# An exterior calculus framework for polytopal methods

Francesco<EMAIL_ADDRESS>IMAG, Univ. Montpellier,
CNRS, Montpellier, France  LAMPS, Université de Perpignan Via Domitia, France
Daniele A. Di<EMAIL_ADDRESS>IMAG, Univ.
Montpellier, CNRS, Montpellier, France  Jérôme
<EMAIL_ADDRESS>School of Mathematics, Monash University,
Australia  Kaibo<EMAIL_ADDRESS>Mathematical Institute,
University of Oxford, UK

###### Abstract

We develop in this work the first polytopal complexes of differential forms.
These complexes, inspired by the Discrete De Rham and the Virtual Element
approaches, are discrete versions of the de Rham complex of differential forms
built on meshes made of general polytopal elements. Both constructions benefit
from the high-level approach of polytopal methods, which leads, on certain
meshes, to leaner constructions than the finite element method. We establish
commutation properties between the interpolators and the discrete and
continuous exterior derivatives, prove key polynomial consistency results for
the complexes, and show that their cohomologies are isomorphic to the
cohomology of the continuous de Rham complex.
Key words. Discrete de Rham complex, Virtual Element Method, differential
forms, exterior calculus, polytopal methods
MSC2020. 65N30, 65N99, 14F40

##  1 Introduction

This work is a first step towards merging two extremely successful avenues of
research in numerical analysis: finite element differential forms and
arbitrary-order polytopal methods.

The well-posedness of important classes of partial differential equations
(PDEs), and the development of stable approximations thereof, hinges on the
properties of underlying Hilbert complexes [23]. The best-known example is
provided by the de Rham complex which, for an open connected polyhedral domain
$\Omega\subset\mathbb{R}^{3}$, reads

${\\{0\\}}$${H^{1}(\Omega)}$${\boldsymbol{H}(\operatorname{\bf
curl};\Omega)}$${\boldsymbol{H}(\operatorname{div};\Omega)}$${L^{2}(\Omega)}$${\\{0\\},}$$\scriptstyle{\operatorname{\bf
grad}}$$\scriptstyle{\operatorname{\bf
curl}}$$\scriptstyle{\operatorname{div}}$ (1.1)

where $H^{1}(\Omega)$ is the space of scalar-valued functions over $\Omega$
that are square-integrable along with their gradient, while
$\boldsymbol{H}(\operatorname{\bf curl};\Omega)$ and
$\boldsymbol{H}(\operatorname{div};\Omega)$ are the spaces of vector-valued
functions over $\Omega$ that are square-integrable along with their curl and
divergence, respectively. Using the framework of differential forms (see
Appendix A), the de Rham complex (1.1) can be generalised to a domain $\Omega$
of any dimension $n$ as:

${\\{0\\}}$${H\Lambda^{0}(\Omega)}$${{\cdots}}$${H\Lambda^{k}(\Omega)}$${{\cdots}}$${H\Lambda^{n}(\Omega)}$${\\{0\\}.}$$\scriptstyle{\mathrm{d}^{0}}$$\scriptstyle{\mathrm{d}^{k-1}}$$\scriptstyle{\mathrm{d}^{k}}$$\scriptstyle{\mathrm{d}^{n-1}}$
(1.2)

In what follows, we shall possibly omit the index $k$ from exterior
derivatives and spaces in (1.2) when no ambiguity can arise.

The de Rham complex enters the well-posedness analysis of PDEs through its
cohomology spaces
$\operatorname{Ker}\mathrm{d}^{k}/\operatorname{Im}\mathrm{d}^{k-1}$. A
classical result links these spaces to the topological features of the domain
and their dimensions to its Betti numbers. Preserving such homological
structures at the discrete level leads to _compatible_ methods and is key to
the design of stable numerical schemes.

The compatible finite element approximation of the vector-valued spaces
appearing in the de Rham complex (1.1) arose as a research subject in the late
70s [54, 53]. In the late 80s, links with Whitney forms were identified [17].
More recently, the development of Finite Element Exterior Calculus (FEEC) [4,
5, 2] has provided a unified perspective on the generation and analysis of
finite element approximations of the de Rham complex (1.2). Finite Element
Systems (FES) are a generalisation of FEEC covering spaces which are not
necessarily piecewise polynomial inside mesh elements (but can be, for
example, piecewise polynomial on subdivisions of these elements); see [29, 30,
31]. FEEC and FES led to the unification of several families of finite
elements and heavily hinge on the notion of subcomplex, which makes them
naturally geared towards conforming approximations.

While conforming methods are still widely used, their construction can only be
carried out on conforming meshes, typically made of elements with simple
shapes (e.g., tetrahedra or hexahedra); extensions to more general meshes,
such as the barycentric dual of a simplicial mesh, have been considered, e.g.,
in [26]. In recent years, significant efforts have been made to develop and
analyse numerical methods that support more general meshes including, e.g.,
general polytopal elements and non-matching interfaces; a representative but
by no means exhaustive list of contributions includes [21, 40, 46, 48, 6, 41,
18, 1, 7, 16, 22, 13, 42, 24, 39, 45, 36, 38, 33]. Polytopal technologies
typically introduce some degree of non-conformity, either because they are
formulated in a fully discrete setting (like Hybrid High-Order [42, 36] or
Discrete de Rham – DDR methods [38, 33]) or through the use of projections (as
in Virtual Element Methods – VEM [7]).

Despite their non-conformity, polytopal technologies can be used to develop
compatible frameworks. Polytopal discretisations of the de Rham complex (1.1)
have been proposed, e.g., in [10, 38, 33], and applied to a variety of models
, such as magnetostatics [8, 34], the Stokes equations [11], and the
Yang–Mills equations [47]; they have also inspired further developments, based
on the same principles, for other complexes of interest such as variants of
the de Rham complex with increased regularity [55, 32], elasticity complexes
[44, 19], and the Stokes complex [14, 49, 12]. Polytopal complexes have
additionally been used to construct methods that are robust with respect to
the variations of physical parameters, in particular for the Stokes problem
[11], for the Reissner–Mindlin equation [43], or the Brinkman model [33]. Many
of these models have also been tackled using finite element complexes and
related methods (see, e.g., [2, 31, 27, 3]). However, due to their higher-
level design, which does not require explicit expressions for the basis
functions, polytopal methods offer distinctive advantages over finite
elements. These include, in addition to the support of general meshes, the
possibility to reduce the dimension of discrete spaces, sometimes below their
finite element counterparts [35, Table 3], through systematic processes such
as enhancement or serendipity [9, 35]

The purpose of the present work is to take one step further and show how
exterior calculus can be used to generalise the construction and analysis of
polytopal complexes. More specifically, we present two discrete de Rham
complexes in arbitrary dimension and with arbitrary approximation degree that
generalise those introduced in [33] (DDR) and [8] (VEM). Three key features
set these constructions apart from Finite Element complexes:

  * •

No explicit spaces of globally conforming differential forms (i.e., subspaces
of $H\Lambda(\Omega)$) are sought. Instead, we work with _fully discrete
spaces_ made of vectors of polynomial components on the mesh cells (of various
dimensions). The meaning of these components is provided by the interpolators
on the fully discrete spaces.

  * •

Due to the absence of explicit underlying conforming spaces, the differential
operator of the complex cannot be the exterior derivative. Instead, a
_discrete exterior derivative_ is constructed combining the polynomial
components to mimic the Stokes formula.

  * •

_Discrete potentials_ are also designed, again mimicking the Stokes formula.
They are piecewise (discontinuous) polynomial forms on the mesh used, in
particular, to define an $L^{2}$-structure on the discrete spaces (an
essential tool to discretise PDEs written in weak form).

The choice of the polynomial components in the spaces and the design of
discrete exterior derivatives and potentials revolve around two key
properties: _polynomial consistency_ , which is related to the ability to
reproduce polynomial differential forms up to a selected polynomial degree,
and _compatibility_ , linked to the existence of an isomorphism between the
cohomology of the discrete and continuous de Rham complexes. While both the
DDR- and VEM-inspired constructions heavily rely on discrete versions of the
Stokes formula, they do so in a radically different spirit: in the DDR
construction, the choice of components in the discrete spaces is inspired by
the formula to reconstruct a discrete exterior derivative, which is then used
to construct discrete potentials. In the VEM construction, on the other hand,
the space components (and, in particular, those associated with differentials)
are chosen based on the formula used to define a discrete potential. While the
choice in the DDR construction leads to leaner spaces, the study of its
properties is more elaborated. Notice that, at this early stage, we haven’t
tried to identify the virtual (conforming) spaces that underlie the VEM-
inspired construction and we have made no effort whatsoever in trying to
reduce the dimension of the discrete spaces through serendipity.

The rest of this work is organised as follows. In Section 2 we establish the
setting. In Section 3 we present and analyse the discrete complex generalising
the DDR construction of [33]. Section 4 contains the definition and analysis
of the complex generalising the VEM construction of [8]. In Section 5, we
discuss in greater detail similarities and differences with respect to the
FEEC, FES and Distributional Differential Forms frameworks.

Differential forms of any degree in dimensions 2 and 3 have interpretations in
terms of vector fields. To make the exposition self-contained and improve the
legibility for the reader not used to differential forms, we recall some facts
on these so-called vector proxies in Appendix A, and we include throughout the
exposition a series of examples to illustrate the development in the
differential forms framework through vector calculus operators.

##  2 Setting

We present here the main notions used in the construction of the polytopal
complexes of differential forms. For the reader not used to the framework of
differential forms, we recall in Appendix A some basic concepts and
definitions.

###  2.1 Spaces of differential forms

Let $M$ denote an $n$-dimensional manifold. In what follows, $M$ will
typically be a cell of a polytopal mesh (see Section 2.5 below), and thus a
relatively open set in a subspace of $\mathbb{R}^{m}$ for some $m\geq n$. For
any natural number $\ell$ such that $0\leq\ell\leq n$, we will denote by
$\Lambda^{\ell}(M)$ the space of differential $\ell$-forms (often just called
$\ell$-forms) on $M$ without explicit regularity requirements. When relevant,
regularity is made explicit by prepending the appropriate space (e.g.,
$L^{2}\Lambda^{\ell}(M)$ stands for square-integrable $\ell$-forms).

###  2.2 Integration by parts

We recall the following integration by parts (Stokes) formula:

$\int_{M}\mathrm{d}\omega\wedge\mu=(-1)^{\ell+1}\int_{M}\omega\wedge\mathrm{d}\mu+\int_{\partial
M}\operatorname{tr}_{\partial M}\omega\wedge\operatorname{tr}_{\partial
M}\mu\qquad\forall(\omega,\mu)\in C^{1}\Lambda^{\ell}(\overline{M})\times
C^{1}\Lambda^{n-\ell-1}(\overline{M}),$ (2.1)

where, for any form degree $m$, $\operatorname{tr}_{\partial
M}:C^{0}\Lambda^{m}(\overline{M})\to C^{0}\Lambda^{m}(\partial M)$ is the
trace operator. Formula (2.1) will provide the starting point to define
discrete counterparts of the exterior derivative and of the corresponding
potentials on mesh cells. It will also drive the choice of the components in
the discrete spaces, geared at ensuring that the reconstructions preserve
certain polynomial differential forms.

###  2.3 Hodge star

Assume now that $M$ is an open set in a subspace of $\mathbb{R}^{m}$. We
denote by $\star:\Lambda^{\ell}(M)\to\Lambda^{n-\ell}(M)$ the Hodge star
operator, and we set

$\star^{-1}\coloneq(-1)^{\ell(n-\ell)}\star,$ (2.2)

a notation justified observing that, for any $\omega\in{\rm Alt}^{\ell}(V)$,
$\star^{-1}\star\omega=\omega$ (see (A.3) in the appendix).

###  2.4 $L^{2}$-orthogonal projectors

Integrating the inner product of ${\rm Alt}^{\ell}(V)$ over $M$ yields the
inner product of $L^{2}\Lambda^{\ell}(M)$. For any closed subspace
$\mathcal{X}$ of $L^{2}\Lambda^{\ell}(M)$, we therefore have an
$L^{2}$-orthogonal projector
$\pi_{\mathcal{X}}:L^{2}\Lambda^{\ell}(M)\to\mathcal{X}$ on $\mathcal{X}$,
defined by the following relation: For all $\omega\in L^{2}\Lambda^{\ell}(M)$,
$\pi_{\mathcal{X}}\omega\in\mathcal{X}$ satisfies

$\int_{M}\pi_{\mathcal{X}}\omega\wedge\star\mu=\int_{M}\omega\wedge{\star\mu}\qquad\forall\mu\in\mathcal{X}.$
(2.3)

To improve legibility, we will introduce in the following sections specific
notations to $\pi_{\mathcal{X}}$ for some polynomial subspaces $\mathcal{X}$
that are particularly relevant to our construction.

For future use, we note the following property.

######  Lemma 1 (Projectors on subspaces of differential forms).

Let $(k,d)$ be integers such that $k\leq d\leq n$,
$f\in\Delta_{d}(\mathcal{M}_{h})$ and $\mathcal{X}$ be a closed subspace of
$L^{2}\Lambda^{d-k}(f)$. Then, it holds: For all $\omega\in
L^{2}\Lambda^{k}(f)$ and all $\mu\in\mathcal{X}$,

$\int_{f}\star^{-1}\pi_{\mathcal{X}}(\star\omega)\wedge\mu=\int_{f}\mu\wedge\star\pi_{\mathcal{X}}(\star\omega)=\int_{f}\omega\wedge\mu.$
(2.4)

###### Proof.

The first relation in (2.4) follows from (A.4). To prove the second relation,
we write

$\displaystyle\int_{f}\mu\wedge\star\pi_{\mathcal{X}}(\star\omega)$
$\displaystyle=\int_{f}\cancel{\pi_{\mathcal{X}}}(\star\omega)\wedge\star\mu=\int_{f}\mu\wedge(\star\star\omega)=\int_{f}\omega\wedge\mu,$

where the first equality follows from (A.4) (with
$(\omega,\mu)\leftarrow(\pi_{\mathcal{X}}(\star\omega),\mu)$), the
cancellation of the projector is justified by its definition (2.3), the second
equality is obtained using (A.4) again, and the conclusion follows from (A.3)
and the anticommutativity (A.1) of $\wedge$. ∎

###  2.5 Polytopal mesh

From this point on, $\Omega$ will denote a polytopal domain of
$\mathbb{R}^{n}$. We let $\mathcal{M}_{h}$ denote a _polytopal mesh_ of
$\Omega$, i.e., a collection of disjoint polytopal sets (mesh entities) of
dimensions in $[0,n]$, relatively open in their spanned affine space, such
that the boundary of each $d$-cell (polytopal set of dimension $d$) is the
union of mesh entities of dimension $<d$, and such that any $d$-cell for $d<n$
is contained in the boundary of some $(d+1)$-cell. For any $d\in[0,n]$, the
set collecting all $d$-cells of $\mathcal{M}_{h}$ is denoted by
$\Delta_{d}(\mathcal{M}_{h})$. Notice that this notion of polytopal mesh
essentially coincides with that of CW-complex in algebraic topology. Thus,
when $\Omega$ is a domain in dimension $n=3$, $\mathcal{M}_{h}$ gathers the
vertices collected in the set
$\mathcal{V}_{h}\coloneq\Delta_{0}(\mathcal{M}_{h})$, the edges collected in
the set $\mathcal{E}_{h}\coloneq\Delta_{1}(\mathcal{M}_{h})$, the faces
collected in the set $\mathcal{F}_{h}\coloneq\Delta_{2}(\mathcal{M}_{h})$, and
the elements collected in the set
$\mathcal{T}_{h}\coloneq\Delta_{3}(\mathcal{M}_{h})$. For all
$f\in\mathcal{M}_{h}$, we select a point $\boldsymbol{x}_{f}\in f$ which, when
$\mathcal{M}_{h}$ belongs to a refined mesh sequence, is assumed at a distance
from the boundary of $f$ comparable to the meshsize.

If $f\in\Delta_{d}(\mathcal{M}_{h})$ and $d^{\prime}\leq d$ is an integer, we
denote by $\Delta_{d^{\prime}}(f)$ the set of subcells of $f$ of dimension
$d^{\prime}$. Hence, if $n=d=3$, so that $f=T\in\mathcal{T}_{h}$ is a
polyhedral element of the mesh, $f\in\Delta_{d^{\prime}}(T)$ is a vertex of
$T$ if $d^{\prime}=0$, an edge of $T$ if $d^{\prime}=1$, a polygonal face of
$T$ if $d^{\prime}=2$, or $T$ itself if $d^{\prime}=3$.

###  2.6 Local polynomial spaces of differential forms

Let $f\in\Delta_{d}(\mathcal{M}_{h})$, $0\leq d\leq n$. For any integer $r\geq
0$, we denote by $\mathcal{P}_{r}\Lambda^{\ell}(f)$ the space of polynomial
$\ell$-forms of total degree $\leq r$ on $f$. We also adopt the standard
convention $\mathcal{P}_{-1}\Lambda^{\ell}(f)\coloneq\\{0\\}$. We denote by
$\pi_{r,f}^{\ell}:L^{2}\Lambda^{\ell}(f)\to\mathcal{P}_{r}\Lambda^{\ell}(f)$
the $L^{2}$-orthogonal projector on $\mathcal{P}_{r}\Lambda^{\ell}(f)$,
defined by (2.3) with $\mathcal{X}=\mathcal{P}_{r}\Lambda^{\ell}(f)$.

The Koszul differential on $f$ (translated by $\boldsymbol{x}_{f}$) is denoted
by $\kappa$ so that, for all $\omega\in\Lambda^{\ell}(f)$,
$\kappa\omega\in\Lambda^{\ell-1}(f)$ satisfies
$(\kappa\omega)_{\boldsymbol{x}}(\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{\ell-1})=\omega_{\boldsymbol{x}}(\boldsymbol{x}-\boldsymbol{x}_{f},\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{\ell-1})$
for all vectors $\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{\ell-1}$ tangent to
$f$. For any $f\in\Delta_{d}(\mathcal{M}_{h})$, $1\leq d\leq n$, any integer
$\ell\in[0,d]$, and any polynomial degree $r\geq 0$, we define the Koszul
complement space as

$\mathcal{K}_{r}^{\ell}(f)\coloneq\kappa\mathcal{P}_{r-1}\Lambda^{\ell+1}(f).$
(2.5)

The indices $r$ and $\ell$ in this notation serve as a reminder that elements
in $\mathcal{K}_{r}^{\ell}(f)$ are polynomial $\ell$-forms of polynomial
degree $r$. Note also that, since $\mathcal{P}_{-1}\Lambda^{\ell}(f)=\\{0\\}$
and $\Lambda^{d+1}(f)=\\{0\\}$, we have

$\mathcal{K}_{0}^{\ell}(f)=\mathcal{K}_{r}^{d}(f)=\\{0\\}$ for all $\ell$ and
all $r$, respectively.  (2.6)

Moreover, since $\kappa\Lambda^{0}(f)=\\{0\\}$, we adopt the convention
$\mathcal{K}_{r}^{-1}(f)\coloneq\\{0\\}$ for all $r$. We denote by
$\pi_{r,f}^{\mathcal{K},\ell}$ the $L^{2}$-orthogonal projector
$L^{2}\Lambda^{\ell}(f)\to\mathcal{K}_{r}^{\ell}(f)$, defined by (2.3) with
$\mathcal{X}=\mathcal{K}_{r}^{\ell}(f)$.

For all integers $r\geq 0$ and $\ell\in[0,d]$, the following direct
decomposition holds (see [4, Eq. (3.11)] for $\ell\geq 1$, the case $\ell=0$
can be directly checked):

$\displaystyle\mathcal{P}_{r}\Lambda^{0}(f)$
$\displaystyle=\mathcal{P}_{0}\Lambda^{0}(f)\oplus\mathcal{K}_{r}^{0}(f),$
(2.7a) $\displaystyle\mathcal{P}_{r}\Lambda^{\ell}(f)$
$\displaystyle=\mathrm{d}\mathcal{P}_{r+1}\Lambda^{\ell-1}(f)\oplus\mathcal{K}_{r}^{\ell}(f)\quad\mbox{
if $\ell\geq 1$}.$ (2.7b)

Since $\mathrm{d}\circ\mathrm{d}=0$ and
$\mathrm{d}\mathcal{P}_{0}\Lambda^{0}(f)=\\{0\\}$, this shows that

$\mathrm{d}\mathcal{P}_{r}\Lambda^{\ell}(f)=\mathrm{d}\mathcal{K}_{r}^{\ell}(f).$
(2.8)

Applying this relation to $(r+1,\ell-1)$ instead of $(r,\ell)$ and recalling
that $\mathrm{d}$ is one-to-one on $\mathcal{K}_{r+1}^{\ell-1}(f)$ (see [4,
Theorem 3.2]), this shows that, for $\ell\geq 1$, the following mapping is an
isomorphism:

$\displaystyle\mathcal{K}_{r+1}^{\ell-1}(f)\times\mathcal{K}_{r}^{\ell}(f)\xrightarrow{\cong}{}$
$\displaystyle\mathcal{P}_{r}\Lambda^{\ell}(f),$ (2.9)
$\displaystyle(\mu,\nu)\mapsto{}$ $\displaystyle\mathrm{d}\mu+\nu.$

######  Example 2 (Interpretation in terms of vector proxies).

In the case $n=3$, thanks to the links between differential forms and vector
proxies (see Appendix A), we can associate to each space of polynomial
differential forms a space of (vector- or scalar-valued) polynomial fields.
Let us consider decomposition (2.7b). We denote by $f_{d}$ a $d$-cell of
$\mathcal{M}_{h}$, and we use a notation analogous to that of [33] for
polynomial spaces and vector calculus differential operators (with the
exception that polynomial degrees are in subscripts instead of superscripts).
Then, by definition (2.5) of the Koszul space, when
$f_{3}=T\in\mathcal{T}_{h}=\Delta_{3}(\mathcal{M}_{h})$ is a mesh element, we
have

$\displaystyle\mathrm{d}\mathcal{P}_{r+1}\Lambda^{0}(f_{3})\leftrightarrow\boldsymbol{\mathcal{G}}_{r}(T)\coloneq\operatorname{\bf
grad}\mathcal{P}_{r+1}(T),$
$\displaystyle\mathcal{K}_{r}^{1}(f_{3})\leftrightarrow\boldsymbol{\mathcal{G}}^{\rm
c}_{r}(T)\coloneq(\boldsymbol{x}-\boldsymbol{x}_{T})\times\boldsymbol{\mathcal{P}}_{r-1}(T),$
$\displaystyle\mathrm{d}\mathcal{P}_{r+1}\Lambda^{1}(f_{3})\leftrightarrow\boldsymbol{\mathcal{R}}_{r}(T)\coloneq\operatorname{\bf
curl}\boldsymbol{\mathcal{P}}_{r+1}(T),$
$\displaystyle\mathcal{K}_{r}^{2}(f_{3})\leftrightarrow\boldsymbol{\mathcal{R}}^{\rm
c}_{r}(T)\coloneq(\boldsymbol{x}-\boldsymbol{x}_{T})\mathcal{P}_{r-1}(T),$
$\displaystyle\mathrm{d}\mathcal{P}_{r+1}\Lambda^{2}(f_{3})\leftrightarrow\operatorname{div}\mathcal{P}_{r+1}(T)=\mathcal{P}_{r}(T),$
$\displaystyle\mathcal{K}_{r}^{3}(f_{3})=\\{0\\},$

where the first identity in the last line results from the surjectivity of the
divergence operator.

On the other hand, when
$f_{2}=F\in\mathcal{F}_{h}=\Delta_{2}(\mathcal{M}_{h})$ is a mesh face, we
obtain the following pair of possible correspondences:

$\displaystyle\mathrm{d}\mathcal{P}_{r+1}\Lambda^{0}(f_{2})\leftrightarrow\boldsymbol{\mathcal{G}}_{r}(F)\coloneq\operatorname{\bf
grad}_{F}\mathcal{P}_{r+1}(F),$
$\displaystyle\mathcal{K}_{r}^{1}(f_{2})\leftrightarrow\boldsymbol{\mathcal{G}}^{\rm
c}_{r}(F)\coloneq(\boldsymbol{x}-\boldsymbol{x}_{F})^{\perp}\mathcal{P}_{r-1}(F)$
(2.10)

or

$\displaystyle\mathrm{d}\mathcal{P}_{r+1}\Lambda^{0}(f_{2})\leftrightarrow\boldsymbol{\mathcal{R}}_{r}(F)\coloneq\operatorname{\bf
rot}_{F}\mathcal{P}_{r+1}(F),$
$\displaystyle\mathcal{K}_{r}^{1}(f_{2})\leftrightarrow\boldsymbol{\mathcal{R}}^{\rm
c}_{r}(F)\coloneq(\boldsymbol{x}-\boldsymbol{x}_{F})\mathcal{P}_{r+1}(F),$
(2.11)

where, for any $\boldsymbol{v}\in\mathbb{R}^{2}$,
$\boldsymbol{v}^{\perp}=\varrho_{-\pi/2}\boldsymbol{v}$ is the clockwise
rotation of $\boldsymbol{v}$ with respect to the orientation of $F$. The
existence of two possible correspondences between polynomial $1$-forms and
polynomial vector fields is to due to the fact that, when $d=2$, one can
identify a $1$-form either with a vector field $\boldsymbol{v}=(v_{1},v_{2})$
or with its rotation through a right angle (cf. [2, Chapter 6]); in
particular, we choose to identify it with the clockwise rotation
$\boldsymbol{v}^{\perp}=(v_{2},-v_{1})$ (see Appendix A for further details).
By (2.6), we have $\mathcal{K}_{r}^{2}(f_{2})=\\{0\\}$ and, according to
whether we consider the vector proxy leading to (2.10) or (2.11),

$\mathrm{d}\mathcal{P}_{r+1}\Lambda^{1}(f_{2})\leftrightarrow\operatorname{rot}_{F}\boldsymbol{\mathcal{P}}_{r+1}(F)=\mathcal{P}_{r}(F)\quad\text{or}\quad\mathrm{d}\mathcal{P}_{r+1}\Lambda^{1}(f_{2})\leftrightarrow\operatorname{div}_{F}\boldsymbol{\mathcal{P}}_{r+1}(F)=\mathcal{P}_{r}(F).$

Hence, since both $1$-forms and $2$-forms in $\mathbb{R}^{3}$ can be
identified with vector fields, and accounting for the two-fold identification
of $1$-forms in $\mathbb{R}^{2}$, the decomposition (2.7b) reads, in terms of
proxies,

$\boldsymbol{\mathcal{P}}_{r}(f_{d})=\boldsymbol{\mathcal{G}}_{r}(f_{d})\oplus\boldsymbol{\mathcal{G}}^{\rm
c}_{r}(f_{d})=\boldsymbol{\mathcal{R}}_{r}(f_{d})\oplus\boldsymbol{\mathcal{R}}^{\rm
c}_{r}(f_{d}),\quad d\in\\{2,3\\},$

i.e., the same expressions as [33, Eqs. (2.4) and (2.6)]. On the other hand,
concerning $0$-forms, the decomposition (2.7a) reads, in terms of proxies,

$\mathcal{P}_{r}(f_{d})=\mathcal{P}_{0}(f_{d})\oplus\mathcal{P}_{r}^{\flat}(f_{d}),\quad
d\in\\{0,\ldots,3\\},$

where we have introduced the notation
$\mathcal{P}_{r}^{\flat}(f)\coloneq(\boldsymbol{x}-\boldsymbol{x}_{f})\cdot\boldsymbol{\mathcal{P}}_{r-1}(f)$
for any $f\in\Delta_{d}(\mathcal{M}_{h})$.

###  2.7 Trimmed local polynomial spaces

We recall the following local trimmed polynomial spaces (see e.g. [4, Theorem
3.5]): For any $f\in~{}\Delta_{d}(\mathcal{M}_{h})$, $1\leq d\leq n$,

$\displaystyle\mathcal{P}_{r}^{-}\Lambda^{0}(f)$
$\displaystyle=\mathcal{P}_{r}\Lambda^{0}(f),$ (2.12a)
$\displaystyle\mathcal{P}_{r}^{-}\Lambda^{\ell}(f)$
$\displaystyle=\mathrm{d}\mathcal{P}_{r}\Lambda^{\ell-1}(f)\oplus\mathcal{K}_{r}^{\ell}(f)\qquad\text{for
$\ell\geq 1$}.$ (2.12b)

In (2.12b), comparing with the decompositions (2.7), we have decreased by one
the polynomial degree of the first space in the direct sum. Note that this
definition leads to the choice

$\mathcal{P}_{r}^{-}\Lambda^{0}(f)\coloneq\mathcal{P}_{r}\Lambda^{0}(f)\cong\mathbb{R}\qquad\forall
f\in\Delta_{0}(\mathcal{M}_{h}).$ (2.13)

The $L^{2}$-orthogonal projector
$L^{2}\Lambda^{\ell}(f)\to\mathcal{P}_{r}^{-}\Lambda^{\ell}(f)$ is denoted by
$\pi_{r,f}^{-,\ell}$, and is defined by (2.3) with
$\mathcal{X}=\mathcal{P}_{r}^{-}\Lambda^{\ell}(f)$.

Let us note a few properties of trimmed polynomial spaces. For $r=0$, only the
space (2.12a) is non-trivial, that is,
$\mathcal{P}_{0}^{-}\Lambda^{\ell}(f)=\\{0\\}$ if $\ell\in[1,d]$. Applying, if
$r\geq 1$ and $\ell\geq 1$, (2.7b) with $r-1$ instead of $r$ and noting that
$\mathcal{K}_{r-1}^{\ell}(f)\subset\mathcal{K}_{r}^{\ell}(f)$, we obtain the
equality

$\mathcal{P}_{r}^{-}\Lambda^{\ell}(f)=\mathcal{P}_{r-1}\Lambda^{\ell}(f)+\mathcal{K}_{r}^{\ell}(f).$
(2.14)

This equality, which obviously also holds for $\ell=0$ (see (2.7a)), shows
that trimmed polynomial spaces sit between full polynomial spaces:

$\mathcal{P}_{r-1}\Lambda^{\ell}(f)\subset\mathcal{P}_{r}^{-}\Lambda^{\ell}(f)\subset\mathcal{P}_{r}\Lambda^{\ell}(f).$

Recalling that $\mathcal{K}_{r}^{d}(f)=\\{0\\}$ and that
$\mathrm{d}\mathcal{P}_{r}\Lambda^{d-1}(f)=\mathcal{P}_{r-1}\Lambda^{d}(f)$
(by exactness of the tail of the polynomial de Rham sequence [2, Corollary
7.3]), it holds

$\mathcal{P}_{r}^{-}\Lambda^{d}(f)=\mathcal{P}_{r-1}\Lambda^{d}(f).$ (2.15)

Applying (2.8) with $\ell-1$ instead of $\ell$, we moreover have

$\mathcal{P}_{r}^{-}\Lambda^{\ell}(f)=\mathrm{d}\mathcal{K}_{r}^{\ell-1}(f)+\mathcal{K}_{r}^{\ell}(f)\qquad\text{for
$\ell\geq 1$}.$ (2.16)

Since $\mathrm{d}$ is one-to-one on $\mathcal{K}_{r}^{\ell-1}(f)$, this gives
the following isomorphism, whenever $\ell\geq 1$:

$\displaystyle\mathcal{K}_{r}^{\ell-1}(f)\times\mathcal{K}_{r}^{\ell}(f)\xrightarrow{\cong}{}$
$\displaystyle\mathcal{P}_{r}^{-}\Lambda^{\ell}(f),$ (2.17)
$\displaystyle(\mu,\nu)\mapsto{}$ $\displaystyle\mathrm{d}\mu+\nu.$

######  Example 3 (Interpretation of (2.12b) in terms of vector proxies).

Let $n=3$. For $d\in\\{2,3\\}$, denoting again by $f_{d}$ a $d$-cell, we
define the (local) Nédélec and Raviart–Thomas spaces

$\boldsymbol{\mathcal{N}}_{r}(f_{d})\coloneq\boldsymbol{\mathcal{G}}_{r-1}(f_{d})+\boldsymbol{\mathcal{G}}^{\rm
c}_{r}(f_{d}),\qquad\boldsymbol{\mathcal{RT}}_{r}(f_{d})\coloneq\boldsymbol{\mathcal{R}}_{r-1}(f_{d})+\boldsymbol{\mathcal{R}}^{\rm
c}_{r}(f_{d}).$

Notice that, when $d=3$, the Nédélec and Raviart–Thomas spaces can be obtained
as polynomial spaces of vector proxies of (2.12b) for $\ell=1$ and $\ell=2$,
respectively. On the other hand, when considering $d=2$, both spaces can be
obtained by taking the same value $\ell=1$ in (2.12b). Again, this is linked
to the two-fold interpretation of $1$-forms in terms of vector proxies in
$\mathbb{R}^{2}$, as already mentioned in Example 2, and corresponds to the
well-known fact that two-dimensional Nédélec elements coincide with two-
dimensional Raviart–Thomas elements rotated by a right angle.

The following result generalises [33, Proposition 8].

######  Lemma 4 (Traces of trimmed polynomial spaces).

The trace preserves trimmed spaces: For all integers $d\in[0,n]$,
$d^{\prime}\in[0,d]$ and $\ell\in[0,d^{\prime}]$, all
$f\in\Delta_{d}(\mathcal{M}_{h})$ and all
$f^{\prime}\in\Delta_{d^{\prime}}(f)$, we have

$\operatorname{tr}_{f^{\prime}}\mathcal{P}_{r}^{-}\Lambda^{\ell}(f)\subset\mathcal{P}_{r}^{-}\Lambda^{\ell}(f^{\prime}).$

###### Proof.

We first notice that the case $\ell=0$ is obvious since, in this case, trimmed
spaces are full polynomial spaces (see (2.12a)), and the trace preserves full
polynomial spaces. We therefore assume in the rest of the proof that $\ell\geq
1$. As the Koszul operators on differential forms on $f$ and $f^{\prime}$ are
not the same (due to the translation by $\boldsymbol{x}_{f}$ and
$\boldsymbol{x}_{f^{\prime}}$, respectively), we temporarily denote them in
this proof by $\kappa_{f}$ and $\kappa_{f^{\prime}}$.

The trace is a pullback, so it commutes with $\mathrm{d}$, and we thus have

$\operatorname{tr}_{f^{\prime}}(\mathrm{d}\mathcal{P}_{r}\Lambda^{\ell-1}(f))=\mathrm{d}(\operatorname{tr}_{f^{\prime}}\mathcal{P}_{r}\Lambda^{\ell-1}(f))\subset\mathrm{d}\mathcal{P}_{r}\Lambda^{\ell-1}(f^{\prime}),$

where the inclusion holds since the trace preserves full polynomial spaces.
Given the definition (2.12b) of the trimmed spaces, the lemma follows if we
show that

$\operatorname{tr}_{f^{\prime}}\mathcal{K}_{r}^{\ell}(f)\subset\mathcal{P}_{r}^{-}\Lambda^{\ell}(f^{\prime})=\mathcal{P}_{r-1}\Lambda^{\ell}(f^{\prime})+\mathcal{K}_{r}^{\ell}(f^{\prime})$
(2.18)

(where the equality follows from (2.14) applied to $f^{\prime}$ instead of
$f$). Let $\omega\in\mathcal{P}_{r-1}\Lambda^{\ell+1}(f)$. The definitions of
$\operatorname{tr}_{f^{\prime}}$ and $\kappa_{f}$ give, for any
$\boldsymbol{x}\in f^{\prime}$ and
$\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{\ell}$ tangent to $f^{\prime}$,

$\displaystyle\operatorname{tr}_{f^{\prime}}(\kappa_{f}\omega)_{\boldsymbol{x}}(\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{\ell})={}$
$\displaystyle\omega_{\boldsymbol{x}}(\boldsymbol{x}-\boldsymbol{x}_{f},\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{\ell})$
$\displaystyle={}$
$\displaystyle\omega_{\boldsymbol{x}}(\boldsymbol{x}_{f^{\prime}}-\boldsymbol{x}_{f},\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{\ell})+\omega_{\boldsymbol{x}}(\boldsymbol{x}-\boldsymbol{x}_{f^{\prime}},\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{\ell})$
$\displaystyle={}$
$\displaystyle\alpha_{\boldsymbol{x}}(\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{\ell})+(\kappa_{f^{\prime}}\operatorname{tr}_{f^{\prime}}\omega)_{\boldsymbol{x}}(\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{\ell}),$

where we have used the linearity of $\omega_{\boldsymbol{x}}$ with respect to
its first argument to obtain the second equality, and introduced the
differential form
$\alpha\coloneq\omega(\boldsymbol{x}_{f^{\prime}}-\boldsymbol{x}_{f},\cdot)$
in the third equality. Hence,
$\operatorname{tr}_{f^{\prime}}(\kappa_{f}\omega)=\alpha+\kappa_{f^{\prime}}\operatorname{tr}_{f^{\prime}}\omega$,
which proves (2.18) since
$\alpha\in\mathcal{P}_{r-1}\Lambda^{\ell}(f^{\prime})$ (as
$\boldsymbol{x}_{f^{\prime}}-\boldsymbol{x}_{f}$ is constant) and
$\operatorname{tr}_{f^{\prime}}\omega\in\mathcal{P}_{r-1}\Lambda^{\ell+1}(f^{\prime})$.
∎

##  3 Discrete de Rham complex

We define in this section a discrete counterpart of the de Rham complex of
differential forms (1.2) in the spirit of [38, 33]. Let, from this point on,
an integer $r\geq 0$ be fixed corresponding to the polynomial degree of the
discrete sequence. The general idea is, for each form degree $k\in[0,n]$, to
select the polynomial components of the discrete spaces in order to
reconstruct, on each $d$-cell $f$ and iteratively on the dimension $d$:

  * •

A _discrete exterior derivative_ in the full polynomial space
$\mathcal{P}_{r}\Lambda^{k+1}(f)$ that can reproduce exactly the exterior
derivative of differential forms in $\mathcal{P}_{r+1}^{-}\Lambda^{k}(f)$;

  * •

Based on this discrete exterior derivative and on traces on $(d-1)$-cells
(either directly available or reconstructed), a _discrete potential_ in
$\mathcal{P}_{r}\Lambda^{k}(f)$ that can reproduce exactly differential forms
belonging to this same space.

###  3.1 Definition

####  3.1.1 Discrete spaces

The discrete counterpart $\underline{X}^{k}_{r,h}$ of the space
$H\Lambda^{k}(\Omega)$, $0\leq k\leq n$, is defined as

$\underline{X}_{r,h}^{k}\coloneq\bigtimes_{d=k}^{n}\bigtimes_{f\in\Delta_{d}(\mathcal{M}_{h})}\mathcal{P}_{r}^{-}\Lambda^{d-k}(f).$
(3.1)

We define the restrictions of the global space (3.1) to a mesh entity or its
boundary as follows: For all integers $k$ and $d$ such that $0\leq k\leq d\leq
n$ and all $f\in\Delta_{d}(\mathcal{M}_{h})$,

$\underline{X}_{r,f}^{k}\coloneq\bigtimes_{d^{\prime}=k}^{d}\bigtimes_{f^{\prime}\in\Delta_{d^{\prime}}(f)}\mathcal{P}_{r}^{-}\Lambda^{d^{\prime}-k}(f^{\prime})$
and $\underline{X}_{r,\partial
f}^{k}\coloneq\bigtimes_{d^{\prime}=k}^{d-1}\bigtimes_{f^{\prime}\in\Delta_{d^{\prime}}(f)}\mathcal{P}_{r}^{-}\Lambda^{d^{\prime}-k}(f^{\prime})$
if $d\geq 1$.

We shall use the notation
$\underline{\omega}_{h}=(\omega_{f})_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k,n]}\in\underline{X}_{r,h}^{k}$
for a generic element of the global discrete space of $k$-forms and
$\underline{\omega}_{f}=(\omega_{f^{\prime}})_{f^{\prime}\in\Delta_{d^{\prime}}(f),\,d^{\prime}\in[k,d]}\in\underline{X}_{r,f}^{k}$
(resp., $\underline{\omega}_{\partial
f}=(\omega_{f^{\prime}})_{f^{\prime}\in\Delta_{d^{\prime}}(f),\,d^{\prime}\in[k,d-1]}\in\underline{X}_{r,\partial
f}^{k}$) for its restriction to $f$ (resp., $\partial f$), obtained collecting
the components on the mesh entities $f^{\prime}\in\Delta_{d^{\prime}}(f)$,
$d^{\prime}\in[k,d]$ (resp., $d^{\prime}\in[k,d-1]$). As a generic convention
in this article, underlined letters denote spaces or vectors made of
polynomial components on mesh entities. Table 1 gives an overview of the
polynomial unknowns in $\underline{X}_{r,f}^{k}$, along with their vector
proxies, in dimensions 0 to 3.

$k$ $d$ | $0$ | $1$ | $2$ | $3$
---|---|---|---|---
0 | $\mathbb{R}=\mathcal{P}_{r}\Lambda^{0}(f_{0})$ | $\mathcal{P}_{r-1}\Lambda^{1}(f_{1})$ | $\mathcal{P}_{r-1}\Lambda^{2}(f_{2})$ | $\mathcal{P}_{r-1}\Lambda^{3}(f_{3})$
1 |  | $\mathcal{P}_{r}\Lambda^{0}(f_{1})$ | $\mathcal{P}_{r}^{-}\Lambda^{1}(f_{2})$ | $\mathcal{P}_{r}^{-}\Lambda^{2}(f_{3})$
2 |  |  | $\mathcal{P}_{r}\Lambda^{0}(f_{2})$ | $\mathcal{P}_{r}^{-}\Lambda^{1}(f_{3})$
3 |  |  |  | $\mathcal{P}_{r}\Lambda^{0}(f_{3})$
$k$ $d$ | $0$ | $1$ | $2$ | $3$
0 | $\mathbb{R}=\mathcal{P}_{r}(f_{0})$ | $\mathcal{P}_{r-1}(f_{1})$ | $\mathcal{P}_{r-1}(f_{2})$ | $\mathcal{P}_{r-1}(f_{3})$
1 |  | $\mathcal{P}_{r}(f_{1})$ | $\boldsymbol{\mathcal{RT}}_{r}(f_{2})$ | $\boldsymbol{\mathcal{RT}}_{r}(f_{3})$
2 |  |  | $\mathcal{P}_{r}(f_{2})$ | $\boldsymbol{\mathcal{N}}_{r}(f_{3})$
3 |  |  |  | $\mathcal{P}_{r}(f_{3})$
Table 1: Polynomial components attached to each mesh entity $f_{d}$ of
dimension $d\in\\{0,\ldots,3\\}$ for the space $\underline{X}_{r,h}^{k}$ for
$k\in\\{0,\ldots,3\\}$ (top) and counterpart through vector proxies (bottom).

####  3.1.2 Interpolators and interpretation of the polynomial components

The precise meaning of the components in each DDR space is provided by the
corresponding interpolator. For $f\in\Delta_{d}(\mathcal{M}_{h})$ and $k\leq
d$, the interpolator
$\underline{I}_{r,f}^{k}:C^{0}\Lambda^{k}(\overline{f})\to\underline{X}_{r,f}^{k}$
is defined by: For all $\omega\in C^{0}\Lambda^{k}(\overline{f})$,

$\underline{I}_{r,f}^{k}\omega\coloneq(\pi_{r,f^{\prime}}^{-,d^{\prime}-k}(\star\operatorname{tr}_{f^{\prime}}\omega))_{f^{\prime}\in\Delta_{d^{\prime}}(f),\,d^{\prime}\in[k,d]}.$
(3.2)

In other words, a discrete $k$-form on the mesh is made of polynomial forms
attached to each mesh entity of dimension $d\geq k$; on each such entity, the
form is of degree $d-k$ as it corresponds to the Hodge star of an underlying
$k$-form. The Hodge star operator is used in the definition of the polynomial
components to ensure that the full space $\mathcal{P}_{r}\Lambda^{0}(f)$ (see
(2.12a)) is attached to the lowest-dimensional cells
$f\in\Delta_{k}(\mathcal{M}_{h})$.

####  3.1.3 Local discrete potentials and discrete exterior derivative

Let $0\leq k\leq n$ be a fixed integer. For all
$f\in\Delta_{d}(\mathcal{M}_{h})$ with $d\geq k$, we define the _discrete
potential_
$P_{r,f}^{k}:\underline{X}_{r,f}^{k}\to\mathcal{P}_{r}\Lambda^{k}(f)$ and, if
$d\geq k+1$, the _discrete exterior derivative_
$\mathrm{d}_{r,f}^{k}:\underline{X}^{k}_{r,f}\to\mathcal{P}_{r}\Lambda^{k+1}(f)$
recursively on the dimension $d$ as follows:

  * •

If $d=k$, then the discrete potential on $f$ is directly given by the
component of $\underline{\omega}_{f}$ on $f$:

$P_{r,f}^{k}\underline{\omega}_{f}\coloneq\star^{-1}\omega_{f}\in\mathcal{P}_{r}\Lambda^{d}(f).$
(3.3)

  * •

If $k+1\leq d\leq n$:

    1. 1.

First, the discrete exterior derivative is defined by: For all
$\underline{\omega}_{f}\in\underline{X}^{k}_{r,f}$,

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\wedge\mu=(-1)^{k+1}\int_{f}\star^{-1}\omega_{f}\wedge\mathrm{d}\mu+\int_{\partial
f}P_{r,\partial f}^{k}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial f}{\mu}\\\
\forall\mu\in\mathcal{P}_{r}\Lambda^{d-k-1}(f),$ (3.4)

where we have introduced the piecewise polynomial boundary potential
$P_{r,\partial f}^{k}:\underline{X}_{r,\partial f}^{k}\to\Lambda^{k}(\partial
f)$ such that $(P_{r,\partial f}^{k})_{|f^{\prime}}\coloneq
P_{r,f^{\prime}}^{k}$ for all $f^{\prime}\in\Delta_{d-1}(f)$
($P_{r,f^{\prime}}^{k}$ being the discrete potential on the $(d-1)$-cell
$f^{\prime}$ defined at the previous step).

    2. 2.

Then, the discrete potential on the $d$-cell $f$ is given by: For all
$\underline{\omega}_{f}\in\underline{X}^{k}_{r,f}$,

$(-1)^{k+1}\int_{f}P_{r,f}^{k}\underline{\omega}_{f}\wedge(\mathrm{d}\mu+\nu)\\\
=\int_{f}\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\wedge\mu-\int_{\partial
f}P_{r,\partial f}^{k}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial
f}{\mu}+(-1)^{k+1}\int_{f}\star^{-1}\omega_{f}\wedge\nu\\\
\forall(\mu,\nu)\in\mathcal{K}_{r+1}^{d-k-1}(f)\times\mathcal{K}_{r}^{d-k}(f).$
(3.5)

Some remarks are in order.

######  Remark 5 (Definitions (3.4) and (3.5)).

The fact that condition (3.4) defines
$\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}$ uniquely is an immediate
consequence of the Riesz representation theorem for
$\mathcal{P}_{r}\Lambda^{k+1}(f)$ equipped with the $L^{2}$-product
$(\rho,\beta)\ni L^{2}\Lambda^{k+1}(f)\times
L^{2}\Lambda^{k+1}(f)\mapsto\int_{f}\rho\wedge\star\beta\in\mathbb{R}$, after
observing that (3.4) can be equivalently reformulated as follows (notice the
change in the degree of the test differential form, with $\beta$ below
corresponding to $\star^{-1}\mu$ in (3.4)):

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\wedge\star\beta=(-1)^{k+1}\int_{f}\omega_{f}\wedge\star\mathrm{d}\star\beta+\int_{\partial
f}P_{r,\partial f}^{k}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial
f}\star\beta\qquad\forall\beta\in\mathcal{P}_{r}\Lambda^{k+1}(f),$

where we have additionally used (A.4) for the first term in the right-hand
side. Similar considerations apply to the definition (3.5) of $P_{r,f}^{k}$,
applying the isomorphism (2.9) with $\ell=d-k\geq 1$.

######  Remark 6 (Validity of (3.5)).

For $k+1\leq d\leq n$, equation (3.5) actually holds for all
$\mu\in\mathcal{P}_{r+1}^{-}\Lambda^{d-k-1}(f)$. To prove this assertion,
since (3.5) holds for $\mu\in\mathcal{K}_{r+1}^{d-k-1}(f)$, it suffices to
show that it also holds for $\nu=0$ and $\mu$ belonging to
$\mathcal{P}_{0}\Lambda^{0}(f)$ if $d=k+1$ (see (2.12a) and (2.7a)) or
$\mathrm{d}\mathcal{P}_{r+1}\Lambda^{d-k-2}(f)$ if $d\geq k+2$ (see (2.12b)).
In both cases, we have $\mathrm{d}\mu=0$, so that the left-hand side of (3.5)
vanishes; since $\mu\in\mathcal{P}_{r}\Lambda^{d-k-1}(f)$, the right-hand side
of (3.5) also vanishes due to the definition (3.4) of the discrete exterior
derivative, which concludes the argument.

######  Remark 7 (Potential for $k=0$).

In the case $k=0$, we can define an improved potential
$P_{r+1,f}^{0}:\underline{X}_{r,f}^{0}\to\mathcal{P}_{r+1}\Lambda^{0}(f)$ of
polynomial degree $r+1$ (instead of $r$) as follows: For all
$\underline{\omega}_{f}\in\underline{X}_{r,f}^{0}$,

  * •

If $d=0$, then
$P_{r+1,f}^{0}\underline{\omega}_{f}=\star^{-1}\omega_{f}\in\mathcal{P}_{r}\Lambda^{0}(f)\cong\mathbb{R}\cong\mathcal{P}_{r+1}\Lambda^{0}(f)$
(since $f$ has dimension $0$);

  * •

If $1\leq d\leq n$,

$-\int_{f}P_{r+1,f}^{0}\underline{\omega}_{f}\wedge\mathrm{d}\mu=\int_{f}\mathrm{d}_{r,f}^{0}\underline{\omega}_{f}\wedge\mu-\int_{\partial
f}P_{r+1,\partial f}^{0}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial
f}{\mu}\qquad\forall\mu\in\mathcal{K}_{r+2}^{d-1}(f).$ (3.6)

This definition is justified by the isomorphism (2.9) with $\ell=d$ and $r+1$
instead of $r$ (recalling that $\mathcal{K}_{r+1}^{d}(f)=\\{0\\}$), and it can
easily be checked, testing (3.5) and (3.6) with
$\mu\in\mathcal{K}_{r+1}^{d-1}(f)$, that
$\pi_{r,f}^{0}P_{r+1,f}^{0}\underline{\omega}_{f}=P_{r,f}^{0}\underline{\omega}_{f}$.
We will moreover see in Remark 18 that $P_{r+1,f}^{0}$ enjoys optimal
consistency properties.

######  Remark 8 (Space of DDR potentials).

The space of DDR reconstructed potentials, that is,

$\\{(P_{r,f}^{k}\underline{\omega}_{h})_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k,n]}\,:\,\underline{\omega}_{h}\in\underline{X}_{r,h}^{k}\\}$

cannot be considered as a space of differential forms with global regularity,
as the reconstructed polynomials do not have any compatibility condition of
the traces; they are inherently piecewise discontinuous polynomials.

######  Example 9 (Interpretation in terms of vector proxies).

We start by considering $k=0$. In this case, formula (3.3) means that
(constant) real values are attached to the vertices
$f_{0}=V\in\mathcal{V}_{h}=\Delta_{0}(\mathcal{M}_{h})$ of the mesh, so that
an iterative procedure can be initialised to reconstruct discrete gradients
and related traces/potentials over higher-dimensional cells. Indeed, formula
(3.4) reconstructs a (scalar) gradient over edges
$f_{1}=E\in\mathcal{E}_{h}=\Delta_{1}(\mathcal{M}_{h})$ (i.e., the derivative
along the direction given by the orientation of $E$) based on the values at
the vertices and the value on the edge itself. This edge gradient, in turn,
enters (3.5) to define a scalar edge trace over $E$. When $d$ takes the values
$2$ and $3$, the successive application of formulas (3.4)–(3.5) defines,
respectively, the pairs (face gradient, scalar face trace) on mesh faces
$f_{2}=F\in\mathcal{F}_{h}=\Delta_{2}(\mathcal{M}_{h})$, and (element
gradient, scalar element potential) on mesh elements
$f_{3}=T\in\mathcal{T}_{h}=\Delta_{3}(\mathcal{M}_{h})$.

Let us now turn to the case $k=1$, for which we provide more details. The
vector proxy for the space $\underline{X}_{r,h}^{1}$ is the space

$\underline{\boldsymbol{X}}_{\operatorname{\bf
curl},h}^{r}=\bigtimes_{E\in\mathcal{E}_{h}}\mathcal{P}_{r}(E)\times\bigtimes_{F\in\mathcal{F}_{h}}\boldsymbol{\mathcal{RT}}_{r}(F)\times\bigtimes_{T\in\mathcal{T}_{h}}\boldsymbol{\mathcal{RT}}_{r}(T)$

and, with standard DDR notation, we denote by
$\underline{\boldsymbol{X}}_{\operatorname{\bf curl},Y}^{r}$ its restriction
to a mesh element or face $Y\in\mathcal{T}_{h}\cup\mathcal{F}_{h}$. By (3.3)
with $d=k=1$, the reconstruction process is initialised by $1$-forms, whose
vector proxies are scalar-valued polynomials of degree $r$ over edges
$f_{1}=E\in\mathcal{E}_{h}$ that play the role of edge tangential traces.

Then, for each mesh face $f_{2}=F\in\mathcal{F}_{h}$, we sequentially
reconstruct a scalar face curl
$C_{F}^{r}:\underline{\boldsymbol{X}}_{\operatorname{\bf
curl},F}^{r}\to\mathcal{P}_{r}(F)$ by (3.4) with $d=k+1=2$ and a vector face
tangential trace $\boldsymbol{\gamma}_{{\rm
t},F}^{r}:\underline{\boldsymbol{X}}_{\operatorname{\bf
curl},F}^{r}\to\boldsymbol{\mathcal{P}}_{r}(F)$ by (3.5). $C_{F}^{r}$ is such
that, for all
$\underline{\boldsymbol{v}}_{F}=\big{(}(v_{E})_{E\in\mathcal{E}_{F}},\boldsymbol{v}_{F}\big{)}\in\underline{\boldsymbol{X}}_{\operatorname{\bf
curl},F}^{r}$,

$\int_{F}C_{F}^{r}\underline{\boldsymbol{v}}_{F}~{}q=\int_{F}\boldsymbol{v}_{F}\cdot\operatorname{\bf
rot}_{F}q+\sum_{E\in\mathcal{E}_{F}}\varepsilon_{FE}\int_{E}v_{E}~{}q\qquad\forall
q\in\mathcal{P}_{r}(F),$

where, for all $E\in\mathcal{E}_{F}$ (the set of edges of $F$),
$\varepsilon_{FE}\in\\{-1,+1\\}$ denotes the orientation of $E$ relative to
$F$, while $\boldsymbol{\gamma}_{{\rm t},F}^{r}$ satisfies, for all
$\underline{\boldsymbol{v}}_{F}\in\underline{\boldsymbol{X}}_{\operatorname{\bf
curl},F}^{r}$,

$\int_{F}\boldsymbol{\gamma}_{{\rm
t},F}^{r}\underline{\boldsymbol{v}}_{F}\cdot(\operatorname{\bf
rot}_{F}q+\boldsymbol{w})=\int_{F}C_{F}^{r}\underline{\boldsymbol{v}}_{F}~{}q-\sum_{E\in\mathcal{E}_{F}}\varepsilon_{FE}\int_{E}v_{E}~{}q+\int_{F}\boldsymbol{v}_{F}\cdot\boldsymbol{w},\\\
\forall(q,\boldsymbol{w})\in\mathcal{P}_{r+1}^{\flat}(F)\times\boldsymbol{\mathcal{R}}^{\rm
c}_{r}(F).$

The alternative interpretation of $1$-forms in dimension $d=2$ results in a
rotation of $\underline{\boldsymbol{X}}_{\operatorname{\bf curl},F}^{r}$ by a
right angle. Correspondingly, (3.4) yields a face divergence (see (A.9) and
(A.10) in Appendix A.2).

Next, for each mesh element $f_{3}=T\in\mathcal{T}_{h}$, (3.4) defines the
element curl
$\boldsymbol{C}_{T}^{r}:\underline{\boldsymbol{X}}_{\operatorname{\bf
curl},T}^{r}\to\boldsymbol{\mathcal{P}}_{r}(T)$ such that, for all
$\underline{\boldsymbol{v}}_{T}=\big{(}(v_{E})_{E\in\mathcal{E}_{T}},(\boldsymbol{v}_{F})_{F\in\mathcal{F}_{T}},\boldsymbol{v}_{T}\big{)}\in\underline{\boldsymbol{X}}_{\operatorname{\bf
curl},T}^{r}$,

$\int_{T}\boldsymbol{C}_{T}^{r}\underline{\boldsymbol{v}}_{T}\cdot\boldsymbol{w}=\int_{T}\boldsymbol{v}_{T}\cdot\operatorname{\bf
curl}\boldsymbol{w}+\sum_{F\in\mathcal{F}_{T}}\varepsilon_{TF}\int_{F}\boldsymbol{\gamma}_{{\rm
t},F}^{r}\underline{\boldsymbol{v}}_{F}\cdot(\boldsymbol{w}\times\boldsymbol{n}_{F})\qquad\forall\boldsymbol{w}\in\boldsymbol{\mathcal{P}}_{r}(T),$

where, for all $F\in\mathcal{F}_{T}$ (the set of faces of $T$),
$\varepsilon_{TF}\in\\{-1,+1\\}$ denotes the orientation of $F$ relative to
$T$, while (3.5) defines the vector potential
$\boldsymbol{P}_{\operatorname{\bf
curl},T}^{r}:\underline{\boldsymbol{X}}_{\operatorname{\bf
curl},T}^{r}\to\boldsymbol{\mathcal{P}}_{r}(T)$ such that, for all
$\underline{\boldsymbol{v}}_{T}\in\underline{\boldsymbol{X}}_{\operatorname{\bf
curl},T}^{r}$,

$\int_{T}\boldsymbol{P}_{\operatorname{\bf
curl},T}^{r}\underline{\boldsymbol{v}}_{T}\cdot(\operatorname{\bf
curl}\boldsymbol{w}+\boldsymbol{z})=\int_{T}\boldsymbol{C}_{T}^{r}\underline{\boldsymbol{v}}_{T}\cdot\boldsymbol{w}-\sum_{F\in\mathcal{F}_{T}}\varepsilon_{TF}\int_{F}\boldsymbol{\gamma}_{{\rm
t},F}^{r}\underline{\boldsymbol{v}}_{F}\cdot(\boldsymbol{w}\times\boldsymbol{n}_{F})+\int_{T}\boldsymbol{v}_{T}\cdot\boldsymbol{z}\\\
\forall(\boldsymbol{w},\boldsymbol{z})\in\boldsymbol{\mathcal{G}}^{\rm
c}_{r+1}(T)\times\boldsymbol{\mathcal{R}}^{\rm c}_{r}(T).$

When $k=2$, (3.4) reconstructs on mesh elements $f_{3}=T\in\mathcal{T}_{h}$ a
discrete divergence of order $r$ based on the polynomial scalar trace defined
by (3.3), which plays the role of a normal trace on the face
$f_{2}=F\in\mathcal{F}_{T}$. Then, (3.5) defines a vector potential of degree
$r$ over $T$.

Finally, in the case $k=3$, (3.3) simply yields a polynomial over mesh
elements $f_{3}=T\in\mathcal{T}_{h}$.

####  3.1.4 Global discrete exterior derivative and DDR complex

To arrange the spaces $\underline{X}_{r,h}^{k}$ into a sequence that mimics
the continuous de Rham complex, for any form degree $k$ such that $0\leq k\leq
n-1$, we introduce the _global discrete exterior derivative_
$\underline{\mathrm{d}}_{r,h}^{k}:\underline{X}^{k}_{r,h}\to\underline{X}_{r,h}^{k+1}$
defined as follows:

$\underline{\mathrm{d}}_{r,h}^{k}\underline{\omega}_{h}\coloneq\big{(}\pi_{r,f}^{-,d-k-1}(\star\mathrm{d}_{r,f}^{k}\underline{\omega}_{f})\big{)}_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k+1,n]}.$
(3.7)

In what follows, given a $d$-cell $f\in\Delta_{d}(\mathcal{M}_{h})$ with
$d\in[k+1,n]$, we denote by $\underline{\mathrm{d}}_{r,f}^{k}$ the _local
discrete exterior derivative_ collecting the components of
$\underline{\mathrm{d}}_{r,h}^{k}$ on $f$ and its boundary. The DDR sequence
reads

${\mathrm{DDR}(r)\coloneq\\{0\\}}$${\underline{X}_{r,h}^{0}}$${\underline{X}_{r,h}^{1}}$${\cdots}$${\underline{X}_{r,h}^{n-1}}$${\underline{X}_{r,h}^{n}}$${\\{0\\}.}$$\scriptstyle{\underline{\mathrm{d}}_{r,h}^{0}}$$\scriptstyle{\underline{\mathrm{d}}_{r,h}^{n-1}}$
(3.8)

The main results concerning this sequence are stated hereafter.

######  Theorem 10 (Cohomology of the Discrete de Rham complex).

The DDR sequence (3.8) is a complex and its cohomology is isomorphic to the
cohomology of the continuous de Rham complex (1.2).

###### Proof.

See Section 3.5. ∎

######  Theorem 11 (Polynomial consistency of the discrete potential and
exterior derivative).

For all integers $0\leq k\leq d\leq n$ and all
$f\in\Delta_{d}(\mathcal{M}_{h})$, it holds

$P_{r,f}^{k}\underline{I}_{r,f}^{k}\omega=\omega\qquad\forall\omega\in\mathcal{P}_{r}\Lambda^{k}(f),$
(3.9)

and, if $d\geq k+1$,

$\mathrm{d}_{r,f}^{k}\underline{I}_{r,f}^{k}\omega=\mathrm{d}\omega\qquad\forall\omega\in\mathcal{P}_{r+1}^{-}\Lambda^{k}(f).$
(3.10)

###### Proof.

See Section 3.4. ∎

######  Remark 12 (Consistency of traces).

The above theorem actually implies that, for any $d$-face
$f\in\Delta_{d}(\mathcal{M}_{h})$, $d\in[k,n]$, any
$\omega\in\mathcal{P}_{r}\Lambda^{k}(f)$, and any integer
$d^{\prime}\in[k,d]$,

$P_{r,f^{\prime}}^{k}\underline{I}_{r,f^{\prime}}^{k}\omega=\operatorname{tr}_{f^{\prime}}\omega\qquad\forall
f^{\prime}\in\Delta_{d^{\prime}}(f).$

This can be easily seen noticing that
$\underline{I}_{r,f^{\prime}}^{k}\omega=\underline{I}_{r,f^{\prime}}^{k}\operatorname{tr}_{f^{\prime}}\omega$
and
$\operatorname{tr}_{f^{\prime}}\omega\in\mathcal{P}_{r}\Lambda^{k}(f^{\prime})$,
and invoking (3.9) with
$(f,\omega)\leftarrow(f^{\prime},\operatorname{tr}_{f^{\prime}}\omega)$.

####  3.1.5 Discrete $L^{2}$-products

Using the potentials built in Section 3.1.3, we can define, for all
$k\in[0,n]$, an inner product
$(\cdot,\cdot)_{k,h}:\underline{X}_{r,h}^{k}\times\underline{X}_{r,h}^{k}\to\mathbb{R}$
that induces an $L^{2}$-structure on $\underline{X}_{r,h}^{k}$. Specifically,
we set: For all
$(\underline{\omega}_{h},\underline{\mu}_{h})\in\underline{X}_{r,h}^{k}\times\underline{X}_{r,h}^{k}$,

$\begin{gathered}(\underline{\omega}_{h},\underline{\mu}_{h})_{k,h}\coloneq\sum_{f\in\Delta_{n}(\mathcal{M}_{h})}(\underline{\omega}_{f},\underline{\mu}_{f})_{k,f}\\\
\text{ with
$(\underline{\omega}_{f},\underline{\mu}_{f})_{k,f}\coloneq\int_{f}P_{r,f}^{k}\underline{\omega}_{f}\wedge\star
P_{r,f}^{k}\underline{\mu}_{f}+s_{k,f}(\underline{\omega}_{f},\underline{\mu}_{f})$
for all $f\in\Delta_{n}(\mathcal{M}_{h})$, }\end{gathered}$ (3.11)

where
$s_{k,f}:\underline{X}_{r,f}^{k}\times\underline{X}_{r,f}^{k}\to\mathbb{R}$ is
the stabilisation bilinear form such that

$s_{k,f}(\underline{\omega}_{f},\underline{\mu}_{f})=\sum_{d^{\prime}=k}^{n-1}h_{f}^{n-d^{\prime}}\sum_{f^{\prime}\in\Delta_{d^{\prime}}(f)}\int_{f^{\prime}}(\operatorname{tr}_{f^{\prime}}P_{r,f}^{k}\underline{\omega}_{f}-P_{r,f^{\prime}}^{k}\underline{\omega}_{f^{\prime}})\wedge\star(\operatorname{tr}_{f^{\prime}}P_{r,f}^{k}\underline{\mu}_{f}-P_{r,f^{\prime}}^{k}\underline{\mu}_{f^{\prime}}),$

with $h_{f}$ denoting the diameter of $f$. The first term in the right-hand
side of $(\cdot,\cdot)_{k,f}$ is responsible for consistency, while the second
one ensures the positivity of this bilinear form. More specifically, by
Theorem 11 and Remark 12 it holds, for all $f\in\Delta_{n}(\mathcal{M}_{h})$,

$(\underline{I}_{r,f}^{k}\omega,\underline{\mu}_{f})_{k,f}=\int_{f}\omega\wedge\star
P_{r,f}^{k}\underline{\mu}_{f}\qquad\forall\omega\in\mathcal{P}_{r}\Lambda^{k}(f)\,,\;\forall\underline{\mu}_{f}\in\underline{X}_{r,f}^{k}.$

Additionally, by (3.20) below, the mapping
$\underline{X}_{r,f}^{k}\ni\underline{\omega}_{f}\mapsto\|\underline{\omega}_{f}\|_{k,f}\coloneq(\underline{\omega}_{f},\underline{\omega}_{f})_{k,f}^{\nicefrac{{1}}{{2}}}\in\mathbb{R}$
defines a norm on $\underline{X}_{r,f}^{k}$. Numerical schemes for linear PDEs
related to the de Rham complex are typically obtained replacing continuous
spaces and $L^{2}$-products with their discrete counterparts, according to the
principles illustrated, e.g., in [33, Section 7].

######  Remark 13 (Stabilisation).

A more general expression for the local $L^{2}$-product in (3.11) is obtained
replacing $s_{k,f}$ with

$s_{\mathcal{B},k,f}(\underline{\omega}_{f},\underline{\mu}_{f})=\mathcal{B}_{f}(\underline{I}_{r,f}^{k}P_{r,f}^{k}\underline{\omega}_{f}-\underline{\omega}_{f},\underline{I}_{r,f}^{k}P_{r,f}^{k}\underline{\mu}_{f}-\underline{\mu}_{f}),$

with
$\mathcal{B}_{f}:\underline{X}_{r,f}^{k}\times\underline{X}_{r,f}^{k}\to\mathbb{R}$
denoting a symmetric positive definite bilinear form inducing a norm that
scales in $h_{f}$ as $\|{\cdot}\|_{k,f}$ defined above. Crucially,
$s_{\mathcal{B},k,f}$ depends on its arguments only through the difference
operator
$\underline{X}_{r,f}^{k}\ni\underline{\omega}_{f}\mapsto\underline{I}_{r,f}^{k}P_{r,f}^{k}\underline{\omega}_{f}-\underline{\omega}_{f}\in\underline{X}_{r,f}^{k}$,
which guarantees that it vanishes whenever one of its arguments is the
interpolate of a differential form in $\mathcal{P}_{r}\Lambda^{k}(f)$ (as can
be checked in the same spirit as [36, Lemma 2.11]).

###  3.2 Complex property

We introduce, for all integers $d\in[1,n]$, the piecewise polynomial boundary
exterior derivative $\mathrm{d}^{k}_{r,\partial f}:\underline{X}_{r,\partial
f}^{k}\to\Lambda^{k+1}(\partial f)$ such that $(\mathrm{d}^{k}_{r,\partial
f})_{|f^{\prime}}\coloneq\mathrm{d}^{k}_{r,f^{\prime}}$ for all
$f^{\prime}\in\Delta_{d-1}(f)$ ($\mathrm{d}^{k}_{r,f^{\prime}}$ being the
discrete exterior derivative on the $(d-1)$-cell $f^{\prime}$ defined by
(3.4)). The following lemma is a generalisation of the links, in the DDR
framework based on vector proxies, between element gradients (resp., curls)
and face gradients (resp., curls), see [33, Propositions 1 and 4].

######  Lemma 14 (Link between discrete exterior derivatives on subcells).

It holds, for all $d\geq k+2$, all $f\in\Delta_{d}(\mathcal{M}_{h})$, and all
$\underline{\omega}_{f}\in\underline{X}_{r,f}^{k}$,

$\int_{f}\mathrm{d}^{k}_{r,f}\underline{\omega}_{f}\wedge\mathrm{d}\alpha=(-1)^{k+1}\int_{\partial
f}\mathrm{d}^{k}_{r,\partial f}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial
f}\alpha\qquad\forall\alpha\in\mathcal{P}_{r+1}^{-}\Lambda^{d-k-2}(f).$ (3.12)

###### Proof.

Take $\mu=\mathrm{d}\alpha\in\mathcal{P}_{r}\Lambda^{d-k-1}(f)$ in (3.4) and
use $\mathrm{d}\circ\mathrm{d}=0$ and $\operatorname{tr}_{\partial
f}\mathrm{d}=\mathrm{d}\operatorname{tr}_{\partial f}$ (since the trace is a
pullback, it commutes with the exterior derivative) to get

$\int_{f}\mathrm{d}^{k}_{r,f}\underline{\omega}_{f}\wedge\mathrm{d}\alpha=\int_{\partial
f}P^{k}_{r,\partial f}\underline{\omega}_{\partial
f}\wedge\mathrm{d}\operatorname{tr}_{\partial f}\alpha.$ (3.13)

For each $f^{\prime}\in\Delta_{d-1}(f)$ forming $\partial f$, by Lemma 4 we
have
$\operatorname{tr}_{f^{\prime}}\alpha\in\mathcal{P}_{r+1}^{-}\Lambda^{d-k-2}(f^{\prime})$
so, by (3.5) applied to $f^{\prime}$ instead of $f$ with test function
$(\mu,\nu)=(\operatorname{tr}_{f^{\prime}}\alpha,0)$ (see Remark 6), we have

$(-1)^{k+1}\int_{f^{\prime}}P^{k}_{r,f^{\prime}}\underline{\omega}_{f^{\prime}}\wedge\mathrm{d}\operatorname{tr}_{f^{\prime}}\alpha=\int_{f^{\prime}}\mathrm{d}^{k}_{r,f^{\prime}}\underline{\omega}_{f^{\prime}}\wedge\operatorname{tr}_{f^{\prime}}\alpha-\int_{\partial
f^{\prime}}P^{k}_{r,\partial f^{\prime}}\underline{\omega}_{\partial
f^{\prime}}\wedge\operatorname{tr}_{\partial
f^{\prime}}(\operatorname{tr}_{f^{\prime}}\alpha).$

Use $\operatorname{tr}_{\partial
f^{\prime}}\circ\operatorname{tr}_{f^{\prime}}=\operatorname{tr}_{\partial
f^{\prime}}$, sum these relations over $f^{\prime}\in\Delta_{d-1}(f)$, invoke
$\partial(\partial f)=0$ (which implies that the integrals over $(\partial
f^{\prime})_{f^{\prime}\in\Delta_{d-1}(f)}$ cancel out due to compatible
orientations), and plug the result into (3.13) to conclude. ∎

######  Theorem 15 (Link between discrete potentials and exterior derivatives,
complex property).

It holds, for all integers $k\in[1,n]$ and $d\geq k$, all
$f\in\Delta_{d}(\mathcal{M}_{h})$, and all
$\underline{\omega}_{f}\in\underline{X}_{r,f}^{k-1}$,

$P^{k}_{r,f}(\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f})=\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f},$
(3.14)

and, if $d\geq k+1$,

$\mathrm{d}^{k}_{r,f}(\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f})=0.$
(3.15)

As a consequence, the sequence (3.8) defines a complex.

###### Proof.

The proof is done by induction on $\rho\coloneq d-k$.

If $\rho=0$ (i.e., $d=k$), by the definitions (3.3) of the discrete potential
and (3.7) of the global discrete exterior derivative with $k-1$ instead of
$k$, we have
$P^{k}_{r,f}(\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f})=\star^{-1}(\star\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f})=\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f}$
(notice that, in the first passage, we can omit the projector found in the
definition of $\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f}$ in
front of $\star\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f}$ since this
quantity sits in
$\mathcal{P}_{r}\Lambda^{0}(f)=\mathcal{P}_{r}^{-}\Lambda^{0}(f)$, and is
therefore left unchanged by $\pi_{r,f}^{-,0}$). This proves (3.14), and the
relation (3.15) is irrelevant here since $d=k$.

Let us now assume that (3.14) and (3.15) hold for a given $\rho\geq 0$, and
let us consider $d$ and $k$ such that $d-k=\rho+1$. We start by considering
(3.15) (which we need to prove since $d\geq k+1$ in the present case). Let us
take $f\in\Delta_{d}(\mathcal{M}_{h})$. Applying (3.4) with
$\underline{\mathrm{d}}^{k-1}_{r,f}\underline{\omega}_{f}$ instead of
$\underline{\omega}_{f}$ and a generic
$\mu\in\mathcal{P}_{r}\Lambda^{d-k-1}(f)$, we have, expanding the local
discrete exterior derivative
$\underline{\mathrm{d}}^{k-1}_{r,f}\underline{\omega}_{f}$ according to its
definition (i.e., the restriction to $f$ of (3.7) with $k-1$ instead of $k$),

$\int_{f}\mathrm{d}^{k}_{r,f}(\underline{\mathrm{d}}^{k-1}_{r,f}\underline{\omega}_{f})\wedge\mu={}(-1)^{k+1}\int_{f}\star^{-1}(\pi_{r,f}^{-,d-k}(\star\mathrm{d}^{k-1}_{r,f}\underline{\omega}_{f}))\wedge\mathrm{d}\mu\\\
+\int_{\partial f}P^{k}_{r,\partial
f}(\underline{\mathrm{d}}^{k-1}_{r,\partial f}\underline{\omega}_{\partial
f})\wedge\operatorname{tr}_{\partial f}\mu.$ (3.16)

By the induction hypothesis, (3.14) holds on each
$f^{\prime}\in\Delta_{d-1}(f)$ (since $(d-1)-k=\rho$), and thus

$P^{k}_{r,\partial f}(\underline{\mathrm{d}}^{k-1}_{r,\partial
f}\underline{\omega}_{\partial f})=\mathrm{d}_{r,\partial
f}^{k-1}\underline{\omega}_{\partial f}.$ (3.17)

Invoking then (2.4) with
$(\mathcal{X},\omega,\mu)\leftarrow(\mathcal{P}_{r}^{-}\Lambda^{d-k}(f),\mathrm{d}^{k-1}_{r,f}\underline{\omega}_{f},\mathrm{d}\mu)$,
noticing that
$\mathrm{d}\mu\in\mathcal{P}_{r-1}\Lambda^{d-k}(f)\subset\mathcal{P}_{r}^{-}\Lambda^{d-k}(f)$
(by (2.14) with $\ell=d-k$) to handle the first term in the right-hand side of
(3.16), we infer

$\int_{f}\mathrm{d}^{k}_{r,f}(\underline{\mathrm{d}}^{k-1}_{r,f}\underline{\omega}_{f})\wedge\mu=(-1)^{k+1}\int_{f}\mathrm{d}^{k-1}_{r,f}\underline{\omega}_{f}\wedge\mathrm{d}\mu+\int_{\partial
f}\mathrm{d}^{k-1}_{r,\partial f}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial f}\mu=0,$ (3.18)

where the conclusion follows from the link (3.12) between discrete exterior
derivatives on subcells applied with $k-1$ instead of $k$ and
$\alpha=\mu\in\mathcal{P}_{r}\Lambda^{d-k-1}(f)\subset\mathcal{P}_{r+1}^{-}\Lambda^{d-(k-1)-2}(f)$.
Since $\mu$ is arbitrary in $\mathcal{P}_{r}\Lambda^{d-k-1}(f)$, (3.18) proves
(3.15).

We next prove (3.14). For any
$(\mu,\nu)\in\mathcal{K}_{r+1}^{d-k-1}(f)\times\mathcal{K}_{r}^{d-k}(f)$, the
definition (3.5) of the potential applied to
$\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f}$ gives

$(-1)^{k+1}\int_{f}P^{k}_{r,f}(\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f})\wedge(\mathrm{d}\mu+\nu)=\int_{f}\mathrm{d}_{r,f}^{k}(\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f})\wedge\mu\\\
-\int_{\partial f}P_{r,\partial f}^{k}(\underline{\mathrm{d}}_{r,\partial
f}^{k-1}\underline{\omega}_{\partial f})\wedge\operatorname{tr}_{\partial
f}{\mu}+(-1)^{k+1}\int_{f}\star^{-1}\pi_{r,f}^{-,d-k}(\star\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f})\wedge\nu,$

where we have additionally used, in the last term, the definition of the local
discrete exterior derivative
$\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f}$, corresponding to
the restriction to $f$ of (3.7) with $k-1$ instead of $k$. Using the complex
property (3.15) that we have just proved, we have
$\mathrm{d}_{r,f}^{k}(\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f})=0$.
Moreover, the induction hypothesis (3.17) yields $P_{r,\partial
f}^{k}(\underline{\mathrm{d}}_{r,\partial f}^{k-1}\underline{\omega}_{\partial
f})=\mathrm{d}_{r,\partial f}^{k-1}\underline{\omega}_{\partial f}$. Hence,
invoking (3.12) with $k-1$ instead of $k$ and $\alpha=\mu$ (notice that
$\mu\in\mathcal{K}_{r+1}^{d-k-1}(f)\subset\mathcal{P}_{r+1}^{-}\Lambda^{d-(k-1)-2}(f)$
by (2.12)) and applying (2.4) with
$(\mathcal{X},\omega,\mu)\leftarrow(\mathcal{P}_{r}^{-}\Lambda^{d-k}(f),\mathrm{d}^{k-1}_{r,f}\underline{\omega}_{f},\nu)$,
which is valid since
$\nu\in\mathcal{K}_{r}^{d-k}(f)\subset\mathcal{P}_{r}^{-}\Lambda^{d-k}(f)$ by
(2.12b) with $\ell=d-k\geq 1$, we obtain

$\displaystyle(-1)^{k+1}\int_{f}P^{k}_{r,f}(\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f})\wedge(\mathrm{d}\mu+\nu)={}$
$\displaystyle-(-1)^{k}\int_{f}\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f}\wedge\mathrm{d}\mu+(-1)^{k+1}\int_{f}\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f}\wedge\nu.$

Simplifying by $(-1)^{k+1}$ and recalling the isomorphism (2.9) concludes the
proof of (3.14). ∎

###  3.3 Commutation

The following lemma shows that the reconstructed potential
$P_{r,f}^{k}\underline{\omega}_{f}$ on a $d$-cell $f$ is built by adding a
high-order correction to $\star^{-1}\omega_{f}$; this correction is designed
to obtain a polynomial consistency unachievable by the component alone (see
(3.9)).

######  Lemma 16 (Links between component and potential reconstruction).

For all integers $d\in[0,n]$ and $k\leq d$, if
$f\in\Delta_{d}(\mathcal{M}_{h})$ and
$\underline{\omega}_{f}\in\underline{X}_{r,f}^{k}$, then it holds

$(-1)^{k+1}\int_{f}P_{r,f}^{k}\underline{\omega}_{f}\wedge(\mathrm{d}\mu+\nu)=(-1)^{k+1}\int_{f}\star^{-1}\omega_{f}\wedge(\mathrm{d}(\pi_{r,f}^{d-k-1}\mu)+\nu)\\\
+\int_{f}\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\wedge(\mu-\pi_{r,f}^{d-k-1}\mu)-\int_{\partial
f}P_{r,\partial f}^{k}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial f}{(\mu-\pi_{r,f}^{d-k-1}\mu)}\\\
\forall(\mu,\nu)\in\mathcal{K}_{r+1}^{d-k-1}(f)\times\mathcal{K}_{r}^{d-k}(f).$
(3.19)

As a consequence,

$\pi_{r,f}^{-,d-k}(\star P^{k}_{r,f}\underline{\omega}_{f})=\omega_{f}.$
(3.20)

###### Proof.

If $d=k$, the relation (3.19) follows from
$\mathcal{K}_{r+1}^{d-k-1}(f)=\mathcal{K}_{r+1}^{-1}(f)=\\{0\\}$ and
$P_{r,f}^{k}\underline{\omega}_{f}=\star^{-1}\omega_{f}$ (see (3.3)), which
also establishes (3.20) since $\pi_{r,f}^{-,0}={\rm Id}$ on
$\mathcal{P}_{r}\Lambda^{0}(f)=\mathcal{P}_{r}^{-}\Lambda^{0}(f)$.

Consider now $d\geq k+1$ and take
$(\mu,\nu)\in\mathcal{K}_{r+1}^{d-k-1}(f)\times\mathcal{K}_{r}^{d-k}(f)$.
Inserting $\pm\pi_{r,f}^{d-k-1}\mu$ into the definition (3.5) of
$P_{r,f}^{k}\underline{\omega}_{f}$ we have

$\displaystyle(-1)^{k+1}{}$
$\displaystyle\int_{f}P_{r,f}^{k}\underline{\omega}_{f}\wedge(\mathrm{d}\mu+\nu)$
(3.21) $\displaystyle={}$
$\displaystyle\int_{f}\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\wedge\pi_{r,f}^{d-k-1}\mu+\int_{f}\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\wedge(\mu-\pi_{r,f}^{d-k-1}\mu)-\int_{\partial
f}P_{r,\partial f}^{k}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial f}{\mu}$
$\displaystyle+(-1)^{k+1}\int_{f}\star^{-1}\omega_{f}\wedge\nu.$

On the other hand, the definition (3.4) of
$\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}$ applied to $\pi_{r,f}^{d-k-1}\mu$
yields

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\wedge\pi_{r,f}^{d-k-1}\mu=(-1)^{k+1}\int_{f}\star^{-1}\omega_{f}\wedge\mathrm{d}(\pi_{r,f}^{d-k-1}\mu)+\int_{\partial
f}P_{r,\partial f}^{k}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial f}(\pi_{r,f}^{d-k-1}\mu).$

Substituting this relation into (3.21) yields (3.19).

To prove (3.20) we apply (3.19) with
$(\mu,\nu)\in\mathcal{K}_{r}^{d-k-1}(f)\times\mathcal{K}_{r}^{d-k}(f)$ and
notice that $\mu=\pi_{r,f}^{d-k-1}\mu$ since
$\mathcal{K}_{r}^{d-k-1}(f)\subset\mathcal{P}_{r}\Lambda^{d-k-1}(f)$, to get

$\int_{f}P^{k}_{r,f}\underline{\omega}_{f}\wedge(\mathrm{d}\mu+\nu)=\int_{f}\star^{-1}\omega_{f}\wedge(\mathrm{d}\mu+\nu).$
(3.22)

The isomorphism (2.17) with $\ell=d-k\geq 1$ shows that $\mathrm{d}\mu+\nu$
spans $\mathcal{P}_{r}^{-}\Lambda^{d-k}(f)$ when $(\mu,\nu)$ span
$\mathcal{K}_{r}^{d-k-1}(f)\times\mathcal{K}_{r}^{d-k}(f)$. Hence, (3.22)
gives

$\int_{f}\star^{-1}\omega_{f}\wedge\alpha=\int_{f}P^{k}_{r,f}\underline{\omega}_{f}\wedge\alpha\stackrel{{\scriptstyle\eqref{eq:remove.projector}}}{{=}}\int_{f}\star^{-1}\pi_{r,f}^{-,d-k}(\star
P^{k}_{r,f}\underline{\omega}_{f})\wedge\alpha\qquad\forall\alpha\in\mathcal{P}_{r}^{-}\Lambda^{d-k}(f),$

proving (3.20) since
$\star^{-1}\omega_{f}\in\mathcal{P}_{r}^{-}\Lambda^{d-k}(f)$ and $\star^{-1}$
is an isomorphism. ∎

######  Theorem 17 (Commutation property for the local discrete exterior
derivative).

For all integers $d\in[1,n]$ and $k\leq d-1$, and for all
$f\in\Delta_{d}(\mathcal{M}_{h})$, recalling the definition (3.2) of the
interpolators, it holds

$\underline{\mathrm{d}}_{r,f}^{k}(\underline{I}_{r,f}^{k}\omega)=\underline{I}_{r,f}^{k+1}(\mathrm{d}\omega)\qquad\forall\omega\in
C^{1}\Lambda^{k}(\overline{f}),$ (3.23)

expressing the commutativity of the following diagram:

${C^{1}\Lambda^{k}(\overline{f})}$${C^{0}\Lambda^{k+1}(\overline{f})}$${\underline{X}_{r,f}^{k}}$${\underline{X}_{r,f}^{k+1}.}$$\scriptstyle{\mathrm{d}}$$\scriptstyle{\underline{I}_{r,f}^{k}}$$\scriptstyle{\underline{I}_{r,f}^{k+1}}$$\scriptstyle{\underline{\mathrm{d}}_{r,f}^{k}}$

###### Proof.

Given the definitions of the interpolator and of the local discrete exterior
derivative (see (3.2) and (3.7)), we have to prove that, for all
$f^{\prime}\in\Delta_{d^{\prime}}(f)$ with $d^{\prime}\in[k+1,d]$,
$\pi_{r,f^{\prime}}^{-,d^{\prime}-k-1}(\star\mathrm{d}_{r,f^{\prime}}^{k}\underline{I}_{r,f^{\prime}}^{k}\omega)=\pi_{r,f^{\prime}}^{-,d^{\prime}-k-1}(\star\operatorname{tr}_{f^{\prime}}(\mathrm{d}\omega))$.
Recalling the definition of the projector
$\pi_{r,f^{\prime}}^{-,d^{\prime}-k-1}$ (i.e., (2.3) with
$\mathcal{X}=\mathcal{P}_{r}^{-}\Lambda^{d^{\prime}-k-1}(f^{\prime})$), we
need to prove that, for any
$\mu\in\mathcal{P}_{r}^{-}\Lambda^{d^{\prime}-k-1}(f^{\prime})$

$\int_{f^{\prime}}\star\mathrm{d}_{r,f^{\prime}}^{k}\underline{I}_{r,f^{\prime}}^{k}\omega\wedge\star\mu=\int_{f^{\prime}}\star\operatorname{tr}_{f^{\prime}}(\mathrm{d}\omega)\wedge\star\mu.$

Applying (A.4), this amounts to proving that

$\int_{f^{\prime}}\mathrm{d}_{r,f^{\prime}}^{k}\underline{I}_{r,f^{\prime}}^{k}\omega\wedge\mu=\int_{f^{\prime}}\operatorname{tr}_{f^{\prime}}(\mathrm{d}\omega)\wedge\mu.$
(3.24)

Using the definitions (3.4) of the discrete exterior derivative on
$f^{\prime}$ and (3.2) of $\underline{I}_{r,f^{\prime}}^{k}$, we have

$\int_{f^{\prime}}\mathrm{d}_{r,f^{\prime}}^{k}\underline{I}_{r,f^{\prime}}^{k}\omega\wedge\mu=(-1)^{k+1}\int_{f^{\prime}}\cancelto{\operatorname{tr}_{f^{\prime}}\omega}{\star^{-1}\pi_{r,f^{\prime}}^{-,d^{\prime}-k}(\star\operatorname{tr}_{f^{\prime}}\omega)}\wedge\mathrm{d}\mu+\int_{\partial
f^{\prime}}P^{k}_{r,\partial f^{\prime}}\underline{I}_{r,\partial
f^{\prime}}^{k}\omega\wedge\operatorname{tr}_{\partial f^{\prime}}\mu,$ (3.25)

where the substitution is justified by (2.4) with
$(\mathcal{X},\omega,\mu)\leftarrow(\mathcal{P}_{r}^{-}\Lambda^{d^{\prime}-k}(f^{\prime}),\operatorname{tr}_{f^{\prime}}\omega,\mathrm{d}\mu)$,
since
$\mathrm{d}\mu\in\mathrm{d}\mathcal{P}_{r}^{-}\Lambda^{d^{\prime}-k-1}(f^{\prime})\subset\mathrm{d}\mathcal{P}_{r}\Lambda^{d^{\prime}-k-1}(f^{\prime})\subset\mathcal{P}_{r}^{-}\Lambda^{d^{\prime}-k}(f^{\prime})$
(see (2.12b)). For all $f^{\prime\prime}\in\Delta_{d^{\prime}-1}(f^{\prime})$
we have
$\operatorname{tr}_{f^{\prime\prime}}\mu\in\mathcal{P}_{r}^{-}\Lambda^{d^{\prime}-1-k}(f^{\prime\prime})$
(see Lemma 4), so

$\displaystyle\int_{f^{\prime\prime}}P^{k}_{r,f^{\prime\prime}}\underline{I}_{r,f^{\prime\prime}}^{k}\omega\wedge\operatorname{tr}_{f^{\prime\prime}}\mu$
$\displaystyle=\int_{f^{\prime\prime}}\star^{-1}\pi_{r,f^{\prime\prime}}^{-,d^{\prime}-1-k}(\star
P^{k}_{r,f^{\prime\prime}}\underline{I}_{r,f^{\prime\prime}}^{k}\omega)\wedge\operatorname{tr}_{f^{\prime\prime}}\mu$
Eq. (2.4)
$\displaystyle=\int_{f^{\prime\prime}}\star^{-1}\pi_{r,f^{\prime\prime}}^{-,d^{\prime}-1-k}(\star\operatorname{tr}_{f^{\prime\prime}}\omega)\wedge\operatorname{tr}_{f^{\prime\prime}}\mu$
Eqs. (3.20), (3.2)
$\displaystyle=\int_{f^{\prime\prime}}\operatorname{tr}_{f^{\prime\prime}}\omega\wedge\operatorname{tr}_{f^{\prime\prime}}\mu.$
Eq. (2.4)

Summing this relation over
$f^{\prime\prime}\in\Delta_{d^{\prime}-1}(f^{\prime})$ and substituting the
result into (3.25) we obtain

$\int_{f^{\prime}}\mathrm{d}_{r,f^{\prime}}^{k}\underline{I}_{r,f^{\prime}}^{k}\omega\wedge\mu=(-1)^{k+1}\int_{f^{\prime}}\operatorname{tr}_{f^{\prime}}\omega\wedge\mathrm{d}\mu+\int_{\partial
f^{\prime}}\operatorname{tr}_{\partial
f^{\prime}}\omega\wedge\operatorname{tr}_{\partial f^{\prime}}\mu.$

The proof of (3.24) is concluded invoking the integration by part formula
(2.1) and writing
$\mathrm{d}\operatorname{tr}_{f^{\prime}}=\operatorname{tr}_{f^{\prime}}\mathrm{d}$
(since the trace is a pullback). ∎

###  3.4 Polynomial consistency

###### Proof of Theorem 11.

The proof is made, as in Theorem 15, by induction on $\rho\coloneq d-k$.

If $\rho=0$, then $d=k$ and the definitions (3.3) of the discrete potential
and (3.2) of the interpolator give
$P_{r,d}^{k}\underline{I}_{r,f}^{k}\omega=\star^{-1}\pi_{r,f}^{-,0}(\star\omega)=\star^{-1}\star\omega=\omega$,
where, to remove the projector, we have used the fact that
$\star\omega\in\mathcal{P}_{r}\Lambda^{0}(f)=\mathcal{P}_{r}^{-}\Lambda^{0}(f)$
(cf. (2.12a)).

Let us now assume that the lemma holds for a given $\rho\geq 0$, and let us
consider $d$ and $k$ such that $d-k=\rho+1$. We first look at (3.10). By the
link (3.14) applied to $k+1$ instead of $k$ and the commutation property
(3.23), we have, for $\omega\in\mathcal{P}_{r+1}^{-}\Lambda^{k}(f)$,

$\mathrm{d}_{r,f}^{k}\underline{I}_{r,f}^{k}\omega=P_{r,f}^{k+1}(\underline{\mathrm{d}}_{r,f}^{k}\underline{I}_{r,f}^{k}\omega)=P_{r,f}^{k+1}\underline{I}_{r,f}^{k+1}(\mathrm{d}\omega).$

We have
$\mathrm{d}\omega\in\mathrm{d}\mathcal{P}_{r+1}^{-}\Lambda^{k}(f)\subset\mathcal{P}_{r}\Lambda^{k+1}(f)$,
and the pair $(d,k+1)$ satisfies $d-(k+1)=\rho$. We can therefore apply the
induction hypothesis to see that (3.9) holds for this pair and
$\mathrm{d}\omega$ instead of $\omega$; this gives
$\mathrm{d}_{r,f}^{k}\underline{I}_{r,f}^{k}\omega=\mathrm{d}\omega$ and
proves (3.10).

We now turn to (3.9). For $\omega\in\mathcal{P}_{r}\Lambda^{k}(f)$, applying
the property (3.10) that we have just proved to $\omega$ and recalling the
definitions (3.5) and (3.2) of the potential and of the interpolator, we find,
for all
$(\mu,\nu)\in\mathcal{K}_{r+1}^{d-k-1}(f)\times\mathcal{K}_{r}^{d-k}(f)$,

$(-1)^{k+1}\int_{f}P_{r,f}^{k}\underline{I}_{r,f}^{k}\omega\wedge(\mathrm{d}\mu+\nu)\\\
=\int_{f}\mathrm{d}\omega\wedge\mu-\int_{\partial f}P_{r,\partial
f}^{k}\underline{I}_{r,\partial f}^{k}\operatorname{tr}_{\partial
f}\omega\wedge\operatorname{tr}_{\partial
f}{\mu}+(-1)^{k+1}\int_{f}\cancelto{\omega}{(\star^{-1}\pi_{r,f}^{-,d-k}\star\omega)}\wedge\nu,$

the replacement being justified by (2.4) and
$\nu\in\mathcal{P}_{r}^{-}\Lambda^{d-k}(f)$ (see (2.12)). We can then apply
the polynomial consistency (3.9) on each $f^{\prime}\in\Delta_{d-1}(f)$ (as
$(d-1)-k=\rho$) to write $P_{r,\partial f}^{k}\underline{I}_{r,\partial
f}^{k}\operatorname{tr}_{\partial f}\omega=\operatorname{tr}_{\partial
f}\omega$ , and then integrate by parts to conclude, since $\mu,\nu$ are
generic elements, that $P_{r,f}^{k}\underline{I}_{r,f}^{k}\omega=\omega$. ∎

######  Remark 18 (Consistency property of the improved potential for $k=0$).

In the case $k=0$, the improved potential defined in Remark 7 satisfies the
following consistency property:

$P_{r+1,f}^{0}\underline{I}_{r,f}^{0}\omega=\omega\qquad\forall\omega\in\mathcal{P}_{r+1}\Lambda^{0}(f).$

To see this, first notice that when $d=k=0$ we have
$P_{r+1,f}^{0}=P_{r,f}^{0}$ since
$\mathcal{P}_{r+1}\Lambda^{0}(f)=\mathcal{P}_{r}\Lambda^{0}(f)\cong\mathbb{R}$,
and then, for $d\geq k+1$, invoke the definition (3.6) of
$P_{r+1,f}^{0}\underline{I}_{r,f}^{0}\omega$, apply (3.10) (since
$\mathcal{P}_{r+1}^{-}\Lambda^{0}(f)=\mathcal{P}_{r+1}\Lambda^{0}(f)$) and a
recursion argument on $d$.

###  3.5 Cohomology

A strategy to establish the exactness of the de Rham complex (for a domain
with trivial topology) is to design a Poincaré operator
$p:C^{1}\Lambda^{k}(\overline{\Omega})\to
C^{1}\Lambda^{k-1}(\overline{\Omega})$, that satisfies
$\mathrm{d}p+p\mathrm{d}={\rm Id}$. The Poincaré operator is built integrating
a certain flow of contracted differential forms; see [31, 51] for details and
applications to the design of finite element complexes. Extending such a
construction to the context of fully discrete spaces is not trivial, as it is
not clear how the discrete polynomial components on cells should evolve with
such a flow. We therefore select an alternative approach, more suited to
hierarchical discrete spaces.

The starting point is the following idea: if $\eta\in
C^{1}\Lambda^{k}(\overline{\Omega})$ satisfies $\mathrm{d}\eta=0$ and we have
$\omega\in C^{2}\Lambda^{k-1}(\overline{\Omega})$ such that
$\mathrm{d}\omega=\eta$, then (2.1) shows that, for any $d$-cell $f$,

$(-1)^{k}\int_{f}\omega\wedge\mathrm{d}\mu=\int_{f}\eta\wedge\mu-\int_{\partial
f}\operatorname{tr}_{\partial f}\omega\wedge\operatorname{tr}_{\partial
f}\mu\qquad\forall\mu\in C^{1}\Lambda^{d-k}(\overline{\Omega}).$ (3.26)

In the discrete setting, $\omega$ is built starting from the lowest-
dimensional cells, and (3.26) thus gives a condition on $\omega$ over $f$
based on the already constructed $\operatorname{tr}_{\partial f}\omega$. To
start this process, we must fix the values of $\omega$ on the lowest-
dimensional cells, which is not an easy task in general. Actually, from the
point of view of differential forms, the lowest-dimensional cells encode the
topology of the domain, and thus the cohomology of the complex; for a generic
$\eta$, the recursive construction of $\omega$ can therefore only be fully
complete if the complex is exact, and thus the topology trivial.

This limitation is circumvented by using the following idea: if $\eta$ has
zero average on $k$-cells, then $\omega$ can be set to zero on $(k-1)$-cells,
which completes the construction above (see Lemma 19 below). This result is
then exploited, through the extension/reduction strategy developed in [35,
37], to compare the cohomology of the arbitrary-order $\mathrm{DDR}(r)$
complex to that of the lowest-order $\mathrm{DDR}(0)$ complex, which is
trivially isomorphic to the CW complex based on the mesh.

We therefore start by considering the subspace $\underline{X}_{r,h,\flat}^{k}$
of $\underline{X}_{r,h}^{k}$ made of vectors of differential forms whose
integrals over cells of dimension $d=k$ vanish:

$\underline{X}_{r,h,\flat}^{k}\coloneq\left\\{\underline{\omega}_{h}=(\omega_{f})_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k,n]}\,:\,\int_{f}\star^{-1}\omega_{f}=0\quad\forall
f\in\Delta_{k}(\mathcal{M}_{h})\right\\}.$

######  Lemma 19 (Exactness property for $\underline{X}_{r,h,\flat}^{k}$).

For any integer $k\in[0,n]$, if
$\underline{\eta}_{h}\in\underline{X}_{r,h,\flat}^{k}$ satisfies
$\underline{\mathrm{d}}_{r,h}^{k}\underline{\eta}_{h}=\underline{0}$, then
there exists $\underline{\omega}_{h}\in\underline{X}_{r,h,\flat}^{k-1}$ such
that
$\underline{\eta}_{h}=\underline{\mathrm{d}}_{r,h}^{k-1}\underline{\omega}_{h}$,
where, in accordance with (3.8), we have set
$\underline{\mathrm{d}}_{r,h}^{-1}=\underline{\mathrm{d}}_{r,h}^{n}\coloneq
0$.

######  Remark 20 (Exact sub-complex).

It can easily be checked that
$\underline{\mathrm{d}}_{r,h}^{k}:\underline{X}_{r,h,\flat}^{k}\to\underline{X}_{r,h,\flat}^{k+1}$.
As a consequence, the previous lemma shows that
$(\underline{X}_{r,h,\flat}^{k},\underline{\mathrm{d}}_{r,h}^{k})_{k}$ is an
exact sub-complex of $\mathrm{DDR}(r)$ (even if the latter complex is not
exact).

###### Proof.

We first notice that the case $r=0$ is trivial since, for all $k$,
$\underline{X}_{0,h,\flat}^{k}=\\{(0)_{f\in\Delta_{k}(\mathcal{M}_{h})}\\}$.
This comes from the fact that the space $\underline{X}_{0,h}^{k}$ only has
non-zero components (which are moreover constant) on cells of dimension $d=k$;
to check this, notice that the spaces (2.12b) are all trivial since the first
component vanishes for $k$-forms with constant coefficients, while the second
is zero by (2.6). We can therefore assume that $r\geq 1$. The cases $k=0$ and
$k\geq 1$ have to be handled separately.
Case $k=0$. We prove that, if
$\underline{\eta}_{h}\in\underline{X}_{r,h,\flat}^{0}$ and
$\underline{\mathrm{d}}_{r,h}^{0}\underline{\eta}_{h}=\underline{0}$, then
$\eta_{f}=0$ for all $f\in\Delta_{d}(\mathcal{M}_{h})$, $d\in[0,n]$. This is
done by induction on $d$. The case $d=0$ follows immediately from the
definition of $\underline{X}_{r,h,\flat}^{0}$ which shows that the value of
$\star^{-1}\eta_{f}$ on any vertex $f\in\Delta_{0}(\mathcal{M}_{h})$ is zero.
Assuming that all components of $\underline{\eta}_{h}$ on cells of dimension
$d-1\geq 0$ vanish, we now prove that $\eta_{f}=0$ for all
$f\in\Delta_{d}(\mathcal{M}_{h})$. Note first that, by (3.14), the property
$\underline{\mathrm{d}}_{r,f}^{0}\underline{\eta}_{f}=\underline{0}$ implies
$\mathrm{d}_{r,f}^{0}\underline{\eta}_{f}=0$. Enforcing then
$\underline{\eta}_{\partial f}=\underline{0}$ (by induction hypothesis) in the
definition (3.4) of $\mathrm{d}_{r,f}^{0}\underline{\eta}_{f}$ gives

$\int_{f}\star^{-1}\eta_{f}\wedge\mathrm{d}\mu=0\qquad\forall\mu\in\mathcal{P}_{r}\Lambda^{d-1}(f).$

By definition (2.12b) of the trimmed space with $\ell=d$, and accounting for
(2.6), we have
$\mathrm{d}\mathcal{P}_{r}\Lambda^{d-1}(f)=\mathcal{P}_{r}^{-}\Lambda^{d}(f)$,
so the relation above and (A.4) with
$(\omega,\mu)\leftarrow(\eta_{f},\mathrm{d}\mu)$ and $\rho=\mathrm{d}\mu$ show
that $\int_{f}\eta_{f}\wedge\star\rho=0$ for all
$\rho\in\mathcal{P}_{r}^{-}\Lambda^{d}(f)$. Since $\eta_{f}$ belongs to that
space, we conclude that $\eta_{f}=0$.
Case $k\geq 1$. Let $\underline{\eta}_{h}\in\underline{X}_{r,h,\flat}^{k}$ be
such that
$\underline{\mathrm{d}}_{r,h}^{k}\underline{\eta}_{h}=\underline{0}$, and let
us construct $\underline{\omega}_{h}\in\underline{X}_{r,h,\flat}^{k-1}$ such
that
$\underline{\mathrm{d}}_{r,h}^{k-1}\underline{\omega}_{h}=\underline{\eta}_{h}$.
This construction of $\underline{\omega}_{h}$ is done by increasing dimension
$d\in[k-1,n]$ of the cells. For all $f\in\Delta_{k-1}(\mathcal{M}_{h})$, we
set $\omega_{f}=0$ (which ensures, in particular, that the zero-average
condition embedded in the space $\underline{X}_{r,h,\flat}^{k-1}$ is
fulfilled). Assume now that the components of $\underline{\omega}_{h}$ have
been constructed up to cells of dimension $d-1\geq k-1$, and consider
$f\in\Delta_{d}(\mathcal{M}_{h})$. We choose
$\omega_{f}\in\mathcal{P}_{r}^{-}\Lambda^{d-k+1}(f)$ such that the following
relation holds:

$(-1)^{k}\int_{f}\star^{-1}\omega_{f}\wedge\mathrm{d}\mu=\int_{f}P_{r,f}^{k}\underline{\eta}_{f}\wedge\mu-\int_{\partial
f}P_{r,\partial f}^{k-1}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial
f}\mu\qquad\forall\mu\in\mathcal{K}_{r}^{d-k}(f).$ (3.27)

Notice that, since the construction is recursive on the dimension of the
cells, $\underline{\omega}_{\partial f}$ has already been constructed at this
stage. Owing to the isomorphism (2.17) with $\ell=d-k+1\geq 1$, this relation
completely defines the projection of $\omega_{f}$ on
$\mathrm{d}\mathcal{K}_{r}^{d-k}(f)\subset\mathcal{P}_{r}^{-}\Lambda^{d-k+1}(f)$.
The projection of $\omega_{f}$ on the remaining component
$\mathcal{K}_{r}^{d-k+1}(f)$ of $\mathcal{P}_{r}^{-}\Lambda^{d-k+1}(f)$ is not
relevant to the rest of the proof and can be set to 0.

Let us now prove that
$\underline{\mathrm{d}}_{r,h}^{k-1}\underline{\omega}_{h}=\underline{\eta}_{h}$.
It suffices to show that

$\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f}=P_{r,f}^{k}\underline{\eta}_{f}\qquad\forall
f\in\Delta_{d}(\mathcal{M}_{h})\,,\;d\in[k,n].$ (3.28)

Indeed, applying $\pi_{r,f}^{-,d-k}\star$ to this relation and using (3.20)
yields
$\pi_{r,f}^{-,d-k}(\star\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f})=\eta_{f}$;
using this relation for all cells $f$, and recalling the definition (3.7) of
the global discrete exterior derivative (with $k-1$ instead of $k$), then
gives
$\underline{\mathrm{d}}_{r,h}^{k-1}\underline{\omega}_{h}=\underline{\eta}_{h}$
as claimed.

The relation (3.28) is a direct consequence of the following property, for the
same $f$ and $d$:

$\int_{f}\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f}\wedge\mu=\int_{f}P_{r,f}^{k}\underline{\eta}_{f}\wedge\mu\qquad\forall\mu\in\mathcal{P}_{r}\Lambda^{d-k}(f).$
(3.29)

Owing to (2.7), we only need to prove this relation for
$\mu\in\mathcal{K}_{r}^{d-k}(f)$, and $\mu\in\mathcal{P}_{0}\Lambda^{0}(f)$ if
$d=k$ or $\mu\in\mathrm{d}\mathcal{P}_{r+1}\Lambda^{d-k-1}(f)$ if $d\geq k+1$.

If $\mu\in\mathcal{K}_{r}^{d-k}(f)$, the definition (3.4) of
$\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f}$ together with the property
(3.27) immediately give (3.29).

Let us consider the case $d=k$ and $\mu\in\mathcal{P}_{0}\Lambda^{0}(f)$. Then
$\mathrm{d}\mu=0$, so the definition (3.4) of
$\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f}$ and
$\underline{\omega}_{\partial f}=0$ (by construction, $\underline{\omega}_{h}$
vanishes on cells of dimension $d-1=k-1$) show that the left-hand side of
(3.29) vanishes. Since $P_{r,f}^{k}\underline{\eta}_{f}=\star^{-1}\eta_{f}$
(see (3.3)) and $\int_{f}\star^{-1}\eta_{f}=0$ as
$\underline{\eta}_{h}\in\underline{X}_{r,h,\flat}^{k}$, the right-hand side of
(3.29) vanishes as well, and this relation holds.

Finally, we turn to the case $d\geq k+1$ and
$\mu\in\mathrm{d}\mathcal{P}_{r+1}\Lambda^{d-k-1}(f)$, which is proved by
induction on $d$ (the base case $d=k$ having already been covered). By (2.8)
with $(\ell,r)\leftarrow(d-k-1,r+1)$, we have
$\mu\in\mathrm{d}\mathcal{K}_{r+1}^{d-k-1}(f)$, and we can therefore write
$\mu=\mathrm{d}\alpha$ with
$\alpha\in\mathcal{K}_{r+1}^{d-k-1}(f)\subset\mathcal{P}_{r+1}^{-}\Lambda^{d-k-1}(f)$
(see (2.12)). Invoking the link (3.12) between discrete exterior derivatives
on subcells (notice that $d\geq(k-1)+2$), we obtain

$\int_{f}\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f}\wedge\mu=(-1)^{k}\int_{\partial
f}\mathrm{d}_{r,\partial f}^{k-1}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial f}\alpha=(-1)^{k}\int_{\partial
f}P_{r,\partial f}^{k}\underline{\eta}_{\partial
f}\wedge\operatorname{tr}_{\partial f}\alpha,$

where the second equality follows from the induction hypothesis that (3.29)
holds on subcells of $f$. We have
$\underline{\mathrm{d}}_{r,f}^{k}\underline{\eta}_{f}=0$ and $d\geq k+1$, so
we can apply (3.14) with $k+1$ instead of $k$ to get
$\mathrm{d}_{r,f}^{k}\underline{\eta}_{f}=0$; the definition (3.5) of
$P_{r,f}^{k}\underline{\eta}_{f}$ (with $(\mu,\nu)\leftarrow(\alpha,0)$, see
Remark 6 for the validity of this choice of $\mu$) allows us to continue with

$\int_{f}\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f}\wedge\mu=-(-1)^{k}\times(-1)^{k+1}\int_{f}P_{r,f}^{k}\underline{\eta}_{f}\wedge\mathrm{d}\alpha.$

Recalling that $\mathrm{d}\alpha=\mu$ concludes the proof of (3.29). ∎

###### Proof of Theorem 10.

As in [37, Lemma 4], it is straightforward to see that the (discrete) de Rham
map establishes a chain isomorphism between the lowest-degree complex
$\mathrm{DDR}(0)$ and the CW complex defined by $\mathcal{M}_{h}$. Since this
CW complex has the same cohomology as the de Rham complex (1.2), the proof is
complete if we show that the cohomology of $\mathrm{DDR}(r)$ is isomorphic to
the cohomology of $\mathrm{DDR}(0)$. This obviously means that we can assume
$r\geq 1$ in the following.
_Step 1: Reductions and extensions._ With the goal of applying [35,
Proposition 2], we define reduction and extension maps between
$\mathrm{DDR}(r)$ and $\mathrm{DDR}(0)$ as in (3.30).

${\mathrm{DDR}(r):\quad\cdots}$${\underline{X}_{r,h}^{k}}$${\underline{X}_{r,h}^{k+1}}$${\cdots}$${\mathrm{DDR}(0):\quad\cdots}$${\underline{X}_{0,h}^{k}}$${\underline{X}_{0,h}^{k+1}}$${\cdots}$$\scriptstyle{\underline{\mathrm{d}}_{r,h}^{k}}$$\scriptstyle{\underline{R}_{h}^{k}}$$\scriptstyle{\underline{R}_{h}^{k+1}}$$\scriptstyle{\underline{\mathrm{d}}_{0,h}^{k}}$$\scriptstyle{\underline{E}_{h}^{k}}$$\scriptstyle{\underline{E}_{h}^{k+1}}$
(3.30)

The reduction
$\underline{R}_{h}^{k}:\underline{X}_{r,h}^{k}\to\underline{X}_{0,h}^{k}$ is
defined taking the average of components on the cells of dimension $k$ (recall
that vectors in $\underline{X}_{0,h}^{k}$ only have components on these
cells): For all $\underline{\omega}_{h}\in\underline{X}_{r,h}^{k}$,

$\underline{R}_{h}^{k}\underline{\omega}_{h}=(\pi_{0,f}^{0}\omega_{f})_{f\in\Delta_{k}(\mathcal{M}_{h})}.$
(3.31)

The extension
$\underline{E}_{h}^{k}:\underline{X}_{0,h}^{k}\to\underline{X}_{r,h}^{k}$ is
defined by induction on the cell dimension: For all
$\underline{\eta}_{h}\in\underline{X}_{0,h}^{k}$,
$\underline{E}_{h}^{k}\underline{\eta}_{h}\coloneq(E_{f}^{k}\underline{\eta}_{f})_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k,n]}$,
where

  * •

If $d=k$,

$E_{f}^{k}\underline{\eta}_{f}=\eta_{f}\in\mathcal{P}_{0}^{-}\Lambda^{0}(f)\subset\mathcal{P}_{r}^{-}\Lambda^{0}(f);$
(3.32a)

  * •

If $d\geq k+1$,
$E_{f}^{k}\underline{\eta}_{f}\in\mathcal{P}_{r}^{-}\Lambda^{d-k}(f)$
satisfies

$(-1)^{k+1}\int_{f}\star^{-1}E_{f}^{k}\underline{\eta}_{f}\wedge(\mathrm{d}\mu+\nu)\\\
=\int_{f}\mathrm{d}_{0,f}^{k}\underline{\eta}_{f}\wedge\mu-\int_{\partial
f}P_{r,\partial f}^{k}\underline{E}_{\partial f}^{k}\underline{\eta}_{\partial
f}\wedge\operatorname{tr}_{\partial
f}\mu+(-1)^{k+1}\int_{f}P_{0,f}^{k}\underline{\eta}_{f}\wedge\nu\\\
\forall(\mu,\nu)\in\mathcal{K}_{r}^{d-k-1}(f)\times\mathcal{K}_{r}^{d-k}(f),$
(3.32b)

where $\underline{E}_{\partial f}^{k}\underline{\eta}_{\partial
f}=(\underline{E}_{f^{\prime}}^{k}\underline{\eta}_{f^{\prime}})_{f^{\prime}\in\Delta_{d-1}(f)}$
gathers the extensions already built at previous steps on the subcells of
dimension $d-1$ of $f$. The isomorphism (2.17) with $\ell=d-k\geq 1$ ensures
that the relation above fully and properly defines
$E_{f}^{k}\underline{\eta}_{f}$.

In passing, we notice that, for all $f\in\Delta_{d}(\mathcal{M}_{h})$ with
$d\geq k+1$, combining the definitions (3.4) of
$\mathrm{d}_{r,f}^{k}\underline{E}_{f}^{k}\underline{\eta}_{f}$ and (3.32b) of
$E_{f}^{k}\underline{\eta}_{f}$ (with $\nu=0$), we have

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{E}_{f}^{k}\underline{\eta}_{f}\wedge\mu=\int_{f}\mathrm{d}_{0,f}^{k}\underline{\eta}_{f}\wedge\mu\qquad\forall\mu\in\mathcal{K}_{r}^{d-k-1}(f).$

Since
$\mathrm{d}_{0,f}^{k}\underline{\eta}_{f}=P_{0,f}^{k+1}\underline{\mathrm{d}}_{0,f}^{k}\underline{\eta}_{f}$
by (3.14), using the definition of
$E_{f}^{k+1}\underline{\mathrm{d}}_{0,f}^{k}\underline{\eta}_{f}$ (namely,
(3.32a) if $d=k+1$, or (3.32b) with $(\mu,\nu)\leftarrow(0,\mu)$ if $d\geq
k+2$) we deduce that

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{E}_{f}^{k}\underline{\eta}_{f}\wedge\mu=\int_{f}\star^{-1}E_{f}^{k+1}\underline{\mathrm{d}}_{0,f}^{k}\underline{\eta}_{f}\wedge\mu\qquad\forall\mu\in\mathcal{K}_{r}^{d-k-1}(f).$
(3.33)

_Step 2: Proof of the theorem._ To apply [35, Proposition 2], we need to prove
that

$\underline{\mathrm{d}}_{0,h}^{k}=\underline{R}_{h}^{k+1}\underline{\mathrm{d}}_{r,h}^{k}\underline{E}_{h}^{k}$
(3.34)

and that [35, Assumption 1] holds, that is:

  * (C1)

$\underline{R}_{h}^{k}\underline{E}_{h}^{k}={\rm Id}$ on
$\operatorname{Ker}\underline{\mathrm{d}}_{0,h}^{k}$;

  * (C2)

$(\underline{E}_{h}^{k}\underline{R}_{h}^{k}-{\rm
Id})(\operatorname{Ker}\underline{\mathrm{d}}_{r,h}^{k})\subset\operatorname{Im}\underline{\mathrm{d}}_{r,h}^{k-1}$;

  * (C3)

The graded maps $\underline{E}_{h}^{\bullet}$ and
$\underline{R}_{h}^{\bullet}$ are cochain maps.

We start by noticing that, since $\mathrm{DDR}(0)$ is already known to be a
complex, (C1) and (C3) imply (3.34). Indeed, (C3) gives
$\underline{R}_{h}^{k+1}\underline{\mathrm{d}}_{r,h}^{k}\underline{E}_{h}^{k}=\underline{R}_{h}^{k+1}\underline{E}_{h}^{k+1}\underline{\mathrm{d}}_{0,h}^{k}$
and, by the complex property,
$\operatorname{Im}\underline{\mathrm{d}}_{0,h}^{k}\subset\operatorname{Ker}\underline{\mathrm{d}}_{0,h}^{k+1}$,
so (C1) applied to $k+1$ instead of $k$ yields (3.34).
1\. _Proof of (C1)._ The definitions (3.31) and (3.32a) of the reduction and
the extension components on the lowest dimensional cells directly shows that
$\underline{R}_{h}^{k}\underline{E}_{h}^{k}={\rm Id}$ on
$\underline{X}_{r,h}^{k}$, which establishes a stronger result than (C1).
2\. _Proof of (C3) for the extension._ We now turn to (C3), considering first
the case of the extension. We have to show that, for all
$\underline{\eta}_{h}\in\underline{X}_{0,h}^{k}$ it holds
$\underline{\mathrm{d}}_{r,h}^{k}\underline{E}_{h}^{k}\underline{\eta}_{h}=\underline{E}_{h}^{k+1}\underline{\mathrm{d}}_{0,h}^{k}\underline{\eta}_{h}$.
Given the definitions (3.7) of the global discrete exterior derivative and of
the extension, this boils down to showing that

$\star^{-1}\pi_{r,f}^{-,d-k-1}(\star\mathrm{d}_{r,f}^{k}\underline{E}_{f}^{k}\underline{\eta}_{f})=\star^{-1}E_{f}^{k+1}\underline{\mathrm{d}}_{0,f}^{k}\underline{\eta}_{f}\qquad\forall
f\in\Delta_{d}(\mathcal{M}_{h})\mbox{ with }d\geq k+1$

which, testing against $\rho\in\mathcal{P}_{r}^{-}\Lambda^{d-k-1}(f)$ and
recalling the relation (2.4), can be recast as

$\displaystyle\int_{f}\mathrm{d}_{r,f}^{k}\underline{E}_{f}^{k}\underline{\eta}_{f}\wedge\rho=\int_{f}\star^{-1}E_{f}^{k+1}\underline{\mathrm{d}}_{0,f}^{k}\underline{\eta}_{f}\wedge\rho{}$
$\displaystyle\forall f\in\Delta_{d}(\mathcal{M}_{h})\mbox{ with }d\geq
k+1\,,$ (3.35)
$\displaystyle\forall\rho\in\mathcal{P}_{r}^{-}\Lambda^{d-k-1}(f).$

We start by noticing that, by (3.33), the relation (3.35) holds for
$\rho\in\mathcal{K}_{r}^{d-k-1}(f)$. The decompositions (2.7a) of
$\mathcal{P}_{r}\Lambda^{0}(f)=\mathcal{P}_{r}^{-}\Lambda^{0}(f)$ (if $d=k+1$)
and (2.16) of $\mathcal{P}_{r}^{-}\Lambda^{d-k-1}(f)$ (if $d\geq k+2$) then
show that we only have to prove (3.35) for
$\rho\in\mathcal{P}_{0}\Lambda^{0}(f)$ (if $d=k+1$) or
$\rho\in\mathrm{d}\mathcal{K}_{r}^{d-k-2}(f)$ (if $d\geq k+2$). This fact is
proved by induction on $d$:

  * •

Let us first consider $d=k+1$ and take $\rho\in\mathcal{P}_{0}\Lambda^{0}(f)$.
We can use this polynomial form as a test function in the definition (3.4) of
$\mathrm{d}_{0,f}^{k}\underline{\eta}_{f}$ to get

$\int_{\partial f}P_{0,\partial f}^{k}\underline{\eta}_{\partial
f}\wedge\operatorname{tr}_{\partial
f}\rho=\int_{f}\mathrm{d}_{0,f}^{k}\underline{\eta}_{f}\wedge\rho=\int_{f}\star^{-1}E_{f}^{k+1}\underline{\mathrm{d}}_{0,f}^{k}\underline{\eta}_{f}\wedge\rho\qquad\forall\rho\in\mathcal{P}_{0}\Lambda^{0}(f),$
(3.36)

where the second equality follows from (3.32a) with
$(k,\underline{\eta}_{f})\leftarrow(k+1,\underline{\mathrm{d}}_{0,f}^{k}\underline{\eta}_{f})$.
For all $f^{\prime}\in\Delta_{k}(f)$, by definition (3.3) of
$P_{0,f^{\prime}}^{k}$ and (3.32a) of $E_{f^{\prime}}^{k}$, we have
$P_{0,f^{\prime}}^{k}\underline{\eta}_{f^{\prime}}=\star^{-1}\eta_{f^{\prime}}=\star^{-1}E_{f^{\prime}}^{k}\underline{\eta}_{f^{\prime}}=P_{r,f^{\prime}}^{k}\underline{E}_{f^{\prime}}^{k}\underline{\eta}_{f^{\prime}}$,
where the last relation follows applying the definition (3.3) of
$P_{r,f^{\prime}}^{k}$. We infer from this equality and (3.36) that

$\int_{\partial f}P_{r,\partial f}^{k}\underline{E}_{\partial
f}^{k}\underline{\eta}_{\partial f}\wedge\operatorname{tr}_{\partial
f}\rho=\int_{f}\star^{-1}E_{f}^{k+1}\underline{\mathrm{d}}_{0,f}^{k}\underline{\eta}_{f}\wedge\rho\qquad\forall\rho\in\mathcal{P}_{0}\Lambda^{0}(f).$

Applying the definition (3.4) of
$\mathrm{d}_{r,f}^{k}\underline{E}_{f}^{k}\underline{\eta}_{\partial f}$ with
$\mu=\rho$ (which satisfies $\mathrm{d}\rho=0$) to the left-hand side then
concludes the proof of (3.35).

  * •

We now take $d\geq k+2$ and $\rho\in\mathrm{d}\mathcal{K}_{r}^{d-k-2}(f)$,
which we write $\rho=\mathrm{d}\alpha$ with
$\alpha\in\mathcal{K}_{r}^{d-k-2}(f)\subset\mathcal{P}_{r}^{-}\Lambda^{d-k-2}(f)$.
Applying the link (3.12) between discrete exterior derivatives on $f$ and
$\partial f$, we have

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{E}_{f}^{k}\underline{\eta}_{f}\wedge\rho=(-1)^{k+1}\int_{\partial
f}\mathrm{d}_{r,\partial f}^{k}\underline{E}_{\partial
f}^{k}\underline{\eta}_{\partial f}\wedge\operatorname{tr}_{\partial
f}\alpha.$ (3.37)

By Lemma 4, for all $f^{\prime}\in\Delta_{d-1}(f)$,
$\operatorname{tr}_{f^{\prime}}\alpha\in\mathcal{P}_{r}^{-}\Lambda^{d-k-2}(f^{\prime})$,
so we can apply (3.35) on $f^{\prime}$ (by the induction hypothesis) to get

$\int_{f^{\prime}}\mathrm{d}_{r,f^{\prime}}^{k}\underline{E}_{f^{\prime}}^{k}\underline{\eta}_{f^{\prime}}\wedge\operatorname{tr}_{f^{\prime}}\alpha=\int_{f^{\prime}}\star^{-1}E_{f^{\prime}}^{k+1}\underline{\mathrm{d}}_{0,f^{\prime}}^{k}\underline{\eta}_{f^{\prime}}\wedge\operatorname{tr}_{f^{\prime}}\alpha=\int_{f^{\prime}}P_{r,f^{\prime}}^{k+1}\underline{E}_{f^{\prime}}^{k+1}\underline{\mathrm{d}}_{0,f^{\prime}}^{k}\underline{\eta}_{f^{\prime}}\wedge\operatorname{tr}_{f^{\prime}}\alpha,$

the second equality being justified by (3.20) and (2.4) (with
$(\mathcal{X},f,d,k)\leftarrow(\mathcal{P}_{r}^{-}\Lambda^{(d-1)-(k+1)}(f^{\prime}),f^{\prime},d-1,k+1)$)
and the fact that
$\operatorname{tr}_{f^{\prime}}\alpha\in\mathcal{P}_{r}^{-}\Lambda^{(d-1)-(k+1)}(f^{\prime})$.
Plugging this relation into (3.37) yields

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{E}_{f}^{k}\underline{\eta}_{f}\wedge\rho=(-1)^{k+1}\int_{\partial
f}P_{r,\partial f}^{k+1}\underline{E}_{\partial
f}^{k+1}\underline{\mathrm{d}}_{0,\partial f}^{k}\underline{\eta}_{\partial
f}\wedge\operatorname{tr}_{\partial f}\alpha.$

Invoking then the definition (3.32b) of
$E_{f}^{k+1}\underline{\mathrm{d}}_{0,f}^{k}\underline{\eta}_{f}$ with
$(k,\mu,\nu,\underline{\eta}_{f})\leftarrow(k+1,\alpha,0,\underline{\mathrm{d}}_{0,f}\underline{\eta}_{f})$,
and using the property
$\mathrm{d}_{0,f}^{k+1}\circ\underline{\mathrm{d}}_{0,f}^{k}=0$ (consequence
of (3.15) with $k+1$ instead of $k$) we infer

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{E}_{f}^{k}\underline{\eta}_{f}\wedge\rho=\int_{f}\star^{-1}E_{f}^{k+1}\underline{\mathrm{d}}_{0,f}^{k}\underline{\eta}_{f}\wedge\mathrm{d}\alpha$

and (3.35) follows by recalling that $\rho=\mathrm{d}\alpha$.

3\. _Proof of (C3) for the reduction._ To conclude the proof of (C3), it
remains to show that
$\underline{R}_{h}^{k+1}\underline{\mathrm{d}}_{r,h}^{k}\underline{\omega}_{h}=\underline{\mathrm{d}}_{0,h}^{k}\underline{R}_{h}^{k}\underline{\omega}_{h}$
for all $\underline{\omega}_{h}\in\underline{X}_{r,h}^{k}$. Since vectors in
$\underline{X}_{0,h}^{k+1}$ only have constant components on cells of
dimension $k+1$, and since $\underline{R}_{h}^{k+1}$ is defined by (3.31), we
only have to show that

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\wedge\rho=\int_{f}\mathrm{d}_{0,f}^{k}\underline{R}_{f}^{k}\underline{\omega}_{f}\wedge\rho\qquad\forall
f\in\Delta_{k+1}(\mathcal{M}_{h})\,,\;\forall\rho\in\mathcal{P}_{0}\Lambda^{0}(f).$
(3.38)

Let $\rho$ as above and apply the definition (3.4) of
$\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}$ to $\mu=\rho$; accounting for
$\mathrm{d}\rho=0$, we obtain

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\wedge\rho=\int_{\partial
f}P_{r,\partial f}^{k}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial f}\rho.$ (3.39)

For each $f^{\prime}\in\Delta_{k}(f)$, by definition (3.3) of
$P_{r,f^{\prime}}^{k}$, we can write

$\int_{f^{\prime}}P_{r,f^{\prime}}^{k}\underline{\omega}_{f}\wedge\operatorname{tr}_{f^{\prime}}\rho=\int_{f^{\prime}}\star^{-1}\omega_{f}\wedge\operatorname{tr}_{f^{\prime}}\rho=\int_{f^{\prime}}\star^{-1}\pi_{0,f}^{0}\omega_{f}\wedge\operatorname{tr}_{f^{\prime}}\rho=\int_{f^{\prime}}P_{0,f^{\prime}}^{k}\underline{R}_{f^{\prime}}^{k}\underline{\omega}_{f^{\prime}}\wedge\operatorname{tr}_{f^{\prime}}\rho,$
(3.40)

where we have used the fact that
$\operatorname{tr}_{f^{\prime}}\rho\in\mathcal{P}_{0}\Lambda^{0}(f^{\prime})$
to insert the projector in the second equality and the definitions (3.31) of
$\underline{R}_{f^{\prime}}^{k}$ and (3.3) of $P_{0,f^{\prime}}^{k}$ to
conclude. Combining (3.39) and (3.40), we find

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\wedge\rho=\int_{\partial
f}P_{0,\partial f}^{k}\underline{R}_{\partial
f}^{k}\underline{\omega}_{\partial f}\wedge\operatorname{tr}_{\partial
f}\rho.$

Applying the definition (3.4) of
$\mathrm{d}_{0,f}^{k}\underline{R}_{f}^{k}\underline{\omega}_{f}$ then
concludes the proof of (3.38).
4\. _Proof of (C2)._ Finally, to prove (C2), we notice that if
$\underline{\omega}_{h}\in\underline{X}_{r,h}^{k}$, then by (3.31) and (3.32a)
the components of
$\underline{E}_{h}^{k}\underline{R}_{h}^{k}\underline{\omega}_{h}$ on the
lowest dimensional cells $f\in\Delta_{k}(\mathcal{M}_{h})$ are just the
averages of the components of $\underline{\omega}_{h}$ on these cells; hence,
$\underline{E}_{h}^{k}\underline{R}_{h}^{k}\underline{\omega}_{h}-\underline{\omega}_{h}\in\underline{X}_{r,h,\flat}^{k}$.
Moreover, by the cochain map property (C3),
$\underline{\mathrm{d}}_{r,h}^{k}(\underline{E}_{h}^{k}\underline{R}_{h}^{k}\underline{\omega}_{h}-\underline{\omega}_{h})=\underline{E}_{h}^{k}\underline{R}_{h}^{k}\underline{\mathrm{d}}_{r,h}^{k}\underline{\omega}_{h}-\underline{\mathrm{d}}_{r,h}^{k}\underline{\omega}_{h}=\underline{0}$
whenever
$\underline{\omega}_{h}\in\operatorname{Ker}\underline{\mathrm{d}}_{r,h}^{k}$.
We can thus, for such an $\underline{\omega}_{h}$, apply Lemma 19 with
$\underline{\omega}_{h}\leftarrow\underline{E}_{h}^{k}\underline{R}_{h}^{k}\underline{\omega}_{h}-\underline{\omega}_{h}$
to see that this element belongs to
$\operatorname{Im}\underline{\mathrm{d}}_{r,h}^{k-1}$, establishing (C2). ∎

##  4 A VEM-inspired complex

In this section we consider an alternative construction inspired by the
Virtual Element complex of [8]. Notice that we make here no effort to reduce
the polynomial degree of certain components of the discrete spaces, which is
known to be possible; see, e.g., [9] and also [35] for a general framework
with application to DDR methods. Notice also that we work in a fully discrete
spirit, without attempting to identify the underlying virtual spaces (which
are not needed for the purposes of the present work).

Let again a polynomial degree $r\geq 0$ be fixed. The general principle to
design the VEM-inspired sequence is to select polynomial components that make
it possible to reconstruct, for each $d$-cell and inductively on the dimension
$d$, a discrete potential capable of reproducing polynomial forms in
$\mathcal{P}_{r+1}^{-}\Lambda^{k}(f)$. The main difference with respect to the
DDR approach illustrated in Section 3 is that, with the exception of
$(k+1)$-cells, the required information on the discrete exterior derivative is
directly encoded in the discrete spaces.

Adopting this approach has several, far-reaching, consequences. The first one
is that the discrete spaces contain a mix of both traces and exterior
derivatives (which, in passing, requires higher regularity in the definition
of the interpolators). The components on $k$\- and $(k+1)$-cells in the
discrete space of $k$-forms play a slightly different role than the others
(and are, as a result, treated separately in the definition of the space). The
second consequence is that the proofs of key properties (polynomial
consistency, cohomology, etc.) are carried out by induction on the dimension
(and not on the difference between the dimension and the form degree, as in
Theorems 11 and 15). This leads to somewhat simpler arguments, at the cost of
larger discrete spaces. Also, the commutation property is essentially obtained
by definition of the local discrete exterior derivative (with the exception of
lowest-dimensional cells).

###  4.1 Definition

####  4.1.1 Discrete spaces

We define the following discrete counterpart of $H\Lambda^{k}(\Omega)$, $0\leq
k\leq n$:

$\underline{V}_{r,h}^{k}\coloneq\bigtimes_{f\in\Delta_{k}(\mathcal{M}_{h})}\mathcal{P}_{r}\Lambda^{0}(f)\times\bigtimes_{f\in\Delta_{k+1}(\mathcal{M}_{h})}\mathcal{K}_{r+1}^{1}(f)\times\mathcal{K}_{r}^{0}(f)\\\
\times\bigtimes_{d=k+2}^{n}\bigtimes_{f\in\Delta_{d}(\mathcal{M}_{h})}\mathcal{K}_{r+1}^{d-k}(f)\times\mathcal{K}_{r+1}^{d-k-1}(f).$
(4.1)

Notice that, on $(k+1)$-cells, the second component has polynomial degree
reduced by one compared to $d$-cells with $d\geq k+2$, i.e., we have
$\mathcal{K}_{r}^{0}(f)$ instead of $\mathcal{K}_{r+1}^{0}(f)$. A generic
element of $\underline{V}_{r,h}^{k}$ will be denoted by

$\underline{\omega}_{h}=\big{(}(\omega_{f})_{f\in\Delta_{k}(\mathcal{M}_{h})},(\omega_{f},D_{\omega,f})_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k+1,n]}\big{)}.$
(4.2)

The notation $D_{\omega,f}$ is reminescent of the fact that these polynomial
components are interpreted as Hodge stars of exterior derivatives. We refer to
Table 2 for an overview of the polynomial unknowns in
$\underline{V}_{r,f}^{k}$ in dimensions 0 to 3, as well as their vector
proxies.

$k$ $d$ | $0$ | $1$ | $2$ | $3$
---|---|---|---|---
0 | $\mathbb{R}=\mathcal{P}_{r}\Lambda^{0}(f_{0})$ | $\\{0\\}\times\mathcal{K}_{r}^{0}(f_{1})$ | $\\{0\\}\times\mathcal{K}_{r+1}^{1}(f_{2})$ | $\\{0\\}\times\mathcal{K}_{r+1}^{2}(f_{3})$
1 |  | $\mathcal{P}_{r}\Lambda^{0}(f_{1})$ | $\mathcal{K}_{r+1}^{1}(f_{2})\times\mathcal{K}_{r}^{0}(f_{2})$ | $\mathcal{K}_{r+1}^{2}(f_{3})\times\mathcal{K}_{r+1}^{1}(f_{3})$
2 |  |  | $\mathcal{P}_{r}\Lambda^{0}(f_{2})$ | $\mathcal{K}_{r+1}^{1}(f_{3})\times\mathcal{K}_{r}^{0}(f_{3})$
3 |  |  |  | $\mathcal{P}_{r}\Lambda^{0}(f_{3})$
$k$ $d$ | $0$ | $1$ | $2$ | $3$
0 | $\mathbb{R}=\mathcal{P}_{r}(f_{0})$ | $\\{0\\}\times\mathcal{P}_{r}^{\flat}(f_{2})$ | $\\{0\\}\times\boldsymbol{\mathcal{R}}^{\rm c}_{r+1}(f_{2})$ | $\\{0\\}\times\boldsymbol{\mathcal{R}}^{\rm c}_{r+1}(f_{3})$
1 |  | $\mathcal{P}_{r}(f_{1})$ | $\boldsymbol{\mathcal{R}}^{\rm c}_{r+1}(f_{2})\times\mathcal{P}_{r}^{\flat}(f_{2})$ | $\boldsymbol{\mathcal{R}}^{\rm c}_{r+1}(f_{3})\times\boldsymbol{\mathcal{G}}^{\rm c}_{r+1}(f_{3})$
2 |  |  | $\mathcal{P}_{r}(f_{2})$ | $\boldsymbol{\mathcal{G}}^{\rm c}_{r+1}(f_{3})\times\mathcal{P}_{r}^{\flat}(f_{3})$
3 |  |  |  | $\mathcal{P}_{r}(f_{3})$
Table 2: Polynomial components attached to each mesh entity $f_{d}$ of
dimension $d\in\\{0,\ldots,3\\}$ for the space $\underline{V}_{r,h}^{k}$ for
$k\in\\{0,\ldots,3\\}$ (top) and counterparts through vector proxies (bottom).

####  4.1.2 Interpolators

For all integers $0\leq k\leq d\leq n$ and any
$f\in\Delta_{d}(\mathcal{M}_{h})$, the local interpolator is such that, for
all $\omega\in C^{1}\Lambda^{k}(\overline{f})$,

$\displaystyle\underline{I}_{r,f}^{k}\omega\coloneq\Big{(}$
$\displaystyle\big{(}\pi_{r,f^{\prime}}^{0}(\star\operatorname{tr}_{f^{\prime}}\omega)\big{)}_{f^{\prime}\in\Delta_{k}(f)},$
(4.3)
$\displaystyle\big{(}\pi_{r+1,f^{\prime}}^{\mathcal{K},1}(\star\operatorname{tr}_{f^{\prime}}\omega),\pi_{r,f^{\prime}}^{\mathcal{K},0}(\star\operatorname{tr}_{f^{\prime}}\mathrm{d}\omega)\big{)}_{f^{\prime}\in\Delta_{k+1}(f)}\big{)},$
$\displaystyle\big{(}\pi_{r+1,f^{\prime}}^{\mathcal{K},d^{\prime}-k}(\star\operatorname{tr}_{f^{\prime}}\omega),\pi_{r+1,f^{\prime}}^{\mathcal{K},d^{\prime}-k-1}(\star\operatorname{tr}_{f^{\prime}}\mathrm{d}\omega)\big{)}_{f^{\prime}\in\Delta_{d^{\prime}}(f),\,d^{\prime}\in[k+2,d]}\Big{)}.$

######  Remark 21 (Domain of the interpolator).

Owing to the presence of polynomial components that are interpreted as
exterior derivatives (compare (3.2) with (4.3)), the interpolator in the VEM-
inspired construction requires higher regularity of the interpolated functions
compared to the DDR complex presented in Section 3, namely
$C^{1}\Lambda^{k}(\overline{f})$ instead of $C^{0}\Lambda^{k}(\overline{f})$.

####  4.1.3 Global discrete exterior derivative and VEM complex

For all $f\in\Delta_{k+1}(\mathcal{M}_{h})$, we define the _discrete exterior
derivative_
$\mathrm{d}_{r,f}^{k}:\underline{V}_{r,f}^{k}\to\mathcal{P}_{r}\Lambda^{k+1}(f)$
such that, for all $\underline{\omega}_{f}\in\underline{V}_{r,f}^{k}$,

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\wedge(\mu+\nu)=\int_{\partial
f}\star^{-1}\omega_{\partial f}\wedge\operatorname{tr}_{\partial
f}\mu+\int_{f}\star^{-1}D_{\omega,f}\wedge\nu\\\
\forall(\mu,\nu)\in\mathcal{P}_{0}\Lambda^{0}(f)\times\mathcal{K}_{r}^{0}(f),$
(4.4)

where, as before, $\omega_{\partial f}$ is defined by $(\omega_{\partial
f})_{|f^{\prime}}=\omega_{f^{\prime}}\in\mathcal{P}_{r}\Lambda^{0}(f^{\prime})$
for all $f^{\prime}\in\Delta_{k}(f)$. Notice that the above equation defines
$\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}$ uniquely as, by (2.7a), $\mu+\nu$
spans $\mathcal{P}_{r}\Lambda^{0}(f)$ as $(\mu,\nu)$ spans
$\mathcal{P}_{0}\Lambda^{0}(f)\times\mathcal{K}_{r}^{0}(f)$. Moreover, taking
$\mu=0$ and letting $\nu$ span $\mathcal{K}_{r}^{0}(f)$, we infer, using (2.4)
with
$(\mathcal{X},\omega,\mu)\leftarrow(\mathcal{K}_{r}^{0}(f),\mathrm{d}_{r,f}^{k}\underline{\omega}_{f},\nu)$,

$D_{\omega,f}=\pi_{r,f}^{\mathcal{K},0}(\star\mathrm{d}_{r,f}^{k}\underline{\omega}_{f})\qquad\forall
f\in\Delta_{k+1}(\mathcal{M}_{h}).$ (4.5)

Unlike the DDR complex, the construction of a _global discrete exterior
derivative_ for the VEM complex does not require to first reconstruct traces
on lower-dimensional cells, as all the necessary information is encoded in the
polynomial components
$(D_{\omega,f})_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k+2,n]}$ supplemented
by
$(\mathrm{d}_{r,f}^{k}\underline{\omega}_{f})_{f\in\Delta_{k+1}(\mathcal{M}_{h})}$.
More specifically, for all integers $k\in[0,n-1]$, we let
$\underline{\mathrm{d}}_{r,h}^{k}:\underline{V}_{r,h}^{k}\to\underline{V}_{r,h}^{k+1}$
be such that, for all $\underline{\omega}_{h}\in\underline{V}_{r,h}^{k}$,

$\underline{\mathrm{d}}_{r,h}^{k}\underline{\omega}_{h}\coloneq\big{(}(\star\mathrm{d}_{r,f}^{k}\underline{\omega}_{f})_{f\in\Delta_{k+1}(\mathcal{M}_{h})},(D_{\omega,f},0)_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k+2,n]}\big{)}$
(4.6)

(compare with (4.2) and notice the different positioning, compared to
$\underline{\omega}_{h}$, of the polynomial components $D_{\omega,f}$). As for
the DDR complex, we will denote by $\underline{\mathrm{d}}_{r,f}^{k}$ the
restriction of $\underline{\mathrm{d}}_{r,h}^{k}$ to
$f\in\Delta_{d}(\mathcal{M}_{h})$ with $d\in[0,n]$ such that $k\leq d-1$.

The VEM sequence of spaces and operators then reads

${\mathrm{VEM}(r)\coloneq\\{0\\}}$${\underline{V}_{r,h}^{0}}$${\underline{V}_{r,h}^{1}}$${\cdots}$${\underline{V}_{r,h}^{n-1}}$${\underline{V}_{r,h}^{n}}$${\\{0\\}.}$$\scriptstyle{\underline{\mathrm{d}}_{r,h}^{0}}$$\scriptstyle{\underline{\mathrm{d}}_{r,h}^{n-1}}$
(4.7)

####  4.1.4 Local discrete potentials and discrete exterior derivatives

Given a form degree $k\in[0,n]$, for all $f\in\Delta_{d}(\mathcal{M}_{h})$,
$k\leq d\leq n$, we define the _local discrete potential_
$P_{r,f}^{k}:\underline{V}_{r}^{k}(f)\to\mathcal{P}_{r+1}^{-}\Lambda^{k}(f)$
by induction on $d$ as follows: For all
$\underline{\omega}_{f}\in\underline{V}_{r,f}^{k}$,

  * •

If $d=k$, we simply set

$P_{r,f}^{k}\underline{\omega}_{f}\coloneq\star^{-1}\omega_{f}\in\mathcal{P}_{r}\Lambda^{d}(f)=\mathcal{P}_{r+1}^{-}\Lambda^{d}(f),$
(4.8)

where the last equality follows from (2.13) if $d=0$ (after noticing that
$\mathcal{P}_{r}\Lambda^{d}(f)\cong\mathbb{R}\cong\mathcal{P}_{r+1}^{-}\Lambda^{d}(f)$)
and from (2.15) if $d\geq 1$;

  * •

If $k+1\leq d\leq n$, using the isomorphism (2.17) with $\ell=d-k\geq 1$ and
$r$ replaced by $r+1$, we define
$P_{r,f}^{k}\underline{\omega}_{f}\in\mathcal{P}_{r+1}^{-}\Lambda^{k}(f)$ as
the unique solution of the following equation:

$(-1)^{k+1}\int_{f}P_{r,f}^{k}\underline{\omega}_{f}\wedge(\mathrm{d}\mu+\nu)=\int_{f}\star^{-1}\widetilde{D}_{\omega,f}\wedge\mu-\int_{\partial
f}P_{r,\partial f}^{k}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial
f}\mu+(-1)^{k+1}\int_{f}\star^{-1}\omega_{f}\wedge\nu\\\
\forall(\mu,\nu)\in\mathcal{K}_{r+1}^{d-k-1}(f)\times\mathcal{K}_{r+1}^{d-k}(f),$
(4.9)

where

$\widetilde{D}_{\omega,f}\coloneq\begin{cases}\star\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}&\text{if
$d=k+1$,}\\\ D_{\omega,f}&\text{if $d\geq k+2$,}\end{cases}$ (4.10)

and we have introduced the piecewise polynomial boundary potential
$P_{r,\partial f}^{k}:\underline{V}_{r,\partial f}^{k}\to\Lambda^{k}(\partial
f)$ such that $(P_{r,\partial f}^{k})_{|f^{\prime}}\coloneq
P_{r,f^{\prime}}^{k}$ for all $f^{\prime}\in\Delta_{d-1}(f)$.

Leveraging the above-defined discrete potentials, we can define the _discrete
exterior derivative_
$\mathrm{d}_{r,f}^{k}:\underline{V}_{r,f}^{k}\to\mathcal{P}_{r}\Lambda^{k+1}(f)$
for all $f\in\Delta_{d}(\mathcal{M}_{h})$, $k+2\leq d\leq n-1$ (this object
was previously only defined for $d=k+1$, see (4.4)), setting:

$\mathrm{d}_{r,f}^{k}\underline{\omega}_{f}\coloneq
P_{r,f}^{k+1}\underline{\mathrm{d}}_{r,f}^{k}\underline{\omega}_{f}\qquad\forall\underline{\omega}_{f}\in\underline{V}_{r,f}^{k}.$
(4.11)

Notice that these discrete exterior derivatives are not relevant in the
definition of the VEM complex, but may be useful in practical applications.

###  4.2 Main properties of the VEM complex

The main results for the VEM complex are stated below.

######  Theorem 22 (Cohomology of the VEM complex).

The VEM sequence (4.7) is a complex and its cohomology is isomorphic to the
cohomology of the continuous de Rham complex (1.2).

###### Proof.

See Section 4.6. ∎

######  Theorem 23 (Polynomial consistency of the discrete potential and
exterior derivative).

For all integers $0\leq k\leq d\leq n$ and all
$f\in\Delta_{d}(\mathcal{M}_{h})$, it holds

$P_{r,f}^{k}\underline{I}_{r,f}^{k}\omega=\omega\qquad\forall\omega\in\mathcal{P}_{r+1}^{-}\Lambda^{k}(f),$
(4.12)

and, if $d\geq k+1$,

$\mathrm{d}_{r,f}^{k}\underline{I}_{r,f}^{k}\omega=\mathrm{d}\omega\qquad\forall\omega\in\mathcal{P}_{r+1}^{-}\Lambda^{k}(f).$
(4.13)

###### Proof.

See Section 4.5. ∎

###  4.3 Complex property

######  Lemma 24 (Complex property).

The sequence (4.7) defines a complex, i.e., for all integers $k\in[1,n-1]$ and
all $\underline{\omega}_{h}\in\underline{V}_{r,h}^{k-1}$,

$\underline{\mathrm{d}}_{r,h}^{k}(\underline{\mathrm{d}}_{r,h}^{k-1}\underline{\omega}_{h})=\underline{0}.$

###### Proof.

Applying the definition (4.6) of the global discrete exterior derivative first
for $k-1$ then for $k$, we obtain

$\underline{\mathrm{d}}_{r,h}^{k-1}\underline{\omega}_{h}=\left((\star\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f})_{f\in\Delta_{k}(\mathcal{M}_{h})},(D_{\omega,f},0)_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k+1,n]}\right)\in\underline{V}_{r,h}^{k},$
(4.14)

which shows that, for all $d\in[k+1,n]$ and all
$f\in\Delta_{d}(\mathcal{M}_{h})$, the exterior derivative components of
$\underline{\mathrm{d}}_{r,h}^{k-1}\underline{\omega}_{h}$ are zero, and thus
that

$\underline{\mathrm{d}}_{r,h}^{k}(\underline{\mathrm{d}}_{r,h}^{k-1}\underline{\omega}_{h})=\left(\big{(}\star\mathrm{d}_{r,f}^{k}(\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f})\big{)}_{f\in\Delta_{k+1}(\mathcal{M}_{h})},(0,0)_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k+2,n]}\right)\in\underline{V}_{r,h}^{k+1}.$

The assertion is therefore proved if we show that
$\mathrm{d}_{r,f}^{k}(\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f})=0$
for all $f\in\Delta_{k+1}(\mathcal{M}_{h})$. Applying the definition of the
local discrete exterior derivative (see (4.4)) with $\underline{\omega}_{f}$
replaced by $\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f}$
obtained by restricting (4.14) to $f$, we get: For all
$(\mu,\nu)\in\mathcal{P}_{0}\Lambda^{0}(f)\times\mathcal{K}_{r}^{0}(f)$,

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{\mathrm{d}}_{r,f}^{k-1}\underline{\omega}_{f}\wedge(\mu+\nu)=\int_{\partial
f}\mathrm{d}_{r,\partial f}^{k-1}\underline{\omega}_{\partial
f}\wedge\operatorname{tr}_{\partial f}\mu=0,$

where the conclusion follows using the definition (4.4) of
$\mathrm{d}_{r,f^{\prime}}^{k-1}\underline{\omega}_{f^{\prime}}$ with
$(\mu,\nu)\leftarrow(\operatorname{tr}_{f^{\prime}}\mu,0)$ for all
$f^{\prime}\in\Delta_{k}(f)$ and noticing, as at the end of the proof of Lemma
14, that the sum over $f^{\prime}$ of the integrals over $\partial f^{\prime}$
is zero. ∎

###  4.4 Commutation

######  Proposition 25 (Commutation property for the discrete exterior
derivative in dimension $d=k+1$).

For all $f\in\Delta_{k+1}(\mathcal{M}_{h})$, it holds

$\mathrm{d}_{r,f}^{k}\underline{I}_{r,f}^{k}\omega=\star^{-1}\pi_{r,f}^{0}(\star\mathrm{d}\omega)\qquad\forall\omega\in
C^{1}\Lambda^{k}(\overline{f}),$ (4.15)

expressing the commutativity of the following diagram:

${C^{1}\Lambda^{k}(\overline{f})}$${C^{0}\Lambda^{k+1}(\overline{f})}$${\underline{V}_{r}^{k}(f)}$${\mathcal{P}_{r}\Lambda^{k+1}(f).}$$\scriptstyle{\mathrm{d}}$$\scriptstyle{\underline{I}_{r,f}^{k}}$$\scriptstyle{\star^{-1}\pi_{r,f}^{0}\star}$$\scriptstyle{\mathrm{d}_{r,f}^{k}}$

###### Proof.

Plugging the definition (4.3) of the interpolator into (4.4) we get, for all
$(\mu,\nu)\in\mathcal{P}_{0}\Lambda^{0}(f)\times\mathcal{K}_{r}^{0}(f)$,

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{I}_{r,f}^{k}\omega\wedge(\mu+\nu)=\int_{\partial
f}\star^{-1}\pi_{r,\partial f}^{0}(\star\operatorname{tr}_{\partial
f}\omega)\wedge\operatorname{tr}_{\partial
f}\mu+\int_{f}\star^{-1}\pi_{r,f}^{\mathcal{K},0}(\star\mathrm{d}\omega)\wedge\nu,$

where $\pi_{r,\partial f}^{0}$ denotes the piecewise $L^{2}$-orthogonal
projector obtained patching together the $\pi_{r,f^{\prime}}^{0}$,
$f^{\prime}\in\Delta_{k}(f)$. Using (2.4) with
$(\mathcal{X},\omega,\mu)\leftarrow(\mathcal{K}_{r}^{0}(f^{\prime}),\mathrm{d}\omega,\nu)$
for the second term and, for each $f^{\prime}\in\Delta_{k}(f)$,
$(\mathcal{X},d,f)\leftarrow(\mathcal{P}_{r}\Lambda^{0}(f^{\prime}),k,f^{\prime})$
for the first term, the projectors can be removed. The Stokes formula (2.1)
along with $\mathrm{d}\mu=0$ (since $\mu$ is constant) then yields

$\int_{f}\mathrm{d}_{r,f}^{k}\underline{I}_{r,f}^{k}\omega\wedge(\mu+\nu)=\int_{f}\mathrm{d}\omega\wedge\mu+\int_{f}\mathrm{d}\omega\wedge\nu=\int_{f}\star^{-1}\pi_{r,f}^{0}(\star\mathrm{d}\omega)\wedge(\mu+\nu),$

where the conclusion follows from (2.4) with
$(\mathcal{X},\omega,\mu)\leftarrow(\mathcal{P}_{r}\Lambda^{0}(f),\mathrm{d}\omega,\mu+\nu)$.
Since, by (2.7a), $\mu+\nu$ spans $\mathcal{P}_{r}\Lambda^{0}(f)$ as
$(\mu,\nu)$ spans $\mathcal{P}_{0}\Lambda^{0}(f)\times\mathcal{K}_{r}^{0}(f)$,
this concludes the proof. ∎

######  Proposition 26 (Commutation property for the local discrete exterior
derivative).

For all integers $d\in[1,n]$ and $k\leq d-1$, and all
$f\in\Delta_{d}(\mathcal{M}_{h})$, it holds

$\underline{\mathrm{d}}_{r,f}^{k}(\underline{I}_{r,f}^{k}\omega)=\underline{I}_{r,f}^{k+1}(\mathrm{d}\omega)\qquad\forall\omega\in
C^{2}\Lambda^{k}(\overline{f}),$ (4.16)

expressing the commutativity of the following diagram:

${C^{2}\Lambda^{k}(\overline{f})}$${C^{1}\Lambda^{k+1}(\overline{f})}$${\underline{V}_{r}^{k}(f)}$${\underline{V}_{r}^{k+1}(f).}$$\scriptstyle{\mathrm{d}}$$\scriptstyle{\underline{I}_{r,f}^{k}}$$\scriptstyle{\underline{I}_{r,f}^{k+1}}$$\scriptstyle{\underline{\mathrm{d}}_{r,f}^{k}}$

###### Proof.

Immediate consequence of (4.15) along with the definition (4.3) of the
interpolator, and the property $\mathrm{d}\circ\mathrm{d}=0$. ∎

###  4.5 Polynomial consistency

###### Proof of Theorem 23.

The proof proceeds by induction on the dimension $d$. When $d=k$, (4.12) is a
direct consequence of the definitions (4.8) of the potential and (4.3) of the
interpolator, which give
$P_{r,f}^{k}\underline{I}_{r,f}^{k}\omega=\star^{-1}\pi_{r,f}^{0}(\star\omega)=\omega$,
where, to remove the projector, we have used the fact that
$\star\omega\in\mathcal{P}_{r}\Lambda^{0}(f)$, since
$\omega\in\mathcal{P}_{r+1}^{-}\Lambda^{d}(f)=\mathcal{P}_{r}\Lambda^{d}(f)$
(see (2.15) with $r+1$ instead of $r$).

We next prove (4.12) for $d\geq k+1$ assuming that it holds for $d-1$. Writing
the definition (4.9) of the potential for
$\underline{\omega}_{f}=\underline{I}_{r,f}^{k}\omega$, we get, for all
$(\mu,\nu)\in\mathcal{K}_{r+1}^{d-k-1}(f)\times\mathcal{K}_{r+1}^{d-k}(f)$,

$\displaystyle(-1)^{k+1}\int_{f}P_{r,f}^{k}(\underline{I}_{r,f}^{k}\omega)\wedge(\mathrm{d}\mu+\nu)$
(4.17)
$\displaystyle\qquad=\int_{f}\star^{-1}\widetilde{D}_{\omega,f}\wedge\mu-\int_{\partial
f}P_{r,\partial f}^{k}(\underline{I}_{r,\partial
f}^{k}\operatorname{tr}_{\partial f}\omega)\wedge\operatorname{tr}_{\partial
f}\mu+(-1)^{k+1}\int_{f}\star^{-1}(\pi_{r+1,f}^{\mathcal{K},d-k}\star\omega)\wedge\nu$
$\displaystyle\qquad=\int_{f}\star^{-1}\widetilde{D}_{\omega,f}\wedge\mu-\int_{\partial
f}\operatorname{tr}_{\partial f}\omega\wedge\operatorname{tr}_{\partial
f}\mu+(-1)^{k+1}\int_{f}\omega\wedge\nu,$

where we have used the induction hypothesis for the second term in the right-
hand side after noticing that, by Lemma 4 with $\ell=k$,
$\operatorname{tr}_{f^{\prime}}\omega\in\mathcal{P}_{r+1}^{-}\Lambda^{k}(f^{\prime})$
for all $f^{\prime}\in\Delta_{d-1}(f)$, together with (2.4) for the third one.
Recalling the definition (4.10) of $\widetilde{D}_{\omega,f}^{k}$, we
distinguish two cases for the first term in the right-hand side. If $d=k+1$,
(4.13) (immediate consequence of (4.15) after observing that
$d\mathcal{P}_{r+1}^{-}\Lambda^{k}(f)\subset\mathcal{P}_{r}\Lambda^{k+1}(f)$)
gives
$\star^{-1}\widetilde{D}_{\omega,f}=\star^{-1}\star\mathrm{d}_{r,f}^{k}(\underline{I}_{r,f}^{k}\omega)=\mathrm{d}\omega$.
If, on the other hand, $d\geq k+2$, recalling the definition (4.3) of the
interpolator, we have
$\int_{f}\star^{-1}\widetilde{D}_{\omega,f}\wedge\mu=\int_{f}\star^{-1}(\pi_{r+1,f}^{\mathcal{K},d-k-1}\star\mathrm{d}\omega)\wedge\mu\stackrel{{\scriptstyle\eqref{eq:remove.projector}}}{{=}}\int_{f}\mathrm{d}\omega\wedge\mu$.
Plugging these relations into (4.17), using the Stokes formula (2.1), and
simplifying, we get

$\int_{f}P_{r,f}^{k}(\underline{I}_{r,f}^{k}\omega)\wedge(\mathrm{d}\mu+\nu)=\int_{f}\omega\wedge(\mathrm{d}\mu+\nu),$

which yields (4.12) for $d\geq k+1$ since, by (2.17) with $\ell=d-k\geq 1$,
$\mathrm{d}\mu+\nu$ spans $\mathcal{P}_{r+1}^{-}\Lambda^{d-k}(f)$ as
$(\mu,\nu)$ spans
$\mathcal{K}_{r+1}^{d-k-1}(f)\times\mathcal{K}_{r+1}^{d-k}(f)$.

We have already seen above that (4.13) holds for $d=k+1$. To prove this
relation for $d\geq k+2$, it suffices to recall (4.11) and (4.16) to write

$\mathrm{d}_{r,f}^{k}\underline{I}_{r,f}^{k}\omega=P_{r,f}^{k+1}(\underline{\mathrm{d}}_{r,f}^{k}\underline{I}_{r,f}^{k}\omega)=P_{r,f}^{k+1}(\underline{I}_{r,f}^{k+1}\mathrm{d}\omega)=\mathrm{d}\omega,$

where the conclusion follows from (4.12) after observing that
$\mathrm{d}\omega\in\mathcal{P}_{r}\Lambda^{k+1}(f)\subset\mathcal{P}_{r+1}^{-}\Lambda^{k+1}(f)$.
∎

###  4.6 Cohomology

As in Section 3.5, given a form degree $k\in[0,n]$, we first consider the
following subspace of $\underline{V}_{r,h}^{k}$:

$\underline{V}_{r,h,\flat}^{k}\coloneq\left\\{\underline{\omega}_{h}=\big{(}(\omega_{f})_{f\in\Delta_{k}(\mathcal{M}_{h})},(\omega_{f},D_{\omega,f})_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k+1,n]}\big{)}\,:\,\int_{f}\star^{-1}\omega_{f}=0\quad\forall
f\in\Delta_{k}(\mathcal{M}_{h})\right\\}.$

######  Lemma 27 (Exactness property for $\underline{V}_{r,h,\flat}^{k}$).

For all $k\in[0,n]$, if $\underline{\eta}_{h}\in\underline{V}_{r,h,\flat}^{k}$
satisfies
$\underline{\mathrm{d}}_{r,h}^{k}\underline{\eta}_{h}=\underline{0}$, then
there exists $\underline{\omega}_{h}\in\underline{V}_{r,h,\flat}^{k-1}$ such
that
$\underline{\eta}_{h}=\underline{\mathrm{d}}_{r,h}^{k-1}\underline{\omega}_{h}$,
where, in accordance with the sequence (4.7), we have set
$\underline{\mathrm{d}}_{r,h}^{-1}=\underline{\mathrm{d}}_{r,h}^{n}\coloneq
0$.

###### Proof.

Recalling the definition (4.6) of
$\underline{\mathrm{d}}_{r,h}^{k}\underline{\eta}_{h}$, we have

$\underline{\mathrm{d}}_{r,h}^{k}\underline{\eta}_{h}=\big{(}(\star\mathrm{d}_{r,f}^{k}\underline{\eta}_{f})_{f\in\Delta_{k+1}(\mathcal{M}_{h})},(D_{\eta,f},0)_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k+2,n]}\big{)}.$

If $k=0$, then $\int_{f}\star^{-1}\eta_{f}=0$ implies $\eta_{f}=0$ for all
$f\in\Delta_{0}(\mathcal{M}_{h})$; moreover, $\eta_{f}=0$ for all
$f\in\Delta_{d}(\mathcal{M}_{h})$, $d\in[1,n]$, by definition (4.1) of
$\underline{V}_{r,h}^{0}$ (recall that $\mathcal{K}_{r}^{d}(f)=\\{0\\}$ for
all $r$, cf. (2.6)). The condition
$\underline{\mathrm{d}}_{r,h}^{k}\underline{\eta}_{h}=\underline{0}$ together
with (4.5) yields $D_{\omega,f}=0$ for all $f\in\Delta_{d}(\mathcal{M}_{h})$,
$d\geq k+1$, and thus

$\underline{\eta}_{h}=\big{(}(0)_{f\in\Delta_{0}(\mathcal{M}_{h})},(0,0)_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k+1,n]}\big{)}=\underline{\mathrm{d}}_{r,h}^{-1}0.$

If $1\leq k\leq n-1$, on the other hand, from
$\underline{\mathrm{d}}_{r,h}^{k}\underline{\eta}_{h}=\underline{0}$ and (4.5)
we infer

$\underline{\eta}_{h}=\big{(}(\eta_{f})_{f\in\Delta_{k}(\mathcal{M}_{h})},(\eta_{f},0)_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k+1,n]}\big{)},$
(4.18a) while, if $k=n$, we simply have
$\underline{\eta}_{h}=(\eta_{f})_{f\in\Delta_{n}(\mathcal{M}_{h})}.$ (4.18b)

Let now

$\underline{\omega}_{h}=\big{(}(0)_{f\in\Delta_{k-1}(\mathcal{M}_{h})},(0,\pi_{r,f}^{\mathcal{K},0}\eta_{f})_{f\in\Delta_{k}(\mathcal{M}_{h})},(0,\eta_{f})_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\in[k+1,n]}\big{)}\in\underline{V}_{r,h,\flat}^{k-1}.$

To check that this $\underline{\omega}_{h}$ is well defined, it suffices to
notice that, if $f\in\Delta_{d}(\mathcal{M}_{h})$ with $d\geq k+1=(k-1)+2$,
then $\eta_{f}\in\mathcal{K}_{r+1}^{d-k}(f)=\mathcal{K}_{r+1}^{d-(k-1)-1}(f)$
is a suitable choice for the corresponding component of
$\underline{\omega}_{h}$. By definition (4.4) of $\mathrm{d}_{r,f}^{k-1}$, we
have: For all $f\in\Delta_{k}(\mathcal{M}_{h})$ and all
$(\mu,\nu)\in\mathcal{P}_{0}\Lambda^{0}(f)\times\mathcal{K}_{r}^{0}(f)$, since
$\omega_{f^{\prime}}=0$ for all $f^{\prime}\in\Delta_{k}(f)$,

$\int_{f}\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f}\wedge(\mu+\nu)=\int_{f}\star^{-1}\cancel{\pi_{r,f}^{\mathcal{K},0}}\eta_{f}\wedge\nu=\int_{f}\star^{-1}\eta_{f}\wedge(\mu+\nu),$

where the cancellation of $\pi_{r,f}^{\mathcal{K},0}$ is made possible by
(2.4) with
$(\mathcal{X},\omega,\mu)\leftarrow(\mathcal{K}_{r}^{0}(f),\star^{-1}\eta_{f},\nu)$,
while the introduction of $\mu$ in the last passage is justified observing
that $\underline{\eta}_{h}\in\underline{V}_{r,h,\flat}^{k}$ implies
$\int_{f}\star^{-1}\eta_{f}=0$ for all $f\in\Delta_{k}(\mathcal{M}_{h})$. This
relation gives
$\mathrm{d}_{r,f}^{k-1}\underline{\omega}_{f}=\star^{-1}\eta_{f}$ for all
$f\in\Delta_{k}(\mathcal{M}_{h})$ which, combined with the definition (4.6) of
the global discrete exterior derivative and the expression (4.18) of
$\underline{\eta}_{h}$, readily yields
$\underline{\eta}_{h}=\underline{\mathrm{d}}_{r,h}^{k-1}\underline{\omega}_{h}$
and concludes the proof. ∎

###### Proof of Theorem 22.

Contrary to the $\mathrm{DDR}(0)$ complex, the $\mathrm{VEM}(0)$ complex is
not isomorphic to the CW complex (the VEM spaces for $r=0$ do not have only
constant polynomial components on the lowest-dimensional cells). As a
consequence, designing extensions and reductions between the $\mathrm{VEM}(r)$
and $\mathrm{VEM}(0)$ complexes would not directly allow us, as it was the
case for the DDR complex in the proof of Theorem 10, to analyse the cohomology
of the VEM complex. To circumvent this difficulty, we will instead design
extensions
$\underline{E}_{h}^{k}:\underline{X}_{0,h}^{k}\to\underline{V}_{r,h}^{k}$ and
reductions
$\underline{R}_{h}^{k}:\underline{V}_{r,h}^{k}\to\underline{X}_{0,h}^{k}$
between the $\mathrm{VEM}(r)$, $r\geq 0$, and the $\mathrm{DDR}(0)$ complexes,
in order to show that their cohomologies are isomorphic. By Theorem 10, this
will prove that the cohomology of $\mathrm{VEM}(r)$ is isomorphic to the
continuous de Rham cohomology.

Throughout the rest of this proof, $(P_{0,f}^{k},\mathrm{d}^{k}_{0,f})$ and
$(P_{r,f}^{k},\mathrm{d}^{k}_{r,f})$ denote, respectively, the couple
(potential reconstruction, discrete exterior derivative) of the
$\mathrm{DDR}(0)$ and $\mathrm{VEM}(r)$ complexes. We do not need to
differentiate these notations, as the argument removes all ambiguity. For all
form degrees $k\in[0,d]$, the reduction is obtained setting

$\underline{R}_{h}^{k}\underline{\omega}_{h}\coloneq\big{(}(\pi_{0,f}^{0}\omega_{f})_{f\in\Delta_{k}(\mathcal{M}_{h})}\big{)}\qquad\forall\underline{\omega}_{h}\in\underline{V}_{r,h}^{k},$
(4.19)

while the extension is given by

$\underline{E}_{h}^{k}\underline{\eta}_{h}\coloneq\begin{aligned}
\Big{(}&(\eta_{f})_{f\in\Delta_{k}(\mathcal{M}_{h})},\\\
&\big{(}\pi_{r+1,f}^{\mathcal{K},1}(\star
P_{0,f}^{k}\underline{\eta}_{f}),\pi_{r,f}^{\mathcal{K},0}(\star\mathrm{d}^{k}_{0,f}\underline{\eta}_{f})\big{)}_{f\in\Delta_{k+1}(\mathcal{M}_{h})},\\\
&\big{(}\pi_{r+1,f}^{\mathcal{K},d-k}(\star
P_{0,f}^{k}\underline{\eta}_{f}),\pi_{r+1,f}^{\mathcal{K},d-k-1}(\star\mathrm{d}^{k}_{0,f}\underline{\eta}_{f})\big{)}_{f\in\Delta_{d}(\mathcal{M}_{h}),\,d\geq
k+2}\Big{)}\qquad\forall\underline{\eta}_{h}\in\underline{X}_{0,h}^{k}.\end{aligned}$
(4.20)

As in the proof of Theorem 10, we need to establish the properties (C1)–(C3)
of [35, Assumption 1] to obtain the desired isomorphism in cohomology (also in
this case, the relation (3.34) is an immediate consequence of (C1) and (C3)).
_Proof of (C1)._ An inspection of the definitions (4.19) of the reduction and
(4.20) of the extension shows that
$\underline{R}_{h}^{k}\underline{E}_{h}^{k}\underline{\eta}_{h}=\underline{\eta}_{h}$
for all $\underline{\eta}_{h}\in\underline{X}_{0,h}^{k}$, and thus (C1) holds
a fortiori.
_Proof of (C3)._ We need to prove that both the reduction and extension are
cochain maps.

Let us start with the extension. We have to prove that, for any integer
$k\in[0,n-1]$ and all $\underline{\eta}_{h}\in\underline{X}_{r,h}^{k}$,
$\underline{E}_{h}^{k+1}(\underline{\mathrm{d}}^{k}_{0,h}\underline{\eta}_{h})=\underline{\mathrm{d}}^{k}_{r,h}(\underline{E}_{h}^{k}\underline{\eta}_{h})$.
Owing to the definitions (4.20) of the extension, (3.7) of
$\underline{\mathrm{d}}^{k}_{0,f}$, and (4.6) of
$\underline{\mathrm{d}}^{k}_{r,f}$, and since
$\mathrm{d}^{k+1}_{0,f}\circ\underline{\mathrm{d}}^{k}_{0,f}=0$ (by (3.15)
with $r=0$ and $k+1$ instead of $k$) this amounts to proving that