 # Finite Element Systems for vector bundles : elasticity and curvature

We develop a theory of Finite Element Systems, for the purpose of discretizing sections of vector bundles, in particular those arizing in the theory of elasticity. In the presence of curvature we prove a discrete Bianchi identity. In the flat case we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress-displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes.

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## Introduction

In this paper we generalize the previously introduced framework of finite element systems (FES)  so that it can treat, in particular, elasticity problems, and provide concrete examples of finite element spaces, some old and some new, that fit the framework.

The general framework provides an approach to finite element discretizations of sections of vectorbundles, and complexes thereof, in particular differential forms with values in a given vector bundle. We make some comments about curvature, but most of the paper concerns the case of flat bundles. For applications in elasticity, the fiber can be identified as the space rigid motions.

In space dimension 2, one can distinguish two differential complexes related to elasticity, which are formal adjoints of each other and give priority to stresses or strains, respectively. For the stress complex (61), we can check that the spaces defined in  and  fit the framework. For the strain complex (63), we introduce, also within the framework, some new finite element spaces. They model symmetric -tensors (metrics) with a good Saint-Venant operator (linearized curvature).

In the obtained finite element complexes, rigid motion like degrees of freedom play a key role, at every index. The FES framework stresses this design principle, and relates it to the interpretation of elasticity in terms of rigid motion valued fields.

Defining discrete spaces of metrics with good curvature in dimension 2 should be useful, in view of the importance of curved surfaces in both pure and applied mathematics, whatever the distinction is. Such applications will be explored elsewhere. Another motivation for this work was to prepare the way for similar constructions in higher dimensions, especially 3 (for classical elasticity) and 4 (for general relativity).

#### Previous work on FES.

Until now, the FES framework has been formulated in order to discretize de Rham complexes. It has been used to define finite element complexes of differential forms on polyhedral meshes , accommodate upwinded finite element complexes containing exponentials , give new presentations of known elements  and to define elements with minimal dimension  under various constraints (such as containing given polynomials).

The regularity of the differential forms, in the above mentioned works, was with exterior derivative in , and the defined finite elements were natural generalizations of, in particular, the Raviart-Thomas-Nédélec (RTN) spaces . The continuity is thus partial, and can be expressed as singlevaluedness of pullbacks to interfaces, corresponding, for vector fields, to continuity in either tangential or normal directions.

In  we extended the FES framework so as to be able to impose stronger interelement continuity. For instance, for a conforming discretization of the Stokes equation, one would like to have spaces of fully continuous vector fields, satisfying a commuting diagram with respect to the divergence operator. For de Rham sequences of higher regularity ( and, if desired, exterior derivative in ), the required continuity can be expressed as singlevaluedness of all components of the differential form and, if desired, of its exterior derivative too, on interfaces. This led us, in , to define FE complexes starting with the Clough-Tocher element, which is of class , instead of, say, Lagrange elements, which are of class . This provided the first conforming polynomial composite Stokes element in dimension 3 (and higher), with piecewise constant divergence and the degrees of freedom of . The latter seem to be the natural ones for lowest order approximations.

#### FE, MFE, FEEC and FES.

Recall Ciarlet’s definition of a finite element (FE), as a space equipped with degrees of freedom .

For mixed finite element methods (MFE), pairs of finite element spaces that are compatible in the sense of Brezzi  should be identified. A particularly convenient tool for this purpose, has been the so-called commuting diagram property, see for instance  page 552 and 570 and compare with 

§8.4 and §8.5. It can sometimes be derived from a commutation property of the interpolators associated with the degrees of freedom. In particular, in

, finite element -- complexes were presented with degrees of freedom providing commuting diagrams.

Arbitrary order finite element complexes of differential forms were defined in . Whitney forms  and the RTN spaces appear as special cases (lowest order – arbitrary dimension, and arbitrary order – low dimension, respectively). This connection between numerical methods and differential topology was first pointed out in . Computational electromagnetics has been one of the main motivations . Its interpretation in terms of differential forms is quite clearcut compared with the case for, say, computational fluid dynamics.

Systematically developing the theory of finite elements in terms of differential complexes equipped with commuting projections was advocated in . Relating de Rham complexes to differential complexes appearing in elasticity, and viewing both as special cases of complexes of Hilbert spaces, has lead to the finite element exterior calculus (FEEC) .

Stability of numerical methods is, in many cases, equivalent to the existence of projections onto the finite element spaces, satisfying commuting diagrams . Uniformly bounded commuting projections can often be obtained from the interpolator associated with degrees of freedom, by a smoothing procedure .

The FES framework downplays the role of degrees of freedom and stresses that, for a finite element space on a cell, there are implicit finite element spaces on the subcells. The claim is that making these spaces explicit has numerous benefits. Interelement continuity is expressed through certain restriction operators, from spaces on cells to spaces on subcells. The spaces on the subcells can also be arranged in complexes, for certain induced differential operators. The restriction operators and the induced differential operators must satisfy commutation and exactness relations. For a given FES, a condition of compatibility, expressed in terms of the restrictions and the induced differentials, ensures the existence of good degrees of freedom and in particular that the harmonic interpolator is well defined and commutes with the differential operators.

In , concerning de Rham complexes of low regularity ( with exterior derivative in ), the relevant restriction operators were pullback by inclusion maps, and the induced differential operators were, again, the exterior derivative. In  the restriction operators could remember all components of the differential forms on the subcell, and possibly of the exterior derivative. The induced differential operators now acted on all this information. Thus appeared some new vectorbundles on subcells, linked by differential operators that were not exactly the exterior derivative on the subcell : they retain additional information about the exterior derivative on the ambient cell. We therefore, for the framework, considered general complexes of spaces, not just complexes of differential forms.

General degrees of freedom are not essential in FES, but they are certainly accomodated and sometimes very convenient. On the other hand certain degrees of freedom are paramount for the development of the theory. For de Rham complexes these degrees of freedom are the integration of -forms on the -dimensional cells of the mesh. This gives rise to the de Rham map, which maps from differential forms to (real valued) cellular cochains ; it commutes with the differentials.

For elasticity complexes, we contend that cellular cochains with coefficients in rigid motions are the right analogue. More precisely we introduce, for each cell of each dimension, a space which is naturally isomorphic to the space of rigid motions. Cochains with coefficients in these spaces form a complex. A generalized de Rham map from elasticity fields to such cochains with coefficients, is then defined and shown to commute.

#### Finite element elasticity complexes.

In dimension 2, for elasticity problems, the stress complex is implicitly used in  and made explicit in . Since then, many more discrete stress complexes have been defined, both conforming  and non-conforming . See  and the references therein for more examples. Notice that the stress complexes in  and  are composite and start with Clough-Tocher elements, whereas those in  are polynomial and start with Argyris elements.

For the systematic design of discrete elasticity complexes, a link between de Rham complexes and elasticity complexes, known as the BGG construction , has been developed . In , known finite element de Rham complexes were tensorized with vectors in order to get vector valued de Rham complexes. Under a surjectivity condition (see page 58), the diagram chase then yielded new spaces for the elasticity complexes.

The finite element complexes defined here behave naturally with respect to the BGG diagram chase. That is, we can define finite element spaces for some interlinked vector valued de Rham complexes, such that the diagram chase at the discrete level works exactly as at the continuous level : isomorphisms at the continuous level correspond to isomorphisms at the discrete level. One thus needs a large supply of discrete de Rham sequences, corresponding to different regularities, that match at different indices. While this can be dispensed of in the presentation of our elasticity elements, it was an important guiding principle towards their design and we have included remarks to this effect.

Finite element de Rham complexes of higher regularity have been constructed . See also  for related Stokes elements. A motivation behind  was to have enough such sequences to address elasticity through diagram chasing.

For elasticity complexes and related BGG diagram chases, the case of regularity is well established in the literature, but the choice of Sobolev spaces is often not explicit. We introduce several Sobolev spaces for our complexes, many of which are not simple tensor products. There are several possible choices for each smooth complex. To obtain the stress complex one can do the chase in (67) or (72). In the latter, the regularity is expressed with a differential operator that acts on columns, whereas the differentials of the complex act by rows. For the strain complex we study two different regularities, corresponding to two different regularities in the diagram chase. Here also the regularity is expressed in terms of differential operators acting on columns as well as rows. A rationale behind our choice of Sobolev spaces is given in Remark 4.8.

Another tool we develop for the purposes of constructing elasticity elements are Poincaré - Koszul operators for elasticity complexes . The Cesaro - Volterra path integral is but one example.

#### FES, sheaves and differential geometry.

The FES framework can be interpreted as a discrete sheaf theory. The main novelty here, compared with , is that we introduce some generalizations of the de Rham maps. We are interested in discretizing sections of vector bundles. These are equipped with a connection. For applications in (linear) elasticity this connection is flat. We have implicitly linearized around the Euclidean metric, for which the Levi Civita connection is flat, as well as other associated connections. But, for the definition of discrete vector bundles, we have also been mindful of situations where non-zero curvature is centre stage.

Regge Calculus , a discrete approach to general relavivity, can be interpreted in a finite element context  and extended to higher orders . One then obtains, in dimension 2, strain complexes of low regularity : they end with discrete spaces containing measures, typically Dirac deltas at vertices. Here, on the contrary, the finite element fields are at least square integrable throughout the complexes. While the regularity of Regge elements seems adapted to general relativity theory, higher regularity, such as that achieved here, could be important to other PDEs in Riemannian geometry, such as Ricci flow.

Lattice Gauge Theory (LGT) , as extended to a finite element context , was also at the back of our minds during this work. In LGT one defines discrete connections and curvature, as well as a discrete Yang-Mills functional, but it is less clear what the discrete covariant exterior derivative and Bianchi identity should be. By contrast, for the discrete theory we develop here, both of these are explicit, see in particular Theorem 2.3.

For the flat case, for which we introduce the FES framework, we prove a variant of the de Rham theorem (e.g.  §V.3.) : the de Rham map induces isomorphisms on cohomology groups, from the space of gobal sections to the cochains with coefficients, see Theorem 3.2.

A role for covariant exterior derivatives in a FES context was also indicated in . Such constructions have applications to PDEs describing for instance convection diffusion problems or band gaps in photonic cristals, see Remark 3.5.

Notice that what we refer to here as elasticity complexes corresponds to Calabi complexes in differential geometry, whereas rigid motions correspond to Killing fields. Sheaf theoretic approaches to such, have been considered in . Discrete sheaf theory for various applications is also developed in .

#### Outline.

The paper is organized as follows. In §1 we develop a theory for discrete flat vector bundles. Results concerning vector bundles with curvature are relegated to §2. In §3 we detail the framework of finite element systems, for a given discrete flat vector bundle. In §4 we provide background on elasticity, including relevant differential operators, differential complexes, the BGG diagram chase and Poincaré operators. In §5 we detail FES for the stress complex, detailing the main example of the Johnson-Mercier element. In §6 we detail FES for the strain complex, with two different regularities, providing the new examples of finite elements for strain tensors (metrics) with compatible Saint Venant operator (linearized curvature).

## 1 Discrete flat vector bundles

### 1.1 Cellular complexes and cochains

Let be a cellular complex. If are cells in we write to signify that is a subcell of . Each cell of dimension at least one is supposed oriented. Given two cells and in , their relative orientation is denoted . It is unless is a codimension one subcell of , in which case it is . The subset of consisting of -dimensional cells is denoted . The space of -cochains is denoted and consists of the maps from to . In other words a -cochain assigns a real number to each cell of dimension . Notice that has a canonical basis indexed by .

The cellular cochain complex is denoted . Its differential, also called the coboundary map, is denoted . Its matrix in the canonical basis is given by relative orientations.

All complexes considered in this paper are cochain complexes, in the sense that the differential increases the index.

### 1.2 Discrete vectorbundles with connection

###### Definition 1.1 (Discrete vectorbundle with connection).

• For each we suppose that we have a vectorspace . We call this a discrete vectorbundle. We call the fiber of at .

• Moreover, when is a codimension 1 face of , we suppose that we have an isomorphism , called the transport map from to . We call this a discrete connection.

###### Remark 1.1.

This setup is at variance with the choices made in Lattice Gauge Theory (LGT). LGT was initially defined for cubical complexes

. An analogue for simplicical complexes was developed in . There, a discrete vector bundle corresponds to a choice of vector space attached to vertices only, whereas we here associate a vectorspace to each cell, of every dimension, in . Moreover, in LGT, a discrete connection is defined only on edges, as a choice of isomorphim between the vectorspaces attached to its two vertices ; here the discrete connection has many more variables.

###### Definition 1.2 (Flatness of discrete connections).

Whenever is a codim-2 face of , if we let and be the two codimension 1 faces of which have as a common codimension 1 face, we require that the following diagram commutes:

 (1)

Or, if one prefers:

 tTT′0tT′0T′′=tTT′1tT′1T′′. (2)

For reasons that will appear later we say that a discrete connection having this property is flat.

For instance we can choose a fixed vector space and let . This defines a discrete vectorbundle with a flat discrete connection. Discrete vectorbundles of this form, for a choice of vectorspace , will be called trivial discrete vectorbundles. We may speak of the trivial discrete vectorbundle modelled on , to make the choice of explicit.

In this setting one defines a cochain complex with coefficients in , denoted , as follows:

###### Definition 1.3 (Cochains with coefficients).

• The space is nothing but , whose elements will be families such that for each , . Such a will be called a -cochain with coefficients in .

• The differential is defined by:

 (δktu)(T)=∑T′⊴To(T,T′)tTT′u(T′). (3)

The operator will be called the discrete covariant exterior derivative.

We notice that we do indeed have a complex:

###### Lemma 1.1.

The operators on , satisfy:

 δk+1tδkt=0. (4)
###### Proof.

This is a direct consequence of (1), given what we know about relative orientations. ∎

This definition can be used in particular when a vector space has been chosen and we let for all and for all . This cochain complex will be denoted . With this notation we have in particular that , the standard cellular cochain complex introduced previously.

###### Remark 1.2.

In Definition 1.1 we could allow to be just a morphism (not necessarily an isomorphism), and still get at complex from (3). However, we have in mind situations where the transport operators mimick the parallel transport associated with a connection on a vectorbundle, and these are isomorphisms. Compare with  §I.4.7.

### 1.3 Transport along some paths within a cell

We notice that when the discrete connection is flat, transport along paths within a given cell just depends on the endpoints. We will use the following more precise statement:

###### Lemma 1.2.

Suppose , and that is a codimension- face of , for some . Then all sequences , where each term in this sequence is a codimension- subcell of the next, give the same map from to . This map will be denoted .

We could therefore make the following alternative description of a discrete vectorbundle with connection:

###### Definition 1.4 (equivalent definition of flat discrete vectorbundles).

For each we suppose that we have, as before, a vectorspace . Moreover, when , we suppose we have an isomorphim . We require that and also that whenever we have .

### 1.4 Discrete gauge transformations

Suppose we have, for each , two choices of vectorspaces denoted and . For a codim-1 face of we suppose that we have transport maps as well as . Under these circumstances we define an isomorphism from to to be a family of isomorphims , one for each , such that:

 θTtTT′=t′TT′θT′. (5)
###### Lemma 1.3.

Under the above circumstances induces an isomorphism of complexes , defined simply by:

 θ:(u(T))T∈Tk↦(θTu(T))T∈Tk. (6)
###### Proof.

Bijectivity is obvious. We prove that is a cochain morphism (i.e. commutes with the differentials). We let be the cochain-map of , while denotes that of . We have, for :

 (δkt′θu)(T) =∑T′⊴To(T,T′)t′TT′θT′u(T′), (7) =∑T′⊴To(T,T′)θTtTT′u(T′), (8) =(θδktu)(T), (9)

as required. ∎

A family of isomorphisms as above may also be referred to as a discrete gauge transformation. This terminology is used in particular when we have one discrete vectorbundle , but with two different choices of discrete connections and ; in this case will be an automorphism of , for each .

###### Lemma 1.4.

For each cell in , the complex (with coefficients) is isomorphic to (with constant fiber). Explicitely, for each subcell we let be the map . Then gives a gauge transformation, from to .

###### Proof.

The requirement is that the following commutation relation holds for subcells of such that :

 θStSS′=t′SS′θS′. (10)

This can also be written:

 tTStSS′=idL(T)tTS′. (11)

This identity holds according to Lemma 1.2. ∎

###### Corollary 1.5.

The sequence is exact, except at index , where the kernel is isomorphic to .

###### Remark 1.3.

The cohomology of the global space could be different from that of (for any choice of a fixed ).

###### Remark 1.4.

Flat vector bundles over a manifold , modulo gauge transformations, correspond to representations of the fundamental group of , modulo conjugacy. For a precise statement, see for instance Theorem 13.2 in  (in the context of principal bundles). This seems to carry over to the discrete setting. In the definition of the fundamental group of a cellular complex one can restrict attention to the so-called edge paths.

## 2 Discrete vector bundles with curvature

### 2.1 Cubes in the barycentric refinement

Consider now an -dimensional cube. For definiteness we consider the unit cube in and denote it as . We let be the canonical basis of .

The vertices of can be indexed by the subsets of . For any subset of , we let be the vertex of defined by:

 pU=∑i∈Uei. (12)

Any face of is uniquely determined by two vertices and , with , such that the vertices of are exactly those of the form for . Then we also have:

 S′={pU+∑i∈V∖Uξiei : ∀i∈V∖Uξi∈[0,1]}. (13)

Subsets of are partially ordered by inclusion, and this uniquely determines a partial ordering of the vertices. Then is the smallest vertex of and the largest.

Let be a simplex. For each face of we let be the isobarycenter of or, more generally, a point in the interior of (so a barycenter with respect to some strictly positive weights), referred to as the inpoint of . Recall that the barycentric refinement of is the simplicial complex whose -dimensional simplices are those of the form such that the are two by two distinct subsimplices of satisfying . We call such subsimplices of barycentric simplices.

The barycentric refinement may be coarsened as follows: for any two subsimplices and of such that , we consider the cell which is the union of all the barycentric simplices such that and .

When we start with a simplex , the cells of the form form a cellular complex, where each cell is, combinatorially, a cube. The same holds true if the cell we start with is a cube. We may consider that is the smallest vertex of and that is the largest. The vector points towards the center of . When is an -dimensional simplex, this procedure will divided it into cubes of dimension , each one of the form , where is a vertex of . We call this the cubical refinement of .

### 2.2 Discrete curvature and Bianchi identity

If we relax condition (1) we model the parallel transport associated with connections with curvature, as opposed to flat connections. We still define the discrete covariant exterior derivative by 3. When we now compute we get an operator , which one would like to interpret in terms of a curvature.

###### Definition 2.1 (Discrete curvature).

We suppose that and are to cells of , such that is a codimension 2 face of , and we let and be the two codimension 1 faces of which have as a common codimension 1 face. The curvature of is then defined by:

 ct(T,T′′)=±(tTT′0tT′0T′′−tTT′1tT′1T′′). (14)

which is associated with the square (element of the cubical refinement of ), whose set of vertices is associated with . The sign in this definition is given by the orientation of the square, which is chosen such that the orientations of the two transverse cells and induce the orientation of .

The definitions were chosen so as to have the trivial:

###### Lemma 2.1.

A discrete connection is flat according to Definition 1.2 iff its curvature according to Definition 2.1 is .

Definition 2.1 also gives:

###### Proposition 2.2.

With the notations of the Definition 2.1 we have:

 δk+1tδktu(T)=±ct(T,T′′)u(T′′). (15)

We have a Bianchi identity in this setting, which we now detail. Recall that the usual Bianchi identity says that the covariant exterior derivative of the curvature 2-form, which is a priori a certain endomorphism valued 3-form, is . In this identity, the relevant covariant exterior derivative is the one associated with the induced connection on the bundle of endomorphisms. The discrete identity will assert that certain linear operators attached to the 3-dimensional cubes, in the cubical refinement, is .

###### Definition 2.2 (Discrete bundle of endomorphisms and its connection).

• For we consider the previously introduced cubical refinement of , whose -dimensional cells are (combinatorial) cubes of the form , for subcells and of , where has codimension in . We let denote the cubical refinement of . We then define to be the discrete vectorbundle on , whose fiber at the cube is the space of linear maps from to .

• The discrete vectorbundle on inherits a discrete connection from the discrete connection of on , as follows.

Consider a -dimensional cube , where we say that is the smallest vertex and is the largest. When is a codimension face of there are two possibilities : either is the smallest vertex of and then we let be the largest, or is the largest vertex of and then we let be the smallest. We define the transport operator on through:

 ⎧⎪⎨⎪⎩End(L)(S′)→End(L)(S)u↦{tT′T0∘u if bT′′∈S′u∘tT0T′′ if bT′∈S′ (16)

The spaces are defined as before and the discrete covariant exterior derivative linking these spaces, is defined as in (3) from the given induced discrete connection on .

###### Remark 2.1.

A discrete connection for the discrete bundle over is thus an element of , where the element of attached to each edge of is bijective. Compare with the fact that, in the continuous setting, the difference between two connections is an endomorphism valued 1-form.

###### Theorem 2.3 (discrete Bianchi identity).

The curvature of , which is defined as an element of by (14), has a covariant exterior derivative (element of ) which is zero.

###### Proof.

That the discrete covariant exterior derivative of the curvature is zero expresses that for each 3-dimensional cube a certain linear map from to is zero. This linear map is a sum of maps of the form:

 ±tT′T1tT1T0tT0T′′, (17)

where the cells represent vertices of the cube, in increasing order. The sum consists of two such contributions from the curvature of each of the six faces of the cube. We thus get twelve maps of the form (17). They cancel two by two ; in fact each map appears twice in the sum, with different signs. ∎

###### Remark 2.2 (consistency).

One would like the discrete covariant exterior derivative to be in some sense consistent with a continuous one. Recall that the coboundary operator acting on (realvalued) simplicial cochains is isomorphic to the exterior derivative acting on Whitney form, via the de Rham map. One would like a similar interpretation of the discrete covariant exterior derivative.

For cellular complexes, an analogue of Whitney forms was provided in , by solving recursively, the PDE system and , or a discrete analogue. One motivation for identifying induced operators on subcells is to extend this construction to other differential complexes, where one wants to find preimages of cochains with coefficients. This connects with a broader theme of defining finite elements as solutions of local PDEs (possibly discretized at a subgrid scale).

This also raises the question of, to which extent, from a discrete vector bundle, one can reconstruct a continuous vector bundle. In the flat case this seems unproblematic. In the presence of curvature, a condition of small curvature might be necessary, to mimick that fibers vary continuously in the continuous setting. For instance, one could require that the maps in Lemma 1.2 should be close to each other, in some sense. As interesting and perhaps simpler special cases, one could consider the reconstruction of line bundles, and bundles over two-dimensional manifolds.

###### Remark 2.3 (gauge transformations and curvature).

Discrete gauge transformations are defined as in §1.4, also in the presence of curvature.

Notice that the discrete curvature transforms naturally under discrete gauge tranformations. Indeed, consider a discrete connection for and a discrete connection for , as well as gauge transformations such that (5) holds. Then we have:

 θTct(T,T′′)=ct′(T,T′′)θT′′. (18)

Also, given a gauge transformation from to , there is an induced gauge transformation from to , equipped with their induced connections. For a cube it maps to . The above transformation of curvature can be seen as a special case.

## 3 Finite element systems

###### Definition 3.1.

We fix a flat discrete vector bundle on in the above sense. A finite element system on consists of the following data, which includes both spaces and operators:

• We suppose that for each , and each we are given a vector space . For we suppose .

• For every and , we have an operator called differential. Often we will denote it just as . We require . This makes into a complex.

• Given in with we suppose we have restriction maps:

 rkT′T:Ak(T)→Ak(T′), (19)

subject to:

• .

• .

This makes the family , for , into an inverse system of complexes.

• For any -dimensional cell in we suppose we have an evaluation map . We suppose that the following formula holds, for :

 eTdTu=∑T′∈∂To(T,T′)tTT′eT′rT′Tu. (20)

###### Remark 3.1.

Identity (20) is a generalization of Stokes’ theorem. Indeed Stokes’ theorem may be regarded as the case where consists of realvalued -forms on , , , and is integration of a -form on a -dimensional cell.

If is a cellular subcomplex of , the spaces with constitute an inverse system. The inverse limits can be identified as:

 lim←−T∈T′Ak(T)={ (uT)T∈T′∈⨁T∈T′Ak(T) : T′⊴T⇒uT′=rT′TuT}. (21)

We notice that, in the special case where is a cell and denotes the cellular complex consisting of all the subcells of in , then the restriction maps provide an isomorphism:

 r:A∙(T)→lim←−T∈S(T)A∙(T). (22)

For this reason it seems safe to use the notation:

 A∙(T′)=lim←−T∈T′A∙(T), (23)

in the sense that if there is no ambiguity.

The notation (23) will be used in particular when is the (set of cells in included in the) boundary of a given cell . In that case denotes also the cellular complex consisting of the strict subcells of , which are precisely those included in its boundary.

###### Remark 3.2.

Another way of formulating (20) is that for any cellular subcomplex of , the evaluations (for ) provide a cochain morphism:

 e:A∙(T′)→C∙(T′,L). (24)

We will later provide conditions under which the evaluation morphism (24) induces isomorphisms on cohomology groups. This would be an analogue of de Rham’s theorem which asserts that the de Rham map, from differential forms to cellular cochains, gives isomorphisms between the respective cohomologies. We therefore refer to the map in (24) as the de Rham map.

We denote by the kernel of the induced map . We consider that the boundary of a point is empty, so that if is a point .

###### Definition 3.2 (Flabby finite element systems).

We say that admits extensions on , if the restriction map induces a surjection:

 r:A∙(T)→Ak(∂T). (25)

We say that admit extensions on or is flabby, if it admits extensions on each .

This notion corresponds to that of flabby sheaves (faisceaux flasques in French ), due to the following result

###### Proposition 3.1.

The FES admits extensions on if an only if, for any cellular complexes such that , the restriction is onto.

###### Proof.

If one can extend from the boundary of a cell to the cell, the one can extend from subcomplexes to complexes, step by step, incrementing dimension by one each time. ∎

###### Remark 3.3.

In particular if admits extensions, then, when is a subcell of , the restriction is onto. However this is, in general, a strictly weaker condition than the extension property. To see this, consider for instance the finite element spaces consisting of functions on a quadrilateral , on its edges and on its vertices . Then the restriction from to each edge is onto, as are the other restrictions from faces to subfaces, but the restriction from to is not onto, since the latter has dimension 4 but the former had dimension only 3. In practice therefore, finite element spaces on a square therefore include, in addition to the affine functions, a bilinear function.

###### Remark 3.4 (kernels).

We define . We notice that we have induced maps , whenever . We also notice that we have a welldefined map : starting with an element in , restrict it to a vertex to get an element of , evaluate it to get an element of and parallel transport it to get an element of (the composition of these steps is independent of the choice of vertex and path from the vertex to ).

###### Definition 3.3 (Exactness of a FES).

• We say that is exact on a cell when the following, equivalent, conditions hold:

• The following sequence is exact:

 (26)

and moreover the map defined in Remark 3.4 is an isomorphism.

• The de Rham map induces isomorphisms on cohomology.

• We say that is locally exact on when is exact on each .

###### Proof.

The equivalence holds by Lemma 1.4. ∎

###### Definition 3.4.

We say that is compatible when it is flabby and is locally exact.

###### Remark 3.5 (The flat connections of upwinding and band gap computations).

Some PDEs, such as convection diffusion equations, can be expressed with covariant derivatives , see also .

Consider real valued differential forms. Choose , called the connection -form. Define the covariant exterior derivative by:

 dAu=du+A∧u. (27)

Then we have:

Then is identified as the curvature of . We suppose , so that the operators constitute a complex. For any contractible subdomain of we may choose such that . Then we have, for :

 dAu=exp(−ϕ)d(exp(ϕ)u). (29)

Furthermore is uniquely determined, if we impose, in addition, the value of for some point .

For every cell , choose an interior point . Let be the unique function such that and . For with , we notice:

 ∫Texp(ϕT)dAu =∫Tdexp(ϕT)u, (30) =∑T′⊴To(T,T′)∫T′exp(ϕT)u|′T, (31) =∑T′⊴To(T,T′)exp(ϕT(xT′))∫T′exp(ϕT′)u|′T. (32)

When is -dimensional, we let be the map:

 eT:u↦∫Texp(ϕT)u. (33)

And, when is a codimension face of we put:

 tTT′=exp(ϕT(xT′)). (34)

Then the preceding identity becomes:

 (35)

as required in (20).

As a slight variant, consider complex-valued differential forms (but still real valued), and let:

 dAu=du+iA∧u, (36)

with comparable consequences. One can even restrict attention to constant. On a torus, is not in general globally of the form , but of course locally this still holds and can be used in particular on individual cells. Such observations were made in  in the context of band gap computations for photonic cristals. The corresponding numerical methods are variants of exponential fitting.

### 3.1 de Rham type theorems.

The following theorem extends Proposition 5.16 in :

###### Theorem 3.2.

Suppose that the element system is compatible. Then the evaluation maps induces isomorphisms on cohomology groups.

###### Proof.

Exactness gives that the map induces isomorphims on cohomology groups.

From there the proof proceeds as in . ∎

We also have the following extension of Proposition 5.17 in :

###### Theorem 3.3.

Suppose that has extensions. Then is compatible if and only if the following condition holds :

For each the sequence has nontrivial cohomology only at index , and there the induced map:

 e:HkA∙0(T)→L(T), (37)

is an isomorphism (it is well defined by (20)).

###### Proof.

(i) For cellular complexes consisting only of vertices, the equivalence trivially holds because, when is a point, .

(ii) We suppose now that and that the equivalence has been proved for cellular complexes consisting of cells of dimension at most . Consider a cellular complex consisting of cells of dimension at most .

Let be a cell of dimension . We suppose that the finite element system is compatible on the boundary of . Since the boundary is -dimensional we may apply the de Rham theorem 3.2 there. In other words induces isomorphisms on cohomology.

We write the following diagram:

 (38)

• The complex is rather trivial. The terms consist of cochains that are zero on the boundary of . In other words the only non-zero space in the complex is at index , where it is .

• On the rows, the second map is inclusion and the third arrow restriction. Both rows are short exact sequences of complexes.

• The vertical maps are the de Rham map.

• The diagram commutes.

We write the two long exact sequences corresponding to the two rows, and connect them by the map induced by the de Rham map.

 (39)

The equivalence is now proved in two steps:

• Suppose that (26) is exact. Then the first and fourth vertical maps are isomorphisms. By the induction hypothesis the second and fifth are isomorphisms. By the five lemma, the third one is an isomorphism. This can be stated by the condition formulated in the theorem.

• Suppose that the stated condition holds. One applies again the five lemma to the long exact sequence, and obtains now that has the same cohomology as .

### 3.2 Extensions, dimension counts and harmonic interpolation.

The following proposition almost exactly reproduces Proposition in .

###### Proposition 3.4.

Suppose that is an element system and that . We are interested only in a fixed index . Suppose that, for each cell , each element of can be extended to an element of in such a way that, and for each cell with the same dimension as , but different from , we have . Then admits extensions on .

###### Proof.

In the situation described in the proposition we denote by a chosen extension of (from to ).

Pick . Define .

Pick and suppose that we have a such that and have the same restrictions on all -dimensional cells in . Put . For each -dimensional cell in , remark that , so we may extend it to the element . Then put:

 ul+1=ul+∑U:dimU=l+1extUrU∂Twl. (40)

Then and have the same restrictions on all -dimensional cells in .

We may repeat until and then is the required extension of . ∎

###### Proposition 3.5.

Let be a FES on a cellular complex . Then:

• We have:

 dimAk(T)≤∑T∈TdimAk0(T). (41)
• Equality holds in (41) if and only if admits extensions on each .

###### Proof.

The proof in  works verbatim. ∎

###### Definition 3.5.

Given a FES on a cellular complex , a system of degrees of freedom is a choice of subspace , for each and . In that situation we can define maps by, for :

 Φk(T)u=(⟨⋅,rT′Tu⟩)T′⊴T (42)

where the brackets denote the canonical bilinear pairing . We say that the system is unisolvent on if is an isomorphism for each .

We will use the following result:

###### Proposition 3.6.

Suppose that is a FES on a cellular complex . Suppose that is a system of degrees of freedom for . Suppose that for each , the canonical map is injective. Suppose that is such that:

 dimAk(T)≥∑T′⊴TdimFk(T). (43)

Then is unisolvent on on the cellular complex , is flabby on and equality holds in (43).

###### Proof.

See Propositions 2.1 and 2.5 in . ∎

###### Example 3.1.

For each cell , equip with a continuous scalar product , typically a variant of the product. We define a system of degrees of freedom as follows. For each , we consider the following space of linear forms on :

 Fk(T) ={⟨⋅|v⟩ : v∈dAk−10(T)}⊕{⟨d⋅|v⟩ : v∈dAk0(T)} (44) ⊕{l∘eT : l∈L(T)⋆}, (45)

where the last space in the direct sum should be included only for .

We call these the harmonic degrees of freedom. For compatible finite element systems these degrees of freedom are unisolvent.

If one considers that the linear forms in are defined on more general fields, these DoFs yield a commuting interpolator onto , which we call the harmonic interpolator. For more on this topic see §2.4 of , in particular Proposition 2.8 of that paper.

###### Remark 3.6.

If the degrees of freedom consisting only of the spaces are unisolvent, then the FES is minimal, and the fields provide an analogue of Whitney forms. One can obtain such a FES inside any compatible FES by imposing the degrees of freedom (44) to be zero. This generalizes the construction of Whitney forms on cellular complexes given in .

### 3.3 Discrete vector bundles : a dual picture

Notice that the degrees of freedom for appearing in (44) play a special role. In practice they often appear in a slightly different way, namely as integration against certain fields, forming a space which is more tangible than (the parallel transport operators acting on can be more natural for instance).

We now make some remarks on this alternative point of view.

We suppose that we have for each a vector space and a bilinear form on . Moreover, when has codimension in we suppose we have a bijective (linear) restriction map , subject to the condition that . The generalized Stokes theorem takes the form, for and :

 ⟨du,ϕ⟩T=∑T′o(T,T′)⟨rT′Tu,sT′Tϕ⟩T′. (46)

In practice, this formula often arises as follows. The bilinear form on is the