SMOOTHED PROJECTIONS OVER WEAKLY LIPSCHITZ DOMAINS

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1 SMOOHED PROJECIONS OVER WEAKLY LIPSCHIZ DOMAINS MARIN WERNER LICH Abstract. We evelop finite element exterior calculus over weakly Lipschitz omains. Specifically, we construct commuting projections from L p e Rham complexes over weakly Lipschitz omains onto finite element e Rham complexes. hese projections satisfy uniform bouns for finite element spaces with boune polynomial egree over shape-regular families of triangulations. hus we exten the theory of finite element ifferential forms to polyheral omains that are weakly Lipschitz but not strongly Lipschitz. As new mathematical tools, we use the collar theorem in the Lipschitz category, an we show that the egrees of freeom in finite element exterior calculus are flat chains in the sense of geometric measure theory. 1. Introuction he aim of this article is to contribute to the unerstaning of finite element methos for partial ifferential equations over omains of low regularity. For partial ifferential equations associate to a ifferential complex, projections that commute with the relevant ifferential operators are central to the analysis of mixe finite element methos. In particular, smoothe projections from Sobolev e Rham complexes to finite element e Rham complexes are use in finite element exterior calculus (FEEC) [1, 3]. his was researche for the case that the unerlying omain is a Lipschitz omain. In this article, we stuy finite element exterior calculus more generally when the unerlying omain is merely a weakly Lipschitz omain. Specifically, we construct an analyze smoothe projections. hus we enable the abstract Galerkin theory of finite element exterior calculus within that generalize geometric setting. It is easy to motivate the class of weakly Lipschitz omains in the context of finite element methos. A omain is calle weakly Lipschitz if its bounary can be flattene locally by a Lipschitz coorinate transformation. his generalizes the classical notion of (strongly) Lipschitz omains, whose bounaries, by efinition, can be written locally as Lipschitz graphs. Although Lipschitz omains are a common choice for the geometric ambient in the theoretical an numerical analysis of partial ifferential equations, they exclue several practically relevant omains. It is easy to fin three-imensional polyheral omains that are not Lipschitz omains, such as the crosse bricks omain [24, p.39, Figure 3.1]. But as we show 2010 Mathematics Subject Classification. Primary 65N30; Seconary 58A12. Key wors an phrases. Finite element exterior calculus, smoothe projection, weakly Lipschitz omain, Lipschitz collar, geometric measure theory. his research was supporte by the European Research Council through the FP7-IDEAS-ERC Starting Grant scheme, project SUCCOFIELDS.. 1

2 2 MARIN WERNER LICH in this article, every three-imensional polyheral omain is still a weakly Lipschitz omain (see heorem 4.1 for the precise statement). Moreover, weakly Lipschitz omains have attracte interest in the theory of partial ifferential equations because basic results in vector calculus, well-known for strongly Lipschitz omains, are still available in this geometric setting [19, 20, 14, 6, 5]. For example, one can show that the ifferential complex (1.1) H 1 (Ω) gra H(curl, Ω) curl H(iv, Ω) iv L 2 (Ω) over a boune three-imensional weakly Lipschitz omain Ω satisfies Poincaré- Frierichs inequalities, an realizes the Betti numbers of the omain on cohomology. Furthermore, a vector fiel version of a Rellich-type compact embeing theorem is vali, an the scalar an vector Laplacians over Ω have a iscrete spectrum. Recasting this in the calculus of ifferential forms, one can more generally establish the analogous properties for the L 2 e Rham complex (1.2) HΛ 0 (Ω) HΛ 1 (Ω) HΛ n (Ω) over a boune weakly Lipschitz omain Ω R n. It is therefore of interest to evelop finite element analysis over weakly Lipschitz omains. Since the analytical theory is formulate within the calculus of ifferential forms, we wish to aopt this calculus on the iscrete level. Specifically, we use the framework of finite element exterior calculus; our agena is to exten that framework to numerical analysis on weakly Lipschitz omains. he founational iea is to mimic the L 2 e Rham complex by a finite element e Rham complex (1.3) PΛ 0 ( ) PΛ 1 ( ) PΛ n ( ). Here, each PΛ k ( ) is a subspace of HΛ k (Ω) whose members are piecewise polynomial with respect to a triangulation of the omain. Arnol, Falk, an Winther [1] have etermine specific finite element e Rham complexes that ai the construction an analysis of stable mixe finite element methos. A central component of finite element exterior calculus are uniformly boune smoothe projections. hese are an instance of commuting finite element projection operators, of which various examples are known in the literature [7, 26, 17, 10]. Our main contribution in this article is to evise such a projection when the omain is merely weakly Lipschitz (see heorem 7.11). As an immeiate consequence, the a priori error estimates of finite element exterior calculus are applicable over weakly Lipschitz omains. he following theorem is a conense version of the main result. heorem 1.1. Let Ω R n be a boune weakly Lipschitz omain, an let be a simplicial triangulation of Ω. Let (1.3) be a ifferential complex of finite element spaces of ifferential forms as in finite element exterior calculus [1]. hen there exist boune linear projections π k : L 2 Λ k (Ω) PΛ k ( ) L 2 Λ k (Ω) such that (1.4) HΛ 0 (Ω) HΛ 1 (Ω) π 0 π 1 PΛ 0 ( ) PΛ 1 ( ) HΛ n (Ω) π n PΛ n ( ) is a commuting iagram. Moreover, π k ω = ω for ω PΛ k ( ). he operator norm of each π k is uniformly boune in terms of the maximum polynomial egree of (1.3), the shape measure of the triangulation, an geometric properties of Ω.

3 SMOOHED PROJECIONS WEAKLY LIPSCHIZ 3 Let us outline the construction of the smoothe projection an the new tools which we employ in this article. We largely follow ieas in the publishe literature [1, 9] but introuce significant technical moifications. Given a ifferential form over the omain, the smoothe projection is compose in several steps. We first exten the ifferential form beyon the original omain by reflection along the bounary, using a parametrize tubular neighborhoo of the bounary. For strongly Lipschitz omains, such a parametrization can be constructe using the flow along a smooth vector fiel transversal to the bounary [1, 9], but for weakly Lipschitz omains such a transversal vector fiel oes not necessarily exist. Instea we obtain the esire parametrize tubular neighborhoo via a variant of the collaring theorem in Lipschitz topology [22]. Next, a regularizaiton operator smooths the extene ifferential form. Local control of the smoothing raius by a smoothe mesh size function guarantees uniform bouns for shape-regular families of meshes. his is similar to [9], but we elaborate the etails of the construction an make a minor correction; see also Remark he smoothe ifferential form has well-efine egrees of freeom. We then apply the canonical finite element interpolant to the smoothe ifferential form. he resulting smoothe interpolant commutes with the exterior erivative an satisfies uniform bouns but is not iempotent generally. We can, however, control the interpolation error over the finite element space. If the smoothe interpolant is sufficiently close to the ientity over the finite element space, then a commuting an uniformly boune iscrete inverse exists. Following an iea of Schöberl [25], the composition of this iscrete inverse with the smoothe interpolant yiels the esire smoothe projection. In orer to erive the aforementione interpolation error estimate over the finite element space, we call on geometric measure theory [12, 27]. he principal motivation in utilizing geometric measure theory is the low regularity of the bounary, which requires new techniques in finite element theory. A key observation, which we believe to be of inepenent interest, is the ientification of the egrees of freeom as flat chains in the sense of geometric measure theory. he esire estimate of the interpolation error over the finite element space is proven eventually with istortion estimates on flat chains. Moreover, we ientify a non-trivial gap in the corresponing proofs of previous works [1, 9]; see also Remark his gives further motivation for our recourse to geometric measure theory. Most of the literature on commuting projections focuses on the L 2 theory (but see also [8, 11]). We consier ifferential forms with coefficients in general L p spaces, following [15]. his article moreover prepares future research on smoothe projections which preserve partial bounary conitions. he remainer of this work is structure as follows. In Section 2, we introuce weakly Lipschitz omains an a collar theorem. We recapitulate the calculus of ifferential forms in Section 3. We briefly review triangulations in Section 4. he relevant backgroun in geometric measure theory is given in Section 5. hen we introuce finite element spaces, egrees of freeom, an interpolation operators in Section 6. In Section 7, we finally construct the smoothe projection. Acknowlegments. he author woul like to thank Douglas N. Arnol an Snorre H. Christiansen for stimulating iscussion. Parts of this article have appeare in the author s PhD thesis. Some of the research of this paper was one

4 4 MARIN WERNER LICH while the author was visiting the School of Mathematics at the University of Minnesota, whose kin hospitality an financial support is gratefully acknowlege. his research was supporte by the European Research Council through the FP7- IDEAS-ERC Starting Grant scheme, project SUCCOFIELDS. 2. Geometric Setting We begin by establishing the geometric backgroun. We review the notion of weakly Lipschitz omains an prove the existence of a close two-sie Lipschitz collar along the bounaries of such omains. We refer to [22] for further backgroun in the area of Lipschitz topology. hroughout this article, an unless state otherwise, we let finite-imensional real vector spaces R n an their subsets be equippe with the canonical Eucliean norms, which we write as. We let B r (U) be the close Eucliean r-neighborhoo, r > 0, of any set U R n, an we write B r (x) := B r ({x}). We introuce some basic notions of Lipschitz analysis. Let X R n an Y R m, an let f : X Y be a mapping. For a subset U X, we let the Lipschitz constant Lip(f, U) [0, ] of f over U be the minimal L [0, ] that satisfies x, x U : f(x) f(x ) L x x. We simply write Lip(f) := Lip(f, U) if U is unerstoo. We call f Lipschitz if Lip(f, X) <. We call f locally Lipschitz or LIP if for each x X there exists a relatively open neighborhoo U X of x such that f U : U Y is Lipschitz. If f is invertible, then we call f bi-lipschitz if both f an f 1 are Lipschitz, an we call f a lipeomorphism if both f an f 1 are locally Lipschitz. If f : X Y is locally Lipschitz an injective such that f : X f(x) is a lipeomorphism, then we call f a LIP embeing. he composition of Lipschitz mappings is again Lipschitz, an the composition of locally Lipschitz mappings is again locally Lipschitz. If X is compact, then every locally Lipschitz mapping is also Lipschitz. Let Ω R n be open. We call Ω a weakly Lipschitz omain if for all x Ω there exist a close neighborhoo U x of x in R n an a bi-lipschitz mapping ϕ x : U x [ 1, 1] n such that ϕ x (x) = 0 an (2.1a) (2.1b) (2.1c) ϕ x (Ω U x ) = [ 1, 1] n 1 [ 1, 0), ϕ x ( Ω U x ) = [ 1, 1] n 1 {0}, ( ) ϕ x Ω c U x = [ 1, 1] n 1 (0, 1]. Note that Ω is a weakly Lipschitz omain if an only if Ω c is a weakly Lipschitz omain. he close sets { Ω U x x Ω} constitute a covering of Ω an the mappings ϕ x Ω Ux : Ω U x [ 1, 1] n 1 are bi-lipschitz. Remark 2.1. In other wors, a weakly Lipschitz omain is a omain whose bounary can be flattene locally by a bi-lipschitz coorinate transformation. he notion of weakly Lipschitz omain contrasts with the classical notion of Lipschitz omain, then also calle strongly Lipschitz omain. A strongly Lipschitz omain is an open subset Ω of R n whose bounary Ω can be written locally as the graph of a Lipschitz function in some orthogonal coorinate system. Strongly Lipschitz omains are weakly Lipschitz omains, but the converse is generally false.

5 SMOOHED PROJECIONS WEAKLY LIPSCHIZ 5 We also note that a ifferent access towars the iea originates from ifferential topology: a weakly Lipschitz omain is a locally flat Lipschitz submanifol of R n in the sense of [22]. Weakly Lipschitz omains insie Lipschitz manifols are efine similarly [14]. Example 2.2. Every boune omain Ω R 3 with a finite triangulation is a weakly Lipschitz omain. We will specify an prove this statement in Section 4 after having formally efine triangulations. At this point, we review a concrete an well-known example, namely the crosse bricks omain Ω CB, which has alreay been mentione in the introuction. Let (2.2) Ω CB := ( 1, 1) (0, 1) (0, 1) (0, 1) (0, 1) ( 1, 1) (0, 1) {0} (0, 1); see also the left part of Figure 1. he omain Ω CB is not a Lipschitz omain because at the origin it is not possible to write Ω CB as the graph of a Lipschitz function in any orthogonal coorinate system. Inee, the contrary woul imply the existence of a vector that has positive angle both with the secon coorinate vector (0, 1, 0) an its negative, corresponing to Ω CB being partly the lower sie of the upper brick an the upper sie of the lower brick near the origin. But Ω CB is a weakly Lipschitz omain. his follows from heorem 4.1 later in this chapter, but it is easy to verify in the particular example of Ω CB. We first observe that near every non-zero x Ω CB we can write Ω CB as a Lipschitz graph, from which we can easily construct a suitable Lipschitz coorinate chart aroun x. But this approach oes not work at the origin. It is possible, however, to eform Ω CB into a strongly Lipschitz omain Ω CB by a bi-lipschitz mapping; see the right part of Figure 1. By the efinition of strongly Lipschitz omains, we can write Ω CB as the hypograph of a Lipschitz function near the origin in an orthogonal coorinate system. From this observation, the existence of a U 0 an ϕ 0 : U 0 [ 1, 1] with ϕ 0 (0) = 0 an (2.1) is easily euce. For technical completeness, we escribe the relevant mappings explicitly. We first we efine ϕ CB : R 3 R 3 by setting ϕ(x 1, x 2, x 3 ) := (x 1, x 2, x 3 ), where x 2 = x 2 3 /4(1 x 2 ) min(1, x 3 ) if x 2 [0, 1], x 2 3 /4(1 + x 2 ) min(1, x 3 ) if x 2 [ 1, 0], x 2 if x 2 / [ 1, 1]. his mapping is obviously Lipschitz. Its inverse ϕ 1 CB is easily seen to be Lipschitz too, being given by ϕ 1 CB (x 1, x 2, x 3 ) = (x 1, x 2, x 3 ) with x 2 = x (1 x 2) min(1, x 3 ) if x 2 [ 3 /4, 1], x 2 + 3(1 + x 2 ) min(1, x 3 ) if x 2 [ 1, 3 /4], x 2 if x 2 / [ 1, 1]. he bi-lipschitz mapping ϕ CB transforms Ω CB onto Ω CB, as isplaye in Figure 1. As a next step, we efine the vectors ( e x := 1 ) ( ) (2.3), 0,, e y := (0, 1, 0), e z := 2, 0,, an for fixe δ > 0 to be etermine below we efine the Minkowski sum V 0 := [ 3δ, 3δ] e x + [ δ, δ] e y + [ δ, δ] e z.

6 6 MARIN WERNER LICH We write Ω CB as the hypograph of a Lipschitz function χ near the origin in the orthogonal coorinate frame (2.3), with the graph varying in the e x irection. We introuce χ : R 2 R by z if y 0, z χ(y, z) := y if 3 2 z y 0, z y if 2 3 z y 0, z otherwise. By visual inspection of Figure 1 an a moment of reflection we see that for δ > 0 chosen small enough the intersection V 0 Ω CB coincies with the set { xex + ye y + ze z 3δ x < χ(y, z), (y, z) [ δ, δ] 2 }. In other wors, we have shown that Ω CB is a Lipschitz graph in an orthogonal coorinate system near the origin. Next we efine ϕ 0 : [ 1, 1] 3 V 0 by ϕ 0 (z, y, x) = x e x + δy e y + δz e z, { where x χ(δy, δz)(1 x) 3δ(x + 1) if x [ 1, 0], := χ(δy, δz)(1 x) + 3δx if x [0, 1]. One can see that ϕ 0 is bi-lipschitz an maps [ 1, 1] 2 [ 1, 0) onto V 0 Ω CB. Now U 0 := ϕ 1 CB ϕ ( 0 [ 1, 1] 3 (2.4) ), ϕ 0 := ϕ 1 0 ϕ CB U 0 is the esire bi-lipschitz coorinate chart aroun the origin in which Ω CB is flattene. A variant of the crosse bricks omain is isplaye in the monograph of Monk [24, Figure 3.1, p.39], an another variant is iscusse in [6]. For a generalization of this example, we refer to Example 2.2 in [4]. Figure 1. Left: polyheral three-imensional omain Ω CB that is not the graph of a Lipschitz function at the marke point. he upper brick extens into the x-irection an the lower brick extens into the y-irection. Right: bi-lipschitz transformation of that omain into a omain Ω CB that is strongly Lipschitz omain, as can be seen by visual inspection. he remainer of this section buils up a key notion of this article. As a motivation, we recall that strongly Lipschitz omains have parametrize tubular neighborhoos, which can be constructe with a transversal vector fiel near the bounary

7 SMOOHED PROJECIONS WEAKLY LIPSCHIZ 7 [26, 8, 18]. Generalizing this iea, the following theorem shows that weakly Lipschitz omains allow for two-sie Lipschitz collars. heorem 2.3. Let Ω R n be a boune weakly Lipschitz omain. hen there exists a LIP embeing Ψ : Ω [ 1, 1] R n such that Ψ(x, 0) = x for x Ω an such that (2.5) Ψ ( Ω, [ 1, 0)) Ω, Ψ ( Ω, (0, 1]) Ω c. Without loss of generality, for every t (0, 1) the sets Ω \ Ψ ( Ω, [ t, 0)) an Ω Ψ ( Ω, (0, t)) are weakly Lipschitz omains. Proof. We first prove a one-sie version of the result. From efinitions we euce that there exist a collection {V i } i N of relatively open subsets of Ω that constitute a covering of Ω an a collection {ψ i } i N of LIP embeings ψ i : V i [0, 1) Ω such that for each i N we have ψ i (x, 0) = x for each x Ω. It follows that {(V i, ψ i )} i N is a local LIP collar in the sense of Definition 7.2 in [22]. By heorem 7.4 in [22], an a successive reparametrization, there exists a LIP embeing Ψ (x, t) : Ω [0, 1] Ω such that Ψ (x, 0) = x for all x Ω. We see that Ω c is a weakly Lipschitz omain too. Analogous arguments give a LIP embeing Ψ + (x, t) : Ω [0, 1] Ω c such that Ψ + (x, 0) = x for all x Ω. We combine these two LIP embeings an let Ψ : Ω [ 1, 1] R n, (x, t) Ψ (x, t) if x Ω, t [ 1, 0), x if x Ω, t = 0, Ψ + (x, t) if x Ω, t (0, 1]. hen Ψ is well-efine, bijective, an (2.5) hols. o prove that Ψ is a LIP embeing, we show the existence of a constant C > 0 such that (2.6) (2.7) Ψ(x 1, t 1 ) Ψ(x 2, t 2 ) C ( x 1 x 2 + t 2 t 1 ), x 1 x 2 + t 2 t 1 C Ψ(x 1, t 1 ) Ψ(x 2, t 2 ) for all x 1, x 2 Ω an t 1, t 2 [ 1, 1]. If t 1 an t 2 are both non-negative or both non-positive, then the both inequalities follow irectly from Ψ + or Ψ being LIP embeings with a constant C 1 that epens only on Ψ + an Ψ. Hence it remains to consier the case t 1 < 0 < t 2. Here, (2.6) follows from Ψ(x 1, t 1 ) Ψ(x 2, t 2 ) Ψ(x 1, t 1 ) x 1 + x 1 x 2 + x 2 Ψ(x 2, t 2 ) C t 1 + C t 2 + x 1 x 2 C t 1 t 2 + x 1 x 2 since Ψ + an Ψ are LIP embeings. Furthermore, there exists z Ω on the straight line segment from Ψ(x 1, t 1 ) to Ψ(x 2, t 2 ). We then fin (2.7) via t 1 t 2 + x 1 x 2 t 1 + x 1 z + z x 2 + t 2 C 1 Ψ(x 1, t 1 ) z + C 1 z Ψ(x 2, t 2 ) = C 1 Ψ(x 1, t 1 ) Ψ(x 2, t 2 ), again using that Ψ + an Ψ are LIP embeings. We conclue that Ψ is a LIP embeing. Restricting an reparametrizing Ψ completes the proof. Remark 2.4. Our heorem 2.3 realizes an iea from ifferential topology in a Lipschitz setting: any locally bi-collare surface is also globally bi-collare. Such

8 8 MARIN WERNER LICH results are well-known in the topological or smooth sense, but it seems to be only folklore in the Lipschitz sense. Notably, the result is mentione in the unpublishe preprint [13]. We have provie a proof for formal completeness. 3. Differential forms In this section we review the calculus of ifferential forms in a setting of low regularity. Particular attention is given to ifferential forms with coefficients in L p spaces an their transformation properties uner bi-lipschitz mappings. We aopt the notion of W p,q ifferential form of [15], to which we also refer for further etails on Lebesgue spaces of ifferential forms. An elementary introuction to the calculus of ifferential forms is given in [21]. Let U R n be an open set. We let M(U) enote the vector space of Lebesgue measurable functions over U up to equivalence almost everywhere. For k Z we let MΛ k (U) be the vector space ifferential k-forms over U with Lebesgue measurable coefficients. We enote by ω η MΛ k+l (U) the exterior prouct of ω MΛ k (U) an η MΛ l (U), an we recall that ω η = ( 1) kl η ω. Let e 1,..., e n be the canonical orthonormal basis of R n. he constant 1-forms x 1,..., x n MΛ 1 (U) are uniquely efine by x i (e j ) = δ ij, where δ ij {0, 1} enotes the Kronecker elta. In the sequel, we let Σ(k, n) enote the set of strictly ascening mappings from {1,..., k} to {1,..., n}. Note that Σ(0, n) = { }. he basic k-alternators are the exterior proucts x σ := x σ(1) x σ(k) MΛ k (U), σ Σ(k, n), an x := 1. he canonical volume n-form vol n MΛ n (U) is vol n := x 1... x n. For every ω MΛ k (U) an σ Σ(k, n) we efine ω σ = ω(e σ(1),..., e σ(k) ) M(U) an note that ω can be written as (3.1) ω = ω σ x σ. σ Σ(k,n) For every n-form ω MΛ n (U) there exists a unique ω vol M(U) such that ω = ω vol vol n. We efine the integral of ω MΛ n (U) over U as (3.2) ω := ω vol x U whenever ω vol M(U) is integrable. Note that this efinition of the integral presumes that R n carries the canonical orientation. For ω, η MΛ k (U) we efine the pointwise l 2 prouct ω, η M(U) by (3.3) ω, η := ω σ η σ. U σ Σ(k,n) For ω MΛ k (U) we let ω = ω, ω M(U) be the pointwise l 2 norm. We let L p (U) enote the Lebesgue space with exponent p [1, ], an let L p Λ k (U) enote the Banach space of ifferential k-forms with coefficients in L p (U). he topology of L p Λ k (U) is generate by the norm ω Lp Λ k (U) := L ω, ω, ω L p Λ k (U). p (U)

9 SMOOHED PROJECIONS WEAKLY LIPSCHIZ 9 We let CΛ k (U) be the Banach space of boune continuous ifferential k-forms over U, equippe with the maximum norm. We let C Λ k (U) be the space of smooth ifferential k-forms over U, we let C Λ k (U) be the subspace of C Λ k (U) whose members can be extene smoothly onto R n, an we let Cc Λ k (U) be the subspace of C Λ k (U) whose members have compact support in U. he exterior erivative : C Λ k (U) C Λ k+1 (U) is efine by (3.4) ω = n ( i ω σ x i) x σ, ω C Λ k (U), σ Σ(k,n) i=1 where we use the representation (3.1). One can show that is linear, satisfies the ifferential property = 0, an relates to the exterior prouct via (3.5) (ω η) = ω η + ( 1) k ω η, ω C Λ k (U), η C Λ l (U). We are intereste in efining the exterior erivative in a weak sense over ifferential forms of low regularity. If ω MΛ k (U) an ξ MΛ k+1 (U) are locally integrable such that (3.6) ξ η = ( 1) k+1 ω η, η Cc Λ n k 1 (U), U U then ξ is the only member of MΛ k+1 (U) with this property, up to equivalence almost everywhere, an we call ω := ξ the weak exterior erivative of ω. Note that ω has vanishing weak exterior erivative, since (3.7) ω η = ( 1) k ω η = 0, η Cc Λ n k 1 (U). U U Moreover, (3.5) generalizes in the obvious manner to the weak exterior erivative, provie that all expressions are well-efine. Next we introuce a notion of Sobolev ifferential forms. For p, q [1, ], we let W p,q Λ k (U) be the space of ifferential k-forms in L p Λ k (U) whose members have a weak exterior erivative in L q Λ k+1 (U). We equip W p,q Λ k (U) with the norm (3.8) ω W p,q Λ k (U) := ω Lp Λ k (U) + ω Lq Λ k+1 (U), ω W p,q Λ k (U). It is obvious that W p,q Λ k (U) is a Banach space. Moreover, as a consequence of [15, Lemma 1.3] we know that C Λ k (U) is ense in W p,q Λ k (U) for p, q [1, ). Note that W p,q Λ k (U) W q,r Λ k+1 (U) for p, q, r [1, ] by efinition. Hence one may stuy e Rham complexes of the form W p,q Λ k (U) W q,r Λ k+1 (U) he choice of the Lebesgue exponents etermines analytical an algebraic properties of these e Rham complexes. his is not a subject of the present article, but we refer to [16] for corresponing results over smooth manifols. De Rham complexes of the above form with a Lebesgue exponent p fixe are known as L p e Rham complexes (e.g. [23]). wo examples of such e Rham complexes are of specific relevance to us.

10 10 MARIN WERNER LICH Example 3.1. he space HΛ k (U) := W 2,2 Λ k (U), consisting of those L 2 ifferential k-forms that have a weak exterior erivative with L 2 integrable coefficients, is a Hilbert space whose topology is inuce by the scalar prouct ω, η HΛ k (U) := ω, η L 2 Λ k (U) + k ω, k η L 2 Λ k+1 (U), ω, η HΛ k (U). In particular, the norms W 2,2 Λ k (U) an HΛ k (U) are equivalent. hese spaces constitute the L 2 e Rham complex HΛ k (U) HΛ k+1 (U) which has receive consierable attention in global an numerical analysis. Example 3.2. he space W, Λ k (U) of flat ifferential forms is spanne by those ifferential forms with essentially boune coefficients whose exterior erivative has essentially boune coefficients. hese spaces constitute the flat e Rham complex W, Λ k (U) W, Λ k+1 (U) Flat ifferential forms have been stuie extensively in geometric integration theory [27]; see also heorem 1.5 of [15]. We conclue this section with some basic results on the behavior of ifferential forms an their integrals uner pullback by bi-lipschitz mappings. For the remainer of this section, we let U, V R n be open sets, an let Φ : U V be a bi-lipschitz mapping. We first gather some facts on the Jacobians of bi-lipschitz mappings. It follows from Raemacher s theorem [12, heorem 3.1.6] that the Jacobians D Φ : U R n n, exist almost everywhere. One can show that (3.9) D Φ 1 : V R n n D Φ L (U) Lip(Φ, U), D Φ 1 L (V ) Lip(Φ 1, V ). Accoring to [12, Lemma 3.2.8], the ientities (3.10) D Φ 1 Φ(x) D Φ x = I, D Φ Φ 1 (y) D Φ 1 y = I hol true almost everywhere over U an V, respectively. In particular, these Jacobians have full rank almost everywhere an by [12, Corollary ] the signs of the Jacobians are essentially constant: uner the conition that U an V are connecte, there exists o(φ) { 1, 1} such that (3.11) o(φ) = sgn et D Φ almost everywhere over U. It follows from [12, heorem 3.2.3] that (3.12) (ω Φ) et D Φ x = ω(y) y U for ω M(V ) if at least one of the integrals exists. he pullback Φ ω MΛ k (U) of ω MΛ k (V ) uner Φ is efine as Φ ω x (ν 1,..., ν k ) := ω Φ(x) (D Φ x ν 1,..., D Φ x ν k ), ν 1,..., ν k R n, x U. By the iscussion at the beginning of Section 2 of [15], the algebraic ientity Φ (ω η) = Φ ω η + ( 1) k ω Φ η hols for ω MΛ k (V ) an η MΛ l (V ). Next we show how the integral of n-forms transforms uner pullback by bi-lipschitz mappings: V

11 SMOOHED PROJECIONS WEAKLY LIPSCHIZ 11 Lemma 3.3. If Φ : U V is a bi-lipschitz mapping between connecte open subsets of R n, then for every Lebesgue integrable function ω M(V ) we have (3.13) Φ (ω vol n ) = o(φ) ω vol n. U Proof. Using (3.11), (3.12), an the efinition of the pullback, we fin Φ (ω vol n ) = (ω Φ) et D Φ vol n = (ω Φ) et D Φ x U U U = (ω Φ) sgn et D Φ et D Φ x U = o(φ) (ω Φ) et D Φ x = o(φ) ω x = o(φ) ω vol n. U V V his shows the esire ientity. It can be shown that the pullback uner bi-lipschitz mappings commutes with the exterior erivative an preserves the L p an W p,q classes of ifferential forms. Lemma 3.4 (heorem 2.2 of [15]). Let Φ : U V be a bi-lipschitz mapping between open subsets of R n. If p [1, ] an ω L p Λ k (V ), then Φ ω L p Λ k (U). If p, q [1, ] an ω W p,q Λ k (V ), then Φ ω W p,q Λ k (U) an Φ ω = Φ ω. We refine the preceing statement an give an explicit estimate for the operator norm of the pullback operation. Here an in the sequel, n/ = 0 for n N. heorem 3.5. Let U, V R n be open sets an let Φ : U V be a bi-lipschitz mapping. For every p [1, ] an every ω L p Λ k (U) we then have (3.14) Φ ω L p Λ k (U) D Φ k L (U) et D Φ 1 1 p L (V ) ω L p Λ k (V ) V D Φ k L (U) D Φ 1 n p L (V ) ω L p Λ k (V ) Proof. Let Φ : U V an p [1, ] be as in the statement of the theorem, an let u L p Λ k (U). For almost every x U we observe Φ ω x D Φ x k ( ) 2 ωσ Φ(x) = D Φ x k ω Φ(x). σ Σ(k,n) From this we easily get Φ ω L p Λ k (U) D Φ k L (U) ω Φ L p (U). he esire statement follows trivially if p =, an (3.12) gives ( ω Φ) p x et D Φ 1 L (V ) ( ω Φ) p et D Φ x U U et D Φ 1 L (V ) ω p x if p [1, ). his shows the first estimate of (3.14). he secon estimate in (3.14) follows by Haamar s inequality, which estimates the eterminant of a matrix by the prouct of the norms of its columns. Φ(U)

12 12 MARIN WERNER LICH 4. riangulations In this section we review simplicial triangulations of omains an relate notions, most of which is stanar in the literature. We assume that Ω R n is a boune open set such that Ω is a topological manifol with bounary an Ω is its interior. A finite triangulation of Ω is a finite set of close simplices such that the union of the elements of equals Ω, such that for any an any subsimplex S we have S, an such that for all, the set is either empty or a common subsimplex of both an. We write ( ) := { S S }, ( ) := { S S }. With some abuse of notation, we let ( ) also enote the close set that is the union of the simplices of ajacent to. We write m for the set of m-imensional simplices in. Having formally introuce triangulations, we make precise an prove the introuction s claim that all polyheral omains in R 3 are weakly Lipschitz omains. heorem 4.1. Let Ω R 3 be an open set. Assume that Ω is a topological submanifol of R 3 with bounary whose interior submanifol equals Ω. If there exists a finite triangulation of Ω, then Ω is a weakly Lipschitz omain. Proof. Let x Ω an be a finite triangulation of Ω. We seek a compact neighborhoo U x R 3 of x an a bi-lipschitz mapping ϕ x : U x [ 1, 1] 3 such that ϕ x (x) = 0 an (2.1) hols. If x is not a vertex of, then x is either containe in the interior of a bounary triangle of, or in the interior of an ege between two ajacent bounary triangles of. In both cases, we may choose U x := B r (x) for r > 0 small enough, an ϕ x : U x [ 1, 1] 3 is easily constructe. It remains to consier the case x 0. Let r > 0 be so small that B r (x) intersects 3 if an only if x, so B r (x) Ω is a simple close curve in B r (x). Hence B r (x) Ω is locally flat in the sense of [22, p.100]. By the Schoenflies theorem in the Lipschitz category (see heorem 7.8 of [22]), there exists a bi-lipschitz mapping ϕ 0 x : B r (x) B 1 (0) R 3 which maps B r (x) Ω onto B 1 (0) {x R 3 x 3 = 0}. By raial continuation, we exten this to a bi-lipschitz mapping ϕ I x : B r (x) B 1 (0) R 3 which maps B r (x) Ω onto B 1 (0) {x R 3 x 3 < 0} an which satisfies ϕ I x(x) = 0. Moreover, there exists a bi-lipschitz mapping ϕ II : B 1 (0) [ 1, 1] 3 which maps B 1 (0) {x R 3 x 3 < 0} onto [ 1, 1] 2 [ 1, 0) an satisfies ϕ II (0) = 0. Specifically, we may set ϕ II (0) = 0 an ϕ II (y) := y 1 l 2 y l y, y B 1 (0) \ {0}. Since all norms on R 3 are equivalent, ϕ II is a bi-lipschitz mapping from the unit ball in the Eucliean norm to the unit ball in the maximum norm. he theorem follows with U x := B r (x) an ϕ x := ϕ II ϕ I x.

13 SMOOHED PROJECIONS WEAKLY LIPSCHIZ 13 Remark 4.2. he class of weakly Lipschitz omains is large but still exclues several omains that are common in finite element literature. For example, the slit omain Ω S = ( 1, 1) 2 \[0, 1) {0} is not a weakly Lipschitz omain. Note that Ω S is not the interior of Ω S, so heorem 4.1 oes not apply. Defining a smoothe projection over such a omain remains for future research. Another example is a cube issecte into two omains by a plane through the origin, which we exclue for analogous reasons. In the latter example, however, we may still apply the results of this article over each subomain separately. he remainer of this section is evote to notions of regularity of triangulations. Let us fix a finite triangulation of Ω. When m is any simplex of the triangulation, then we write h = iam( ) for the iameter of, an = vol m ( ) for the m-imensional volume of. If V 0, then V = 1, an h V is efine, by convention, as the average length of all n-simplices of that are ajacent to V. We efine the shape constant of as the minimal C mesh > 0 that satisfies (4.1) (4.2) n : h n C mesh,, S ( ) : h C mesh h S. Intuitively, (4.1) escribes a boun on the flatness of the simplices, while (4.2) escribes that the iameter of ajacent simplices are comparable. In applications, we consier families of triangulations, such as generate by successive uniform refinement or newest vertex bisection, whose shape constants are uniformly boune. We can boun some important quantities in terms of C mesh an the geometric ambient. here exists a constant C N > 0, epening only on C mesh an the ambient imension n, such that (4.3) : ( ) C N. his bouns the numbers of neighbors of any simplex. Furthermore, there exists a constant ɛ h > 0, epening only on C mesh an Ω, such that (4.4) : B ɛh h ( ) Ω ( ). In the sequel, we use affine transformations to a reference simplex. Let n = convex{0, e 1,..., e n } R n be the n-imensional reference simplex. For each n-simplex n of the triangulation, we fix an affine transformation ϕ (x) = M x + b where b R n an M R n n are such that ϕ ( n ) =. Each matrix M is invertible, an (4.5) M c M h, M 1 C Mh 1 for constants c M, C M > 0 that epen only on C mesh an n. 5. Elements of Geometric Measure heory his section gives an outline of relevant ieas from geometric measure theory, for which we use Whitney s monograph [27] as our main reference. Our motivation for stuying geometric measure theory lies in proving heorem 7.9 later in this article. he key observation is that finite element ifferential forms are flat ifferential forms, an that the egrees of freeom are flat chains (see Lemma 5.2). his allows us to estimate Lipschitz eformations of egrees of freeom (Lemma 5.4), which is of critical importance in the construction of the smoothe projection.

14 14 MARIN WERNER LICH We begin with basic notions of chains an cochains in geometric measure theory, which can be foun in Sections 1-3 of Chapter V in [27]. hroughout this section, we fix for each simplex S R n an orientation. We may ientify each positively oriente simplex S with the inicator function χ S : R n R. Let k Z. o each finite formal sum i a is i of (oriente) k-simplices S i with real coefficients a i we may associate the function i a iχ Si. We call two such finite formal sums i a is i an j b j j equivalent, written i a is i j b j j, if the associate functions i a iχ Si an j b jχ j agree almost everywhere with respect to the k-imensional Hausorff measure. he space C pol k (Rn ) of polyheral k-chains in R n is the vector space of finite formal sums of positively oriente k-simplices with the equivalence relation factore out. If S C pol k (Rn ), then we write S i a is i if the latter formal sum represents S. We may ientify a polyheral k-chain S i a is i in R n with the function χ S = i a iχ Si whenever convenient. he bounary S of a positively oriente k-simplex S R n is efine as (5.1) S = F (S) k 1 F, where each F (S) k 1 carries the orientation inuce by S. We efine a linear operator on the finite formal sums of positively oriente k-simplices by linear extension: i a is i = i a i S i. Furthermore, it is apparent that this operation preserves the equivalence relation. he bounary operator (5.1) gives rise to a linear mapping : C pol k (Rn ) C pol k 1 (Rn ) that satisfies = 0. he mass S k of a polyheral k-chain S in R n is efine as the L 1 norm of the associate function χ S with respect to the k-imensional Hausorff measure. 1 Hence, if S i a is i with the simplices S i being essentially isjoint with respect to the k-imensional Hausorff measure, then S k = i a i vol k (S i ). It is easy to see that k is a norm on the polyheral chains, calle mass norm. We write Ck mass (R n ) for the Banach space that results from taking the completion of the polyheral chains with respect to the mass norm. he flat norm S k, of a polyheral k-chain S C pol k (Rn ) is efine as ) (5.2) S k, := inf ( S Q k + Q k+1. Q C pol k+1 (Rn ) As the name alreay suggest, one can show that k, is a norm on the polyheral chains. he Banach space Ck (Rn ) is efine as the completion of C pol k (Rn ) with respect to the flat norm. It is apparent from the efinition that S k, S k, S C pol k (Rn ). In particular, Ck mass (R n ) is ensely embee in Ck (Rn ). he bounary operator is boune with respect to the flat norm. o see this, let S C pol k (Rn ), let ɛ > 0, an let Q C pol k+1 (Rn ) such that S Q k + Q k+1 1 We assume the convention that the k-imensional Hausorff volume of a k-simplex S equals its k-imensional volume vol k (S).

15 SMOOHED PROJECIONS WEAKLY LIPSCHIZ 15 S k, + ɛ. We then observe that S k 1, S (S Q) k 1 + S Q k = S Q k S k, + ɛ. aking ɛ to zero gives S k 1, S k,. Using the ensity of C pol k (Rn ) in C k (Rn ), we eventually verify that α k 1, α k,, α C k(r n ). We remark that the bounary operator is generally not boune with respect to the mass norm. his can be seen by shrinking a single simplex: the surface measure scales ifferently than the volume. Remark 5.1. he space Ck mass (R n ) is a close subspace of the Banach space of functions over R n integrable with respect to the k-imensional Hausorff measure. he members of C pol k (Rn ) play a similar role as the simple functions in the theory of the Lebesgue measure. Every polyheral k-chain can be represente as the finite linear combination of k-simplices that are essentially isjoint with respect to the k-imensional Hausorff measure. he Banach space Ck (Rn ) can be motivate by the following example: for r (0, 1), consier the two opposing longer sies of the rectangle [0, r] [0, 1]. he mass norm of these two eges is 2 regarless of r, but their flat norm equals r, corresponing to (areal) mass of the original rectangle. In this sense, the flat norm takes into account the istance between simplices. he chains in the space Ck mass (R n ) are the most important ones in this chapter. We iscuss the space Ck (Rn ) to utilize some technical tools in geometric measure theory that are state for flat chains in the literature. he Banach space C k (Rn ) of flat chains has a ual space, which is calle the Banach space of flat cochains. he space of flat cochains can be represente by a class of ifferential forms: to every cochain we associate a ifferential form such that evaluating the cochain on a simplex is equal to integrating the associate ifferential form over that simplex. his is another instance of a recurrent iea throughout ifferential geometry. Specifically, the space of flat cochains can be represente by the space of flat ifferential forms. Flat forms were stuie in Whitney s monograph [27], there mainly as representations of flat cochains, an in functional analysis (see [15]). For the following facts, we refer to Section 2 of [15] an Chapters IX an X of Whitney s monograph [27]. Flat ifferential forms have well-efine traces on simplices. More precisely, for each m-simplex S R n there exists a boune linear mapping tr S : W, Λ k (R n ) W, Λ k (S), which extens the trace of smooth forms. In particular, for ω W, Λ k (R n ) the trace tr S ω epens only on the values of ω near S. We write (5.3) ω := tr S ω S for the integral of ω W, Λ k (R n ) over a k-simplex S. By linearity, (5.3) gives a bilinear pairing between C pol k (Rn ) an W, Λ k (R n ). Via the ensity of the polyheral k-chains in Ck mass (R n ) this furthermore inuces a bilinear pairing between S

16 16 MARIN WERNER LICH (R n ) an W, Λ k (R n ). We have (5.4) ω S k ω W, Λ k (R ), n C mass k S S C mass k (R n ), ω W, Λ k (R n ). his pairing furthermore extens to flat chains. We have (5.5) ω α k, ω W, Λ k (R ), α Ck(R n ), ω W, Λ k (R n ). n α he exterior erivative between spaces of flat forms is ual to the bounary operator between spaces of flat chains (see Paragraph 12 of Chapter IX of [27]), an consequently we have (5.6) ω = ω, α Ck(R n ), ω W, Λ k (R n ), α as a generalize Stokes theorem. α Many results in geometric measure theory are invariant uner Lipschitz mappings. We recall some basic facts about pushforwars of chains an pullbacks of ifferential forms along Lipschitz mappings. Here we refer to Paragraph 7 in Chapter X of Whitney s monograph [27]. Let ϕ : R m R n be a Lipschitz mapping. hen there exists a mapping (5.7) ϕ : C k(r m ) C k(r n ), calle the pushforwar along ϕ, which commutes with the bounary operator, (5.8) an which satisfies the norm estimates (5.9) (5.10) ϕ α = ϕ α, α C k(r m ), ϕ α k, max { Lip(ϕ, R m ) k, Lip(ϕ, R m ) k+1} α k,, α C k(r m ), ϕ S k Lip(ϕ, R m ) k S k, S C mass k (R m ). he pushforwar of chains is ual to the pullback of ifferential forms. We recall that the latter is a mapping (5.11) ϕ : W, Λ k (R n ) W, Λ k (R m ) which commutes with the exterior erivative, (5.12) an satisfies the norm estimate (5.13) ϕ ω = ϕ ω, ω W, Λ k (R n ), ϕ ω L Λ k (R m ) Lip(ϕ, R n ) k ω L Λ k (R n ), ω W, Λ k (R n ). he pushforwar an the pullback are relate by the ientity (5.14) ω = ϕ ω, ω W, Λ k (R n ), α Ck(R m ). ϕ α α Lastly, if ϕ : R m R n an ψ : R l R m are Lipschitz mappings, then ϕψ : R l R n is a Lipschitz mapping, an we have (ϕψ) = ϕ ψ an (ϕψ) = ψ ϕ over the corresponing spaces of chains an ifferential forms, respectively. Having outline basic concepts of geometric measure theory, we provie a new result which makes these notions interesting for the theory of finite element methos: the egrees of freeom in finite element exterior calculus are flat chains.

17 SMOOHED PROJECIONS WEAKLY LIPSCHIZ 17 Lemma 5.2. Let F R n be a close oriente m-simplex an η C Λ m k (F ). hen there exists α(f, η) Ck (Rn ) such that for all ω W, Λ k (R n ) we have (5.15) tr F ω η = ω. F α(f,η) Moreover, α(f, η) Ck mass (R n ) an α(f, η) Ck 1 mass(rn ). Proof. We first assume that im F = n an F is positively oriente. We use heorem 15A of [27, Chapter IX] to euce the existence of α(f, η) Ck (Rn ) such that tr F ω η = ω, ω W, Λ k (R n ), F α(f,η) an such that α(f, η) k = η L1 Λ m k (F ). In particular, α(f, η) Ck mass (R n ). Now assume that im F = m n. here exist a positively oriente simplex F 0 R m an an isometric inclusion ϕ : R m R n which maps F 0 onto F. Recall that the pullback of a flat form along a Lipschitz mapping is well-efine. For ω W, Λ k (R n ) we have tr F ω η = tr F ω η = ϕ tr F ω ϕ η F ϕ F 0 F 0 = ϕ tr F ω = ω. α(f 0,ϕ η) ϕ α(f 0,ϕ η) hus we may choose α(f, η) = ϕ α(f 0, ϕ η) Ck mass (R n ). It remains to show that k 1 α(f, η) Ck 1 mass(rn ). For ω W, Λ k 1 (R n ), we erive ω = ω = tr F ω η α(f,η) α(f,η) = ( 1) k = ( 1) k F F tr F ω η + α(f,η) ω + f (F ) m 1 f (F ) m 1 f α(f,tr F,f η) tr f ω tr F,f η Here, tr F,f η enotes the trace of η C (F ) onto a subsimplex f (F ) m 1, an each such f is assume to carry the orientation inuce by F. Moreover, we have use the generalize Stokes theorem (5.6). We conclue that the action of the flat chain α(f, η) on W, Λ k 1 (R n ) can be represente as the finite sum of integrals against smooth ifferential forms over simplices. Hence α Ck 1 mass(rn ) by the previous observations. he proof is complete. Remark 5.3. In the next section we review how the egrees of freeom in finite element exterior calculus can be escribe in terms of integrals over simplices weighte against polynomial ifferential forms. Hence Lemma 5.2 can be applie to ientify the egrees of freeom with flat chains. We finish this excursion into geometric measure theory with an estimate on the eformation of flat chains by Lipschitz mappings. his result is applie later in this article an provies the rationale for consiering geometric measure theory. ω.

18 18 MARIN WERNER LICH Lemma 5.4. Let F R n be an m-simplex an η C Λ m k (F ). Let α(f, η) C k (Rn ) be the associate flat chain in the manner of Lemma 5.2. Let r > 0 be fixe an let ϕ : B 2r (F ) B 3r (F ) be a Lipschitz mapping that maps B r (F ) into B 2r (F ). Writing L := max{lip(ϕ, B 2r (F )), 1}, we then have (5.16) ϕ α α k, ϕ I L (B 2r(F ),R n ) ( L k α k + L k 1 α k 1 ). Proof. o prove this result, we gather several aitional notions of Whitney s monograph. For any open set U R n, a polyheral chain S i a is i C pol k (Rn ) is in U if all S i are containe in U, an S is of U if there exists an open set V R n compactly containe in U such that S is a chain in V (see [27]). he support of a flat chain α Ck (Rn ) is the set of all points x R n such that for all ɛ > 0 there exists ω C Λ k (R n ) with support in B ɛ (x) such that α S 0. It follows from Definition (1) in Section I.13 of [27, p.52] an the iscussion in Section V.10 of [27] up to heorem V.10A that our efinition of support agrees with the efinition of support in [27, Section VII.3] Having establishe these aitional notions, the claim follows by heorem 13A in Chapter X in [27] together with Equation VIII.1.(7) in [27, p.233]. 6. Finite Element Spaces, Degrees of Freeom, an Interpolation In this section we outline the iscretization theory of finite element exterior calculus. We summarize basic facts on the finite element spaces an their spaces of egrees of freeom. he most important construction is the canonical finite element interpolant I k. Moreover we consier several inverse inequalities. he reaer is assume to be familiar with the backgroun in [2] an [1, Section 3 5]. We outline this backgroun an aitionally apply geometric measure theory in the perspective of the preceing section. For the uration of this section, we fix a boune weakly Lipschitz omain Ω R n an a finite triangulation of Ω. he essential iea is to consier a ifferential complex of finite element spaces that mimics the e Rham complex on a iscrete level. he finite element spaces are finite-imensional spaces of piecewise polynomial ifferential forms. Let n be an n-simplex, an let r, k Z. We efine P r Λ k ( ) as the space of ifferential k-forms whose coefficients are polynomials over of egree at most r. We efine P r Λ k ( ) := P r 1 Λ k ( ) + X P r 1 Λ k+1 ( ), where X enotes contraction with the source vector fiel X(x) = x. One can show that P r Λ k ( ) an P r Λ k ( ) are invariant uner pullback by affine automorphisms of. Some basic properties of these spaces are P r Λ k ( ) P r+1 Λk ( ), P r Λ k ( ) P r Λ k ( ), P r Λ k ( ) P r 1 Λ k+1 ( ), P r Λ k ( ) = P r Λ k ( ), P r Λ 0 ( ) = P r Λ 0 ( ), P r Λ n ( ) = P r+1 Λn ( ). For any subsimplex F ( ) of we let tr,f : C Λ k ( ) C Λ k (F ) enote the trace mapping from onto F, an we set (6.1) P r Λ k (F ) := tr,f P r Λ k ( ), P r Λ k (F ) := tr,f P r Λ k ( ).

19 SMOOHED PROJECIONS WEAKLY LIPSCHIZ 19 Note that these two spaces o not epen on. We efine the finite element spaces P r Λ k ( ) := { ω W, Λ k (Ω) n : ω P r Λ k ( ) }, P r Λ k ( ) := { ω W, Λ k (Ω) n : ω P r Λ k ( ) }. hese are spaces of piecewise polynomial ifferential forms. Membership of piecewise polynomial ifferential forms in W, Λ k (Ω) enforces tangential continuity along simplex bounaries. In particular, if ω P r Λ k ( ) an, are neighboring simplices, then the restrictions of ω to an have the same trace on their common subsimplex. hus our efinition recovers precisely the finite element spaces of finite element exterior calculus [1]. From P r Λ k ( ) an Pr Λ k ( ) we can construct finite element e Rham complexes, but the combination of spaces is not arbitrary. We single out a class of ifferential complexes that has been iscusse by Arnol, Falk an Winther [1] an that we call FEEC-complexes in this article. A FEEC-complex is a ifferential complex (6.2) 0 PΛ 0 ( ) PΛ 1 ( ) such that for all k Z there exists r Z with (6.3) PΛ k ( ) { P r Λ k ( ), Pr Λ k ( ) } PΛ n ( ) 0 an that for all k Z we have PΛ k ( ) { P r Λ k ( ), Pr Λ k ( ) } (6.4) = PΛ k+1 ( ) { P r 1 Λ k+1 ( ), Pr Λ k+1 ( ) }. Next we introuce the egrees of freeom of finite element exterior calculus. hey are represente by taking the trace of a ifferential form onto a simplex of an then integrating against a smooth ifferential form. By virtue of Lemma 5.2, we introuce the egrees of freeom as chains of finite mass. Specifically, when F an m = im(f ), then we efine { } P r Ck F := S Ck(R n ) η S P r+k m Λm k (F ) : = η S, S F { } Pr Ck F := S Ck(R n ) η S P r+k m 1 Λ m k (F ) : = η S. We furthermore obtain by Lemma 5.2 that the egrees of freeom are flat chains of finite mass with bounaries of finite mass. One can show that we have irect sums P r C k ( ) := F P r C F k, P r C k ( ) := F S P r C F k. Moreover, these flat chains have bounaries of finite mass. One can show that P r C k ( ) P r+1 C k 1( ), P r C k ( ) P r+1 C k( ) P r+1 C k ( ). Remark 6.1. o see the first inclusion, let F an η P r Λ k (F ). hese efine an element S P r Ck F, an similar as in the proof of Lemma 5.2 the generalize Stokes theorem shows for all ω W, Λ k 1 (R n ) that tr F ω η = ( 1) k ω η + tr,f η tr f ω. F F f (F ) m 1 f F

20 20 MARIN WERNER LICH Here, each f (F ) m 1 carries the orientation inuce by F. he first inclusion has been use implicitly in the proof of Lemma 4.24 of [1]. he secon chain of inclusions follows immeiately from the efinition of P r C k ( ) an P r C k ( ). With respect to a given FEEC-complex (6.2), we then efine { Pr C PC k ( ) = k ( ) if PΛ k ( ) = P r Λ k ( ), (6.5) Pr C k ( ) if PΛ k ( ) = Pr Λ k ( ). for k Z. Note that PC k+1 ( ) PC k ( ) by construction. One can show that (6.6a) S PC k ( ) : S 0 = ω PΛ k ( ) : ω 0, S (6.6b) ω PΛ k ( ) : ω 0 = S PC k ( ) : ω 0. We conclue that PC k ( ), restricte to PΛ k ( ), spans the ual space of PΛ k ( ). Notably, the last implication can be strengthene to the following local result. When an ω PΛ k ( ), then (6.7) ω = 0 F ( ) : S PCk F : ω = 0. So the value of ω PΛ k ( ) is etermine uniquely by the values of the egrees of freeom associate with that simplex. We introuce the canonical finite element interpolant. his linear mapping is well-efine an boune both over CΛ k (Ω) an W, Λ k (Ω). We efine (6.8) by requiring that (6.9) ω = S S I k : CΛ k (Ω) + W, Λ k (Ω) PΛ k ( ) I k ω, S PC k ( ), ω CΛ k (Ω) + W, Λ k (Ω). he finite element interpolant commutes with the exterior erivative, which follows easily from (6.9) an (5.6). We have (6.10) I k+1 ω = ω = ω = I k ω = I k ω S for all ω W, Λ k (Ω) an S PC k+1 ( ). In particular, (6.11) S S W, Λ k (Ω) W, Λ k+1 (Ω) I k I k+1 PΛ k ( ) S PΛ k+1 ( ) is a commuting iagram. Furthermore I k is iempotent, which means (6.12) as follows irectly from (6.7). I k ω = ω, ω PΛ k ( ), In the remainer of this section, we introuce a number of inverse inequalities. hese rely on the equivalence of norms over finite-imensional vector spaces. We note that, by construction, the pullbacks ϕ ω lie in a common finiteimensional vector space as ω PΛ k ( ) an n vary. his is a fixe space S S S