LOCAL L2-BOUNDED COMMUTING PROJECTIONS IN FEEC

We construct local projections into canonical finite element spaces that appear in the finite element exterior calculus. These projections are bounded in L and commute with the exterior derivative. Mathematics Subject Classification. 65N30. Received April 26, 2021. Accepted August 31, 2021.

in [9]. That paper makes use of weight functions ([9], pg. 2642) which belong to the finite element spaces. Here, instead, we use weight functions Z ( ) that are not finite element functions. Their use allows us to avoid an extra correction step that seemed to be required in [9]. In order to define Z ( ) we use the formal adjoint of the exterior derivative and bubble functions to guarantee smoothness across interelement boundaries. Our functions Z ( ) rely on the existence of regular potentials for closed forms on contractible domains, for which we rely on the work of Costabel and McIntosh [5]. In particular, we use these results on the extended patch of a subsimplex.
As in [9], we also first construct the projection onto the lowest-order space (e.g., the Whitney forms [16]). Then our projection for higher order elements uses the lowest order projection. An important difference is that we use alternative degrees of freedom for higher order elements. The degrees of freedom are essentially the ones used in the projection-based commuting interpolants developed by Demkowicz and collaborators [6]. In fact, the degrees of freedom which we use are exactly the generalization of the degrees of freedom of the ones used be Melenk et al. [14]. These degrees of freedom we allow us to define the higher order projections more efficiently.
The paper is organized as follows. In the next section we give some preliminaries. We then state the main result, Theorem 3.1, in the following section. In Section 4 we assume the existence of the weight functions Z ( ) satisfying certain requirements, and use them to construct the projections onto the lowest-order spaces, i.e., the spaces of the Whitney forms [16]. In the next section, we build on the lowest-order case to construct the projection onto higher order finite elements, concluding with the proof of the main result of the paper. Finally, in Section 6 we give the deferred construction of the weight functions Z ( ).
The Hodge star operator ⋆ maps 2 Λ isomorphically onto 2 Λ − for each . Using it we define the formal adjoint of −1 by: for ∈ (Ω), and the spaces the latter incorporating boundary conditions. The adjoint relation between and may be expressed as . We simply write ⟨︀ · , · ⟩︀ when the domain is understood from the context.

Simplicial complexes and co-boundaries
Let Ω ⊂ R be a bounded domain and let T ℎ be a simplicial triangulation of Ω consisting of -simplices. We assume the shape regularity condition ℎ ≤ , where is the diameter of the largest inscribed ball in , ℎ is the diameter of , and > 0 is the shape regularity constant. Closely related to the triangulation T ℎ is the associated simplicial complex (T ℎ ) consisting of all the simplices of T ℎ and all their subsimplices of dimension 0 through . We denote by (T ℎ ), or simply when the triangulation is clear, the collection of all the simplices in (T ℎ ) of dimension . If 0 , . . . , ∈ R are the vertices of ∈ , we may write [ 0 , . . . , ] for , the closed convex hull of the vertices. Often we need to endow a simplex with an orientation. This is a choice of ordering of the vertices with two orders differing by an even permutation giving the same orientation. If we select an ordering of all the vertices of T ℎ , this implies a default orientation for each of the simplices in (T ℎ ).
Let 1 , . . . , be such an enumeration of the vertices of the mesh and let 1 , . . . , be the continuous piecewise linear functions such that ( ) = . For = [ 0 , . . . , ] ∈ we define the bubble function a non-negative piecewise polynomial with support equal to . To any simplex = [ 0 , . . . , ] ∈ (T ℎ ) we also associate the Whitney -form ∈ (Ω) defined as For ∈ (T ℎ ) we define the star of as i.e., the union of all -simplices containing . The extended star of is given by the union of -simplices intersecting . As in [9], we assume that es( ) is contractible for all in (T ℎ ), as is usually the case. Associated with the simplicial complex are a chain complex and cochain complex. The space of -chains is the vector space where is given the default orientation and the same simplex with the opposite orientation is identified with − . The boundary map : The dual space of C is the space C of cochains. The basis for chains, leads to the dual basis * , ∈ , for cochains, defined by * ( ) = , , ∈ (using the Kronecker delta). The coboundary operator d : C → C +1 is defined by duality in the usual way: For the coboundary operator applied to a basis cochain we find that (2.3)

The FEEC forms
For integers 0 ≤ ≤ and > 0, the space of trimmed polynomial -forms of degree on R is where is the Kozul operator (see [2]). For a simplex in R of any dimension, the trimmed space on is given by restriction: P − ( ) = { tr : ∈ P − (R ) }. Associated to any triangulation T ℎ of R and to the integers and we then have the global trimmed finite element space P − (T ℎ ), which is defined as The Whitney forms , ∈ , form a basis of P − 1 (T ℎ ). The space P − (T ℎ ) decomposes into a kernel portion and its orthogonal complement: For a -form that is smooth enough to admit an 1 trace on some ∈ (T ℎ ), the de Rham map defines the -cochain R by From Stokes theorem we easily see that The Whitney interpolant : C → (Ω) is defined in term of the Whitney forms by ( * ) = , so The following properties of the Whitney interpolant are crucial (see [16], (4), (5) in pg. 139) (2.5c) In particular, (2.5a) gives for ∈ It is easily shown that where ℎ is the local mesh size near . (We may define ℎ precisely as the diameter of if > 0 and as the diameter of st( ) if is a vertex.) Now that we have introduced the Whitney and de Rham maps we can define the canonical projection, Π 1 onto the Whitney forms, which is given by Since we are assuming that es( ) is contractible for all ∈ (T ℎ ), the local spaces P − ℓ (es( )) form an exact sequence; see [1,2,9]. Proposition 2.1. Assume es( ) is contractible. For any ∈ (T ℎ ) and ≥ 1 the following sequence is exact: We will also need a discrete Poincaré inequality on the extended star es( ).
There exists a constant such that for all ∈ (T ℎ ) one has To prove this one uses the equivalence of norms on a finite dimensional space, together with a compactness argument and scaling by dilation. See ( [2], Sect. 5.4) and ( [9], Sect. 5) for similar arguments.
The following proposition can be found Costabel and McIntosh ([5], Thm. 4.9(c)). It is proven using a generalized Bogovskii operator.
Using Friedrich's inequality (see [12]) we have that ‖ ‖ 2 ( ) ≤ diam( ) ‖ ‖ 2 ( ) . In [5] the authors do not track the constant . However, in [13] it is shown that if is star-shaped with respect to a ball of similar diameter then the constant can be bounded. Moreover, in Theorem 33 of [13] bounds for the constants in slightly more general cases are given. However, for arbitrary ∈ (T ℎ ), the patch es( ) need not be star-shaped with respect to a ball and we cannot show that in general it satisfies the conditions of ( [13], Thm. 33). Therefore, we assume that the constants es( ) are uniformly bounded.
Using that diam(es( )) ≤ ℎ , we obtain the following result, which will use below.
Proposition 2.5. Let ∈ (T ℎ ). Assume the hypotheses of Proposition 2.3 with = es( ) and also Assumption 2.4. Then there exists a constant > 0 such that where is the function defined in Proposition 2.3.

Main result
We now present the main result of the paper.
Theorem 3.1. The operator is a projection. It commutes with the exterior derivative: Moreover, if the mesh is shape regular and Assumption 2.4 holds, then we have the local 2 estimates where The construction of the operator and proof of the theorem is carried out in the remaining part of the paper. In the next section we begin with the construction in the lowest order case.

Projection for the lowest order case = 1
In order to motivate our construction of an 2 -bounded projection, we recall the canonical projection which maps an element ∈ (Ω) to In order that R ( ) be well defined, tr must be defined and integrable. This is not the case for general ∈ 2 (Ω) when < . To obtain a projection that is well defined for ∈ 2 (Ω), we replace R ( ) with ⟨︀ Z ( ), ⟩︀ for a suitable Z ( ) ∈ 2 (Ω). (The subscript , which refers to the polynomial degree, is introduced for the higher-order projections introduced in the next section.) In this section we state the properties required of the differential form Z ( ) and, assuming that such a form exists, develop an 2 -bounded projection into the Whitney forms. We will verify the existence of a suitable form Z ( ) in Section 6.
Precisely, we shall show that for each ≥ 1 and 0 ≤ ≤ there exist a linear operator Z : C →˚(Ω) which satisfies It follows directly from (4.2a) and (4.1) that is an extension of Π 1 | P − (T ℎ ) : For any ≥ 1, the operator Moreover, the operators form bounded commuting projections: is a projection and the following commuting property holds: Moreover, if the mesh is shape-regular and Assumption 2.4 holds, we obtain the following bound: Proof. The fact that is a projection follows from (4.3) and the fact that Π 1 is a projection.
We see that 1 := 1 is our desired projection in the lowest-order case. In the next section we will obtain the projection in the higher order case > 1 as a correction to .

Idea of the construction
Next we discuss the strategy for constructing the projection in the general case. The first step is to decompose the space P − (T ℎ ) using the projection Π 1 . For each ≥ 1 we have In particular, 1 = 0. Also, using Stokes theorem we easily see that the spaces with fixed and increasing form a sub-complex of the complex formed by the P − (T ℎ ). The key step is to construct a projection : 2 (Ω) → that is local, 2 -bounded and commutes with the exterior derivative = +1 , ∈ (Ω).
Then we define : 2 (Ω) → P − (T ℎ ) for all ≥ 1 as If ∈ P − (T ℎ ) then − ∈ by (4.6) and hence = , so is indeed a projection. Moreover, one can easily show that it commutes with the exterior derivative.

Alternative degrees of freedom for ∈ P − ( )
We now turn to the key step of constructing the projection . For this, it is useful to use degrees of freedom (dofs) for the space P − (T ℎ ) different than the canonical degrees of freedom described in [2]. Instead we will use dofs developed by Demkowicz and collaborators [6], a generalization of the ones found in Melenk et al. [14].
Let be any simplex and consider the polynomial differential form spaces P − ( ) andP − ( ) = { ∈ P − ( ) : tr = 0 } where ≤ dim . We have the following exact sequence we obtain the exact sequence: Next, we decomposeP − ( ) into the kernel of and the space orthogonal to the kernel: In order to introduce the dofs efficiently we define the bilinear form where P is the 2 -orthogonal projection onto ZP − ( ) given by Note that ⟨⟨ · , · ⟩⟩ is an inner product on the spaceP − ( ). We now give the dofs for P − ( ) and prove their unisolvence. Proof. By (5.7) the total number of dofs in (5.8) is the same as the dimension of P − ( ). Suppose that the dofs (5.8) of vanish. We must show that = 0. We can do this by induction on dim . The base case dim = 0 is trivial. By the induction step tr = 0 for all ⊂ ( ) with ̸ = which in particular implies that ∈P − ( ). Thus, choosing = and = in (5.8) gives that ⟨⟨ , ⟩⟩ = 0. Since ⟨⟨ · , · ⟩⟩ is an inner-product onP − ( ), this implies that = 0 As a corollary we immediately obtain dofs for the global finite element space P − (T ℎ ). In fact, based on this remark we have the following corollary.

Discrete extensions
In this subsection we define some key spaces and extension operators. A differential form ∈ P − (T ℎ ) is determined by the dofs given in Corollary 5.2. For ∈ (T ℎ ), we define ( ) as the space of all ∈ P − (T ℎ ) for which all those dofs vanish except those associated to the simplex . We note that ( ) = 0 if dim < or dim ≥ + . In any case, if ∈ ( ), then supp ⊂ st( ). A simple consequence of Lemma 5.1 is the following: Lemma 5.5. Let ∈ ( ), and suppose that ∈ (T ℎ ) does not contain . Then, In particular, this occurs when dim < dim or dim = dim but ̸ = .
Next we define an operator from The next result shows that the operator is an extension operator if we restrict ourselves toP − ( ) and that it commutes the exterior derivative if we further restrict ourselves toP − ( ). From this we conclude that ≡ 0 which proves (5.12).

The projection
In this section we construct the projection : 2 (Ω) → and show that it is bounded and commutes with the exterior derivative. Following (5.19) it takes the form: Therefore we first construct the forms U ( , ) and establish their key properties. As usual, let 0 ≤ ≤ and ≥ 1 be integers. The -forms U are linear operators of the second argument which is the superposition of the bubble functions on the -simplices comprising st ℎ ( ). Clearly it is supported in st( ) and vanishes on any simplex in T ℎ of dimension less than .
Turning to the definition of U ( ), we note that if dim < , thenP − ( ) vanishes and so U ( ) = 0. For ≤ dim we define U ( ) separately on ZP − ( ) and Z ⊥P− ( ), namely we define where ∈ (st ℎ ( )) is the unique solution to and The following lemma establishes the properties of U ( ).
Lemma 5.8. For every ∈ (T ℎ ) the following properties hold: Proof. We see that (5.22b) and (5.22c) follow immediately from the definition of U. Let us prove (5.22a) first in the case ∈ ZP − ( ). In this case = 0 and = P and so where we used (5.21). On the other hand, suppose that ∈ Z ⊥P− ( ). Then we note that ∈ ZP − +1 ( ) and so by the previous case ⟨︀ , U ( , ) ⟩︀ = ⟨⟨tr , ⟩⟩ for any ∈ since ∈ +1 . Thus, for ∈ we can use the definition of U and integration by parts to get Using that ∘ = 0, P = and P = 0 we get This proves (5.22a). The estimate (5.22d) follows from a scaling argument.
Before proving the main result of this section we will need the following estimate that follows from the definition of and a scaling argument Proof. The fact that is a projection follows from (5.19), (5.22a). To prove (5.24) we use (5.13) and (5.17) to get = ∑︁ On the other hand, using integration integration by parts we obtain If use (5.22b) then we see that U +1 ( , ) = 0 when ∈ z +1,⊥ ( ). Hence, We now see that (5.24) follows from another application of (5.22b). Finally, one can easily establish (5.25) using (5.23) and (5.22d).

The final projection
Having defined , the projection : 2 (Ω) → P − (T ℎ ) is defined by (5.2). As pointed out there, it is indeed a projection operator. Moreover, since 1 = 0, the operator 1 vanishes and so 1 coincides with 1 . We can now prove our main result, Theorem 3.1.

Construction of Z
It remains to construct the linear operators Z : C →˚(Ω) satisfying the properties (4.2a)-(4.2d) whose existence was asserted in Section 4. We will again use a superposition of bubble functions as in (5.20), but now defined with respect to the extended star of a simplex: Thus b is supported in es( ) and vanishes on any simplex in T ℎ of dimension less than .
Lemma 6.1. Let ∈ and let : Z ⊥ P − (es ℎ ( )) → R be a linear functional. Then the following problem has a unique solution: Find ∈ Z ⊥ P − (es ℎ ( )) satisfying Proof. This is a square linear system and therefore we only have to prove uniqueness. Suppose that ⟨︀ b , ⟩︀ = 0 for some ∈ Z ⊥ P − (es ℎ ( )). Then, given the property of b we have that for each ∈ with ⊂ es( ) we have that vanishes on . Hence, vanishes on es( ) or ∈ ZP − (es ℎ ( )). Thus, must be zero.
We also need the following bound that follows from inverse estimates: Finally, we will need the following result which follows from a scaling argument: Theorem 6.2. Assume that the mesh is shape regular and assume that Assumption (2.4) holds. For 0 ≤ ≤ , there exists a linear operator Z : C →˚(Ω) satisfying (4.2a)-(4.2d).
Proof. The proof is by induction on . To initialize we need to define Z 0 ( ) ∈˚(Ω) for ∈ 0 and ≥ 1.
By another inverse estimate we have This combined with (6.7) shows (4.2d) for the case = 0.
If we now use an inverse estimate we get Combining (6.12) and (6.13) gives (4.2d).