Regular Decompositions of Vector Fields-Continuous, Discrete, and Structure-Preserving

We elaborate so-called regular decompositions of vector fields on a threedimensional Lipschitz domain where the field and its rotation/divergence belong to L and where the tangential/normal component of the field vanishes on a sufficiently smooth “Dirichlet” part of the boundary. We impose no restrictions on the topology of the domain, its boundary, or the Dirichlet boundary parts. The field is split into a regular vector field, whose Cartesian components lie in H and vanish on the Dirichlet boundary, and a remainder contained in the kernel of the rotation/divergence operator. The decomposition is proved to be stable not only in the natural norms, but also with respect to the L norm. Besides, for special cases of mixed boundary conditions, we show the existence of H-regular potentials that characterize the range of the rotation and divergence operator. We conclude with results on discrete counterparts of regular decompositions for spaces of low-order discrete differential forms on simplicial meshes. Essentially, all results for function spaces carry over, though local correction terms may be necessary. These discrete regular decompositions have become an important tool in finite element exterior calculus (FEEC) and for the construction of preconditioners.


Introduction
If the domain Ω is convex then the respective complementary space, X N (Ω) or X T (Ω), is continuously embedded in the space H 1 (Ω) of vector fields with Cartesian components in H 1 (Ω), cf. [1]. Then one can, for instance, write any u ∈ H(curl, Ω) as with p ∈ H 1 (Ω) and z ∈ H 1 (Ω). Since ∇ p L 2 (Ω) ≤ u L 2 (Ω) one obtains (using the continuous embedding) the stability property 1 A similar decomposition can be found for u ∈ H 0 (curl, Ω).
Generally, a decomposition of form (1) with the stability property (2) is called regular decomposition, even if L 2 -orthogonality does not hold. Actually, it turns out that (1)-(2) can be achieved even in cases where Ω is non-convex, in particular on non-smooth domains, or in cases where Ω or its boundary have non-trivial topology; only the L 2 -orthogonality has to be sacrificed, cf. [23].
The second stability estimate states that if u is already in the kernel of the curl operator, then z is zero. Hence, (i) the operator mapping u to h is a projection onto the kernel space and (ii) the complement operator projects u to the function z of higher regularity H 1 (Ω). For trivial topology of Ω and ∂Ω, the two decompositions (1)- (2) and (3)-(4) coincide. As a few among many more [20,Sect. 1.5], we would like to highlight two important applications of these regular decompositions.

(i) The second form (3)-(4), in the sequel called rotation-bounded decomposition,
can be used to show that the operator underlying a certain boundary value prob-lem for Maxwell's equations is a Fredholm operator. The key point is that the complement space of the kernel (from the view of the mentioned projections) is H 1 (Ω) which is compactly embedded in L 2 (Ω), see e.g., [16,18] and references therein. (ii) The first form (1)- (2), in the sequel called gradient-based decomposition, has been used to generate stable three-term splittings of a finite element subspace of H(curl, Ω), cf. [21,22,25,23], which allows the construction of so-called fictitious or auxiliary space preconditioners for the ill-conditioned system matrix underlying the discretized Maxwell equations.
In both applications, it is desirable to obtain the decompositions for minimal smoothness of the domain, e.g., Lipschitz domains, which are not necessarily convex. Moreover, it is also desirable to go beyond decompositions of the entire space H(curl, Ω) and extend them to subspaces for which the appropriate trace vanishes on a "Dirichlet part" Γ D of the boundary. In this case traces of the two summands should also vanish on Γ D .
In the present paper, we provide regular decompositions of both types for subspaces of H(curl, Ω) (in Section 3) and H(div, Ω) (in Section 4) comprising functions with vanishing trace on a part Γ D of the boundary ∂Ω for Lipschitz domains Ω of arbitrary topology. In particular, Ω is allowed to have handles, and ∂Ω and Γ D may have several connected components. The Dirichlet boundary Γ D must satisfy a certain smoothness assumption that we shall introduce in Section 2.1. In addition to the stability estimtes (2), (4), we show that the decompositions are stable even in L 2 (Ω).
In the final part of the manuscript, in Section 5, we establish regular decompositions of spaces of Whitney forms, which are lowest-order conforming finite element subspaces of H(curl, Ω) and H(div, Ω), respectively, built upon simplicial triangulations of Ω.
We point out that parts of this work are based on [20] and some results have already been obtained there and will be referenced precisely throughout the text.

Geometric Setting
Since subtle geometric arguments will play a major role for parts of the theory, we start with a precise characterization of the geometric setting: Let Ω ⊂ R 3 be an open, bounded, connected Lipschitz domain 2 . We write d(Ω) for its diameter. Its boundary Γ := ∂Ω is partitioned according to Γ = Γ D ∪ Σ ∪ Γ N , with relatively open sets Γ D and Γ N . We assume that this provides a piecewise C 1 dissection of ∂Ω in the sense of [14,Definition 2.2]. Sloppily speaking, this means that Σ is the union of closed curves that are piecewise C 1 .
Under the above assumptions on Ω and Γ D , [14,Lemma 4.4] guarantees the existence of an open Lipschitz neighborhood Ω Γ ("Lipschitz collar") of Γ and of a continuous vector field n on Ω Γ with n(x) = 1 that is transversal to Γ : where n is the outward unit normal on Γ . Extrusion of Γ D by the local flow induced by n spawns the "bulge" Υ D ⊂ Ω Γ \ Ω, see Section 2.   denote the standard Sobolev spaces where the distributional gradient, curl, or divergence is in L 2 and where the pointwise trace γu, the tangential trace γ τ u, or the normal trace γ n u, respectively, vanishes on the Dirichlet boundary Γ D , see e.g. [7,29,3]. These space are linked via the de Rham complex,

De Rham Complex with Boundary Conditions
where The range of each operator in (6) Barring topological obstructions these kernels can be represented through potentials: Let β 1 (Ω) denote the first Betti number of Ω (the number of "handles") and β 2 (Ω) the second Betti number (the number of connected components of ∂Ω minus one). By the very definition of the Betti numbers as dimensions of co-homology spaces we have cf. [29]. We call Ω topologically trivial if β 1 (Ω) = β 2 (Ω) = 0.

Regular Decompositions and Potentials related to H(curl)
Throughout we rely on the properties of Ω and Γ D as introduced in Section 2.1 and use the notations from Theorem 1. We write C for positive "generic constants" and say that a constant "depends only on the shape of Ω and Γ D ", if it depends on the geometric setting alone, but is invariant with respect to similarity transformations. To achieve this the diameter of Ω will have to enter the estimates; we denote it by d(Ω).
Theorem 2 (Gradient-Based Regular Decomposition of H(curl)) Let (Ω, Γ D ) satisfy the assumptions of Section 2.1. Then for each u ∈ H ΓD (curl, Ω) there exist z ∈ H 1 ΓD (Ω) and p ∈ H 1 ΓD (Ω) depending linearly on u such that with constants depending only on the shape of Ω and Γ D , but not on d(Ω).
For completeness, we give the proof, which makes use of three auxiliary results that we quote without proof: and E curl D u has compact support.
We follow the proof as in [23,Thm. 5.9], which is based on the ideas in [6,Prop. 5.1], and establish the L 2 -stability using the ideas from [31, Lemma 2.2]. Let u ∈ H ΓD (curl, Ω) be arbitrary but fixed.
Step 1: We extend u by zero to a function in H(curl, Ω e ), where Ω e is the extended domain from Theorem 1, and then to u ∈ H(curl, R 3 ) using the operator E curl Ω e from Lemma 2. We observe u |ΥD = 0 and that Lemma 2 implies Step 2: Let B ⊇ Ω e be a ball such that 1 ≤ d(B) ≤ 2 and define w := (L curl u) |B .
Due to Lemma 3, (LC 1 ), curl w = curl u in B. Since B has trivial topology, there exists a scalar potential ψ ∈ H 1 (B) with zero average B ψ dx = 0 such that Lemma 3 together with (11) implies where the last estimate is due to Poincaré's inequality on the convex ball B [4].
Step 3: we conclude that ψ |ΥD ∈ H 2 (Υ D ). We define the extension ψ := (E ∇,Stein ΥD ψ) |B ∈ H 2 (B). Note that if the bulge domain Υ D has multiple connected components, we have to define E ∇,Stein ΥD by putting together individual extension operators using a partition of unity. From Lemma 1, we obtain where ∇ ∇ designates the Hessian.
Step 4: In B, it holds that ).
It is easy to see that p = 0 in Υ D and so p ∈ H 1 ΓD (Ω). Correspondingly, ∇ p = 0 and u = 0 in Υ D , and so z ∈ H 1 ΓD (Ω). Combining (12) and (13) yields the desired estimates for z and p.
Remark 1 An early decomposition of a subspace of H(curl, Ω) ∩ H(div, Ω) into a regular part in H 1 (Ω) and a singular part in ∇H 1 (Ω) can be found in [5] and in [6,Proposition 5.1], see also [9,Sect. 3] and references therein. Theorem 2 was proved in [16,Lemma 2.4] for the case of Γ D = ∂Ω and without the L 2 -stability estimate, following [

Remark 2
The constant C in Theorem 2 depends mainly on the stability constants of the extension operators E curl Ω e and E ∇,Stein

ΥD
. If the bulge Υ D has multiple components Υ D,k , the final estimate will depend on the relative distances between Υ D,k , Υ D,ℓ , k = ℓ and the ratios d(Υ D,k )/ d(Ω) due to the construction of E ∇,Stein ΥD using the partition of unity.

Remark 3
If Γ D = ∂Ω, we can use the trivial extension by zero instead of E curl Ω e . In that case, one may modify the construction of the scalar potentials ψ to ψ k ∈ Results on regular decompositions in this special case can be found in [31,26].

Regular Potentials for Some Divergence-Free Functions
Let the domain Ω and the Dirichlet boundary part Γ D be as introduced in Section 2.1 and let Γ i , i = 0, . . . , β 2 (Ω), denote the connected components of ∂Ω, where β 2 (Ω) is the second Betti number of Ω.
We define the space 4 Above γ n denotes the normal trace operator, and the duality pairing is that between The next result identifies the above space as the range of the curl operator.
Theorem 3 (Regular potential of range(curl)) Let (Ω, Γ D ) be as in Section 2.1 and assume in addition that each connected component Υ D,k of the bulge has vanishing first Betti number, β 1 (Υ D,k ) = 0. Then and for each q ∈ H ΓD (div 00, Ω) there exists ψ ∈ H 1 ΓD (Ω) depending linearly on q such that where C depends only on the shape of Ω and Γ D , but not on d(Ω) The proof is along the lines of [ Proof (Theorem 3, part 1) Let u ∈ H ΓD (curl, Ω) be arbitrary but fixed, and observe that curl u ∈ H ΓD (div, Ω). For fixed i = 0, . . . , β 2 (Ω), let µ i ∈ C ∞ (Ω) be a smooth cut-off function that is one in a neighborhood of Γ i and zero on the other components, cf. [13, p. 45]. Then µ i u ∈ H(curl, Ω) and by Gauss' theorem

This proves that
In the second part, we show that H ΓD (div 00, Ω) ⊂ curl H 1 ΓD (Ω), for which we need two auxiliary results about the divergence-free extension of vector fields and H 1 -regular vector potentials.

Lemma 4 (Divergence-free extension)
Let D be a bounded Lipschitz domain. There exists a bounded extension operator , Proof This result is a by-product of the proof of [13,Thm. 3.1]. For each connected component of ∂D, one solves a Neumann problem for the Laplace equation with Neumann data γ n q and uses the gradient of the solution as the extension.
Proof On R 3 it is natural to use Fourier techniques, cf. the proof of [20,Lemma 2.7].
Let q = F q be the Fourier transform of q and set It turns out that w is the Fourier transform of a function w ∈ H 1 (R 3 ), cf. [13, p. 46], and it is easily seen that curl w = q and div w = 0. By construction, w depends linearly on q. Plancherel's theorem allows for the estimate which finishes the proof.
Step 1: Using Lemma 4 we extend q by zero from Ω to Ω e , resulting in q e ∈ H(div 00, Ω e ). We define q := E div,0 Ω e q e and the estimate of Lemma 4 shows that q ∈ H(div 0, R 3 ) with q |Ω = q and q 0,R 3 ≤ C q 0,Ω .
Step 3: Let B be the smallest ball containing Ω e such that d(B) = 2 d(Ω), see Sect. 2.1. We define w 1 ∈ H 1 (B) by where w B is the constant vector field that agrees with the average of w over B. Thus, w 1 B = 0 and still curl w 1 = q in B.
Step 4: Recall that q |ΥD = 0. Hence, Since we have assumed that each connected component Υ D,k of Υ D has vanishing first Betti number, due to (8), there exist scalar potentials where N is the number of connected components of Υ D . This shows that ϕ k ∈ H 2 (Υ D,k ). Using an adaptation E ∇,Stein ΥD of the Stein extension operator from Lemma 1, we obtain a function ϕ ∈ H 2 (R 3 ) such that ϕ |Υ D,k = ϕ k . Moreover, thanks to the stability of the extension, Note that the L 2 norm of ϕ k on Υ D,k has been estimated in terms of its gradient by Poincaré's inequality, making use of ϕ k Υ D,k = 0.
where we have used Poincaré's inequality and the fact that w 1 B = 0. Finally, (16) and the same arguments yield This concludes the proof of Theorem 3.

Remark 4
For the case that Γ D = ∅, we reproduce the classical result see [13,Thm. 3.4]. In that case, Step 4 of the proof can be left out and ψ = w 1 which is why div ψ = 0 in Ω. This property, however, is lost in the general case.

Remark 5
The constant C in Theorem 4 depends essentially on the stability constants of the divergence-free extension operator E div,0

Rotation-Bounded Regular Decomposition of H(curl)
We can now formulate another new variety of regular decompositions, for which the H 1 -component will vanish for curl-free fields.

Theorem 4 (Rotation-Bounded Regular Decomposition of H(curl) (I))
Let (Ω, Γ D ) be as in Section 2.1 and assume, in addition, that each connected component Υ D,k of the bulge has vanishing first Betti number, β 1 (Υ D,k ) = 0. Then, for each u ∈ H ΓD (curl, Ω) there exist z ∈ H 1 ΓD (Ω) and a curl-free vector field h ∈ H ΓD (curl 0, Ω), depending linearly on u such that where C depends only on the shape of Ω and Γ D , but not on d(Ω).
Proof Let u ∈ H ΓD (curl, Ω) be arbitrary but fixed. Due to Theorem 3, curl u ∈ H ΓD (div 00, Ω) and there exists a regular potential z ∈ H 1 ΓD (Ω) depending linearly on curl u such that curl z = curl u,

Remark 6
The constant C in Theorem 4 depends essentially on the stability constants of the divergence-free extension operator E div,0 where C depends only on the shape of Ω and Γ D , but not on d(Ω).
For the proof of Theorem 5 we need a Friedrichs type inequality.
Lemma 6 (Friedrichs-Type inequality in H(curl)) Let (Ω, Γ D ) be as introduced in Section 2.1 and write Π curl 0,ΓD for the L 2orthogonal projector from H ΓD (curl, Ω) onto H ΓD (curl 0, Ω). Then with a constant C depending only on the shape of Ω and Γ D , but not on d(Ω).
Note that if each connected component of the bulge Υ D has vanishing first Betti number, then Lemma 6 can be derived directly from Theorem 4 using that Π curl 0,ΓD is L 2 -orthogonal: In general, one obtains Lemma 6 from the more general Gaffney inequality and by Lemma 6, As a next step, we apply Theorem 2 to u 1 to obtain with z ∈ H 1 ΓD (Ω) and p ∈ H 1 ΓD (Ω) and ∇ z 0,Ω ≤ C curl u 1 0,Ω + d(Ω) −1 u 1 0,Ω .

Remark 8
We would like to emphasize that both in Theorem 2 and Theorem 5, the domain Ω may be non-convex, non-smooth, and may have non-trivial topology: It may have handles and its boundary may have multiple components. Also the Dirichlet boundary Γ D may have multiple components, each of which with nontrivial topology. Moreover, we have the pure L 2 (Ω)-stability in both theorems. In this sense, the results of Theorem 2 and Theorem 5 are superior to those found, e.g., in [9, Thm 3.4], [22] or the more recent ones in [10, Thm. 2.3], [24]. This in turn implies that range(N) = H ΓD (curl 0, Ω), which is a closed subspace of H ΓD (curl, Ω). It also implies that N 2 = N and so N is a projection (but different from the L 2 -orthogonal projection onto H ΓD (curl 0, Ω)). Hence, the operator R = id − N is a projection too and range(R) = ker(N) is closed in H ΓD (curl, Ω).
Since N is a projection, one can easily see that F 2 = id, so F is an isomorphism.

Remark 11
The L 2 -estimates from Theorem 4 then show that the corresponding operator R can be extended to a continuous operator mapping from L 2 (Ω) to L 2 (Ω).

Regular Decompositions and Potentials Related to H(div)
The developments of this section are largely parallel to those of Section 3 with some new aspects concerning extensions and topological considerations.

Rotation-Based Regular Decomposition of H(div)
The following theorem is the H(div)-counterpart of Theorem 2.
Theorem 6 (Rotation-Based Regular Decomposition of H(div)) Let (Ω, Γ D ) satisfy the assumptions made in Section 2.1. Then for each v ∈ H ΓD (div, Ω) there exist z ∈ H 1 ΓD (Ω) and q ∈ H 1 ΓD (Ω) depending linearly on v such that v = z + curl q, with constant C depending only on the shape of Ω and Γ D , but not on d(Ω).
We also need a counterpart to Lemma 2: an extension operator in H(div) that is continuous in L 2 and H(div). The proof essentially follows that of [20,Lemma 2.6] and is left to the reader.

Let D be a bounded Lipschitz domain with d(D) = 1. Then there exists a bounded linear extension operator E div
D : L 2 (D) → L 2 (R 3 ) such that, with constants depending only on D, and E div D v has compact support.
Finally, we need a counterpart to the projection introduced in Lemma 3. The proof of the following lemma is very close to that of [20, Lemma 2.7] and we skip it.

Lemma 9 (Fourier-based Projection for H(div))
There exists a bounded linear operator L div : H(div, Proof (Theorem 6) Without loss of generality, we may assume that d(Ω) = 1. Let v ∈ H ΓD (div, Ω) be arbitrary but fixed.
Step 1: We extend v by zero to a function in H(div, Ω e ), where Ω e is the extended domain as defined in Section 2.1, and then to v ∈ H(div, R 3 ) using E div Ω e . We observe that v |ΥD = 0 and that Lemma 8 implies Step 2: Let B ⊃ Ω e be a ball such that 1 ≤ d(B) ≤ 2 and define w := (L div v) |B .
Due to Lemma 9, div w = div v in B. Since B has trivial topology, we find from (9) that v |B − w ∈ H(div 0, B) = H(div 00, B), and conclude from Theorem 3 that there exists a vector potential and (using Lemma 9) Step 3: we conclude that ψ ∈ H 1 (curl, Υ D ). We define ψ := (E curl,Stein ΥD ψ) |B ∈  H 1 (curl, B). Note that if the bulge domain Υ D has multiple components, we have to define E curl,Stein ΥD by blending individual extension operators using a partition of unity. From Lemma 7, we obtain Step 4: ) .
It is easy to see that q = 0 in Υ D and so q |Ω ∈ H 1 ΓD (Ω). By the same argument, curl q = 0 and v = 0 in Υ D and so z ∈ H 1 ΓD (Ω). Combining (26) and (27) yields the desired estimates for z and q.

Regular Potential with Prescribed Divergence
The next result carries Theorem 3 over to H(div).
Moreover, for each v ∈ L 2 (Ω) there exists q ∈ H 1 ΓD (Ω) depending linearly on v such that, with a constant C depending on Ω and Γ D but not on d(Ω), div q = v and Note that the assumption on Υ D rules out the case Γ D = ∂Ω even for a domain with trivial topology, since at least one component of the bulge would have a boundary with multiple components. Necessarily so, because div H ∂Ω (div, Ω) contains only functions with vanishing mean.
Step 1: We extend v by zero to a function v ∈ L 2 (R 3 ).
Step 2: We denote by v := F v the Fourier transform of v and define One can show that w := F −1 w ∈ H 1 (R 3 ) and that curl w = 0 and div w = v. 5 By construction w depends linearly on v. Due to Plancherel's theorem, Step 3: Let B ⊃ Ω e a ball with 1 ≤ d(B) ≤ 2. We define w 1 ∈ H 1 (B) by where w B is the component-wise mean value over B, such that Step 4: Recall that v is zero outside of Ω. In particular, div w 1 = 0 in Υ D,k ∀k = 1, . . . , N.
Since each connected component Υ D,k of Υ D is assumed to have a connected boundary, we obtain H(div 0, Υ D,k ) = H(div 00, Υ D,k ) and so Theorem 3 guarantees the existence of a vector potential ψ k ∈ H 1 (Υ D,k ) depending linearly on w 1 such that This shows that ψ k ∈ H 1 (curl, Υ D,k ). We extend {ψ k } N k=1 to a function ψ ∈ H 1 (curl, R 3 ) by patching together the extensions E curl,Stein Υ D,k ψ k by a partition of unity such that (using Lemma 7, (34), (31), and the Poincaré inequality) 5 Alternatively, one can set w := ∇∆ −1 v, where ∆ −1 corresponds to solving the variational Laplace problem with right-hand side in L 2 (R 3 ) and the energy space L 2 loc (Ω) with gradient in L 2 (R 3 ). By a standard regularity argument [28], the gradient of the solution is even in H 1 (R 3 ).

Remark 12
The constant C in Theorem 7 depends essentially on the stability constants of the H k (curl)-extension operator from Lemma 7 and the partition of unity. It therefore involves the relative diameters d(Υ D,k )/ d(Ω) and the relative distances between Υ D,k and Υ D,ℓ for k = ℓ.

Divergence-Bounded Regular Decompositions of H(div)
We can now formulate other variants of regular decompositions of H(div) in analogy to what we did in Section 3.3.

Theorem 8 (Divergence-Bounded Regular Decomposition of H(div) (I))
Let (Ω, Γ D ) be as in Section 2.1. In addition, assume that each connected component Υ D,k of the bulge has a connected boundary, i.e., β 2 (Υ D,k ) = 0. Then, for each v ∈ H ΓD (div, Ω) there exists z ∈ H 1 ΓD (Ω) and a divergence-free vector field h ∈ H ΓD (div 0, Ω) depending linearly on v such that where C depends only on the shape of Ω and Γ D , but not on d(Ω).
Proof Let v ∈ H ΓD (div, Ω) be arbitrary but fixed. Then div v ∈ L 2 ΓD (Ω) and, due to Theorem 7, there exists a regular potential z ∈ H 1 ΓD (Ω) depending linearly on div v such that div z = div v,

Remark 13
The constant C is the same as in Theorem 7.
The last variant of H(div) regular decomposition of H(div) dispenses with the assumptions on the topology of the Dirichlet boundary and has better stability properties than the splitting from Theorem 8 (though with less explicit constants).
Theorem 9 (Divergence-Bounded Regular Decomposition of H(div) (II)) Let (Ω, Γ D ) be as in Section 2.1. Then, for each v ∈ H ΓD (div, Ω) there exists z ∈ H 1 ΓD (Ω) and a divergence-free vector field h ∈ H ΓD (div 0, Ω) depending linearly on v such that where C depends only on the shape of Ω and Γ D , but not on d(Ω).
For the proof of Theorem 9 we need a Friedrichs type inequality. Let Π div 0,ΓD be the L 2 -orthogonal projector from H ΓD (div, Ω) to H ΓD (div 0, Ω).

Lemma 10 (Friedrichs-Type Inequality for H(div))
Let (Ω, Γ D ) be as introduced in Section 2.1. Then with a constant C depending only on the shape of Ω and Γ D , but not on d(Ω).
Note that, if each connected component of the bulge Υ D has connected boundary, then Lemma 10 can be derived directly from Theorem 7 using that Π div 0,ΓD is L 2orthogonal: In general, one obtains Lemma 10 from the Gaffney inequality

Proof (Theorem 9)
Let v ∈ H ΓD (div, Ω) be arbitrary but fixed and set and by Lemma 10, As a next step, we apply Theorem 6 to v 1 to obtain with z ∈ H 1 ΓD (Ω) and q ∈ H 1 ΓD (Ω) and Using that div v 1 = div v and combining all the stability estimates obtained so far yields the desired result.

Discrete Counterparts of the Regular Decompositions
The discrete setting to which we want to extend the concept of regular decompositions is provided by finite element exterior calculus (FEEC, [2]) which introduces finite element subspaces of H(curl) and H(div) as special instances of spaces of discrete differential forms. In this section we confine ourselves to the lowest-order case of piecewise linear finite element functions. Throughout, we assume that (Ω, Γ D ) is as in Section 2.1, and, additionally, that Ω is a polyhedron and that ∂Γ D consists of straight line segments. All considerations take for granted a shape-regular family of meshes {T h } h of Ω, consisting of tetrahedral elements, and resolving Γ D in the sense that Γ D is a union of faces of some of the tetrahedra.
The following finite element spaces will be relevant:

Discrete Regular Decompositions for Edge Elements
Commuting projectors, also known as co-chain projectors, are the linchpin of FEEC theory [2,Sect. 7], and it is not different with our developments. Thus, let and the local stability estimates for all mesh elements T , where ω T is the element patch of T , the union of neighboring elements, and h T the element diameter. The constant C is uniform in T and depends only on the shape regularity of the mesh T h (Ω).
and satisfying the stability estimates where C is a generic constant that depends only on the shape of (Ω, Γ D ), but not on d(Ω), and on the shape regularity constant of T h (Ω). Above, h −1 is the piecewise constant function that is equal to h −1 T on every element T .
Obviously, this is a discrete counterpart of the regular decomposition of H(curl) from Theorem 2. The following theorem appears to be new and it corresponds to the rotation-bounded regular decomposition of Theorem 5. For the sake of brevity define the discrete nullspace of the curl operator If Ω and Γ D have simple topology, X h = ∇ W 0 h,ΓD (Ω), but if the first Betti number of Ω is non-zero, or if Γ D has multiple components, then a finite-dimensional cohomology space has to be added [2, Sect. 5.6].
where C is a uniform constant that depends only on the shape of (Ω, Γ D ), but not on d(Ω), and on the shape regularity constant of T h (Ω).
For the proof of Theorem 11 we need a discrete Friedrichs inequality, otherwise it runs parallel to the proof of Theorem 5. The discrete Friedrichs inequality can elegantly be derived using the co-chain projector R 1 h,ΓD .

Lemma 11 (discrete Friedrichs inequality for
with a uniform constant C that depends on the constant in the continuous Friedrichs inequality (Lemma 6) and on the shape-regularity constant of T h (Ω).
Proof Recall the L 2 -orthogonal projector Π curl 0,ΓD mapping from H ΓD (curl, Ω) to H ΓD (curl 0, Ω). Due to the L 2 -minimization property of where we have used the projection property R 1 h,ΓD and the stability estimate (51), and h max is the maximal element diameter. The proof is concluded by observing that curl Π curl,ΓD = 0 and h max ≤ d(Ω), and by the continuous Friedrichs inequality Lemma 6.

Proof (Theorem 11)
Let v h ∈ W 1 0,ΓD (Ω) be arbitrary but fixed and set v h,0 : and by Lemma 11, We now apply Theorem 10 to v h,1 and obtain Setting h h := ∇p h + v h,0 and combining the stability estimates from Theorem 10 with (55)-(56) concludes the proof.
We stress that the statements of Theorem 10 and Theorem 11 do not hinge on any assumptions on the topological properties of Ω and Γ D .

Discrete Regular Decompositions for Face Elements
For face elements, the construction of a boundary-aware co-chain projection operator that commutes with R 1 h,ΓD and the curl-operator has not yet been accomplished. Fortunately, in the case Γ D = ∅, this operator is available from [12]. Thus, in the following, we treat only the case Γ D = ∅ and just omit the subscript Γ D . Then, from [12] we can borrow a linear operator R 2 h : H(div, Ω) → W 2 h (Ω) such that and The next result takes Theorem 6 to the discrete setting.
Finally, we present a counterpart to the divergence-bounded regular decomposition of Theorem 9. For convenience we introduce the space of divergence-free face element functions The constants C depend only on the shape of Ω, but not on d(Ω), and the shape regularity of T h (Ω).
For the proof, we need a discrete Friedrichs type inequality: with a uniform constant C that depends on the constant in the continuous Friedrichs inequality (Lemma 10) and on the shape-regularity constant of T h (Ω).
Proof Recall the L 2 -orthogonal projector Π div 0 mapping H(div, Ω) to H(div 0, Ω). Due to the L 2 -minimization property of where we have used the projection property of R 2 h and the stability estimate (58). The proof is concluded by observing that div Π div 0 = 0 and h max ≤ d(Ω), and using the continuous Friedrichs inequality Lemma 10.

Proof (Theorem 13)
Let v h ∈ W 2 h (Ω) be arbitrary but fixed and let We apply Theorem 12 to v h,1 such that with which concludes the proof.

Remark 14
The result of Theorem 13 can be viewed as an improvement of the decompositions in [27] which are elaborated for the case of essential boundary conditions on ∂Ω.
Proof If the second Betti number of Ω vanishes, then N 2 h = curl W 1 h (Ω). So h h = curl q h for some q h ∈ W 1 h (Ω). Setting q h := q h − Π 1 h q h and using Lemma 11 concludes the proof.

Remark 15
The result of Corollary 2 is an improvement of [22, Lemma 5.2] which assumes a domain Ω that is smooth enough to allow H 2 -regularity of the Laplace problem (2-regular case, for details see [22,Sect. 3]). This lemma is used in [30] in a domain decomposition framework, where convex subdomains are assumed. With our improved version, this assumption can be weakened considerably.