1 Introduction
This article constructs a family of finite element cochain complexes (FECC) of differential forms in any dimension with arbitrary continuity on Cartesian meshes. The building block is a onedimensional FECC based on modified Hermite interpolation. It is conforming, meaning that, for all , there holds . The FECC in arbitrary dimension is then derived by adhering to a strict tensor product construction.
Finite element cochain complexes have developed as an important tool to construct and analyze discretization methods for vector fields and operators of the de Rham complex, see e. g.
[2]. By applying the BernsteinGelfandGelfand construction, they can also be used to obtain mixed discretization schemes for more complex situations like elasticity [3]. There, a sequence of de Rham sequences of varying continuity is employed.A first step in the direction of FECC with arbitrary continuity was accomplished in [5], where the authors introduce finite element (FE) spaces forming exact conforming cochain subcomplexes of the de Rham complex. The present contribution generalizes [5] from conformity to arbitrary regularity.
A fundamental property of an FECC is the commutation between the exterior derivative and the FE interpolation operator, since it immediately implies that the FECC is a subcomplex of the continuous complex. We prove this commutativity using a general lemma, which we introduce in Section 2. In Section 3, we develop FEs that admit this commutation property in one dimension. Tensor products of these elements are used in Section 4 to construct the FECC in higher dimensions.
2 Cochain Complexes of Finite Element Differential Forms
The construction of FE spaces of differential forms follows the general construction principle of FE spaces, namely first subdividing the domain of computation into a mesh of simple mesh cells —typically simplices or hypercubes— and then defining a shape function space on each cell. The continuity between mesh cells is established by the choice of node functionals at the interfaces between cells. These node functionals must also be chosen so that each function in the shape function space is uniquely determined by the values of the node functionals, the so called unisolvence.
The construction of an FECC differs only by the fact that the shape function spaces and node functionals are not chosen independently, but in relation to each other. Therefore, let
be an exact sequence of differential forms with polynomial coefficients forming the shape function spaces. Let denote the exterior derivative, and define as well as for . Now assume that on sets of basis functions and node functionals are chosen with the following properties:
(1a)  
(1b)  
(1c) 
Namely, the last basis functions span the kernel of and the node functionals separate between kernel and cokernel in the sense that the matrix of node functionals applied to basis functions has the structure
(2) 
For the space we assume that sets of basis functions and node functionals are chosen so that span the range of , and the node functionals separate the range of and its complement, namely
(3a)  
(3b)  
(3c) 
Thus, we have the corresponding structure
Note though that the subspaces of the two partitionings are defined in a different way. Nevertheless, both choices are always possible by basic results of linear algebra.
Lemma 1
Proof: Given the node functionals, the FE interpolation operators , for , are defined as
(6) 
where the coefficients are the entries of the matrix . We thus have by (3b)
where we shortened the second sum to terms using the block structure of . Similarly, by (1a)
Thus, applying the commutation conditions in (4), the commutation result (5) will hold if we can show that there holds
namely, if . Applying again (4), we have
hence the commutation property (5) is proved.
3 The Finite Element Cochain Complex in One Dimension
A polynomial cochain complex in one dimension has the form
(7) 
Introducing vector proxies, a natural correspondence can be established between the spaces of polynomial 0forms and polynomial functions of degree , as well as the spaces of polynomial 1forms and polynomial functions of degree . In this view, the exterior derivative mapping forms to forms corresponds to the derivative in one dimension. The spaces fulfill the property . In order to construct a FE method of such forms, we must complement the spaces of polynomial 0 and 1forms by suitable node functionals which serve to establish continuity of piecewise polynomials across the interfaces between intervals. We note that a version of (7) for regular functions reads
(8) 
Thus, piecewise polynomials in for 0forms must be paired with such in in order to be consistent. In the next sections, we establish such continuity for polynomial differential forms on the interval
by a mixture of node functionals for Hermite interpolation and integral moments.
3.1 Finite elements for 0forms
We equip the space with the following set of node functionals
. First, we use the standard derivative degrees of freedom of twopoint Hermitian interpolation:
(9) 
They are complemented by interior moments
(10) 
where denotes the Legendre polynomial of degree on the interval , normalized with the condition . Finally, the last degree of freedom is
(11) 
Note that in (10)
(12) 
Hence, and are linear combinations of the function value degrees of freedom of Hermite interpolation, which together with (11) ensure continuity at the end points of the interval .
For and let be the standard polynomial basis used for Hermite interpolation such that for there holds
We obtain basis functions for the space of polynomial 0forms as follows: first,
We refer to this set of basis functions as the subset of Hermite interpolating basis functions. Except for , they are all polynomials of degree . Note in particular that has been defined following the alternative expression of given in (12).
Let denote the polynomials obtained by subsequent integration of Legendre polynomials. They are recursively defined by and
The remaining basis functions are then defined as
(13) 
Note that has an fold root at both interval ends and is of polynomial degree .
3.2 Finite elements for 1forms
We construct node functionals and basis functions for 1forms on such that the commuting interpolation result of the Section 2 holds. First, we observe that according to (7) the dimension is reduced by one. Thus, we define basis functions for by
Note that is the constant function which is mapped to zero by the derivative. The linear independence of the basis functions will follow from the unisolvence proven in the next section.
Similarly, we define node functionals with commutation in mind: first, let
(14) 
Hence, we have for any :
The remaining node functionals are
(15) 
Comparing with (10), we obtain for any :
A particular consequence of this construction is
(16) 
3.3 Unisolvence
In order to prove unisolvence of the FE for , we study the structure of nonzero entries of the matrix . To this end, we split node functionals and basis functions into three groups:

Hermite interpolation: basis functions and node functionals with index .

Additional basis functions (bubble functions) and node functionals with indices to .

The basis function in the kernel of and according node functional with index .
By construction, there holds immediately
Furthermore, since the function values and the first derivatives of vanish in the end points, there holds
What is missing are the node functionals in the second group. If applied to the functions in the first group, there may be nonzero entries anywhere. For the functions in the second group, we have to expand the integrand with respect to Legendre polynomials. Since there holds
(17) 
we obtain by recursion that
(18) 
where are coefficients computable from (17). Using that is the derivative of and is the lowest order polynomial in its expansion, we note for
This yields a lower triangular matrix. Ignoring the finer structure in formula (18), we obtain the following block matrix structure:
(19) 
Since this matrix is lower triangular with nonzero diagonal entries, it has full rank and thus the FE is unisolvent. In particular, the set of basis functions and the set of node functionals are both linearly independent.
Furthermore, it has the same block structure as in (2), the kernel being the one dimensional subspace corresponding to the last row and column, while the matrix corresponds to the first block system. Due to (16), the matrix is obtained by deleting the last row and column from (19). Therefore, is unisolvent as well.
3.4 Commutation property of interpolation operators
The node functionals and shape functions in subsections 3.1 and 3.2 serve to define interpolation operators according to (6). They are constructed precisely to fulfil the assumptions of Lemma 1. Thus, by the use of this construction, we obtain a commuting interpolation operator.
Remark 2
While the node functionals in (9) are standard degrees of freedom for Hermite interpolation, those in (12) and (11) are linear combinations of the usual ones. More so, they make the implementation considerably more complicated, since their usual purpose of ensuring continuity is not easily accomplished. Therefore, we point out that these degrees of freedom serve for the analysis of interpolation operators only, but that implementations should use and as in Hermite interpolation.
Remark 3
The node functionals and are welldefined for regular functions. Following the same procedure as in [5], it is possible to weaken such conditions, and introduce weighted node functionals and which are bounded in . This construction yields the stable and commuting quasiinterpolation operators given by:
(20) 
for .
4 Tensorization of the Finite Element Complex
In the present section we apply the tensor product construction of cochain complexes (see [1]) to the FE de Rham subcomplex (7). Moreover, tensor product interpolation operators are introduced.
4.1 Tensor complex in two dimensions
The tensor product of complex (7) on with itself is the following de Rham complex on the Cartesian product :
(21) 
where the spaces of polynomial differential forms are defined as
and the exterior derivative , , is the linear operator acting on tensor product functions as follows:
(22)  
We note that (21) is a subcomplex of the de Rham complex of smooth forms on , as .
A basis for , , can be obtained by tensorization of the basis and of and , respectively, namely
(23)  
The node functionals for , , are defined by tensor product and apply to tensor product functions as follows:
(24) 
for all compatible pairs of indices and . For example, in the particular case , the regular tensor product FEs are complemented with the following node functionals:

for and

for and ,

for and
Tensor product node functionals apply to twodimensional functions in a straightforward way. For example, for , there holds:
Finally, the tensor product interpolation operator , , is defined as follows:
It applies to rankone functions of the form as
and can be naturally applied to twodimensional functions.
4.2 Tensor complex in dimensions
The tensor product construction generalizes to any tensorization order , giving rise to the following de Rham subcomplex on the Cartesian product :
The tensor product spaces are defined as
(25) 
where and is the characteristic vector of a combination out of given by
(26) 
In analogy to formula (23), there holds
(27) 
with . The tensor product basis is also known as rankone basis.
The exterior derivative acts on the rankone basis function as
(28) 
with . Formula (28) extends by linearity to any element of .
The tensor product construction yields node functionals for that are defined on rankone basis functions as
(29) 
and then extend by linearity to any element of , with the convention for . The node functionals apply to higherdimensional regular functions in a straightforward way.
Having introduced node functionals and rankone basis functions for , it is natural to define the dimensional interpolation operator, which coincides with the tensor product interpolation operator
(30) 
once we extend on 1forms and on 0forms as the nulloperator.
Exploiting the tensor product construction, and using Lemma 1, the following result can be proved.
Lemma 4
The tensor product interpolation operator defined in (30) is a cochain operator, namely, for all smooth forms , there holds
(31) 
5 Conclusions and Further Developments
We have developed tensor product FEs for the de Rham complex of arbitrary smoothness and dimension. Their construction is based on one dimensional Hermite interpolation and the commutation property of the finite element interpolation operators is proven using a general commutation lemma.
This construction plays an important role in the framework of BernsteinGelfandGelfand (BGG) sequences [3], where for instance de Rham complexes of different smoothness are combined to yield new complexes. The construction here yields the onedimensional building blocks for tensor product FE BGG sequences as well as the tensor product base complexes, as we point out in [6].
The general framework of de Rham complex embeds several problems important from the application point of view, like the Darcy and the Maxwell problem. To discretize those problems, it is then possible to employ the proposed FE spaces. In particular, they can be combined with model order reduction techniques, to handle parametric boundary value problems. Some works in this direction [11, 12, 13, 14, 15] consider the parametric Helmholtz equation, and they could represent the starting point towards the parametric Maxwell equation. Moreover, following the ideas in [4], they can be employed in combination with uncertainty quantification techniques to deal with the lognormal Darcy problem modeling the fluid flow in bounded heterogeneous media [9, 10, 8, 7].
Acknowledgements
The first author acknowledge support from the HGS MathComp through the Distinguished Romberg Guest Professorship Program. Moreover, the work of the first author is part of a project that has received funding from the European Research Council ERC under the European Union’s Horizon 2020 research and innovation program (Grant agreement No. 865751). The second author was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1  390900948 (the Heidelberg STRUCTURES Excellence Cluster).
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