FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES

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1 MATHEMATICS OF COMPUTATION Volume 83, Number 288, July 2014, Pages S (2013) Article electronically publishe on October 17, 2013 FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES DOUGLAS N. ARNOLD AND GERARD AWANOU Abstract. We evelop a family of finite element spaces of ifferential forms efine on cubical meshes in any number of imensions. The family contains elements of all polynomial egrees an all form egrees. In two imensions, these inclue the serenipity finite elements an the rectangular BDM elements. In three imensions they inclue a recent generalization of the serenipity spaces, an new H(curl) an H(iv) finite element spaces. Spaces in the family can be combine to give finite element subcomplexes of the e Rham complex which satisfy the basic hypotheses of the finite element exterior calculus, an hence can be use for stable iscretization of a variety of problems. The construction an properties of the spaces are establishe in a uniform manner using finite element exterior calculus. 1. Introuction In this paper we evelop a family of finite element spaces S r Λ k (T h ) of ifferential forms, where T h is a mesh of cubes in n 1 imensions, r 1 is the polynomial egree, an 0 k n is the form egree. Thus, in 3 imensions, the space S r Λ k (T n ) is a finite element subspace of the Hilbert space H 1, H(curl), H(iv), or L 2,accoring to whether k =0,1,2,or3. Forn = 1 or 2, the spaces were previously known, while in three (or more) imensions, they are mostly new. Specifically, our construction yiels a new family of H(curl) elements an a new family of H(iv) elements on cubical meshes in three imensions. Our treatment in an exterior calculus framework allows all the spaces an their properties to be evelope together. The spaces combine together in complexes satisfying the basic hypotheses of the finite element exterior calculus [5]. This means that, in aition to their use iniviually, they can be use in pairs, S r+1 Λ k 1 (T h ) S r Λ k (T h )inavariety of mixe finite element applications, with stability an convergence following from the abstract theory of [5]. Element iagrams for some of the spaces are shown in Figure 1. The imension of the shape function spaces are given in Theorem 3.6 below an are tabulate for n 4anr 7in Table 1. Since the evelopment of the first stable mixe finite elements for the Poisson equation by Raviart an Thomas in 1977, such elements have proven to be powerful tools for numerical computation. Their paper [11] introuces a family of finite element iscretizations of the space H(iv, Ω) for a two-imensional omain Ω, one Receive by the eitor April 11, 2012 an, in revise form, December 21, Mathematics Subject Classification. Primary 65N30. Key wors an phrases. Mixe finite elements, finite element ifferential forms, finite element exterior calculus, cubical meshes, cubes. The work of the first author was supporte in part by NSF grant DMS The work of the secon author was supporte in part by NSF grant DMS an the Sloan Founation c 2013 American Mathematical Society

2 1552 DOUGLAS N. ARNOLD AND GERARD AWANOU r =1 r =2 r =3 k =0 S 1 Λ 0 S 2 Λ 0 S 3 Λ 0 k =1 S 1 Λ 1 S 2 Λ 1 S 3 Λ 1 k =2 S 1 Λ 2 S 2 Λ 2 S 3 Λ 2 k =3 S 1 Λ 3 S 2 Λ 3 S 3 Λ 3 Legen: symbols represent the value or moment of the inicate quantities: scalar(1dof); tangential vector fiel (2 DOFs); vector fiel (3 DOFs); tangential component of vector fiel on ege or normal component on face (1 DOF). Figure 1. Element iagrams for S r Λ k in three imensions for r 3. for each polynomial egree. Together with a corresponing iscontinuous piecewise polynomial iscretization of L 2 (Ω), these RT spaces stably iscretize the mixe variational formulation of the Poisson equation on Ω. In [11], versions of the RT elements were given both for meshes of Ω by triangles an by rectangles. Besies the original application of mixe finite elements for the Poisson equation, the RT elements can be use together with the stanar Lagrange finite element iscretization of H 1 (Ω) to give a stable mixe finite element iscretization of the vector Poisson equation curl curl u gra iv u = f, in which the vector variable u is sought in H(iv) an approximate by RT elements, an the scalar variable σ =curlu is sought in H 1 an approximate by Lagrange elements. The RT elements were generalize to three imensions by Néélec [9], with separate generalizations giving iscretizations of H(iv, Ω) an of H(curl, Ω), incluing both tetraheral an cubic mesh generalizations of each.

3 FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES 1553 Table 1. Dimension of S r Λ k (I n ). r k n = n = n = n = In [7], Brezzi, Douglas, an Marini introuce a secon family of finite element iscretizations of H(iv, Ω) in two imensions, for both triangular an rectangular meshes. These BDM elements have also proven to be very useful. Néélec [10] generalize the BDM family to tetraheral meshes in three imensions, giving analogues both for H(iv) an H(curl). He also efine H(iv) an H(curl) elements on cubic meshes in [10]. However, these cannot really be consiere analogues of the BDM elements, as they o not seem to lea to stable mixe finite element pairs. The generalization of the BDM elements to H(iv) on three-imensional omains (but not H(curl)), was also mae by Brezzi, Douglas, Durán, an Fortin in [6]. 1 The paper [6] also introuce an analogue of the rectangular BDM elements to cubic meshes in three imensions. The finite element exterior calculus [4, 5] has greatly clarifie the relation of many of these mixe finite element methos. In exterior calculus the space H(iv) is viewe as a space of ifferential forms of egree n 1inn imensions, while H(curl) is a space of 1-forms, H 1 a space of 0-forms, an L 2 aspaceofn-forms. These spaces are connecte via the e Rham complex, in which the funamental ifferential operators gra, curl, iv (an others in higher imensions) are unifie as the exterior erivative. We refer the reaer to Table 2.2 in [4] for a summary 1 The finite element iscretization of H(iv) in [6] is ientical with that of [10] in the case of tetraheral meshes. However, the interior egrees of freeom iffer. This oes not affect the stability analysis for the mixe Poisson equation, but only the ones given by [10] can be use to establish stability when iscretizing a mixe formulation of the vector Laplacian.

4 1554 DOUGLAS N. ARNOLD AND GERARD AWANOU of corresponences between ifferential forms an vector fiels. In [4,5] two funamental families of finite element ifferential forms are efine on simplicial meshes in n imensions. Tables 5.1 an 5.2 in [4] summarize the corresponences between these spaces of finite element ifferential forms an classical finite element spaces in two an three imensions. The P r Λ k family specializes to the Lagrange elements, the RT elements, an the fully iscontinuous polynomial elements, for k = 0, 1, an 2, respectively, in two imensions, an to the Lagrange elements, Néélec s generalizations of the RT elements to H(curl) an to H(iv), an the iscontinuous elements for k =0,...,3 in three imensions. Taken together, these spaces form a complex which is a finite element subcomplex of the e Rham complex: 0 P r Λ 0 P r Λ 1 P r Λ n 0. The secon family of finite elements iscusse in [4, 5] is the P r Λ k family. For 0- forms an n-forms this brings nothing new, just recapturing the Lagrange an fully iscontinuous polynomial elements, but for 0 <k<n, this is a ifferent family of finite element spaces. For n =2,k = 1 it gives the BDM triangular elements, an for n =3,k =1an2,itgivesNéélec s generalizations of these to H(iv) an H(curl). They combine into a secon finite element e Rham subcomplex 0 P r Λ 0 Pr 1 Λ 1 Pr n Λ n 0. Note that the egree ecreases in this complex, in contrast to the preceeing one. We now turn to the construction of analogous spaces an complexes for cubical meshes. An analogue to the Pr Λ k complex of elements for cubical meshes may be easily constructe via a tensor prouct construction. For an explicit escription in n imensions an for all form egrees k, see [3]. This inclues the tensor prouct Lagrange, or Q r, elements for 0-forms, the rectangular RT elements for 1-forms in 2-D, an the 3-D generalizations of them given in [9]. The space for n-forms (L 2 ) is the fully iscontinuous space of tensor prouct polynomials (shape functions in Q r ). This paper evelops a secon family of finite element spaces for cubical meshes. This family may be viewe as an analogue of the P r Λ k family for cubical meshes. We enote the new family of elements by S r Λ k an efine such a space for all imensions n 1, all polynomial egrees r 1, an all form egrees 0 k n. For 0-forms (iscretization of H 1 ), S r Λ 0 is not equal to Q r, but rather to the serenipity elements in 2-D, their three-imensional extension (which can be foun in many places for small values of r an for general r in [12] an [8]), an to a recent extension to all imensions [2]. For n-forms, S r Λ n uses fully iscontinuous elements with shape functions in P r (not Q r ). For 1-forms in 2-D, the space S r Λ 1 coincies with the rectangular BDM elements of [7]. For 2-forms in 3-D, we believe that the space S r Λ 2, which has not appeare before as far as we know, is the correct analogue of the BDM elements on cubic meshes. It has the same egrees of freeom as the space given in [6] but the shape functions have better symmetry properties. For 1-forms in 3-D, S r Λ 1 is a finite element iscretization of H(curl). To the best of our knowlege, neither the egrees of freeom nor the shape functions for this space have been propose previously. Even for 0-forms, the spaces S r Λ 0 were only iscovere very recently in higher imensions. They are the generalization of the serenipity spaces given by the present authors in [2]. That work was motivate by the search for a finite element

5 FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES 1555 iscretization of the e Rham complex on cubical meshes which is complete in this paper. In the remainer of the paper we will evelop the S r Λ k spaces an their properties in the setting of ifferential forms on cubical meshes of arbitrary imensions. Here we explicitly escribe the spaces which arise in three imensions in traitional finite element terminology by giving their shape functions an egrees of freeom. See also Figure 1. The space S r Λ 3. This is simply the space of piecewise polynomials of egree at most r with no continuity requirements. Obviously this gives a finite element subspace of L 2 (which, unlike the remaining spaces, is efine also for r =0). Its imension is (r +1)(r +2)(r +3)/6. The space S r Λ 2. The shape functions for the space S r Λ 2 on the unit cube I 3 in three imensions are vector polynomials of the form (1) u =(v 1,v 2,v 3 )+curl(x 2 x 3 (w 2 w 3 ),x 3 x 1 (w 3 w 1 ),x 1 x 2 (w 1 w 2 )), with v i,w i P r (I 3 )anw i inepenent of x i. The imension of this space is (r +1)(r 2 +5r + 12)/2. As egrees of freeom for u S r Λ 2 we take u u np, p P r (f), fafaceofi 3 ; u u p, p [P r 2 (I 3 )] 3. f I 3 The space S r Λ 1. The shape functions for the space S r Λ 1 in three imensions are vector polynomials of the form (2) u =(v 1,v 2,v 3 )+(x 2 x 3 (w 2 w 3 ),x 3 x 1 (w 3 w 1 ),x 1 x 2 (w 1 w 2 )) + gra s, with v i P r (I 3 ), w i P r 1 (I 3 ) inepenent of x i,ansis a polynomial of I 3 with superlinear egree at most r + 1, where the superlinear egree of a polynomial is the orinary egree ignoring variables which enter linearly (e.g., the superlinear egree of x 2 1x 2 x 3 3 is 5). The imension of this space is (r +1)(r 2 +5r + 18)/2. As egrees of freeom for u S r Λ 1 we take u u tp, p P r (e), ean ege of I 3 ; e u u np, p [P r 2 (f)] 2,fafaceofI 3 ; u u p, p [P r 4 (I 3 )] 3. f I 3 The space S r Λ 0. Finally, the space S r Λ 0 is the generalize serenipity space of [2]. The shape functions are all polynomials of superlinear egree at most r, an the egrees of freeom are the values at the vertices an the moments of egree at most r 2, r 4, an r 6 on the eges, faces, an interior, respectively. The imension of this space is 8 for r =1,20forr =2,an(r +1)(r 2 +5r + 24)/6 for r 3. As a consequence of the general theory below, the egrees of freeom given are unisolvent for all these spaces, an for any cubical ecomposition T of Ω, the assemble finite element spaces S r Λ k (T ) have exactly the continuity require to belong to H 1 (Ω), H(iv, Ω), H(curl, Ω), an L 2 (Ω), for k =0,...,3, respectively. In other wors, S r Λ k (T ) belongs to the omain of the exterior erivative on k-forms. Moreover, exterior erivative maps S r Λ k (T )intos r 1 Λ k+1 (T ), so we obtain a finite element subcomplex 0 S r Λ 0 (T ) gra S r 1 Λ 1 (T ) curl S r 2 Λ 2 (T ) iv S r 3 Λ 3 (T ) 0

6 1556 DOUGLAS N. ARNOLD AND GERARD AWANOU of the e Rham complex. Finally, if we efine projection operators π k from smooth fiels into the finite element spaces S r Λ k using the egrees of freeom, these commute with exterior ifferentiation. That is, the following iagram commutes: gra 0 C (Ω) [C (Ω)] 3 curl [C (Ω)] 3 iv C (Ω) 0 π 0 π 1 π 2 π 3 0 S r Λ 0 (T ) gra S r 1 Λ 1 (T ) curl S r 2 Λ 2 (T ) iv S r 3 Λ 3 (T ) 0 where, for simplicity, we assume r 3 (otherwise some of the spaces are unefine an some parts of the iagram are not applicable). The remainer of the paper consists of four sections. In Section 2, we recall some key concepts from exterior calculus, particularly the Koszul ifferential an Koszul complex of polynomial ifferential forms, which will be crucial to our construction. We introuce the concept of linear egree, an show that the subcomplex obtaine from the Koszul complex by placing a lower boun on linear egree is exact. This is a key step in the unisolvence proof in Section 3. In Section 3, we efine the spaces S r Λ k on an n-imensional cube by giving the shape functions an egrees of freeom. We erive a number of properties from these efinitions, leaing to a formula for the imension of the space of shape functions an a proof of unisolvence of the egrees of freeom. We then efine projection operators πr k mapping smooth fiels into the finite element spaces S r Λ k an show that the projections commute with the exterior erivative. This accomplishe, the new elements fit squarely into the framework of the finite element exterior calculus given in [5]. Therefore the application of the elements to PDE problems an their numerical analysis oes not require new ieas, an so we o not iscuss that here. The proof of unisolvence in Section 3 hinges on the special case of functions with vanishing trace. The proof in that case is the topic of Section Notation an preliminaries For n 1, the number of imensions, let N n = {1,...,n}, an let Σ(k), 0 k n, enote the set of subsets of N n consisting of k elements. For σ Σ(k) we enote by σ Σ(n k) its complement N n \ σ. For p σ we write σ p for σ \{p} Σ(k 1), an for q σ we write σ + q for σ {q} Σ(k +1). If σ N n an q σ we let ɛ(q, σ) =( 1) l where l =#{p σ p <q}, anset ɛ(q, p, σ) :=ɛ(q, σ)ɛ(p, σ + q p) forq σ, p σ. For later reference, we note that (3) ɛ(q, p, σ) = ɛ(p, σ p)ɛ(q, σ p), which is easily verifie by consiering the cases q<pan q>pseparately. We now recall some basic tools an results of exterior algebra an exterior calculus. These can be foun, for example, in [4, Section 2]. For each σ Σ(k) we enote by σ 1,...,σ k its elements in increasing orer, an by x σ = x σ1 x σk the corresponing basic alternator. A ifferential k-form ω on a omain Ω R n may be written as (4) ω = ω σ x σ, σ Σ(k)

7 FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES 1557 with coefficients ω σ belonging to any esire space of functions on Ω, e.g., L 2 (Ω). A 0-form is simply such a function. The exterior erivative of the ifferential form (4) is ω = q ω σ x q x σ, σ Σ(k) q N n where q = / x q. A ifferential k-form may be contracte with a vector fiel on Ω to give a ifferential (k 1)-form (or zero if k = 0). When the vector fiel is simply the ientity, the resulting operator is the Koszul ifferential. Equivalently, we may efine the Koszul ifferential on the basic alternators by κ(x σ1 x σk )= k ( 1) i+1 x σi x σ1 x σi x σk, i=1 an then exten it to a general ifferential form by linearity: (5) κ( ω σ x σ )= k ω σ ( 1) i+1 x σi x σ1 x σi x σk. σ σ i=1 The operator κ is a grae ifferential, meaning that κ(κω) =0, κ(ω η) =κω η +( 1) k ω κη, if ω is a k-form an η an l-form. The following lemma collects formulas for κω, ω, anκω. Lemma 2.1. If ω is given by (4), then (6) κω = η ζ x ζ, where η ζ = ɛ(q, ζ)x q ω ζ+q, q ζ (7) (8) κω = ω = σ Σ(k) Proof. By efinition so ζ Σ(k 1) ɛ(q, ρ q) q ω ρ q, ν ρ x ρ, where ν ρ = ρ Σ(k+1) q ρ μ σ x σ, where μ σ = [ ] xq q ω σ + ɛ(q, p, σ)x q p ω σ+q p. q σ p σ κω = κx σ = q σ σ Σ(k) q σ Making the change of variable ζ = σ q, so ɛ(q, σ q)x q x σ q, ɛ(q, σ q)x q ω σ x σ q. σ Σ(k), q σ ζ Σ(k 1), q ζ, we obtain (6). The secon result is proven similarly, an the thir follows from the first two. We now turn to ifferential forms with polynomial coefficients. A monomial in n variables is etermine by a multi-inex α of n nonnegative integers: x α = x α 1 1 xα n n. By a form monomial in n variables, we mean the prouct of a monomial

8 1558 DOUGLAS N. ARNOLD AND GERARD AWANOU with a basic alternator: m = x α x σ for some multi-inex α an σ N n. The polynomial egree an the linear egree of m are efine as eg m = α := i α i, leg m =#{ i σ α i =1}. Thus the linear egree of m is the egree of its polynomial coefficient counting only those variables which enter linearly, an excluing variables which enter the alternator. For orinary monomials, i.e., 0-forms, the linear egree is equal to the ifference between the polynomial egree an the superlinear egree which appeare in the introuction. We efine H r Λ k = H r Λ k (R n ) to be the span of the k-form monomials m with eg m = r, anp r Λ k = P r Λ k (R n )= r s=0 H sλ k to be the span of those with eg m r. If k = 0, we may simply write H r an P r. If Ω is a subomain of R n, we efine P r Λ k (Ω) to be the space of restrictions to Ω of the elements of P r Λ k (an similarly for other function spaces). Note that maps H r Λ k into H r 1 Λ k+1 while κ maps H r Λ k into H r+1 Λ k 1. An extremely useful ientity is the homotopy formula ([4, Theorem 3.1]): (9) (κ + κ)ω =(r + k)ω, ω H r Λ k (R n ). We exten the linear egree for form monomials to polynomial ifferential forms by efining leg μ for any μ H r Λ k to be the minimum of the linear egree among all the monomials m in μ. Wesaythatμ is of homogeneous linear egree equal to l if leg m = l for every monomial of μ. WeenotebyH r,l Λ k the space of forms in H r Λ k of linear egree at least l. Obviously, (10) H r,l Λ k (R n )=0, l > min(r, n k). The exterior erivative may ecrease the linear egree of a polynomial ifferential form, but κ an κκ o not. Lemma 2.2. For any ω P r Λ k, leg κω leg ω an leg κκω leg ω. Proof. If m is monomial of ω an l =legm, then it follows irectly from the efinition (5) that the monomials of κm are of linear egree l an/or l +1, so leg κm l leg ω. Since every monomial of κω is a monomial of κm for some monomial m of ω, this implies the first inequality. For the secon we use the ifferential property of κ an the homotopy formula to see that κκm =(κ + κ)κm is a multiple of κm. Therefore leg κκm =legκm leg ω, whichgives the secon inequality. In view of Lemma 2.2, for each l 0, we obtain a complex (11) κ H r 1,l Λ k+1 κ Hr,l Λ k κ. When l = 0 this is the Koszul complex, an exactness follows from the homotopy formula. In fact, the complex (11) is exact for all l 0. Theorem 2.3. For r 1, 0 l<r, 0 k<n, the sequence H r 1,l Λ k+1 κ Hr,l Λ k κ Hr+1,l Λ k 1 is exact. Equivalently, im κ(h r 1,l Λ k+1 )+imκ(h r,l Λ k )=imh r,l Λ k. The proof is ue to Scot Aams an Victor Reiner [1]. Its main ingreient is containe in the following lemma.

9 FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES 1559 Lemma 2.4. Let r 1, l 1, an0 k<n. Suppose that μ H r Λ k is of linear egree at least l 1 an κμ is of linear egree at least l. Then there exists ν H r 1 Λ k+1 such that μ κν is of linear egree at least l. Further, if μ H r Λ n is nonzero, then κμ is of linear egree 0. Proof. For the final statement, concerning n-forms, we write μ = px 1 x n where p H r.sincer 1, p is not constant. But then κμ = i ( 1)i+1 px i x 1 x i x n is easily seen to be of linear egree 0. For 0 k<n, the proof hinges on a canonical form for an element of μ H r Λ k which we establish before proceeing. Let us say that a form monomial x α x σ is full if σ supp(α), the support of α. To each of the monomials m = x α x σ of μ H r Λ k, we associate the increasing sequences ρ an τ with ρ = σ supp(α) an τ = σ \ supp(α). Then m = ±η x τ where η = x α x ρ is a full form monomial which is inepenent of the τ variables (that is, supp(α) ρ is isjoint from τ). Note that leg η =legm. Finally, in the expansion of μ as a linear combination of its monomials, we gather together the terms with the same τ = σ \ supp(α), an in this way write (12) μ = η τ x τ, τ N n where η τ H r Λ k #τ is inepenent of the τ variables, an has all of its monomials full. The expression on the right-han sie of (12) is the esire canonical form of μ. Now we procee with the proof of the lemma. We consier first the special case in which μ is of homogeneous linear egree l 1, an, in this special case, we use inuction on l, thecasel = 0 being known (exactness of the Koszul complex). Expressing μ in the canonical form (12), we have leg η τ = l 1foreachτ (for which the coefficient η τ oes not vanish). Now (13) κμ = κη τ x τ + ±η τ κx τ. The first sum is of homogeneous linear egree l 1 an the secon of homogeneous linear egree l. Since we assume that κμ is of linear egree at least l, the first sum must vanish. But this sum is in canonical form, so we conclue that κη τ =0 for each τ. Invoking the inuctive hypothesis, we can write η τ = κν τ where ν τ has linear egree at least l 1. Let ν = ν τ x τ H r 1 Λ k+1. Then μ κν = η τ x τ κν τ x τ ±ν τ κx τ = ±ν τ κx τ. Expaning the right-han sie into monomials, we see that each has leg at least l, so this completes the proof uner the assumption that μ is of homogeneous linear egree l 1. Next we turn to the general case, in which μ is of linear egree at least l 1. We may split μ as μ + μ with μ of homogeneous linear egree l 1anμ of linear egree at least l. Thenκμ splits into a part of homogeneous linear egree l 1 an a part of homogeneous linear egree l, while leg(κμ ) l. Since, by assumption, leg(κμ) l, the part of κμ with linear egree equal to l 1 must vanish. That is, leg(κμ ) l. Therefore, we may apply the result of the preceing special case to μ

10 1560 DOUGLAS N. ARNOLD AND GERARD AWANOU to obtain ν H r 1 Λ k+1 such that leg(μ κν) l. Thenμ κν =(μ κν)+μ is of linear egree at least l. Finally, we give the proof of Theorem 2.3. Proof. The result is certainly true for l = 0, so we may assume 1 l<r(an so r 2). Suppose ω H r,l Λ k with κω = 0. We must show that there exists η H r 1 Λ k+1 with linear egree at least l, such that κη = ω. Nowμ := ω/(r + k) H r 1 Λ k+1 is of linear egree at least l 1 an satisfies κμ =(κ+κ)ω/(r +k) =ω by (9). If k = n 1, the final sentence of the lemma insures that μ =0. For 0 k<n 1, we apply the lemma with r an k replace by r 1ank +1, respectively, an conclue that there exists ν H r 2 Λ k+2 such that η := μ κν is of linear egree at least l. Clearly, κη = κμ = ω. 3. The S r Λ k spaces Here, the main section of the paper, we efine the polynomial spaces S r Λ k (R n ) we shall use as shape functions (see (17) below) an the egrees of freeom for these (see (21)). We erive a number of properties of these polynomial spaces in Theorems 3.2 through 3.5 an use them to verify unisolvence in Theorem 3.6. The space of shape functions will consist of polynomials of a given egree plus certain aitional terms of higher egree which will be efine in terms of the following auxilliary space: J r Λ k (R n )= l 1 κ H r+l 1,l Λ k+1 (R n ). In view of (10), the sum is finite an (14) J r Λ k (R n ) P r+n k 1 Λ k (R n ). Moreover, the sum is irect, since the polynomial egrees of the summans iffer. The following proposition, which follows irectly from the efinitions, helps to clarify the meaning of this space. Proposition The space l 1 H r+l 1,lΛ k (R n ) is the span of all k-form monomials m with eg m r an eg m leg m r The space J r Λ k (R n ) is the span of κm for all (k +1)-form monomials m with eg m r an eg m leg m r 1. For several values of k, this space can be escribe more explicitly. By (10), (15) J r Λ k (R n )=0, for k = n or n 1, while J r Λ n 2 (R n )=κh r,1 Λ n 1 (R n ). Now, H r,1 Λ n 1 (R n ) is the span of the monomials x i w i θ i,wherew i H r 1 (R n )is inepenent of x i an θ i := ( 1) i 1 x 1... x i... x n.wethenhave κθ i = x j θ j,i x j θ i,j, j<i j>i with θ i,j =( 1) i+j x 1... x i... x j... x n.

11 FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES 1561 Therefore, (16) J r Λ n 2 (R n )={ i<j x i x j (w i w j )θ i,j w i H r 1 (R n ) inepenent of x i }. Finally, we ientify J r Λ 0 (R n ). By Theorem 2.3 in the case k =0,weseethat J r Λ 0 (R n )= l 1 H r+l,lλ 0 (R n ). By Proposition 3.1, this space is the span of monomials of egree >rwhose superlinear egree, that is, eg leg, is at most r. We can now efine the space of polynomial k-forms which we use for shape functions, (17) S r Λ k (R n ):=P r Λ k (R n )+J r Λ k (R n )+J r+1 Λ k 1 (R n ), efine for all r 1anall0 k n. From (14), (18) S r Λ k (R n ) P r+n k Λ k (R n ). Note that, in case k = 0, the final term in (17) vanishes, an, by the characterization of J r Λ 0 (R n )justerive,s r Λ 0 (R n ) consists precisely of the span of all monomials of superlinear egree at most r. This is exactly the serenipity space as efine in [2]. In this case, (14) gives the sharper egree boun (19) S r Λ 0 (R n ) P r+n 1 Λ 0 (R n ). Another case in which the expression for S r Λ k (R n ) can be simplifie is when k = n. By (15), S r Λ n (R n )=P r Λ n (R n ). When k = n 1, we have, by (15), S r Λ n 1 (R n ):=P r Λ n 1 (R n )+J r+1 Λ n 2 (R n ), where the last space is characterize in (16). In the case of three imensions, this is formula (1) given in the introuction, state in the language of exterior calculus. In a similar way, we recover formula (2) for the 3-D H(curl) elements iscusse in the introuction. We now erive several properties of these polynomial spaces. The first limits the monomials that appear in the polynomials in S r Λ k (R n ). Theorem 3.2 (Degree property). For any n, r 1 an 0 k n, the space S r Λ k (R n ) is containe in the span of the k-form monomials m of egree at most r + n k δ k0 for which (20) eg m leg m r +1 δ k0. Proof. The boun r + n k δ k0 on the egree is given in (18) for k>0an in (19) for k = 0, so we nee only show (20). If m is a monomial of an element of P r Λ k (R n ), then eg m r an leg m 0, so (20) hols with r on the righthan sie. If m is a monomial of an element of J r Λ k (R n ), then m occurs in the expansion of κp, wherep is a (k + 1)-form monomial with eg p leg p r 1 (Proposition 3.1). Then eg m =egp + 1 an, by Lemma 2.2, leg m leg p, so again eg m leg m r. Finally, if k>0anm is a monomial of an element of J r+1 Λ k 1 (R n ), then by the argument just given, m is a monomial of q where q is a (k 1)-form monomial with eg q leg q r + 1. Since eg m =egq 1 an leg m leg q 1, we get (20).

12 1562 DOUGLAS N. ARNOLD AND GERARD AWANOU A crucial property of these polynomial form spaces, is that they can be combine to form a subcomplex of the e Rham complex. Theorem 3.3 (Subcomplex property). Let n, r 1, anlet0 <k n. Then S r+1 Λ k 1 (R n ) S r Λ k (R n ). Proof. Withreferenceto(17),wenotethatP r+1 Λ k 1 (R n ) P r Λ k (R n ), an (J r+2 Λ k (R n )) vanishes, so it suffices to prove that J r+1 Λ k 1 (R n ) S r Λ k (R n ), which is immeiate from (17). We next observe that the spaces increase with increasing polynomial egree. Theorem 3.4 (Inclusion property). Let n, r 1, anlet0 k n. Then S r Λ k (R n ) S r+1 Λ k (R n ). Proof. We must show that each of the three summans on the right-han sie of (17) is inclue in S r+1 Λ k (R n ). Clearly, P r Λ k (R n ) P r+1 Λ k (R n ) S r+1 Λ k (R n ), which establishes the first inclusion. Next, we show that J r Λ k (R n ) P r+1 Λ k (R n )+J r+1 Λ k (R n ) S r+1 Λ k (R n ). By Proposition 3.1, elements of J r Λ k (R n ) are of form κm with m a(k +1)-form monomial with eg m r, eg m leg m r 1. By the homotopy formula (9), κm is a constant multiple of κp with p = κm. Wehaveegp =egman leg p leg m 1. Then eg p leg p eg m leg m +1 r. Ifegp = r, eg κm = r +1 an κm P r+1 Λ k (R n ). On the other han, if eg p r + 1, by Proposition 3.1, κm J r+1 Λ k (R n ). This establishes the secon inclusion. To complete theproof,weshowthatj r+1 Λ k 1 (R n ) S r+1 Λ k (R n ). Since J r+1 Λ k 1 (R n ) S r+2 Λ k 1 (R n ) (by the inclusion just establishe), we infer from the subcomplex property that J r+1 Λ k 1 (R n ) S r+2 Λ k 1 (R n ) S r+1 Λ k (R n ). The thir property of the S r Λ k spaces that we establish concerns traces on hyperplanes. Consier a hyperplane f of R n of the form x i = c for some 1 i n an some constant c. The variables x j, j i, form a coorinate system for f, so we may ientify f with R n 1 an consier the space S r Λ k (f). It is a space of polynomial k-forms on f, an so vanishes if k = n. Next, we consier the trace on f of a ifferential form in n variables (efine as the pullback of the form through the inclusion map f R n ). Let σ Σ(k), an let ω σ be a function of n variables. Then { 0, i σ, tr f (ω σ x σ )= (tr f ω σ ) x σ, i / σ. In the last expression, tr f ω σ enotes the function of n 1 variables obtaine by setting x i = c an we view x σ as a basic alternator in the n 1 variablesx j, j i. The trace property states that if u S r Λ k (R n ), then tr f u, which is a polynomial k-form on f, belongs to S r Λ k (f). Theorem 3.5 (Trace property). Let n, r 1, 0 k n, anletf be a hyperplane of R n obtaine by fixing one coorinate. Then tr f S r Λ k (R n ) S r Λ k (f). (This inclusion will be shown to be an equality in (27) below.)

13 FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES 1563 Proof. Without loss of generality, we assume that f = { x R n x 1 = c }. First let us comment on the Koszul operator applie to a polynomial ifferential form on f. Such a form may be written as a linear combination of monomials x α x σ where α 1 =0an1/ σ. Referring to (5) we see that, if we view x α x σ as a form monomial in n variables an take the Koszul ifferential, the result is the same as if we view it as form monomial in n 1 variables an take the Koszul ifferential. Thus we nee not istinguish between the Koszul ifferential on R n an that on f. We will prove the theorem by inuction on k. Fork = 0 we recall that S r Λ 0 (R n ) is the serenipity space spanne by the monomials of superlinear egree at most r, an, of course, the superlinear egree oes not increase when taking the trace. Hence, tr f S r Λ 0 (R n ) S r Λ 0 (f). To prove the theorem for k>0, assume that it hols with k replace by k 1. In light of (17) we nee to show that the traces of each of the three spaces P r Λ k (R n ), J r Λ k (R n ), an J r+1 Λ k 1 (R n )arecontainein S r Λ k (f) =P r Λ k (f)+j r Λ k (f)+j r+1 Λ k 1 (f). For the P r Λ k (R n ), this is evient, since tr f P r Λ k (R n ) P r Λ k (f). Next, we establish that tr f J r+1 Λ k 1 (R n ) S r Λ k (f). Inee, tr f J r+1 Λ k 1 (R n )=tr f J r+1 Λ k 1 (R n ) tr f S r+1 Λ k 1 (R n ) S r+1 Λ k 1 (f) S r Λ k (f), where we have use, in turn, the commutativity of the trace with exterior ifferentiation, (17), the inuctive hypothesis, an Theorem 3.3. By Proposition 3.1, in orer to show that tr f J r Λ k (R n ) S r Λ k (f), an to complete the proof, it suffices to show that tr f κm S r Λ k (f) whenever m is a (k + 1)-form monomial with eg m leg m r 1. We write m as x α x σ,an consier separately the cases 1 / σ an 1 σ. Assuming 1 / σ, letp be the (k + 1)-form monomial obtaine restricting m to f, i.e., by setting x 1 = c in the coefficient x α. Then tr f κm = κp. If m is linear in x 1,thenegp =egm 1anlegp =legm 1. Otherwise eg p eg m an leg p =legm. In either event, egp leg p eg m leg m r 1, so κp S r Λ k (f) (again using Proposition 3.1). Assuming, instea, that 1 σ, we may write m = x α x σ = x α 1 1 xβ x 1 x τ, where β is a multi-inex with β 1 =0anτ Σ(k) hasτ 1 > 1. Then p := tr f κm = c α 1+1 x β x τ is a k-form monomial inepenent of x 1 with eg p eg m an leg p =legm. Wearetryingtoshowthatp S r Λ k (f). This is obvious if eg p r, sowemay assume that eg p>r. We shall show that both κp an κp belong to S r Λ k (f), which suffices by (9). Now eg p =egp 1 eg m 1anlegp leg p 1=legm 1, so eg p leg p eg m leg m r 1. Therefore κp J r Λ k (f) S r Λ k (f), as require. Finally, we show that κp J r+1 Λ k 1 (f), whence κp S r Λ k (f) aswell. By Proposition 3.1 this hols, since eg p r+1 an eg p leg p eg m leg m r (even r 1). This conclues the proof.

14 1564 DOUGLAS N. ARNOLD AND GERARD AWANOU Having efine the space S r Λ k (R n ) of polynomial ifferential forms, we turn now to the efinition of the associate finite element space on a cubical mesh. As usual the finite element space is efine element by element, by specifying a space of shape functions an a set of egrees of freeom on each cube T in the mesh. (More generally the element T may be a right rectangular prism, that is, the Cartesian prouct of n close intervals of positive finite length.) As shape functions on T we use S r Λ k (T ), the restriction of the above polynomial space to the cube. The egrees of freeom have a very simple expression. Writing Δ (T ) for the set of -imensional faces of T,theyaregivenby (21) μ tr f μ ν, ν P r 2( k) Λ k (f), f Δ (T ), k min(n, r/2 + k). Since f im P r 2( k) Λ k (f) =imp r 2( k) (f) ( ) = k ( r +2k )( k for f a face of imension, an since there are 2 n ( n ) -imensional faces of an n-cube, the number of egrees of freeom in (21) is given by (22) min(n, r/2 +k) =k ( )( n r +2k 2 n )( k We now turn to one of the main results of this paper, the proof that the egrees of freeom (21) are unisolvent for S r Λ k (T ). Theorem 3.6 (Unisolvence). Let n, r 1 an 0 k n, anlett be a cube in R n.then (1) im S r Λ k (T ) is given by (22). (2) If μ S r Λ k (T ) an all the egrees of freeom in (21) vanish, then μ 0. Using the trace property, we will reuce the proof of (2) to the case where μ belongs to the space S r Λ k (T ):={ μ S r Λ k (T ) tr f μ = 0 on each face f Δ n 1 (T ) }, the subspace with vanishing traces. This case is given in the following proposition. Proposition 3.7. If μ S r Λ k (T ) an (23) μ ν =0, ν P r 2(n k) Λ n k (T ), then μ vanishes. T We efer the proof of this proposition to Section 4. Now, assuming this result, we prove Theorem 3.6. Proof. We begin with the first statement of the theorem. Since J r Λ k (T )isinthe range of κ, an since the homotopy formula implies that no nonzero ifferential form is in the range of both κ an, the sum on the left of (17) is irect. The homotopy formula implies as well that is injective on the range of κ. Therefore im S r Λ k (T )=imp r Λ k (T )+ im κh r+l 1,l Λ k+1 (T )+ im κh r+l,l Λ k (T ). l 1 l 1 ). )

15 FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES 1565 Applying Theorem 2.3, this becomes im S r Λ k (T )=imp r Λ k (T )+ l 1 im H r+l,l Λ k (T ) = im[p r Λ k (T )+ l 1 H r+l,l Λ k (T )]. The space in brackets is exactly the span of the k-form monomials m with eg m leg m r, an hence we nee only count these monomials. This gives ( ) n (24) im S r Λ k (T )= #A(r, k, n), k where ( n k) is the number of basic alternators an A(r, n, k) isthesetofmonomials p in n variables which are linear in some number l of the first n k variables, x 1,...,x n k,withegp l r. Now we count the elements of A(r, n, k). For any monomial p in n variables let J N n k be the set of inices for which x i enters p superlinearly, let 0 be the carinality of J, anfori N n k \ J, leta i =0or 1 accoring to whether p is of egree 0 or 1 in x i.then p = ( j J x 2 j ) q ( i N n k \J where q is a monomial in the variables inexe by J an the last k variables. With l the number of a i equalto1,wehaveegp =2 +egq + l. Thusegp l r if an only if eg q r 2. Thus we may uniquely specify an element of A(r, k, n) by choosing 0, choosing the set J consisting of of the n k variables (for ) possibilities), choosing the monomial q of egree at most r 2 in the + k variables ( ( ) r +k +k possibilities), an choosing the exponent ai to be either 0 or 1 for the n k remaining inices (2 n k possibilities). Thus min(n k, r/2 ) ( )( ) n k r + k #A(r, k, n) = 2 n k + k =0 (25) min(n, r/2 +k) ( )( ) n k r +2k = 2 n, k which there are ( n k =k where the secon sum comes from a change of the summation inex ( k). Substituting (25) into (24) an using the binomial ientity ( )( ) ( )( ) n n k n =, k k k we conclue that min(n, r/2 +k) ( )( )( ) n r +2k (26) im S r Λ k (T )= 2 n. k =k This completes the proof of the imension formula for S r Λ k (R n ). The proof of unisolvence is easily complete base on the trace property an Proposition 3.7. We use inuction on the imension n, the one-imensional case being trivial. Suppose μ S r Λ k (T ) an all its egrees of freeom vanish. For any face f of imension n 1, tr f μ S r Λ k (f) an all the egrees of freeom for it x a i i ),

16 1566 DOUGLAS N. ARNOLD AND GERARD AWANOU vanish. By inuction tr f μ 0onf. This implies that μ S r Λ k (T ), an we invoke Proposition 3.7 to conclue that μ vanishes ientically. We remark that, as a corollary of unisolvence, we may strengthen the result of Theorem 3.5 to equality (27) tr f S r Λ k (R n )=S r Λ k (f). Inee, let T be a cube with one face containe in the hyperplane f. ThenT f is an (n 1)-imensional cube an any ν S r Λ k (f) is uniquely etermine by the egrees of freeom for the S r Λ k (T f). Now we may etermine an element μ S r Λ k (R n ) by assigning the egrees of freeom for the space S r Λ k (T ) arbitrarily. In particular, we may choose the values of those egrees of freeom associate to the face T f an its subfaces to be the same as those for ν. Thentr f μ S r Λ k (f) (by Theorem 3.5), an tr f μ an ν have ientical egrees of freeom, an so they are equal (by Theorem 3.6). With the efinition of the spaces complete, we use the subcomplex property, Theorem 3.3, to efine a subcomplex of the e Rham complex on the cube R S r Λ 0 (T ) S r 1 Λ 1 (T ) S r n Λ n (T ) 0. To show that this complex is exact, we efine the canonical projection πr k : CΛk (T ) S r Λ k (T ), associate to the unisolvent egrees of freeom. That is, πr kμ S rλ k (T )isetermine by the equations tr f (πr k μ) ν = tr f μ ν, ν P r 2( k) Λ k (f), f Δ (T ), f Then, the following iagram commutes: f R C Λ 0 (T ) C Λ 1 (T ) πr 0 πr 1 1 R S r Λ 0 (T ) S r 1 Λ 1 (T ) k min(n, r/2 + k). C Λ n (T ) 0 πn r n S r n Λ n (T ) 0 The proof of commutativity is base on two basic properties of ifferential forms: (1) the commutativity of trace an exterior ifferentiation, tr f ω = f tr f ω,an (2) integration by parts, which for ifferential forms can be written as ω η =( 1) k 1 ω η + tr Ω ω tr Ω η, Ω Ω for a k-form ω anan(n k 1)-form η on an n-imensional omain Ω. See [4, Lemma 4.24] for the same argument applie to simplicial elements. Since the top row of the iagram, the e Rham complex on the cube, is exact, the commutativity of the iagram implies that the bottom row is exact as well. Having efine the finite element space S r Λ k (T )onasinglecubet an establishe its properties, the space S r Λ k (T h ) associate to a cubical mesh T h is efine through the usual finite element assembly. In view of the unisolvence result Theorem 3.6 an the trace result (27), the egrees of freeom associate to a face of the cube an its subfaces etermine the trace of the finite element ifferential form on the face. It follows that S r Λ k (T h ) HΛ k (Ω) (see [4, Section 5.1]). Ω

17 FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES Unisolvence over the space with vanishing traces We conclue the paper with the proof of Proposition 3.7, which is base on the following lemma. Lemma 4.1. Suppose that η = σ Σ(k) η σ x σ where, for each σ Σ(k), η σ is a homogeneous polynomial which is superlinear in all the σ variables. Further suppose that leg κη 1 an leg κη 1. Thenη =0. Proof of Lemma 4.1. By (8) we have κη = σ Σ(k) μ σ x σ,where (28) μ σ = [ ] xq q η σ + ɛ(q, p, σ)x q p η σ+q p. q σ p σ Since leg κη 1, each monomial of the polynomial μ σ is linear in at least one σ variable. For any σ Σ(k) an any subset τ of σ, letsσ τ be the span of the (orinary, 0-form) monomials which are inepenent of the τ variables but epen on all of the other σ variables, an let Sσ enote the span of the monomials which are superlinear in all the σ variables. Denote by Pσ τ the projection onto Sσ.Thatis, τ if p = p m m where the sum is over all monomials m an the coefficients p m are real numbers, all but finitely many zero, then Pσ τ p := m S p mm. Similarly, we σ τ enote by Q τ σ the projection onto Sσ τ Sσ. We now calculate the result of applying Q τ σ to both sies of (28). First we note that (29) Q τ σμ σ =0, since every monomial of μ σ is linear in at least one σ variable, an so none of them belong to Sσ.Next,foreachq σ, (30) Q τ σ(x q q η σ )=Pσ τ (x q q η σ )=x q q (Pσ τ η σ ), with the first inequality holing since each monomial of η σ, an therefore also of x q q η σ, is superlinear in all of the σ variables. Now we etermine the action of Q τ σ on the terms of the secon sum on the right-han sie of (28). For any q σ an p σ, we claim that { (31) Q τ σ (x 0, p τ, q p η σ+q p )= x q p (Pσ+q pη τ σ+q p ), p σ \ τ. Inee, η σ+q p is superlinear in x p, so every monomial of x q p η σ+q p epens on x p. This implies that the projection is 0 in the case p τ. In case p σ \ τ, we write x q p η σ+q p = x q p m, where the sum is over the monomials m of η σ+q p. Since neither p or q belongs to τ, the monomial m := x q p m is inepenent of the τ variables if an only if the same is true of the monomial m, an, since m always epens on x p, it epens on all of the σ \ τ variables if an only if m epens on all of the (σ p) \ τ variables. Further, m is always superlinear in all of the σ variablesexceptpossiblyx q, an it is superlinear in x q if an only if m epens on x q. In short, m belongs to Sσ τ Sσ if an only if m belongs to Sσ+q p. τ This completes the verification of (31). Thus the application of Q τ σ to (28) gives, in light of (29), (30), an (31), that [ (32) xq q (Pσ τ η σ )+ ɛ(q, p, σ)x q p (Pσ+q pη τ σ+q p ) ] =0. q σ p σ\τ

18 1568 DOUGLAS N. ARNOLD AND GERARD AWANOU We now claim that, for any σ Σ(k) anτ σ, that (33) ɛ(q, p, σ)x q p (Pσ+q pη τ σ+q p )= ( c + ) x i i (P τ σ η σ ), q σ p σ\τ i σ\τ where c =#(σ \ τ). Assuming this, we have from (32), 0= ( c + x i i + ) x q q (P τ σ η σ )= ( ) c + x i i (P τ σ η σ )=(c + r +1)(Pσ τ η σ ), i σ\τ q σ i τ where we have use Euler s formula for homogeneous polynomials (i.e., the homotopy formula for 0-forms) in the last step. Thus, Pσ τ η σ vanishes, an so η σ = τ σ P σ τ η σ also vanishes. Since σ Σ(k) isarbitrary,thisimpliesthatη =0. Thus it remains only to prove (33). By the first formula of Lemma 4, κη = ζ Σ(k 1) ω ζ x ζ where ω ζ = ɛ(q, ζ)x q η ζ+q. q ζ Now let ζ Σ(k 1) an τ ζ, an enote by Rζ τ the projection onto Sτ ζ S ζ (the span of monomials which epen on all the ζ variablesexcepttheτ variables an which are superlinear in the ζ variables). By the hypothesis that leg κη 1, Rζ τ ω ζ =0. Next we compute Rζ τ (x qη ζ+q )forq ζ. If m is a monomial of η ζ+q,thenm = x q η ζ+q epens on all the ζ variablesexceptfortheτ variables if an only if the same is true of m. Moreover, m is superlinear in all the ζ variables if an only if m epens on x q (since m is superlinear in all the ζ variables with the possible exception of x q ). Thus, Rζ τ (x q η ζ+q )=x q Pζ+qη τ ζ+q. Combining the last three isplaye equations, we obtain ɛ(q, ζ)x q Pζ+qη τ ζ+q =0. q ζ Now choose some p ζ an ifferentiate this equation with respect to x p to get ɛ(p, ζ) p (x p Pζ+pη τ ζ+p )+ ɛ(q, ζ)x q p (Pζ+qη τ ζ+q )=0, q ζ p or, after rearranging, (34) P τ ζ+pη ζ+p + x p p (P τ ζ+pη ζ+p )= q ζ p ɛ(p, ζ)ɛ(q, ζ)x q p (P τ ζ+qη ζ+q ). For any σ Σ(k) ananyp σ set ζ = σ p Σ(k 1). Then we can rewrite (34) in terms of σ as Pσ τ η σ + x p p (Pσ τ η σ )= ɛ(p, σ p)ɛ(q, σ p)x q p (Pσ+q pη τ σ+q p ). q σ Finally, taking any τ σ Σ(k), we sum over p σ \ τ to obtain cpσ τ η σ + x p p (Pσ τ η σ )= ɛ(p, σ p)ɛ(q, σ p)x q p (Pσ+q pη τ σ+q p ), q σ p σ\τ p σ\τ where c =#(σ \τ). In light of (3), this establishes (33), an so completes the proof of the lemma.

19 FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES 1569 Finally, we give the proof of Proposition 3.7. Proof. By ilating an translating, it suffices to prove the result when T = I n with I =[ 1, 1]. Let μ = μ σ x σ σ Σ(k) be a polynomial ifferential form on the cube I n.thentr f μ vanishes on the faces x i = ±1 if an only if for each σ such that i/ σ, 1 x 2 i ivies μ σ. Now suppose that μ S r Λ k (I n ), so that tr f μ vanishes on all the faces of the cube. It follows that μ σ = μ σ (1 x 2 i ) i σ for some polynomial μ σ. The monomial expansion of μ then contains the form monomial m σ i σ x 2 i x σ,wherem σ is any monomial of highest egree of μ σ. The linear egree of this form monomial is 0, so, by the egree property (20), its egree is at most r +1. Having establishe that μ is of egree at most r+1, let η be its homogeneous part of egree r +1. Wehave legη = 0. Now we may match terms in the efinition (17) of S r Λ k (Ω) to obtain that η = κυ + κω, forsomeυ H r,1 Λ k+1 (I n )an ω H r+1,1 Λ k (I n ). Therefore, leg κη =legκκω leg ω 1anlegκη = leg κκυ leg υ 1, where we have use Lemma 2.2. By Lemma 4.1, η = 0, hence the monomial of highest orer in the expansion of μ σ is of egree at most r. It follows that μ σ is of egree r 2(n k) foreach σ Σ(k). We can then choose the test function ν = σ Σ(k) ( 1)sgn(σ,σ ) μ σ x σ in (23) to conclue that μ vanishes. References [1] Scot Aams an Victor Reiner, private communication. [2] Douglas N. Arnol an Gerar Awanou, The serenipity family of finite elements, Foun. Comput. Math. 11 (2011), no. 3, , DOI /s MR (2012i:65249) [3] Douglas N. Arnol, Daniele Boffi, an Francesca Bonizzoni, Tensor prouct finite element ifferential forms an their approximation properties, preprint 2012, arxiv: [math.na]. [4] Douglas N. Arnol, Richar S. Falk, an Ragnar Winther, Finite element exterior calculus, homological techniques, an applications, Acta Numer. 15 (2006), 1 155, DOI /S MR (2007j:58002) [5] Douglas N. Arnol, Richar S. Falk, an Ragnar Winther, Finite element exterior calculus: from Hoge theory to numerical stability, Bulletin of the American Mathematical Society 42 (2010), no. 2, MR (2011f:58005) [6] Franco Brezzi, Jim Douglas Jr., Ricaro Durán, an Michel Fortin, Mixe finite elements for secon orer elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, , DOI /BF MR (88f:65190) [7] Franco Brezzi, Jim Douglas Jr., an L. D. Marini, Two families of mixe finite elements for secon orer elliptic problems, Numer. Math. 47 (1985), no. 2, , DOI /BF MR (87g:65133) [8] Runchang Lin an Zhimin Zhang, Natural superconvergence points in three-imensional finite elements, SIAM J. Numer. Anal. 46 (2008), no. 3, , DOI / MR (2009a:65321) [9] Jean-Claue Néélec, Mixe finite elements in R 3, Numerische Mathematik 35 (1980),

20 1570 DOUGLAS N. ARNOLD AND GERARD AWANOU [10] J.-C. Néélec, A new family of mixe finite elements in R 3, Numer. Math. 50 (1986), no. 1, 57 81, DOI /BF MR (88e:65145) [11] Pierre-Arnau Raviart an Jean-Marie Thomas, A mixe finite element metho for 2n orer elliptic problems, Mathematical aspects of finite element methos (Proc. Conf., Consiglio Naz. elle Ricerche (C.N.R.), Rome, 1975), Vol. 606 of Lecture Notes in Mathematics (Berlin), Springer, 1977, pp MR (58 #3547) [12] Barna Szabó anivobabuška, Finite element analysis, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, MR (93f:73001) Department of Mathematics, University of Minnesota, Minneapolis, Minnesota aress: arnol@umn.eu Department of Mathematics, Statistics, an Computer Science, M/C 249, University of Illinois at Chicago, Chicago, Illinois aress: awanou@uic.eu