Whitney forms and their extensions

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ Whitney forms and their extensions © 2021 The Author(s). Published by Elsevier B.V. Published version Lohi, Jonni; Kettunen, Lauri Lohi, J., & Kettunen, L. (2021). Whitney forms and their extensions. Journal of Computational and Applied Mathematics, 393, Article 113520. https://doi.org/10.1016/j.cam.2021.113520 2021 Journal of Computational and Applied Mathematics 393 (2021) 113520 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Whitney forms and their extensions✩ Jonni Lohi ∗, Lauri Kettunen Faculty of Information Technology, University of Jyväskylä, PO Box 35, FI 40014, Finland article info Article history: Received 20 August 2020 Received in revised form 15 January 2021 MSC: primary 65N30 secondary 58A10 Keywords: Whitney forms abstract Whitney forms are widely known as finite elements for differential forms. Whitney’s original definition yields first order functions on simplicial complexes, and a lot of research has been devoted to extending the definition to nonsimplicial cells and higher order functions. As a result, the term Whitney forms has become somewhat ambiguous in the literature. Our aim here is to clarify the concept of Whitney forms and explicitly explain their key properties. We discuss Whitney’s initial definition with more depth than usually, giving three equivalent ways to define Whitney forms. We give a comprehensive exposition of their main properties, including the proofs. Understanding of these properties is important as they can be taken as a guideline on how to extend Whitney forms to nonsimplicial cells or higher order functions. We discuss several generalisations of Whitney forms and check which of the properties can be preserved. ©2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents 1. Introduction............................................................................................................................................................................................... 2 2. Preliminaries and notation...................................................................................................................................................................... 2 3. Three equivalent definitions of Whitney forms ................................................................................................................................... 3 3.1. Proxy fields................................................................................................................................................................................... 5 4. Properties of Whitney forms................................................................................................................................................................... 7 5. Generalisations of Whitney forms.......................................................................................................................................................... 12 5.1. Whitney forms on a manifold.................................................................................................................................................... 12 5.2. Higher order Whitney forms...................................................................................................................................................... 13 5.3. Whitney forms on other cells than simplices.......................................................................................................................... 15 5.3.1. Construction based on generalised barycentric functions....................................................................................... 16 5.3.2. Construction based on conation and extrusion........................................................................................................ 17 5.3.3. Formulas on cubes, triangular prisms, and pyramids.............................................................................................. 17 Acknowledgements .................................................................................................................................................................................. 18 References ................................................................................................................................................................................................. 18 ✩Funding: University of Jyväskylä, Finland. ∗Corresponding author. E-mail addresses: [email protected] (J. Lohi), [email protected] (L. Kettunen). https://doi.org/10.1016/j.cam.2021.113520 0377-0427/©2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons. org/licenses/by-nc-nd/4.0/). J. Lohi and L. Kettunen Journal of Computational and Applied Mathematics 393 (2021) 113520 1. Introduction Whitney forms first appeared in the book of Hassler Whitney [1], which did not originally have any relation to numerical mathematics or to finite element and finite difference kind of approaches. Instead, Whitney formulated a theory of p-dimensional integration in n-dimensional affine space with chains and cochains. In a proof relating the cohomology of flat cochains to simplicial cohomology, he introduced elementary flat cochains and the corresponding differential forms [1, VII, §11]. Jozef Dodziuk used these forms (or their generalisations onto a manifold, to be precise) to approximate continuous Hodge theory with combinatorial Hodge theory and introduced the name Whitney forms in his thesis [2]. Unlike Whitney’s work, Dodziuk’s ideas were closely related to finite difference approaches. Whitney forms became popular within the computational electromagnetics community in late 1980s and early 1990s after the pioneering work of Alain Bossavit [3–8]. He revealed their immediate relation to mixed finite elements [9,10] and emphasised the benefits of presenting the field equations in terms of differential forms instead of scalar and vector fields. Thereafter cochains and Whitney forms were shown to yield a natural framework to explain the finite difference method and its relation to the finite element method [11–14]. Differential forms have since been accepted as an appropriate tool to present both of these methods [15–20], and Whitney forms are widely used to build finite-dimensional subspaces of differential forms; for more examples of the use of Whitney forms (or their proxy fields) in the literature, see e.g. [21–27]. Whitney’s original definition yields first order functions on simplicial complexes. In practice, the assumption of simplices behind Whitney forms is restrictive. Hence, in the literature one can find extensions to other cell types [18,28,29]. Furthermore, there have also been attempts to generalise them to higher order functions [30–32]. While the literature recognises several extensions of Whitney forms, the usage of the term ‘‘Whitney forms’’ is not unambiguous. The term is used for different type of objects by different authors, and the other way around, some instances of Whitney forms are sometimes called with a completely different name. In this paper we clarify the concept of Whitney forms and create a synthesis of papers published on them. Our aim is to explain explicitly the key properties of Whitney forms and provide foundations for extending Whitney forms beyond their original assumptions. For this, in Section 3, we discuss Whitney’s initial definition in more depth than usually and give three equivalent definitions, each emphasising a certain aspect of Whitney forms. In Section 4we give a comprehensive exposition of their main properties, including the proofs. To further clarify the concept of Whitney forms, in Section 5we consider generalisations that are called Whitney forms in the literature and check which of the properties are preserved. This reveals the trade-offs involved in extending Whitney forms to non-simplicial complexes and higher order functions. That is, to bypass assumptions involved in Whitney’s initial setting, one also has to give up on some properties. Regarding our contribution to the scientific literature, there is no prior paper which systematically lists all the key properties of Whitney forms with proofs. Although the results included in this paper can be considered as known, there are new aspects and some technical details that have not been published before. Our definitions and results are given in the spirit of Whitney’s book and do not require Lebesgue theory or Sobolev spaces. This includes Theorems 4.9 and 5.1, which bring the approximation property of finite element theory into Whitney’s setting. The proof of Theorem 4.3 has also not appeared elsewhere. This theorem could also be shown using Proposition 4.4 and the known fact that constants are in the span of Whitney forms, but the authors are not aware of such a proof – or even the proof of Proposition 4.4 – in the literature. 2. Preliminaries and notation In this section some of the prerequisite concepts are briefly recalled. We expect the reader is familiar with exterior algebra and differential forms (see e.g. [1, Chapters I–III]). Standard Whitney forms are differential forms in a simplicial complex. Simplicial complex K is a finite set of simplices such that •each face of every simplex in Kis also in K. •The intersection of two simplices in Kis either a common face of theirs or the empty set. Complexes consisting of more general cells can be defined similarly. As in the initial context of Whitney forms [1], we assume that the simplices are embedded in affine space and tile a domain Ω. For simplicity, we may take Rnas the affine space, keeping in mind that only the affine structure of Rnis required, so that Ωis a polyhedron in Rn. The general case where Ωis a manifold is covered in Section 5, which discusses generalisations of Whitney forms. We denote simplices by labels σand τ, and σ=x0. . . xpmeans that σis the oriented p-simplex whose vertices are x0,...,xpand whose orientation is implied by this order of vertices. Spdenotes the set of p-simplices and vect(σ) the vectorial volume of σ (i.e. the p-vector of σ, see [1, III, §1]). Recall that to each 0-simplex xiof Kcorresponds a barycentric function λi— it is the unique function which is affine in each simplex and whose value is one at xiand zero at other 0-simplices. Barycentric functions are the main building block for Whitney forms. We remark that they are exclusive to simplicial complexes, but we will discuss the generalisation of barycentric coordinates for other cells than simplices when considering extensions of Whitney forms. Differential p-form in a complex K [1, p. 226] is a set of smooth p-forms ωσin the cells σof Ksatisfying the following patch condition: if τis a face of σ, then the trace ωσ|τof ωσequals ωτin τ. In other words, ⟨ωσ(x), α⟩=⟨ωτ(x), α⟩for all 2 J. Lohi and L. Kettunen Journal of Computational and Applied Mathematics 393 (2021) 113520 x∈τand all p-vectors αin the plane of τ. (Here and throughout the paper, we denote the action of a p-covector ωon a p-vector αby ⟨ω, α⟩.) This means that if τis the cell for which x∈τ−∂τ and αis in the plane of τ, then ⟨ωσ(x), α⟩is the same for all σcontaining x. Hence the set of p-forms ωσinduces a single p-form ωsuch that ⟨ω(x), α⟩is single-valued (i.e. well-defined) for such p-vectors α. The patch condition ensures that differential p-forms in Kcan be integrated over p-cells in K. Denote by Fp(K) the space of differential p-forms in K. Note that since the exterior derivative d commutes with trace, we have d ω∈Fp+1(K) if ω∈Fp(K), but the Hodge star ⋆ω is not necessarily in Fn−p(K). When Kis a simplicial complex, formal sums ∑σi∈Spaiσiof oriented p-simplices with real coefficients are called p-chains of K [1, App. II, §6]. These form a vector space Cp(K) for which the p-simplices σiconstitute a natural basis (here σi=1σi, the sum in which aj=δij, the Kronecker delta). The elements of the dual space C∗ p(K) are p-cochains of K. Following [1], we use σito denote also the cochain whose value is δij at the chain σj. Then the p-simplices σiconstitute the dual basis for C∗ p(K), and also cochains can be written as formal sums of simplices. Negative coefficients indicate change of orientation so that −σis the simplex σwith opposite orientation. Chains and cochains for more general cell complexes are defined similarly. Since p-forms can be integrated over p-cells, each p-form ωyields a p-cochain whose values on chains are determined by integration of ω. Namely, the de Rham map C:Fp(K)→C∗ p(K) is a linear map defined by Cω(∑ i aiσi)=∫∑iaiσi ω=∑ i ai∫σi ω, where the second equality is the definition of integration on p-chains. Coboundary operator d:C∗ p(K)→C∗ p+1(K) is a linear map defined by d X(c)=X(∂c). We use the same notation d as for the exterior derivative of forms. Stokes’ theorem then implies that Cd=dC. 3. Three equivalent definitions of Whitney forms Whitney p-forms are a finite-dimensional subspace of differential p-forms in a simplicial complex K. To each p-simplex σcorresponds a Whitney p-form Wσ. Since σalso denotes a basis cochain of C∗ p(K) (and linear maps are uniquely determined by their action on basis elements), this correspondence defines a linear map W:C∗ p(K)→Fp(K). Wis known as the Whitney map, and Whitney forms are its images. This is made precise in the following definition. Definition 3.1. The Whitney 0-form corresponding to the 0-simplex xiis the barycentric function Wxi=λi. For p>0, the Whitney p-form corresponding to the p-simplex x0. . . xpis [1, VII, 11.16] W(x0. . . xp)=p! p ∑ i=0 (−1)iλidλ0∧···∧ˆ dλi∧···∧dλp,(3.1) where ˆindicates a term omitted from the product. For each p, the Whitney map W:C∗ p(K)→Fp(K) is defined by setting W(∑ σi∈Sp aiσi)=∑ σi∈Sp aiW(σi). The image W(C∗ p(K)) =span{Wσ|σ∈Sp} ⊂ Fp(K) is the space of Whitney p-forms and denoted by Wp. Note that although the λiare not globally smooth, they are smooth in each simplex, so (3.1) defines a p-form in each simplex of K. The patch condition holds because barycentric functions in σrestrict to barycentric functions on the faces of σ(and trace commutes with ∧and d). Hence (3.1) yields a well-defined differential form in K. Note also that the right hand side of (3.1) changes sign when the orientation changes, so W(−σ)= −Wσand the Whitney map is well-defined. Since the definition of Whitney forms is the main issue here, we cover it in more detail than usually and from different viewpoints. First, we give an alternative but equivalent definition. Set Wσ=0 in τif σis not a face of τ. If it is, say σ=x0. . . xpand τhas vertices {x0,...,xp,xp+1. . . , xq}, set [1, VII, 11.12] ⟨Wσ(x), α⟩ = p!α∧(xp+1−x)∧(xp+2−xp+1)∧···∧(xq−xp+1) (x1−x0)∧···∧(xq−x0)in τ;(3.2) that is, the value of the p-form Wσat point x∈τis the p-covector whose value on a p-vector αis defined as the ratio of the two q-vectors in the plane of τ. This can be written equivalently as ⟨Wσ(x), α⟩ = p!(q−p)! q! α∧vect(xxp+1. . . xq) vect(x0. . . xp. . . xq), from which we see that Wσin τdoes not depend on the orientation of τbut changes sign when the orientation of σ changes. For x∈y0. . . yp⊂τ,(3.2) becomes ⟨Wσ(x),vect(y0. . . yp)⟩ = vect(y0. . . ypxp+1. . . xq) vect(x0. . . xq).(3.3) 3 J. Lohi and L. Kettunen Journal of Computational and Applied Mathematics 393 (2021) 113520 Fig. 1. Illustration of (3.3) in tetrahedron τ=x0x1x2x3for the cases σ=x0,σ=x0x1,σ=x0x1x2, and σ=x0x1x2x3. In each case, ⟨Wσ(x),vect(y0. . . yp)⟩is the ratio of the highlighted volume and the volume of the whole tetrahedron. This holds for all x∈y0. . . yp. To see this, note that vect(y0. . . yp)∧(xp+1−x)=vect(y0. . . yp)∧(xp+1−yp−(x−yp)) =vect(y0. . . yp)∧(xp+1−yp) since x−ypis in the plane of y0. . . yp. Although volumes depend on the metric, their ratios do not, and the above formula is meaningful in affine space. This definition beautifully illustrates the geometric character of Whitney forms (see Fig. 1), while Definition 3.1 offers an explicit formula in terms of barycentric functions. Whitney showed that these two definitions are indeed equivalent. Proposition 3.2. The definition with the geometric formula (3.2) is equivalent to Definition 3.1. Proof. Let σ=x0. . . xp∈Spand τ∈Sq, and denote by W1σthe Whitney form of σgiven by (3.1) and by W2σthat given by (3.2). To show that W1σ=W2σin τ, we first note that both W1σand W2σzero in τif σis not a face of τ. Moreover, both are affine in τ, are zero at those vertices of τthat are not in σ, and change sign when the orientation of σchanges. Hence it suffices to consider the case τ=x0. . . xpxp+1. . . xqand show that W1σ(x0)=W2σ(x0). Since the edge vectors xi−x0span the plane of τ, all p-vectors in τcan be written as linear combinations of their wedge products. Hence it suffices to show ⟨W1σ(x0), α⟩=⟨W2σ(x0), α⟩for p-vectors αof the form α=(xi1−x0)∧···∧(xip−x0) for i1<··· <ip. Since λi(x0)=0 and ⟨dλi(x0),xj−x0⟩ = δij if i= 0, we have ⟨W1σ(x0),(xi1−x0)∧···∧(xip−x0)⟩ = 0 if any of the indices ijare not in {1,...,p} ⟨W1σ(x0),(x1−x0)∧···∧(xp−x0)⟩ = p! By (3.2) the same is true for W2σ(x0); hence W1σ(x0)=W2σ(x0). □ At this point, it is instructive to briefly discuss Whitney’s book [1] and the role of Whitney forms there. The book is about p-dimensional integration in n-dimensional space. What we call chains (and cochains) of Kare called algebraic chains (and cochains) in [1] where p-chains have a more general meaning as p-dimensional domains of integration. Whitney starts from polyhedral p-chains – formal sums of polyhedral p-cells with real coefficients and invariance under subdivision – which form an infinite-dimensional vector space. This space can be equipped with a norm and then completed with respect to that norm; for example, the flat norm [1, V, §3] yields the space of flat p-chains C♭ p. Its (continuous) dual space C♭∗ pis the space of flat p-cochains and consists of bounded linear functionals C♭ p→R. Similarly, the sharp norm [1, V, §6] yields the spaces of sharp p-chains C♯ pand sharp p-cochains C♯∗ p. We saw that Whitney forms correspond to (algebraic) cochains of a simplicial complex K, but they also correspond to certain flat cochains in K. This explains why Whitney forms are sometimes called flat forms. A correspondence between flat forms and flat cochains is made precise in Wolfe’s theorem [1, IX, Theorem 7C]. Without going into details, p-form ω and p-cochain Xcorrespond if ∫σω=X(σ) for all p-cells σ. In his work [1, VII, §11], Whitney defined a linear injection φ from the algebraic cochains of Kto flat cochains in K, which he used to prove that the cohomology ring of flat cochains is isomorphic to that of algebraic cochains. The images of φhe called elementary flat cochains in K, and these are in correspondence with Whitney forms. Whitney’s theory of p-chains as p-dimensional domains of integration had some shortcomings. For instance, sharp chains do not have a continuous boundary operator, while the Hodge star of a flat form is not flat. The theory has since been extended by Jenny Harrison [33]. We need not go deeper into this. However, now that we have mentioned chains, we can briefly discuss another way to look at the definition of Whitney forms, as emphasised by Alain Bossavit [14,29,34]: approximating p-chains with algebraic p-chains. To explain this, we extend the notation ⟨ω, c⟩ := ∫cωfor differential forms ωand chains c. This expression is bilinear and can be interpreted either as the evaluation of ωon cor (by duality) as the evaluation of con ω. Similarly, denote ⟨X,c⟩ = X(c) for cochains Xand chains c. Whitney forms have the property ⟨Wσj, σi⟩ = δij and hence enable one to approximate a p-form ωin Wpwith ˜ω=∑σi∈Sp⟨ω, σi⟩Wσi. The approximation ˜ωhas the property that ⟨˜ω, c⟩=⟨ω, c⟩ — not for all p-chains c, but for algebraic chains, namely those in Cp(K). This has a dual viewpoint: one can approximate ap-chain cin Cp(K) with ˜ c=∑σi∈Sp⟨Wσi,c⟩σi, and the approximation ˜ chas the property that ⟨ω, c⟩=⟨ω, ˜ c⟩— not for all p-forms ω, but for those in Wp. Letting Wtdenote the map c↦→ ˜ c, we have ⟨WX,c⟩=⟨X,Wtc⟩for all p-chains cand all X∈C∗ p(K). 4 J. Lohi and L. Kettunen Journal of Computational and Applied Mathematics 393 (2021) 113520 On the other hand, if we have such a map Wtto approximate p-chains in Cp(K), this defines a map Wfrom C∗ p(K) to Fp(K) by requiring that ⟨WX,c⟩=⟨X,Wtc⟩hold for all p-chains cand all X∈C∗ p(K). This approach to the definition of Whitney forms is used e.g. in [14,29,31,34,35]. When suitable conditions are imposed for the map Wt, this approach leads to the following, yet another equivalent definition of Whitney forms, which first appeared in [34]. Setting Wxi=λifor p=0, the Whitney form corresponding to p-simplex σfor p>0 is obtained recursively by Wσ=∑ τ∈Sp−1 dσ τλσ−τdWτ, (3.4) where dσ τis the incidence number relating τand σ(which is 0 if τis not a face of σand ±1 if it is, the sign depending on whether the orientations agree or not) and σ−τdenotes the vertex opposite to the (p−1)-face τof σ. It is easy to show that this definition is equivalent to Definition 3.1, after we first note that the exterior derivative of the Whitney p-form W(x0. . . xp) for any p-simplex x0. . . xp∈Spis dW(x0. . . xp)=p! p ∑ i=0 (−1)idλi∧dλ0∧···∧ˆ dλi∧···∧dλp=(p+1)!dλ0∧···∧dλp.(3.5) Proposition 3.3. The definition with the recursive formula (3.4) is equivalent to Definition 3.1. Proof. First note that writing σ=x0. . . xpwe get ∑ τ∈Sp−1 dσ τλσ−τdWτ= p ∑ i=0 (−1)iλidW(x0. . .ˆ xi. . . xp). For σ=x0x1this becomes λ0dWx1−λ1dWx0=λ0dλ1−λ1dλ0, which is the same as Wx0x1of Definition 3.1, proving the claim for 1-simplices. Suppose as induction hypothesis that it holds for (p−1)-simplices, and let σ=x0. . . xpbe a p-simplex. By (3.5) we get ∑ τ∈Sp−1 dσ τλσ−τdWτ= p ∑ i=0 (−1)iλidW(x0. . .ˆ xi. . . xp) = p ∑ i=0 (−1)iλip!dλ0∧···∧ˆ dλi∧. . . dλp=W(x0. . . xp).□ 3.1. Proxy fields The definition of Whitney forms does not require the notion of metric; only the affine structure of the ambient space is invoked. However, metric structure allows one to identify certain differential forms with scalar or vector fields, so-called proxy fields. Indeed, Whitney forms are often presented in terms of these proxy fields. To clarify such seemingly different definitions, let us look at the 3-dimensional case with Euclidean metric and standard orientation (so that right-hand rule is used for cross product). 0-forms are scalar functions, so there is no distinction between a 0-form and its proxy field. In each simplex of K, flat map ♭from vector fields to 1-forms is defined by ⟨♭u(x), v⟩ = u(x)·v; that is, the value of ♭uat point xis the covector whose value on vector vis the dot product u(x)·v. This is an isomorphism with inverse ♯, and the proxy field of a 1-form ωis the vector field ♯ω. Similarly, if uis a vector field, the rule v1∧v2↦→ u(x)·v1×v2defines a 2-form, and this yields a correspondence between vector fields and 2-forms. The proxy field of a 2-form ωcan be written as ♯ ⋆ ω, where ⋆is the Hodge star. Finally, a scalar field fdefines a 3-form by the rule v1∧v2∧v3↦→ f(x)det(v1, v2, v3), and any 3-form is obtained this way from a scalar field f, its proxy field. When considered globally in K, the proxy fields of 1and 2-forms in Khave a well-defined tangential and normal component on inter-element boundaries (respectively). In this case the proxy fields are perhaps more easily explained in terms of standard coordinates. The proxy field of the 1-form ω1dx1+ω2dx2+ω3dx3is the vector field (ω1, ω2, ω3), the proxy field of the 2-form ω12 dx1∧dx2+ω13 dx1∧ dx2+ω23 dx2∧dx3is the vector field (ω23,−ω13, ω12), and the proxy field of the 3-form ω123 dx1∧dx2∧dx3is the scalar field ω123. (This holds more generally when Ωis an oriented Riemannian manifold and {x1,x2,x3}is any positively oriented orthonormal frame.) When ωis a differential form, denote by ω♯its proxy field. Theorem 3.4. In a tetrahedron x0x1x2x3, the proxy fields of Whitney forms are (Wx0x1)♯=λ0∇λ1−λ1∇λ0 (Wx0x1x2)♯=2(λ0∇λ1×∇λ2−λ1∇λ0×∇λ2+λ2∇λ0×∇λ1) 5 J. Lohi and L. Kettunen Journal of Computational and Applied Mathematics 393 (2021) 113520 (Wx0x1x2x3)♯=6(λ0(∇λ1×∇λ2)·∇λ3−λ1(∇λ0×∇λ2)·∇λ3 +λ2(∇λ0×∇λ1)·∇λ3−λ3(∇λ0×∇λ1)·∇λ2) and their values at x ∈x0x1x2x3can be written as (Wx0x1)♯(x)=a×x+b (Wx0x1x2)♯(x)=cx +d (Wx0x1x2x3)♯(x)= ± 1 |x0x1x2x3|, where the vectors a = ± x3−x2 6|x0x1x2x3|, b = ∓ x3−x2 6|x0x1x2x3|×x2, and d = ± 1 3|x0x1x2x3|x3and the scalar c = ∓ 1 3|x0x1x2x3|are constants and the signs depend on whether {x1−x0,x2−x0,x3−x0}is a right-handed frame or not. Proof. Since the gradient ∇fof a function fis (d f)♯and for 1-forms ω,η, and ξwe have (ω∧η)♯=ω♯×η♯,(ω∧η∧ξ)♯=(ω♯×η♯)·ξ♯, the first part follows from Definition 3.1. Since the gradients of barycentric functions are constants, we omit the variable xfrom them and write (Wx0x1)♯(x)=λ0(x)∇λ1−λ1(x)∇λ0= ∇λ0·(x−x2)∇λ1−∇λ1·(x−x2)∇λ0 =(∇λ0·x)∇λ1−(∇λ1·x)∇λ0−(∇λ0·x2)∇λ1+(∇λ1·x2)∇λ0 =(∇λ0×∇λ1)×x−(∇λ0×∇λ1)×x2. Here we used the identity (a×b)×c=(a·c)b−(b·c)a.(3.6) Note that in place of x2in the vector bwe could use any point of x2x3. For any permutation i1i2i3i4of 1234, the vector (xi2−xi3)×(xi4−xi3) is orthogonal to xi2xi3xi4and has length equal to 2|xi2xi3xi4|. On the other hand, ∇λi1is orthogonal to xi2xi3xi4and has length equal to the reciprocal of the height of the tetrahedron with respect to the face xi2xi3xi4. Hence we have ∇λi1= ±(xi2−xi3)×(xi4−xi3) 6|x0x1x2x3|. The sign is +if {xi2−xi3,xi4−xi3,xi1−xi3}is a right-handed frame and −otherwise. Using (3.6) again we get ∇λi1×∇λi2= ±(xi2−xi3)×(xi4−xi3) 6|x0x1x2x3|×∇λi2= ± xi4−xi3 6|x0x1x2x3|, (∇λi1×∇λi2)·∇λi3= ± xi4−xi3 6|x0x1x2x3|·∇λi3=∓1 6|x0x1x2x3|, the signs depending as above. Using the handedness of {x1−x0,x2−x0,x3−x0}to determine the signs for each permutation, these formulas yield (Wx0x1)♯(x)=(∇λ0×∇λ1)×x−(∇λ0×∇λ1)×x2=a×x+b, (Wx0x1x2)♯(x)=2(λ0(x)∇λ1×∇λ2−λ1(x)∇λ0×∇λ2+λ2(x)∇λ0×∇λ1) =2(λ0(x)(±x3−x0 6|x0x1x2x3|)−λ1(x)(±x1−x3 6|x0x1x2x3|)+λ2(x)(±x3−x2 6|x0x1x2x3|)) = ± 1 3|x0x1x2x3|((λ0(x)+λ1(x)+λ2(x))x3−λ0(x)x0−λ1(x)x1−λ2(x)x2) = ± 1 3|x0x1x2x3|((1−λ3(x))x3−λ0(x)x0−λ1(x)x1−λ2(x)x2)= ± x3−x 3|x0x1x2x3|=cx +d, (Wx0x1x2x3)♯(x)=6(λ0(x)(∇λ1×∇λ2)·∇λ3−λ1(x)(∇λ0×∇λ2)·∇λ3 +λ2(x)(∇λ0×∇λ1)·∇λ3−λ3(x)(∇λ0×∇λ1)·∇λ2)=6(λ0(x)±1 6|x0x1x2x3| −λ1(x)∓1 6|x0x1x2x3|+λ2(x)±1 6|x0x1x2x3|−λ3(x)∓1 6|x0x1x2x3|)= ± 1 |x0x1x2x3|.□ 6 J. Lohi and L. Kettunen Journal of Computational and Applied Mathematics 393 (2021) 113520 The proxy fields of Whitney forms first appeared in [10] and are sometimes called Whitney elements or Nedelec elements; 1-forms correspond to ‘‘edge elements’’. Be aware that in some places the proxy fields are called just Whitney forms and are given as the definition of Whitney forms. We make the distinction that Whitney forms are always differential forms and Whitney elements mean their proxy fields. 4. Properties of Whitney forms In this section we discuss the main properties of Whitney forms. Although these are mostly well-known, the kind of list that we have compiled is not easily found in the literature. In particular, we include proofs for all properties that are not evident from the discussion of Section 3. We also try to put emphasis on why these properties are relevant, to explain why one would like to preserve them for generalisations of Whitney forms. Property 1: Whitney forms are differential forms in a complex ‘‘Whitney forms are differential forms in a complex’’ concisely summarises their conformity properties on interelement boundaries. Whitney p-form is an element of the space Fp(K), so it is a set of p-forms ωσin the cells σof K. Thanks to the patch condition in the definition of Fp(K), we can consider this set of p-forms as a single p-form ωsuch that ⟨ω(x), α⟩is well-defined for p-vectors αin the plane of the cell τfor which x∈τ−∂τ. This reflects how finite element spaces for differential forms are built in FEEC theory [19,36] by first constructing them in each cell and then assembling the local constructions together. This property ensures that Whitney forms can be used as conforming finite elements and p-forms can be integrated over p-cells in K. Perhaps most importantly, it prescribes what type of objects Whitney forms are in the first place. Hence we propose that all generalisations of Whitney forms should at the very least be differential forms in a complex to be called Whitney forms. Property 2: Wpis isomorphic to C∗ p(K) That Whitney forms correspond to the cells of Kcan already be seen from Definition 3.1: to the cochain σcorresponds the Whitney form Wσ, and Whitney p-forms are the images of the map W:C∗ p(K)→Fp(K). The following proposition makes the correspondence more precise. Proposition 4.1. The map W:C∗ p(K)→Fp(K)is an isomorphism onto its image Wp. Moreover, CWX=X for all X ∈C∗ p(K). Proof. For the first claim it suffices to show that Wis injective, which follows from the second claim. To prove CWX=X for all X∈C∗ p(K) it suffices to show that ∫σi Wσj=δij, whence the claim follows by linearity. That ∫σi Wσj=δij is perhaps most easily seen using (3.2) or (3.3) and Proposition 3.2.□ Because of this property, integrals on p-cells of Kserve as unisolvent degrees of freedom for Whitney p-forms. This means that values of the integrals are in one-to-one correspondence with elements of Wp. Moreover, this correspondence is the simplest possible since ∫σi Wσj=δij. Note that ∑σi∈SpaiWσi=0 implies aj=∫σj∑σi∈SpaiWσi=0∀j, so the set {Wσi|σi∈Sp}is linearly independent. Since it also spans Wp, it constitutes a basis for Wp. There are two consequences. Firstly, we can interpolate the cochain X∈C∗ p(K) with the p-form WX, and the integrals of this interpolant match with the values of the cochain on p-simplices: CWX=X. Secondly, we can approximate the p-form ω∈Fp(K) with the Whitney form WCω, and the integrals of this approximation match with those of ωon psimplices: CWCω=Cω. Indeed, Whitney p-forms are commonly considered as a tool for either interpolating p-cochains or approximating differential p-forms. Property 3: Whitney forms are first order polynomials in each cell In each cell, barycentric functions are affine and hence their exterior derivatives are constant, so we see from Definition 3.1 that Whitney forms are affine. Hence they are at most first order polynomials in each cell. (That is, if ω∈Wp, the function x↦→ ⟨ω(x), α⟩is a first order polynomial for each p-vector α.) This of course implies that they are also smooth in each cell. Property 4: Whitney forms are affine invariant In addition to being affine in each cell, Whitney forms are affine objects in the following two senses. First, their definition is meaningful in affine space without any choice of metric. Furthermore, they are invariant under affine transformations. 7 J. Lohi and L. Kettunen Journal of Computational and Applied Mathematics 393 (2021) 113520 Proposition 4.2. Let σ=x0. . . xnand τ=y0. . . ynbe two n-simplices and ϕ:σ→τaffine map such that ϕ(xi)=yi. Then W(x0. . . xp)=ϕ∗(W(y0. . . yp)) in σ . Proof. Let λidenote the barycentric coordinates in σand µithose in τ. Since µi◦ϕis affine in σand (µi◦ϕ)(xj)=δij, it follows that ϕ∗(µi)=µi◦ϕ=λi. Hence by the naturality of pullback with respect to wedge product and exterior derivative we have ϕ∗(W(y0. . . yp)) =ϕ∗(p! p ∑ i=0 (−1)iµidµ0∧···∧ ˆ dµi∧···∧dµp) =p! p ∑ i=0 (−1)iµidϕ∗(µ0)∧···∧ ˆ dϕ∗(µi)∧···∧dϕ∗(µp)=W(x0. . . xp).□ This property is useful because computations done in a reference simplex transfer to all simplices by affine transformations and hence need be done only once. For example, using (3.3), ∫ϕ(z0z1) Wy0y1=∫z0z1 ϕ∗(Wy0y1)=∫z0z1 Wx0x1=vect(z0z1x2. . . xn) vect(x0. . . xn)for z0z1⊂σ. This equality is also seen from (3.3), since volume ratios are preserved by affine transformations. Property 5: locality Whitney form Wσis nonzero only on those simplices that include σas a face. Locality is needed to make system matrices sparse in numerical methods that utilise Whitney forms. Property 6: Whitney forms constitute a partition of unity Barycentric functions sum up to one, forming a partition of unity. The following theorem generalises this property for other Whitney forms. Theorem 4.3. In any q-simplex τ∈Sq, for all points x and all p-vectors αin τ, ∑ σi∈Sp⟨Wσi(x), α⟩vect(σi)=α. Proof. Suppose τ=x0. . . xq∈Sqand x∈τ. Since the edge vectors xi−x0span the plane of τ, all p-vectors in τcan be written as linear combinations of their wedge products. Hence it suffices to consider the case α=(x1−x0)∧···∧(xp−x0), whereafter the claim follows by linearity. At all points of τ ⟨dλi,xk−xj⟩ = {0 if i/∈ {j,k} 1 if i=k −1 if i=j and hence for i1<··· <ipwe have ⟨dλi1∧···∧dλip,(x1−x0)∧···∧(xp−x0)⟩ = {0 if {i1,...,ip} ⊂ {0,...,p} (−1)kif {i1,...,ip}⊂{0,...,ˆ k,...,p} Using this we see that ⟨W(xi0. . . xip)(x), α⟩ = 0 if at least two of the indices ijare not in {0,...,p} ⟨W(x0. . . xk−1xikxk+1. . . xp)(x), α⟩ = p!(−1)kλik(x)(−1)k=p!λik(x) for ik/∈ {0,...,p} ⟨W(x0. . . xp)(x), α⟩ = p! p ∑ j=0 (−1)jλj(x)(−1)j=p! p ∑ j=0 λj(x) 8 J. Lohi and L. Kettunen Journal of Computational and Applied Mathematics 393 (2021) 113520 Theorem 5.1. Let V:C∗ p(Kk)→Wp kbe the linear map obtained with a choice of kth order small simplices as explained above, and let ωbe a smooth p-form in Ω. There exists a constant Cω,ksuch that |VCω(x)−ω(x)| ≤ Cω,k Cp Θ hkfor all x∈τin all τ∈Sn whenever h >0, CΘ>0, and K is a simplicial complex in Ωsuch that diam(σ)≤h and Θ(σ)≥CΘfor all simplices σof K. Proof. It suffices to prove this for ω=ωIdxi1∧ ··· ∧ dxipwhere 1 ≤i1<··· <ip≤n. Denote by Tythe (k−1)th order Taylor polynomial of ωIat y. Since ωis smooth in the polyhedron Ω, we can find a constant CIsuch that |ωI(x)−Ty(x)| ≤ CI|x−y|kwhenever yx ⊂Ω. Fix τ∈Snand y∈τ. We can write ω(x)=ωI(x) d xi1∧···∧dxip=(Ty(x)+g(x)) d xi1∧···∧dxip,where g(x)=ωI(x)−Ty(x),|g(x)| ≤ CI|x−y|k≤CIhkif x∈τ. Since the constant d xi1∧···∧dxipis in Wp(by Corollary 4.5) and Tyis in the span of the products λkwith k∈I(n+1,k−1) (it is a polynomial of order k−1), we see from (5.1) that Tydxi1∧···∧dxipis in Wp k. Hence VC(Tydxi1∧···∧dxip)= Tydxi1∧···∧dxipand VCω(x)=Ty(x) d xi1∧···∧dxip+VC(gdxi1∧···∧dxip)(x), VCω(x)−ω(x)=VC(gdxi1∧···∧dxip)(x)−g(x) d xi1∧···∧dxip. Denote by ˆ Sp kthe chosen subset of Sp kand by ˆ Sp k(τ) its restriction to those small simplices that are in τ. The interpolant VC(gdxi1∧···∧dxip) is a linear combination ∑υi∈ˆ Sp kαiw(υi) of the spanning forms w(υi). Since w(υ)=0 in τif υ⊂ τ, it suffices to consider ∑υi∈ˆ Sp k(τ)αiw(υi). Each coefficient αiis a linear combination of the integrals ∫υjgdxi1∧···∧dxip, υj∈ˆ Sp k(τ). The coefficients of this latter linear combination are constant and affine-invariant quantities (determined by the inverse of the matrix Awith components Aij =∫υiw(υj)). Hence there exists a constant Cαsuch that |αi| ≤ Cα∑ υj∈ˆ Sp k(τ)|∫υj gdxi1∧···∧dxip| ≤ Cα∑ υj∈ˆ Sp k(τ) CIhk|υj| holds for all of the coefficients αi. Using the facts that diam(τ)≥diam(υj) and |υj| ≤ 1 p!diam(υj)pand denoting by Ckthe cardinality of ˆ Sp k(τ), we get |αi| ≤ CαCkCIhk1 p!diam(τ)p. To find a bound for the |w(υi)|, suppose that υiis the image of the p-face σ⊂τ. Then clearly |w(υi)(x)|≤|Wσ(x)| ∀x∈ τ, and hence using the affine map to the standard n-simplex exactly in the same way as in the proof of Theorem 4.9 we find |w(υi)(x)| ≤ (√2 n! n+1Θ(τ) diam(τ))p p!√1+n−pfor all x∈τ . Combining these estimates yields |VC(gdxi1∧···∧dxip)(x)|=| ∑ υi∈ˆ Sp k(τ) αiw(υi)(x)| ≤ ∑ υi∈ˆ Sp k(τ)|αi||w(υi)(x)| ≤C2 kCαCI(√2 n! n+1Θ(τ))p√1+n−p·hkfor all x∈τ . This holds for all τ∈Sn, so we may choose Cω,k=C2 kCαCI(√2 n! n+1)p√1+n−p+CI, and then |VCω(x)−ω(x)|≤|VC(gdxi1∧···∧dxip)(x)|+|g(x) d xi1∧···∧dxip| ≤C2 kCαCI(√2 n! n+1Θ(τ))p√1+n−p·hk+CIhk≤Cω,k Cp Θ hkfor all x∈τin all τ∈Sn.□ 5.3. Whitney forms on other cells than simplices Standard Whitney forms are differential forms in a simplicial complex. For flexibility in modelling and mesh generation, also other kind of cells should be allowed, and there have been several approaches to generalising Whitney forms for nonsimplicial cells. In this subsection, we consider the case where Kis a cell complex of convex polyhedral cells. 15 J. Lohi and L. Kettunen Journal of Computational and Applied Mathematics 393 (2021) 113520 When moving to nonsimplicial cells, we would like to preserve at least properties 1 and 2 of Whitney forms, so we take these as a guideline. Firstly, as stated earlier, all Whitney forms should be differential forms in the complex K— elements of Fp(K). Secondly, there should be a Whitney p-form Wσcorresponding to each p-cell σof K, so that we get a linear map W:C∗ p(K)→Fp(K) whose image is the space of Whitney p-forms Wp. In addition, Wshould be an isomorphism onto its image, so that integrals over p-cells uniquely determine an element of Wpand serve as degrees of freedom. Without loss of generality, we may then also require that ∫σi Wσj=δij, which is probably the best-known property of Whitney forms. Property 5, locality, will be fulfilled by all constructions without further mention. In general, properties 3 and 4 as such will be lost. This is inevitable: a first order polynomial would already be fixed by its values on n+1 vertices, and there is no affine map like in Proposition 4.2 between more general cells. However, for some cell types there is a same kind of canonical map (maybe not affine) and Whitney forms on one cell move onto another through taking pullback. For example, any cube is obtained from the reference cube [0,1]3with the obvious map after the image of one vertex is fixed, and to define Whitney forms on cubes it suffices to consider the reference cube. The same applies to for example triangular prisms and pyramids. To define Whitney forms for a convex polyhedral cell, we may consider the cell and its faces as the cell complex K so that there is only one n-cell. In doing so, we must ensure that traces on faces depend only on the face itself, so that the same Whitney forms belong to Fp(K) also in the case when Khas many n-cells. The complex Kmay even contain different kind of cells, as long as traces on faces shared by two such cells are the same according to both constructions. From the literature, we have chosen two constructions that we believe best preserve the properties of Whitney forms. These will be discussed below. More options can be found in the literature if one is willing to give up on more of the properties (see e.g. [53–56]). In particular we would like to mention [54], where the author shows a way to construct finite-dimensional spaces of differential forms on arbitrary polytopes in any dimension such that the basis p-forms correspond to the p-cells and the spaces fulfil the exact sequence property. It requires auxiliary spaces on a simplicial refinement of the complex, and as these one can use Whitney forms. However, the resulting forms are in Fp(K′) with respect to the refinement K′and not necessarily in Fp(K) with respect to the initial complex K(discontinuities are allowed in the cells of K). Another downside is that explicit expressions for the basis forms are not given on general polytopes, so they might not be easily computable. The rest of this subsection is divided into parts as follows. First we briefly discuss two relevant approaches to generalising Whitney forms. The first approach is based on the construction of [57] and generalised barycentric coordinates. The second approach [29] is based on geometric conation and extrusion operations and constructs Whitney forms for cells obtained with these operations recursively. Finally, we summarise the Whitney forms resulting from these approaches on cubes, triangular prisms, and pyramids in 3D. 5.3.1. Construction based on generalised barycentric functions Whitney forms in a simplicial complex were built using barycentric functions. These are exclusive to simplicial complexes, but for nonsimplicial cells there are generalised barycentric coordinates, which are no longer unique. Suppose σis a convex polyhedral p-cell in Rnwith mvertices x1,...,xm. Any set of mnonnegative functions λi:σ→Rare called generalised barycentric coordinates in σif for all x∈σ m ∑ i=1 λi(x)=1, m ∑ i=1 λi(x)xi=x.(5.4) Note that generalised barycentric coordinates in σrestrict to generalised barycentric coordinates on its faces. The functions λiare not uniquely determined by (5.4) for general cells, and there are different kind of generalised barycentric coordinates (see the references in [56] and [57]). On simplices, these all reduce to the standard barycentric coordinates. Generalised barycentric functions in Kare defined after we choose barycentric coordinates in each cell such that their restrictions agree on inter-element faces. This is typically ensured by using the same kind of coordinates on incident cells [56]. In [56] and [57], Whitney forms are generalised for nonsimplicial cells by taking generalised barycentric functions as Whitney 0-forms and using the same formula (3.1) (without the multiplier p!) for 1and 2-forms. This gives the 1-form λidλj−λjdλifor any two vertices xiand xjand the 2-form λidλj∧dλk−λjdλi∧dλk+λkdλi∧dλjfor any three vertices xi,xj, and xk. (In [56] and [57], the forms are given in terms of their proxy fields.) These do not correspond to the cells of K, but they are used in [57] to construct finite elements in 2D and 3D that (although not called Whitney forms in [57]) actually better fulfil the properties of Whitney forms. The construction of [57] uses Wachspress coordinates [58]. In both two and three dimensions, we get linear maps W:C∗ p(K)→Fp(K) such that the spaces of Whitney p-forms Wp=W(C∗ p(K)) constitute an exact sequence. Moreover, we have ∫σi Wσj=δij, and integrals over p-cells serve as degrees of freedom. In 2D any convex nondegenerate polygons are allowed, but in 3D the complex Kis restricted by the additional requirement that the faces of the polyhedral cells be triangles or parallelograms. As Whitney 0-forms we take the generalised barycentric functions resulting from Wachspress coordinates. In 2D, define the Whitney 1-forms corresponding to the edges of Ksuch that their proxy fields are the qiin Lemma 3.1 of [57] rotated 90 degrees counterclockwise and divided by the edge length. In 3D, define the Whitney 1and 2-forms corresponding to 16 J. Lohi and L. Kettunen Journal of Computational and Applied Mathematics 393 (2021) 113520 the edges and the faces of Ksuch that their proxy fields are the peand the qfin Lemmas 4.7 and 4.6 of [57] divided by the edge length and the face area, respectively. Define the n-forms corresponding to the polygons/polyhedra of Ksuch that their proxy fields equal the reciprocal of the area/volume in the corresponding polygon/polyhedron and zero elsewhere. Then ∫σi Wσj=δij by Lemmas 3.1, 4.7, and 4.6 of [57]. In 2D, it follows from Lemma 3.4 of [57] that constants are in Wp, and hence the partition of unity property holds by Proposition 4.4. In 3D this holds for certain types of cells by Lemma 4.14 of [57]. The counterpart to property 8 in 2D is Lemma 3.10 of [57], but we do not know if this has been proved in 3D yet. As discussed, properties 3 and 4 are lost. In general, Wachspress coordinates are rational functions. However, we remark that in the case of simplices everything reduces to normal Whitney forms. Thus, the construction of [57] truly generalises Whitney forms while preserving many of their properties. 5.3.2. Construction based on conation and extrusion To present how Whitney forms for polytopal cells are obtained systematically, one approach is to first consider a systematic construction of the cells themselves. In [29] Whitney forms are defined recursively for cells that are obtained through conation and extrusion operations. Consider an n-dimensional cell σwith plane Pin Rn+1, a point a∈Rn+1 outside P, and a vector vnot parallel to P. Conation yields the (n+1)-dimensional cell cone(σ)= {λa+(1 −λ)x|x∈σ, 0≤λ≤1}, and extrusion yields the (n+1)-dimensional cell extr(σ)= {x+λv |x∈σ, 0≤λ≤1}. In [29], it is shown how Whitney forms lift up onto either of these (n+1)-dimensional cells, supposing we know them on σ. The requirements (1)–(3) on page 1570 of [29] ensure ∫σi Wσj=δij, the exact sequence property, and the inclusion of constant p-forms in Wp(which by Proposition 4.4 implies the partition of unity property). Properties 3 and 4 are again understandably lost. In the case of simplices, this construction yields the usual Whitney forms (with repeated conation starting from a 0-cell). In 3D, other cell types that fit this approach are parallelepipeds (conation, extrusion, extrusion), pyramids (conation, extrusion, conation), and triangular prisms (conation, conation, extrusion). Recently in [59], the authors combined these conation and extrusion techniques with their earlier construction [57] to define Whitney forms on polygon-based prisms and cones. The work [59] covers both theoretical analysis and implementation instructions. As mentioned in [59], any convex polyhedral cell can be divided into polygon-based cones by connecting the vertices with a chosen interior point. Hence one could define Whitney forms for cell complexes of arbitrary convex polyhedra by refining the complex this way — if one does not mind that the resulting forms are in Fp(K′) only with respect to the refined complex K′. 5.3.3. Formulas on cubes, triangular prisms, and pyramids Finally, to show examples of Whitney forms on other cells than simplices, we give formulas of Whitney forms on cubes, triangular prisms, and pyramids. These three cell types are suitable for examples since the Whitney forms on them have sufficiently simple explicit formulas. In addition, both of the approaches we considered in this subsection yield these Whitney forms. In all of the examples, we use Cartesian xyz-coordinates. The cell σis defined by giving its vertices xiin R3. Its edges are oriented so that i<jfor any edge xixj, and its facets are oriented such that the normal vector (prescribed by the right hand rule) points outward. Example 5.2 (Cubes).Consider the cube σwith vertices x1=(0,0,0) x2=(1,0,0) x3=(0,1,0) x4=(1,1,0) x5=(0,0,1) x6=(1,0,1) x7=(0,1,1) x8=(1,1,1) The Whitney forms on σare Wx1=(1 −x)(1 −y)(1 −z)Wx2=x(1 −y)(1 −z)Wx3=(1 −x)y(1 −z) Wx4=xy(1 −z)Wx5=(1 −x)(1 −y)zWx6=x(1 −y)z Wx7=(1 −x)yz Wx8=xyz Wx1x2=(1 −y)(1 −z) d xWx3x4=y(1 −z) d xWx5x6=(1 −y)zdx Wx7x8=yz dxWx1x3=(1 −x)(1 −z) d yWx2x4=x(1 −z) d y Wx5x7=(1 −x)zdyWx6x8=xz dyWx1x5=(1 −x)(1 −y) d z Wx2x6=x(1 −y) d zWx3x7=(1 −x)ydzWx4x8=xy dz Wx5x6x8x7=zdx∧dyWx1x3x4x2= −(1 −z) d x∧dyWx1x2x6x5=(1 −y) d x∧dz Wx3x7x8x4= −ydx∧dzWx1x5x7x3= −(1 −x) d y∧dzWx2x4x8x6=xdy∧dz Wσ=dx∧dy∧dz The proxy fields of the 1and 2-forms above first appeared in [10]. 17 J. Lohi and L. Kettunen Journal of Computational and Applied Mathematics 393 (2021) 113520 Example 5.3 (Triangular Prisms).Consider the triangular prism σwith vertices x1=(0,0,0) x2=(1,0,0) x3=(0,1,0) x4=(0,0,1) x5=(1,0,1) x6=(0,1,1) The Whitney forms on σare Wx1=(1 −y−x)(1 −z)Wx2=x(1 −z)Wx3=y(1 −z) Wx4=(1 −y−x)z,Wx5=xz,Wx6=yz Wx1x4=(1 −y−x) d zWx2x5=xdzWx3x6=ydz Wx1x2=(1 −y)(1 −z) d x+x(1 −z) d yWx2x3= −y(1 −z) d x+x(1 −z) d y Wx1x3=y(1 −z) d x+(1 −x)(1 −z) d yWx4x5=(1 −y)zdx+xz dy Wx5x6= −yz dx+xz dyWx4x6=yz dx+(1 −x)zdy Wx1x2x5x4=(1 −y) d x∧dz+xdy∧dzWx2x3x6x5= −ydx∧dz+xdy∧dz Wx1x4x6x3= −ydx∧dz−(1 −x) d y∧dzWx1x3x2= −2(1 −z) d x∧dy Wx4x5x6=2zdx∧dyWσ=2 d x∧dy∧dz The proxy fields of the 1and 2-forms above first appeared in [60]. Example 5.4 (Pyramids).Consider the pyramid σwith vertices x1=(0,0,0) x2=(1,0,0) x3=(0,1,0) x4=(1,1,0) x5=(0,0,1) The Whitney forms on σare Wx1=(1−z−x)(1−z−y) 1−zWx2=x(1−z−y) 1−zWx3=(1−z−x)y 1−zWx4=xy 1−zWx5=z Wx1x2=(1 −z−y) d x+x(1−z−y) 1−zdzWx2x4=xdy+xy 1−zdz Wx3x4=ydx+xy 1−zdzWx1x3=(1 −z−x) d y+(1−z−x)y 1−zdz Wx1x5=(z−yz 1−z) d x+(z−xz 1−z) d y+(1 −x−y+xy 1−z−xyz (1−z)2) d z Wx2x5=(−z+yz 1−z) d x+xz 1−zdy+(x−xy 1−z+xyz (1−z)2) d z Wx3x5=yz 1−zdx+(−z+xz 1−z) d y+(y−xy 1−z+xyz (1−z)2) d z Wx4x5= − yz 1−zdx−xz 1−zdy+(xy 1−z−xyz (1−z)2) d z Wx1x2x5=zdx∧dy+(2 −y−y 1−z) d x∧dz−xz 1−zdy∧dz Wx1x5x3=zdx∧dy+yz 1−zdx∧dz+(−2+x+x 1−z) d y∧dz Wx2x4x5=zdx∧dy+yz 1−zdx∧dz+(x+x 1−z) d y∧dz Wx4x3x5=zdx∧dy+(−y−y 1−z) d x∧dz−xz 1−zdy∧dz Wx1x3x4x2= −(1 −z) d x∧dy−ydx∧dz+xdy∧dz Wσ=3 d x∧dy∧dz Whitney forms on pyramids first appeared in [28]. Acknowledgements The authors thank Sanna Mönkölä, Tuomo Rossi, and two anonymous reviewers for their helpful comments on the manuscript. References [1] H. 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