# Distributional Hessian and divdiv complexes on triangulation and cohomology

Kaibo Hu  School of Mathematics, the University of Edinburgh, James Clerk
Maxwell Building, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, UK.
<EMAIL_ADDRESS>,  Ting Lin  School of Mathematics, Peking University,
Beijing 100871, P.R.China<EMAIL_ADDRESS>and  Qian Zhang  Department
of Mathematical Sciences, Michigan Technological University, Houghton, MI
49931, USA<EMAIL_ADDRESS>
###### Abstract.

In this paper, we construct discrete versions of some Bernstein-Gelfand-
Gelfand (BGG) complexes, i.e., the Hessian and the divdiv complexes, on
triangulations in 2D and 3D. The sequences consist of finite elements with
local polynomial shape functions and various types of Dirac measure on
subsimplices. The construction generalizes Whitney forms (canonical conforming
finite elements) for the de Rham complex and Regge calculus/finite elements
for the elasticity (Riemannian deformation) complex from discrete topological
and Discrete Exterior Calculus perspectives. We show that the cohomology of
the resulting complexes is isomorphic to the continuous versions, and thus
isomorphic to the de Rham cohomology with coefficients.

Keywords. Bernstein-Gelfand-Gelfand sequences, cohomology, finite element
exterior calculus, discrete exterior calculus, Regge calculus

##  1\. Introduction

Preserving cohomological structures is crucial for reliable and efficient
numerical solutions of a large class of numerical PDEs [1, 2, 3]. For problems
involving vector-valued functions and differential forms, important structures
are encoded in the de Rham complex. There are canonical discretizations of the
de Rham complex as discrete differential forms in finite element exterior
calculus [1, 2, 3, 31]. In the lowest order case, this coincides with the
Whitney forms [7, 32], and the degrees of freedom of $k$-forms are distributed
on $k$-simplices, reflecting a discrete topological structure. See Figure 1
for an illustration of the 3D case.

Figure 1. The lowest-order finite element de Rham complex, consisting of
Whitney forms (the lowest-order Lagrange, Nédélec [46], and Raviart-Thomas
[50] elements).

Moreover, discretizations of differential forms are of fundamental interest.
Finite element de Rham complexes lead to Discrete Exterior Calculus (DEC)
schemes in some cases when the mass matrix is approximated [24, 33] and the
idea of discrete forms and cochains can be extended to graphs [44]. These
finite elements are conforming in the sense that the $k$-th space is a
subspace of $H(d^{k}):=\\{u\in L^{2}\Lambda^{k}:d^{k}u\in
L^{2}\Lambda^{k+1}\\}$, where $d^{k}$ is the $k$-th exterior derivative.

Motivated by Equilibrated Residual Error Estimators, Braess and Schöberl [8]
extended the concept of finite elements by permitting distributions as shape
functions. Specifically, they proposed a complex that, in 3D, comprises
piecewise constants, Dirac delta in the face normal direction, Dirac delta in
the edge tangential direction, and vertex deltas, respectively. See Figure 2.

Figure 2. Distributional finite element de Rham complex in 3D.

The distributional complex (Figure 2) can be viewed as the dual of the lowest
order Whitney forms (Figure 1). The construction has been systematically
extended to a double complex in [43] and the cohomology is also provided.

There is significant interest in identifying and discretizing differential
structures in other problems. A systematic approach is inspired by the
Bernstein-Gelfand-Gelfand (BGG) construction [4, 12, 10], where one derives
new complexes from existing ones (mostly de Rham complexes), and the
cohomology of the resulting complexes is isomorphic to the input.

For the three basic examples of the BGG complexes in 3D, i.e., the Hessian,
elasticity, and divdiv complexes, conforming finite elements on simplicial
meshes in 2D and 3D have been discussed in, e.g., [14, 17, 15, 13, 21, 26, 34,
38, 39]. There have also been results in nD [6, 16]. These results focus on
conforming finite elements and can be viewed as an extension of the study of
multivariate (simplicial) splines [41] from a homological perspective. Due to
the conformity requirement, the results unavoidably involve either high-order
polynomials and supersmoothness or special meshes. Discretizations for BGG
complexes involving Dirac delta also exist. For the elasticity complex,
Christiansen [20] interpreted Regge calculus as a finite element fitting in a
discrete complex (see Figure 3). The metric is in the Regge space, which is a
piecewise constant symmetric field with continuous tangential-tangential
components. The linearized curvature then consists of Dirac deltas along
hinges (edges). The Regge finite element has been extended in [42] to an
arbitrary polynomial degree, and was investigated for discretizing various
kinds of curvature tensors [25, 5, 28, 25, 29] and solving problems from
continuum mechanics and general relativity [42, 47, 48]. For elasticity
problems, the TDNNS discretization involving distributional spaces leads to a
convergent scheme for the Hellinger-Reissner principle [49]. For fluid
problems, analogous schemes can be found in [27]. For plate problems, spaces
in the Hellan-Herrmann-Johnson (HHJ) method can be viewed as a rotation of the
2D Regge element [48, 42].

Figure 3. Regge-elasticity complex [20].

As the Whitney forms both lead to successful finite element schemes and have a
neat discrete topological interpretation, it will be natural to ask the
question of what are the analogy of the Whitney forms for the BGG complexes.
Answers to this question may extend the discrete topological structures
encoded in the de Rham complexes to other differential structures encoded in
BGG, such as discrete Riemannian geometry. The Regge element and complex seem
to be a natural candidate for the elasticity complex due to the canonical
degrees of freedom, discrete geometric interpretation (in terms of Regge
metric) and the duality (the elasticity complex is formally self-adjoint on
the continuous level, which is preserved in the Regge complex).

In this paper, we generalize the Regge complex to the other two BGG complexes
in 3D, i.e., the Hessian complex and the divdiv complex, and show that their
cohomologies are isomorphic to the continuous versions. This construction is
inspired by a Discrete Exterior Calculus observation for the Regge complex. We
also obtain the corresponding results in 2D. This thus establishes the
cohomology for the Regge complex in 2D.

The rest of the paper will be organized as follows. In Section 2, we introduce
notations of topology, spaces and complexes at the continuous level. In
Section 3, we summarize the motivation and the main results. The technical
details are provided in Section 4 and Section 5. In Section 6, we provide
concluding remarks and an outlook.

##  2\. Function spaces and BGG complexes

###  2.1. Domain and topology

Throughout this paper, we assume that $\Omega$ is a bounded Lipschitz domain
following the definition of [23], i.e., $\Omega$ is a connected bounded open
set in $\mathbb{R}^{n}$ which is strong Lipschitz. Let $\Delta$ be a
triangulation (simplicial complex) of $\Omega$, and we use
$\mathsf{V},\mathsf{E},\mathsf{F},\mathsf{K}$ to denote the set of vertices,
edges, faces, and 3D cells, respectively. Note that in 2D, $\mathsf{F}$
denotes the top (two) dimensional cells. Their cardinality is denoted as
$\sharp_{V},\sharp_{E},\sharp_{F},\sharp_{K}$, respectively. We use
$\mathsf{V}_{0}$ and $\mathsf{V}_{\partial}$ to denote the set of internal
vertices and boundary vertices, respectively. Their cardinality is denoted as
$\sharp_{V_{0}}$ and $\sharp_{V_{\partial}}$, respectively. Similar notations
are used for edges and faces.

We refer to standard texts (e.g., [1, 30]) for definitions of homology and
cohomology. But we provide a brief reminder on simplicial homology
$\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)$ and its relative version
$\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta;\partial\Delta)$, together
with their relations to the de Rham cohomologies, which will be used in the
rest of the paper.

We first introduce the orientation function. Given a $k$-simplex
$\sigma=[x_{1},x_{2},\cdots,x_{k}]$ and a $(k-1)-$simplex $\tau$, define

(2.1)
$\mathcal{O}^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)}(\tau,\sigma)=\left\\{\begin{aligned}
(-1)^{j}&\text{ if
}\tau=[x_{1},x_{2},\cdots,\widehat{x_{j}},\cdots,x_{k}]\text{ for some index
}j,\\\ 0&\text{ otherwise. }\end{aligned}\right.$

Here, the superscript means that this orientation is used to compute the
simplicial homology $\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)$
(distinguished from the relative version below). Now, we fix a vector space
$V$, and consider the free abelian group $C_{k}(\Delta,V)$ with basis
$\Delta_{k}$, the $k$-simplices of the complex $\Delta$, and coefficients in
$V$. Then the simplicial boundary operator $\partial_{k}:C_{k}(\Delta,V)\to
C_{k-1}(\Delta,V)$ can be written as

(2.2)
$\partial^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)}_{k}\sigma=\sum_{\tau\in\Delta_{k-1}}\mathcal{O}^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)}(\tau,\sigma)\tau,$

for any $k$-simplex $\sigma$. A straightforward calculation yields that
$\partial_{k-1}\circ\partial_{k}=0$, and the standard simplicial homology is
then defined by

$\mathcal{H}_{k}(\Delta,V):=\mathcal{N}(\partial_{k}^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)})/\mathcal{R}(\partial_{k-1}^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)}).$

In this paper, we will also use a special kind of relative homology,
$\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta;\partial\Delta).$ Here, the
chain group $C_{k}(\Delta,V;\partial\Delta)$ is generated by $k$-simplices in
$\Delta\setminus\partial\Delta$. Given a $k$-simplex
$\sigma=[x_{1},x_{2},\cdots,x_{k}]$ and a $(k-1)-$simplex $\tau$, define

(2.3)
$\mathcal{O}^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta;\partial\Delta)}(\tau,\sigma)=\left\\{\begin{aligned}
\mathcal{O}^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)}(\tau,\sigma)&\text{
if }\tau\not\in\partial\Delta,\\\ 0&\text{ otherwise. }\end{aligned}\right.$

Similarly, we define

(2.4)
$\partial^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta;\partial\Delta)}_{k}\sigma=\sum_{\tau\in\Delta_{k-1}}\mathcal{O}^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta;\partial\Delta)}(\tau,\sigma)\tau,$

and

$\mathcal{H}_{k}(\Delta,V;\partial\Delta)=\mathcal{N}(\partial_{k}^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta;\partial\Delta)})/\mathcal{R}(\partial_{k-1}^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta;\partial\Delta)}).$

We simply use the notations
$\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)$ and
$\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta;\partial\Delta)$ when
$V=\mathbb{Z}$. Two theorems from algebraic topology will be relevant for the
discussions below. The first is the universal coefficient theorem, computing
the homology with coefficients in $V$.

######  Theorem 2.1 (Universal coefficient theorem, simplified version).

For any vector space $V$, it holds that

(2.5)
$\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta,V)\cong\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)\otimes
V,$

and

(2.6)
$\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta,V;\partial\Delta)\cong\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta;\partial\Delta)\otimes
V.$

The next celebrated result is the Lefschetz duality, which relates the
relative homology to the de Rham cohomology.

######  Theorem 2.2 (Lefschetz duality theorem).

Let $\Delta$ be a $n$-complex with boundary $\partial\Delta$. Then it holds
that

(2.7) $\mathcal{H}_{k}(\Delta;\partial\Delta)\cong\mathcal{H}^{n-k}(\Delta),$

and

(2.8) $\mathcal{H}_{k}(\Delta)\cong\mathcal{H}^{n-k}(\Delta;\partial\Delta),$

Here $\mathcal{H}^{n-k}$ is the simplicial cohomology.

Now we consider the relation to de Rham cohomology. We use
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR}(\Omega)$ to denote the
standard de Rham cohomology, and
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR,c}(\Omega)$ to denote the
compactly supported version. Using the de Rham theorem, we get the following
result.

######  Theorem 2.3.

The following results hold.

  1. (1)

$\mathcal{H}_{k}(\Delta;\partial\Delta)\cong\mathcal{H}^{n-k}_{dR}(\Omega).$

  2. (2)

$\mathcal{H}_{k}(\Delta)\cong\mathcal{H}^{n-k}_{dR,c}(\Omega).$

Unless otherwise specified, we use $\mathcal{O}$ to denote the orientation
with respect to the relative homology
$\mathcal{O}^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta;\partial\Delta)}$,
use $\mathcal{O}_{0}$ to denote
$\mathcal{O}^{\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)}$, in
accordance with the compact supported de Rham cohomology.

In the rest of this paper, we will relate functions and distributions to an
element in the chain group $C_{k}(\Delta)$. To this end, we denote by
$\|\sigma\|$ the free element in $C_{k}(\Delta)$ associated with $k$-th
simplex $\sigma$. This does not lead to confusion since we will not use
$\|\sigma\|$ as a norm.

###  2.2. Function spaces

In this subsection, we define various spaces of vector-valued and matrix-
valued functionals.

We introduce some scalar spaces following [23]. Recall that $\Omega$ is a
bounded Lipschitz domain. Let $s$ be any real number. Then $H^{s}(\Omega)$
denotes the quotient of $H^{s}(\mathbb{R}^{n})$ with distributions vanishing
in $\Omega$, which is isomorphic to the standard definition as a space of
distributions in $\Omega$. Throughout this paper, we will simply consider
$H^{s}(\Omega)$ as a space of distributions over $\Omega$.

Let $H_{\overline{\Omega}}^{s}(\mathbb{R}^{n})$ be the subspace of
$H^{s}(\mathbb{R}^{n})$ consisting of all the distributions with support in
$\overline{\Omega}$. For any $s$, $H_{\overline{\Omega}}^{s}(\mathbb{R}^{n})$
can be identified with the closure of $C_{c}^{\infty}$ in
$H^{s}(\mathbb{R}^{n})$, and also the dual space of $H^{-s}(\Omega)$, i.e.,

(2.9)
$H_{\overline{\Omega}}^{s}(\mathbb{R}^{n})=\stackon[-1.5pt]{C_{c}^{\infty}}{\rule[2.0pt]{0.0pt}{0.4pt}_{H^{s}(\mathbb{R}^{n})}}=[H^{-s}(\Omega)]^{\ast},\quad\forall
s\in\mathbb{R}.$

We also have

$H_{\overline{\Omega}}^{s}(\mathbb{R}^{n})=H^{s}_{0}(\Omega),\quad\forall\mbox{
$s>0$ and $s-\frac{1}{2}$ is not an integer,}$

where $H^{s}_{0}(\Omega)$ denotes the closure of $C^{\infty}_{c}(\Omega)$ in
$H^{s}(\Omega)$.

Let $H^{s}\Lambda^{k}$ (or $H^{s}_{\overline{\Omega}}\Lambda^{k}$) be the
space of differential $k$-forms with coefficients in $H^{s}(\Omega)$ (or
correspondingly, $H^{s}_{\overline{\Omega}}(\mathbb{R}^{n})$). Following [23],
we have two complexes:

(2.10)
${0}$${H^{s}\Lambda^{0}}$${H^{s-1}\Lambda^{1}}$${\cdots}$${H^{s-n}\Lambda^{n}}$${0,}$$\scriptstyle{d^{0}}$$\scriptstyle{d^{1}}$$\scriptstyle{d^{n-1}}$
(2.11)
${0}$${H^{s}_{\overline{\Omega}}\Lambda^{0}}$${H^{s-1}_{\overline{\Omega}}\Lambda^{1}}$${\cdots}$${H^{s-n}_{\overline{\Omega}}\Lambda^{n}}$${0.}$$\scriptstyle{d^{0}}$$\scriptstyle{d^{1}}$$\scriptstyle{d^{n-1}}$

The complexes (2.10) and (2.11) both have uniform cohomology in the sense that
the cohomology is isomorphic to the smooth versions.

######  Remark 2.1.

Note that the differential operators in the complexes actually have different
meanings. In (2.10), we consider distributions in $\Omega$, where the test
function is chosen as $C_{c}^{\infty}(\Omega)$. While in (2.11), we consider
distributions in $\mathbb{R}^{n}$, where the test function is in
$C_{c}^{\infty}(\mathbb{R}^{n}).$

Most of the examples in this paper will be given in vector/matrix forms. We
introduce some notation following [4, 10].

$\mathbb{V}$ | $\mathbb{R}^{n}$
---|---
$\mathbb{M}$ | the space of all $n\times n$-matrices
$\mathbb{S}$ | symmetric matrices
$\mathbb{K}$ | skew symmetric matrices
$\mathbb{T}$ | trace-free matrices
$\operatorname{skw}:\mathbb{M}\to\mathbb{K}$ | skew symmetric part of a matrix
$\operatorname{sym}:\mathbb{M}\to{\mathbb{S}}$ | symmetric part of a matrix
$\operatorname{tr}:\mathbb{M}\to\mathbb{R}$ | matrix trace
$\iota:\mathbb{R}\to\mathbb{M}$ | the map $\iota u:=uI$ identifying a scalar with a scalar matrix
$\operatorname{dev}:\mathbb{M}\to\mathbb{T}$ | deviator (trace-free part of a matrix) given by $\operatorname{dev}u:=u-1/n\operatorname{tr}(u)I$

We use $\mathcal{D}^{\prime}$ to denote the space of distributions and
$\langle\cdot,\cdot\rangle$ for dual pairs. We use $(\cdot,\cdot)$ to denote
the $L^{2}$ inner product. This can be naturally extended to vector- or
matrix-valued functions. For example, $(\sigma,\tau):=\int\sigma:\tau\,dx$
when $\sigma$ and $\tau$ are both matrix fields. The above matrix operations
can be extended to functions and distributions naturally. For example, we
define
$\operatorname{sym}:\mathcal{D}^{\prime}(\mathbb{M})\to\mathcal{D}^{\prime}(\mathbb{S})$
by
$\langle\operatorname{sym}\xi,\sigma\rangle=\langle\xi,\operatorname{sym}\sigma\rangle,\quad\forall\sigma\in\mathcal{D}(\mathbb{M}),$
and define
$\operatorname{dev}:\mathcal{D}^{\prime}(\mathbb{M})\to\mathcal{D}^{\prime}(\mathbb{T})$
by
$\langle\operatorname{dev}\xi,\sigma\rangle=\langle\xi,\operatorname{dev}\sigma\rangle,\quad\forall\sigma\in\mathcal{D}(\mathbb{M}).$

Let $\mathbb{X}_{i}$ be one of the above vector or matrix spaces
($\mathbb{V},\mathbb{M},\mathbb{S}$, etc.) and $D^{i}$ be a linear
differential operator with constant coefficients of order $\alpha_{i}$.
Suppose that we have the following short sequence

(2.12)
${\cdots}$${H^{s_{0},s_{1}}(D^{1},\mathbb{X}_{1})}$${H^{s_{1},s_{2}}(D^{2},\mathbb{X}_{2})}$${H^{s_{2},s_{3}}(D^{3},\mathbb{X}_{3})}$${\cdots,}$$\scriptstyle{D^{1}}$$\scriptstyle{D^{2}}$

where

$H^{s_{1},s_{2}}(D;\mathbb{X}_{i}):=\\{u\in
H^{s_{1}}\otimes\mathbb{X}_{i}:Du\in H^{s_{2}}\otimes\mathbb{X}_{i+1}\\}.$

Similarly, we define

$H_{0}^{s_{1},s_{2}}(D;\mathbb{X}_{i}):=\\{u\in
H_{\overline{\Omega}}^{s_{1}}\otimes\mathbb{X}_{i}:Du\in
H_{\overline{\Omega}}^{s_{2}}\otimes\mathbb{X}_{i+1}\\},$

and they also form a complex. When $s_{1}=s_{2}$, we denote
$H^{s}(D;\mathbb{X}_{i}):=H^{s,s}(D;\mathbb{X}_{i})$, and when $s=0$, we
further denote $H(D;\mathbb{X}_{i}):=H^{0}(D;\mathbb{X}_{i})$ and
$H_{0}(D;\mathbb{X}_{i}):=H^{0}_{0}(D;\mathbb{X}_{i})$.

To state regularity and duality results, we assume that the sequence

(2.13)
${\cdots}$${H^{q}\otimes\mathbb{X}_{1}}$${H^{q-\alpha_{1}}\otimes\mathbb{X}_{2}}$${H^{q-\alpha_{1}-\alpha_{2}}\otimes\mathbb{X}_{3}}$${\cdots,}$$\scriptstyle{D^{1}}$$\scriptstyle{D^{2}}$$\scriptstyle{D^{3}}$

is a Hilbert scale in the sense that for any $q\in\mathbb{R}$, we have

(2.14)
$\mathcal{N}(D^{2},H^{q-\alpha_{1}}\otimes\mathbb{X}_{2})=(D^{1}H^{q}\otimes\mathbb{X}_{1})\oplus\mathscr{H}^{2},$

and

(2.15)
$\mathcal{N}(D^{3},H^{q-\alpha_{1}-\alpha_{2}}\otimes\mathbb{X}_{3})=(D^{2}H^{q-\alpha_{1}}\otimes\mathbb{X}_{2})\oplus\mathscr{H}^{3},$

where $\mathscr{H}^{2}$ and $\mathscr{H}^{3}$ are finite dimensional spaces
not depending on $q$ (thus containing smooth functions) [4].

Let $(D^{1})^{\ast}$ be the formal adjoint of $D^{1}$. Related to (2.12), we
have the following, generalizing the regular decomposition argument [4,
Theorem 3] and the duality result in [49, Lemma 1].

######  Theorem 2.4.

We have

(2.16)
$H^{s_{1},s_{2}}(D^{2};\mathbb{X}_{2})=H^{s_{2}+\alpha_{2}}\otimes\mathbb{X}_{2}+D^{1}H^{s_{1}+\alpha_{1}}(D^{1},\mathbb{X}_{1}),$

and the duality

(2.17)
$H_{0}^{s_{1},s_{2}}(D^{2},\mathbb{X}_{2})^{\ast}=H^{-s_{2}-\alpha_{2},-s_{1}-\alpha_{1}}((D^{1})^{\ast},\mathbb{X}_{2}).$

Similarly, for another type of boundary conditions, we have

$H_{0}^{s_{1},s_{2}}(D^{2};\mathbb{X}_{2})=H_{0}^{s_{2}+\alpha_{2}}\otimes\mathbb{X}_{2}+D^{1}H_{0}^{s_{1}+\alpha_{1}}(D^{1},\mathbb{X}_{1}),$

and the duality

$H^{s_{1},s_{2}}(D^{2},\mathbb{X}_{2})^{\ast}=H_{0}^{-s_{2}-\alpha_{2},-s_{1}-\alpha_{1}}((D^{1})^{\ast},\mathbb{X}_{2}).$

###### Proof.

For any $u\in H^{s_{1},s_{2}}(D^{2};\mathbb{X}_{2})$, $D^{2}u\in
H^{s_{2}}\otimes\mathbb{X}_{3}$. Due to (2.15), there exists $\varphi\in
H^{s_{2}+\alpha_{2}}\otimes\mathbb{X}_{2}$ such that $D^{2}\varphi=D^{2}u$,
i.e., $D^{2}(u-\varphi)=0$. Using (2.14), we have

$u-\varphi=D^{1}\beta+h,$

where $\beta\in H^{s_{1}+\alpha_{1}}\otimes\mathbb{X}_{1}$ and $h$ is smooth.
This proves (2.16).

For $w\in H_{0}^{s_{1},s_{2}}(D^{2},\mathbb{X}_{2})^{\ast}$,

$\displaystyle\|w\|_{H_{0}^{s_{1},s_{2}}(D^{2},\mathbb{X}_{2})^{\ast}}$
$\displaystyle=\sup_{v\in
H_{0}^{s_{1},s_{2}}(D^{2},\mathbb{X}_{2})}\frac{\langle
v,w\rangle}{\|v\|_{H_{0}^{s_{1},s_{2}}(D^{2},\mathbb{X}_{2})}}$
$\displaystyle\cong\sup_{y\in
H^{s_{1}+\alpha_{1}}_{0}\otimes\mathbb{X}_{1},z\in
H^{s_{2}+\alpha_{2}}_{0}\otimes\mathbb{X}_{2}}\frac{\langle
D^{1}y+z,w\rangle}{\|y\|_{s_{1}+\alpha_{1}}+\|z\|_{s_{2}+\alpha_{2}}}$
$\displaystyle\cong\sup_{y\in
H^{s_{1}+\alpha_{1}}_{0}\otimes\mathbb{X}_{1},z\in
H^{s_{2}+\alpha_{2}}_{0}\otimes\mathbb{X}_{2}}\frac{\langle
D^{1}y,w\rangle}{\|y\|_{s_{1}+\alpha_{1}}}+\frac{\langle
z,w\rangle}{\|z\|_{s_{2}+\alpha_{2}}}$ $\displaystyle=\sup_{y\in
H^{s_{1}+\alpha_{1}}_{0}\otimes\mathbb{X}_{1},z\in
H^{s_{2}+\alpha_{2}}_{0}\otimes\mathbb{X}_{2}}\frac{\langle
y,(D^{1})^{\ast}w\rangle}{\|y\|_{s_{1}+\alpha_{1}}}+\frac{\langle
z,w\rangle}{\|z\|_{s_{2}+\alpha_{2}}}$
$\displaystyle=\|(D^{1})^{\ast}w\|_{-s_{1}-\alpha_{1}}+\|w\|_{-s_{2}-\alpha_{2}}.$

∎

As special cases of Theorem 2.4, we recover results from the literature (c.f.
[49])

$H_{0}(\operatorname{curl})^{\ast}=H^{-1}(\operatorname{div}),\quad
H_{0}(\operatorname{div})^{\ast}=H^{-1}(\operatorname{curl}),$

and obtain new results. For example, associated with the Hessian complex, we
have

$H_{0}(\operatorname{curl};\mathbb{S})^{\ast}=H^{-1,-2}(\operatorname{div}\operatorname{div};\mathbb{S}):=\\{w\in
H^{-1}\otimes\mathbb{S}:\operatorname{div}\operatorname{div}w\in H^{-2}\\},$

and associated with the elasticity complex,

$H_{0}(\operatorname{inc};\mathbb{S})^{\ast}=H^{-2,-1}(\operatorname{div};\mathbb{S}):=\\{w\in
H^{-2}\otimes\mathbb{S}:\operatorname{div}w\in H^{-1}\\}.$

###  2.3. BGG complexes

The BGG construction is a machinery to derive other complexes from the de Rham
complexes. We follow and review the constructions in [4, 10]. Even though we
only focus on complexes in 2D and 3D in this case, the more general
differential form point of view shall provide discrete topological
inspirations and interpretations for our construction in this paper.

For $i\geq 0$, let $\mathrm{Alt}^{i}\mathbb{R}^{n}$ be the space of algebraic
$i$-forms, that is, of alternating $i$-linear maps on $\mathbb{R}^{n}$. We
also set
$\mathrm{Alt}^{i,j}\mathbb{R}^{n}=\mathrm{Alt}^{i}\mathbb{R}^{n}\otimes\mathrm{Alt}^{j}\mathbb{R}^{n}$,
the space of $\mathrm{Alt}^{j}\mathbb{R}^{n}$-valued $i$-forms or,
equivalently, the space of $(i+j)$-linear maps on $\mathbb{R}^{n}$ which are
alternating in the first $i$ variables and also in the last $j$ variables. A
major example in [4] is constructed from the following diagram:

(2.18)
${0}$${H^{q}\otimes\mathrm{Alt}^{0,0}}$${H^{q-1}\otimes\mathrm{Alt}^{1,0}}$${\cdots}$${H^{q-n}\otimes\mathrm{Alt}^{n,0}}$${0}$${0}$${H^{q-1}\otimes\mathrm{Alt}^{0,1}}$${H^{q-2}\otimes\mathrm{Alt}^{1,1}}$${\cdots}$${H^{q-n-1}\otimes\mathrm{Alt}^{n,1}}$${0}$${\vdots}$${\vdots}$${\vdots}$${0}$${H^{q-n+1}\otimes\mathrm{Alt}^{0,n-1}}$${H^{q-n}\otimes\mathrm{Alt}^{1,n-1}}$${\cdots}$${H^{q-2n+1}\otimes\mathrm{Alt}^{n,n-1}}$${0}$${0}$${H^{q-n}\otimes\mathrm{Alt}^{0,n}}$${H^{q-n-1}\otimes\mathrm{Alt}^{1,n}}$${\cdots}$${H^{q-2n}\otimes\mathrm{Alt}^{n,n}}$${0.}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{S^{0,1}}$$\scriptstyle{d}$$\scriptstyle{S^{1,1}}$$\scriptstyle{d}$$\scriptstyle{S^{n-1,1}}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{S^{0,n}}$$\scriptstyle{d}$$\scriptstyle{S^{1,n}}$$\scriptstyle{d}$$\scriptstyle{S^{n-1,n}}$

Here
$s^{i,j}:\mathrm{Alt}^{i,j}\mathbb{R}^{n}\to\mathrm{Alt}^{i+1,j-1}\mathbb{R}^{n}$
are algebraic maps, and $S^{i,j}=I\otimes
s^{i,j}:H^{q}\otimes\mathrm{Alt}^{i,j}\to H^{q}\otimes\mathrm{Alt}^{i+1,j-1}$
for any Sobolev order $q$. The $k$-th row of (2.18) is obtained by tensoring a
de Rham complex with alternating $k$-forms.

The BGG diagram (2.18) has the following vector/matrix proxies for $n=3$:

(2.19)
${0}$${H^{q}\otimes\mathbb{R}}$${H^{q-1}\otimes\mathbb{V}}$${H^{q-2}\otimes\mathbb{V}}$${H^{q-3}\otimes\mathbb{R}}$${0}$${0}$${H^{q-1}\otimes\mathbb{V}}$${H^{q-2}\otimes\mathbb{M}}$${H^{q-3}\otimes\mathbb{M}}$${H^{q-4}\otimes\mathbb{V}}$${0}$${0}$${H^{q-2}\otimes\mathbb{V}}$${H^{q-3}\otimes\mathbb{M}}$${H^{q-4}\otimes\mathbb{M}}$${H^{q-5}\otimes\mathbb{V}}$${0}$${0}$${H^{q-3}\otimes\mathbb{R}}$${H^{q-4}\otimes\mathbb{V}}$${H^{q-5}\otimes\mathbb{V}}$${H^{q-6}\otimes\mathbb{R}}$${0,}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{I}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{2\operatorname{vskw}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{\operatorname{tr}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{-\operatorname{mskw}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{2\operatorname{vskw}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\iota}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{-\operatorname{mskw}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{I}$

where, for any matrix $u$, $\mathcal{S}u=u^{T}-\operatorname{tr}(u)I$.

In 3D, there are three basic examples of BGG complexes [4, 10]: the Hessian
complex

(2.20)
${0}$${H^{q}\otimes\mathbb{R}}$${H^{q-2}\otimes\mathbb{S}}$${H^{q-3}\otimes\mathbb{T}}$${H^{q-4}\otimes\mathbb{V}}$${0,}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

where $\operatorname{hess}:=\operatorname{grad}\operatorname{grad}$; the
elasticity complex

(2.21)
${0}$${H^{q-1}\otimes\mathbb{V}}$${H^{q-2}\otimes\mathbb{S}}$${H^{q-4}\otimes\mathbb{S}}$${H^{q-5}\otimes\mathbb{V}}$${0,}$$\scriptstyle{{\operatorname{def}}}$$\scriptstyle{\operatorname{inc}}$$\scriptstyle{\operatorname{div}}$

where $\operatorname{def}:=\operatorname{sym}\operatorname{grad}$ is the
linearized deformation (symmetric part of gradient) and
$\operatorname{inc}=\operatorname{curl}\mathcal{S}^{-1}\operatorname{curl}$
leads to the linearized Einstein tensor; and the
$\operatorname{div}\operatorname{div}$ complex

(2.22)
${0}$${H^{q-2}\otimes\mathbb{V}}$${H^{q-3}\otimes\mathbb{T}}$${H^{q-4}\otimes\mathbb{S}}$${H^{q-6}\otimes\mathbb{V}}$${0.}$$\scriptstyle{\operatorname{dev}\operatorname{grad}}$$\scriptstyle{\operatorname{sym}\operatorname{curl}}$$\scriptstyle{\operatorname{div}\operatorname{div}}$

The three basic examples (2.20)-(2.22) are derived from (2.19). The first two
rows imply the Hessian complex (2.20); the middle two rows derive the
elasticity complex (2.21); the last two rows yield the
$\operatorname{div}\operatorname{div}$ complex (2.22). The complexes
(2.20)-(2.22) are resolutions of finite dimensional spaces

$\mathcal{P}_{1}=\mathcal{N}(\operatorname{hess}),\quad\mathcal{RM}:=\mathcal{N}(\operatorname{def}),\quad\mathcal{RT}:=\\{\bm{a}+b\bm{x}:\bm{a}\in\mathbb{R}^{n},b\in\mathbb{R}\\}=\mathcal{N}(\operatorname{dev}\operatorname{grad}),$

respectively. Here $\mathcal{RM}$ and $\mathcal{RT}$ refer to infinitesimal
rigid body motions and the Raviart-Thomas [50] shape functions. The cohomology
of (2.20)-(2.22) is isomorphic to the de Rham cohomology with
$\mathcal{P}_{1}$, $\mathcal{RM}$, and $\mathcal{RT}$ coefficients,
respectively.

The general idea of deriving a BGG complex from a diagram is to eliminate
spaces connected by the algebraic operators as much as possible. We use the
elasticity complex (2.21) to explain this construction. The first connecting
map $-\operatorname{mskw}$ is surjective. Therefore we eliminate
$H^{q-2}\otimes\mathbb{V}$ and the image of $-\operatorname{mskw}$ (as a
subspace of $H^{q-2}\otimes\mathbb{M}$). The remaining part in
$H^{q-2}\otimes\mathbb{M}$ is thus symmetric matrices. The second connecting
map, $\mathcal{S}$, is bijective. Hence we eliminate the two spaces connected
by $\mathcal{S}$ from the diagram and connect the two complexes by a zig-zag
$\operatorname{curl}\circ\mathcal{S}^{-1}\circ\operatorname{curl}$. Finally,
the last operator $2\operatorname{vskw}$ is surjective. We eliminate
$H^{q-4}\otimes\mathbb{V}$ and the corresponding part in
$H^{q-4}\otimes\mathbb{M}$. Again, this leaves symmetric matrices in the
derived complex.

In $\mathbb{R}^{2}$, a skew-symmetric matrix can be identified with a scalar.
Using the same notation as in 3D, define
$\operatorname{mskw}:\mathbb{R}\to\mathbb{K}$ by

$\operatorname{mskw}(u):=\left(\begin{array}[]{cc}0&u\\\
-u&0\end{array}\right)\quad\mbox{in }\mathbb{R}^{2}.$

Let
$\operatorname{sskw}=\operatorname{mskw}^{-1}\circ\operatorname{skw}:\mathbb{M}\to\mathbb{R}$
be the map taking the skew part of a matrix and identifying it with a scalar
(see [4]).

The 2D version of the diagram (2.19) is

(2.23)
${0}$${H^{q}\otimes\mathbb{R}}$${H^{q-1}\otimes\mathbb{V}}$${H^{q-2}\otimes\mathbb{R}}$${0}$${0}$${H^{q-1}\otimes\mathbb{V}}$${H^{q-2}\otimes\mathbb{M}}$${H^{q-3}\otimes\mathbb{V}}$${0}$${0}$${H^{q-2}\otimes\mathbb{R}}$${H^{q-3}\otimes\mathbb{V}}$${H^{q-4}\otimes\mathbb{R}}$${0.}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{rot}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{I}$$\scriptstyle{\operatorname{rot}}$$\scriptstyle{-2\operatorname{sskw}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{mskw}}$$\scriptstyle{\operatorname{rot}}$$\scriptstyle{\mathrm{id}}$

The derived complexes are a rotated stress complex

(2.24)
${0}$${H^{q}}$${H^{q-2}\otimes\mathbb{S}}$${H^{q-3}\otimes\mathbb{V}}$${0,}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{rot}}$

and the strain complex

(2.25)
${0}$${H^{q-1}\otimes\mathbb{V}}$${H^{q-2}\otimes\mathbb{S}}$${H^{q-4}}$${0.}$$\scriptstyle{\operatorname{sym}\operatorname{grad}}$$\scriptstyle{\operatorname{rot}\operatorname{rot}}$

The BGG framework provides a differential form interpretation for the Hessian,
elasticity, and $\operatorname{div}\operatorname{div}$ complexes. In 3D, they
can be sketched as follows:

(2.26)
${0}$${\Lambda^{0,0}}$${\Lambda^{1,1}}$${\Lambda^{2,1}}$${\Lambda^{3,1}}$${0,}$$\scriptstyle{d\circ
I\circ d}$$\scriptstyle{d}$$\scriptstyle{d}$ (2.27)
${0}$${\Lambda^{0,1}}$${\Lambda^{1,1}}$${\Lambda^{2,2}}$${\Lambda^{3,2}}$${0,}$$\scriptstyle{d}$$\scriptstyle{d\circ\mathcal{S}^{-1}\circ
d}$$\scriptstyle{d}$ (2.28)
${0}$${\Lambda^{0,2}}$${\Lambda^{1,2}}$${\Lambda^{2,2}}$${\Lambda^{3,3}}$${0.}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{d\circ
I\circ d}$

In this paper, we mainly consider the following Sobolev versions of the BGG
complexes:

(2.29)
${0}$${H^{1}}$${H^{-1}(\operatorname{curl},\mathbb{S})}$${H^{-1}(\operatorname{div},\mathbb{T})}$${H^{-1}\otimes\mathbb{V}}$${0,}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$
(2.30)
${0}$${H^{1}\otimes\mathbb{V}}$${H^{0,-1}(\operatorname{inc},\mathbb{S})}$${H^{-1}(\operatorname{div},\mathbb{S})}$${H^{-1}\otimes\mathbb{V}}$${0,}$$\scriptstyle{\operatorname{def}}$$\scriptstyle{\operatorname{inc}}$$\scriptstyle{\operatorname{div}}$
(2.31)
${0}$${H^{1}}$${H(\operatorname{sym}\operatorname{curl},\mathbb{S})}$${H^{0,-1}(\operatorname{div}\operatorname{div},\mathbb{T})}$${H^{-1}\otimes\mathbb{V}}$${0.}$$\scriptstyle{\operatorname{dev}\operatorname{grad}}$$\scriptstyle{\operatorname{sym}\operatorname{curl}}$$\scriptstyle{\operatorname{div}\operatorname{div}}$

Note that (2.29)-(2.31) have a symmetric pattern. First, we observe that
$H^{1}=H^{-1}(\operatorname{hess})$ (by the Lions lemma [45, 22],
$\operatorname{hess}u\in H^{-1}$ implies that $u\in H^{1}$), and thus all the
spaces in (2.29) has the form of $H^{-1}(D)$, where $D$ is the corresponding
BGG differential operator. Second, due to Theorem 2.4, the divdiv complex
(2.31) is dual (adjoint) to the compactly supported version of (2.29).
Finally, the elasticity complex (2.30) has a self-dual pattern:
$H^{1}(\mathbb{V})^{\ast}=H_{0}^{-1}(\mathbb{V})$, and
$H^{0,-1}(\operatorname{inc},\mathbb{S})^{\ast}=H^{-1}_{0}(\operatorname{div},\mathbb{S})$.

All these complexes also have compactly supported versions. For example,
another version of (2.29) is

(2.32)
${0}$${H^{1}_{0}}$${H^{-1}_{0}(\operatorname{curl},\mathbb{S})}$${H^{-1}_{0}(\operatorname{div},\mathbb{T})}$${H^{-1}_{0}\otimes\mathbb{V}}$${0.}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

To clarify notations in 2D, we define $\operatorname{curl}u=[-\frac{\partial
u}{\partial y},\frac{\partial u}{\partial x}]$, and
$\operatorname{rot}([w_{x},w_{y}])=\frac{\partial w_{y}}{\partial
x}-\frac{\partial w_{x}}{\partial y}.$ By definition, $\operatorname{rot}$ and
$-\operatorname{curl}$ are formal adjoints to each other. Moreover, we define
$([x,y])^{\perp}=[-y,x]$. Therefore,
$\operatorname{curl}u=(\operatorname{grad}u)^{\perp}$ and
$\operatorname{rot}(\bm{w})=\operatorname{div}(\bm{w}^{\perp})$. For each edge
we assume that $\bm{n}_{e}=\bm{t}_{e}^{\perp}$. It is easy to see that
$\operatorname{curl}u\cdot\bm{n}_{e}=\operatorname{grad}u\cdot\bm{t}_{e}$. For
matrices, we define $\operatorname{curl}$ and $\operatorname{rot}$ rowwise.

In 2D, we consider the (rotated) stress complex

(2.33)
${0}$${H^{1}}$${H^{-1}(\operatorname{curl},\mathbb{S})}$${H^{-1}(\mathbb{V})}$${0,}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{rot}}$

and the stain complex

(2.34)
${0}$${H^{1}}$${H^{0,-1}(\operatorname{rot}\operatorname{rot},\mathbb{S})}$${H^{-1}}$${0.}$$\scriptstyle{\operatorname{def}}$$\scriptstyle{\operatorname{rot}\operatorname{rot}}$

Now the two complexes (2.33) and (2.34) are dual (adjoint) to each other when
one of them has compact support.

##  3\. Main results

In this section, we summarize the main results of this paper.

We first introduce the Dirac delta located on (sub)simplices. Given a simplex
$\sigma$, we define the scalar delta $\delta_{\sigma}$ by

$\langle\delta_{\sigma},u\rangle=\int_{\sigma}u,$

and the vector delta $\delta_{\sigma}[\bm{a}]$ by

$\langle\delta_{\sigma}[\bm{a}],\bm{v}\rangle=\int_{\sigma}\bm{a}\cdot\bm{v},$

and the tensor (matrix) delta $\delta_{\sigma}[\bm{A}]$ by

$\langle\delta_{\sigma}[\bm{A}],\bm{W}\rangle=\int_{\sigma}\bm{A}:\bm{W},$

for smooth function $u$, vector field $\bm{v}$, and tensor field $\bm{W}$,
respectively. Here $\bm{a}\in\mathbb{R}^{n}$ and $\bm{A}\in\mathbb{R}^{n\times
n}$ for $n=2,3$. We say that $\delta_{\sigma}[\bm{A}]$ is symmetric
(traceless) if $\bm{A}$ is symmetric (traceless). We will often use
$\delta_{\sigma}[\bm{u}\otimes\bm{v}]$, which by definition is
$\bm{W}\mapsto\int_{\sigma}\bm{W}:(\bm{u}\otimes\bm{v})=\int_{\sigma}\bm{u}\cdot\bm{W}\cdot\bm{v}$.
It is symmetric if $\bm{u}=\bm{v}$, and it is traceless if
$\bm{u}\perp\bm{v}$.

###  3.1. Motivation: Discrete Exterior Calculus (DEC) interpretations of the
distributional de Rham and Regge complexes

We first provide some remarks for the de Rham complex as a motivation. The
finite element de Rham complex (Whitney forms, Figure 1) can be viewed as a
discretization for

(3.1)
${0}$${H^{1}}$${H(\operatorname{curl})}$${H(\operatorname{div})}$${L^{2}}$${0,}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

while the distributional complex (Figure 2) is a slightly nonconforming
discretization for

(3.2)
${0}$${L^{2}}$${H^{-1}_{0}(\operatorname{curl})}$${H_{0}^{-1}(\operatorname{div})}$${H_{0}^{-1}}$${0,}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

or equivalently, due to the duality (2.17),

(3.3)
${0}$${{[L^{2}]}^{\ast}}$${{[H(\operatorname{div})]}^{\ast}}$${{[H(\operatorname{curl})]}^{\ast}}$${{[H^{1}]}^{\ast}}$${0.}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

The nonconformity is due to the fact that the canonical degrees of freedom
(integrating $k$-forms on $k$-chains) is not well-defined for $L^{2}$ spaces.

We observe that the distributional de Rham complex (Figure 2) is formally a
reversed version of the standard de Rham finite element complex, i.e., the
$k$-th space is discretized on $(n-k)$-cells. In the Discrete Exterior
Calculus (DEC) perspective, this means that $k$-forms are discretized on dual
$k$-chains and the operators become discrete exterior derivatives on the dual
mesh in the usual sense (see Figure 4). This leads to another perspective of
the distributional complexes: DEC uses dual of meshes, while distributional
finite elements use dual of spaces and operators. This establishes a link
between DEC and finite elements. In DEC, there are different ways to define
dual meshes, which might affect the convergence; while for distributional
finite elements, there is a canonical definition for the distributional
derivatives. Note that the dual of a simplicial mesh is not simplicial.
Therefore, there are no trivial generalizations of Whitney forms (shape
functions) on the dual mesh.

Figure 4. Discrete Exterior Calculus perspective of a distributional de Rham
complex in 2D. The black cells form the primal mesh, and the red dashed lines
form the dual mesh. The degrees of freedom and Dirac delta in the
distributional complex correspond to DEC discrete forms on the dual mesh.
Alternatively, one may view the red cells as primal and black as dual. Then
the distributional spaces correspond to discrete forms on the primal mesh.

To motivate the construction in this paper for other BGG complexes, we make
some observations on the Regge complex (Figure 3) from a discrete form point
of view. We denote the Regge complex (Figure 3, [20]) as follows (compared
with (2.27)):

(3.4)
${0}$${\Lambda_{h}^{0,1}}$${\Lambda_{h}^{1,1}}$${\Lambda_{h}^{2,2}}$${\Lambda_{h}^{3,2}}$${0.}$$\scriptstyle{\operatorname{def}}$$\scriptstyle{\operatorname{inc}}$$\scriptstyle{\operatorname{div}}$

Here, $\Lambda_{h}^{0,1}$ is a vector-valued version of the Lagrange finite
element; $\Lambda_{h}^{1,1}$ consists of Regge metrics, which are piecewise
constant symmetric matrix fields, and the degrees of freedom are given by the
lengths of edges $\int_{e}\bm{t}_{e}\cdot\bm{g}\cdot\bm{t}_{e}$;
$\Lambda_{h}^{2,2}$ is the dual of $\Lambda_{h}^{1,1}$, which consists of
tangential-tangential Dirac delta on the edges; and $\Lambda_{h}^{3,2}$ is the
dual of the Lagrange element (vertex evaluation). The Regge complex (3.4) is a
slightly nonconforming discretization of (2.30), and is self-adjoint:
$\Lambda_{h}^{3,2}$ is the dual of $\Lambda_{h}^{0,1}$ and $\Lambda_{h}^{2,2}$
is the dual of $\Lambda_{h}^{1,1}$. This is consistent with the structure in
(2.30).

On the continuous level, the sequence (2.27) starts with $\Lambda^{0,1}$
(1-form-valued 0-form). In the Regge complex, this is discretized by a vector-
valued Lagrange element and can be viewed as attaching an alternating 1-form
at each 0-chain (recalling that in de Rham sequences, one attaches a scalar
value, or alternating 0-form, to each 0-chain). The discrete version of
$\Lambda^{1,1}$ in (2.27) is the Regge metric. The shape functions are
$\bm{t}_{e}\odot\bm{t}_{e},e\in{\mathcal{E}}$, and the degrees of freedom can
be viewed as $\omega_{e}\odot\omega_{e}$, where $\omega_{e}$ is the dual
1-form of the constant vector field $\bm{t}_{e}$. This is a natural extension
of discrete 1-forms in the de Rham case. The third space, $\Lambda^{2,2}$
seems rather different from the de Rham case, as 1) we have distributions
here, rather than piecewise functions; 2) the degrees of freedom are located
on edges, rather than on faces as de Rham 2-forms in 3D are. Our
interpretation of this difference is that, in the extended picture for (2.27),
following the zig-zag in the BGG diagram (which leads to the second-order
operator $\operatorname{inc}$), we move to the dual mesh. Tangential-
tangential components on the primal edges correspond to the normal-normal
components on the dual mesh. Therefore, we interpret their space in the Regge
complex (Figure 3) as $\tilde{\bm{n}}_{e}\odot\tilde{\bm{n}}_{e}$, where
$\tilde{\bm{n}}_{e}=\bm{t}_{e}$ is the normal vector on the dual faces or the
tangent vector on the primal edges. This thus has a neat correspondence to the
de Rham 2-forms in 3D, which are discretized in the normal direction of faces.
Moreover, we can further interpret $\tilde{\bm{n}}_{e}\odot\tilde{\bm{n}}_{e}$
as
$(\tilde{\bm{t}}_{e}^{1}\wedge\tilde{\bm{t}}_{e}^{2})\odot(\tilde{\bm{t}}_{e}^{1}\wedge\tilde{\bm{t}}_{e}^{2})$,
where $\tilde{\bm{t}}_{e}^{1}$ and $\tilde{\bm{t}}_{e}^{2}$ are the two
tangent vectors of the dual face of $e$, and
$[\tilde{\bm{t}}_{e}^{1},\tilde{\bm{t}}_{e}^{2},\tilde{\bm{n}}_{e}]$ forms an
orthonormal frame on the face dual to $e$. This interpretation allows us to
view the third space in the Regge complex as a 2-form-valued 2-forms, or a
4-tensor which is skew-symmetric with respect to the first two legs and with
respect to the last two legs, with the additional symmetry that we can
exchange the first two indices with the last two. This is exactly the symmetry
of the Riemannian tensor. In fact, this corresponds to the identification of
the Riemannian tensor with the Ricci or Einstein tensor in 3D.

###  3.2. Hessian complex in 3D

Inspired by these observations, we establish Hessian and divdiv complexes. For
the Hessian complex, the Lagrange element is a natural discretization for
$\Lambda^{0,0}$ as it assigns a scalar value to each vertex (0-chain). In the
Regge complex, we see that one turns to the dual mesh following the zig-zag
($\operatorname{inc}=\operatorname{curl}\circ\mathcal{S}^{-1}\circ\operatorname{curl}$).
Correspondingly, in the Hessian complex, a natural guess is that
$\operatorname{hess}=\operatorname{grad}\circ I\circ\operatorname{grad}$ takes
us to the dual mesh, leading to discretizing $\Lambda^{1,1}$ on the tangential
direction of dual 1-chain, i.e., the normal direction of the primal 2-chains.
In fact, we verify below that this is the case. As a summary, our discrete
Hessian complex is the following (see Figure 5):

(3.5)
${0}$${V^{0}}$${\bm{V}^{1}}$${\bm{V}^{2}}$${\bm{V}^{3}}$${0.}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

This is a discrete version of (2.29).

Figure 5. Hessian complex in 3D (3.5).

Now we construct the spaces $V^{0}$, $\bm{V}^{1}$, $\bm{V}^{2}$, and
$\bm{V}^{3}$ using vector and tensor deltas.

The space $V^{0}$ is chosen as the standard Lagrange space, namely,

(3.6) $V^{0}:=\\{u\in C^{0}(\Omega):u|_{K}\in\mathcal{P}_{1}(K),\text{ for
each cell }K\in\mathsf{K}\\}.$

The space $\bm{V}^{1}$ consists of normal-normal deltas on faces:

$\bm{V}^{1}:=\operatorname{span}\\{\delta_{f}[\bm{n}_{f}\otimes\bm{n}_{f}];f\in\mathsf{F}_{0}\\},$

where $\bm{n}_{f}$ is the normal vector of $f$. Clearly, the dimension of
$\bm{V}^{1}$ is $\dim\bm{V}^{1}=\sharp_{\mathsf{F}_{0}}.$ The space
$\bm{V}^{2}$ consists of normal-tangential deltas on edges:

(3.7)
$\bm{V}^{2}:=\operatorname{span}\\{\delta_{e}[\bm{m}_{e}\otimes\bm{t}_{e}]:~{}\bm{m}_{e}\text{
is normal to }e,~{}e\in\mathsf{E}_{0}\\}.$

Note that for each internal edge $e$, the normal vector (with respect to $e$)
forms a two-dimensional space. We choose a basis $\bm{n}_{e,+}$ and
$\bm{n}_{e,-}$ of this space. The space $\bm{V}^{2}$ can be equivalently
reformulated as

(3.8)
$\bm{V}^{2}=\operatorname{span}\\{\delta_{e}[\bm{n}_{e,+}\otimes\bm{t}_{e}],\,\delta_{e}[\bm{n}_{e,-}\otimes\bm{t}_{e}]:~{}e\in\mathsf{E}_{0}\\}.$

As a consequence, the dimension of $\bm{V}^{2}$ is
$\dim\bm{V}^{2}=2\sharp_{\mathsf{E}_{0}}$.

Finally, $\bm{V}^{3}$ consists of the vector-valued delta at interior
vertices:

$\bm{V}^{3}:=\operatorname{span}\\{\delta_{x}[\bm{a}_{x}],\bm{a}_{x}\in\mathbb{R}^{3}:x\in\mathsf{V}_{0}\\}.$

We have explicit forms of the differential operators on these function or
distributional spaces, which we will verify using definitions of
distributional derivatives.

We can identify $u$ with $\sum_{K\in\mathsf{K}}u|_{K}\|K\|$. For $u\in V^{0}$,

(3.9)
$\operatorname{hess}u=\sum_{f\in\mathsf{F}_{0}}\delta_{f}\big{[}[\\![\nabla
u]\\!]_{f}\otimes\bm{n}_{f}\big{]}.$

Here $[\\![\nabla u]\\!]_{f}=\sum_{K:K\supset f}\mathcal{O}(f,K)\nabla u|_{K}$
is the jump of $\nabla u$ on $f$, which has vanishing tangential components.
For
$\bm{\sigma}=\sum_{f\in\mathsf{f}_{0}}\bm{\delta}_{f}[\bm{m}_{f}\otimes\bm{n}_{f}]\in\bm{V}^{1}$
with $\bm{m}_{f}$ normal to $f$, we have

(3.10)
$\operatorname{curl}\bm{\sigma}=\sum_{e\in\mathsf{E}_{0}}\sum_{f:f\supset
e}\mathcal{O}(e,f)\delta_{e}[\bm{m}_{f}\otimes\bm{t}_{e}].$

The condition $f\supset e$ implies that $\bm{m}_{f}$ is normal to the edge
$e$. Consequently, we have $\operatorname{curl}\bm{V}^{1}\subseteq\bm{V}^{2}$.
Let
$\bm{v}=\sum_{e\in\mathsf{E}_{0}}\bm{\delta}_{e}[\bm{m}_{e}\otimes\bm{t}_{e}]\in\bm{V}^{2}$,
with $\bm{m}_{e}$ normal to the edge $e$. Then, it holds that

(3.11) $\operatorname{div}\bm{v}=-\sum_{x\in\mathsf{V}_{0}}\sum_{e:e\ni
x}\mathcal{O}(x,e)\delta_{x}[\bm{m}_{e}].$

This implies that, $\operatorname{div}\bm{V}^{2}\subseteq\bm{V}^{3}.$

######  Theorem 3.1.

The sequence (3.5) is a complex, and its cohomology is isomorphic to the
$\mathcal{P}_{1}$-valued de Rham cohomology
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR}(\Omega)\otimes\mathcal{P}_{1}$.

We now give a geometric interpretation of the distribution spaces defined
above. The space $\bm{V}^{1}$ can be regarded as a direct sum of
$\mathcal{P}_{0}^{n}(f)\|f\|$, where
$\mathcal{P}_{0}^{n}(f):=\\{\bm{a}\in\mathbb{R}^{3}:\bm{a}\text{ is normal to
}f\\}.$ Similarly, we can define $\mathcal{P}_{0}^{n}(e)$ as
$\mathcal{P}_{0}^{n}(e):=\\{\bm{a}\in\mathbb{R}^{3}:\bm{a}\text{ is normal to
}e\\}$, and $\mathcal{P}_{0}^{n}(x)=\mathcal{P}_{0}\otimes\mathbb{R}^{3}.$
Therefore, there is an isomorphism from the distribution spaces $\bm{V}^{i}$
and the direct sum of $\mathcal{P}_{0}^{n}$’s.

(3.12)
${0}$${\bm{V}^{1}}$${\bm{V}^{2}}$${\bm{V}^{3}}$${0}$${0}$${\bigoplus_{f\in\mathsf{F}_{0}}\mathcal{P}_{0}^{n}(f)}$${\bigoplus_{e\in\mathsf{E}_{0}}\mathcal{P}_{0}^{n}(e)}$${\bigoplus_{x\in\mathsf{V}_{0}}\mathcal{P}_{0}^{n}(x)}$${0.}$$\scriptstyle{\partial}$$\scriptstyle{\partial}$

Here the boundary operators in the bottom row are the standard boundary
operators (see Section 2.1 and [1, 30]). For the bottom row of (3.12), we note
the fact $\mathcal{P}_{0}^{n}(f)\subset\mathcal{P}_{0}^{n}(e)$ and identify
$\mathcal{P}_{0}^{n}(f)$ as subspaces of $\mathcal{P}_{0}^{n}(e)$ in the
definition of the boundary operators, see Remark 2.1.

We also have another version of the Hessian complex with homogeneous boundary
conditions:

(3.13)
${0}$${V^{0}_{0}}$${\bm{V}^{1}_{0}}$${\bm{V}^{2}_{0}}$${\bm{V}^{3}_{0}}$${0.}$$\scriptstyle{\operatorname{hess}_{0}}$$\scriptstyle{\operatorname{curl}_{0}}$$\scriptstyle{\operatorname{div}_{0}}$

Here the differential operators call for more explanations. Note that (3.13)
is a discrete version of (2.32), and the distributions in (2.32) are defined
as a space of distributions in $\mathbb{R}^{n}$. Therefore, the differential
operators are defined with test functions from
$C_{c}^{\infty}(\mathbb{R}^{n})$. To emphasize the difference, we will use the
subscript $0$ to represent the differential operators in
$H_{0}^{s}=H_{\overline{\Omega}}^{s}$ sequence. More specifically, we define

$\langle\operatorname{hess}_{0}u,\bm{\sigma}\rangle=\langle
u,\operatorname{div}\operatorname{div}\bm{\sigma}\rangle,\quad\forall\bm{\sigma}\in
C_{c}^{\infty}(\mathbb{R}^{3};\mathbb{S}),$

and similarly we define $\operatorname{curl}_{0}$ and $\operatorname{div}_{0}$
using test functions in $C_{c}^{\infty}(\mathbb{R}^{3};\mathbb{T})$ and
$C_{c}^{\infty}(\mathbb{R}^{3};\mathbb{R}^{3})$.

A significant difference is that functions in $C_{c}^{\infty}(\Omega)$ have
vanishing traces in $\partial\Omega$ while the traces of functions in
$C_{c}^{\infty}(\mathbb{R}^{3})$ are not necessarily zero. This implies that
the operators $\operatorname{hess}_{0}$, $\operatorname{curl}_{0}$ and
$\operatorname{div}_{0}$ might lead to boundary terms, which disappear in the
spaces in (3.5).

The observation above motivates us to define $\bm{V}_{0}^{1}$,
$\bm{V}_{0}^{2}$, and $\bm{V}_{0}^{3}$ via modifications of boundary terms.
Let

$\bm{V}^{1}_{0}:=\operatorname{span}\\{\delta_{f}[\bm{n}_{f}\otimes\bm{n}_{f}]:f\in\mathsf{F}\\},$
$\bm{V}^{2}_{0}:=\operatorname{span}\\{\delta_{e}[\bm{m}_{e}\otimes\bm{t}_{e}]:~{}\bm{m}_{e}\text{
is normal to }e,~{}e\in\mathsf{E}\\},$

and

$\bm{V}^{3}_{0}:=\operatorname{span}\\{\delta_{x}[\bm{a}_{x}]:\bm{a}_{x}\in\mathbb{R}^{3},x\in\mathsf{V}\\}.$

Finally, $V_{0}^{0}$ is the piecewise linear Lagrange element space with
homogeneous boundary conditions.

######  Theorem 3.2.

The sequence (3.13) is a complex, and its cohomology is isomorphic to
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR,c}(\Omega)\otimes\mathcal{P}_{1}$.

###  3.3. Divdiv complex in 3D

The $\operatorname{div}\operatorname{div}$ complex (see Figure 6) is inspired
by the dual of the Hessian complex (3.13), which we denote as

(3.14)
${0}$${\bm{U}^{0}}$${\bm{U}^{1}}$${\bm{U}^{2}}$${U^{3}}$${0.}$$\scriptstyle{\operatorname{dev}\operatorname{grad}}$$\scriptstyle{\operatorname{sym}\operatorname{curl}_{h}}$$\scriptstyle{\widehat{\operatorname{div}\operatorname{div}}}$

Figure 6. Divdiv complex in 3D (3.14).

The space $U^{3}$ is defined as the dual of $V^{0}_{0}$. Note that $V^{0}_{0}$
is the Lagrange finite element space with homogeneous boundary conditions, and
therefore $U^{3}$ consists of Dirac delta at interior vertices. Clearly, there
is a dual pairing between $U^{3}$ and $V^{0}_{0}$: For given function $v\in
V^{0}_{0}$ and $\hat{v}=\sum_{x\in\mathsf{V}}u_{x}\delta_{x}\in U^{3}$, we set

(3.15) $\langle v,\hat{v}\rangle_{V^{0}_{0}\times
U^{3}}=\sum_{x\in\mathsf{V}}u_{x}v(x).$

The space $\bm{U}^{0}$ is defined as the dual of $\bm{V}_{0}^{3}$. Recall that
$\bm{V}_{0}^{3}$ is the (vector) vertex delta function. As a consequence, we
choose $\bm{U}^{0}$ to be the standard Lagrange space.

The space $\bm{U}^{2}$ is inspired by the dual of $\bm{V}^{1}_{0}$. Recall
that $\bm{V}^{1}_{0}$ consists of face normal-normal delta (including the
boundary modes). This motivates us to choose $\bm{U}^{2}$ as a finite element
function space with normal-normal continuity. A natural choice is the stress
element of the TDNNS formulation (see [49]):

(3.16) $\bm{U}^{2}=\\{\bm{\sigma}\in
L^{2}(\Omega;\mathbb{S}):\bm{\sigma}|_{K}\in\mathcal{P}_{0}(\mathbb{S}),K\in\mathsf{K},\bm{n}_{f}\cdot\bm{\sigma}\cdot\bm{n}_{f}\text{
is continuous across faces}\\}.$

The shape function space of $\bm{U}^{2}$ is constant symmetric matrices, which
has six dimensions. In the context of the divdiv complex, $\bm{U}^{2}$ and
$U^{3}$ are related to the Hellan-Herrmann-Johnson (HHJ) formulation for the
biharmonic problem [18, 42, 48].

The degrees of freedom are given by the evaluation of the normal-normal
component on each face, i.e.,
$\int_{f}\bm{n}_{f}\cdot\bm{\sigma}\cdot\bm{n}_{f}$, for any $f\in\mathsf{F}$,
plus interior degrees of freedom. More precisely, we first decompose constant
symmetric matrices as follows:

(3.17) $\mathbb{S}=\mathbb{B}_{K}\oplus\mathcal{I}_{K},$

where $\mathbb{B}_{K}$ is the normal-normal bubble, namely,

(3.18)
$\mathbb{B}_{K}=\\{\bm{\sigma}\in\mathbb{S}:\bm{n}_{f}\cdot\bm{\sigma}\cdot\bm{n}_{f}=0\text{
for each face }f\text{ of }K\\},$

and the $\mathcal{I}_{K}$ is the orthogonal complement of $\mathcal{B}_{K}$
with respect to the Frobenius inner product. The degree of freedom of
$\bm{U}^{2}$ can be given as the following: for a given $\sigma\in\mathbb{S}$,

  1. (1a)

The normal-normal component on each face $f$:
$\int_{f}\bm{n}_{f}\cdot\bm{\sigma}\cdot\bm{n}_{f}$;

  2. (1b)

The inner product against the normal-normal bubble
$\int_{K}\bm{\sigma}:\bm{\tau}$ for $\bm{\tau}\in\mathbb{B}_{K}$.

The dimension of the global finite element space is thus
$\sharp_{F}+2\sharp_{K}$.

Similarly, $\bm{U}^{1}$ is inspired by the dual of $\bm{V}_{0}^{2}$. As we
have enriched $\bm{U}^{2}$ with two interior degrees of freedom for
unisolvency, we should also enrich the local shape function space of
$\bm{U}^{1}$. More precisely, we note that the following is an exact sequence:

(3.19)
${\cdots}$${\mathcal{P}_{1}(\mathbb{V})}$${\mathcal{P}_{0}(\mathbb{T})+\bm{x}\times\mathcal{P}_{0}(\mathbb{S})}$${\mathcal{P}_{0}(\mathbb{S})}$${\cdots.}$$\scriptstyle{\operatorname{dev}\operatorname{grad}}$$\scriptstyle{\operatorname{sym}\operatorname{curl}}$

The exactness follows from the existence of null-homotopy operators [11], as

$\mathcal{P}_{0}(\mathbb{T})+\bm{x}\times\mathcal{P}_{0}(\mathbb{S})=\operatorname{dev}\operatorname{grad}\mathcal{P}_{1}(\mathbb{V})\oplus\kappa\mathcal{P}_{0}(\mathbb{S}),$

where $\kappa$ is the Poincaré operator from [11]. Note that
$\bm{x}\times\mathbb{S}$ is automatically trace-free (the same algebraic
structure as the fact that $\operatorname{curl}$ of a symmetric matrix is
trace-free).

######  Remark 3.1.

To show the exactness of (3.19), it suffices to follow the construction in
[11] with smooth functions. The formulas in [11] involve derivative terms
(like the Cesàro-Volterra formula), but these terms vanish on constants. The
exactness of the above complexes can be also checked with direct computation,
see [17].

The degrees of freedom for $\bm{U}^{1}$ on $K$ are given by, for any
$\bm{\sigma}\in\bm{U}^{1}$,

  1. (2a)

$\int_{e}\bm{n}_{e,+}\cdot\bm{\sigma}\cdot\bm{t}_{e},\,\,\int_{e}\bm{n}_{e,-}\cdot\bm{\sigma}\cdot\bm{t}_{e},\quad\forall
e\in\mathsf{E}(K)$,

  2. (2b)

$\int_{K}\operatorname{curl}\bm{\sigma}:\bm{b},\quad\forall\bm{b}\in\mathbb{B}_{K}$.

######  Theorem 3.3.

The above degrees of freedom are unisolvent for $\bm{U}^{1}$.

Traceless finite elements with tangential-normal continuity have been used in
Mass Conserving mixed Stress (MCS) formulations for the Stokes equations [27].
Here $U^{1}$ is different from the construction in [27, Section 5.1]. To see
this, it suffices to note that in the lowest order case, the finite element in
[27, Section 5.1] has dimension 16 (see [27, Theorem 5.4]). Nevertheless,
$U^{1}$ has dimension 14. Moreover, the local shape function space of $U^{1}$
has a construction based on the Poincaré map
${\scriptscriptstyle\bullet}\times x$. This is important for characterising
the kernel of $\operatorname{sym}\operatorname{curl}_{h}$ (as a special case
of the cohomology result).

Next, we explain the operators in (3.14). The
$\operatorname{dev}\operatorname{grad}$ operator is a canonical one. The
operator $\operatorname{sym}\operatorname{curl}_{h}$ means the piecewise
$\operatorname{sym}\operatorname{curl}$ operator. Finally, as $\bm{U}^{2}$ is
a piecewise function, $\operatorname{div}\operatorname{div}$ in the sense of
distributions will map $\bm{U}^{2}$ to Dirac deltas on edges (codimension two
cells). However, we introduce a new operator
$\widehat{\operatorname{div}\operatorname{div}}$, which is defined with a
similar idea as the TDNNS method [49]. More precisely, the first
$\operatorname{div}$ maps a normal-normal continuous matrix field to the edge
deltas (dual of the Nédélec element) as in TDNNS. Then the second
$\operatorname{div}$ has the usual definition in the sense of distributions.
Alternatively, one may understand
$\widehat{\operatorname{div}\operatorname{div}}$ as the distributional
$\operatorname{div}\operatorname{div}$ restricted to vertices. The rigorous
definition is provided in (3.21).

######  Theorem 3.4.

The sequence (3.14) is a complex, and the cohomology is isomorphic to
$\mathcal{H}_{dR}^{{\scriptscriptstyle\bullet}}\otimes\mathcal{RT}$.

The two spaces in the middle of (3.14) contain interior degrees of freedom. To
obtain a precise dual version of (3.5) with a neat discrete form
interpretation, we can eliminate the interior degrees of freedom from the two
spaces simultaneously. Let $\widehat{\bm{U}}^{1}$ and $\widehat{\bm{U}}^{2}$
be the space spanned by the dual basis of (1a) and (1b), respectively. This
leads to

(3.20)
${0}$${\bm{U}^{0}}$${{\widehat{\bm{U}}}^{1}}$${\widehat{\bm{U}}^{2}}$${U^{3}}$${0.}$$\scriptstyle{\operatorname{dev}\operatorname{grad}}$$\scriptstyle{\operatorname{sym}\operatorname{curl}_{h}}$$\scriptstyle{\widehat{\operatorname{div}\operatorname{div}}}$

Here $\widehat{\operatorname{div}\operatorname{div}}$ is defined as

(3.21)
$\langle\widehat{\operatorname{div}\operatorname{div}}\bm{\sigma},v\rangle:=\sum_{f\in\mathsf{F}}\sum_{K:K\supset
f}\int_{f}\Big{[}\mathcal{O}(f,K)\bm{n}_{f}\times\bm{\sigma}|_{K}\cdot\bm{n}_{f}\Big{]}\cdot(\bm{n}_{f}\times\nabla
v).$

The definition comes from the following observation. For the distributional
$\operatorname{div}\operatorname{div}$ operator, it follows that for smooth
$\bm{u}$,

(3.22)
$\begin{split}\langle\operatorname{div}\operatorname{div}\bm{\sigma},\bm{u}\rangle:=\langle\bm{\sigma},\operatorname{hess}u\rangle=&\sum_{K\in\mathsf{K}}\int_{K}\bm{\sigma}:\operatorname{hess}u\\\
=&\sum_{K\in\mathsf{K}}\int_{\partial K}\bm{\sigma}\bm{n}\cdot\nabla u\\\
=&\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\Big{[}\mathcal{O}(f,K)\bm{\sigma}|_{K}\cdot\bm{n}_{f}\Big{]}\cdot\nabla
u\\\ =&\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\Big{[}\mathcal{O}(f,K)\bm{n}_{f}\times\bm{\sigma}|_{K}\cdot\bm{n}_{f}\Big{]}\cdot(\bm{n}_{f}\times\nabla
u).\end{split}$

The right-hand side makes sense for continuous, piecewise smooth function.
Restricting the distributional $\operatorname{div}\operatorname{div}$ operator
to the Lagrange space $\bm{U}^{3}$ leads to the
$\widehat{\operatorname{div}\operatorname{div}}$ operator.

The trimmed space $\widehat{\bm{U}}^{2}$ gains a precise duality to
$\bm{V}^{1}_{0}$ and a canonical form interpretation (analogous to the Whitney
forms) by sacrificing some approximation property since it does not contain
piecewise constant. But further applications and analysis, e.g., in solving
PDEs, are beyond the scope of this paper.

We also propose a compactly supported version of the
$\operatorname{div}\operatorname{div}$ complex as a dual of (3.5),

(3.23)
${0}$${\bm{U}^{0}_{0}}$${\bm{U}^{1}_{0}}$${\bm{U}^{2}_{0}}$${U^{3}_{0}}$${0.}$$\scriptstyle{\operatorname{dev}\operatorname{grad}}$$\scriptstyle{\operatorname{sym}\operatorname{curl}_{h}}$$\scriptstyle{\widehat{\operatorname{div}\operatorname{div}}_{0}}$

Here $U^{3}_{0}$ are spanned by vertex deltas of all the vertices
$x\in\mathsf{V}$. Finally, $\bm{U}^{i}_{0}$ are the subspace of $\bm{U}^{1}$
consisting of functions such that the degrees of freedom on the boundary
vanish.

######  Theorem 3.5.

The sequence (3.23) is a complex, and the cohomology is isomorphic to
$\mathcal{H}_{dR,c}^{{\scriptscriptstyle\bullet}}\otimes\mathcal{RT}$.

###  3.4. Complexes in 2D

We construct discrete versions of the Hessian and divdiv complexes and show
that their cohomology is isomorphic to the continuous version.

With a slight abuse of notation, we adopt the same notation as the 3D version
and denote the 2D discrete Hessian complex as

(3.24)
${0}$${V^{0}}$${\bm{V}^{1}}$${\bm{V}^{2}}$${0.}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{rot}}$

Figure 7. Distributional Hessian complex in 2D (3.24).  Figure 8.
Distributional divdiv complex in 2D (3.26).

The space $V^{0}$ is the first-order Lagrange finite element space consisting
of continuous piecewise linear functions. The space $\bm{V}^{1}$ consists of
normal-normal deltas on each interior edge, i.e.,

$\bm{V}^{1}:=\left\\{\sum_{e\in\mathsf{E}_{0}}u_{e}\delta_{e}[\bm{n}_{e}\bm{n}_{e}^{T}]:u_{e}\in\mathbb{R}\right\\}.$

The space $\bm{U}^{2}$ consists of Dirac deltas at each interior vertex:

$\bm{V}^{2}:=\left\\{\sum_{x\in\mathsf{V}_{0}}\delta_{x}[\bm{a}_{x}]:\bm{a}_{x}\in\mathbb{R}^{2}\right\\}.$

######  Theorem 3.6.

The sequence (3.24) is a complex. The cohomology of (3.24) is isomorphic to
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR}(\Omega)\otimes\mathcal{P}_{1}$.

Similarly, we can introduce the version of homogeneous boundary conditions:

(3.25)
${0}$${V^{0}_{0}}$${\bm{V}^{1}_{0}}$${\bm{V}^{2}_{0}}$${0.}$$\scriptstyle{\operatorname{hess}_{0}}$$\scriptstyle{\operatorname{rot}_{0}}$

The first space $V^{0}_{0}$ is the piecewise linear Lagrange finite element
space with zero boundary condition, and correspondingly,

$\bm{V}^{1}_{0}:=\Big{\\{}\sum_{e\in\mathsf{E}}u_{e}\delta_{e}[\bm{n}_{e}\otimes\bm{n}_{e}]\Big{\\}},\text{
and
}\bm{V}^{2}_{0}:=\Big{\\{}\sum_{x\in\mathsf{V}}\delta_{x}[\bm{a}_{x}]\Big{\\}}.$

Compared to $\bm{V}^{1}$ and $\bm{V}^{2}$, the only difference here is that
the distribution spaces $\bm{V}^{1}_{0}$ and $\bm{V}^{2}_{0}$ consist of Dirac
deltas on the boundary. The cohomology of (3.25) is isomorphic to the
compactly supported version
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR,c}(\Omega)\otimes\mathcal{P}_{1}$.

The discrete divdiv complex reads as follows:

(3.26)
${0}$${\bm{U}^{0}}$${\bm{U}^{1}}$${U^{2}}$${0.}$$\scriptstyle{\operatorname{sym}\operatorname{curl}}$$\scriptstyle{\operatorname{div}\operatorname{div}}$

Here, the space $\bm{U}^{0}$ is the vector Lagrange finite element space

(3.27) $\bm{U}^{0}:=\\{\bm{u}\in
C^{0}(\Omega;\mathbb{R}^{2}):\bm{u}|_{f}\in\mathcal{P}_{1}\otimes\mathbb{R}^{2}\text{
for each face }\bm{f}\in\mathsf{F}\\},$

the space $\bm{U}^{1}$ is the rotated Regge element space:

(3.28) $\bm{U}^{1}:=\\{\bm{\sigma}\in L^{2}(\Omega;\mathbb{S}^{2\times
2}):\bm{\sigma}|_{f}\in\mathbb{S}^{2\times
2},\bm{n}_{e}\cdot\bm{\sigma}\cdot\bm{n}_{e}\text{ is continuous on each
edge}.\\}$

The last space $U^{2}$ is the scalar vertex delta :

(3.29) $U^{2}:=\operatorname{span}\\{\delta_{x}:x\in\mathsf{V}_{0}\\}.$

Similarly, we obtain the compactly supported version by alternating the
boundary conditions.

######  Theorem 3.7.

The cohomology of (3.26) is isomorphic to the de Rham cohomology
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR}(\Omega)\otimes\mathcal{RT}$,
while the cohomology of the compactly supported version is isomorphic to
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR,c}(\Omega)\otimes\mathcal{RT}$.

We remark that (3.26) (with $U^{2}$ replaced by its dual, the Lagrange finite
element space) has appeared in [chen2018multigrid]. Nevertheless, to the best
of our knowledge, the cohomology was open.

###  3.5. Tensor product construction

Although in this paper we mostly focus on simplicial meshes and discrete
structures on them, we provide some brief remarks on a tensor product
construction in this section.

For the diagram (2.18) and thus the complexes derived from it, there is a
canonical construction on cubical grids [6]. This was done by a tensor product
of de Rham and BGG complexes in 1D. The focus of [6] was on conforming finite
elements. Therefore, both the first row and the first column of (2.18) were
discretized by a standard conforming finite element sequence. However, much of
the algebraic structure does not rely on this conformity and can be
generalized once we have other patterns in 1D. In particular, we can start
from distributional complexes in 1D and derive $n$D versions with a tensor
product construction. The generalization is relatively straightforward.
Therefore, we will not present the construction in full detail, but rather
refer to Figure 9-11 for an illustration of the idea.

Figure 9. Discrete BGG diagram and complex in 1D. In the diagram (figure on
the left), the top row is the canonical finite element de Rham complex
consisting of the Lagrange element (continuous piecewise linear) and piecewise
constant. The bottom row starts with piecewise constants and ends up with
Dirac delta at interior vertices. It is straightforward to check that the
cohomology of both rows is isomorphic to the continuous version. Connecting
the two rows, we derive the BGG (1D Hessian) complex (figure on the right).
Figure 10. BGG diagram for deriving the Hessian complex in 2D. The top row is
a canonical finite element de Rham complex (see, e.g., [9, 19]) obtained by a
tensor product of the first row in Figure 9 (left). The bottom row starts from
a tensor product of constants and linear functions (the first space of the top
and bottom rows in the diagram in Figure 9). Figure 11. Tensor product
distributional Hessian complex in 2D derived from the diagram in Figure 10,
which also follows from a generalization of [6].

##  4\. Complexes in 2D

To show the ideas of deriving cohomology in a relativity simple case, we start
with complexes in 2D.

###  4.1. 2D Hessian complex

In this section, we focus on the Hessian complex in 2D (3.24), showing that
the sequence is a complex (Theorem 3.6, part I) and the cohomology is
isomorphic to
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR}(\Omega)\otimes\mathcal{P}_{1}$
(Theorem 3.6, part II).

###### Proof of Theorem 3.6 (part I).

We identify $u$ with $\sum_{f}u|_{f}\|f\|.$ For $u\in V^{0}$ and $\varphi\in
C_{c}^{\infty}(\Omega;\mathbb{R}^{2\times 2})$, it holds that

$\displaystyle\langle\operatorname{grad}\operatorname{grad}u,\varphi\rangle=\langle
u,\operatorname{div}\operatorname{div}\varphi\rangle=\sum_{f\in\mathsf{F}}\int_{f}u\operatorname{div}\operatorname{div}\varphi$
$\displaystyle=$
$\displaystyle\sum_{f\in\mathsf{F}}\int_{f}\operatorname{grad}\operatorname{grad}u:\varphi-\int_{\partial
f}\operatorname{grad}u\cdot\varphi\bm{n}={-}\sum_{e\in\mathsf{E}_{0}}\sum_{f:f\supset
e}\mathcal{O}(e,f)\int_{e}\nabla u|_{f}\cdot\varphi\bm{n}_{e}$

which implies

(4.1)
$\operatorname{grad}\operatorname{grad}u={-}\sum_{e\in\mathsf{E}_{0}}\sum_{f\in\mathsf{F}}\mathcal{O}(e,f)(\nabla
u|_{f}\cdot\bm{n}_{e})\delta_{e}[\bm{n}_{e}\bm{n}_{e}^{\mathrm{T}}]\in\bm{V}^{1}.$

Here we have used the facts that $u$ is piecewise linear (thus
$\operatorname{grad}\operatorname{grad}u$ vanishes on each cell), $u$ is
continuous (thus integrating by parts the first $\operatorname{div}$ does not
lead to jump terms), and $\varphi$ vanishes on the boundary (thus no boundary
terms appear).

For
$\sigma=\sum_{e\in\mathsf{E}_{0}}\sigma_{e}\delta_{e}[\bm{n}_{e}\bm{n}_{e}^{\mathrm{T}}]\in\bm{V}^{1}$
and $\varphi\in C_{c}^{\infty}(\Omega;\mathbb{R}^{2})$,

$\displaystyle{-}\langle\operatorname{rot}\sigma,\varphi\rangle=\langle\sigma,\operatorname{curl}\varphi\rangle=\left\langle\sum_{e\in\mathsf{E}_{0}}\sigma_{e}\delta_{e}[\bm{n}_{e}\bm{n}_{e}^{\mathrm{T}}],\operatorname{curl}\varphi\right\rangle=\sum_{e\in\mathsf{E}_{0}}\sigma_{e}\int_{e}\bm{n}_{e}\bm{n}_{e}^{\mathrm{T}}:\operatorname{curl}\varphi\mathrm{d}s$
$\displaystyle=$
$\displaystyle\sum_{e\in\mathsf{E}_{0}}\sigma_{e}\int_{e}\bm{n}_{e}\bm{n}_{e}^{\mathrm{T}}:\operatorname{curl}\varphi\mathrm{d}s=\sum_{e\in\mathsf{E}_{0}}\sigma_{e}\int_{e}\bm{n}_{e}\cdot\operatorname{curl}\varphi\bm{n}_{e}\mathrm{d}s$
$\displaystyle=$
$\displaystyle\sum_{e\in\mathsf{E}_{0}}\sigma_{e}\bm{n}_{e}\cdot\int_{e}\operatorname{grad}\varphi\bm{\tau}_{e}\mathrm{d}s=\left\langle\sum_{v\in\mathsf{V}_{0}}\sum_{e:v\subset
e}\sigma_{e}(\bm{\tau}_{e}\cdot\bm{\tau}_{e}^{v})\delta_{v}[\bm{n}_{e}],\varphi\right\rangle,$

where $\bm{\tau}_{e}^{v}$ a unit vector parallel to $\bm{\tau}_{e}$ pointing
to $v$ (thus $\bm{\tau}_{e}\cdot\bm{\tau}_{e}^{v}=\pm 1$ reflects the
orientation). Therefore,

$\operatorname{rot}\sigma={-}\sum_{v\in\mathsf{V}_{0}}\sum_{e:v\subset
e}\mathcal{O}(v,e)\sigma_{e}\delta_{v}[\bm{n}_{e}]\in\bm{V}^{2}.$

This proves that the sequence (3.24) is a complex. ∎

In the rest of this section, we compute the cohomology of (3.24).

The proof of Theorem 3.6 (part II) follows two steps. First, we construct an
auxiliary sequence starting with piecewise polynomials without interelement
continuity, see (4.2) below. A straightforward calculation shows that there is
a correspondence between this sequence and the chain complex of $\Delta$. This
thus identifies the cohomology of (4.2) with the homology (with
$\mathcal{P}_{1}$ coefficients) of the domain. Second, we form short exact
sequences using (4.2) and (3.24). A diagram chase allows us to conclude with
the cohomology of (3.24).

Next, we provide details of the above sketched proof. Consider an auxiliary
sequence

(4.2)
${0}$${V^{0}_{-}}$${\bm{V}^{1}_{-}}$${\bm{V}^{2}_{-}}$${0.}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{rot}}$

The space $V^{0}_{-}$ consists of piecewise linear, but discontinuous
functions:

(4.3) $V^{0}_{-}=C^{-1}\mathcal{P}_{1}:=\\{u\in
L^{2}(\Omega):u|_{f}\in\mathcal{P}_{1}(f)\text{ for all }f\in\mathsf{F}\\}.$

To proceed, we first consider $\operatorname{hess}u$ for $u\in
C^{-1}\mathcal{P}_{1}$ in the distributional sense. This resembles the
procedure of (3.6). Actually, for $\bm{\sigma}\in
C_{c}^{\infty}(\Omega;\mathbb{S})$, we have

(4.4) $\begin{split}\langle\operatorname{hess}u,\bm{\sigma}\rangle:=&\langle
u,\operatorname{div}\operatorname{div}\bm{\sigma}\rangle\\\
=&\sum_{f\in\mathsf{F}}\int_{f}u\cdot\operatorname{div}\operatorname{div}\bm{\sigma}\\\
=&\sum_{f\in\mathsf{F}}\int_{\partial
f}u\cdot(\operatorname{div}\bm{\sigma}\bm{n}_{\partial f})-\int_{f}\nabla
u\cdot\operatorname{div}\bm{\sigma}\\\
=&\sum_{e\in\mathsf{E}_{0}}\sum_{f\in\mathsf{F}}\mathcal{O}(e,f)\Big{[}u|_{f}\cdot(\operatorname{div}\bm{\sigma}\cdot\bm{n}_{e})-(\nabla
u|_{f})\cdot(\bm{\sigma}\bm{n}_{e})\Big{]}.\end{split}$

This invokes us to define the following functionals: for each edge
$e\in\mathsf{E}$ and $p\in\mathcal{P}_{1}$, we define the distribution
$\hat{\bm{v}}_{e}^{1}[p]$ as

(4.5)
$\langle\hat{\bm{v}}_{e}^{1}[p],\bm{\sigma}\rangle=\int_{e}(\operatorname{div}\operatorname{sym}(\bm{\sigma})\cdot\bm{n}_{e})p-\operatorname{sym}\bm{\sigma}:{(\nabla
p\otimes{\bm{n}}_{e})}.$

######  Remark 4.1.

By our convention of notation, $\mathcal{P}_{1}$ is a bivariate function.
Therefore, $\nabla p$ is well-defined. In evaluating the integrals, we
restrict $p$ and $\nabla p$ to $e$, with a slight abuse of notation. Recall
that in standard notation of Discontinuous Galerkin method,
$\sum_{f}\mathcal{O}(e,f)u|_{f}$ refers to the jump of $u$ on edge $e$, while
$\sum_{f}\mathcal{O}(e,f)\nabla u|_{f}$ refers to the jump of $\nabla u$.

We define the space $\bm{V}^{1}_{-}$ to be the span of all the distributions
$\hat{\bm{v}}_{e}^{1}[p]$, namely,

(4.6)
$\bm{V}^{1}_{-}=\Big{\\{}\sum_{e\in\mathsf{E}_{0}}\hat{\bm{v}}_{e}^{1}[p_{e}]:p_{e}\in\mathcal{P}_{1},\forall
e\in\mathsf{E}_{0}\Big{\\}}.$

######  Lemma 4.1.

For $u\in V^{0}_{-}$, it holds that

$\operatorname{hess}u=\sum_{e\in\mathsf{E}_{0}}\sum_{f:f\supset
e}\mathcal{O}(e,f)\hat{\bm{v}}_{e}^{1}[u|_{f}].$

######  Remark 4.2.

We note that Lemma 4.1 is consistent with (4.1) if $u\in\bm{V}^{1}$, i.e., $u$
is continuous. In fact, the continuity of $u$ implies $p=\sum_{f:f\supset
e}\mathcal{O}(e,f)u|_{f}$ vanishes at $e$. Using the fact that
$p\in\mathcal{P}_{1}$ and $p|_{e}=0$, we obtain $\nabla p=\alpha\bm{n}_{e}$
for some (scalar) constant $\alpha$. Therefore,

(4.7)
$\langle\hat{\bm{v}}_{e}^{1}[p],\bm{\sigma}\rangle=\alpha\int_{e}\bm{\sigma}:(\bm{n}_{e}\otimes\bm{n}_{e})=\langle\alpha\delta_{e}[\bm{n}_{e}\otimes\bm{n}_{e}],\bm{\sigma}\rangle$

This has the form of (4.1).

Finally, we consider $\operatorname{rot}$ of $\hat{\bm{v}}_{e}^{1}[p]$. Again,
a direct calculation yields

(4.8)
$\begin{split}\langle\operatorname{rot}\hat{\bm{v}}_{e}^{1}[p],\bm{v}\rangle=&-\langle\hat{\bm{v}}_{e}^{1}[p],\operatorname{curl}\bm{v}\rangle\\\
=&\int_{e}\operatorname{curl}\bm{v}:\operatorname{sym}(\nabla
p\otimes\bm{n}_{e}){-}(\operatorname{div}\operatorname{sym}\operatorname{curl}\bm{v}\cdot\bm{n}_{e})p.\end{split}$

It holds that

(4.9)
$\begin{split}\int_{e}\operatorname{curl}\bm{v}:\operatorname{sym}(\nabla
p\otimes\bm{n}_{e})&=\int_{e}\operatorname{sym}\operatorname{curl}\bm{v}:(\nabla
p\otimes\bm{n}_{e})\\\ &=\int_{e}\operatorname{curl}\bm{v}:(\nabla
p\otimes\bm{n}_{e})-\frac{1}{2}\begin{bmatrix}0&\operatorname{div}\bm{v}\\\
{-}\operatorname{div}\bm{v}&0\end{bmatrix}:(\nabla p\otimes\bm{n}_{e})\\\
&=\int_{e}\frac{\partial}{\partial\bm{t}_{e}}(\nabla
p\cdot\bm{v})-\frac{1}{2}\int_{e}\operatorname{div}\bm{v}\cdot\frac{\partial
p}{\partial\bm{t}_{e}},\end{split}$

and

(4.10)
$\begin{split}\int_{e}(\operatorname{div}\operatorname{sym}\operatorname{curl}\bm{v}\cdot\bm{n}_{e})p&=\frac{1}{2}\int_{e}(\operatorname{curl}(\operatorname{div}\bm{v})\cdot\bm{n}_{e})p=\frac{1}{2}\int_{e}\frac{\partial}{\partial\bm{t}_{e}}(\operatorname{div}\bm{v})p.\end{split}$

Therefore,

(4.11)
$\begin{split}\langle\operatorname{rot}\hat{\bm{v}}_{e}^{1}[p],\bm{v}\rangle&=\int_{e}\frac{\partial}{\partial\bm{t}_{e}}(\bm{v}\cdot\nabla
p-\frac{1}{2}\operatorname{div}\bm{v}\cdot p)=(\bm{v}\cdot\nabla
p-\frac{1}{2}\operatorname{div}\bm{v}\cdot p)|_{x_{1}}^{x_{2}}.\end{split}$

We then define the final space $\bm{V}^{2}_{-}$ as follows:

(4.12)
$\bm{V}^{2}_{-}=\Big{\\{}\sum_{x\in\mathsf{V}_{0}}\hat{\bm{v}}_{x}^{2}[p_{x}]:p_{x}\in\mathcal{P}_{1}\Big{\\}},$

where $\hat{\bm{v}}_{x}^{2}[p_{x}]$ is defined as

(4.13) $\langle\hat{\bm{v}}_{x}^{2}[p_{x}],\bm{w}\rangle=(\bm{w}\cdot\nabla
p-\frac{1}{2}\operatorname{div}\bm{w}\cdot p)(x).$

The above calculation yields the following lemma.

######  Lemma 4.2.

For $\hat{\bm{v}}_{e}^{1}[p]\in\bm{V}^{1}_{-}$,

(4.14)
$\operatorname{rot}\hat{\bm{v}}_{e}^{1}[p]=\sum_{x\in\mathsf{V}_{0}}\mathcal{O}(x,e)\hat{\bm{v}}_{x}^{2}[p].$

As a result, $\operatorname{rot}\bm{V}^{1}_{-}\subset\bm{V}^{2}_{-}.$

Combining both Lemma 4.1 and Lemma 4.2, we have

######  Proposition 4.1.

The sequence (4.2) is a complex, and its cohomology is isomorphic to
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR}(\Omega)\otimes\mathcal{P}_{1}$.

###### Proof.

The above calculation can be summarized in the following diagram.

(4.15)
${0}$${V^{0}_{-}}$${\bm{V}^{1}_{-}}$${\bm{V}^{2}_{-}}$${0}$${0}$${C_{2}(\Delta,\mathcal{P}_{1};\partial\Delta)}$${C_{1}(\Delta,\mathcal{P}_{1};\partial\Delta)}$${C_{0}(\Delta,\mathcal{P}_{1};\partial\Delta)}$${0.}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\kappa^{0}_{-}}$$\scriptstyle{\operatorname{rot}}$$\scriptstyle{\kappa^{1}_{-}}$$\scriptstyle{\kappa^{2}_{-}}$$\scriptstyle{\partial}$$\scriptstyle{\partial}$

Here $C_{k}(\Delta,\mathcal{P}_{1};\partial\Delta)$ is the space of simplicial
$k$-chains with $\mathcal{P}_{1}$ coefficients, see Section 2.1. For each
element in $C_{2}(\Delta,\mathcal{P}_{1};\partial\Delta)$, we can represent it
as

(4.16) $\omega=\sum_{f\in\mathsf{F}}p_{f}\|f\|,$

where $\|f\|$ is the free element associated with the face $f$. Similar
expressions will be used later for 1-chain and 0-chain.

The vertical mapping $\kappa^{0}_{-}$ maps $v\in V^{0}_{-}$ to

$\kappa^{0}_{-}(v)=\sum_{f\in\mathsf{F}}v|_{f}\|f\|.$

We can similarly define linear maps $\kappa^{1}_{-}$ and $\kappa^{2}_{-}$ by
evaluating the coefficients, i.e.,

$\kappa^{1}_{-}(\hat{\bm{v}}_{e}^{1}[p]):=p\|e\|,\quad\kappa^{2}_{-}(\hat{\bm{v}}_{x}^{2}[q]):=q\|x\|.$

Obviously, $\kappa_{-}^{i},=0,1,2$ are bijective and the diagram (4.15)
commutes. As a consequence, $\kappa_{-}^{{\scriptscriptstyle\bullet}}$ induces
bijective maps on cohomology, and therefore the cohomology
$\mathcal{H}^{k}(V^{{\scriptscriptstyle\bullet}}_{-})$ is isomorphic to
$\mathcal{H}_{2-k}(\Delta,\mathcal{P}_{1};\partial\Delta)$, which is further
isomorphic to
$\mathcal{H}_{2-k}(\Delta;\partial\Delta)\otimes\mathcal{P}_{1}\cong\mathcal{H}^{k}_{dR}(\Omega)\otimes\mathcal{P}_{1}$
by Theorems 2.1 and 2.3. ∎

######  Remark 4.3.

In fact, the complex (4.2) can be regarded as the “skeleton” of the finite
element divdiv complex [37], playing the role of Whitney forms for high-order
de Rham complexes. Similar structures have been used in [35, 36].

Now we consider the cohomology of the original complex (3.24). The key is to
regard (3.24) as the kernel of (4.2) under certain maps. This leads to short
exact sequences and the following diagram:

(4.17)
${0}$${0}$${\bm{V}^{1}}$${\bm{V}^{2}}$${0}$${0}$${V^{0}_{-}}$${\bm{V}^{1}_{-}}$${\bm{V}^{2}_{-}}$${0}$${0}$${\bigoplus\limits_{f\in\mathsf{F}}\mathcal{P}_{1}(f)}$${\bigoplus\limits_{e\in\mathsf{E}_{0}}\mathcal{P}_{1}(e)}$${\bigoplus\limits_{x\in\mathsf{V}_{0}}\mathcal{P}_{1}(x)}$${0.}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{rot}}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{g^{0}}$$\scriptstyle{\operatorname{rot}}$$\scriptstyle{g^{1}}$$\scriptstyle{g^{2}}$$\scriptstyle{\widetilde{\partial}}$$\scriptstyle{\widetilde{\partial}}$

First, we introduce the last row of (4.17):

(4.18)
${0}$${\bigoplus\limits_{f\in\mathsf{F}}\mathcal{P}_{1}(f)}$${\bigoplus\limits_{e\in\mathsf{E}_{0}}\mathcal{P}_{1}(e)}$${\bigoplus\limits_{x\in\mathsf{V}_{0}}\mathcal{P}_{1}(x)}$${0.}$$\scriptstyle{\widetilde{\partial}}$$\scriptstyle{\widetilde{\partial}}$

Here $\widetilde{\partial}$ is a composition of the boundary operator and the
restriction operators, defined as

(4.19)
$\widetilde{\partial}(\sum_{f\in\mathsf{F}}p_{f}\|f\|)=\sum_{e\in\mathsf{E}_{0}}\sum_{f\in\mathsf{F}}\mathcal{O}(e,f)(p_{f})|_{e}\|e\|,$
(4.20)
$\widetilde{\partial}(\sum_{e\in\mathsf{E}_{0}}p_{e}\|e\|)=\sum_{x\in\mathsf{V}_{0}}\sum_{e\in\mathsf{E}_{0}}\mathcal{O}(x,e)(p_{e})|_{x}\|x\|.$

The following results were proved in [43, Lemma 4.9].

######  Proposition 4.2.

The sequence (4.18) is a complex, and its cohomology is $V^{0}$, $0$ and $0$,
respectively.

For completeness, we provide a proof here. The proof is based on the following
lemma.

######  Lemma 4.3.

The complex (4.18) can be identified with a direct sum of the following vertex
patch complexes for each vertex $v$,

(4.21) ${0}$${\bigoplus\limits_{f\in\mathsf{F},f\ni
v}\mathbb{R}}$${\bigoplus\limits_{e\in\mathsf{E}_{0},e\ni
v}\mathbb{R}}$${\mathbb{R}}$${0.}$$\scriptstyle{{\partial}}$$\scriptstyle{{\partial}}$

Here $\partial$ is induced by the boundary operator (the relative homology
version).

###### Proof.

The key observation is that for each simplex $f\in\mathsf{F}$, the space
$\mathcal{P}_{1}(f)$ is generated by the barycenter coordinates $\lambda_{v}$,
where $v$ is a vertex of $f$. By this, we can rewrite
$\mathcal{P}_{1}(f)\cong\bigoplus_{v\in f}\mathbb{R}$. Therefore,

(4.22)
$\bigoplus_{f\in\mathsf{F}}\mathcal{P}_{1}(f)\cong\bigoplus_{f\in\mathsf{F}}\bigoplus_{v\in
f}\mathbb{R}\cong\bigoplus_{v\in\mathsf{V}}\bigoplus_{f\in\mathsf{F},f\ni
v}\mathbb{R}.$

For $f\in\mathsf{F}$ and $\lambda_{v}\in\mathcal{P}_{1}(f)$ with $v\in f$, by
definition it holds that

(4.23) $\widetilde{\partial}\lambda_{v}|_{f}=\sum_{e\in\mathsf{E}_{0},f\ni
e}\mathcal{O}(e,f)\lambda_{v}|_{e}.$

For $e$ such that $v\not\in e$, $\lambda_{v}$ vanishes on $e$. Therefore, the
right-hand side of (4.23) contains only terms that are in the patch of $v$.
This proves that (4.18) can be identified as a complex on each vertex patch.∎

###### Proof of Proposition 4.2.

Note that the homology of (4.21) is identical to the relative homology of the
vertex patch of $v$, where $v$ is either an interior or a boundary vertex. As
a vertex patch is contractible, the homology vanishes except for at index
zero, where $\mathcal{H}_{0}\cong\operatorname{span}\\{\lambda_{v}\\}$, the
span of the Lagrange basis (hat) function at $v$. ∎

Now we are ready to finish the proof of Theorem 3.6.

###### Proof of Theorem 3.6 (part II).

Recall the diagram (4.17). We now define the vertical maps in (4.17) as the
restriction to subcells, i.e., $g^{0}=I$,
$g^{1}:\hat{\bm{v}}_{e}^{1}[p]\mapsto p|_{e}\|e\|$, and
$g^{2}:\hat{\bm{v}}_{x}^{2}[p]\mapsto p|_{x}\|x\|$. It is not difficult to see
that each $g^{k}$ is onto. Now we prove that the space
$\bm{V}^{1}=\mathcal{N}(g^{1})$ and $\bm{V}^{2}=\mathcal{N}(g^{2})$. The
former is implied by Remark 4.2. To see the latter, note that, by (4.13),
$p|_{x}=0$ implies

$\langle\hat{\bm{v}}_{x}^{2}[p_{x}],\bm{w}\rangle=(\bm{w}\cdot\nabla p)(x),$

which is exactly all the vertex delta. Therefore, the kernel of $g^{2}$ is
$\bm{V}^{2}$.

The above argument shows that columns of (4.17) are short exact sequences.
Thus, it induces a long exact sequence of homologies:

(4.24)
${0}$${0}$${\mathcal{N}(\operatorname{rot}:\bm{V}^{1}\to\bm{V}^{2})}$${\mathcal{H}^{2}(V^{{\scriptscriptstyle\bullet}})}$${0}$${0}$${\mathcal{H}^{0}_{dR}(\Omega)\otimes\mathcal{P}_{1}}$${\mathcal{H}^{1}_{dR}(\Omega)\otimes\mathcal{P}_{1}}$${\mathcal{H}^{2}_{dR}(\Omega)\otimes\mathcal{P}_{1}}$${0}$${0}$${V^{0}}$${0}$${0}$${0.}$

With a diagram chase, we can verify that the connecting map
$V^{0}\to\mathcal{N}(\operatorname{rot}:\bm{V}^{1}\to\bm{V}^{2})$ is Hessian.

Now we compute the cohomology. For
$\mathcal{H}^{0}(V^{{\scriptscriptstyle\bullet}})=\mathcal{N}(\operatorname{hess}:V^{0}\to\bm{V}^{1})$,
by the exactness of the long sequence,
$\mathcal{H}^{0}(V^{{\scriptscriptstyle\bullet}})\cong\mathcal{H}^{0}_{dR}(\Omega)\otimes\mathcal{P}_{1}$.
For
$\mathcal{H}^{1}(V^{{\scriptscriptstyle\bullet}})={\mathcal{N}(\operatorname{rot}:\bm{V}^{1}\to\bm{V}^{2})}/{\operatorname{hess}V^{0}}$,
note that the following part from the long sequence

${V^{0}}$${\mathcal{N}(\operatorname{rot}:\bm{V}^{1}\to\bm{V}^{2})}$${\mathcal{H}^{1}_{dR}(\Omega)\otimes\mathcal{P}_{1}\to
0}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{j}$

is exact. Therefore,
${\mathcal{N}(\operatorname{rot}:\bm{V}^{1}\to\bm{V}^{2})}/{\operatorname{hess}V^{0}}\cong\mathcal{R}(j)=\mathcal{H}^{1}_{dR}(\Omega)\otimes\mathcal{P}_{1}.$
Here we used the fact that $f:X\to Y$ induces an isomorphism
$X/\mathcal{N}(f)\cong\mathcal{R}(f).$

Finally, it holds that
$\mathcal{H}^{2}(V^{{\scriptscriptstyle\bullet}})\cong\mathcal{H}^{2}_{dR}(\Omega)\otimes\mathcal{P}_{1}$.
∎

###  4.2. 2D Hessian complex with homogeneous boundary condition

In this section, we introduce the discrete Hessian complex with homogeneous
boundary conditions (HBCs) and compute its cohomology.

######  Lemma 4.4.

The sequence (3.25) is a complex.

###### Proof.

We check the complex property by definition. For $u\in V^{0}_{0}$ and
$\bm{\sigma}\in C^{\infty}_{c}(\mathbb{R}^{2};\mathbb{S})$, we have

(4.25)
$\begin{split}\langle\operatorname{hess}_{0}u,\bm{\sigma}\rangle&=\langle
u,\operatorname{div}\operatorname{div}\bm{\sigma}\rangle\\\
&=\sum_{f\in\mathsf{F}}\int_{f}u\operatorname{div}\operatorname{div}\bm{\sigma}\\\
&=-\sum_{f\in\mathsf{F}}\int_{f}\nabla u\cdot\operatorname{div}\bm{\sigma}\\\
&=-\sum_{e\in\mathsf{E}}\sum_{f:f\supset
e}\int_{e}\Big{[}\mathcal{O}_{0}(e,f)(\nabla
u|_{f})\cdot\bm{n}_{e}\Big{]}\left(\bm{n}_{e}\cdot\bm{\sigma}\cdot\bm{n}_{e}\right).\end{split}$

The last line uses the fact that the jump of $\nabla u$ only has the normal
component. Note that $v|_{\partial\Omega}=0$ implies that $\frac{\partial
v}{\partial\bm{t}}|_{\partial\Omega}=0.$ The remaining proof is similar to
those in Theorem 3.6. ∎

Now we consider the cohomology of (3.25), and intuitively we can guess the
result is $\mathcal{H}_{2-k}(\Delta)\otimes\mathcal{P}_{1}$, where
$\mathcal{H}_{{\scriptscriptstyle\bullet}}(\Delta)$ is the simplicial
homology. We verify this claim below.

######  Theorem 4.1.

The cohomology of (3.25) is isomorphic to
$\mathcal{H}_{2-k}(\Delta)\otimes\mathcal{P}_{1}$, therefore to
$\mathcal{H}^{k}_{dR,c}(\Omega)\otimes\mathcal{P}_{1}.$

The proof is analogous to the Hessian complex without homogeneous boundary
conditions. We first introduce $V^{0}_{0,-}:=V^{0}_{-}$,

$\bm{V}^{1}_{0,-}:=\Big{\\{}\sum_{e\in\mathsf{E}}\hat{\bm{v}}^{1}_{e}[p_{e}]:p_{e}\in\mathcal{P}_{1}\Big{\\}},\text{
and
}\bm{V}^{2}_{0,-}:=\Big{\\{}\sum_{x\in\mathsf{V}}\hat{\bm{v}}^{2}_{x}[p_{x}]:p_{x}\in\mathcal{P}_{1}\Big{\\}}.$

Using the same technique, we can prove that

######  Proposition 4.3.

The sequence

(4.26)
${0}$${V^{0}_{0,-}}$${\bm{V}^{1}_{0,-}}$${\bm{V}^{2}_{0,-}}$${0}$$\scriptstyle{\operatorname{hess}_{0}}$$\scriptstyle{\operatorname{rot}_{0}}$

is a complex, and the cohomology is
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR,c}(\Omega)\otimes\mathcal{P}_{1}.$

The detailed proof of Proposition 4.3 can be found in Appendix.

To complete, we now introduce the boundary version of (4.18):

(4.27)
${0}$${\bigoplus_{f\in\mathsf{F}}\mathcal{P}_{1}(f)}$${\bigoplus_{e\in\mathsf{E}}\mathcal{P}_{1}(e)}$${\bigoplus_{x\in\mathsf{V}}\mathcal{P}_{1}(x)}$${0.}$$\scriptstyle{\widetilde{\partial}_{0}}$$\scriptstyle{\widetilde{\partial}_{0}}$

Here $\widetilde{\partial}$ is defined as

(4.28)
$\widetilde{\partial}_{0}(\sum_{F\in\mathsf{F}}p_{f}\|f\|)=\sum_{f\in\mathsf{F}}\sum_{e\in\mathsf{E}}\mathcal{O}_{0}(e,f)(p_{f})|_{e}\|e\|,$
(4.29)
$\widetilde{\partial}_{0}(\sum_{e\in\mathsf{E}}p_{e}\|e\|)=\sum_{x\in\mathsf{V}}\sum_{e\in\mathsf{E}}\mathcal{O}_{0}(x,e)(p_{e})|_{x}\|x\|.$

######  Proposition 4.4.

The sequence (4.18) is a complex, and its cohomology is $V_{0}^{0}$, $0$ and
$0$, respectively.

###### Proof.

The proof is similar to that of Proposition 4.2, and can be found in Appendix.
∎

With these two results, we can prove Theorem 4.1 by investigating the diagram
and its cohomology below:

(4.30)
${0}$${0}$${\bm{V}^{1}_{0}}$${\bm{V}^{2}_{0}}$${0}$${0}$${V^{0}_{0,-}}$${\bm{V}^{1}_{0,-}}$${\bm{V}^{2}_{0,-}}$${0}$${0}$${\bigoplus\limits_{f\in\mathsf{F}}\mathcal{P}_{1}(f)}$${\bigoplus\limits_{e\in\mathsf{E}}\mathcal{P}_{1}(e)}$${\bigoplus\limits_{x\in\mathsf{V}}\mathcal{P}_{1}(x)}$${0.}$$\scriptstyle{\operatorname{hess}_{0}}$$\scriptstyle{\operatorname{rot}_{0}}$$\scriptstyle{\operatorname{hess}_{0}}$$\scriptstyle{g^{0}}$$\scriptstyle{\operatorname{rot}_{0}}$$\scriptstyle{g^{1}}$$\scriptstyle{g^{2}}$$\scriptstyle{\widetilde{\partial}_{0}}$$\scriptstyle{\widetilde{\partial}_{0}}$

Here $g^{k}$’s are the same as those in the proof of Theorem 3.6. The only
difference comes from the homology of the lower row. The desired result
follows from a chase on the following diagram:

(4.31)
${0}$${0}$${\mathcal{N}(\operatorname{rot}_{0}:\bm{V}^{1}_{0}\to\bm{V}^{2}_{0})}$${\mathcal{H}^{2}(V^{{\scriptscriptstyle\bullet}}_{0})}$${0}$${0}$${\mathcal{H}^{0}_{dR,c}(\Omega)\otimes\mathcal{P}_{1}}$${\mathcal{H}^{1}_{dR,c}(\Omega)\otimes\mathcal{P}_{1}}$${\mathcal{H}^{2}_{dR,c}(\Omega)\otimes\mathcal{P}_{1}}$${0}$${0}$${V^{0}_{0}}$${0}$${0}$${0.}$

###  4.3. 2D divdiv complex

In this section, we focus on the divdiv complex in 2D, showing that the
sequence (3.26) is a complex and proving the cohomology is isomorphic to
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR}(\Omega)\otimes\mathcal{RT}$
(Theorem 3.7, part I).

######  Proposition 4.5.

The sequence (3.26) is a complex.

###### Proof.

It is straightforward to see that $\operatorname{sym}\operatorname{curl}$ maps
$\bm{U}^{0}$ to $\bm{U}^{1}$. For the $\operatorname{div}\operatorname{div}$
part, it suffices to note that for $u\in C_{c}^{\infty}(\Omega)$,

(4.32)
$\begin{split}\langle\operatorname{div}\operatorname{div}\bm{\sigma},u\rangle=&\langle\bm{\sigma},\nabla^{2}u\rangle\\\
=&\sum_{f\in\mathsf{F}}\int_{f}\bm{\sigma}:\nabla^{2}u\\\
=&\sum_{e\in\mathsf{E}_{0}}\sum_{f:f\supset
e}\int_{e}\Big{[}\mathcal{O}(e,f)[\bm{\sigma}|_{f}\bm{n}_{e}]\Big{]}\cdot\nabla
u\\\ =&\sum_{e\in\mathsf{E}_{0}}\sum_{f:f\supset
e}\int_{e}\Big{[}\mathcal{O}(e,f)[\bm{t}_{e}\cdot\bm{\sigma}|_{f}\cdot\bm{n}_{e}]\Big{]}\frac{\partial
u}{\partial\bm{t}_{e}}\\\ =&\sum_{e\in\mathsf{E}_{0}}\sum_{f:f\supset
e}\Big{[}\mathcal{O}(e,f)[\bm{t}_{e}\cdot\bm{\sigma}|_{f}\cdot\bm{n}_{e}]\Big{]}(u(x_{2})-u(x_{1})),\end{split}$

where $x_{1}$ and $x_{2}$ are the two vertices of $e$. The third line is due
to the normal-normal continuity of $\bm{\sigma}$. This implies that
$\operatorname{div}\operatorname{div}\bm{U}^{1}\subset U^{2}$. ∎

It is nontrivial to compute the cohomology of (3.26) directly. Nevertheless,
we can regard the divdiv complex (3.26) as the dual of the Hessian complex
with homogeneous boundary conditions (3.25). This allows us to conclude the
cohomology of (3.26) from the results for the Hessian complex that we have
obtained. The duality is explained in the following diagram and will be
explained below:

(4.33)
${0}$${V^{0}_{0}}$${\bm{V}^{1}_{0}}$${\bm{V}^{2}_{0}}$${0}$${0}$${U^{2}}$${\bm{U}^{1}}$${\bm{U}^{0}}$${0,}$$\scriptstyle{\operatorname{hess}_{0}}$${\ast}$$\scriptstyle{\operatorname{rot}_{0}}$${\ast}$${\ast}$$\scriptstyle{\operatorname{div}\operatorname{div}}$$\scriptstyle{\operatorname{sym}\operatorname{curl}}$

where $\ast$ denotes duality. Observe that in each of the vertical lines in
(4.33), the degrees of freedom and the delta functions change their roles:

  * -

$V^{0}_{0}$ is the scalar Lagrange finite element, whose degrees of freedom
are at each vertex. Correspondingly, the distribution space $U^{2}$ consists
of the vertex deltas. The relationship of $\bm{V}^{2}_{0}$ and $\bm{U}^{0}$
shares the same idea, but is in a vector setting.

  * -

For the tensor case, the distribution space $\bm{V}^{1}_{0}$ is spanned by the
normal-normal deltas, while $\bm{U}^{1}$ is the piecewise constant function
with normal-normal continuity.

This invokes us using dual pairing to show that the two horizontal complexes
of (4.33) have the same homology. With slight abuse of notation, we still use
$\langle\cdot,\cdot\rangle$ to denote the family of dual pairing, but we do
not require that the functions are smooth. For example, for
$\hat{v}=\sum_{x\in\mathsf{V}_{0}}a_{x}\delta_{x}$, we define the dual pairing
$\langle v,\hat{v}\rangle=\sum_{x\in\mathsf{V}_{0}}a_{x}v(x).$ The expression
is well-defined whenever $v$ is single-valued at each internal vertex.
Restricting the dual pairing to the spaces $\bm{U}^{2}\times V^{0}_{0}$ will
give a non-degenerate one. Similarly, we can define the dual pairing between
$\bm{U}^{0}$ and $\bm{V}^{2}_{0}$. For the tensor case, for
$\hat{\bm{\sigma}}=\sum_{e\in\mathsf{E}}a_{e}\delta_{e}[\bm{n}_{e}\otimes\bm{n}_{e}]$,
we define

(4.34)
$\langle\bm{\sigma},\hat{\bm{\sigma}}\rangle=\sum_{e\in\mathsf{E}}a_{e}\int_{e}(\bm{n}_{e}\cdot\bm{\sigma}\cdot\bm{n}_{e})|_{e}.$

The dual pairing is well-defined since
$\bm{n}_{e}\cdot\bm{\sigma}\cdot\bm{n}_{e}$ is single-valued.

The next proposition shows that with the pairs specified above, the two
complexes are adjoint to each other.

######  Proposition 4.6.

The divdiv complex (3.26) is adjoint to the Hessian complex with boundary
conditions in (3.25), namely,

(4.35) $\langle
v,\operatorname{div}\operatorname{div}\bm{\sigma}\rangle_{V^{0}_{0}\times\bm{U}^{2}}=\langle\operatorname{hess}_{0}v,\bm{\sigma}\rangle_{\bm{V}^{1}_{0}\times\bm{U}^{1}},\quad
v\in V^{0}_{0},\bm{\sigma}\in\bm{U}^{1},$

and

(4.36)
$\langle\bm{\sigma},\operatorname{sym}\operatorname{curl}\bm{u}\rangle_{V^{1}_{0}\times\bm{U}^{1}}=\langle{-}\operatorname{rot}_{0}\sigma,\bm{u}\rangle_{V^{2}_{0}\times\bm{U}^{0}},\quad\bm{\sigma}\in\bm{V}^{1}_{0},\bm{u}\in\bm{U}^{0}.$

######  Remark 4.4.

These identities resemble the definition of distributional derivatives. The
difference is that the differential operators on both sides here are in the
sense of distributions. As a result, the identities hold valid only on the
given finite/distributional element pair, which relies on the fact that the
functions are piecewise constant or linear.

###### Proof.

We first show (4.35). For $u\in V^{0}_{0}$ and $\bm{\sigma}\in\bm{U}^{1}$,
note that $\int_{f}\operatorname{div}\operatorname{div}(u\bm{\sigma})=0$ on
each face $f$ as $u$ is linear and $\sigma$ is constant (here
$\operatorname{div}\operatorname{div}$ is defined on $f$ for smooth functions
in the usual sense). This implies

(4.37)
$\begin{split}0&=\sum_{e\in\mathsf{E}}\sum_{f:f\supset\mathsf{F}}\int_{e}\Big{[}\mathcal{O}_{0}(e,f)\operatorname{div}(u|_{f}\bm{\sigma})\Big{]}\cdot\bm{n}_{e}\\\
&=\sum_{e\in\mathsf{E}}\int_{e}\Big{[}\mathcal{O}_{0}(e,f)\frac{\partial}{\partial\bm{n}_{e}}(u|_{f}\bm{\sigma}\bm{n}_{e})\cdot\bm{n}_{e}\Big{]}+\Big{[}\mathcal{O}_{0}(e,f)\frac{\partial}{\partial\bm{t}_{e}}(u\bm{\sigma}\bm{t}_{e})\cdot\bm{n}_{e}\Big{]}\\\
&=\sum_{e\in\mathsf{E}}\sum_{f:f\supset
e}\int_{e}\Big{[}\mathcal{O}_{0}(e,f)\frac{\partial}{\partial\bm{n}_{e}}u|_{f}\Big{]}(\bm{n}_{e}\cdot\bm{\sigma}\cdot\bm{n}_{e})+\int_{e}\Big{[}\mathcal{O}_{0}(e,f)\bm{t}_{e}\cdot\bm{\sigma}|_{f}\cdot\bm{n}_{e}\Big{]}\frac{\partial}{\partial\bm{t}_{e}}u.\end{split}$

Here the last line is due to the fact that $u$ and
$\bm{n}_{e}\cdot\bm{\sigma}\cdot\bm{n}_{e}$ are continuous across the edge. It
then follows from (4.32) (together with the fact that $u$ vanishes on
$\mathsf{E}_{\partial}$), and (4.1) that

$0=\langle
v,\operatorname{div}\operatorname{div}\bm{\sigma}\rangle_{V_{0}^{0}\times\bm{U}^{2}}-\langle\operatorname{hess}_{0}v,\bm{\sigma}\rangle_{\bm{V}_{0}^{1}\times\bm{U}^{1}}.$

This proves (4.35).

Next, we have for $\bm{u}\in\bm{U}^{0}$, and $e\in\mathsf{E}$,

(4.38)
$\begin{split}\langle\operatorname{sym}\operatorname{curl}\bm{u},\delta_{e}[\bm{n}_{e}\otimes\bm{n}_{e}]\rangle=&\int_{e}\bm{n}_{e}\cdot\operatorname{curl}\bm{u}\cdot\bm{n}_{e}\\\
=&\int_{e}\bm{n}_{e}\cdot\nabla\bm{u}\cdot\bm{t}_{e}\\\
=&\bm{u}\cdot\bm{n}_{e}(x_{2})-\bm{u}\cdot\bm{n}_{e}(x_{1})\\\
=&\langle\bm{u},{-}\operatorname{rot}\delta_{e}[\bm{n}_{e}\otimes\bm{n}_{e}]\rangle,\end{split}$

where $e=[x_{1},x_{2}]$. This proves the second identity. ∎

###### The proof of Theorem 3.7 (part I).

We first define $\pi_{\bm{V}^{0}_{0}\to U^{2}}$ such that
$\pi_{\bm{V}^{0}_{0}\to U^{2}}(v)=\sum_{x\in\mathsf{V}}v(x)\delta_{x}.$
Similarly, we define $\pi_{\bm{V}^{1}_{0}\to\bm{U}^{1}}$,
$\pi_{\bm{V}^{2}_{0}\to\bm{U}^{0}}$. Define
$\pi_{\bm{U}^{k}\to\bm{V}^{2-k}_{0}},k=0,1,2,$ to be their inverses.

Next, we define suitable inner products on the spaces $U^{k}$ and
$V^{2-k}_{0}$. For $V^{0}_{0}$, we define the inner product

(4.39) $(u,u^{\prime})_{V^{0}}=\sum_{x\in\mathsf{V}_{0}}u(x)u^{\prime}(x).$

For $U^{2}$, we define the inner product

(4.40)
$(\sum_{x\in\mathsf{V}_{0}}u_{x}\delta_{x},\sum_{x\in\mathsf{V}_{0}}u^{\prime}_{x}\delta_{x})_{U^{2}}=\sum_{x\in\mathsf{V}_{0}}u_{x}u^{\prime}_{x}.$

Therefore, we have

(4.41) $(u,u^{\prime})_{V^{0}_{0}}=\langle
u,\pi_{V^{0}_{0}\to\bm{U}^{2}}u^{\prime}\rangle_{V^{0}_{0}\times\bm{U}^{2}}=(\pi_{V^{0}_{0}\to\bm{U}^{2}}u,\pi_{V^{0}_{0}\to\bm{U}^{2}}u^{\prime})_{U^{2}}.$

Similarly, we can define the inner product and the one-to-one mappings for the
remaining spaces. Now we can regard the complexes (4.33) as two Hilbert
complexes (with finite dimensional spaces) to compute the (co)homology.

Our goal is to show that
$\mathcal{H}^{2}(U^{{\scriptscriptstyle\bullet}})\cong\mathcal{H}^{2-k}(V_{0}^{{\scriptscriptstyle\bullet}})$.
The key is that when specifying an inner product, the cohomology can be
represented by the harmonic forms. For example, the cohomology
$\mathcal{H}^{1}(V^{{\scriptscriptstyle\bullet}}_{0})$ can be represented via
$\bm{\sigma}\in\bm{V}_{0}^{1}$, satisfying
$\operatorname{rot}_{0}\bm{\sigma}=0$ and
$(\bm{\sigma},\operatorname{hess}u)_{V^{1}_{0}}=0.$ Such a $\bm{\sigma}$ is
called a harmonic form of $\bm{V}^{1}_{0}$.

Now we show that $\bm{\sigma}$ is a harmonic form of $\bm{V}^{1}_{0}$ if and
only if $\pi_{\bm{V}^{1}_{0}\to\bm{U}^{1}}\bm{\sigma}$ is a harmonic form of
$\bm{U}^{1}$. This follows from direct calculations.

(4.42) $\begin{split}&\operatorname{rot}_{0}\bm{\sigma}=0\\\
\Longleftrightarrow&\langle\operatorname{rot}_{0}\bm{\sigma},\bm{\xi}\rangle_{\bm{V}^{1}_{0}\times\bm{U}^{1}}=0,\quad\forall\bm{\xi}\in\bm{U}^{1}\\\
\Longleftrightarrow&\langle\bm{\sigma},\operatorname{sym}\operatorname{curl}\bm{\xi}\rangle_{\bm{V}^{1}_{0}\times\bm{U}^{1}}=0,\quad\forall\bm{\xi}\in\bm{U}^{1}\\\
\Longleftrightarrow&(\pi_{\bm{V}^{1}_{0}\to\bm{U}^{1}},\bm{\sigma},\operatorname{sym}\operatorname{curl}\bm{U}^{1})_{\bm{U}^{1}}=0.\end{split}$

And

(4.43) $\begin{split}&(\bm{\sigma},\operatorname{hess}_{0}u)=0,\quad\forall
u\in V^{0}_{0}\\\
\Longleftrightarrow&\langle\pi_{\bm{V}^{1}_{0}\to\bm{U}^{1}}\bm{\sigma},\operatorname{hess}_{0}u\rangle_{\bm{U}^{1}\times\bm{V}^{1}_{0}}=0,\quad\forall
u\in V^{0}_{0}\\\
\Longleftrightarrow&\langle\operatorname{div}\operatorname{div}\pi_{\bm{V}^{1}_{0}\to\bm{U}^{1}}\bm{\sigma},u\rangle_{U^{2}\times
V^{0}_{0}}=0,\quad\forall u\in V^{0}_{0}\\\
\Longleftrightarrow&\operatorname{div}\operatorname{div}\pi_{\bm{V}^{1}_{0}\to\bm{U}^{1}}\bm{\sigma}=0.\end{split}$

This indicates that $\pi_{\bm{V}^{1}_{0}\to\bm{U}^{1}}$ induces an
isomorphism. We can show that the other $\pi_{\scriptscriptstyle\bullet}$
operators are isomorphisms similarly.

It follows that
$\mathcal{H}^{k}(U^{{\scriptscriptstyle\bullet}})\cong\mathcal{H}^{2-k}(V^{{\scriptscriptstyle\bullet}}_{0})\cong\mathcal{H}^{2-k}_{dR,c}(\Omega)\otimes\mathcal{P}_{1}$.
The theorem is proved by noticing that
$\mathcal{H}^{2-k}_{dR,c}(\Omega)\cong\mathcal{H}^{k}_{dR}(\Omega).$ ∎

###  4.4. 2D divdiv complex with homogeneous boundary condition

We can also consider a divdiv complex with homogeneous boundary condition:

(4.44)
${0}$${\bm{U}^{0}_{0}}$${\bm{U}^{1}_{0}}$${U^{2}_{0}}$${0,}$$\scriptstyle{\operatorname{sym}\operatorname{curl}_{0}}$$\scriptstyle{\operatorname{div}\operatorname{div}_{0}}$

where $\bm{U}^{0}_{0}$ is the (vector) piecewise Lagrange element with zero
boundary conditions,

(4.45)
$\bm{U}^{1}_{0}:=\\{\bm{\sigma}\in\bm{U}^{1}:(\bm{n}_{e}\cdot\bm{\sigma}\cdot\bm{n}_{e})|_{\partial\Omega}=0\\},$

and

(4.46) $U^{2}_{0}:=\operatorname{span}\\{\delta_{x}:x\in\mathsf{V}\\}.$

Regarding (4.44) as the dual complex of (3.24), we can prove Theorem 3.7 (part
II), which is re-summarized in the following theorem. The detailed proof can
be found in Appendix.

######  Theorem 4.2.

The sequence (4.44) is a complex, with cohomology isomorphic to
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR,c}(\Omega)\otimes\mathcal{RT}$.

##  5\. Complexes in 3D

###  5.1. 3D Hessian complex

Note that $\delta_{f}[\bm{A}]$ is linear with respect to the tensor $\bm{A}$,
therefore, we can rewrite
$c_{f}\delta_{f}[\bm{n}_{f}\otimes\bm{n}_{f}]=\delta_{f}[c_{f}\bm{n}_{f}\otimes\bm{n}_{f}]$
for any real number $c_{f}\in\mathbb{R}$. Moreover, the element
$\bm{\sigma}\in\bm{V}^{1}$ can be expressed as

(5.1)
$\bm{\sigma}=\sum_{f\in\mathsf{F}_{0}}\delta_{f}[\bm{m}_{f}\otimes\bm{n}_{f}],$

where $\bm{m}_{f}\in\mathbb{R}^{3}$ is a normal vector to the face $f$ (not
necessarily unit).

######  Proposition 5.1.

The sequence (3.5) is a complex.

###### Proof.

Let $u\in V^{0}$. For any $\bm{\varphi}\in
C_{c}^{\infty}(\Omega;\mathbb{R}^{3\times 3})$, it holds that

(5.2) $\begin{split}\langle\operatorname{hess}u,\bm{\varphi}\rangle&=\langle
u,\operatorname{div}\operatorname{div}\bm{\varphi}\rangle=\int_{\Omega}u\operatorname{div}\operatorname{div}\bm{\varphi}\\\
&=-\int_{\Omega}\operatorname{grad}u\cdot\operatorname{div}\bm{\varphi}\\\
&=-\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\Big{[}\mathcal{O}(f,K)\nabla
u|_{K}\Big{]}\cdot(\bm{\varphi}\cdot\bm{n}_{f})\\\
&=-\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\Big{[}\mathcal{O}(f,K)\frac{\partial
u|_{K}}{\partial\bm{n}}\Big{]}\cdot(\bm{n}_{f}\otimes\bm{n}_{f}:\bm{\varphi}).\end{split}$

Here, the second line uses the fact that $u$ is continuous, and the third line
uses the fact that the piecewise gradient of $\operatorname{grad}u$ is zero.
The last line is due to that $u$ is continuous, and thus, the jump of $u$
across the face has vanishing tangential components. Therefore, we have

(5.3) $\operatorname{hess}u=-\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\Big{[}\mathcal{O}(f,K)\frac{\partial
u|_{K}}{\partial\bm{n}}\Big{]}\,\delta_{f}[\bm{n}_{f}\otimes\bm{n}_{f}].$

Let $\bm{\sigma}\in\bm{V}^{1}$, which has the expression

(5.4)
$\bm{\sigma}=\sum_{f\in\mathsf{F}_{0}}\delta_{f}[\bm{m}_{f}\otimes\bm{n}_{f}],$

where $\bm{m}_{f}$ is normal to face $f$, $f\in\mathsf{F}_{0}$. Then it holds
that

(5.5)
$\operatorname{curl}\bm{\sigma}=\sum_{e\in\mathsf{E}_{0}}\sum_{f:f\supset
e}\mathcal{O}(e,f)\delta_{e}[\bm{m}_{f}\otimes\bm{t}_{e}].$

Consequently, we have $\operatorname{curl}\bm{V}^{1}\subseteq\bm{V}^{2}$.

We now show (5.5). By linearity, it suffices to show that the identity (5.5)
holds for a single term $\delta_{f}[\bm{m}_{f}\otimes\bm{n}_{f}]$. We follow
the definition: for $\bm{\sigma}\in
C^{\infty}(\mathbb{R}^{3};\mathbb{R}^{3\times 3})$, it holds that,

(5.6)
$\begin{split}\langle\operatorname{curl}\delta_{f}[\bm{m}_{f}\otimes\bm{n}_{f}],\bm{\sigma}\rangle=&\langle\delta_{f}[\bm{m}_{f}\otimes\bm{n}_{f}],\operatorname{curl}\bm{\sigma}\rangle=\int_{f}\bm{m}_{f}\cdot\operatorname{curl}\bm{\sigma}\cdot\bm{n}_{f}\\\
=&\int_{\partial f}\bm{m}_{f}\cdot\bm{\sigma}\cdot\bm{t}_{\partial
f}.\end{split}$

Taking $\bm{\sigma}\in C_{c}^{\infty}(\Omega;\mathbb{R}^{3\times 3})$
completes the proof of (5.5).

Let $\bm{\tau}\in\bm{V}^{2}$, and denote

$\bm{\tau}=\sum_{e\in\mathsf{E}_{0}}\delta_{e}[\bm{m}_{e}\otimes\bm{t}_{e}],$

where $\bm{m}_{e}$ is normal to the edge $e$. Then, it holds that

(5.7) $\operatorname{div}\bm{\tau}=-\sum_{x\in\mathsf{V}_{0}}\sum_{e:e\ni
x}\mathcal{O}(x,e)\delta_{x}[\bm{m}_{e}].$

This implies that, $\operatorname{div}\bm{V}^{2}\subseteq\bm{V}^{3}.$

It suffices to check the case for
$\operatorname{div}\delta_{e}[\bm{m}_{e}\otimes\bm{t}_{e}]$. For $\bm{u}\in
C^{\infty}_{c}(\Omega)$, it holds that

(5.8)
$\begin{split}\langle\operatorname{div}\delta_{e}[\bm{m}_{e}\otimes\bm{t}_{e}],\bm{u}\rangle=&\langle\delta_{e}[\bm{m}_{e}\otimes\bm{t}_{e}],-\nabla\bm{u}\rangle\\\
=&-\int_{e}\bm{m}_{e}\cdot\frac{\partial\bm{u}}{\partial\bm{t}_{e}}\\\
=&(\bm{m}_{e}\cdot\bm{u})(x_{1})-(\bm{m}_{e}\cdot\bm{u})(x_{2}).\end{split}$

Here $e=[x_{1},x_{2}]$. This completes the proof.

∎

Here we use the following lemma, which is a direct consequence of the Stokes’
formula.

######  Lemma 5.1.

For an oriented face $f$ and a vector-valued function $\bm{u}$, we have

(5.9) $\int_{f}\operatorname{curl}\bm{u}\cdot\bm{n}=\int_{\partial
f}\bm{u}\cdot\bm{t},$

where the direction of the unit normal vector $\bm{n}$ of $f$, and that of the
unit tangential vector $\bm{t}$ of $\partial f$, are determined by the given
orientation.

Analogous to the two-dimensional case, we introduce the following auxiliary
Hessian complex starting with a discrete $L^{2}$ space:

(5.10)
${0}$${V^{0}_{-}}$${\bm{V}^{1}_{-}}$${\bm{V}^{2}_{-}}$${\bm{V}^{3}_{-}}$${0.}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

Here, the superscript $-$ denotes that this complex starts with a
discontinuous element.

#### Construction of $V^{0}_{-}$

The space is chosen as the discontinuous piecewise linear function space,
i.e.,

(5.11) $V^{0}_{-}=C^{-1}\mathcal{P}_{1}:=\\{u\in
L^{2}(\Omega):u|_{K}\in\mathcal{P}_{1}(K)\text{ for all cells
}K\in\mathsf{K}\\}.$

We can identify the space to
$\bigoplus_{K\in\mathsf{K}}\mathcal{P}_{1}=C_{3}(\Delta,\mathcal{P}_{1};\partial\Delta)$
by $\kappa^{0}_{-}(u):=\sum_{K\in\mathsf{K}}u|_{K}\|K\|.$

#### Construction of $\bm{V}^{1}_{-}$

For each face $f\in\mathsf{F}$ and $p\in\mathcal{P}_{1}$, we first define the
following distribution:

(5.12)
$\langle\hat{\bm{v}}^{1}_{f}[p],\bm{\sigma}\rangle=\int_{f}(\operatorname{div}\operatorname{sym}(\bm{\sigma})\cdot\bm{n}_{f})p-\int_{f}\operatorname{sym}\bm{\sigma}:(\nabla
p\otimes\bm{n}_{f}),$

for $\bm{\sigma}\in C^{\infty}_{c}(\Omega;\mathbb{R}^{3\times 3})$.

We now define the space $\bm{V}^{1}_{-}$ to be the span of all the
distribution $\hat{\bm{v}}_{f}^{1}[p]$, namely,

(5.13)
$\bm{V}^{1}_{-}=\Big{\\{}\sum_{f\in\mathsf{F}_{0}}\hat{\bm{v}}^{1}_{f}[p_{f}]:f\in\mathsf{F}_{0},p_{f}\in\mathcal{P}_{1}\Big{\\}}.$

The following proposition states that
$\operatorname{hess}V^{0}_{-}\subseteq\bm{V}^{1}_{-}$.

######  Proposition 5.2.

For $u\in V^{0}_{-}$, we have

(5.14) $\operatorname{hess}u=\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\mathcal{O}(f,K)\hat{\bm{v}}^{1}_{f}[u|_{K}]$

in the sense of distributions. Consequently, we have
$\operatorname{hess}V^{0}_{-}\subseteq\bm{V}^{1}_{-}.$

###### Proof.

By definition, for symmetric-matrix valued function $\bm{\sigma}\in
C_{c}^{\infty}(\Omega;\mathbb{S})$ we have

(5.15) $\begin{split}\langle\operatorname{hess}u,\bm{\sigma}\rangle=&\langle
u,\operatorname{div}\operatorname{div}\bm{\sigma}\rangle=\int_{\Omega}u\operatorname{div}\operatorname{div}\bm{\sigma}\\\
=&-\int_{\Omega}\nabla
u\cdot\operatorname{div}\bm{\sigma}+\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\Big{[}\mathcal{O}(f,K)u|_{K}\Big{]}\operatorname{div}\bm{\sigma}\cdot\bm{n}_{f}\\\
=&-\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\Big{[}\mathcal{O}(f,K)\nabla
u|_{K}\Big{]}\cdot(\bm{\sigma}\bm{n}_{f})+\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\Big{[}\mathcal{O}(f,K)u|_{K}\Big{]}\operatorname{div}\bm{\sigma}\cdot\bm{n}_{f}\\\
=&-\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\bm{\sigma}:(\Big{[}\mathcal{O}(f,K)\nabla
u|_{K}\Big{]}\otimes\bm{n}_{f})+\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\Big{[}\mathcal{O}(f,K)u|_{K}\Big{]}\operatorname{div}\bm{\sigma}\cdot\bm{n}_{f}.\end{split}$

Since $\operatorname{hess}u$ is symmetric, it then holds that

(5.16)
$\begin{split}\langle\operatorname{hess}u,\bm{\sigma}\rangle=&\langle\operatorname{hess}u,\operatorname{sym}\bm{\sigma}\rangle\\\
=&-\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\bm{\sigma}:\operatorname{sym}(\Big{[}\mathcal{O}(f,K)\nabla
u|_{K}\Big{]}\otimes\bm{n}_{f})\\\
&\quad\quad\quad+\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\Big{[}\mathcal{O}(f,K)\nabla
u|_{K}\Big{]}\operatorname{div}\operatorname{sym}\bm{\sigma}\cdot\bm{n}_{f}.\end{split}$

This completes the proof of (5.14), and therefore
$\operatorname{hess}V^{0}_{-}\subseteq\bm{V}^{1}_{-}.$ ∎

We define the following mapping:

$\kappa^{1}_{-}:\bm{V}^{1}_{-}\to\bigoplus_{f\in\mathsf{F}_{0}}\mathcal{P}_{1}=C_{2}(\Delta,\mathcal{P}_{1};\partial\Delta),$

by

(5.17)
$\kappa^{1}_{-}(\sum_{f\in\mathsf{F}_{0}}\hat{\bm{v}}_{f}^{1}[p_{f}])=\sum_{f\in\mathsf{F}_{0}}p_{f}\|f\|.$

Here $\|f\|$ is the free element associated to the face $f$.

#### Construction of $\bm{V}^{2}_{-}$

The space $\bm{V}^{2}_{-}$ contains the following traceless distribution
$\hat{\bm{v}}_{e}^{2}[p]$ for each internal edge $e\in\mathsf{E}_{0}$ and
$p\in\mathcal{P}_{1}$:

(5.18)
$\langle\hat{\bm{v}}_{e}^{2}[p],\bm{\sigma}\rangle=-\int_{e}\operatorname{dev}\bm{\sigma}:(\nabla
p\otimes\bm{t}_{e})+\frac{1}{2}\int_{e}(\operatorname{div}\operatorname{dev}\bm{\sigma}\cdot\bm{t}_{e})p,$

for any test function $\bm{\sigma}\in
C^{\infty}_{c}(\mathbb{R}^{3};\mathbb{R}^{3\times 3}).$

By definition, $\hat{\bm{v}}_{e}^{2}[p]$ is a traceless distribution.

######  Proposition 5.3.

For $\bm{\sigma}\in\bm{V}^{1}_{-}$, denote that
$\bm{\sigma}=\sum_{f\in\mathsf{F}_{0}}\hat{\bm{v}}^{1}_{f}[p_{f}]$. Then we
have

(5.19)
$\operatorname{curl}\bm{\sigma}=\sum_{e\in\mathsf{E}_{0}}\sum_{f:f\supset
e}\mathcal{O}(e,f)\hat{\bm{v}}_{e}^{2}[p_{f}].$

This implies $\operatorname{curl}\bm{\sigma}\in\bm{V}^{2}_{-}$ in the sense of
distributions.

###### Proof.

It suffices to check a single term $\hat{\bm{v}}^{1}_{f}[p]\in\bm{V}^{1}_{-}$.
By definition, for a matrix-valued function $\bm{\sigma}\in
C^{\infty}_{c}(\Omega;\mathbb{T})$, it follows that

(5.20)
$\begin{split}\langle\operatorname{curl}\hat{\bm{v}}^{1}_{f}[p],\bm{\sigma}\rangle=&\langle\hat{\bm{v}}^{1}_{f}[p],\operatorname{sym}\operatorname{curl}\bm{\sigma}\rangle\\\
=&\int_{f}-(\operatorname{sym}\operatorname{curl}\bm{\sigma}\cdot\bm{n}_{f})\cdot\nabla
p+(\operatorname{div}(\operatorname{sym}\operatorname{curl}\bm{\sigma})\cdot\bm{n}_{f})p\\\
=&\int_{f}-(\operatorname{curl}\bm{\sigma}\cdot\bm{n}_{f})\cdot\nabla
p+\frac{1}{2}\int_{f}(\operatorname{div}\bm{\sigma}\times\bm{n}_{f})\cdot\nabla
p+\frac{1}{2}\int_{f}(\operatorname{curl}\operatorname{div}\bm{\sigma}\cdot\bm{n}_{f})p\\\
=&-\int_{\partial f}(\bm{\sigma}\cdot\bm{t}_{\partial f})\cdot\nabla
p+\frac{1}{2}\int_{f}\operatorname{curl}(\operatorname{div}\bm{\sigma}p)\cdot\bm{n}_{f}\\\
=&-\int_{\partial f}(\bm{\sigma}\cdot\bm{t}_{\partial f})\cdot\nabla
p+\frac{1}{2}\int_{\partial
f}(\operatorname{div}\bm{\sigma}\cdot\bm{t}_{\partial f})p.\end{split}$

Here the third line comes from the identity
$(\operatorname{sym}\operatorname{curl}\bm{\sigma})\bm{n}_{f}=(\operatorname{curl}\bm{\sigma})\bm{n}_{f}-\frac{1}{2}\operatorname{div}\bm{\sigma}\times\bm{n}_{f}$,
and the fourth line is due to the identity
$\operatorname{curl}(p\bm{w})=\nabla
p\times\bm{w}+p\operatorname{curl}\bm{w}$. Consequently, we have
$\operatorname{curl}\hat{\bm{v}}^{1}_{f}[p]\in\bm{V}^{2}_{-}$. ∎

For
$\bm{\tau}=\sum_{e\in\mathsf{E}_{0}}\hat{\bm{v}}^{2}_{e}[p_{e}]\in\bm{V}^{2}_{-}$,
we define
$\kappa^{2}_{-}:\bigoplus_{e\in\mathsf{E}_{0}}\mathcal{P}_{1}(e)=C_{1}(\Delta,\mathcal{P}_{1};\partial\Delta)$
as

(5.21) $\kappa^{2}_{-}(\bm{\tau})=\sum_{e\in\mathsf{E}_{0}}p_{e}\|e\|.$

#### Construction of $\bm{V}^{3}_{-}$

Finally, let us consider the construction of $\bm{V}^{3}_{-}$. The space
$\bm{V}^{3}_{-}$ consists of the following distribution
$\hat{\bm{v}}^{3}_{x}[p]$ for each internal vertex $x\in\mathsf{V}_{0}$ and
$p\in\mathcal{P}_{1}$:

(5.22) $\langle\hat{\bm{v}}^{3}_{x}[p],\bm{q}\rangle=(\bm{q}\cdot\nabla
p)(x)-\frac{1}{3}(\operatorname{div}\bm{q})p(x),$

for all $\bm{q}\in C_{c}^{\infty}(\mathbb{R}^{3};\mathbb{R}^{3})$.

######  Proposition 5.4.

For $\bm{\tau}=\sum_{e}\hat{\bm{v}}^{2}_{e}[p_{e}]\in\bm{V}^{2}_{-}$, we have

$\operatorname{div}\bm{\tau}=\sum_{x\in\mathsf{V}_{0}}\sum_{e:e\ni
x}\mathcal{O}(x,e)\hat{\bm{v}}^{3}_{x}[p_{e}]$

in the distributional sense.

###### Proof.

By definition, we have

(5.23)
$\begin{split}\langle\operatorname{div}\hat{\bm{v}}^{2}_{e}[p_{e}],\bm{q}\rangle&=-\langle\hat{\bm{v}}^{2}_{e}[p_{e}],\operatorname{dev}\nabla\bm{q}\rangle\\\
&=\int_{e}(\operatorname{dev}\nabla\bm{q}\cdot\bm{t}_{e})\cdot\nabla
p-\frac{1}{2}\int_{e}(\nabla\operatorname{div}\operatorname{dev}\bm{q}\cdot\bm{t}_{e})p\\\
&=\int_{e}(\nabla\bm{q}\cdot\bm{t}_{e})\cdot\nabla
p-\frac{1}{3}\int_{e}\operatorname{div}q(\bm{t}_{e}\cdot\nabla
p)-\frac{1}{3}\int_{e}(\nabla\operatorname{div}\bm{q}\cdot\bm{t}_{e})p\\\
&=(\bm{q}\cdot\nabla
p)(x_{2})-\frac{1}{3}(p\operatorname{div}\bm{q})(x_{2})-(\bm{q}\cdot\nabla
p)(x_{1})+\frac{1}{3}(p\operatorname{div}\bm{q})(x_{1}).\end{split}$

Here we use the fact that
$\operatorname{dev}\nabla\bm{q}=\nabla\bm{q}-\frac{1}{3}\operatorname{div}\bm{q}I$
and
$\operatorname{div}\operatorname{dev}\nabla\bm{q}=\frac{2}{3}\operatorname{div}\bm{q}.$
Here $e=[x_{1},x_{2}]$. ∎

For
$\bm{\xi}=\sum_{x\in\mathsf{V}_{0}}\hat{\bm{v}}_{x}^{3}[p_{x}]\in\bm{V}^{3}_{-}$,
define
$\displaystyle\kappa^{3}_{-}(\bm{\xi})=\sum_{x\in\mathsf{V}_{0}}p_{x}\|x\|.$

We have the following result about the cohomology of the auxiliary complex.

######  Theorem 5.1.

The sequence (5.10) is a complex. The cohomology is isomorphic to
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR}(\Omega)\otimes\mathcal{P}_{1}$.

###### Proof.

By Proposition 5.2, Proposition 5.3 and Proposition 5.4, the following diagram
commutes.

(5.24)
${0}$${V^{0}_{-}}$${\bm{V}^{1}_{-}}$${\bm{V}^{2}_{-}}$${\bm{V}^{3}_{-}}$${0}$${0}$${C_{3}(\Delta,\mathcal{P}_{1};\partial\Delta)}$${C_{2}(\Delta,\mathcal{P}_{1};\partial\Delta)}$${C_{1}(\Delta,\mathcal{P}_{1};\partial\Delta)}$${C_{0}(\Delta,\mathcal{P}_{1};\partial\Delta)}$${0.}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\kappa^{0}_{-}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\kappa^{1}_{-}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{\kappa^{2}_{-}}$$\scriptstyle{\kappa^{3}_{-}}$$\scriptstyle{\partial}$$\scriptstyle{\partial}$$\scriptstyle{\partial}$

Clearly, $\kappa^{{\scriptscriptstyle\bullet}}_{-}$ are (cochain)
isomorphisms. Consequently, we have

(5.25)
$\mathcal{H}^{k}(V^{{\scriptscriptstyle\bullet}}_{-})\cong\mathcal{H}_{3-k}(C{{\scriptscriptstyle\bullet}}(\Delta,\mathcal{P}_{1};\partial\Delta)).$

It then follows from the universal coefficient theorem that
$\mathcal{H}_{3-k}(C{{\scriptscriptstyle\bullet}}(\Delta,\mathcal{P}_{1};\partial\Delta))\cong\mathcal{H}_{3-k}(\Delta;\partial\Delta)\otimes\mathcal{P}_{1}\cong\mathcal{H}_{dR}^{k}(\Omega)\otimes\mathcal{P}_{1}.$
∎

Now we consider the cohomology of (3.5). In 3D, consider the following
diagram:

(5.26)
${0}$${0}$${\bm{V}^{1}}$${\bm{V}^{2}}$${\bm{V}^{3}}$${0}$${0}$${V^{0}_{-}}$${\bm{V}^{1}_{-}}$${\bm{V}^{2}_{-}}$${\bm{V}^{3}}$${0}$${0}$${\bigoplus\limits_{K\in\mathsf{K}}\mathcal{P}_{1}}$${\bigoplus\limits_{f\in\mathsf{F}_{0}}\mathcal{P}_{1}(f)}$${\bigoplus\limits_{e\in\mathsf{E}_{0}}\mathcal{P}_{1}(e)}$${\bigoplus\limits_{x\in\mathsf{V}_{0}}\mathcal{P}_{1}(x)}$${0.}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{g^{0}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{g^{1}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{g^{2}}$$\scriptstyle{g^{3}}$$\scriptstyle{\widetilde{\partial}}$$\scriptstyle{\widetilde{\partial}}$$\scriptstyle{\widetilde{\partial}}$

Here the lower row is defined similarly to (4.18). We have the following
result.

######  Proposition 5.5.

The sequence

${0}$${\bigoplus_{K\in\mathsf{K}}\mathcal{P}_{1}}$${\bigoplus_{f\in\mathsf{F}_{0}}\mathcal{P}_{1}(f)}$${\bigoplus_{e\in\mathsf{E}_{0}}\mathcal{P}_{1}(e)}$${\bigoplus_{x\in\mathsf{V}_{0}}\mathcal{P}_{1}(x)}$${0}$$\scriptstyle{\widetilde{\partial}}$$\scriptstyle{\widetilde{\partial}}$$\scriptstyle{\widetilde{\partial}}$

is a complex, and its cohomology is $V^{0}$, $0$ and $0$, respectively.

The proof of Proposition 5.5 can be found in Appendix. Following the proof of
Theorem 3.6, we can prove Theorem 3.1 using the above theorem.The cohomology
can be calculated from the following long exact sequence.

(5.27)
${0}$${0}$${\mathcal{N}(\operatorname{curl}:\bm{V}^{1}\to\bm{V}^{2})}$${\mathcal{H}^{2}(V^{{\scriptscriptstyle\bullet}})}$${\mathcal{H}^{3}(V^{{\scriptscriptstyle\bullet}})}$${0}$${0}$${\mathcal{H}^{0}_{dR}(\Omega)\otimes\mathcal{P}_{1}}$${\mathcal{H}^{1}_{dR}(\Omega)\otimes\mathcal{P}_{1}}$${\mathcal{H}^{2}_{dR}(\Omega)\otimes\mathcal{P}_{1}}$${\mathcal{H}^{3}_{dR}(\Omega)\otimes\mathcal{P}_{1}}$${0}$${0}$${V^{0}}$${0}$${0}$${0}$${0.}$

###  5.2. 3D Hessian complex with homogeneous boundary conditions

This subsection focuses on the cohomology of (3.13). We first introduce an
auxiliary complex, which corresponds to the HBC version of (5.10). We define
$V^{0}_{0,-}=V^{0}_{-}$,

$\bm{V}^{1}_{0,-}=\operatorname{span}\Big{\\{}\hat{\bm{v}}^{1}_{f}[p]:f\in\mathsf{F},p\in\mathcal{P}_{1}\Big{\\}},$
$\bm{V}^{2}_{0,-}=\operatorname{span}\Big{\\{}\hat{\bm{v}}^{2}_{e}[p]:e\in\mathsf{E},p\in\mathcal{P}_{1}\Big{\\}},$

and

$\bm{V}^{3}_{0,-}=\operatorname{span}\Big{\\{}\hat{\bm{v}}^{3}_{x}[p]:x\in\mathsf{V},p\in\mathcal{P}_{1}\Big{\\}}.$

Then we have

######  Theorem 5.2.

The sequence

(5.28)
${0}$${V^{0}_{0,-}}$${\bm{V}^{1}_{0,-}}$${\bm{V}^{2}_{0,-}}$${\bm{V}^{3}_{0,-}}$${0}$$\scriptstyle{\operatorname{hess}_{0}}$$\scriptstyle{\operatorname{curl}_{0}}$$\scriptstyle{\operatorname{div}_{0}}$

is a complex, with cohomology isomorphic to
$\mathcal{H}^{k}_{dR,c}(\Omega)\otimes\mathcal{P}_{1}$.

Similarly to the previous section, we can prove Theorem 3.2, see Appendix for
more details.

###  5.3. 3D divdiv complex

We first prove Theorem 3.3, the unisolvency of the
$\mathbb{T}\oplus\bm{x}\times\mathbb{S}$ element.

###### Proof of Theorem 3.3.

The number of the proposed degrees of freedom is equal to the number of shape
functions. Therefore, to show the unisolvency, we only show that if all the
degrees of freedom (2a) and (2b) vanish on a function
$\bm{u}\in\mathbb{T}\oplus\bm{x}\times\mathbb{S}$, then $\bm{u}$ vanishes.

Let $\bm{u}=\bm{v}+\bm{x}\times\bm{\sigma}$. We have for any face $f$,

(5.29)
$-2\int_{f}\bm{n}_{f}\cdot\bm{\sigma}\cdot\bm{n}_{f}=\int_{f}\bm{n}_{f}\cdot\operatorname{curl}\bm{u}\cdot\bm{n}_{f}=\int_{\partial
f}\bm{n}_{f}\cdot\bm{u}\cdot\bm{n}_{f}\times\bm{\nu}_{\partial
f,f}=\int_{\partial f}\bm{n}_{f}\cdot\bm{u}\cdot\bm{t}_{\partial f}=0.$

Now (5.29) and the fact that $\int_{e}\bm{n}_{e}\cdot\bm{u}\cdot\bm{t}_{e}=0$
imply that $\operatorname{curl}\bm{u}\in\mathbb{B}_{K}$. By the second set of
degrees of freedom (2b), $\operatorname{curl}\bm{u}=0$, which means that
$\bm{u}$ is a constant. Furthermore, we can conclude that $\bm{u}=0$.

∎

We set the bubble function space $\mathbb{B}_{K}^{*}$ to be the span of the
dual basis of (2b).

In Section 3.3, we introduced the divdiv complex (3.14), which consists of a
distributional $\operatorname{dev}\operatorname{grad}$ operator, a piecewise
$\operatorname{sym}\operatorname{curl}$ operator and a discrete
$\operatorname{div}\operatorname{div}$ operator. The first several spaces are
finite elements with polynomial shape functions and locally defined degrees of
freedom. To obtain a neat discrete topological interpretation, we eliminated
the interior degrees of freedom and obtained (3.20). To show the cohomology of
these complexes, in this section, we view (3.20) from a different point of
view as the dual of the distributional Hessian complex (3.13). We will use the
duality structure and the cohomology of the Hessian complex (Theorem 3.2) to
derive the cohomology of (5.30). This will prove Theorem 3.4.

We observe that the spaces in (3.20) are dual to the spaces in the Hessian
complex (3.13). This inspires us to consider the following sequence with dual
operators:

(5.30)
${0}$${\bm{U}^{0}}$${\widehat{\bm{U}}^{1}}$${\widehat{\bm{U}}^{2}}$${U^{3}}$${0.}$$\scriptstyle{\widetilde{\operatorname{dev}\operatorname{grad}}}$$\scriptstyle{\widetilde{\operatorname{sym}\operatorname{curl}}}$$\scriptstyle{\widetilde{\operatorname{div}\operatorname{div}}}$

Here, the operators are defined as the dual of $\operatorname{div}_{0}$,
$\operatorname{curl}_{0}$, and $\operatorname{hess}_{0}$ (in the sense of
distributions) in the Hessian complex (3.13). More specifically,

  * -

For $\bm{u}\in\bm{U}^{0}$, define
$\widetilde{\operatorname{dev}\operatorname{grad}}\bm{u}\in\bm{U}^{1}$ such
that

$\langle\widetilde{\operatorname{dev}\operatorname{grad}}\bm{u},\bm{\tau}\rangle_{\bm{U}^{1}\times\bm{V}^{2}_{0}}:={-}\langle\bm{u},\operatorname{div}_{0}\bm{\tau}\rangle_{\bm{U}^{0}\times\bm{V}^{3}_{0}},\quad\forall\bm{\tau}\in\bm{V}^{2}_{0}.$

  * -

For $\bm{\xi}\in\bm{U}^{1}$, define
$\widetilde{\operatorname{sym}\operatorname{curl}}\bm{\xi}\in\bm{U}^{2}$ such
that

$\langle\widetilde{\operatorname{sym}\operatorname{curl}}\bm{\xi},\bm{\sigma}\rangle_{\bm{U}^{2}\times\bm{V}^{1}_{0}}:=\langle\bm{\xi},\operatorname{curl}_{0}\bm{\sigma}\rangle_{\bm{U}^{1}\times\bm{V}^{2}_{0}},\quad\forall\bm{\sigma}\in\bm{V}^{1}_{0}.$

  * -

For $\bm{\eta}\in\bm{U}^{2}$, define
$\widetilde{\operatorname{div}\operatorname{div}}\bm{\eta}\in\bm{U}^{3}$ such
that

$\langle\widetilde{\operatorname{div}\operatorname{div}}\bm{\eta},u\rangle_{\bm{U}^{3}\times\bm{V}^{0}_{0}}:=\langle\bm{\eta},\operatorname{hess}_{0}u\rangle_{\bm{U}^{2}\times\bm{V}^{1}_{0}},\quad\forall
u\in\bm{V}^{0}_{0}.$

Since these dual pairs are non-degenerate, the dual operators
$\widetilde{\operatorname{dev}\operatorname{grad}}$,
$\widetilde{\operatorname{sym}\operatorname{curl}}$,
$\widetilde{\operatorname{div}\operatorname{div}}$ are well-defined by the
right-hand side. Similar to the proof of Theorem 3.7, we can introduce an
inner product on each space and prove the following result, see Appendix for
more details.

######  Theorem 5.3.

The sequence (5.30), with the spaces and operators are defined above, is a
complex. The cohomology is isomorphic to
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR}(\Omega)\otimes\mathcal{RT}.$

In fact, we can verify that the operators in (5.30) coincide with the
operators in (3.14), i.e.,
$\widetilde{\operatorname{dev}\operatorname{grad}}=\operatorname{dev}\operatorname{grad}$,
$\widetilde{\operatorname{sym}\operatorname{curl}}=\operatorname{sym}\operatorname{curl}_{h}$,
and
$\widehat{\operatorname{div}\operatorname{div}}=\widetilde{\operatorname{div}\operatorname{div}}$.
Thus, the two complexes are identical. The details can be found in the
Appendix. As a consequence, the sequence (3.20) is a complex with the same
cohomology.

Note that the divdiv complex (3.14) is a direct sum of (3.20) and

(5.31)
${0}$${0}$${\mathbb{B}_{K}^{*}}$${\mathbb{B}_{K}}$${0}$${0.}$$\scriptstyle{\operatorname{sym}\operatorname{curl}_{h}}$

This concludes with the cohomology of (3.14) (Theorem 3.4).

For the divdiv complex with homogeneous boundary conditions (3.23), the
construction and the proof of the cohomology (Theorem 3.5) are similar. The
key is to eliminate the interior degrees of freedom and identify the resulting
complex as the dual of the Hessian complex (3.5), see Appendix.

##  6\. Conclusions and outlook

In this paper, we constructed some distributional BGG complexes. The canonical
degrees of freedom of the resulting spaces allow a discrete topological and
geometric interpretation. Although we focus on 2D and 3D, we hope the
discussions will shed light on discretizing high dimensional and high order
tensors on triangulation with potential applications in broad areas such as
numerical geometric PDEs and exterior calculus on graphs and in graphics [44,
51]. This paper also developed a strategy for showing the cohomology of
distributional complexes by constructing auxiliary sequences and using diagram
chase. Consequently, we demonstrated the cohomology of the Regge finite
element/sequence in 2D, which was open to the best of our knowledge. A similar
idea may also be used to compute the cohomology of the 3D Regge sequence.
However, further investigation is beyond the scope of this paper.

Unless Whitney forms for the de Rham complexes, the resulting spaces for the
BGG complexes are less regular. Solving PDEs with these schemes calls for
further numerical analysis.

Another example of the BGG construction is the conformal (deformation) complex
[4, 10]. Although there has been progress on conforming finite element
discretizations [40], distributional versions encoding discrete conformal
geometric structures remain open.

We investigated the connections between distributional finite elements and
Discrete Exterior Calculus. This shift of point of view might provide another
perspective for establishing convergence analysis for DEC and other lattice
methods.

##  Appendix A Technical Proofs

In the Appendix, we include some technical proofs.

###  A.1. Operators in the 3D divdiv complex

We verify that the operators in (5.30) defined by duality are identical to
those in (3.20).

######  Lemma A.1.

In $\bm{U}^{0}$, we have
$\widetilde{\operatorname{dev}\operatorname{grad}}=\operatorname{dev}\operatorname{grad}$.

###### Proof.

Since $\bm{u}$ is $\bm{H}^{1}$ conforming, the
$\operatorname{dev}\operatorname{grad}$ operator is piecewise. It follows that
$\operatorname{dev}\operatorname{grad}\bm{u}\in\operatorname{dev}\operatorname{grad}\mathcal{P}_{1}=\mathbb{T}$,
which is in the local shape function space of $\widehat{\bm{U}}^{1}$. It
follows from (3.11) that
$\langle{\operatorname{dev}\operatorname{grad}}\bm{u},\delta_{e}[\bm{n}_{e,\pm}\otimes\bm{t}_{e}]\rangle_{\widehat{\bm{U}}^{1}\times\bm{V}^{2}_{0}}=\langle\bm{u},-\operatorname{div}\delta_{e}[\bm{n}_{e,\pm}\otimes\bm{t}_{e}]\rangle_{\bm{U}^{0}\times\bm{V}^{3}_{0}}.$
This implies that,
$\widetilde{\operatorname{dev}\operatorname{grad}}=\operatorname{dev}\operatorname{grad}$
on $\bm{U}^{0}$.

∎

######  Lemma A.2.

In $\bm{U}^{1}$, we have
$\widetilde{\operatorname{sym}\operatorname{curl}}=\operatorname{sym}\operatorname{curl}_{h}$,
the piecewise $\operatorname{sym}\operatorname{curl}$ operator.

###### Proof.

It follows from (3.10). ∎

######  Lemma A.3.

For any function $\bm{\sigma}\in\bm{U}^{2}$, it holds that
$\widehat{\operatorname{div}\operatorname{div}}\bm{\sigma}=\widetilde{\operatorname{div}\operatorname{div}}\bm{\sigma}.$

###### Proof.

It follows from

(A.1)
$\begin{split}\langle\bm{\sigma},\operatorname{hess}u\rangle_{\bm{U}^{2}\otimes\bm{V}^{1}_{0}}=&{-}\langle\bm{\sigma},\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\Big{[}\mathcal{O}(f,K)\frac{\partial
u|_{K}}{\partial\bm{n}}\Big{]}\delta_{f}[{\bm{n}_{f}\otimes\bm{n}_{f}}]\rangle_{\bm{U}^{2}\otimes\bm{V}^{1}_{0}}\\\
=&{-}\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\Big{[}\mathcal{O}(f,K)\frac{\partial
u|_{K}}{\partial\bm{n}}\Big{]}\int_{f}(\bm{n}_{f}\cdot\bm{\sigma}\cdot\bm{n}_{f}).\end{split}$

Subtracting the right-hand side of (3.21) from the above equation leads to

(A.2)
$\langle\operatorname{div}\operatorname{div}\sigma,\bm{u}\rangle-\langle\bm{\sigma},\operatorname{hess}u\rangle_{\bm{U}^{2}\otimes\bm{V}^{1}_{0}}=\sum_{f\in\mathsf{F}_{0}}\sum_{K:K\supset
f}\int_{f}\Big{[}\mathcal{O}(f,K)\nabla
u|_{K}\cdot\bm{\sigma}\cdot\bm{n}_{f}\Big{]}=0.$

Here the first identity comes from a tangential-normal decomposition of
$\nabla u$ and $\bm{\sigma}$, and the second is due to
$\operatorname{div}(\bm{\sigma}\cdot\nabla u)=0$ in each element $K$, as $u$
is linear and $\bm{\sigma}$ is constant. ∎

###  A.2. Diagrams illustrating the proof

In the main body of this paper, some theorems follow from a similar argument.
For example, to prove the cohomology of the Hessian complex (3.24), we first
consider an auxiliary Hessian complex (5.10), and relate it to a simplicial
homology. Then, we use diagram chasing to obtain the cohomology of the
original Hessian complex (3.24). For the divdiv complex (3.26), we use a
duality argument, and using isomorphisms between harmonic forms to derive the
cohomology. For brevity, we omitted some details in the main text. In this
appendix, we provide diagrams and a sketch of the omitted proofs.

###### Proof of Proposition 4.3.

The proof is based on the following diagram:

(A.3)
${0}$${V^{0}_{0,-}}$${\bm{V}^{1}_{0,-}}$${\bm{V}^{2}_{0,-}}$${0}$${0}$${C_{2}(\Delta,\mathcal{P}_{1})}$${C_{1}(\Delta,\mathcal{P}_{1})}$${C_{0}(\Delta,\mathcal{P}_{1})}$${0.}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\kappa^{0}_{-}}$$\scriptstyle{\operatorname{rot}}$$\scriptstyle{\kappa^{1}_{-}}$$\scriptstyle{\kappa^{2}_{-}}$$\scriptstyle{\partial_{0}}$$\scriptstyle{\partial_{0}}$

The vertical maps induce an isomorphism between $\mathcal{H}(V^{k}_{0,-})$ and
$\mathcal{H}_{2-k}(\Delta,\mathcal{P}_{1})$, which is isomorphic to
$\mathcal{H}^{{\scriptscriptstyle\bullet}}_{dR,c}(\Omega)\otimes\mathcal{P}_{1}$.
∎

###### Proof of Proposition 4.4.

The homology can be identified with a direct sum of

(A.4) ${0}$${\bigoplus\limits_{f\in\mathsf{F},f\ni
v}\mathbb{R}}$${\bigoplus\limits_{e\in\mathsf{E},e\ni
v}\mathbb{R}}$${\mathbb{R}}$${0.}$$\scriptstyle{{\partial}}$$\scriptstyle{{\partial}}$

For an interior vertex $v$, the homology is the relative homology (with
respect to boundary) of the local patch of $v$, which vanishes except for at
index zero. For a boundary vertex $v$, the homology is the relative homology
(with respect to boundaries that are not the boundary of $\Delta$) of the
local patch of $v$, which vanishes for all indices. Therefore, we conclude the
result. ∎

###### Proof of Proposition 5.5.

Using the fact that the original complex can be identified with the direct sum
of

(A.5) ${0}$${\bigoplus\limits_{K\in\mathsf{K},K\ni
v}\mathbb{R}}$${\bigoplus\limits_{f\in\mathsf{F},f\ni
v}\mathbb{R}}$${\bigoplus\limits_{e\in\mathsf{E},e\ni
v}\mathbb{R}}$${\mathbb{R}}$${0.}$$\scriptstyle{{\partial}}$$\scriptstyle{{\partial}}$

∎

###### Proof of Theorem 4.2.

We introduce the following dual pairs,

(A.6)
${0}$${V^{0}}$${\bm{V}^{1}}$${\bm{V}^{2}}$${0}$${0}$${U^{2}_{0}}$${\bm{U}^{1}_{0}}$${\bm{U}^{0}_{0}}$${0,}$$\scriptstyle{\operatorname{hess}}$${\ast}$$\scriptstyle{\operatorname{rot}}$${\ast}$${\ast}$$\scriptstyle{\operatorname{div}\operatorname{div}_{0}}$$\scriptstyle{\operatorname{sym}\operatorname{curl}_{0}}$

and the inner products. For example, the dual pair of $v\in V^{0}$ and