# A Discrete Exterior Calculus of Bundle-valued Forms

Theo Braune
LIX-Ecole Polytechnique, IP Paris, Inria  Email<EMAIL_ADDRESS>Yiying Tong
Michigan State U.  Email<EMAIL_ADDRESS>François Gay-Balmaz
Nanyang Technological University  Email<EMAIL_ADDRESS>Mathieu
Desbrun
Inria, Ecole Polytechnique Email<EMAIL_ADDRESS>
###### Abstract

The discretization of Cartan’s exterior calculus of differential forms has
been fruitful in a variety of theoretical and practical endeavors: from
computational electromagnetics to the development of Finite-Element Exterior
Calculus, the development of structure-preserving numerical tools satisfying
exact discrete equivalents to Stokes’ theorem or the de Rham complex for the
exterior derivative have found numerous applications in computational physics.
However, there has been a dearth of effort in establishing a more general
discrete calculus, this time for differential forms with values in vector
bundles over a combinatorial manifold equipped with a connection. In this
work, we propose a discretization of the exterior covariant derivative of
bundle-valued differential forms. We demonstrate that our discrete operator
mimics its continuous counterpart, satisfies the Bianchi identities on
simplicial cells, and contrary to previous attempts at its discretization,
ensures numerical convergence to its exact evaluation with mesh refinement
under mild assumptions.

Keywords: Discrete exterior calculus, differential forms, vector bundles,
structure-preserving discretization

Mathematical Subject Classification: 53A70

##  1 Introduction

Over the past few decades, structure-preserving finite-dimensional
approximations of Cartan’s exterior calculus such as Finite-Element Exterior
Calculus (FEEC [2]) or Discrete Exterior Calculus (DEC [15]) have proven
effective in fields as varied as computational physics and geometry processing
[12]. At their heart is a discretization of smooth, _scalar-valued_
differential forms as cochains [6]: the natural pairing of a cochain with a
chain (which, itself, provides a discrete notion of domain) then becomes a
discrete analog of the _integration_ of a continuous form over a domain.
Defining the discrete exterior derivative $\boldsymbol{d}$ as the adjoint of
the boundary operator on chains thus leads to a structure-preserving
discretization of the continuous exterior derivative operator $d$ in the sense
that it preserves discrete counterparts to fundamental differential properties
such as Stokes’ theorem or the de Rham cohomology.

However, to date, far less attention has been dedicated to the discretization
of _differential forms with values in a vector bundle_ — even if Cartan’s
original work made extensive use of such bundle-valued forms. One obvious
difficulty is that bundle-valued forms do not seamlessly admit a simple
discretization: indeed, the mere pointwise expression of a bundle-valued form
requires a pointwise fiber, making it difficult to properly define a discrete
counterpart to the exterior covariant derivative where the integration over a
domain can only be achieved in a common fiber. An exception is the case of
discrete connections on two-dimensional surfaces, which has garnered
significant interest in vector field processing [13, 14] to define parallel
transport and connection curvature for instance; but this 2D case can be
handled with connections that are seen as angle-valued forms (since 2D
rotations are commutative) once a section of the frame bundle has been chosen,
while the general case requires the use of discrete forms with values in the
group of rotation matrices, bringing significant difficulties due in part by
their non-commutativity.

Yet, the use of connection curvature is key in fields such as Yang-Mills
theory and relativity in computational physics, since it provides an
invariant, frame-independent measure. It is therefore natural to hope that the
development of structure-preserving discretizations of the exterior calculus
of bundle-valued forms can be beneficial for a wide range of numerical
endeavors. This paper thus develops a formal, readily-computable
discretization of connections, bundle-valued forms, and connection one-forms,
along with a discrete exterior covariant derivative of bundle-valued forms
that not only satisfies the well-known Bianchi identities in this discrete
realm, but also converges to its smooth equivalent under mesh refinement —
thus fixing a recent attempt at defining such an operator on vector-valued
forms[5], based on synthetic differential geometry [22, 23], which fails to be
numerically convergent as we will discuss.

##  2 Smooth Theory of Connections

We begin by first reviewing the geometric concepts that we will discretize,
highlighting the properties we wish to preserve in the discrete realm.

###  2.1 Preliminaries

In the study of an $n$-dimensional manifold $M$, we often encounter a vector
bundle $\pi:E\rightarrow M$, and a local trivialization
$\phi:E|_{U}\rightarrow U\times\mathbb{R}^{r}$ with $U\subset M$ being an open
subset. This provides us with a local frame field $\\{f_{1},...,f_{r}\\}$ on
$U$ associated with the canonical basis $\\{e_{1},...,e_{r}\\}$ of
$\mathbb{R}^{r}$ given by $f_{a}(x)=\phi^{-1}(x,e_{a})$ for each point $x\in
U$. If we switch to another local trivialization
$\widetilde{\phi}:\pi^{-1}(\widetilde{U})=E|_{\widetilde{U}}\rightarrow\widetilde{U}\times\mathbb{R}^{r}$
(where $\widetilde{U}$ is another open subset of $M$), the resulting local
frame $\\{\widetilde{f}_{a}\\}$ relates to the local frame $\\{f_{a}\\}$ via a
matrix of change of bases $A:U\cap\widetilde{U}\rightarrow GL(r)$ through

$f_{a}=\widetilde{f}_{b}A_{a}^{b}.$ (1)

For a section $s\in\Gamma(E)$ of the vector bundle (i.e., an assignment for
each point $p\in M$ of a vector $s(p)$ in the point’s fiber $\pi^{-1}(p)$),
which we can express locally as
$s=f_{a}s^{a}=\widetilde{f}_{a}\widetilde{s}^{a}$, the components $s^{a}$ and
$\widetilde{s}^{a}$ are, instead, related through

$\widetilde{s}^{a}=A^{a}_{b}s^{b}.$

##### Connection.

Given a vector bundle $\pi:E\\!\to\\!M$, a connection $\nabla$ is a bilinear
map $\nabla:\Gamma(TM)\\!\times\\!\Gamma(E)\rightarrow\Gamma(E)$ such that for
$g\in C^{\infty}(M)$, $s\in\Gamma(E)$, and $X\in\Gamma(TM)$, we have

  * (i)

$\nabla_{X}(gs)=g\nabla_{X}\;s+dg(X)\;s$ (i.e., it satisfies Leibniz Rule)

  * (ii)

$\nabla_{(gX)}\;s=g\nabla_{X}\;s$. (i.e., it is tensorial in $X$)

In local frames, where $s=f_{a}s^{a}$, we can write $\nabla
s=f_{a}\otimes(ds^{a}+\omega_{b}^{a}s^{b})$, i.e.,
$\nabla_{X}s=f_{a}(ds^{a}(X)+\omega_{b}^{a}(X)s^{b})$, for all
$X\in\Gamma(TM)$, where $\omega\\!\in\\!\Omega^{1}(U,\mathfrak{gl}(r))$ is the
local connection $1-$form given by $\nabla f_{a}\\!=\\!f_{b}\omega_{a}^{b}$.
Using Eq. (1) we can express the change of the local connection $1-$form
$\widetilde{\omega}$ for another local trivialization through the _gauge
transformation_ :

$\omega=A^{-1}\widetilde{\omega}A+A^{-1}{\rm
d}A\quad\Leftrightarrow\quad\widetilde{\omega}=A\omega A^{-1}-{\rm
d}A\;A^{-1}.$ (2)

This relationship between the local connection one-forms is central to the
understanding of the behavior of the connection under changes of local
trivializations.

A connection $\nabla$ on the vector bundle $\pi:E\rightarrow M$ naturally
induces a connection $\nabla^{\rm End}$ on the endomorphism bundle ${\rm
End}(E)\rightarrow M$, whose fiber at $p$ is the space ${\rm
End}(E_{p})=\operatorname{Hom}(E_{p},E_{p})$ of endomorphisms of $E_{p}$. It
is defined by enforcing the Leibniz rule on $L(x)$,

$\big{(}\nabla^{\rm End}_{X}L\big{)}(s):=\nabla_{X}(L(s))-L(\nabla_{X}s),$ (3)

for all $L\in\Gamma(\operatorname{End}(E))$.

##### Pullback connection.

Given a smooth map $\Psi\colon M\to N$ between two manifolds and a vector
bundle $\pi:E\rightarrow N$ over $N$, we can define the _pullback bundle_ as
$\pi^{\prime}:\Psi^{\ast}E\to M$, $(\Psi^{\ast}E)_{p}:=E_{\Psi(p)}$. Given a
connection $\nabla$ on $E$, we want to equip the pullback bundle with a
pullback connection $\Psi^{\ast}\nabla$ such that, for any given section
$s\in\Gamma E$, we have

$(\Psi^{\ast}\nabla)\Psi^{\ast}s=\Psi^{\ast}(\nabla s).$

Such a connection exists and is called the _pullback connection_. We refer the
reader to [26] for the proof of its existence and uniqueness.

##### Parallel transport.

Parallel transport is a fundamental concept in differential geometry and plays
a key role in understanding the geometry of manifolds. Given a curve
$\gamma:[0,1]\rightarrow M$ and a vector $v_{0}\in E_{\gamma(0)}$ in the fiber
at $\gamma(0)$, parallel transport along $\gamma$ is the process of
transporting $v_{0}$ to $\gamma(t)$ while maintaining its direction in the
vector bundle. The resulting vector at $\gamma(t)$ is denoted by $v(t)$ and is
a solution to the differential equation

$D_{t}v(t)=0\quad\text{and}\quad v(0)=v_{0},$ (4)

where $D_{t}v$ is the covariant time derivative of curves $v(t)\in E$
associated to a connection $\nabla$ on $E$, given locally by

$D_{t}v=f_{a}(\partial_{t}v^{a}+\omega^{a}_{b}(\dot{\gamma})v^{b}).$ (5)

Here, $v^{a}$ are the components of $v$ in a local frame ${f_{a}}$ for the
fiber, and $\omega^{a}_{b}$ are the connection coefficients associated with
$\nabla$ in $f_{a}$.

The unique solution to the initial value problem (4) along $\gamma$ yields a
one-parameter family of parallel transport maps

$\mathcal{R}_{\gamma,t}\colon E_{\gamma(t)}\to E_{\gamma(0)},$

which is an intrinsic object. We will simply write $\mathcal{R}_{t}$ the
parallel transport maps, when the choice of the curve $\gamma$ is obvious from
the context. If we consider a local frame field $f_{a}(x)$ in a neighborhood
of the curve, the linear map can be represented by a matrix $(R_{t})^{a}_{b}$,
via $\mathcal{R}_{t}f_{a}(\gamma(t))=f_{b}(\gamma(0))(R_{t})_{a}^{b}$.
Moreover, according to Eq. (2), the matrix $R_{t}$ is the solution to
$\frac{d}{dt}R_{t}=R_{t}\omega_{\gamma(t)}(\dot{\gamma}(t))$.

The solution to this last equation can be written as the path-ordered matrix
exponential
$P\left(\operatorname{exp}\int_{0}^{t}\omega_{\gamma(\tau)}(\dot{\gamma}(\tau))d\tau\right)$.
As $\omega$ is applied to the right of $R$, we order the terms with larger
$\tau$ to the right. For instance, the second-order term in the power series
of the matrix exponential is

$P\;\frac{1}{2}\left(\int_{0}^{t}\omega_{\gamma(\tau)}(\dot{\gamma}(\tau))\
d\tau\right)^{2}=\int_{\tau_{2}=0}^{t}\int_{\tau_{1}=0}^{\tau_{2}}\omega_{\gamma(\tau_{1})}(\dot{\gamma}(\tau_{1}))\
\omega_{\gamma(\tau_{2})}(\dot{\gamma}(\tau_{2}))\ d\tau_{1}d\tau_{2}.$

By utilizing the parallel transport matrices associated with $(f_{a})$ along
$\gamma$ as a gauge field, we can derive a new frame field

$\widetilde{f}_{a}(\gamma(t))=f_{b}(\gamma(t))(R_{t}^{-1})^{b}_{a}.$

Consequently, in this new frame, $\tilde{\omega}(\dot{\gamma})=0$ by
construction, and a constant $\widetilde{v}\in\mathbb{R}^{r}$ along the curve
satisfies the initial value problem (4), i.e.,
$\widetilde{v}=R_{t}v(t)=v_{0}$. Alternatively, in the original frame
${f_{a}}$, the parallel transport is given by $v(t)=R_{t}^{-1}v_{0}$.

If $X\\!=\\!\dot{\gamma}(0)\in T_{\gamma(0)}M$ and $\alpha\in\Gamma(E)$ is a
section of $E$, then the parallel transport map $\mathcal{R}_{t}$ can be used
to define the covariant derivative of $\alpha$ with respect to $X$ at
$\gamma(0)$ as

$(\nabla_{X}\alpha)_{\gamma(0)}=\lim_{t\to
0}\frac{\mathcal{R}_{t}\alpha_{\gamma(t)}-\alpha_{\gamma(0)}}{t}=\frac{d}{dt}\bigg{|}_{t=0}\mathcal{R}_{t}\alpha_{\gamma(t)}.$
(6)

##### Integration of bundle-valued forms.

Once we are given parallel transport maps, we can define a _connection-
dependent integration of a bundle valued $1-$form_ quite naturally.

######  Definition 2.1.

Let $\pi:E\to M$ be a vector bundle with connection $\nabla$ and let
$\alpha\in\Omega^{1}(M;E)$ be a bundle-valued $1-$form. For a curve
$\gamma\colon[0,1]\to M$, we define:

$\mathop{\mathrlap{\mkern
0.0mu\raisebox{5.5pt}{$\scriptstyle\nabla$}}\int}\limits_{\gamma}\alpha=\mathop{\mathrlap{\mkern
0.0mu\raisebox{5.5pt}{$\scriptstyle\nabla$}}\int}\limits_{[0,1]}\gamma^{\ast}\alpha=\int_{0}^{1}\mathcal{R}_{\gamma,t}\alpha_{\gamma(t)}\left(\dot{\gamma}(t)\right)dt\in
E_{\gamma(0)},$

where $\mathcal{R}_{\gamma,t}\colon E_{\gamma(t)}\to E_{\gamma(0)}$ denotes
the parallel transport along $\gamma$ for a fixed $t$. We note that the
integral gives a vector in the fiber attached to the initial point of the
curve $\gamma$.

We can further extend this notion of connection-dependent integral
$\mathop{\mathrlap{\mkern-1.5mu\raisebox{2.5pt}{$\scriptscriptstyle\nabla$}}\int}\limits$
to an arbitrary $k$-form over a retractable region $S\subset M$: the existence
of a connection allows us to define a proper integration through parallel
transport to bring the final value to a same “evaluation” vector space, as
defined below.

######  Definition 2.2.

Let $\pi:E\to M$ be a vector bundle with connection $\nabla$ and let
$\alpha\in\Omega^{k}(M;E)$ be a bundle valued $k$-form. For a region $S\subset
M,$ with $S$ homeomorphic to a closed $k$-dimensional ball, let $\varphi\colon
S\to B_{k}(0)\subset\mathbb{R}^{k}$ be a homeomorphism from $S$ to the unit
$k$-dimensional ball centered at 0. Furthermore, let
$\tilde{\gamma}_{\varphi(v),\varphi(p)}\colon[0,1]\to\varphi(S)=B_{k}(0)$ be
the straight joining path from $\varphi(p)$ to a chosen point $\varphi(v)$ and
let
$\gamma_{v,p}:=\varphi^{-1}(\tilde{\gamma}_{\varphi(v),\varphi(p)})\colon[0,1]\to
S$ be the induced curve in $S$. For $p\in S$ let

$\mathcal{R}^{\nabla,\varphi_{v}}_{p}\in\mathrm{Hom}(E_{p},E_{v})$ (7)

be the parallel transport along $\gamma_{v,p}$. We denote by
$\mathcal{R}^{\varphi_{v}}$ the induced transport field, i.e., a function
mapping a point $p$ to a linear map between $E_{p}$ and $E_{v}$. We define the
parameterization-induced connection-dependent integral as

$\mathop{\mathrlap{\mkern-10.0mu\raisebox{5.5pt}{$\underset{\scriptscriptstyle\varphi_{v}}{\scriptstyle\nabla}$}}\int}\limits_{S}\alpha=\int_{S}{\mathcal{R}^{\nabla,\varphi_{v}}\alpha}\,\;\;\in\\!E_{v}.$

If the choice of the parameterization is clear from context, we will suppress
the parameterization $\varphi$ in the notation for the integral and denote the
parallel-transport field as $\mathcal{R}^{\nabla,v}$.

The connection-dependent integral can more generally be defined through a
smooth strong deformation retraction $\varphi_{v}\colon[0,1]\times S\to S$
onto $v\in S,$ with the path $\gamma$ defined through
$\gamma_{p}(t)=\varphi_{v}(1-t,p)$, since $\varphi_{v}$ satisfies
$\varphi_{v}(0,p)=p$, $\varphi_{v}(1,p)=v$, and $\varphi_{v}(t,v)=v$.

###  2.2 Operators and their properties

We end this section with a review of the basic operators (and their
properties) for which we will provide discrete equivalents in the remainder of
this paper.

##### Wedge product.

For scalar-valued differential forms on a smooth manifold $M$ there is a well-
defined product on the space of differential forms: given
$\alpha\\!\in\\!\Omega^{k}(M)$ and $\beta\\!\in\\!\Omega^{\ell}(M)$, the
$(k\\!+\\!\ell)-$form $\alpha\wedge\beta\\!\in\\!\Omega^{k+\ell}(M)$ is the
antisymmetrization of the tensor product of the two forms, i.e., for all
$X_{1},\ldots,X_{k+\ell}\in\Gamma(TM)$,

$\displaystyle\alpha\wedge\beta(X_{1},\ldots,X_{k+\ell})$
$\displaystyle=\frac{(k+\ell)!}{k!\ell!}\mathrm{Alt}(\alpha\otimes\beta)(X_{0},\ldots,X_{k+\ell})$
$\displaystyle=\frac{1}{k!\ell!}\sum_{\sigma\in
S_{k+\ell}}\mathrm{sgn}(\sigma)\alpha(X_{\sigma(0)},\ldots,X_{\sigma(k)})\beta(X_{\sigma(k+1)},\ldots,x_{\sigma(k+\ell+1)}),$

where $S_{k+\ell}$ denotes the permutation of $(k\\!+\\!\ell)$ indices.

This notion can be similarly extended to forms that take values in vector
bundles: the wedge product $\wedge$ between bundle-valued forms and scalar-
valued forms can be defined as:

$\wedge\colon\Omega^{k}(M,E)\times\Omega^{\ell}(M)\to\Omega^{k+\ell}(M,E)\colon(\alpha,\beta)\mapsto\alpha\wedge\beta=\frac{(k+\ell)!}{k!\ell!}\mathrm{Alt}(\alpha\otimes\beta),$

where the tensor product applied to $(k\\!+\\!\ell)$ vectors
$X_{1},\ldots,X_{k+\ell}\\!\in\\!\Gamma(TM)$ is

$\alpha\otimes\beta(X_{1},\ldots,X_{k+\ell})=\alpha(X_{1},\ldots,X_{k})\beta(X_{k+1},\ldots,X_{k+\ell}).$

##### Exterior covariant derivative.

The exterior covariant derivative is a natural extension of the standard
exterior derivative of scalar-valued forms to forms that take values in a
vector bundle. In particular, given a connection $\nabla$ on
$\pi:E\\!\rightarrow\\!M$, the exterior covariant derivative
$d^{\nabla}:\Omega^{k}(M,E)\rightarrow\Omega^{k+1}(M,E)$ obeys

$d^{\nabla}(\alpha\otimes s)=d\alpha\otimes s+(-1)^{k}\alpha\wedge\nabla
s,\quad\forall\alpha\in\Omega^{k}(M),\forall s\in\Gamma(E),$ (8)

where $\otimes$ denotes the tensor product of differential forms and sections
of the bundle $E$. In local coordinates, Eq. (8) is expressed as

$(d^{\nabla}\gamma)^{a}=d\gamma^{a}+\omega^{a}_{b}\wedge\gamma^{b},$ (9)

where $\gamma\\!=\\!f_{a}\gamma^{a}\in\Omega^{k}(U,E)$ in a given local frame
field $\\{f_{a}\\}$ of $E$, while the associated local connection $1-$form for
this frame field has components $\omega^{a}_{b}$.

Note that given a smooth map $\Psi\colon M\to N$ between two manifolds, and a
vector bundle $\pi:E\rightarrow N$ with connection $\nabla$, the exterior
covariant derivative commutes with the pullback: given a form
$\alpha\in\Omega^{k}(N,E)$ one has

$d^{\Psi^{\ast}\nabla}(\Psi^{\ast}\alpha)=\Psi^{\ast}(d^{\nabla}\alpha).$ (10)

Additionally, given a bundle-valued form $\alpha\\!\in\\!\Omega^{k}(M,E)$ and
a scalar-valued form $\lambda\\!\in\\!\Omega^{\ell}(M),$ the _Leibniz rule_
between the wedge product and the exterior covariant derivative holds, i.e.,

$d^{\nabla}(\alpha\wedge\lambda)={\rm
d}^{\nabla}\alpha\wedge\lambda+(-1)^{k}\alpha\wedge d\lambda.$

For a proof of these last two properties, see [25] for instance.

##### Curvature Two form and Bianchi identities.

Connections are geometric structures that capture the idea of parallel
transport along curves, and are an essential tool in the study of fiber
bundles. One important geometric invariant of connections is the _curvature
$2-$form_ $\Omega^{\nabla}\\!\in\\!\Omega^{2}(M,\mathrm{End}(E))$, which
measures the failure of a connection to be flat, or in other words, the
impossibility of endowing the base manifold with a Euclidean structure. This
skew-symmetric tensor is useful in geometry and physics in areas such as
general relativity, gauge theory, and topological quantum field theory. It
quantifies the noncommutativity of parallel transport around closed loops and
encodes the infinitesimal holonomy of the connection, also known as Ambrose-
Singer theorem [1]. The curvature $2-$form is defined as

$\Omega^{\nabla}(X,Y)s:=d^{\nabla}(d^{\nabla}s)(X,Y),\quad
s\in\Gamma(E),\;X,Y\in\Gamma(TM).$

If we are given a local frame field, it can be shown that we have the local
formula

$\Omega^{a}_{b}=d\omega^{a}_{b}+\omega^{a}_{c}\wedge\omega^{c}_{b}=(d\omega+\omega\wedge\omega)^{a}_{b}.$
(11)

This local formula is also sometimes written abusively as
$\Omega^{\nabla}\\!=\\!d^{\nabla}\omega$, to be understood as applying the
exterior covariant derivative for vector-valued forms to the columns of the
connection $1-$form $\omega$ in the local frame.

One can then show that the following general equation holds:

$d^{\nabla}d^{\nabla}\alpha=\Omega^{\nabla}(\cdot,\cdot)\wedge\alpha,\quad\forall\alpha\in\Omega^{k}(M,E),$
(12)

a property proving that unlike the exterior derivative ${\rm d}$ of scalar-
valued differential forms, the operator ${\rm d}^{\nabla}$ is _not_ nilpotent
in general. Note that this property is often referred to as the _algebraic
Bianchi identity_.

Another important property is what is often referred to as the _differential
Bianchi identity_ , a consequence of the Leibniz rule and antisymmetry of the
exterior covariant derivative: it states that

$d^{\nabla^{\rm End}}\Omega^{\nabla}=0,$ (13)

where $\nabla^{\rm End}$ is the induced connection on the endomorphism bundle
$\operatorname{End}(E)\rightarrow M$. More generally it can be shown that for
the induced connection on the endomorphism bundle, Eq. (12) can be expressed
as

$d^{\nabla^{\rm End}}d^{\nabla^{\rm
End}}\beta=[\Omega^{\nabla}\wedge\beta],\quad\forall\beta\in\Omega^{k}(M,\mathrm{End}(E)),$
(14)

where $[\cdot\wedge\cdot]$ is the commutator using the wedge product.

Another usual identity linking curvature tensor and covariant derivatives is
that, given a connection $\nabla$ on a vector bundle $\pi:E\\!\rightarrow\\!M$
and for any 0-form $s\\!\in\\!\Gamma(E)$, one has:

$\Omega^{\nabla}(X,Y)s=\nabla_{X}\nabla_{Y}s-\nabla_{Y}\nabla_{X}s-\nabla_{[X,Y]}s,\quad\forall
X,Y\in\Gamma(TM).$ (15)

The curvature $2-$form and its two associated Bianchi identities are central
geometric notions in the study of connections as they are independent of the
choice of local trivializations of the associated vector bundle $E$.
Consequently, we will ensure that our discretization respects these important
properties.

######  Remark 2.3.

The curvature is _gauge invariant_. This means that given two local frame
fields $\\{f_{a}\\}$ and $\\{\tilde{f}_{a}\\}$, related via a matrix of change
of bases as $f_{a}=\tilde{f}_{b}A^{b}_{a}$, see (1), with associated local
connection $1-$form $\omega$ and $\tilde{\omega}$, then the components of the
matrix-valued curvature in both frame are related as

$\widetilde{\Omega}^{\nabla}=d\tilde{\omega}+\tilde{\omega}\wedge\tilde{\omega}=A(d\omega+\omega\wedge\omega)A^{-1}=A\Omega^{\nabla}A^{-1}.$

That is, a change of basis in the fiber only amounts to a change of basis for
the representation matrix of the curvature endomorphism, see [9, Eq. (1.29),
page 108].

##### Riemannian connections.

Given a smooth manifold $M$, we call $M$ a _Riemannian manifold_ if the
tangent bundle possesses a Riemannian metric, i.e., a section
$g\\!\in\\!\Gamma(\smash{\mathrm{Sym}^{2}(TM)})$ such that $g_{x}:T_{x}M\times
T_{x}M\rightarrow\mathbb{R}$ is a scalar product for each fiber space. A
connection $\nabla$ on the tangent bundle is said to be compatible with the
metric if it preserves the metric structure, which is a natural condition in
Riemannian geometry. In other words, $\nabla$ is compatible with the metric if
the covariant derivative of the metric with respect to a vector field $X$ can
be expressed in terms of the metric and $\nabla$ as

$d\big{(}g(s_{1},s_{2})\big{)}(X)=g(\nabla_{X}s_{1},s_{2})+g(s_{1},\nabla_{X}s_{2})\quad\forall
s_{1},s_{2}\in\Gamma(TM),X\in\Gamma(TM).\vspace*{-2mm}$

If the connection is compatible with the metric, for any two vectors $v,w\in
T_{x}M$, the inner product $g(R_{t}v,R_{t}w)$ is preserved along parallel
transport, where $R_{t}$ denotes the parallel transport map associated with
the connection $\nabla$. In a local orthonormal frame, this implies that the
local expressions of $R_{t}$ belong to the special orthogonal group $SO(n)$.
This compatibility of the connection with the metric structure is a
fundamental requirement in Riemannian geometry: it plays a major role in the
definition of the Levi-Civita connection, which is the unique connection that
is both compatible with the metric and torsion-free — see the next paragraph
for a definition of torsion.

##### Solder form and torsion.

For the special case where $E\\!=\\!TM,$ the tautological $1-$form, or _solder
form_ , $\theta$ is a bundle-valued $1-$form, defined as $\theta(X)\\!=\\!X$
for any vector field $X$ on $M$. For a connection $\nabla$, the _torsion
$2-$form_ is derived from the solder form through

$\Theta^{\nabla}=d^{\nabla}\theta=d\theta+\omega\wedge\theta\
\in\Omega^{2}(M,TM),$ (16)

where the last equality holds only locally. While the curvature $2-$form
$\Omega^{\nabla}$ measures the deviation of the covariant derivative from the
exterior derivative, the torsion $2-$form $\Theta$, on the other hand,
measures the deviation of the covariant derivative from the Lie derivative,
meaning that it measures the failure of the connection to preserve the Lie
bracket of vector fields under parallel transport. Specifically, given a
connection $\nabla$, the torsion $2-$form $\Theta$ is also expressed through

$\Theta^{\nabla}(X,Y)=d^{\nabla}\theta(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y]$

for vector fields $X$ and $Y$ on $M$.

The _algebraic Bianchi identity_ often refers to the differential identity
from Eq. (12) specifically applied to the solder form; it relates the exterior
derivative of the torsion $2-$form to the curvature $2-$form and the solder
form through

$d^{\nabla}\Theta^{\nabla}=d^{\nabla}d^{\nabla}\theta=\Omega^{\nabla}\wedge\theta.$
(17)

######  Remark 2.4 (Bianchi Identities in (Pseudo) Riemannian Geometry).

Suppose we consider the special case of a Riemannian manifold and
$E\\!=\\!TM$. Let $\nabla$ be the Levi-Civita connection, i.e., the unique
torsion free connection compatible with the metric. In this case we usually
write $\Omega^{\nabla}(X,Y)Z=R(X,Y)Z$, with $R$ the Riemann curvature
$(1,3)$-tensor, i.e.,
$R\\!\in\\!TM^{\ast}\\!\otimes\\!TM^{\ast}\\!\otimes\\!TM^{\ast}\\!\otimes\\!TM$.
We have

${\rm d}^{\nabla^{\rm
End}}\Omega^{\nabla}=0\;\Leftrightarrow\;\nabla_{X}R(Y,Z)+\nabla_{Y}R(Z,X)+\nabla_{Z}R(X,Y)=0$

and

$0=d^{\nabla}d^{\nabla}\theta=\Omega^{\nabla}\wedge\theta\;\Leftrightarrow\;R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0,$

which are the familiar Bianchi identities as formuluated in
(pseudo)-Riemannian geometry. A common form found in literature uses tensor
notation. Let $(\partial_{1},\ldots,\partial_{n})$ be a local orthonormal
frame field of $TM$ consisting of vector fields for which the Lie bracket
vanishes. Let $(dx^{1},\ldots,dx^{n})$ the associated dual frame. Using the
Einstein sum convention the coordinates of the Riemannian curvature tensor can
be expressed as the components of the curvature $2-$form as

$[\Omega^{\nabla}]^{\mu}_{\nu}=R^{\mu}_{\nu,\alpha,\beta}dx^{\alpha}\wedge
dx^{\beta},$

where

$R^{\mu}_{\nu,\alpha,\beta}=dx^{\mu}\left(R(\partial_{\alpha},\partial_{\beta})\partial_{\nu}\right).$

The algebraic Bianchi identity then reads

$0=R^{\mu}_{\nu\alpha\beta}+R^{\mu}_{\alpha\beta\nu}+R^{\mu}_{\beta\nu\alpha}.$

As for the endomorphism-valued case, the definition for the covariant
derivative of a tensor field this time becomes:

$R^{\mu}_{\nu\alpha\beta;\lambda}={\partial_{\lambda}R^{\mu}_{\nu\alpha\beta}}+R^{\kappa}_{\nu\alpha\beta}\omega^{\mu}_{\lambda\kappa}-R^{\mu}_{\kappa\alpha\beta}\omega^{\kappa}_{\lambda\nu}-R^{\mu}_{\nu\kappa\beta}\omega^{\kappa}_{\lambda\alpha}-R^{\mu}_{\nu\alpha\kappa}\omega^{\kappa}_{\lambda\beta}.$

Using this definition of covariant tensor derivative, the differential Bianchi
identity for the Levi-Civita connection reads

$0=R^{\mu}_{\nu\alpha\beta;\lambda}+R^{\mu}_{\nu\lambda\alpha;\beta}+R^{\mu}_{\nu\beta\lambda;\alpha}.$

For a detailed derivation of these formulas from Eqs. (13) and (17), see [16]
p.298-301.

##  3 Discrete Connections and Bundle-valued Differential Forms

We now discuss our discretization of bundle-valued forms and connections, for
which related operators will be derived in the next section. Throughout this
section and for the remainder of this paper, we consider an arbitrary discrete
manifold $M$ represented by a _simplicial complex_ embedded in
$\mathbb{R}^{n}$. We denote its vertices by $\mathcal{V}=\\{v_{i}\\}$, its
edges by $\mathcal{E}=\\{e_{ij}\\}$ (where $e_{ij}$ is the oriented edge
between $v_{i}$ and $v_{j}$), its faces by $\mathcal{F}=\\{f_{ijk}\\}$ (whose
boundaries consist of oriented edges), its 3D cells by
$\mathcal{C}=\\{c_{ijkl}\\}$, etc. More generally, we will also refer to the
set of all $k-$simplices (simplices made out of $(k+1)$ vertices) of $M$ as
$\smash{\mathcal{M}^{k}}$. For simplicity of exposition, we will assume that
the manifold is _oriented_ , which is a negligible constraint given the local
nature of this theory.

###  3.1 Discrete Vector Bundles, Frame Bundles, and Connections

In our discrete setup, a number of discrete notions of continuous definitions
are rather natural to define as a collection of discrete objects on $M$,
representing a discretization of their respective continuous notions. While
these definitions alone are not sufficient (in particular, no notion of bundle
topology will be defined yet), we will complete the discrete picture in
upcoming sections.

######  Definition 3.1 (Discrete Vector Bundle).

A discrete vector bundle of rank $r$ over a discrete orientable manifold $M$
is simply defined as a collection of vector spaces
$\\{\mathbf{E}_{v_{i}}\\}_{v_{i}\in\mathcal{V}}\\!\coloneqq\\!\mathbf{E}(M)$
(i.e., one vector space per vertex), with
$\mathrm{dim}(\mathbf{E}_{v_{i}})\\!=\\!r$.

We can then equip each of these vertex-based vector spaces with a frame to
form a _discrete analog of a local section of the frame bundle_ over $M$.

######  Definition 3.2 (Section of Discrete Frame Bundle).

A section of the _discrete frame bundle_ of the rank-$r$ vector bundle
$\mathbf{E}(M)$ consists in a collection of frames
$\\{\mathbf{F}_{v_{i}}\\}_{v_{i}\in\mathcal{V}}\\!\coloneqq\\!\mathbf{F}(M)$
defining an arbitrary choice of frame for each vector space
$\mathbf{E}_{v_{i}}$.

We can now define a discrete connection $\boldsymbol{\nabla}$ as a collection
of maps between the extremities of each edge of $M$.

######  Definition 3.3 (Discrete Connection).

A discrete connection $\boldsymbol{\nabla}$ is defined as an assignment, for
each edge $e_{ij}$ of the discrete orientable manifold $M$, of an invertible
linear map $\mathcal{R}_{ij}\colon\mathbf{E}_{v_{j}}\to\mathbf{E}_{v_{i}}$,
with $\mathcal{R}_{ij}\circ\mathcal{R}_{ji}\\!=\\!\mathbf{Id}_{v_{i}}$
representing the parallel transport induced by a continuous connection
$\nabla$ along $e_{ij}$.

These maps $\\{\mathcal{R}_{ij}\\}_{e_{ij}\in\mathcal{E}}$ can be thought of
as discrete parallel transport maps along edges, where the continuous
definition from Eq. (7) is integrated along an edge to define a map between
two adjacent vertices. In practice, one can express each linear map
$\mathcal{R}_{ij}$ of a connection through a matrix $R_{ij}\\!\in\\!GL(r)$,
stored on vertex $v_{i}$, using a discrete frame field $\mathbf{F}(M)$. Note
that a number of properties in the continuous case apply to these discrete
equivalents. For instance, if each of the fibers $\mathbf{E}_{v_{i}}$
possesses an inner product
$\langle\cdot,\cdot\rangle\smash{{}_{\mathbf{E}_{v_{i}}}}$, we say that the
discrete connection is compatible with the metric if
$\smash{\mathcal{R}_{ij}:(\mathbf{E}_{v_{j}},\left\langle\cdot,\cdot\right\rangle_{v_{j}})\rightarrow(\mathbf{E}_{v_{i}},\left\langle\cdot,\cdot\right\rangle_{v_{i}})}$
are orthonormal maps.

When each frame $\mathbf{F}_{v_{i}}$ is orthonormal, this implies
$R_{ij}\\!\in\\!\mathrm{SO}(r)$: a frame at a vertex is naturally parallel-
transported along an edge through a pure rotation.

Figure 1: Discrete Connections. A discrete vector bundle $\boldsymbol{E}$
assigns a vector space to each vertex on a mesh. To compare quantities in
different fibers, we need to encode how the individual fibers are connected. A
connection consists of a collection of linear maps between neighboring fibers.
In the figure, each box represents a frame for each fiber space over a vertex.
The right part of the figure visualizes the local action of two different
discrete connections. Starting with an initial vector
$v\in\boldsymbol{E}_{p_{0}}$ (gray), we illustrate the different parallel
transports induced by two different connections, $\nabla^{1}$ and
$\nabla^{2}$. Different connections on the same vector bundle result in
distinct notions of parallel transport and curvature.

Given two discrete manifolds ${M,N}$ and a discrete vector bundle $\mathbf{E}$
over $N$, with a discrete connection $\boldsymbol{\nabla}$, we can finally
define the notion of _discrete pullback bundle_ as follows: if $f\colon M\to
N$ is a simplicial map associating simplices in $M$ to simplices in $N$, the
pullback bundle $f^{\ast}\mathbf{E}$ over $M$ is naturally defined as
$(f^{\ast}\mathbf{E})_{v_{i}}\\!=\\!\mathbf{E}_{f(v_{i})}$, $\forall
v_{i}\\!\in\\!\mathcal{V}.$ We then follow [5] for the definition of the
_pullback connection_.

######  Definition 3.4 (Pullback Connection [5]).

Let $f\colon M\to N$ be a simplicial map and $\mathbf{E}$ a discrete vector
bundle over $N$ with discrete connection $\boldsymbol{\nabla}$. A pullback
connection $f^{\ast}\boldsymbol{\nabla}$ is defined as a collection of linear
maps
$\\{\mathcal{R}_{ij}\colon\mathbf{E}_{f(j)}\to\mathbf{E}_{f(i)}\\}_{e_{ij}\in\mathcal{E}(M)}$
with:

$\mathcal{R}_{ij}:=\begin{cases*}\mathcal{R}_{f(v_{i})f(v_{j})}&if
$e_{f(v_{i}),f(v_{j})}\in\mathcal{E}({N})$\\\
\mathrm{Id}_{\mathbf{E}_{f(v_{i})}}=\mathrm{Id}_{\mathbf{E}_{f(v_{j})}}&otherwise,
$f(v_{i})=f(v_{j})$.\end{cases*}$

###  3.2 Discrete Bundle-Valued Differential Forms

Given a discrete vector bundle $\mathbf{E}$ over a discrete manifold $M$, we
now define the notion of _discrete bundle-valued $\ell-$form_ as, for now, a
collection of abstract maps. How these maps relate to the continuous case will
be elucidated later (see Sec. 4) in order to guarantee proper convergence to
continuous forms for increasing mesh resolution.

######  Definition 3.5 (Discrete (1,0)-tensor-valued $\ell-$form).

A discrete vector-valued $\ell-$form $\boldsymbol{\alpha}$ on $M$ is a
collection of maps which, for each $\ell-$simplex $\sigma$ and one of its
vertices $v$, returns a vector in $\mathbf{E}_{v}$, i.e.,

$\boldsymbol{\alpha}\colon\sigma\in\mathcal{M}^{\ell},v\in\sigma\subset\mathcal{V}(M)\mapsto\boldsymbol{\alpha}(\sigma,v)\in\mathbf{E}_{v},$
(18)

such that if $\bar{\sigma}$ is the simplex $\sigma$ with reversed orientation,
one has $\boldsymbol{\alpha}(\bar{\sigma},v)=-\boldsymbol{\alpha}({\sigma},v)$
for all $v\\!\in\\!\sigma$.

Note that this definition means that in contrast to the discretization of
scalar-valued forms in FEEC or DEC, discrete bundle-valued differential forms
are _not_ to be understood as a linear space of cochains, but rather as _maps
for simplex-vertex pairs_ : bundle-valued $\ell-$forms should be regarded as
abstract “sided” maps (as they are attached to a particular vertex of the
simplex) for now. Moreover, we will assume _for now_ that a discrete bundle-
valued $\ell-$form is defined through its values on all simplex-vertex pairs
$(\sigma,v)$ for $\sigma$ is an $\ell-$simplex and $v$ is one of its vertices:
later on (see Sec 4.2), once we precisely define how a discrete bundle-valued
form is sampled from a continuous form, we will restrict this definition to
only one vector value per $\ell-$simplex, like in existing finite-dimensions
(scalar-valued) exterior calculus frameworks.

We also define the notion of _discrete $(1,1)-$tensor-valued $\ell-$forms_ (or
_discrete homomorphism-valued_ $\ell-$forms). Given a vector bundle
$\pi:E\\!\rightarrow\\!M$ with connection $\nabla$, we consider the bundle
$\mathrm{End}(E)\rightarrow M$ with induced connection
$\nabla^{\mathrm{End}}$. One can equivalently describe it as the bundle of
$(1,1)-$tensors of $E$. This latter description allows for a more flexible
representation,

preserving the underlying structure of endomorphism forms by considering
separate evaluation and input fibers for each cell.

######  Definition 3.6 (Discrete $(1,1)-$tensor-valued $\ell-$form).

A discrete $(1,1)-$tensor-valued $\ell-$form $\boldsymbol{\beta}$ on $M$ is a
collection of maps which, for each $\ell-$simplex $\sigma$ and two of its
vertices ($w$, the input [or ’cut’] fiber, and $v$, the output (or
_evaluation_) fiber), returns a homomorphism between $\mathbf{E}_{v}$ and
$\mathbf{E}_{w}$, i.e.,

$\boldsymbol{\beta}\colon\sigma\in\mathcal{M}^{\ell},v\in\sigma,w\in\sigma\mapsto\boldsymbol{\beta}(\sigma,v,w)\in\operatorname{Hom}(\mathbf{E}_{w},\mathbf{E}_{v}),$
(19)

such that if $\bar{\sigma}$ is the simplex $\sigma$ with reversed orientation,
one has
$\boldsymbol{\beta}(\bar{\sigma},v,w)\\!=\\!\\!-\boldsymbol{\beta}({\sigma},v,w)$
for all $v,w\\!\in\\!\sigma$.

######  Remark 3.7.

The discrete curvature $\boldsymbol{\Omega}^{\nabla}$ defined in Def. (3.12)
with evaluation and cut fibers at two different vertices is an example of a
discrete endomorphism-valued $2-$form. Note that we could trivially extend
these previous definitions to $(p,q)-$tensor-valued discrete forms using $p$
input vertices and $q$ output vertices; but we will not explore this extension
in this paper.

Following [5] we can finally define the notion of _pullback of a discrete
bundle-valued form_ as follows:

######  Definition 3.8 (Pullback of a Discrete Vector-Valued Form [5]).

Given a simplicial map $f\colon M\to N$ and a discrete vector bundle
$\mathbf{E}$ over $N$, let $\boldsymbol{\alpha}$ be an $\mathbf{E}-$valued
discrete $\ell-$form. We define the $f^{\ast}\mathbf{E}-$valued pullback form
$f^{\ast}\boldsymbol{\alpha}$ on $M$ as

$f^{\ast}\boldsymbol{\alpha}(\underbrace{s\\!\coloneqq\\![v_{0},\ldots,v_{\ell}]}_{\text{$\in\mathcal{M}^{\ell}$}},{v_{0}})=\begin{cases}\boldsymbol{\alpha}([f(v_{0}),\ldots,f(v_{\ell})],{f(v_{0})})&\text{if
}[f(v_{0}),\ldots,f(v_{\ell})]\in\mathcal{N}^{\ell},\\\
0&\text{otherwise.}\end{cases}$

A similar definition for _discrete endomorphism-/homomorphism-valued forms_
holds trivially.

###  3.3 Discrete Connection One-Form

Recall that in the smooth theory, once we are given a trivialization of the
smooth vector bundle $E\\!\to\\!M$, we can treat the fibers as vector spaces
of rank $r$ (say, copies of $\mathbb{R}^{r}$). The local connection $1-$form
$\omega$ associated with a given connection $\nabla$ is then defined as
$\nabla s^{a}\\!=\\!ds^{a}+\omega^{a}_{b}s^{b}$, where $\omega$ measures how
much the connection deviates from the exterior derivative $d$ (applied to
coordinates); so if $\omega\\!=\\!0$, i.e., $\nabla\\!=\\!d$ in a given frame
field, then the matrix representation of parallel transport in this frame
field satisfies $R_{ij}\\!=\\!\mathrm{Id}_{\mathbb{R}^{r}}$.

We thus formulate the definition of a _discrete connection $1-$form_ as
follows:

######  Definition 3.9.

Let $M$ be a discrete manifold with discrete vector bundle $\mathbf{E}(M)$ of
rank $r$ and let $\mathbf{F}(M)$ be a section of the discrete frame bundle.
For a given discrete connection $\boldsymbol{\nabla}$ defined by a set of edge
matrices $R_{ij}$ when expressed in $\mathbf{F}$, we define its associated
discrete local connection $1-$form $\boldsymbol{\omega}$ as the collection of
$r\\!\times\\!r$ matrices $\boldsymbol{\omega}_{ij}$ (one per edge
$e_{ij}\\!\in\\!M$) with

$\boldsymbol{\omega}_{ij}=R_{ij}-\mathrm{Id}_{v_{i}}\in\mathbb{R}^{r\times
r}.\vspace*{-1mm}$ (20)

Note that a discrete local connection $1-$form thus also measures how much the
discrete connection-induced parallel transport locally deviates from the
identity, mimicking the continuous definition $\nabla
s^{a}\\!=\\!ds^{a}+\omega^{a}_{b}s^{b}$. And just like in the smooth case, the
realizations of the per-coordinate exterior derivative and the connection
$1-$form for a given connection change depending on the chosen frame field
$\mathbf{F}$. We will, in fact, exploit this property in our work.

######  Remark 3.10.

We can also regard Eq. (20) as a linear approximation of the smooth case: if
each parallel-transport edge matrix $R_{ij}$ comes from the integration of a
continuous $1-$form $\omega$ over the edge $e_{ij}$, then it is given by
$R_{ij}\\!=\\!P\exp\smash{\bigl{(}\int_{e_{ij}}\omega\bigr{)}}$. The first-
order Taylor expansion of this path-ordered matrix exponential is
$\smash{\mathrm{Id}_{v_{i}}+\int_{e_{ij}}}\omega$,

ensuring consistency with the continuous case. One could instead define
$\boldsymbol{\omega}_{ij}$ as the integrated value of the continuous
connection $1-$form $\omega$, and then linearize the exponential map by
setting $R_{ij}\\!=\\!\mathrm{Id}_{v_{i}}+\boldsymbol{\omega}_{ij}$; but given
the central role of parallel transport in our work, we find this second way to
discretize connection and parallel transport less appealing.

######  Remark 3.11.

In the context of this discretization approach, we presume the continuity of
the underlying local frame field. However, this assumption encounters
limitations due to the globally assigned frame per vertex. Notably, the
applicability of a continuous local frame encounters constraints, as
illustrated by the hairy ball theorem, which prohibits the existence of a
global frame for the tangent bundle on a sphere. Extending this notion to
bundles with potentially nontrivial characteristic classes, even for a
seemingly straightforward base manifold like a tetrahedron, it may become
infeasible to employ a single bundle chart to parametrize the bundle on the
cell. In our work, we specifically focus on scenarios where a local setup
suffices, and a single chart is adequate to parametrize the bundle. In this
case we want to point out that the parametrization of the bundle is
independent of the chart. However, in cases involving multiple charts, a need
arises for compatibility conditions between them to meet global topology
constraints. Further exploration of these conditions remains a direction for
future research.

###  3.4 Discrete Curvature Two-Form

Given a discrete connection $\boldsymbol{\nabla}$, we now design a notion of
_discrete connection curvature_ akin to the continuous definition of
$\Omega^{\nabla}$. Since the Ambrose-Singer theorem links the curvature of the
connection to the holonomy of an infinitesimal loop, it is tempting to define
the discrete curvature $2-$form $\boldsymbol{\Omega}^{\boldsymbol{\nabla}}$ on
a triangle $\sigma\\!=\\![v_{0},v_{1},v_{2}]$ on, say, fiber
$\mathbf{E}_{v_{0}}$ through the difference between the integrated connection
$1-$form over the triangle edges and the identity at the evaluation fiber via

$\boldsymbol{\Omega^{\nabla}}(\sigma,v_{0})\coloneqq\mathcal{R}_{01}\cdot\mathcal{R}_{12}\cdot\mathcal{R}_{20}-\mathrm{Id}_{0}\in\mathrm{End}(\mathbf{E}_{v_{0}}).\vspace*{-2mm}$
(21)

However, Eq. (21) is not a constructive definition: one cannot meaningfully
_sum_ this definition over two triangles, even if they share the same
evaluation vertex $v_{i}$. In other words, since this curvature is associated
with a loop instead of a simple chain, we cannot use the summability of chains
to induce a notion of curvature integral over a larger region. We need to
leverage the underlying algebraic structure of chains instead.

Inspired by synthetic geometry for infinitesimal cells [22] and the definition
proposed in [28, 5], we propose not to use parallel transport along triangle
boundary loops, but instead, to _compare parallel transport along two
$1-$chains whose difference forms a triangle boundary loop_. This change,
which amounts to comparing the integration of the connection form over two
$1-$chains — and each time parallel-transporting the result back to fiber
$\mathbf{E}_{v_{0}}$ — may seem tautological, but it will allow us to define a
notion of addition of the curvature $2-$form over two $2-$chains sharing a
1-chain. Note that it now casts the curvature two-form on $M$ as a discrete
_homomorphism-valued $2-$form_.

######  Definition 3.12.

Let $M$ be a discrete orientable manifold equipped with a discrete vector
bundle $\mathbf{E}(M)$ and connection
$\boldsymbol{\nabla}\\!=\\!\\{\mathcal{R}_{ij}\\}_{e_{ij}\in\mathcal{E}}.$ Let
$\sigma=[v_{0},v_{1},v_{2}]\in\mathcal{M}^{2}$ be a triangle of $M$. For the
evaluation fiber $\mathbf{E}_{v_{0}}$ at $v_{0}$, the three expressions for
the discrete curvature $2-$form $\smash{\boldsymbol{\Omega^{\nabla}}}$
depending on whether we use $v_{1}$, $v_{2}$, or $v_{0}$ as the cut vertex
(comparing respectively parallel-transports over the oriented $1-$chains
$e_{01}$ vs. $e_{02}e_{21}$, $e_{01}e_{12}$ vs. $e_{02}$, and
$e_{01}e_{12}e_{20}$ vs. $\varnothing$, see Fig. 2) are:

$\boldsymbol{\Omega^{\nabla}}(\sigma,v_{0},v_{1})=\mathcal{R}_{01}-\mathcal{R}_{02}\mathcal{R}_{21}\,\in\operatorname{Hom}(\mathbf{E}_{v_{1}},\mathbf{E}_{v_{0}}),\vspace*{-1mm}$
(22)
$\boldsymbol{\Omega^{\nabla}}(\sigma,v_{0},v_{2})=\mathcal{R}_{01}\mathcal{R}_{12}-\mathcal{R}_{02}\,\in\operatorname{Hom}(\mathbf{E}_{v_{2}},\mathbf{E}_{v_{0}}),$
(23)
$\boldsymbol{\Omega^{\nabla}}(\sigma,v_{0},v_{0})=\mathcal{R}_{01}\mathcal{R}_{12}\mathcal{R}_{20}-\mathrm{Id}_{0}\,\in\operatorname{Hom}(\mathbf{E}_{v_{0}},\mathbf{E}_{v_{0}}).$
(24)

Figure 2: Curvatures. Illustration of the discrete curvatures for different
cut vertices.

From this definition, the meaning of “ _evaluation fiber_ ” and “ _cut fiber_
” should become clear: instead of measuring the traditional holonomy of the
loop starting at $v_{0}$ through
$\smash{\boldsymbol{\Omega^{\nabla}}(\sigma,v_{0})}$, we instead compare
transport along two paths, both from the evaluation fiber to the cut fiber but
on opposite sides of the triangle, with the path difference being the original
triangle boundary.

######  Remark 3.13.

Given a _non-simplicial_ 2-cell with a discrete vector bundle and connection,
we can easily extend the definition of the discrete curvature similarly
through a difference of parallel transports along two paths following the
boundary of the cell from the evaluation vertex to the cut vertex.

Eq. (24) matches the discrete holonomy expression from Eq. (21). In fact, all
three expressions listed in Def. 3.12 are trivial extensions of the definition
in Eq. (21): since
$\mathcal{R}_{ij}\circ\mathcal{R}_{ji}\\!=\\!\operatorname{Id}_{v_{i}}$, they
are all equivalent up to post-multiplication. Similarly, changing the
evaluation fiber to another vertex than $v_{0}$ would simply require a pre-
multiplication by the accumulated parallel transport from $v_{0}$ to the new
evaluation fiber. Despite its apparent redundancy, the value of this encoding
is in its _summability_ : the curvature $2-$form for two adjacent simplices
can be summed if their evaluation and cut fibers are the same, resulting in an
expression of the curvature on the (now non-simplicial) union still
representing the difference in integration of the connection form over two
$1-$chains evaluated at the evaluation fiber. Moreover, we will see in the
next section that the expressions we propose actually correspond to a
particular choice of integration of the curvature $2-$form over $2-$simplices
using an ordering induced by a canonical retraction, _putting on a formal
footing these seemingly arbitrary expressions_.

##  4 Discrete Bundle-valued Exterior Calculus

Given the discretization of bundle-valued forms and connections that we
reviewed, we are now equipped with all the tools needed to describe our
bundle-valued variant of discrete exterior calculus. Along the way, we will
point out major differences with the typical discrete calculus of scalar-
valued forms due to our need to deal with fiber-based evaluations; similarly,
we will point out where a previous attempt at a calculus of bundle-valued
forms [5] failed to provide useful numerical evaluations, despite reproducing
discrete Bianchi identities.

###  4.1 Guiding approach to discretization: parallel-propagated frame fields

The discrete exterior derivative $\boldsymbol{d}$ for scalar-valued forms
finds a simple and exact discretization as the coboundary operator in
numerical versions of exterior calculus by leveraging Stokes’ theorem
$\int_{S}d\cdot=\int_{\partial S}\cdot$, from which ensues a whole discrete de
Rham complex [6, 15, 2]. In our bundle-valued case, finding a discrete version
of the covariant exterior derivative $d^{\nabla}$ is more difficult: for a
bundle-valued form $\alpha$, one has to define the discrete equivalent of —
and eventually, numerically evaluate — integrals of the type:

$\int_{S}d^{\nabla}\alpha=\int_{S}d\alpha+\int_{S}\omega\wedge\alpha=\int_{\partial
S}\alpha+\int_{S}\omega\wedge\alpha.\vspace*{-1mm}$ (25)

Note that such integrals of bundle-valued forms necessitate a chosen frame
field to be well defined, in contrast to the scalar-valued scenario. Moreover,
while Stokes’ theorem can be partially leveraged as indicated in Eq. (25), the
last term of this equation is particularly difficult to handle: it involves
the integration of a wedge product which, even for scalar-based forms, endures
severe limitations in the discrete realm [24]. In order to deal with these two
issues, we define a specific frame field that will enable a consistent
framework for the discretization of bundle-valued exterior calculus with
convergence guarantees.

##### Designing gauge fields to simplify evaluations.

Our approach relies on a simple, but key property afforded by gauge
transformations: _there always exists a local frame field with respect to
which the connection $1-$form is zero at some local point_ (see Fig. 3). As
described for instance in [9, Thm 2.1, page 107], given a local connection
$1-$form $\omega$, one can always construct a local frame field such that the
connection $1-$form $\tilde{\omega}$ after gauge transformation (i.e.,
expressed in this new frame field) vanishes at a given point $c\\!\in\\!M$.
Therefore, an approach to discretizing expressions such as Eq. (25) — and thus
inducing the notion of a discrete (integrated) exterior covariant derivative —
is to design an “as-parallel-as-possible” frame field which will render the
connection $1-$form $\omega$ zero at a point in the simplex (we will use this
point to be one of its vertices, or sometimes, its barycenter): indeed,
assuming the connection $1-$form is Lipschitz continuous, the use of such a
frame field will _bound the value of the last term of Eq. ( 25)_ by the
diameter of the simplex — thus _guaranteeing its convergence to zero in the
limit of mesh refinement_. Consequently, the integral of the covariant
exterior derivative of a bundle-valued form over a simplex evaluated in a
properly chosen frame field will be expressible, just like in the scalar-
valued case, as _boundary integral values_ only. This approach, which recovers
Stokes’ theorem for flat connections, will be key to _ensure correct numerical
evaluation_ in the limit of mesh refinement as it will allow us to recover
actual continuous values of exterior covariant derivatives of bundle-valued
forms.

Figure 3: We investigate two distinct frame fields on the same vector bundle
with an identical connection $(E,\nabla)$. In this context, the boxes
represent the individual fibers of the bundle. Both bundles exhibit a similar
“twisted” structure, whereby the fibers are equally interconnected, implying
that the connection (or parallel transport) of the bundle remains consistent.
With a local frame construction facilitated by a rotation field
$R^{\scriptscriptstyle\nabla}$, we can alter both the differential and the
connection $1-$form. By employing a parallel-propagated frame field (left), we
achieve ${\omega}^{\scriptscriptstyle\nabla}(c)=0$ at a specific point $c$. In
contrast, when using a non-parallel-propagated frame (right), the connection
$1-$form does not vanish at $c$. Intuitively, the parallel-propagated frame
field on the left “follows” the bundle along radial lines emanating from $c$.
Consequently, $\tilde{\omega}$ vanishes along these radial lines. This
behavior highlights the distinctive properties of the two frame fields in
terms of their interaction with the bundle’s connection.

##### Parallel-propagating frame fields.

To explain more concretely how one can design a frame field that bounds the
wedge-product term from Eq. (25), let us go back temporarily to the continuous
case, where we can formally construct a notion of parallel-propagated frame
field and understand its impact on the evaluation of the integrated exterior
covariant derivative.

######  Definition 4.1 (Continuous Parallel-Propagated Frame).

Let $\pi:E\rightarrow M$ be a vector bundle with connection $\nabla$. Let
$s=[v_{0},\ldots,v_{\ell}]\\!\subset\\!M$ be a region in $M$ for which there
exists a diffeomorphism to an $\ell-$simplex
$\sigma\\!=\\![w_{0},\ldots,w_{\ell}]$ where each point $v_{i}$ are mapped to
an associated vertex $w_{i}$ of $\sigma$. Let $f$ be a local, arbitrary frame
field of $E$ over this region $s$. For any given corner
$v\\!\in\\!\\{v_{0},\ldots,v_{\ell}\\}$, we also define a (strong deformation)
retraction $\varphi_{v}:[0,1]\times s\rightarrow s$ derived from a canonical
retraction $\varphi_{w}^{\sigma}:[0,1]\times\sigma\rightarrow\sigma$ of the
simplex $\sigma$ through the aforementioned diffeomorphism, whose retracting
paths are radially joining the vertex $w$ associated to point $v$ (see Fig. 4
for an illustration in 2D); i.e.,

$\begin{split}\varphi_{w}^{\sigma}\colon[0,1]\times\sigma\,&\to\sigma\\\
(t,p)\,&\mapsto t\,w+(1-t)\,p.\\\\[-5.69054pt] \end{split}$ (26)

Moreover, for any point $p\\!\in\\!\sigma$, we denote by
$\mathcal{R}^{\\!\scriptscriptstyle\nabla\\!,v}(p)\colon E_{p}\to E_{v}$ the
$\nabla-$induced parallel transport map from $E_{v}$ to $E_{p}$ along the path
induced by the retraction $\varphi_{v}$ and
$R^{\\!\scriptscriptstyle\nabla\\!,v}\colon\smash{\mathbb{R}^{r}\to\mathbb{R}^{r}}$
the matrix field representing
$\mathcal{R}^{\\!\scriptscriptstyle\nabla\\!,v}(.)$ in the coordinate frame
$f$.

A frame field $\\{f_{a}^{\scriptscriptstyle\nabla\\!,v}\\}$ over the region
$s$ is now called parallel-propagated frame field from $v\\!\in\\!M$ if

$\mathcal{R}^{\scriptscriptstyle\nabla\\!,v}(p)f_{a}^{\scriptscriptstyle\nabla\\!,v}(p)=f_{a}(v),\quad\text{for
all $p\in S$, for all $a=1,\ldots,r$.}$ (27)

i.e., if the frame $\\{f_{a}(v)\\}$ at $v$ has been parallel-transported
throughout $s$ via the connection $\nabla$. Furthermore, we call
$R^{\scriptscriptstyle\nabla\\!,v}$ the _gauge field_ of the parallel-
propagated frame field from $v\\!\in\\!M.$

With this notion of a parallel-propagated field (denoted PPF subsequently for
short), we can now define how a form is expressed in such a PPF field:

######  Definition 4.2.

Using the same setting as in Def. 4.1 we denote the representation
$\alpha^{\scriptscriptstyle\nabla\\!,v}$ in the PPF from $v$ of an E-valued
$\ell-$form $\alpha\\!\in\\!\Omega^{\ell}(M;E)$ through:

$f_{a}^{\scriptscriptstyle\nabla\\!,v}(\alpha^{\scriptscriptstyle\nabla\\!,v})^{a}=\alpha=f_{a}\alpha^{a}=\left(f_{b}^{\scriptscriptstyle\nabla\\!,v}(R^{\scriptscriptstyle\nabla\\!,v})^{b}_{a}\right)\alpha^{a},$

where $\alpha$ denotes at the same time the $\ell-$form and its local
expression with respect to the local frame $f$. We have

$\alpha^{\scriptscriptstyle\nabla\\!,v}=R^{\scriptscriptstyle\nabla\\!,v}\alpha.$
(28)

Similarly, we denote the representation of the local connection $1-$form
$\omega$ (associated with the connection $\nabla$) in the PPF from $v$
undergoing the gauge transformation by

$\omega^{\scriptscriptstyle\nabla\\!,v}=R^{\scriptscriptstyle\nabla\\!,v}(\omega-(R^{\scriptscriptstyle\nabla\\!,v})^{-1}dR^{\scriptscriptstyle\nabla\\!,v})(R^{\scriptscriptstyle\nabla\\!,v})^{-1},\vspace*{-1mm}$

see Eq. (2).

######  Remark 4.3.

Let us note that a connection-dependent integral can be written in terms of
the frames as

$\mathop{\mathrlap{\mkern-10.0mu\raisebox{5.5pt}{$\underset{\scriptscriptstyle\varphi_{v}}{\scriptstyle\nabla}$}}\int}\limits_{s}\alpha=\int_{s}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v}\alpha=\int_{s}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v}(\alpha^{\scriptscriptstyle\nabla\\!,v})^{a}f_{a}^{\scriptscriptstyle\nabla\\!,v}(p)=\int_{s}(\alpha^{\scriptscriptstyle\nabla\\!,v})^{a}f_{a}(v)\in
E_{v},$

where we used Eq. (27). Because of the last expression, in our development
below, this $E_{v}-$valued integral will be often identified with the
$\mathbb{R}^{r}-$valued integral of the local expression
$\alpha^{\scriptscriptstyle\nabla\\!,v}$, namely, with

$\int_{s}\alpha^{\scriptscriptstyle\nabla\\!,v}=\int_{s}R^{\scriptscriptstyle\nabla\\!,v}\alpha,$

see Eq. (28). Here $R^{\scriptscriptstyle\nabla\\!,v}$ the matrix of
$\mathcal{R}^{\scriptscriptstyle\nabla\\!,v}$ with respect to the frame
$\\{f_{a}\\}$. One passes from one to the other representation by using the
frame $\\{f_{a}\\}$ at the point $v$.

Note that with the change of frame field from $\\{f_{a}\\}$ to
$\\{f^{\scriptscriptstyle\nabla\\!,v}_{a}\\}$, forms are not the only entities
changing: the differential $d$ also changes its coordinate-based expression in
the PPF frame accordingly.

It is now a trivial exercise to check that
$\omega^{\scriptscriptstyle\nabla\\!,v}(v)=0$, i.e.,
$\omega^{\scriptscriptstyle\nabla\\!,v}$ vanishes at the point $v$, by
construction. Assuming that the curvature $2-$form is bounded, we can thus
deduce that $\|\omega^{\scriptscriptstyle\nabla\\!,v}\|=\mathcal{O}(h)$ on
$s$, where $h$ is the diameter of the simplicial region $s$. If we now express
the covariant exterior derivative in this novel frame field
$\\{f^{\scriptscriptstyle\nabla\\!,v}_{a}\\}$, as explained in Sec. 2.2, Eq.
(9), we obtain, with this choice of coordinates, the following connection-
dependent expression:

$\left(d^{\nabla}\alpha\right)^{\scriptscriptstyle\nabla\\!,v}=d\alpha^{\scriptscriptstyle\nabla\\!,v}+\omega^{\scriptscriptstyle\nabla\\!,v}\wedge\alpha^{\scriptscriptstyle\nabla\\!,v}.$

We now recall from Def. 2.2 that the connection dependant integral is given by

$\mathop{\mathrlap{\mkern-10.0mu\raisebox{5.5pt}{$\underset{\scriptscriptstyle\varphi_{v}}{\scriptstyle\nabla}$}}\int}\limits_{s}d^{\nabla}\alpha:=\int_{s}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v}d^{\nabla}\alpha\in
E_{v}.$

As commented above, this $E_{v}-$valued integral can be identified with the
$\mathbb{R}^{r}-$valued integral
$\int_{s}(d^{\nabla}\alpha)^{\scriptscriptstyle\nabla\\!,v}$, which yields

$\int_{s}(d^{\nabla}\alpha)^{\scriptscriptstyle\nabla\\!,v}=\int_{s}d\alpha^{\scriptscriptstyle\nabla\\!,v}+\omega^{\scriptscriptstyle\nabla\\!,v}\wedge\alpha^{\scriptscriptstyle\nabla\\!,v}=\int_{\partial
s}\alpha^{\scriptscriptstyle\nabla\\!,v}+\int_{s}\omega^{\scriptscriptstyle\nabla\\!,v}\wedge\alpha^{\scriptscriptstyle\nabla\\!,v}=\int_{\partial
s}R^{\scriptscriptstyle\nabla\\!,v}\alpha+\int_{s}\omega^{\scriptscriptstyle\nabla\\!,v}\wedge\alpha^{\scriptscriptstyle\nabla\\!,v}.$
(29)

Figure 4: Retraction. In this illustration, we consider a simplicial cell
$\sigma$ and demonstrate the retraction of $\sigma$ onto one of its vertices
$v\\!\in\\!\sigma$. To aid visualization, an offset from the initial triangle
is used to depict the shrinkage induced by the retraction function $\varphi$.
For any point $p\\!\in\\!\sigma$, the aforementioned retraction induces radial
joining paths from $p$ to $v$, which can serve as paths for parallel transport
from source point $p$ to $v$. This parallel transport, defined along these
paths, can be used subsequently to define the parallel propagated frame based
in $v$.

Given that the volume of $s$ is of order $h^{\ell+1}$, the last term
$\int_{s}\omega^{\scriptscriptstyle\nabla\\!,v}\wedge
R^{\scriptscriptstyle\nabla\\!,v}\alpha$ is of order $\mathcal{O}(h^{\ell+2})$
due to our choice of frame field — thus negligible when analyzing the
convergence under refinement of the integrated covariant exterior derivative
of the bundle-valued form $\alpha$. Only the boundary integral term is of
order $\mathcal{O}(h^{\ell+1})$ and thus dominates under refinement. It holds

$\int_{s}(d^{\nabla}\alpha)^{\nabla,v}=\int_{\partial
s}\alpha^{\nabla,v}+\mathcal{O}(h^{\ell+2}).$ (30)

##### Consequences.

From Eq. (29), we notice that we can define a discrete bundle-valued exterior
calculus built out of discrete forms which are the integrals of their
continuous counterparts evaluated using parallel-propagated frames: once the
high-order wedge product term is neglected, we get that the PPF-evaluated
integral of the covariant exterior derivative of a form $\alpha$ over a
simplex $s$ can be well approximated by PPF-evaluated integrals of the form
$\alpha$ over the boundary faces of $s$ — a simple extension of the regular
discrete exterior derivative for scalar-based forms defined via Stokes’
theorem. The remainder of this paper describes this approach in detail; we
will even show that notions that we proposed previously, like the discrete
curvature two-form for example, can also be understood in light of this PPF-
based calculus, providing a formal link between discrete and continuous
calculus.

###  4.2 Revisiting the Discretization of Bundle-valued Forms

While Def. 3.5 formalized a discrete bundle-valued differential form as a
collection of abstract maps, we can now express what these maps are in
relation to the continuous case. Once this discretization (often referred to
the de Rham map for scalar-based discrete exterior calculus [29]) is defined,
we can then define a notion of reconstruction which provides a differential
form interpolating the discrete mesh values (referred to as the Whitney map in
scalar-based discrete exterior calculus [29]).

##### De Rham map for bundle-valued forms.

Based on the continuous retraction explained in Sec. 4.1, the value of a
discrete form evaluated on a simplex $\sigma$ at a vertex $v$ could be defined
as the PPF-induced integration of its continuous counterpart form, where a
simplicial region $s$ of a continuous manifold is approximated as the simplex
$\sigma$, i.e.,

$\smash{\boldsymbol{\alpha}(\sigma,v)\coloneqq\int_{s}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v}\alpha}\in
E_{v}.$ (31)

However, having to store values for each pair of simplex $\sigma$ and corner
vertex $v$ grows rapidly with the dimension of the simplex, and it does not
match the number of degrees of freedom of the scalar case of discrete or
finite-element exterior calculus.

Instead, we propose to use a _barycenter-based PPF_ per simplex, leading to a
single value per $\ell-$simplex for an $\ell-$form as conventional
discretizations of exterior calculus of scalar-based forms [6, 15, 2]. Corner
evaluations for this simplex can then be achieved through parallel transport
$\mathcal{R}_{v,c_{s}}$ from its barycenter $c_{s}$ to one of its corners $v$
— which is either evaluated using the continuous parallel transport or
approximated using the discrete $1-$connection (through the integral of a
least-squares estimate of a linear $\omega$ within the simplex or the use of
Whitney basis $1-$forms); in this case, the discretization of a bundle-valued
form evaluated on a simplex $\sigma$ at a vertex $v$ becomes:

$\boldsymbol{\alpha}(\sigma,v)=\mathcal{R}_{v,c_{s}}\int_{s}\mathcal{R}^{\scriptscriptstyle\nabla\\!,c_{s}}\alpha\in
E_{v}.$ (32)

In practice, one could store directly the barycenter-based value, or just one
of its derived corner values per simplex $\sigma$ as all other corner values
can be recovered through parallel transport. While these two definitions
exactly match for $1-$forms as parallel transport along edges is known (while
it has to be interpolated from edge values for higher-order simplices), they
differ by $\mathcal{O}(h^{\ell+3})$ for a general $\ell-$dimensional
simplicial region $s$. Note that both definitions are frame-dependent: they
rely on a canonical frame field, the PPF. This is a specificity of bundle-
valued forms on non-flat manifolds as it involves non-commutative compositions
of parallel transport: discretization must account for a particular choice of
a local frame field to even make sense. Once these initial simplex values are
established via the resulting de Rham maps we just described, the actual
continuous manifold can be forgotten as all computations of our discrete
calculus will solely use these values of discrete forms.

##### Whitney map for bundle-valued forms.

Given the definition of the de Rham map described above, constructing a
Whitney map (mapping all the barycenter-based values of all $\ell-$simplices
of a discrete bundle-valued $\ell-$form $\boldsymbol{\alpha}$ to a
differential bundle-valued form $\tilde{\alpha}$) is rather simple. For a
simplex $\tau$ of dimension $\dim(\tau)\geq\ell$, let $t$ be the simplicial
region that is the image of $\tau$ under a diffeomorphism following the
conventions in Def. 4.1. Let $\sigma^{\ell}(\tau)$ be the set of all simplices
of dimension $\ell$ included in $\tau$. The reconstruction of a differential
$\ell-$form $\tilde{\alpha}$ expressed in the barycenter-based PPF of $\tau$
can be achieved through the weighted sum of the discrete bundle-valued forms
for all $\ell-$dimensional simplices $\sigma\\!\in\\!\sigma^{\ell}(\tau)$. The
pointwise evaluation of $\tilde{\alpha}$ for a given point $p$ then depends on
the lowest-dimensional simplex $\tau$ that contains $p$: the evaluation of the
bundle-valued $\ell-$form $\tilde{\alpha}$ on $\tau$ at the point
$p\\!\in\\!\tau$ given by

$\tilde{\alpha}_{|_{\tau}}(p)=\mathcal{R}^{\scriptscriptstyle\nabla\\!,c_{\tau}}(p)^{-1}\sum_{\sigma\in\sigma^{\ell}(\tau)}\mathcal{R}_{c_{\tau},c_{\sigma}}\,\phi_{\sigma}(p)\,\boldsymbol{\alpha}(\sigma,c_{\sigma})\;\in\Omega^{\ell}(\tau,E),\vspace*{-1mm}$
(33)

where $\phi_{\sigma}$ is the Whitney $\ell-$form associated with $\sigma$
within $\tau$, and $\mathcal{R}_{c_{\tau},c_{\sigma}}$ is barycenter-to-
barycenter parallel transport constructed based on the discrete $1-$form
connection. One can check that the de Rham map of a Whitney map of a discrete
bundled-valued form is the identity, providing a proper link between
continuous and discrete bundle-valued forms: indeed, for a given $\ell-$cell
$\eta$, the Whitney map for all point $p$ inside $\eta$ is given by

$\tilde{\alpha}_{p}=\mathcal{R}^{\scriptscriptstyle\nabla\\!,c_{\eta}}(p)^{-1}\sum_{\sigma\in\sigma^{\ell}(\eta)}\mathcal{R}_{c_{\eta},c_{\sigma}}\,\phi_{\sigma}(p)\,\boldsymbol{\alpha}(\sigma,c_{\sigma}){=}\mathcal{R}^{\scriptscriptstyle\nabla\\!,c_{\eta}}(p)^{-1}\mathcal{R}_{c_{\eta},c_{\eta}}\,\phi_{\eta}(p)\,\boldsymbol{\alpha}(\eta,c_{\eta}),$

since the set of all $\ell-$simplices $\sigma^{\ell}(\eta)$ is reduced to
$\eta$ for each $p\\!\in\\!{\eta}$. Therefore the re-discretization of the
Whitney map through the integral in the parallel-propagated frame field yields
back the original discrete form.

######  Remark 4.4.

Note that we provide, through Eq. (33), a piecewise expression of the Whitney
map to convert the discrete values of $\ell-$simplices into a differential
form. This should not be surprising as transitioning between simplices of
different dimensions means that the local frame field changes. Consequently,
the coordinate representation of the form’s evaluation may exhibit jumps
across simplices. While readers may find this behavior perplexing, it mirrors
the jumping behavior already observed in scalar-valued Whitney forms when
changing cells, as only tangential conitnuity is enforced. Moreover, in our
case we must now also accommodate the varying frame fields, leaving no choice
but to use a piecewise definition.

###  4.3 Discrete Exterior Covariant Derivative for Bundle-valued Forms

In this section, we formally define our discrete operator for the exterior
covariant derivative, in two steps to highlight the similarities and
differences with previous work.

##### Sided Exterior Covariant Derivative.

Guided by Eq. (29), Eq. (30) and our meaning of a discrete form from Eq. (31),
we first define a discrete operator $\mathfrak{d}^{\boldsymbol{\nabla}}$ which
approximates the exterior covariant derivative.

######  Definition 4.5 (PPF-induced Covariant Exterior Derivative).

Let $M$ be a discrete manifold and $\mathbf{E}\\!\to\\!M$ be a discrete vector
bundle with connection $\boldsymbol{\nabla}\\!=\\!\\{\mathcal{R}_{ij}\\}$. Let
$\sigma\\!=\\![v_{0},\ldots,v_{\ell+1}]$ be an $(\ell+1)-$simplex and
$\boldsymbol{\alpha}$ a discrete $\mathbf{E}-$valued $\ell-$form. We define
the _sided_ discrete exterior derivative of the form $\boldsymbol{\alpha}$ as

$\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}([v_{0},\ldots,v_{\ell+1}],v_{0})=\mathcal{R}_{0,1}\
\boldsymbol{\alpha}([v_{1},\ldots,v_{\ell+1}],v_{1})+\sum_{i=1}^{\ell+1}(-1)^{i}\boldsymbol{\alpha}([v_{0},\ldots,\hat{v}_{i},\ldots,v_{\ell+1}],v_{0}).$
(34)

######  Remark 4.6.

The operator $\mathfrak{d}^{\boldsymbol{\nabla}}$ was already introduced in an
equivalent form in [5] as _the_ exterior covariant derivative. One of our
contributions is to show that if the discrete form $\boldsymbol{\alpha}$
arises through integration in a vertex-based PPF and the discrete connection
$\boldsymbol{\nabla}$ arises through integration from a smooth connection
$\nabla$, then this operator approximates the integral in Eq. (29) with

$\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}([v_{0},\ldots,v_{\ell+1}],v_{0})\\!=\\!\int_{s}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v_{0}}\
d^{\nabla}\alpha\\!+\\!\mathcal{O}(h^{\ell+2})$

and, thus, converges under refinement. To see this, consider the case of a
one-form $\alpha$ and a triangle $\sigma:=[v_{a},v_{b},v_{c}]\subset M$. From
Eq. (29), we know that the integral over $\sigma$ of the exterior covariant
derivative in the PPF based at $v_{a}$ turns into

$\displaystyle\int_{\sigma}R^{\scriptscriptstyle\nabla\\!,v_{a}}d^{\nabla}\alpha=\int_{[v_{a},v_{b}]}R^{\scriptscriptstyle\nabla\\!,v_{a}}\alpha+\int_{[v_{b},v_{c}]}R^{\scriptscriptstyle\nabla\\!,v_{a}}\alpha+\int_{[v_{c},v_{a}]}R^{\scriptscriptstyle\nabla\\!,v_{a}}\alpha+\mathcal{O}(h^{3}),$

see Remark 4.3. Def. 4.5 tells us that
$\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}([v_{a},v_{b},v_{c}],v_{a})\coloneqq\mathcal{R}_{v_{a}v_{b}}\boldsymbol{\alpha}([v_{b},v_{c}],{v_{b}})-\boldsymbol{\alpha}([v_{a},v_{c}],{v_{a}})+\boldsymbol{\alpha}([v_{a},v_{b}],{v_{a}}).$
Since our values from the discrete form $\boldsymbol{\alpha}$ were induced
from a $v_{a}-$centered PPF, we have

$\int_{[v_{a},v_{b}]}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v_{a}}\alpha=\boldsymbol{\alpha}([v_{a},v_{b}],{v_{a}})\quad\text{and}\quad\int_{[v_{c},v_{a}]}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v_{a}}\alpha=-\boldsymbol{\alpha}([v_{a},v_{c}],{v_{a}}),$

hence the continuous and discrete definitions match if we can prove that:

$\mathcal{R}_{v_{a},v_{b}}\boldsymbol{\alpha}([v_{b},v_{c}],{v_{b}})-\int_{[v_{b},v_{c}]}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v_{a}}\alpha=\mathcal{O}(h^{3}).$

Indeed, one has:

$\mathcal{R}_{v_{a},v_{b}}\int_{[v_{b},v_{c}]}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v_{b}}\alpha-\int_{[v_{b},v_{c}]}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v_{a}}\alpha=\int_{[v_{b},v_{c}]}\left(\mathcal{R}_{v_{a},v_{b}}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v_{b}}-\mathcal{R}^{\scriptscriptstyle\nabla\\!,v_{a}}\right)\alpha,$
(35)

and since for any point $p$ in $\sigma$ we can build an auxiliary simplex
$[v_{a},v_{b},p]$ such that Eq. (35) turns into an integral of evaluations of
discrete curvature expression in the sense of Def. 3.12, we get

$\left(\mathcal{R}_{v_{a},v_{b}}\mathcal{R}^{\scriptscriptstyle\nabla\\!,v_{b}}-\mathcal{R}^{\scriptscriptstyle\nabla\\!,v_{a}}\right)(p)=\boldsymbol{\Omega^{\nabla}}([v_{a},v_{b},p],v_{a},p)\;\sim\;\mathcal{O}(h^{2}).$

Below, we will delve into further detail regarding how the discrete curvature,
defined as the difference of parallel transport, accurately approximates the
sampled smooth curvature up to order $\mathcal{O}(h^{3})$. This analysis will
provide justification for the estimation
$\boldsymbol{\Omega^{\nabla}}([v_{a},v_{b},p],v_{a},p)\;\sim\;\mathcal{O}(h^{2})$.

Hence, the discrete operator $\mathfrak{d}^{\boldsymbol{\nabla}}$ will
converge in the limit to the correct pointwise value of the exterior covariant
derivative. Note that this argument holds for a form of arbitrary order
$\ell$: each time, it is the opposite face to the evaluation vertex that needs
to be examined to prove the equivalence between discrete and continuous
definitions up to $\mathcal{O}(h^{\ell+2})$.

##### Averaging Derivatives.

While we proved the convergence of $\mathfrak{d}^{\boldsymbol{\nabla}}$ to its
continuous equivalence, one may note that the order of accuracy is not
sufficient to ensure that two consecutive applications of this operator
($\mathfrak{d}^{\boldsymbol{\nabla}}\circ\mathfrak{d}^{\boldsymbol{\nabla}}$)
will still converge, which is the real obstacle to establishing the Bianchi
identities (Eqs. (12) and (13)). We thus further alter our discrete exterior
covariant derivative operator to gain one additional order of accuracy through
local averaging, with an operator we denote as the _alternation operator_.

######  Definition 4.7 (Alternation Operator for Discrete Bundle-Valued
Forms).

Let $\mathbf{E}$ be a discrete vector bundle with connection
$\boldsymbol{\nabla}\\!=\\!\\{\mathcal{R}_{ij}\\}_{(ij)\in\mathcal{E}}$ over a
discrete manifold $M$. Let $\sigma=[v_{0},\ldots,v_{\ell}]\in M$ be an
$\ell-$simplex. For a discrete $\mathbf{E}$ valued $\ell-$form
$\boldsymbol{\alpha}$ we define its “alternated” form
$\smash{\mathrm{Alt}^{\boldsymbol{\nabla}}(\boldsymbol{\alpha})}$ on $\sigma$
evaluated at $v_{0}$ via

$\mathrm{Alt}^{\boldsymbol{\nabla}}(\boldsymbol{\alpha})([v_{0},\ldots,v_{\ell}],{v_{0}})=\frac{1}{(\ell+1)!}\sum_{\pi\in
S_{\ell+1}}\mathrm{sgn}(\pi)\mathcal{R}_{v_{0},v_{\pi(0)}}\boldsymbol{\alpha}([v_{\pi(0)},\ldots,v_{\pi(\ell)}],{v_{\pi(0)}}).$

where $S_{\ell+1}$ is the set of all permutations $\pi$ of the $(\ell+1)$
vertices of $\sigma$.

Note that this operator will allow us to transform the operator
$\mathfrak{d}^{\boldsymbol{\nabla}}$ so that the vertex $v_{0}$ in Eq. (34)
from Def. 4.5 is no longer singled out arbitrarily: the reshuffling from the
alternation operator will average all the similar exterior covariant
derivative estimates using all other vertices. Moreover, one can easily check
that the connection $1-$form is stable under this alternation operator.

######  Remark 4.8.

Note that the _alternation of a discrete bundle-valued form_ requires a
connection to allow for the averaging to be performed in a common fiber, in
contrast to the direct symmetrization of the smooth case. It is thus easy to
show that in general,
$\mathrm{Alt}^{\boldsymbol{\nabla}}(\mathrm{Alt}^{\boldsymbol{\nabla}}(\boldsymbol{\alpha}))(\sigma,v_{0})\neq\mathrm{Alt}^{\boldsymbol{\nabla}}(\boldsymbol{\alpha})(\sigma,v_{0})$,
i.e., the alternation operator is not a projector. However, we preserve the
fact that the alternation operator commutes with the pullback of a simplicial
map in the sense that
$\mathrm{Alt}^{f^{\ast}\boldsymbol{\nabla}}(f^{\ast}\boldsymbol{\alpha})(\sigma,{v_{0}})\\!=\\!\mathrm{Alt}^{\boldsymbol{\nabla}}(\boldsymbol{\alpha})(f(\sigma),{f(v_{0})})$.

##### Discrete Exterior Covariant Derivative.

We now propose to change the formula of the exterior derivative as follows.

######  Definition 4.9 (Discrete Bundle-valued Exterior Covariant Derivative).

Let $\mathbf{E}$ be a discrete vector bundle with connection
$\boldsymbol{\nabla}$ over a discrete manifold $M$. Let $\sigma\subset M$ with
$\sigma=[v_{0},\ldots,v_{\ell+1}]$ be a $(\ell+1)-$simplex. For a discrete
vector-valued differential $\ell-$form $\boldsymbol{\alpha}$ we define the
discrete covariant exterior derivative with the alternation from Def. 4.7 as

$\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}([v_{0},\ldots,v_{\ell+1}],{v_{0}})=\mathrm{Alt}^{\boldsymbol{\nabla}}\mathfrak{d}^{\boldsymbol{\nabla}}(\boldsymbol{\alpha})([v_{0},\ldots,v_{\ell+1}],{v_{0}}).$

Applying the alternation operator after $\mathfrak{d}^{\boldsymbol{\nabla}}$
removes the arbitrariness of picking the second vertex in the simplex $\sigma$
in Eq. (34), thus in effect averaging various _sided_ exterior covariant
derivative estimates of equal approximation order $\mathcal{O}(h^{\ell+2})$.
Through the averaging, the resulting discrete exterior covariant derivative
gains one order of accuracy, becoming accurate to $\mathcal{O}(h^{\ell+3})$.
This accuracy improvement is akin to the accuracy gain one obtains when using
centered vs. sided finite difference approximations of derivatives. More
precisely, we will show in the convergence proof in Thm. 5.9 that if
$\alpha\in\Omega^{\ell}(M,E)$ and $\boldsymbol{\alpha}$ is the associated
discrete form as defined in Eq. (31), then the following holds

$\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}([v_{0},\ldots,v_{\ell+1}],v_{0})\\!=\\!\mathcal{R}_{v_{0},c_{s}}\int_{s}\mathcal{R}^{\scriptscriptstyle\nabla\\!,c_{s}}\,d^{\nabla}\alpha\\!+\\!\mathcal{O}(h^{\ell+3}),$

where $c$ denotes the barycenter of a region. That is, the alternation results
in a barycenter-based PPF evaluation, parallel transported back to the
evaluation vertex $v_{0}$. This barycentric PPF, replacing the vertex-based
PPF, is what leads to an improvement in accuracy, even though the
discretization of $\alpha$ is still achieved through an integral within a
vertex-based PPF.

###  4.4 Discrete Endomorphism-valued Exterior Covariant Derivative

While we defined a discrete exterior covariant derivative applicable to
bundle-valued forms in Sec. 4.3, we need to extend this operator to discrete
two-prong endomorphism-valued forms as well in order to be able to apply it to
discrete curvatures, for instance.

######  Definition 4.10.

(PPF-induced discrete exterior covariant derivative for endomorphism-valued
forms) Let $\mathbf{E}$ be a discrete vector bundle over a discrete manifold
$M$ with connection $\boldsymbol{\nabla}\\!=\\!\\{\mathcal{R}_{ij}\\}$. Let
$\sigma=[v_{0},\ldots,v_{\ell+1}]\subset M$ be a $(\ell+1)-$simplex and
$\boldsymbol{\beta}$ be a discrete $(1,1)-$tensor valued $\ell-$form. Finally,
for $v\\!\in\\!\sigma$, let
$\sigma_{v}=\smash{[v_{0},\ldots,\hat{v},\ldots,v_{\ell+1}]}$ be the
simplicial face of $\sigma$ opposite to vertex $v$. We define the PPF-induced
discrete exterior covariant derivative on simplex $\sigma$ for $(1,1)-$tensor
valued forms given an evaluation fiber $\smash{\mathbf{E}_{v_{0}}}$ and a cut
fiber $\mathbf{E}_{v_{\ell+1}}$ through:

$\mathfrak{d}^{\\!\boldsymbol{\nabla}}\boldsymbol{\beta}(\sigma,v_{0},v_{\ell+1})\\!=\\!\mathcal{R}_{01}\boldsymbol{\beta}(\sigma_{v_{0}},v_{1},v_{\ell+1})+\sum_{i=1}^{\ell}(-1)^{i}\boldsymbol{\beta}(\sigma_{v_{i}},v_{0},v_{\ell+1})+(-1)^{\ell+1}\boldsymbol{\beta}(\sigma_{v_{\ell+1}},v_{0},v_{\ell})\mathcal{R}_{\ell,\ell+1}.\vspace*{-3mm}$

This new definition is simply a variant of Def. 4.5 to account for the two-
prong nature of discrete endomorphism-valued forms. It was already proposed,
as is, in [5]; but for the same numerical issues discussed earlier, we must
complement this sided definition with a post-alternation, which also requires
a new definition for the case of $(1,1)-$tensor-valued forms, as given below.

######  Definition 4.11.

(Alternation Operator for Discrete $(1,1)-$tensor-valued Forms) Let
$\mathbf{E}$ be a discrete vector bundle with connection
$\boldsymbol{\nabla}\\!=\\!\\{\mathcal{R}_{ij}\\}_{(ij)\in\mathcal{E}}$ over a
discrete manifold $M$. Let $\sigma=[v_{0},\ldots,v_{\ell}]\in M$ be an
$\ell-$simplex. For a discrete $(1,1)-$tensor valued $\ell-$form
$\boldsymbol{\beta}$ we define its “alternated” form
$\smash{\mathrm{Alt}^{\boldsymbol{\nabla}}(\boldsymbol{\beta})}$ for an
$\ell-$form $\boldsymbol{\beta}$ on the simplex $\sigma$ evaluated in $v_{0}$
with input in $v_{\ell}$ through

$\displaystyle\mathrm{Alt}^{\boldsymbol{\nabla}}\boldsymbol{\beta}$
$\displaystyle([v_{0},\ldots,v_{\ell}],{v_{0}},v_{\ell})$
$\displaystyle=\frac{1}{(\ell+1)!}\sum_{{\pi\in
S_{\ell+1}}}\left(\frac{1+\mathrm{sgn}(\pi)}{2}\mathcal{R}_{v_{0},v_{\pi(0)}}\boldsymbol{\beta}([v_{\pi(0)},\ldots,v_{\pi(\ell)}],{v_{\pi(0)}},{v_{\pi(\ell)}})\mathcal{R}_{v_{\pi(\ell)},v_{\ell}}\right.$
$\displaystyle\quad\left.+\frac{\mathrm{sgn}(\pi)-1}{2}\mathcal{R}_{v_{0},v_{\pi(0)}}\boldsymbol{\beta}([v_{\pi(0)},\ldots,v_{\pi(\ell)}],{v_{\pi(0)}},{v_{\pi(\ell)}})\mathcal{R}_{v_{\pi(\ell)},v_{\pi(\ell-1)}}\mathcal{R}_{v_{\pi(\ell-1)},v_{\ell}}\right).$

######  Remark 4.12 (Invariance of the alternation for the curvature form).

Note that the curvature form is stable under this definition of the
alternation of $(1,1)-$tensor valued forms: for a simplicial cell
$\sigma=[abc],$ one can directly check that

$\displaystyle\mathrm{Alt}^{\boldsymbol{\nabla}}\boldsymbol{\Omega^{\nabla}}([abc],a,c)$
$\displaystyle=\frac{1}{6}\big{(}\boldsymbol{\Omega^{\nabla}}([abc],a,c)+R_{ab}\boldsymbol{\Omega^{\nabla}}([bca],b,a)R_{ac}+R_{ac}\boldsymbol{\Omega^{\nabla}}([cab],c,b)R_{bc}$
$\displaystyle\;\;\;\;-\boldsymbol{\Omega^{\nabla}}([acb],a,b)R_{bc}-R_{ab}\boldsymbol{\Omega^{\nabla}}([bac],b,c)-R_{ac}\boldsymbol{\Omega^{\nabla}}([cba],c,a)R_{ab}R_{bc}\big{)}$
$\displaystyle=\boldsymbol{\Omega^{\nabla}}([abc],a,c).\vspace*{-3mm}$

This property would not have held if an average with pre- and post-
composition along a joining edge would have been used.

From the two operators above, we can now define our discrete exterior
covariant derivative for endomorphism-valued forms.

######  Definition 4.13 (Discrete endomorphism-valued exterior covariant
derivative).

Let $\mathbf{E}$ be a discrete vector bundle over a discrete manifold $M$ with
connection $\boldsymbol{\nabla}\\!=\\!\\{\mathcal{R}_{ij}\\}$. Let
$\sigma=[v_{0},\ldots,v_{\ell+1}]\subset M$ be a $(\ell+1)-$simplex and
$\boldsymbol{\beta}$ be a discrete $(1,1)-$tensor valued $\ell-$form. We
define the discrete covariant exterior derivative on simplex $\sigma$ for
$(1,1)-$tensor valued forms given an evaluation fiber
$\smash{\mathbf{E}_{v_{0}}}$ and a cut fiber $\mathbf{E}_{v_{\ell+1}}$
through:

$\boldsymbol{d^{\nabla}}\boldsymbol{\beta}([v_{0},\ldots,v_{\ell+1}],{v_{0}},{v_{\ell+1}})=\mathrm{Alt}^{\boldsymbol{\nabla}}\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\beta}([v_{0},\ldots,v_{\ell+1}],{v_{0}},{v_{\ell+1}}).$

As we will discuss at length in Sec. 5.2 when we provide numerical tests, our
proposed operator for endomorphism-valued forms satisfies our initial goals of
proper accuracy order and consistency with the vector-valued case (so that we
obtain a discrete algebraic Bianchi identity). However, it is _not_ the only
definition with these properties. For instance, we average over all possible
permutations, but one can in fact pick only a subset of permutations (which,
in practice, improves the efficiency of the computations). In this case, not
all combinatorial properties will be fulfilled. However, the improved order of
convergence in the limit can still be maintained.

###  4.5 Revisiting Discrete Curvature

The earlier definition of the discrete curvature two-form
$\boldsymbol{\Omega^{\nabla}}$ in Sec. 3.4 was _postulated_ directly from a
discrete connection $\boldsymbol{\nabla}$ (and its associated connection one-
form $\boldsymbol{\omega}$, see Sec. 3.3) as an early concrete example of
discrete (1,1)-tensor-valued form: it was obtained by extending the notion of
holonomy-based curvature by measuring the difference of parallel transport on
two different paths between a “cut” and an “evaluation” fiber. Now that the
link between discrete bundle-valued forms and continuous ones through PPF-
dependent integration has been established, we revisit this key notion to show
that it mirrors the continuous definition of the curvature two-form after a
specific contraction-based integration.

##### Continuous case.

We consider a _contractible_ two-dimensional region $C$ (assumed, without loss
of generality, to be a half-disk of a smooth manifold $M$ and a vector bundle
$\pi:E\rightarrow M$ with connection $\nabla$. Given a local frame field
$\\{f_{a}\\}$, the connection can be expressed with the local connection
$1-$form $\omega$, moreover the curvature $2-$form is locally given as
$\Omega^{\nabla}=d\omega+\omega\wedge\omega$, see Eq. (11). Let us select two
points, $v$ and $w$, on the boundary $\partial C$ of $C$. The point $v$
corresponds to the evaluation vertex discussed above in the discrete case,
while $w$ corresponds to the cut vertex. We denote by
$\gamma_{v,w}\subset\partial C$ the path joining $v$ and $w$ along the
boundary of $C$ with positive orientation, and by $\gamma_{v,w}^{\prime}$ the
corresponding boundary path with negative orientation,

such that $\gamma_{v,w}-\gamma_{v,w}^{\prime}=\partial C.$ We proceed by
generating a new frame field, denoted as $\\{\tilde{f}_{a}\\}$, through
parallel transport. The initial frame, $\\{f_{a}(v)\\}$, situated at vertex
$v$, is transported along two distinct paths: $\gamma_{v,w}$ and
$\gamma_{v,w}^{\prime}$. The discrepancy between the frames, which results at
vertex $w$ after the parallel transports, is intricately linked to the
holonomy. Extending this frame construction to the interior of $C$ involves a
retraction corresponding to a 2D polar $(\rho,\theta)$ parametrization of the
interior of $C$: in this parametrization, $\mathbf{e}_{\rho}$ aligns
consistently with the retracted boundary, as depicted in Fig. 5, with vertex
$w$ fixed. From this parametrization, we can construct a new frame field by
parallel transporting the boundary frames along the $\mathbf{e}_{y}$ direction
towards the interior of $C$. This process forms a new frame field,
$\tilde{f}_{b}R^{b}_{a}=f_{a}$, where $R$ denotes the rotation field
responsible for adapting vector components from the local frame $\\{f_{a}\\}$
into the new frame $\\{\tilde{f}_{a}\\}$.

It now holds for the connection $1-$form $\tilde{\omega}$ in this new frame
field that $\tilde{\omega}=R\omega R^{-1}-dRR^{-1}.$ Consequently, for any
point $p\in C\setminus w$ in the punctured region, one has
$\tilde{\omega}_{p}(e_{\rho})=0.$ The curvature form $\tilde{\Omega}^{\nabla}$
in the new frame field becomes $\tilde{\Omega}^{\nabla}=R\Omega R^{-1}$ (see
Sec. 2). With the components expressed in $\tilde{F}$, one has

$\widetilde{\Omega}^{\nabla}=d\tilde{\omega}+\tilde{\omega}\wedge\tilde{\omega}.$

However in the punctured region $C\setminus w$,
$\tilde{\omega}\wedge\tilde{\omega}\equiv 0,$ since
$\tilde{\omega}\wedge\tilde{\omega}(e_{\rho},e_{\theta})=\tilde{\omega}(e_{\rho})\tilde{\omega}(e_{\theta})-\tilde{\omega}(e_{\theta})\tilde{\omega}(e_{\rho})=0$
except at the singularity $w$. Thus, the integral of the curvature becomes:

$\int_{C}R{\Omega}^{\\!\nabla}R^{-1}=\int_{C}\widetilde{\Omega}^{\\!\nabla}=\int_{C}{d}\tilde{\omega}+\tilde{\omega}\wedge\tilde{\omega}=\int_{\partial
C}\tilde{\omega}.$ (36)

In other words, the fact that we parallel-propagated the frame $\\{f_{a}\\}$
at $v$ through retraction proves that the mismatch at $w$ when we parallel
transport the frame along the two sides of the boundary from $v$ (which was,
in essence, the meaning of our discrete curvature evaluation) is exactly equal
to this retraction-induced connection-dependent _integral of the curvature
$2-$form_.

Figure 5: Retraction-based parameterization. (Left): a retraction of the disk
defines a $(\rho,\theta)$ (polar) parameterization of the (punctured) disk,
where the $\rho$ direction is along the retraction direction, while the
$\theta$ direction corresponds to a transversal direction. (Right): Starting
from a parallel-transported frame field $F_{\text{eval}}$ from
$v_{\text{eval}}$ along the boundary, we can parallel-propagate this boundary
frame field towards the interior of the punctured disk along the curves
corresponding to the $y$ direction of this retraction-based parameterization.
The curvature integral simplifies to the discrepancy between frames at
$v_{\text{cut}}$, transported from $v_{\text{eval}}$ via the retraction, and
evaluated in
$\operatorname{Hom}(\mathbf{E}_{v_{\text{cut}}},\mathbf{E}_{v_{\text{eval}}})$.
Figure 6: Disk to Triangle. A parallel-propagated frame over a disc induces a
parameterization within the triangle once a homeomorphism to transform the
disc into a triangle is defined. We illustrate the resulting parallel-
transport paths as images under the homeomorphism map. The integrated
curvature in this frame field precisely corresponds to the disparity in
parallel transport along the two distinct boundary paths connecting the cut
and evaluation fibers.

##### Consequences for the discrete curvature two form.

The interpretation of the frame mismatch described above brings a series of
consequences. First and foremost, it puts our definition of the two-prong
discrete curvature $2-$form sketched in Sec. 3.4 on solid footing as the
continuous argument above remains valid on a $2-$simplex
$\sigma\\!=\\![v,u,w]$. Additionally, it also justifies the summability of
this definition, in stark contrast to a holonomy-based notion of curvature: if
one considers a second simplex $\sigma^{\prime}\\!=\\![w,z,v]$ sharing the
edge ${v,w}$, then the discrete curvature
$\boldsymbol{\Omega^{\\!\nabla}}(\sigma,v,w)$ and the discrete curvature
$\boldsymbol{\Omega^{\\!\nabla}}(\sigma^{\prime},v,w)$ can be summed to become
the discrete non-simplicial curvature
$\boldsymbol{\Omega^{\\!\nabla}}(\sigma\mathop{\cup}\sigma^{\prime},v,w)$
since the two integrals sum trivially: the two canonical simplicial
retractions form a retraction over their union, which means that the induced
parallel-propagated frame field on each simplex corresponds to the frame
obtained by applying the aforementioned construction for the non-simplicial
union cell (see Fig. 7). Consequently, the discrete notion of two-prong
curvature applies as is for non-simplicial two-dimensional contractible cells,
and the sum of two curvatures over a region joined by a discrete edge path
delimited by the evaluation vertex and the cut vertex happens naturally as the
parallel transports on each side of the common edge path cancel out. Finally,
the interpretation of the two-prong discrete curvature as the discrete
equivalent of a connection-based integral of the continuous form implies that
$\boldsymbol{\Omega^{\\!\nabla}}$ at an evaluation vertex $v$ of a simplex
$\sigma$ of diameter $h$ is equivalent to the pointwise evaluation of the
continuous curvature $2-$form at $v$ (on the two unit tangent vectors along
the two outgoing edges at $v$), up to $\mathcal{O}(h^{3})$. More precisely,
for a simplex $\sigma=[v_{\mathrm{eval}},w,v_{\mathrm{cut}}]$ with evaluation
in $E_{v_{\mathrm{eval}}}$ and cut in $E_{v_{\mathrm{cut}}}$ we obtain that
the curvature integral in the novel frame field reduces to the mismatch of the
frame observed at $v_{\text{eval}}$, expressed in $F_{\text{cut}}$, i.e

$\int_{\sigma}R\Omega^{\nabla}R^{-1}=\int_{\partial\sigma}\tilde{\omega}=R_{v_{\mathrm{eval}}{w}}R_{wv_{\mathrm{cut}}}-R_{v_{\mathrm{eval}},v_{\mathrm{cut}}}.$

By construction it holds for the gauge field at the evaluation vertex
$R_{v_{\mathrm{eval}}}=\mathrm{Id}$. Thus, if we expand the gauge field $R$ at
$v_{\mathrm{eval}}$ (or the respective Lie algebra element) we obtain that

$\int_{\sigma}R\Omega^{\nabla}R^{-1}=\int_{\sigma}\Omega^{\nabla}+\mathcal{O}(h^{3}).$

Additionally, sampling the curvature at $v_{\mathrm{eval}}\in\sigma$ yields

$\int_{\sigma}\Omega^{\nabla}=\Omega^{\nabla}_{v_{\mathrm{eval}}}(v_{\mathrm{cut}}-v_{\mathrm{eval}},w-v_{\mathrm{eval}})+\mathcal{O}(h^{3})$

and therefore

$\displaystyle\int_{\sigma}R\Omega^{\nabla}R^{-1}$
$\displaystyle=R_{v_{\mathrm{eval}}w}R_{wv_{\mathrm{cut}}}-R_{v_{\mathrm{eval}}v_{\mathrm{cut}}}$
$\displaystyle=\boldsymbol{\Omega^{\nabla}}(\sigma,v_{\mathrm{eval}},v_{\mathrm{cut}})=\Omega^{\nabla}_{v_{\mathrm{eval}}}(v_{\mathrm{eval}}-v_{\mathrm{cut}},v_{\mathrm{eval}}-w)+\mathcal{O}(h^{3}).$
(37)

We will show in our section describing our numerical tests that the discrete
curvature, with subsequent pre- and post-composition, converges in fact to the
integral of the curvature $2-$form in a center-based parallel-propagated frame
field with an error decay of $\mathcal{O}(h^{4})$.

Figure 7: Summing Curvatures. For two triangles using the same two prongs
($v_{\mathrm{cut}},v_{\mathrm{eval}}$), the curvature evaluated over their
union will be the sum of the curvatures with the same two prongs because the
parallel-propagated frames derived from these two prongs properly align.

##### PPF-induced Discrete Covariant Exterior Derivative for the Connection
One Form.

When computing the discrete covariant exterior derivative of a vector-valued
form, only an evaluation fiber suffices for the definition of a parallel-
propagated frame field. However, when dealing with the exterior covariant
derivative of the connection $1-$form, it becomes necessary to specify both a
cut and evaluation fibers since the result is a curvature, i.e., a
$(1,1)-$tensor in this particular case. Upon specifying these parameters, we
obtain an induced parallel propagated frame field, as illustrated in Fig. 5.
This reduction leads to the boundary integral:

$\int_{C}R{\Omega}^{\\!\nabla}R^{-1}=\int_{C}\widetilde{\Omega}^{\\!\nabla}=\int_{C}{d}\tilde{\omega}+\tilde{\omega}\wedge\tilde{\omega}=\int_{\partial
C}\tilde{\omega}.$

To evaluate the Lie algebra integral, we can treat the columns of the
connection $1-$form as three independent vector valued $1-$forms. This
motivates the definition of the PPF-induced exterior covariant derivative,
evaluated at $v_{\mathrm{eval}}$ and cut at $v_{\mathrm{cut}}$:

$\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\omega}([v_{\mathrm{eval}},w,v_{\mathrm{cut}}],v_{\mathrm{eval}},v_{\mathrm{cut}}):=\boldsymbol{\omega}_{v_{\mathrm{eval}},w}+\mathcal{R}_{w,v_{\mathrm{eval}}}\boldsymbol{\omega}_{w,v_{\mathrm{cut}}}-\boldsymbol{\omega}_{v_{\mathrm{eval}},v_{\mathrm{cut}}},$

for a given discrete connection $\boldsymbol{\nabla}$ with discrete connection
form $\boldsymbol{\omega}$.

Note that this PPF-induced discrete covariant exterior derivative shares a
resemblance with the vector-valued one, akin to the smooth case. By regarding
the matrix of the discrete connection $1-$form as a set of $r$ vector-valued
differential forms, the PPF-induced discrete covariant exterior derivative
manifests as a vector-valued discrete exterior covariant derivative on these
vector components. The case of the connection $1-$form is, however, special:
the covariant exterior derivative of the connection $1-$form is not a Lie-
algebra valued quantity anymore, but a $(1,1)-$tensor valued $2-$form instead,
necessitating in the discrete case the addition of a choice of “cut fiber”.
Using the definition of the discrete connection $1-$form, see Eq. (20), we
obtain that indeed:

$\displaystyle\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\omega}([v_{\mathrm{eval}},w,v_{\mathrm{cut}}],v_{\mathrm{eval}},v_{\mathrm{cut}})$
(38)
$\displaystyle=(\mathcal{R}_{v_{\mathrm{eval}},w}-\mathrm{Id})+\mathcal{R}_{v_{\mathrm{eval}},w}(\mathcal{R}_{w,v_{\mathrm{cut}}}-\mathrm{Id})-(\mathcal{R}_{v_{\mathrm{eval}},v_{\mathrm{cut}}}-\mathrm{Id})$
$\displaystyle=\mathcal{R}_{v_{\mathrm{eval}},w}\mathcal{R}_{w,\mathrm{cut}}-\mathcal{R}_{{v_{\mathrm{eval}}},v_{\mathrm{cut}}}=\boldsymbol{\Omega^{\nabla}}([v_{\mathrm{eval}},w,v_{\mathrm{cut}}],v_{\mathrm{eval}},v_{\mathrm{cut}}),$
(39)

justifying, a posterior, our definition of discrete curvature.

##  5 Properties and Convergence of our Discrete Operators

In the previous section, we introduced a discrete calculus for discrete
bundle-valued forms in a principled manner, based on parallel-propagated
frames. We now shift our focus towards discussing the resulting properties of
our discrete operators, before analyzing their convergence under refinement.
We will then provide quantitative evidence through numerical tests to verify
our claims, both in terms of properties and of convergence order.

###  5.1 Properties of the Discrete Exterior Derivative

The discrete covariant exterior derivative $\boldsymbol{d^{\nabla}}$
introduced in Def. 4.9 has several intrinsic properties, and as it differs
from the operator introduced by [5] mostly through a post alternation, it also
inherits combinatorial properties proven in [5]. We review them next.

##### Naturality of $\boldsymbol{d^{\nabla}}$.

As mentioned in Eq. (10), it is an important result in the theory of smooth
exterior calculus that the pullback commutes with the exterior derivative. For
the discrete operator $\boldsymbol{d^{\nabla}}$, a similar result holds true
due to a lemma proven in [5].

######  Lemma 5.1 (Pullback commutes with
$\mathfrak{d}^{\boldsymbol{\nabla}}$, from [5]).

Given a simplicial map $f\colon M\\!\to\\!N$ between two discrete manifolds
$M$ and $N$, and a discrete vector bundle with connection
$(\mathbf{E},\boldsymbol{\nabla})$ over $N$, let $\boldsymbol{\alpha}$ be an
$\mathbf{E}-$valued discrete $\ell-$form. For a $(\ell\\!+\\!1)-$simplex
$\sigma=[v_{0},\ldots,v_{\ell+1}]$ of $M$, one has:

$f^{\ast}(\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha})(\sigma,{v_{0}})=\mathfrak{d}^{f^{\ast}\boldsymbol{\nabla}}f^{\ast}\boldsymbol{\alpha}(f(\sigma),{f(v_{0})})$

######  Corollary 5.2 (Naturality of $\boldsymbol{d^{\nabla}}$).

Given a simplicial map $f\colon M\\!\to\\!N$ between two discrete manifolds
$M$ and $N$, and a discrete vector bundle with connection
$(\mathbf{E},\boldsymbol{\nabla})$ over $N$, let $\alpha$ be an
$\mathbf{E}-$valued discrete $\ell-$form. For a $(\ell\\!+\\!1)-$simplex
$\sigma=[v_{0},\ldots,v_{\ell+1}]$ of $M$, one has:

$f^{\ast}(\boldsymbol{d^{\nabla}}\alpha)(\sigma,{v_{0}})=\boldsymbol{d}^{f^{\ast}\boldsymbol{\nabla}}f^{\ast}\alpha(f(\sigma),{f(v_{0})})$

###### Proof.

This is a direct consequence of the previous lemma and the fact that the
pullback commutes with the alternation operator as mentioned in Rem. 4.8. ∎

##### Antisymmetry of $\boldsymbol{d^{\nabla}}\\!$.

In the scalar-valued case, discrete differential forms are skew-symmetric
maps, in the sense that permuting the vertices of an evaluation simplex leads
to multiplication with the sign of the permutation. In the bundle-valued case,
discrete differential forms are defined as abstract maps mapping into a vertex
fiber of a given cell. Therefore, our operator is antisymmetric as long as we
keep the evaluation fiber fixed, since any switch of two vertices will result
in a sign flip in the evaluation of $\boldsymbol{d^{\nabla}}$ due to the
alternation within its definition.

##### Differential Bianchi identity.

The celebrated second Bianchi identity reviewed in Eq. (13) holds exactly in
the discrete setting.

######  Proposition 5.3 (Differential Bianchi identity).

Let $\mathbf{E}$ be a discrete vector bundle with connection
$\boldsymbol{\nabla}$ over a discrete manifold $M$. For any $3-$simplex
$\sigma=[v_{0},v_{1},v_{2},v_{3}]\subset M$. one has:

$\boldsymbol{d^{\nabla}}\boldsymbol{\Omega^{\nabla}}(\sigma,{v_{0}},{v_{3}})=0$
(40)

###### Proof.

As proven in [5], we know that
$\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\Omega^{\nabla}}(\sigma,{v_{0}},{v_{3}})=0$.
Therefore,

$\boldsymbol{d^{\nabla}}\boldsymbol{\Omega^{\nabla}}(\sigma,{v_{0}},{v_{3}})=\mathrm{Alt}^{\boldsymbol{\nabla}}\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\Omega^{\nabla}}(\sigma,{v_{0}},{v_{3}})=0.\vspace*{-7mm}$

∎

##### Algebraic Bianchi identity.

The first Bianchi identity, reviewed in Eq. (17), also holds in our discrete
calculus. While [5] showed that their covariant exterior derivative satisfies
$\mathfrak{d}^{\boldsymbol{\nabla}}\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}([v_{0},\ldots,v_{\ell+2}],v_{0})\\!=\\!\boldsymbol{\Omega^{\nabla}}([v_{0},v_{1},v_{2}],v_{0},v_{2})\
\boldsymbol{\alpha}([v_{2},\ldots,v_{\ell+2}],v_{2})$ for any vector-valued
form $\boldsymbol{\alpha}$, our discrete covariant exterior derivative
satisfies the algebraic Bianchi identity for an implicitly-defined
combinatorial wedge product.

######  Theorem and Definition 5.4 (Algebraic Bianchi Identity).

Let $M$ be a discrete manifold with a discrete bundle $\mathbf{E}$ and
connection $\boldsymbol{\nabla}$. Further let $\boldsymbol{\alpha}$ be a
discrete $\mathbf{E}-$valued $\ell-$form and
$\sigma=[v_{0},\ldots,v_{\ell+2}]$ a $(\ell+2)-$simplex. In this case there
exists a set $K\subseteq C^{2}(\sigma)\times C^{\ell}(\sigma)$ consisting of
$2-$ and $\ell-$subcells of $\sigma$ such that for all $(f,\kappa)\in K$ with
a shared vertex $w_{f,\kappa}$ we have

$\boldsymbol{d^{\nabla}d^{\nabla}\alpha}(\sigma,v_{0})=\frac{1}{(\ell+3)!\,(\ell+2)!}\sum_{(f,\kappa)\in
K}\boldsymbol{\Omega^{\nabla}}(f,v_{0},w_{f,\kappa})\
\boldsymbol{\alpha}(\kappa,w_{f,\kappa})=:\boldsymbol{\Omega^{\nabla}\wedge\alpha}(s,v_{0}),\vspace{-1mm}$

where the sum defines implicitly a _wedge product_ between the curvature
$2-$form and $\boldsymbol{\alpha}$, reminiscent of scalar-based wedge products
defined through cup products [21]. Similarly, for any $(1,1)-$tensor-valued
$\ell-$form $\boldsymbol{\beta}$, two consecutive applications of our discrete
covariant exterior derivative $\boldsymbol{d^{\nabla}}$ yield implicitly a
discrete commutator:

$\boldsymbol{d^{\nabla}d^{\nabla}\beta}=\boldsymbol{[\Omega^{\nabla}\wedge\beta]}.$

###### Proof.

For discrete $(1,0)-$tensor valued forms, see App. A. The case of discrete
$(1,1)-$tensor valued forms is similar — but for an evaluation fiber
$\smash{\mathbf{E}_{v_{0}}}$ and a cut fiber $\mathbf{E}_{v_{\ell+2}}$ one
has:

$\displaystyle\mathfrak{d}^{\boldsymbol{\nabla}}\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\beta}([v_{0},\ldots,v_{\ell+2}],v_{0},v_{\ell+2})$
$\displaystyle=\boldsymbol{\Omega^{\nabla}}([v_{0},v_{1},v_{2}],v_{0},v_{2})\boldsymbol{\beta}([v_{2},\ldots,v_{\ell+2}],v_{2},v_{\ell+2})$
$\displaystyle\qquad-\boldsymbol{\beta}([v_{0},\ldots,v_{\ell}],v_{0},v_{\ell})\boldsymbol{\Omega^{\nabla}}([v_{\ell},v_{\ell+1},v_{\ell+2}],v_{\ell},v_{\ell+2}),$

hence the resulting commutator. ∎

######  Remark 5.5.

In the realm of scalar-valued DEC, [24] established theoretical limitations
regarding the definition of a discrete scalar-valued wedge product. However,
associativity of the wedge product is not an issue in our context, as there is
no inherent product structure on the bundle; and graded anti-commutativity is
irrelevant given the non-commutative nature of the underlying product for
$(1,1)-$tensors, and vectors. We will demonstrate, in Sec. 5.3 that our
discrete wedge product converges under refinement toward the smooth wedge
product when applied to the proper discretization of a smooth form — unlike
the definition given in [5] as they did not properly relate discrete and
continuous bundle-valued forms, see Fig. 11.

######  Remark 5.6.

Given a connection
$\boldsymbol{\nabla}\\!=\\!\\{\mathcal{R}_{ab}\\}_{e_{ab}\in\mathcal{E}}$ and
an arbitrary choice of frame field, the discrete connection $1-$form for the
edge between vertices $a$ and $b$ is given by
$\boldsymbol{\omega}_{ab}={R}_{ab}-\mathrm{Id}$. Furthermore, we saw in Eq.
(39) that

$\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\omega}([abc],a,c)=\boldsymbol{\Omega^{\nabla}}([abc],a,c).$

Hence, with our alternation operator for $(1,1)-$tensor based forms and the
stability of $\boldsymbol{\Omega^{\nabla}}$ under the $\operatorname{Alt}$
operator (see Rem. 4.12), it holds that

$\boldsymbol{d^{\nabla}}\boldsymbol{\omega}([abc],a,c)=\boldsymbol{\Omega^{\nabla}}([abc],a,c),$
(41)

in an exact sense. However it should be noted that the use of
$\boldsymbol{d^{\nabla}}$ above is slightly abusive: it is neither the
discrete covariant exterior derivative of a $(1,0)-$tensor valued form, nor of
a $(1,1)-$tensor valued form. Instead, it is made out of the composition of a
PPF-induced covariant exterior derivative for $(1,0)-$tensor valued form
applied to $\omega$, composed with the alternation operator for $(1,1)-$tensor
valued forms. This _hybrid_ operator is specific to the connection $1-$form
and reflects the special and local nature of the curvature form in the
continuous case.

We should also mention here that our linearized version of the matrix
logarithm used in the definition of the discrete connection $1-$form is
crucial to ensure that this property holds (see Rem. 3.10). Finally, we proved
that $\boldsymbol{d^{\nabla}d^{\nabla}}=\boldsymbol{\Omega^{\nabla}\wedge}$
for any vector-valued form, and one can check that indeed, for a discrete
vector-valued $0-$form $\mathbf{z}$, we have:

$\displaystyle\boldsymbol{d^{\nabla}}\boldsymbol{d^{\nabla}}\mathbf{z}([abc],a)$
$\displaystyle=\frac{1}{6}\Bigl{(}(R_{ab}R_{bc}-R_{ac})z_{c}+R_{ab}(R_{bc}R_{ca}-R_{ba})z_{a}+R_{ac}(R_{ca}R_{ab}-R_{cb})z_{b}$
$\displaystyle\quad-(R_{ac}R_{cb}-R_{ab})z_{b}-R_{ab}(R_{ba}R_{ac}-R_{bc})z_{c}-R_{ac}(R_{cb}R_{ba}-R_{ca})z_{a}\Bigr{)}.$

Note that this wedge product between the curvature $2-$ form and a vector-
valued $0-$form does _not_ correspond to a simple pointwise product as in the
continuous case: this property will only be valid in the limit of mesh
refinement.

###  5.2 Convergence analysis

In Sec. 4.1, we introduced the use of parallel-propagated frame fields to
simplify the high-order terms of the integral of the discrete covariant
exterior derivative. In this section, we discuss how the order of
approximation is affected by the location of the _origin_ of the PPF being
used: indeed, if we integrate the exterior derivative in a center-based
parallel-propagated frame field, we achieve an even higher accuracy order than
using a corner-based PPF.

Figure 8: Parallel-propagated frames. Given a connection, we can use the
parallel propagated frame of $T_{p}M$ (bottom) for integration; integration in
some non-parallel-propagated/non-geometric frame (like the one on top) does
not lead to convergence.

######  Theorem 5.7 (Improved accuracy with center-based PPF).

Let $\pi:E\rightarrow M$ be a vector bundle with connection $\nabla$. Let
$s=[v_{0},\ldots,v_{\ell}]\\!\subset\\!M$ be a region in $M$ for which there
exists a diffeomorphism to a $\ell-$simplex $\sigma$ where each point $v_{i}$
is mapped to an associated vertex $w_{i}$ of $\sigma$, and let us call the
“center” $c_{s}\\!\in\\!s$ the point of $s$ being mapped to the barycenter of
$\sigma$. For $\alpha\in\Omega^{\ell}(M,E)$ it holds that

$\int_{s}{(d^{\nabla}\alpha)}^{\nabla,c_{s}}=\int_{\partial
s}\alpha^{\nabla,c_{s}}+\mathcal{O}(h^{\ell+3})$

(compare with Eq. (30)).

###### Proof.

It holds in the center-based PPF that

$\int_{s}(d^{\nabla}\alpha)^{\nabla,c_{s}}=\int_{\partial
s}\alpha^{\nabla,c_{s}}+\int_{s}\omega^{\nabla,c_{s}}\wedge\alpha^{\nabla,c_{s}}.$

Additionally, one has

$\int_{s}\omega^{\nabla,c_{s}}\wedge\alpha^{{\nabla,c_{s}}}=\mathcal{O}(h^{\ell+3})$

since the integral of the linear part cancels out when the center verifies
$\omega^{\nabla,{c_{s}}}(c_{s})=0$. To see this, we consider a Taylor
expansion of $\omega^{\nabla,{c_{s}}}$ and $\alpha$. It holds for $p\in s$

$\left(\omega^{\nabla,c_{s}}\right)_{p}=\underbrace{\left(\omega^{\nabla,c_{s}}\right)_{c_{s}}}_{=0}+\left(\omega^{\nabla,c_{s}}\right)^{\prime}_{c_{s}}(p-c_{s})+\mathcal{O}(h^{2}),$

where of $(\omega^{\nabla,c_{s}})^{\prime}$ is to be understood as a
derivative applied to the components of $\omega^{\nabla,c_{s}}$. Assuming
sufficient smoothness, we get that $(\omega^{\nabla,c_{s}})^{\prime}$ is still
a smooth $1-$form. Similarly, we obtain

$\alpha^{\nabla,c_{s}}_{p}=\alpha^{\nabla,c_{s}}_{c_{s}}+(\alpha^{\nabla,c_{s}})^{\prime}_{c_{s}}(p-c_{s})+\mathcal{O}(h^{\ell+2}).$

Hence

$\displaystyle\norm{\int_{s}\omega^{\nabla,c_{s}}\wedge\alpha^{\nabla,c_{s}}}$
$\displaystyle\leq\norm{\int_{p\in
s}((\omega^{\nabla,c_{s}})_{c_{s}}^{\prime}\wedge\alpha^{\nabla,c_{s}}_{c_{s}})(c_{s}-p)dp}+\underbrace{\norm{\int_{s}((\omega^{\nabla,c_{s}})^{\prime}_{c_{s}}\wedge(\alpha^{\nabla,c_{s}})^{\prime}_{c_{s}})}}_{=\mathcal{O}(h^{\ell+1})}\cdot\mathcal{O}(h^{2})+\mathcal{O}(h^{\ell+3}).$

Since
$(\omega^{\nabla,c_{s}})_{c_{s}}^{\prime}\wedge\alpha^{\nabla,c_{s}}_{c_{s}}$
is constant over $s$, we get that

$\int_{p\in
s}((\omega^{\nabla,c_{s}})_{c_{s}}^{\prime}\wedge(\alpha^{\nabla,c_{s}})_{c_{s}})(c_{s}-p)dp=0,$
(42)

leaving us with
$\norm{\int_{s}\omega^{\nabla,c_{s}}\wedge\alpha^{\nabla,c_{s}}}=\mathcal{O}(h^{\ell+3})$.
∎

######  Theorem 5.8.

Let $\pi:E\rightarrow M$ be a vector bundle with connection $\nabla$. Let
$s=[v_{0},\ldots,v_{\ell+1}]\\!\subset\\!M$ be a region in $M$ for which there
exists a diffeomorphism to a $(\ell+1)-$simplex $\sigma$ where each point
$v_{i}$ are mapped to an associated vertex $w_{i}$ of $\sigma$. Further let
$\alpha\in\Omega^{\ell}(M,E)$ and $\boldsymbol{\alpha}$ be its discretization
defined in Eq. (31). In this case we have

$R_{v_{0},c_{s}}\int_{s}(d^{\nabla}\alpha)^{\nabla,{c_{s}}}-\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}(s,{v_{0}})=\mathcal{O}(h^{\ell+2}),$
(43)

or expressed intrinsically,

$\mathcal{R}_{v_{0},c_{s}}\;\
\mathop{\mathrlap{\mkern-10.0mu\raisebox{5.5pt}{$\underset{\scriptscriptstyle\varphi_{c_{s}}}{\scriptstyle\nabla}$}}\int}\limits_{s}d^{\nabla}\alpha\;-\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}(s,{v_{0}})=\mathcal{O}(h^{\ell+2}).$
(44)

###### Proof.

We use the identifications stated in Remark 4.3. It holds in the center-based
parallel propagated frame that

$R_{v_{0},c_{s}}\int_{s}(d^{\nabla}\alpha)^{\nabla,{c_{s}}}=R_{v_{0},c_{s}}\int_{\partial
s}R_{c_{s}}\alpha+R_{v_{0},c_{s}}\int_{s}{\omega}^{\nabla,c_{s}}\wedge
R_{c_{s}}\alpha.$ (45)

As discussed above, we know

${\int_{s}{\omega}^{\nabla,c_{s}}\wedge\alpha^{\nabla,c_{s}}}=\mathcal{O}(h^{\ell+3}).$

In the following we denote
$s_{i}=[v_{0},\ldots,\hat{v_{i}},\ldots,v_{\ell+1}]$ the $\ell-$subsimplex of
$s$ without $v_{i}$. By $f_{i}$ we will denote the $\ell-$chain opposite to
$v_{i}$ with _positive_ orientation relative to that of $s,$ i.e.,
$f_{i}=(-1)^{i}s_{i}$. Extending discrete $\boldsymbol{\alpha}$ to $f_{i}$ by
$\boldsymbol{\alpha}(f_{i},v_{0})=(-1)^{i}\ \boldsymbol{\alpha}(s_{i},v_{0}),$
we have

$\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}(s,{v_{0}})=R_{v_{0},v_{1}}\
\boldsymbol{\alpha}(s_{0},v_{1})+\sum_{i=1}^{\ell+1}(-1)^{i}\
\boldsymbol{\alpha}(s_{i},v_{0})=R_{v_{0},v_{1}}\
\boldsymbol{\alpha}(f_{0},v_{1})+\sum_{i=1}^{\ell+1}\
\boldsymbol{\alpha}(f_{i},v_{0}).$

For all faces containing $v_{0}$, we have by definition

$\boldsymbol{\alpha}(f_{i},{v_{0}})=\int_{f_{i}}R^{\nabla,v_{0}}\alpha.$

If we compare the contribution of the PPF-derivative to the contribution from
the boundary integral in Eq. (45) coming from $f_{0}$, we obtain

$R_{v_{0},c_{s}}\int_{f_{0}}R^{\nabla,c_{s}}\alpha-
R_{v_{0},v_{1}}\int_{f_{0}}R^{\nabla,v_{1}}\alpha=\int_{f_{0}}(R_{v_{0},c_{s}}R^{\nabla,c_{s}}-R_{v_{0},v_{1}}R^{\nabla,v_{1}})\alpha.$
(46)

If we do a Taylor expansion for

$g\colon f_{0}\to\mathbb{R}^{r\times r}\colon
p\mapsto(R_{v_{0},c_{s}}R^{\nabla,c_{s}}-R_{v_{0},v_{1}}R^{\nabla,v_{1}})(p)$

at $c_{f_{0}}$, we obtain

$g(c_{f_{0}})=(\underbrace{R_{v_{0},c_{s}}R_{c_{s},c_{f}}}_{R_{v_{0},c_{f}}}-R_{v_{0},v_{1}}R_{v_{1},c_{f}})=-\boldsymbol{\Omega^{\nabla}}([v_{0},v_{1},c_{f}],{v_{0}},{c_{f}})+\mathcal{O}(h^{3})$

and therefore, we get

$g(p)=\boldsymbol{\Omega^{\nabla}}([v_{0},v_{1},c_{f}],{v_{0}},{c_{f}})+\mathcal{O}(h^{3})$

for $p\in f_{0}$. Now consider any face $f\in\partial s$ containing $v_{0}$
with positive orientation. From Sec. 4.5,

$R_{v_{0},c_{s}}\int_{f}R_{c_{s}}\alpha-
R_{v_{0},c_{f}}\int_{f}R_{c_{f}}\alpha=-\Omega_{v_{0}}^{\nabla}(v_{1}-v_{0},c_{f}-v_{0})\alpha_{c_{f}}[f]+\mathcal{O}(h^{\ell+3}).$
(47)

However we have

$\Omega_{v_{0}}^{\nabla}(v_{1}-v_{0},c_{f}-v_{0})\alpha_{c_{f}}[f]=\mathcal{O}(h^{\ell+2}).$

We obtain for the contribution of the error coming from $f$ in Eq. (43), again
with a Taylor expansion at $c_{f}$,

$\int_{f}R_{v_{0}}\alpha-
R_{v_{0},c_{s}}\int_{f}R_{c_{s}}\alpha=\Omega_{v_{0}}^{\nabla}(c_{f}-v_{0},c_{s}-v_{0})\cdot\alpha_{c_{f}}[f]+\mathcal{O}(h^{\ell+3}).$
(48)

Since
$\Omega_{v_{0}}^{\nabla}(c_{f}-v_{0},c_{s}-v_{0})\cdot\alpha_{c_{f}}[f]\\!=\\!\mathcal{O}(h^{\ell+2})$,
this yields the claim. ∎

Since the volume of the simplicial cell is of order $\mathcal{O}(h^{\ell+1})$,
this shows that the operator $\mathfrak{d}^{\boldsymbol{\nabla}}$ converges
under refinement _if the input is obtained through integration in an
appropriate PPF._ However, to ensure convergence of a second application of
$\boldsymbol{d^{\nabla}}$, this is not sufficient since the error after
division with the volume of a $(\ell+1)-$simple can remain arbitrarily large.
This is where the alternation can improve the accuracy order.

######  Theorem 5.9 (Alternation increases accuracy).

Let $\pi:E\rightarrow M$ be a vector bundle with connection $\nabla$. Let
$s=[v_{0},\ldots,v_{\ell+1}]\\!\subset\\!M$ be a region in $M$ for which there
exists a diffeomorphism to a $(\ell+1)-$simplex $\sigma$ where each point
$v_{i}$ is mapped to an associated vertex $w_{i}$ of $\sigma$. Further let
$\alpha\in\Omega^{\ell}(M,E)$ and $\boldsymbol{\alpha}$ be its discretization
defined in Eq. (31). In this case, we have

$R_{v_{0},c_{s}}\int_{s}(d^{\nabla}\alpha)^{(\nabla,c_{s})}=\mathrm{Alt}^{\nabla}\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}(s,{v_{0}})+\mathcal{O}(h^{\ell+3})=\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(s,{v_{0}})+\mathcal{O}(h^{\ell+3}),$
(49)

or expressed intrinsically,

$\mathcal{R}_{v_{0},c_{s}}\ \
\mathop{\mathrlap{\mkern-10.0mu\raisebox{5.5pt}{$\underset{\scriptscriptstyle\varphi_{c_{s}}}{\scriptstyle\nabla}$}}\int}\limits_{s}d^{\nabla}\alpha=\mathrm{Alt}^{\nabla}\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}(s,{v_{0}})+\mathcal{O}(h^{\ell+3})=\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(s,{v_{0}})+\mathcal{O}(h^{\ell+3}).$
(50)

###### Proof.

It holds by Theorem (5.7)

$\int_{s}R_{v_{0},c_{s}}(d^{\nabla}\alpha)^{\nabla,c_{s}}=R_{v_{0},c_{s}}\int_{\partial
s}\alpha^{\nabla,c_{s}}+\mathcal{O}(h^{\ell+3}).$ (51)

Let $\\{f_{i}\\}\subset\partial s$ be the set of $\ell-$cells with positive
orientation forming the boundary of $s$ where $v_{i}\notin f_{i}$. In the
following, we will show that the contributions for every $\ell-$face coming
from Eq. (51) and those of $\boldsymbol{d^{\nabla}}(s,v_{0})$ differ by a term
of order $\mathcal{O}(h^{\ell+3})$. To see this, we rewrite the exterior
covariant derivative as

$\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(s,v_{0})=\mathrm{Alt}^{\boldsymbol{\nabla}}\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}(s,v_{0})=\frac{1}{\ell+2}\sum_{j=0}^{\ell+1}R_{v_{0},v_{j}}\Bigl{(}\frac{1}{(\ell+1)!}\sum_{\begin{subarray}{c}\pi\in
S_{\ell+2}\\\
\pi(0)=j\end{subarray}}\mathrm{sgn}(\pi)\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}(\pi(s),v_{j})\Bigr{)}.$
(52)

Let

$\mathrm{Alt}^{\boldsymbol{\nabla}}_{j}\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}(s,v_{j})\coloneqq\frac{1}{(\ell+1)!}\sum_{\pi\in
S_{\ell+2},\pi(0)=j}\mathrm{sgn}(\pi)\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}(\pi(s),v_{j})$

be the operator that uses an alternation on the face opposite to $v_{j}$ only.
For the boundary cell $f_{i}$ we consider the contributions coming from
$\mathrm{Alt}^{\boldsymbol{\nabla}}_{j}\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}$
to Eq. (52) for $i\neq j$ first. In this case the face $f_{i}$ is not opposite
to $v_{j}$. Hence, its contributions turn into

$R_{v_{0},v_{j}}\boldsymbol{\alpha}(f_{i},v_{j})=R_{v_{0},v_{j}}\int_{f_{i}}R^{\nabla,v_{j}}\alpha.$

For the case $j=i$, $f_{i}$ is the face opposite to the evaluation vertex of
$\mathrm{Alt}^{\boldsymbol{\nabla}}_{j}\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}(s,v_{j})$.
In this case, we can rewrite

$\mathrm{Alt}^{\boldsymbol{\nabla}}_{i}\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{\alpha}(s,v_{i})=\frac{1}{\ell+1}\sum_{v_{k}\in
f_{i}}\frac{1}{\ell!}\sum_{\begin{subarray}{c}\pi\in S_{\ell+2}\\\ \pi(0)=j\\\
\pi(1)=k\end{subarray}}\mathrm{sgn}(\pi)\mathfrak{d}^{\boldsymbol{\nabla}}\alpha(\pi(s),v_{j}).$

This yields that the contributions coming from $f_{i}$ are of the form

$R_{v_{0},v_{i}}\frac{1}{\ell+1}\sum_{k\neq
i}R_{v_{i},v_{k}}\int_{f_{i}}R^{\nabla,v_{k}}\alpha.$ (53)

If we analyze the difference of this expression to the integral of $\alpha$ in
the $v_{i}-$based parallel propagated frame, we obtain

$\frac{1}{\ell+1}\sum_{k\neq
i}R_{v_{i},v_{k}}\int_{f_{i}}R^{\nabla,v_{k}}\alpha-\int_{f_{i}}R^{\nabla,v_{i}}\alpha=\frac{1}{\ell+1}\sum_{k\neq
i}\int_{f_{i}}(R_{v_{i},v_{k}}R^{\nabla,v_{k}}-R^{\nabla,v_{i}})\alpha.$

Similar to the discussion in Theorem (5.7), we can argue that the integral of
the center-based Taylor expansion yields

$\frac{1}{\ell+1}\sum_{k\neq
i}\int_{f_{i}}(R_{v_{i},v_{k}}R^{\nabla,v_{k}}-R^{\nabla,v_{i}})\alpha=\frac{1}{\ell+1}\sum_{k\neq
i}(R_{v_{i},v_{k}}R_{v_{k},c_{f_{i}}}-R_{v_{i},c_{f_{i}}})\alpha_{c_{f_{i}}}[f_{i}]+\mathcal{O}(h^{\ell+3})$

As explained in Sec. 4.5 , it holds

$(R_{v_{i},v_{k}}R_{v_{k},c_{f_{i}}}-R_{v_{i},c_{f_{i}}})=\boldsymbol{\Omega^{\nabla}}([v_{i},v_{k},c_{f_{i}}],v_{i},c_{f_{i}})=\Omega^{\nabla}_{v_{i}}[v_{k}-v_{i},c_{f_{i}}-v_{i}]+\mathcal{O}(h^{3}).$

Therefore,

$\displaystyle\frac{1}{\ell+1}\sum_{k\neq
i}(R_{v_{i},v_{k}}R_{v_{k},c_{f_{i}}}-R_{v_{i},c_{f_{i}}})\alpha_{c_{f_{i}}}[f_{i}]$
$\displaystyle=\frac{1}{\ell+1}\sum_{k\neq
i}\Omega^{\nabla}_{v_{i}}[v_{k}-v_{i},c_{f_{i}}-v_{i}]\alpha_{c_{f_{i}}}[f_{i}]+\mathcal{O}(h^{\ell+3})$
$\displaystyle=\Omega^{\nabla}_{v_{i}}\Bigl{[}\frac{1}{\ell+1}\sum_{k\neq
i}v_{k}-v_{i},c_{f_{i}}-v_{i}\Bigr{]}\alpha_{c_{f_{i}}}[f_{i}]+\mathcal{O}(h^{\ell+3}).$

Since $\frac{1}{\ell+1}\sum_{k\neq i}v_{k}=c_{f_{i}},$ we obtain

$\frac{1}{\ell+1}\sum_{k\neq
i}R_{v_{i},v_{k}}\int_{f_{i}}R^{\nabla,v_{k}}\alpha-\int_{f_{i}}R^{\nabla,v_{i}}\alpha\in\mathcal{O}(h^{\ell+3}).$

Therefore, we can deduce that the contribution of the face $f_{i}$ for the
difference
$\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(s,v_{0})-R_{v_{0},c_{s}}\int_{s}(d^{\nabla}\alpha)^{\nabla,c_{s}}$
is, up to order $\mathcal{O}(h^{\ell+3}),$ of the form

$\frac{1}{\ell+2}\sum_{j=0}^{\ell+1}R_{v_{0},v_{j}}\int_{f_{i}}R^{\nabla,v_{j}}\alpha-
R_{v_{0},c_{s}}\int_{f_{i}}R^{\nabla,c_{s}}\alpha=\frac{1}{\ell+2}\sum_{j=0}^{\ell+1}\int_{f_{i}}\Bigl{(}R_{v_{0},v_{j}}R^{\nabla,v_{j}}-R_{v_{0},c_{s}}R^{\nabla,c_{s}}\Bigr{)}\alpha.$

Integration of the Taylor expansion at $c_{f_{i}}$ yields

$\displaystyle\frac{1}{\ell+2}$
$\displaystyle\sum_{j=0}^{\ell+1}R_{v_{0},v_{j}}\int_{f_{i}}R^{\nabla,v_{j}}\alpha-
R_{v_{0},c_{s}}\int_{f_{i}}R^{\nabla,c_{s}}\alpha$
$\displaystyle=\frac{1}{\ell+2}\sum_{j=0}^{\ell+1}(R_{v_{0},v_{j}}R_{v_{j},c_{f_{i}}}-R_{v_{0},c_{s}}R_{c_{s},c_{f_{i}}})\alpha_{c_{f_{i}}}[f_{i}]+\mathcal{O}(h^{\ell+3})$
$\displaystyle=\frac{1}{\ell+2}\sum_{j=0}^{\ell+1}(R_{v_{0},v_{j}}R_{v_{j},c_{f_{i}}}-R_{v_{0},c_{f_{i}}}-(R_{v_{0},c_{s}}R_{c_{s},c_{f_{i}}}-R_{v_{0},c_{f_{i}}}))\alpha_{c_{f_{i}}}[f_{i}]+\mathcal{O}(h^{\ell+3})$
$\displaystyle=\frac{1}{\ell+2}\sum_{j=0}^{\ell+1}\Omega^{\nabla}_{v_{0}}[v_{j}-v_{0},c_{f_{i}}-v_{0}]\alpha_{c_{f_{i}}}[f_{i}]-\Omega^{\nabla}_{v_{0}}[c_{s}-v_{0},c_{f_{i}}-v_{0}]\
\alpha_{c_{f_{i}}}[f_{i}]+\mathcal{O}(h^{\ell+3})$
$\displaystyle=\Omega^{\nabla}_{v_{0}}\biggl{[}\underbrace{\frac{1}{\ell+2}\sum_{j=0}^{\ell+1}v_{j}}_{=c_{s}}-v_{0},c_{f_{i}}-v_{0}\biggr{]}\alpha_{c_{f_{i}}}[f_{i}]-\Omega^{\nabla}_{v_{0}}[c_{s}-v_{0},c_{f_{i}}-v_{0}]\
\alpha_{c_{f_{i}}}[f_{i}]+\mathcal{O}(h^{\ell+3})$
$\displaystyle=\mathcal{O}(h^{\ell+3}).$

Since $v_{i}$ was chosen arbitrarily, this result yields the claim. ∎

To conclude the convergence of
$\boldsymbol{d^{\nabla}}\boldsymbol{d^{\nabla}}$ under refinement, we need
another lemma.

######  Lemma 5.10.

Let $\pi:E\rightarrow M$ be a vector bundle with connection $\nabla$. Let
$s=[v_{0},\ldots,v_{\ell+1}]\\!\subset\\!M$ be a region in $M$ for which there
exists a diffeomorphism to a $\ell-$simplex $\sigma$ where each point $v_{i}$
are mapped to an associated vertex $w_{i}$ of $\sigma$. Let $f\subset s$ be a
$\ell-$subcell containing $v_{0}$. In this case, it holds that

$R_{v_{0},c_{f}}\int_{f}\alpha^{\nabla,c_{f}}=R_{v_{0},c_{s}}\int_{f}\alpha^{\nabla,c_{s}}+\mathcal{O}(h^{\ell+2}),$
(54)

which can be expressed intrinsically as

$\mathcal{R}_{v_{0},c_{f}}\ \
\mathop{\mathrlap{\mkern-10.0mu\raisebox{5.5pt}{$\underset{\scriptscriptstyle\varphi_{c_{f}}}{\scriptstyle\nabla}$}}\int}\limits_{f}\alpha=\mathcal{R}_{v_{0},c_{s}}\
\
\mathop{\mathrlap{\mkern-10.0mu\raisebox{5.5pt}{$\underset{\scriptscriptstyle\varphi_{c_{s}}}{\scriptstyle\nabla}$}}\int}\limits_{f}\alpha+\mathcal{O}(h^{\ell+2})$
(55)

where two PPFs (one from the center $c_{f}$ of $f$, one from the center
$c_{s}$ of $s$) are used, see Fig. 9.

###### Proof.

It holds

$\displaystyle
R_{v_{0},c_{s}}\int_{f}\alpha^{\nabla,c_{s}}-R_{v_{0},c_{f}}\int_{f}\alpha^{\nabla,c_{f}}$
$\displaystyle=\int_{f}(R_{v_{0},c_{s}}R^{\nabla,c_{s}}-R_{v_{0},c_{f}}R^{\nabla,c_{f}})\alpha$
$\displaystyle=(R_{v_{0},c_{s}}R_{c_{s},c_{f}}-R_{v_{0},c_{f}})\alpha_{c_{f}}[f]+\mathcal{O}(h^{\ell+3})$
$\displaystyle=\underbrace{\boldsymbol{\Omega^{\nabla}}([v_{0},c_{s},c_{f}],v_{0},c_{f})}_{=\mathcal{O}(h^{2})}\underbrace{\alpha_{c_{f}}[f]}_{=\mathcal{O}(h^{\ell})}=\mathcal{O}(h^{\ell+2}).$

∎

Figure 9: Visualization of the two parallel propagated frame fields used in
Lemma 5.10. It is noteworthy that the integral computed on the right-hand side
involves a PPF whose source is not contained within the integration domain
$f$.

This allows us to conclude the convergence of
$\boldsymbol{d^{\nabla}}\boldsymbol{d^{\nabla}}$ under refinement, as
explained next.

######  Corollary 5.11 (Convergence of
$\boldsymbol{d^{\nabla}}\boldsymbol{d^{\nabla}}$).

Let $\pi:E\rightarrow M$ be a vector bundle with connection $\nabla$. Let
$s=[v_{0},\ldots,v_{\ell+2}]\\!\subset\\!M$ be a region in $M$ for which there
exists a diffeomorphism to a $\ell-$simplex $\sigma$ where each point $v_{i}$
are mapped to an associated vertex $w_{i}$ of $\sigma$. Further, let
$\alpha\in\Omega^{\ell}(M,E)$ and $\boldsymbol{\alpha}$ be its discretization
defined in Eq. (31). We have

${\boldsymbol{d^{\nabla}}\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(\sigma,{v_{0}})}={R_{v_{0},c_{s}}\int_{s}(d^{\nabla}d^{\nabla}\alpha)^{\nabla,c_{s}}}+\mathcal{O}(h^{\ell+3}).$
(56)

or equivalently,

${\boldsymbol{d^{\nabla}}\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(\sigma,{v_{0}})}=\mathcal{R}_{v_{0},c_{s}}\
\
\mathop{\mathrlap{\mkern-10.0mu\raisebox{5.5pt}{$\underset{\scriptscriptstyle\varphi_{c_{s}}}{\scriptstyle\nabla}$}}\int}\limits_{s}(d^{\nabla}d^{\nabla}\alpha)+\mathcal{O}(h^{\ell+3}).$
(57)

###### Proof.

It holds

$\displaystyle
R_{v_{0},c_{s}}\int_{s}(d^{\nabla}d^{\nabla}\alpha)^{\nabla,c_{s}}=R_{v_{0},c_{s}}\int_{\partial
s}(d^{\nabla}\alpha)^{\nabla,c_{s}}+\mathcal{O}(h^{\ell+4}).$

Hence, by the previous lemma and since
$d^{\nabla}\alpha\in\Omega^{\ell+1}(M;E)$, we obtain,

$R_{v_{0},c_{s}}\int_{s}(d^{\nabla}d^{\nabla}\alpha)^{\nabla,c_{s}}=\sum_{f\in\partial
s}R_{v_{0},c_{s}}\int_{f}(d^{\nabla}\alpha)^{\nabla,c_{f}}+\mathcal{O}(h^{\ell+3}).$

As in the previous proof, let $f_{i}$ be the $(\ell+1)-$subcell in $\partial
s$ opposite to $v_{i}$ with positive orientation. Thus by Theorem 5.9 we
obtain

$R_{v_{0},c_{s}}\int_{s}(d^{\nabla}d^{\nabla}\alpha)^{\nabla,c_{s}}=R_{v_{0},c_{f_{0}}}\int_{f_{0}}(d^{\nabla}\alpha)^{\nabla,c_{f_{0}}}+\sum_{i=1}^{\ell+2}\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(f_{i},v_{0})+\mathcal{O}(h^{\ell+3}).$

For the first term, note that

$R_{v_{0},c_{f_{0}}}\int_{f_{0}}(d^{\nabla}\alpha)^{\nabla,c_{f_{0}}}=R_{v_{0},v_{1}}R_{v_{1},c_{f_{0}}}\int_{f_{0}}(d^{\nabla}\alpha)^{\nabla,c_{f_{0}}}-\boldsymbol{\Omega^{\nabla}}([v_{0},v_{1},c_{f_{0}}],v_{0},c_{f_{0}})\int_{f_{0}}(d^{\nabla}\alpha)^{\nabla,c_{f_{0}}}.$

Since

$\underbrace{\boldsymbol{\Omega^{\nabla}}([v_{0},v_{1},c_{f_{0}}],v_{0},c_{f_{0}})}_{=\mathcal{O}(h^{2})}\underbrace{\int_{f_{0}}(d^{\nabla}\alpha)^{\nabla,c_{f_{0}}}}_{=\mathcal{O}(h^{\ell+1})}=\mathcal{O}(h^{\ell+3})$

and

$R_{v_{0},v_{1}}R_{v_{1},c_{f_{0}}}\int_{f_{0}}(d^{\nabla}\alpha)^{\nabla,c_{f_{0}}}=R_{v_{0},v_{1}}\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(f_{0},v_{1}),$

we obtain

$\displaystyle{R_{0,c_{s}}\int_{s}(d^{\nabla}d^{\nabla}\alpha)^{\nabla,c_{s}}}$
$\displaystyle=R_{v_{0},v_{1}}\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(f_{0},v_{1})+\sum_{j=1}^{\ell+2}\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(f_{j},v_{0})+\mathcal{O}(h^{\ell+3})$
$\displaystyle=\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(s,v_{0})+\mathcal{O}(h^{\ell+3}).$

Furthermore,

$\displaystyle\mathrm{Alt}^{\boldsymbol{\nabla}}\left(\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(s,v_{0})\right)-R_{v_{0},c_{s}}\int_{s}(d^{\nabla}d^{\nabla}\alpha)^{\nabla,c_{s}}$
$\displaystyle=\mathrm{Alt}^{\boldsymbol{\nabla}}\left({R_{v_{0},c_{s}}\int_{s}(d^{\nabla}d^{\nabla}\alpha)^{\nabla,c_{s}}}\right)-R_{v_{0},c_{s}}\int_{s}(d^{\nabla}d^{\nabla}\alpha)^{\nabla,c_{s}}+\mathcal{O}(h^{\ell+3})$
$\displaystyle=\frac{1}{\ell+3}\sum_{j=0}^{\ell+2}\left(R_{v_{0},v_{j}}R_{v_{j},c_{s}}-R_{v_{0},c_{s}}\right)\left(\int_{s}(d^{\nabla}d^{\nabla}\alpha)^{\nabla,c_{s}}\right)+\mathcal{O}(h^{\ell+3})$
$\displaystyle=\Omega^{\nabla}\Bigl{(}v_{0}-c_{s},\underbrace{\frac{1}{\ell+3}\sum_{j=0}^{\ell+2}v_{j}-c_{s}}_{=0}\Bigr{)}\int_{s}(d^{\nabla}d^{\nabla}\alpha)^{\nabla,c_{s}}+\mathcal{O}(h^{\ell+3})\in\mathcal{O}(h^{\ell+3}).$

Hence
$\boldsymbol{d^{\nabla}}\boldsymbol{d^{\nabla}}\boldsymbol{\alpha}(s,{v_{0}})-R_{v_{0},c_{s}}\int_{s}(d^{\nabla}d^{\nabla}\alpha)^{\nabla,c_{s}}\in\mathcal{O}(h^{\ell+3})$,
which yields the claim. ∎

In the proof, we only show that the operators
$\boldsymbol{d^{\nabla}}\boldsymbol{d^{\nabla}}$ and
$\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{d^{\nabla}}$ are of the same
convergence order, although one might expect that a novel alternation could
augment the accuracy of the PPF-derivative. However, for a second application,
the alternation cannot compensate for the lack of accuracy from the original
input. Also numerical tests show that a novel application of the alternation
operator to $\mathfrak{d}^{\boldsymbol{\nabla}}\boldsymbol{d^{\nabla}}$ cannot
augment the accuracy further, indicating that a numerical realization of a
third application of the exterior covariant derivative will be, in this setup,
impossible to achieve accurately.

######  Remark 5.12.

This corollary confirms the convergence, under mesh refinement, of the
implicitly defined wedge product between the curvature $2-$form and a vector-
valued form, as introduced in Theorem 5.4. It converges towards the smooth
curvature wedge product in Eq. (12). Thus, the discrete algebraic Bianchi
Identity holds both conceptually and numerically in the limit under mesh
refinement.

######  Remark 5.13.

As discussed in the analysis preceding Def. 4.11, a noteworthy reduction in
the number of terms for the alternation can be used and will still ensure
enhanced convergence of the sided discrete covariant exterior derivative. Eq.
53 shows that we can reduce the number of permutations from the full
alternation group down to only $(\ell+1)-$terms, resulting in a substantial
decrease in the computational cost of the alternation.

######  Remark 5.14 (Convergence analysis for general endomorphism-valued
forms).

In the vector-valued case, we have successfully established that the discrete
exterior derivative converges under refinement and satisfies the Bianchi
identities. These results hold great significance within the context of our
study. We could successfully show that the differential Bianchi Identity holds
true in an exact sense. In the definition of the exterior covariant derivative
for $(1,1)-$tensor valued forms, we used an alternation of the PPF-derivative,
although the differential Bianchi Identity is already enforced for the PPF-
derivative exactly. However, for the analysis of convergence under refinement
of general endomorphism-valued forms, it is necessary to use an alternation of
the PPF-derivative as in the vector-valued case. It turns out that the
convergence analysis for arbitrary endomorphism-valued forms can be carried
out similarly to the vector-valued case. We saw that the alternated PPF-
derivative realizes a discrete notion of the identity

$d^{\nabla^{\mathrm{End}}}d^{\nabla^{\mathrm{End}}}\beta=[\Omega^{\nabla}\wedge\beta],$

for $\beta\in\Omega^{\ell}(M,\mathrm{End}(E))$. For the discretization of a