, in higher dimensions).We know the weights for a point : these are the

. Recursively, suppose we know them for a segment,i.e., , and let us try to find . Bylinearity, , where the “join”

is the triangle displayed in Fig. 2, left. So the question is:which 2-chain best represents ? As suggested by Fig. 2, theonly answer consistent with (1) is

(3)

Fig. 1. Representing curvec by a weighted sum of mesh-edges or 1-chain.Graded-line thicknesses are meant to suggest the respective weights. Edges suchase, whose “control domain” (shaded) does not intersectc, get zero weight.Which weights thus to assign is the central issue in our approach to Whitneyforms.

Fig. 2. Left: With edgee = fi; jg and facesfi; j; kg andfi; j; lg orientedas shown, the 2-chain to associate with the “join”x _ e, aliasfi; j; xg, canonly be� (x)fi; j; kg + � (x)fi; j; lg. But this is what (3) says. Right:Same relation between the joinx _ n and the 1-chain� (x)nk + � (x)nl +� (x)nm.

Fig. 3. Just as the barycentric weight of pointx with respect to nodei isvol(xjkl), if one takes vol(ijkl) as unity, the weight of the segmentxy withrespect to edgefi; jg is vol(xykl), and the weight of the trianglexyz withrespect to facefi; j; kg is vol(xyzl).

So ,hence

(4)

On the other hand, ,and the same for , from which we get

for any small triangle , and hence .The reader will easily check that formulas (2)—actually, the

deployment of a unique formula—describe the aforementionedWhitney forms. It is also instructive to recognize in (4) the de-velopment of the 3 3 determinant of the array of barycentriccoordinates of , hence the geometrical interpretation ofthe weights displayed in Fig. 3.

This, let us stress again, was only heuristics: We gave aratio-nale for (2). We now derive from (2) a few formulas, for their

BOSSAVIT: GENERATING WHITNEY FORMS 343

own sake and as a preparation for the proof of the all importantresult, Prop. 4 below.

Proposition 2: .Proof: By (2),

thanks to the relation , because is the samefor all in .

As a corollary, and by using

(5)

and other similar alternatives to (2).Proposition 3: For each -simplex , one has

(i) (ii) (6)

Proof: This is true for . Assume it for . Then

by (6i), hence . Next,

.Now, yet another alternative to (2), but without summation

this time. For any edgesuch that , one has

(7)

This is proved by recursion, using, where and the identity

. We may conclude withProposition 4: .

Proof: Since both terms vanish out of the “star” of, i.e.,the union st() of volumes containing it, one may do as if st()was the whole meshed region. Note thaton st( ). Then,

,by using (6i).

V. “PARTITION OF UNITY” AND HIGHER DEGREEFORMS

The point of what precedes is not only mathematical tidiness.The above results are stepping stones toward those of practicalimport that follow. We now revert to the standard vector analysisframework, where denotes the “proxy” [2] vector field [i.e.,

)] of the form .Recall that barycentric functions sum to 1, thus forming a

“partition of unity”: . Now, using symbol notonly as a label, but also for the vector of modulus area() or-thogonal to face (“vectorial area” of ), one has:

Proposition 5: At all points , for all vectors ,

(8)

This is a case of something true of all simplices, anda consequence of the above construction in which theweights were assigned in order to have

. Replacing by its proxy, and

and by their vectorial areas, we find (8). As a corollary(replace by , by , and integrate in ), one has

(9)

where stands for the vector joining the ends of the dual edgeassociated with in the barycentric dual mesh. We shall arguethat (9) constitutes acriterion to choose , a criterion which isthus satisfied in the Galerkin approach, thanks to (8), and also(by construction) in FIT [3] and the cell method [1], but allowsa much larger choice.

Some notation, first. Recall that stands for the sum. We denote by and the square roots of

the quantities and . The Euclidean norm of anordinary vector is . The generic symbol denotes the mapfrom a field to a DoF array. For instance, .Note that . On the other hand, it’s an old resultabout Whitney forms (see [2] for references) that con-verges to “when ,” i.e., when the mesh is refined ac-cording to some suitable uniformity assumption, described inmore detail in [2]: In brief, only a finite family of local shapescan be produced by successive refinements. Here we shall onlyneed the following, which is a consequence.

Uniformity Hypothesis:There is a constant such that,for all allowed meshes of the domain of interest,

length area volume (10)

Now, let us carry on with the error analysis of the staticproblem , that was tackled in [2]in the case of a diagonal, the starting point of which was theequality

(11)

with convergence if the “consistency error”tends to 0 when . Formula (9) entails a

null consistency error for locally constant (zero degree) fields:Indeed, if , one has and

, hence by (9). Condition (9), if understoodas describing the relation between the “discrete Hodge operator”

and metric properties of the mesh, is thus “almost necessary”[though not necessary in a logical sense: if (9) fails to hold forsome faces, but in vanishingly small proportion with refinement,the right-hand side of (11) can still tend to 0]. We now prove thatit is sufficientas well.

Proposition 6: If (9) and (10) hold, then, assuming piecewisesmooth fields and , the right-hand side of (11) tends to 0 when

.Proof: The right-hand side of (11) is a sum of terms of the

form , whichwe’ll abbreviate as . Local smoothness of thevector proxy of allows us to write it as ,with a constant vector and a residual such that

, where is a constant which dependson the whole field , but not on . The contribution of

344 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002

to , which is , is 0 thanks to(9). The contribution of is, by the mean value theorem,

, for some samplepoints and , whose distance doesn’t exceed the“grain” of the mesh, i.e., the least upper bound of all cells’diameters. This rewrites ,by using (9), which we can bound by area .By a similar reasoning [but now, use the symmetric formula

, which (9) implies], the term isbounded by length . The whole sum over

, now, is therefore bounded by

area length

area length (12)

which tends to 0 with thanks to (10).To summarize, Whitney forms were built in such a way that

the partition of unity property (8) ensues. This property makesthe mass matrix of face elements satisfy a compatibility cri-terion, (9), which we see is the key to the convergence proof.Therefore, we may assert thatWhitney forms of higher polyno-mial degree, too, should satisfy(8), and take this as heuristicguide in the derivation of such forms.

Since the themselves form a partition of unity, we have

(13)

hence the recipe: Attach to edges, faces, etc., the products,, etc., where spans . One can show [thanks to (2)]

that the degree-2 complex thus obtained enjoys the exact se-quence property: If for instance has a di-vergence-free proxy ( ), then there are DoFs such that

.Note however that, because of Proposition 2, these forms are

not linearly independent. For instance, the span of the ,over a tetrahedron, has dimension 20 instead of the apparent24. The main problem with such forms is the interpretation ofDoFs. With standard edge elements, the DoFwas the integralof the 1-form over edge . Here, we cannot

Fig. 4. Left: “Small” edges (some of them are broken lines) associated withforms� w . Right: One of the possible systems of elementary chains in 1–1correspondence with independent DoFs (12 half-edges, four inner small edges,and four three-pronged stars).

expect to find a family of simple 1-chains such that eachwould be the integral of over one of them,and have a null integral over all other chainsof the family. Al-though such a family will exist, the emphasized condition makesit anything but simple. We must be content with less (see [5]):1-cells such that integrals of over them deter-mine the , and in clear 1–1 correspondence with the basisforms (Fig. 4).

Note Added in Proof:The author regrets and warns the readerabout a serious mistake in the proof of Proposition 6, discoveredtoo late in the publishing process to be rectified here. The resultis correct under more stringent conditions, and an updated proofis forthcoming.

REFERENCES

[1] E. Tonti, “A direct formulation of field laws: The cell method,”CMES,vol. 2, pp. 237–258, 2001.

[2] A. Bossavit and L. Kettunen, “Yee-like schemes on a tetrahedral mesh,”Int. J. Numer. Modeling, vol. 12, pp. 129–142, 1999.

[3] T. Weiland, “Time domain electromagnetic field computation with finitedifference methods,”Int. J. Numer. Modeling, vol. 9, pp. 295–319, 1996.

[4] A. Bossavit, “Solving Maxwell’s equations in a closed cavity, and thequestion of spurious modes,”IEEE Trans. Magn., vol. 26, pp. 702–705,Mar. 1990.

[5] A. Kameari, “Symmetric second order edge elements for triangles andtetrahedra,”IEEE Trans. Magn., vol. 35, pp. 1394–1397, May 1999.

[6] Z. Ren and N. Ida, “High-order elements of complete and incompletebases in electromagnetic field computation,”IEEE Trans. Magn., to bepublished.

[7] T. V. Yioultsis and T. D. Tsiboukis, “Development and implementationof second and third order vector finite elements,”IEEE Trans. Magn.,vol. 33, pp. 1812–1815, June 1997.