1. Introduction
The Finite Element (FE) Exterior Calculus (EC) [2, 1] is a powerful technique that combines tools from differential geometry and finite element analysis in constructing discretizations which inherit the natural structure of the underlying physical models. In our work, we consider general convectiondiffusion equations on Hilbert complexes, such as the ones involving , , and in 3D, describing diffusion (by Hodge Laplacian) and corresponding transport driven by different velocity fields.
The design of stable discretizations for convectiondiffusion problems, even in the scalar case, is a challenging task as these are singularly perturbed differential equations with small, and even vanishing, diffusion (see, e.g. [25] and the references therein for discussion on such topics). There is a vast amount of literature on various techniques designed to take care of the numerical instabilities associated with this type of equations. We refer the reader to recent and classical works on the subject focused on some of these techniques: mixed FE methods [11, 8, 19, 7, 10]; discontinuous Galerkin methods [12, 15, 17, 20]; discontinuous PetrovGalerkin methods [5, 14, 13].
Our results show unisolvence for the quasipolynomial (weighted) spaces used in simplexaveraged finite element (SAFE) discretization [27] for convectiondiffusion equations in Hilbert complexes. Such exponentially fitted finite element schemes have been used with success for scalar convectiondiffusion equations, i.e., in our terminology for convectiondiffusion problems in . A rough explanation of the ideas behind SAFE discretizations could be as follows: (1) define a variable representing the flux, as in mixed methods, and use a variable change to symmetrize the equation; (2) discretize the differential operator using discretization for the flux and the primal variable; (3) eliminate the flux (locally) and change the variables to obtain a discretization of the original problem. Such a path for the derivation of discrete problems is seen in the pioneering work on discretizing driftdiffusion models in 1D [26] and later in FE and finite volume schemes in higher spatial dimensions [3, 28, 21]. Recently, a more general FEEC approach has brought mechanisms that can utilize higher degree polynomials and can work in any spatial dimension. In addition to the SAFE discretizations [27], the FEEC approach was an important tool in designing exponentially fitted spacetime discretizations in [4].
In this work we consider one of the key ingredients needed in steps (2) and (3) above, namely, determining a set of unisolvent functionals for the numerical flux. A typical situation in the discretizations discussed above is the following: Given a polynomial vector space
, on an dimensional simplex , we discretize the flux using a quasipolynomial space of differential forms,where denotes the space of forms in with coefficients from . For example, for a convectiondiffusion equation in
in 3D the degrees of freedom (unisolvence functionals) which uniquely determine an element
, are the moments of
on edges of , faces of and itself. The classical works [6, 18, 1, 2] usually use the Stokes’ Theorem, when verifying the unisolvence of such functionals for , and the arguments involve differentiation of . For quasipolynomial spaces, with a nonconstant weight , such differentiation results in terms that have both derivatives of and derivatives of . The standard arguments are, therefore, not applicable except in some special cases, such as the lowest order first kind of NédélecRaviartThomas elements. Our analysis here circumvents the use of Stokes’ Theorem, and we are able to show that the unisolvence functionals for are also unisolvence functionals for for differential forms of all orders , in any spatial dimension , and all polynomial spaces of the first and second kind (Nédélec–Raviart–Thomas [24, 23, 22], Nédélec–Brezzi–Douglas–Marini [23, 22, 9] spaces).The landscape of the paper can be mapped as follows: Preliminaries and FEEC notation is introduced in Section 2. The unisolvence sets of functionals for the first kind (Nédélec–Raviart–Thomas) and second kind (Nédélec–Brezzi–Douglas–Marini) are discussed in Section 3 and Section 4, respectively. Examples for constructing discretizations of the flux are then given in Section 5, and numerical tests are shown in Section 6.
2. Preliminaries
In this section, we present some preliminary results which will be used in the following sections. We begin by a simple result, frequently used in the analysis.
Lemma 2.1.
Let be an open domain and be a Riemann integrable function. If (or ), then implies that almost everywhere.
We denote the spaces of polynomials in variables of degree at most and of homogeneous polynomial functions of degree by and , respectively. We will abbreviate them to and at times. Next, following [1], we present some basic notation commonly used in FEEC when working with polynomial differential forms.
2.1. Simplices and barycentric coordinates
Let denote the set of increasing maps , for . is the complementary map of with . For any , denote by . Similarly, let denote the set of increasing maps for . The map complementary to , denoted by , satisfies such that . Here, represents the range of in ascending order, which is also denoted by if there is no ambiguity. In addition, denotes the cardinality of .
Let be an simplex with the vertices . For each , the set is a subsimplex of dimension . For , is the dimensional subsimplex of opposite to the subsimplex . The set of subsimplices of dimension of is denoted by , and the set of all subsimplices of is denoted by .
We denote by the barycentric coordinates satisfying . Clearly, form a basis of and satisfy . For a subsimplex with , there is an isomorphism between of polynomial functions on and the space . That is, each can be expressed as
for a unique . The extension, denoted by , is defined by extending the righthand side for . It is readily seen that the extension is an injective mapping from to .
Since the vectors form a basis for , the dual basis functions form a basis for . For any face , the restrictions of to the tangent space of at any point of give a basis for .
The algebraic forms , , form a basis for . Hence, a differential form can be uniquely written in the form
By definition of wedge product, we have , which implies that
(2.1) 
where denotes the volume form in .
2.2. Whitney forms
For any and , an associated differential form (called Whitney form) is given by
(2.2) 
where the inverted hat represents a suppressed argument. As shown below, the Whitney form gives an explicit formulation of the basis of .
Theorem 2.2 (Theorem 4.1 in [1]).
The Whitney forms corresponding to form a basis for .
3. Unisolvence for quasipolynomial spaces of first kind
In this section we consider the first type quasipolynomial
(3.1) 
Here, is a general positive weight on .
3.1. Geometrical decomposition of the first kind polynomial spaces
We now introduce the geometrical decomposition of and . The degrees of freedom of as given in [1, Section 4.6] are
(3.2) 
The proof (see, e.g. [1]) that these functionals form a unisolvent set uses induction argument and the Stokes’ Theorem, and the arguments do not carry over to quasipolynomial spaces. An attempt to prove the result for quasipolynomial spaces (3.1), however, reveals that the characterization of the trace free part of the plays a crucial role in showing the unisolvence. Such a characterization is given in the theorem below.
Theorem 3.1 (Theorem 4.16 in [1]).
For , , the map
(3.3) 
where the , defines an isomorphism of onto . Here, .
Next, we use this Theorem to show that the unisolvence functionals for the polynomial space also work for the quasipolynomials.
3.2. Polynomials of first kind with vanishing traces
The main result in this section is the following lemma.
Lemma 3.2.
Let . Suppose that
(3.4) 
Then .
Postponing the proof of this lemma for later, we note that its implications show the desired unisolvence results. Indeed, by induction argument (from low dimension subsimplices to high dimension subsimplices), it can be easily shown that the functionals given in (3.2) also give degrees of freedom for , and we have the following theorem.
Theorem 3.3.
Let , . Suppose that satisfies
Then .
Proof.
Before we proceed the proof of Lemma 3.2 for general case, we first give some examples to fix the ideas as abstractions can often be difficult to grasp. Noting that cases for forms in which or are trivial.
Proof of Lemma 3.2 for 1forms in .
In this case, there are two maps in , namely
In light of (3.3), can be uniquely written as
where . We choose a special test form defined as
We note here that the sign of is in accordance with the isomorphism defined in (3.3). Using that , , and collecting the coefficients of then shows that
On the other hand the polynomial function under the integral is nonnegative as seen below,
Finally, using that the weight is positive, i.e., together with Lemma 2.1, shows that and therefore proves Lemma 3.2 for and . ∎
Proof of Lemma 3.2 for forms and any spatial dimension .
Consider the general case . In a similar way, there are maps in , denoted by , where . Hence, can be uniquely written as
where . Taking a special test form as
which gives
and with
(3.5) 
If we now denote , , we have
or
(3.6) 
Since , for in the interior of , this shows Lemma 3.2 for and any spatial dimension . ∎
Proof of Lemma 3.2 for forms in any spatial dimension .
Consider the case in which . Again, there are maps in , whose complements we denote by with . Next, we rewrite as
where . Choose a special test function
Then, we have , where and with
We then have
(3.7) 
Therefore, we see that Lemma 3.2 holds for any with . ∎
3.3. Summary (spacial cases of Lemma 3.2)
Let us summarize what we have shown so far: Lemma 3.2 holds for any form for . As a consequence, for spatial dimensions , we have proved Lemma 3.2 in all the possible cases.
To generalize the ideas for other values of and , we proceed as in the special cases considered above. For a given , we find a special test form so that does not change sign on . This gives us a “mass” matrix which corresponds to the Whitney form bases in and .
We now follow this plan and generalize the unisolvence result to discrete differential forms of arbitrary order in any spatial dimension and any quasipolynomial Hilbert complex of first kind.
3.4. Calculating the “mass” matrix
Let us fix the spatial dimension and recall that
We now define as the number of inversions of the array corresponding to . For instance, when such that , then associates with the array and hence . It is easy to show that for any .
We first give the following result relating the maps , and the Whitney forms.
Lemma 3.4.
For any ,
(3.8) 
Proof.
We use the definition of the Whitney form (2.2), to obtain that
(3.9) 
Note that for any ,
The result follows by summing up the identities above. ∎
Lemma 3.5.
For and , it holds that
(3.10)  
Proof.
Since , , we easily see that . Moreover, if , we deduce from equation (3.9).
For the case in which , we see that . Notice that and , then
This completes the proof. ∎
Corollary 3.6.
For any , it holds that
(3.11) 
Proof.
We verify the statement case by case:

: obvious.

: both left hand side (LHS) and right hand side (RHS) are zero.

: Recall that , by (3.10),
This completes the proof. ∎
3.5. Proof of Lemma 3.2
Let us consider a differential form . By Theorem 3.1, we can write as
where . We take a special test form in (3.4) as
By Lemma 3.4 and Lemma 3.5, we have
(3.12) 
where is symmetric (by Corollary 3.6) and
(3.13) 
Recall that the exterior algebra has as a basis for , where is the orthogonal basis of . Further, the inner product on is defined as (c.f. [1, pp. 11])
(3.14) 
We have the following result which generalizes (3.6) and (3.7) for arbitrary and .
Lemma 3.7.
Proof.
We write
Next, we verify (3.15) for each component:

, the coefficient of is

: the coefficient of is obviously zero.

: Recalling that , we find the coefficient of as
Here, in the last step, we have used
where represent the common indices, which contribute equally in counting both and .
Combining the three cases above and using (3.13), we obtain (3.15). ∎
The main result in this section is shown next.
4. Unisolvence for the quasipolynomial spaces of second kind
We now prove the unisolvence of the degrees of freedom for quasipolynomials derived from polynomial Hilbert complexes of second kind which are defined as
(4.1) 
4.1. Geometrical decomposition of the polynomial spaces of second kind
We now consider the geometrical decomposition of the polynomial spaces associated with the Hilbert complexes of second kind: and . We point out that there is a little (if any) analogy in the proofs for these spaces.
Following [1, Section 4.5], the degrees of freedom for are
(4.2) 
Further, the characterization of the trace free part of the is stated below.
Theorem 4.1 (Theorem 4.22 in [1]).
For , , the map
(4.3) 
where the , defines an isomorphism of onto .
4.2. Polynomials of second kind with vanishing trace
We now state the main result of this section showing unisolvence of the functionals used for degrees of freedom (4.2).
Lemma 4.2.
Let . Suppose that
(4.4) 
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