Local $L^2$-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 $L^2$ and commute with the exterior derivative.


Introduction
Bounded commuting projections are a primary instrument in the finite element exterior calculus (FEEC) [2,3,1]. In particular, the existence of such projections from the Hilbert variant of the de Rham complex to a finite dimensional subcomplex is the primary requirement for stable Galerkin approximations for the Hodge Laplacian [1,Theorem 3.8]. Error estimates for the Galerkin approximation then follow. For this, the projections must be bounded on the space of L 2 differential forms with exterior derivative in L 2 . A stronger condition is that the projections are bounded on the larger space L 2 , which is a primary requirement to obtain improved error estimates [1,Theorem 3.11] and also to convergence for eigenvalue problems [1,Theorem 3.19].
The first commuting projections were developed by Schöberl [15] and later by Christiansen and Winther [4], who treated non-quasiuniform meshes and spaces with essential boundary conditions. In particular, the projection [4] is bounded in L 2 . However, the projections [15,4] are not local, meaning that the projection of a form u on a simplex T does not depend solely on u on a patch of elements surrounding T . More recently, Falk and Winther [9,10] constructed commuting projections that are local and defined for L 2 bounded k-forms with its exterior derivative also belonging to L 2 . More recently in two and three dimensions, Ern et al. [8] have developed commuting projections for the last part of the de Rham complex that are local and bounded in L 2 . Bounded commuting projections that are local may also be used to obtain error estimates in other norms such as L ∞ [11,7].
In this paper, inspired by the techniques of Falk and Winther [9], we construct local, L 2 -bounded, commuting projections from the de Rham complex in any space dimension onto subcomplexes consisting of finite element subspaces formed with respect to arbitrary simplicial meshes. To keep the presentation as simple as possible we give details in the case of trimmed finite element spaces P − r , although the results will also work for the full polynomials spaces. While there is overlap, there are also important differences between the work here and that in [9]. That paper makes use of weight functions z k f [9, pg. 2642] which belong to the finite element spaces. Here, instead, we use weight functions Z k r (σ) 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 k r (σ) we use the formal adjoint of the exterior derivative and bubble functions to guarantee smoothness across interelement boundaries. Our functions Z k r (σ) 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. In Section 3 we assume the existence of the weight functions Z k r (σ) satifying 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 a statement of the main result of the paper in Theorem 4.10. Finally, in Section 5 we give the deferred construction of the weight functions Z k r (σ).

Preliminaries
2.1. Differential Forms. The space of differential k-forms with smooth coefficients on a domain S is denoted by Λ k (S). The larger space allowing L 2 coefficients is denoted by L 2 Λ k (S) and similarly H ℓ Λ k (S) denotes the space of k-forms with coefficents in the Sobolev space H ℓ (S). The exterior derivative, denoted by d k , maps Λ k (S) → Λ k+1 (S) and extends to the spaces with less regularity. Finally, we define The Hodge star operator ⋆ maps L 2 Λ k isomorphically onto L 2 Λ n−k for each k. Using it we define the formal adjoint of d k−1 by: for w ∈ Λ k (Ω), and the spaces the latter incorporating boundary conditions. The adjoint relation between d and δ may be expressed as where · , · S is the inner-product of L 2 Λ k+1 (S). We let v 2 L 2 (S) = v, v S . We simply write · , · when the domain S is understood from the context.

Simplicial complexes and co-boundaries.
Let Ω ⊂ R n be a bounded domain and let T h be a simplicial triangulation of Ω consisting of n-simplices. We assume the shape regularity condition where ρ σ is the diameter of the largest inscribed ball in σ, h σ is the diameter of σ, and C S > 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 n. We denote by ∆ k (T h ), or simply ∆ k when the triangulation is clear, the collection of all the simplices in ∆(T h ) of dimension k. If x 0 , . . . , x k ∈ R n are the vertices of σ ∈ ∆ k , we may write [x 0 , . . . , x k ] for σ, 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 x 1 , . . . , x N be such an enumeration of the vertices of the mesh and let λ 1 , . . . , λ N be the continuous piecewise linear functions such that λ i (x j ) = δ ij . For σ = [x i0 , . . . , x in ] ∈ ∆ n we define the bubble function b σ := λ i0 λ i1 · · · λ in , a non-negative piecewise polynomial with support equal to σ. To any simplex σ = [x i0 , . . . , x i k ] ∈ ∆(T h ) we also associate the Whitney k-form φ σ ∈ HΛ k (Ω) defined as For σ ∈ ∆(T h ) we define the star of σ as i.e., the union of all n-simplices containing σ. The extended star of σ is given by the union of n-simplices intersecting σ. As in [9], we assume that es(σ) is contractible for all σ in ∆(T h ), as is usually the case. Associated with the simplicial complex are a chain complex and cochain complex. The space of s-chains is the vector space where σ is given the default orientation and the same simplex with the opposite orientation is identified with −σ. The boundary map ∂ k : The dual space of C k is the space C k of cochains. The basis ∆ k for chains, leads to the dual basis σ * , σ ∈ ∆ k , for cochains, defined by σ * (τ ) = δ στ , σ, τ ∈ ∆ k (using the Kronecker delta). The coboundary operator d k : C k → C k+1 is defined by duality in the usual way: For the coboundary operator applied to a basis cochain we find that [3]). For a simplex τ in R n of any dimension, the trimmed space on τ is given by restriction: Associated to any triangulation T h of R n and to the integers k and r we then have the global trimmed finite element space decomposes into a kernel portion and its orthogonal complement: For a k-form v that is smooth enough to admit an L 1 trace on some σ ∈ ∆(T h ), the de Rham map defines the k-cochain R k v by From Stokes theorem we easily see that The following properties of the Whitney interpolant are crucial (see [16, (4), (5) in page 139]) It is easily shown that where h σ is the local mesh size near σ. (We may define h σ precisely as the diameter of σ if s > 0 and as the diameter of st(σ) if σ is a vertex.) Now that we have introduced the Whitney and de Rham maps we can define the canonical projection, Π k 1 onto the Whitney forms, which is given by Π k 1 := W k R k . Since we are assuming that es(σ) is contractible for all σ ∈ ∆(T h ), the local spaces P − r Λ ℓ (es(σ)) form an exact sequence; see [2,3,9]. Proposition 2.1. Assume es(σ) is contractible. For any σ ∈ ∆(T h ) and r ≥ 1 the following sequence is exact: We will also need a discrete Poincaré inequality on the extended star es(σ).
. To prove this one uses the equivalence of norms on a finite dimensional space, together with a compactness argument and scaling by dilation. See [3, Section 5.4] and [9, Section 5] for similar arguments.
The following proposition can be found Costabel and McIntosh [5, Theorem 4.9 (c)]. It is proven using a generalized Bogovskii operator. Using Friedrich's inequality (see [12]) we have that ρ L 2 (D) ≤ Cdiam(D)C D u L 2 (D) . In [5] the authors do not track the constant C D . However, in [13] it is shown that if D is star-shaped with respect to a ball of similar diameter then the constant C D can be bounded. Moreover, in [13,Thm 33] bounds for the constants in slightly more general cases are given. However, for arbitrary σ ∈ ∆(T h ), the patch es(σ) 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 C es(σ) are uniformly bounded.
Using that diam(es(σ)) ≤ Ch σ , we obtain the following result, which will use below.
Proposition 2.5. Let σ ∈ ∆(T h ). Assume the hypotheses of Proposition 2.3 with D = es(σ) and also Assumption 2.4. Then there exists a constant C δ > 0 such that 3. Projection for the lowest order case r = 1 In order to motivate our construction of an L 2 -bounded projection, we recall the canonical projection which maps an element u ∈ Λ k (Ω) to In order that R k u(σ) be well defined, tr σ u must be defined and integrable. This is not the case for general u ∈ L 2 Λ k (Ω) when k < n. To obtain a projection that is well defined for u ∈ L 2 Λ k (Ω), we replace R k u(σ) with Z k r (σ), u for a suitable Z k r (σ) ∈ L 2 Λ k (Ω). (The subscript r, 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 k r (σ) and, assuming that such a form exists, develop an L 2 -bounded projection into the Whitney forms. We will verify the existence of a suitable form Z k r (σ) in Section 5. Precisely, we shall show that for each r ≥ 1 and 0 ≤ k ≤ n there exist a linear operator Z k r : It follows directly from (3.2a) and (3.1) that P k r is an extension of Π k 1 | P − r Λ k (T h ) : Lemma 3.2. For any r ≥ 1, the operator P k r : . Moreover, the operators P k r form bounded commuting projections: Theorem 3.3. The operator P k r : L 2 Λ k (Ω) → P − 1 Λ k (T h ) 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: (3.5) P k r u L 2 (T ) ≤ C u L 2 (es(T )) , T ∈ ∆ n , u ∈ L 2 Λ k (Ω). Finally, Proof. The fact that P k r is a projection follows from (3.3) and the fact that Π k 1 is a projection. To prove (3.5), let T ∈ ∆ n . Since #{ σ ∈ ∆ k : σ ⊂ T } = c 1 := n+1 k+1 , we have P k r u 2 where we used (2.7), (3.2d). To prove (3.4) it suffices to prove To this end, let τ ∈ ∆ k+1 and use (2.6) to re-write the right-hand side as To treat the left-hand side we write where the σ i ∈ ∆ k are the k-faces of τ . By (2.3), d k σ * i is the sum of terms η * where η runs over the (k + 1)-simplices which contain σ i (taken with proper orientation), and, in particular, includes τ . Using again (2.6) we see that (3.9) R k+1 (W k+1 (d k σ * i ))(τ ) = 1, 0 ≤ i ≤ k + 1, while, if σ ∈ ∆ k and σ is not contained in the boundary of τ , then Therefore, Thus, (3.7) holds. Finally, (3.6) follows from (3.3) and the definition of Π k 1 . We see that π k 1 := P k 1 is our desired projectionin the lowest-order case. In the next section we will obtain the projection in the higher order case r > 1 as a correction to P k r .

4.1.
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 − r Λ k (T h ) using the projection Π k 1 . For each r ≥ 1 we have In particular, M k 1 = 0. Also, using Stokes theorem we easily see that the spaces M k r with r fixed and k increasing form a sub-complex of the complex formed by the P − r Λ k (T h ). The key step is to construct a projection Q k r : L 2 Λ k (Ω) → M k r that is local, L 2 -bounded and commutes with the exterior derivative Then we define π k r : (3.6) and hence π k r u = u, so π k r is indeed a projection. Moreover, one can easily show that it commutes with the exterior derivative.

4.2.
Alternative degrees of freedom for v ∈ P − r Λ k (τ ). We now turn to the key step of constructing the projection Q k r . For this, it is useful to use degrees of freedom (dofs) for the space P − r Λ k (T h ) different than the canonical degrees of freedom described in [3]. Instead we will use dofs developed by Demkowicz and collaborators [6], a generalization of the ones found in Melenk et al. [14].
Let τ be any simplex and consider the polynomial differential form spaces We have the following exact sequence Next, we decomposeP − r Λ k (τ ) into the kernel of d and the space orthogonal to the kernel: . We know that [3, Theorem 4.14] In particular,P − r Λ k (τ ) = 0 if dim τ < k or if dim τ ≥ r + k. Therefore, by [3,Theorem 4.13] that (4.7) dim In order to introduce the dofs efficiently we define the bilinear form where P τ is the L 2 -orthogonal projection onto ZP − r Λ k (τ ) given by P τ u, w τ = u, w τ , w ∈ ZP − r Λ k (τ ).
Proof. By (4.7) the total number of dofs in (4.8) is the same as the dimension of P − r Λ k (T ). Suppose that the dofs (4.8) of w vanish. We must show that w = 0. We can do this by induction on dim T . The base case dim T = 0 is trivial. By the induction step tr τ w = 0 for all τ ⊂ ∆(T ) with τ = T which in particular implies that w ∈P − r Λ k (T ). Thus, choosing τ = T and y = w in (4.8) gives that w, w T = 0. Since · , · T is an inner-product onP − r Λ k (T ), this implies that w = 0 As a corollary we immediately obtain dofs for the global finite element space Remark 4.3. Let dim τ = k. Then the volume form of τ , vol τ , belongs toP − r Λ k (τ ) for all r ≥ 1 and thus τ w = w, vol τ τ = tr τ w, vol τ τ is always a dof of w ∈ P − r Λ k (T h ). If r = 1 then these are the only dofs given in Corollary 4.2 and they coincide with the canonical dofs in the case r = 1.
In fact, based on this remark we have the following corollary.
Corollary 4.4. A differential k form w ∈ M k r is uniquely determined by tr τ w, y τ , y ∈P − r Λ k (τ ), τ ∈ ∆(T h ). 4.3. Discrete Extensions. In this subsection we define some key spaces and extension operators. A differential form w ∈ P − r Λ k (T h ) is determined by the dofs given in Corollary 4.2. For σ ∈ ∆(T h ), we define G k r (σ) as the space of all w ∈ P − r Λ k (T h ) for which all those dofs vanish except those associated to the simplex σ. We note that G k r (σ) = 0 if dim σ < k or dim σ ≥ r + k. In any case, if w ∈ G k r (σ), then supp w ⊂ st(σ).
A simple consequence of Lemma 4.1 is the following: Let v ∈ G k r (σ), and suppose that τ ∈ ∆(T h ) does not contain σ. Then, (4.9) tr τ v = 0.
Note that E σ maps a k-form on σ to a piecewise polynomial k-form on Ω. In view of Lemma 4.1, we see that . The next result shows that the operator E σ is an extension operator if we restrict ourselves toP − r Λ k (σ) and that it commutes the exterior derivative if we further restrict ourselves toP − r Λ k (σ). Lemma 4.6. Let σ ∈ ∆(T h ), 0 ≤ k ≤ n and r ≥ 1. Then, Proof. We prove (4.12) first. Using (4.9) we have that tr σ E σ ρ ∈P − r Λ k (σ). Then, φ = tr σ E σ ρ − ρ ∈ P − r Λ k (σ) and it satisfies φ, y σ = 0, y ∈P − r Λ k (σ). From this we conclude that φ ≡ 0 which proves (4.12).
In order to define the projections it is helpful to identify an orthonormal basis forP − r Λ k (σ) which we will denote by p k r (σ). For 0 ≤ k ≤ dim σ we let z k,⊥ r (σ) be a basis of Z ⊥P− r Λ k (σ) satisfying (4. 16) p, q σ = δ pq , p, q ∈ z k,⊥ r (σ).