# Generalized Korn’s Inequalities for Piecewise $H^{2}$ Vector Fields

David M. Williams  Department of Mechanical Engineering, The Pennsylvania
State University, Pennsylvania, United States, 16802<EMAIL_ADDRESS>and
Qingguo Hong  Department of Mathematics, The Pennsylvania State University,
Pennsylvania, United States, 16802<EMAIL_ADDRESS>In memory of Philip Eric
Gino Zwanenburg

###### Abstract.

The purpose of this paper is to construct a new class of discrete generalized
Korn’s inequalities for piecewise $H^{2}$ vector fields in three-dimensional
space. The resulting Korn’s inequalities are different from the standard
Korn’s inequalities, as they involve the trace-free symmetric gradient
operator, in place of the usual symmetric gradient operator. It is anticipated
that the new generalized Korn’s inequalities will be useful for the analysis
of a broad range of finite element methods, including mixed finite element
methods and discontinuous Galerkin methods.

###### 2020 Mathematics Subject Classification:

Primary 65N30, 76M10, 76N06

##  1\. Introduction

Broadly speaking, Korn’s inequalities are an important part of the
mathematical analysis of many partial differential equations and their
associated finite element methods. In particular, the _standard_ Korn’s
inequalities provide an upper bound for the $L_{2}$ norm of the gradient of a
sufficiently-smooth vector field in terms of the $L_{2}$ norm of the symmetric
gradient and (sometimes) an auxiliary norm/seminorm. These types of Korn’s
inequalities can be classified into two general categories: i) continuous
inequalities, and ii) discrete inequalities. The continuous inequalities are
usually formulated for sufficiently smooth vector fields which are well-
defined (with regard to smoothness) over the entirety of an open, bounded,
Lipschitz domain. In contrast, discrete inequalities are usually formulated
for sufficiently smooth vector fields which are well-defined in a piecewise
fashion on Lipschitz subdomains of a larger domain. Naturally, the associated
subdomains are usually convex, polyhedral elements which belong to a
triangulation. The continuous class of inequalities can be used to analyze the
non-discretized behavior of partial differential equations, or the behavior of
numerical methods in which the entire computational domain is covered by a
single element. One may consult [25] for a brief review of continuous Korn
inequalities and their applications in the field of continuum mechanics.
Conversely, the discrete class of inequalities can be used to analyze finite
element methods which operate on the aforementioned partial differential
equations. In particular, the discrete standard Korn’s inequalities are useful
for analyzing finite element methods for applications in linear elasticity [7,
32, 13, 12, 24], and incompressible fluid dynamics [14, 16, 17, 26, 23, 20].

Recently, an interest in _generalized_ Korn’s inequalities has emerged. The
generalized Korn’s inequalities are very similar to their standard
counterparts, with the following distinction: they provide an upper bound for
the gradient of a vector field in terms of the trace-free symmetric gradient
instead of the standard symmetric gradient. The trace-free symmetric gradient
naturally arises in the context of compressible fluid dynamics, where the
trace of the velocity gradient tensor (i.e. the divergence of the velocity
field) is non-zero [2]. In addition, there are applications in the areas of
general relativity, Cosserat elasticity, and geometry [21, 33, 18]. Several
classes of continuous generalized Korn’s inequalities have been previously
developed by Dain [11], Schirra [34], Feireisl and Novotnỳ [19], and Breit et
al. [4]. However, to the author’s knowledge, there have been no rigorous
efforts to develop discrete generalized Korn’s inequalities. In principle, the
development of such discrete inequalities will facilitate the construction of
new, robust finite element methods in the aforementioned application areas. In
addition, we note that there is an existing class of ‘versatile mixed finite
element methods’ for simulating incompressible/compressible fluid dynamics
problems that will immediately benefit from the introduction of such
inequalities [9, 31]. In particular, discrete generalized Korn’s inequalities
are required in order to construct rigorous error estimates for these methods.
For all of the reasons mentioned above, this paper will attempt to derive the
required class of generalized inequalities.

In order to obtain the desired generalized inequalities, we will follow the
approach of Brenner and coworkers [6, 8, 5]. We note that other researchers
have performed extensive work on constructing discrete standard Korn’s
inequalities, including Knobloch and Tobiska [27], Attia and Starke [3],
Mardal and Winther [30], Zhang [35], and Lee [29]; however, we choose to
primarily focus on the work of Brenner in this paper. The interested reader is
encouraged to consult [22] for a compact review of the other work on this
topic.

In [6], Brenner constructs a class of discrete standard Korn’s inequalities
for piecewise $H^{1}$ vector fields in 2D and 3D. In [8], similar techniques
are used to construct Poincaré–Friedrichs discrete inequalities for piecewise
$H^{2}$ vector fields in 2D. Finally, in [5], standard Korn and
Poincaré–Friedrichs discrete inequalities are obtained for classes of meshes
generated by independent refinements, for piecewise $H^{1}$ vector fields in
2D and 3D.

We intend for the present work to be a natural extension of [6] to the case of
generalized Korn’s inequalities, and piecewise $H^{2}$ vector fields in 3D.
The organization of this paper is as follows. In section 2, we provide an
overview of generalized Korn’s inequalities at the continuous level. In
section 3, we present several new generalized Korn’s inequalities at the
discrete level. In section 4, we provide proofs for the discrete inequalities.
Finally, in section 5, we conclude with a brief summary of the present work.

##  2\. The Continuous Case

Let us begin by introducing some important notation and concepts which will be
beneficial during the remainder of our presentation.

One may consider a bounded, connected, open, polyhedral domain $\Omega$ in
$\mathbb{R}^{3}$, with boundary $\partial\Omega$. By construction, the
boundary of this domain is Lipschitz. On this domain, we define a generic
vector field $\bm{u}\in\left[H^{2}(\Omega)\right]^{3}$. In a natural fashion,
we define the following norms and seminorms which produce finite results when
acting upon $\bm{u}$

$\displaystyle\left\|\bm{u}\right\|_{L_{2}\left(\Omega\right)}$
$\displaystyle=\left(\int_{\Omega}|\bm{u}|^{2}\,dx\right)^{1/2},$
$\displaystyle\left|\bm{u}\right|_{H^{1}(\Omega)}^{2}$
$\displaystyle=\sum_{j=1}^{3}\left|u_{j}\right|^{2}_{H^{1}(\Omega)},\qquad\left\|\bm{u}\right\|_{H^{1}(\Omega)}^{2}=\left\|\bm{u}\right\|_{L_{2}\left(\Omega\right)}^{2}+\left|\bm{u}\right|_{H^{1}(\Omega)}^{2},$
$\displaystyle\left|\bm{u}\right|_{H^{2}(\Omega)}^{2}$
$\displaystyle=\sum_{j=1}^{3}\left|u_{j}\right|^{2}_{H^{2}(\Omega)},\qquad\left\|\bm{u}\right\|_{H^{2}(\Omega)}^{2}=\left\|\bm{u}\right\|_{L_{2}\left(\Omega\right)}^{2}+\left|\bm{u}\right|_{H^{1}(\Omega)}^{2}+\left|\bm{u}\right|_{H^{2}(\Omega)}^{2},$

where $|\bm{u}|$ is the standard Euclidean norm of $\bm{u}$, and

$\displaystyle\left|u_{j}\right|_{H^{k}(\Omega)}=\left(\sum_{|\bm{\alpha}|=k}\int_{\Omega}\left|D^{\bm{\alpha}}u_{j}\right|^{2}\,dx\right)^{1/2},$

is the Hilbert seminorm of $u_{j}$ for $H^{k}(\Omega)$. In the equation above,
$\bm{\alpha}=\left(\alpha_{1},\alpha_{2},\alpha_{3}\right)$ is the multi-
index, and $D^{\bm{\alpha}}$ is the generalized, multi-dimensional derivative
operator.

Next, we introduce the following, more specialized, derivative operators

$\displaystyle\epsilon_{ij}\left(\bm{u}\right)$
$\displaystyle=\frac{1}{2}\left(\text{grad}_{ij}(\bm{u})+\text{grad}_{ji}(\bm{u})\right)=\frac{1}{2}\left(\frac{\partial
u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right),$
$\displaystyle\text{div}\left(\bm{u}\right)$
$\displaystyle=\sum_{i=1}^{3}\frac{\partial u_{i}}{\partial x_{i}},$
$\displaystyle\text{curl}\left(\bm{u}\right)$
$\displaystyle=\left(\frac{\partial u_{3}}{\partial x_{2}}-\frac{\partial
u_{2}}{\partial x_{3}},\frac{\partial u_{1}}{\partial x_{3}}-\frac{\partial
u_{3}}{\partial x_{1}},\frac{\partial u_{2}}{\partial x_{1}}-\frac{\partial
u_{1}}{\partial x_{2}}\right)^{t},$

where $\bm{\epsilon}\left(\cdot\right)$ is the symmetric gradient operator,
$\text{grad}(\cdot)$ is the standard gradient operator,
$\text{div}\left(\cdot\right)$ is the divergence operator, and
$\text{curl}\left(\cdot\right)$ is the curl operator. In addition, we
introduce the trace-free symmetric gradient operator, which takes the
following form

$\displaystyle\bm{\epsilon}\left(\bm{u}\right)-\frac{1}{3}\text{div}\left(\bm{u}\right)\mathbb{I},$

where $\mathbb{I}$ is the $3\times 3$ identity matrix. Now, we can define the
following norms which remain finite for all
$\bm{u}\in\left[H^{1}(\Omega)\right]^{3}$

(2.1)
$\displaystyle\left\|\bm{\epsilon}\left(\bm{u}\right)\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}=\left(\int_{\Omega}\left|\bm{\epsilon}\left(\bm{u}\right)\right|^{2}\,dx\right)^{1/2},$
(2.2)
$\displaystyle\left\|\bm{\epsilon}\left(\bm{u}\right)-\frac{1}{3}\text{div}\left(\bm{u}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}=\left(\int_{\Omega}\left|\bm{\epsilon}\left(\bm{u}\right)-\frac{1}{3}\text{div}\left(\bm{u}\right)\mathbb{I}\right|^{2}\,dx\right)^{1/2},$

where the quantity $\left|\bm{\epsilon}\left(\bm{u}\right)\right|$ is the
standard Frobenius inner-product applied to $\bm{\epsilon}(\bm{u})$.

We note that if $\bm{u}$ is pointwise divergence-free, then the norm
definitions in Eqs. (2.1) and (2.2) are identical, and the necessary analysis
has already been previously established in other work, (see the Introduction).
However, for the case of a generic $\bm{u}$ which is not divergence-free,
there are some important results which remain to be investigated.

Towards this end, one may introduce the kernel space of the trace-free
symmetric gradient

$\displaystyle\mathbf{CK}\left(\Omega\right)=\left\\{2\left(\bm{a}\cdot\bm{x}\right)\bm{x}-\left|\bm{x}\right|^{2}\bm{a}+\rho\bm{x}+\bm{Q}\bm{x}+\bm{b}:\bm{a}\in\mathbb{R}^{3},\bm{b}\in\mathbb{R}^{3},\rho\in\mathbb{R},\;\text{and}\;\bm{Q}\in\mathfrak{so}\left(3\right)\right\\},$

where $\bm{x}=\left(x_{1},x_{2},x_{3}\right)$ is a coordinate vector, and
$\mathfrak{so}(3)$ is the space of $3\times 3$ skew-symmetric matrices. The
kernel space (above) is identical to the space of conformal Killing vectors,
which was originally identified in [11] and further studied in [34]. This
space contains infinitesimal rigid body motions, Killing vectors, and
dilatations. By construction, we have that

(2.3)
$\displaystyle\bm{\epsilon}\left(\bm{u}\right)-\frac{1}{3}\text{div}\left(\bm{u}\right)\mathbb{I}=\bm{0}\qquad\Longleftrightarrow\qquad\bm{u}\in\mathbf{CK}\left(\Omega\right).$

It turns out that proving Korn-type inequalities for the trace-free symmetric
gradient is less straightforward than proving the equivalent inequalities for
the standard symmetric gradient. The main challenges arise from the fact that
the kernel space for the symmetric gradient is simpler than
$\mathbf{CK}(\Omega)$, as it is merely the space of infinitesimal rigid body
motions

$\displaystyle\mathbf{RM}(\Omega)=\left\\{\bm{Q}\bm{x}+\bm{b}:\bm{b}\in\mathbb{R}^{3}\;\text{and}\;\bm{Q}\in\mathfrak{so}\left(3\right)\right\\}.$

The key complication is that $\mathbf{RM}(\Omega)$ is linear in $\bm{x}$,
whereas the space $\mathbf{CK}(\Omega)$ is quadratic.

With the above discussion in mind, it is somewhat surprising to find that
Korn’s inequalities for the trace-free symmetric gradient exist. In
particular, in accordance with [19] (Theorem 11.22, pg. 466), we have that

$\displaystyle\left\|\bm{u}\right\|_{H^{1}(\Omega)}\leq
C\left(\left\|\bm{\epsilon}\left(\bm{u}\right)-\frac{1}{3}\text{div}\left(\bm{u}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\int_{\Omega}\left|\bm{u}\right|\,dx\right),\qquad\forall\bm{u}\in\left[H^{1}(\Omega)\right]^{3}.$

Now, after some simple manipulations with Hölder’s inequality, we obtain

(2.4) $\displaystyle\left\|\bm{u}\right\|_{H^{1}(\Omega)}\leq
C_{\Omega}\left(\left\|\bm{\epsilon}\left(\bm{u}\right)-\frac{1}{3}\text{div}\left(\bm{u}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left\|\bm{u}\right\|_{L_{2}(\Omega)}\right),$

and

(2.5) $\displaystyle\left|\bm{u}\right|_{H^{1}(\Omega)}\leq
C_{\Omega}\left(\left\|\bm{\epsilon}\left(\bm{u}\right)-\frac{1}{3}\text{div}\left(\bm{u}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left\|\bm{u}\right\|_{L_{2}(\Omega)}\right).$

Here, the Korn’s inequality has been rewritten in the standard fashion:
namely, it provides a bound for the gradient norm of $\bm{u}$ in terms of the
trace-free gradient norm and the $L_{2}$ norm. It remains for us to specialize
this inequality, and replace the $L_{2}$ norm with a broader class of more
useful seminorms. Towards this end, we define a class of seminorms on
$\left[H^{1}(\Omega)\right]^{3}$ that satisfy the following constraints

(2.6) $\displaystyle\bm{\Phi}\left(\bm{v}\right)\leq
C_{\bm{\Phi}}\left\|\bm{v}\right\|_{H^{1}(\Omega)},\qquad\forall\bm{v}\in\left[H^{1}(\Omega)\right]^{3},$

and

(2.7)
$\displaystyle\bm{\Phi}\left(\bm{m}\right)=0\qquad\text{and}\qquad\bm{m}\in\mathbf{CK}(\Omega)\qquad\Longleftrightarrow\qquad\bm{m}=\text{a
constant vector},$

where $C_{\bm{\Phi}}$ is a constant that depends on our particular choice of
seminorm. It turns out that all seminorms that satisfy the second criterion,
will also be invariant under the addition of a constant vector, such that

(2.8)
$\displaystyle\bm{\Phi}\left(\bm{v}+\bm{c}\right)=\bm{\Phi}\left(\bm{v}\right),$

where $\bm{c}\in\mathbb{R}^{3}$.

Now, based on the seminorm properties given above, and Eq. (2.5), it can be
shown that

(2.9) $\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega)}\leq
C_{\Omega}\left(\left\|\bm{\epsilon}\left(\bm{v}\right)-\frac{1}{3}\text{div}\left(\bm{v}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\bm{\Phi}\left(\bm{v}\right)\right),\qquad\forall\bm{v}\in\left[H^{1}(\Omega)\right]^{3}.$

The proof of Eq. (2.9) uses the fact that $H^{1}$ is compactly embedded into
$L_{2}$. Some details of the proof are given in Appendix A, Lemma A.2.

There are an infinite number of seminorms which satisfy the required
constraints (Eqs. (2.6) and (2.7)). However, we choose to focus on the
following seminorms due to their utility for practical applications

(2.10) $\displaystyle\bm{\Phi}_{1}\left(\bm{v}\right)$
$\displaystyle=\left\|G\bm{v}\right\|_{L_{2}(\Omega)},$ (2.11)
$\displaystyle\bm{\Phi}_{2}\left(\bm{v}\right)$
$\displaystyle=\sup_{\begin{subarray}{c}\bm{m}\in\mathbf{CK}(\Omega)\\\
\left\|\bm{m}\right\|_{L_{2}(\partial\Omega)}=1,\int_{\partial\Omega}\bm{m}\,ds=\bm{0}\end{subarray}}\int_{\partial\Omega}\bm{v}\cdot\bm{m}\,ds,$

where

$\displaystyle G\bm{v}=\bm{v}-\frac{1}{|\Omega|}\int_{\Omega}\bm{v}\,dx,$

is the projection from $\left[L_{2}(\Omega)\right]^{3}$ on to the orthogonal
complement of constant vector fields. Generally speaking, we have that

(2.12) $\displaystyle\left\|G\bm{v}\right\|_{L_{2}(\Omega)}\leq
C\left\|\bm{v}\right\|_{L_{2}(\Omega)},\qquad\forall\bm{v}\in\left[L_{2}(\Omega)\right]^{3}.$

##  3\. The Discrete Case

Our domain $\Omega\subset\mathbb{R}^{3}$ can tessellated by non-overlapping
tetrahedral elements. We denote the individual elements by $T$, and the
overall partition of the domain into these elements by $\mathcal{T}$. The
resulting partition has the following properties:

$\displaystyle\overline{\Omega}=\bigcup_{T\in\mathcal{T}}\overline{T},$

and

$\displaystyle T\cap T^{\prime}=\emptyset\qquad\text{unless}\qquad
T=T^{\prime}.$

We assume that the minimum interior angle of any tetrahedron $T\in\mathcal{T}$
is given by $\theta_{\mathcal{T}}$. In addition, we assume that the partition
is conforming, such that all interior faces are shared by two tetrahedra, and
all boundary faces are associated with only one tetrahedron. We denote the
individual faces in the partition by $\sigma$. The total set of interior faces
is denoted by $S\left(\mathcal{T},\Omega\right)$, and for each $\sigma\in
S\left(\mathcal{T},\Omega\right)$, we can define a “$-$” side and a “$+$”
side. Next, we define a jump operator

$\displaystyle\left[\\!\left[\bm{u}\right]\\!\right]_{\sigma}=\bm{u}_{+}-\bm{u}_{-},$

where $\bm{u}$ is a generic vector that has well-defined values on the
interior faces. It is also convenient to define a quadratic projection
operator $\pi_{\sigma}$ on each face, such that
$\pi_{\sigma}:\left[L_{2}(\sigma)\right]^{3}\rightarrow\left[P_{2}(\sigma)\right]^{3}$.
This operator satisfies the following standard projection inequality

(3.1)
$\displaystyle\left\|\pi_{\sigma}\left[\\!\left[\bm{u}\right]\\!\right]_{\sigma}\right\|_{L_{2}(\sigma)}\leq
C\left\|\left[\\!\left[\bm{u}\right]\\!\right]_{\sigma}\right\|_{L_{2}(\sigma)},\qquad\forall\bm{u}\in\left[L_{2}(\sigma)\right]^{3}.$

The definitions (above) allow us to effectively characterize vector fields
which possess discontinuities along the interior faces. We are particularly
interested in the class of (possibly discontinuous) piecewise $H^{2}$ vector
fields $\bm{u}\in\left[H^{2}(\Omega,\mathcal{T})\right]^{3}$, where

$\displaystyle\left[H^{2}(\Omega,\mathcal{T})\right]^{3}=\left\\{\bm{u}\in\left[L_{2}(\Omega)\right]^{3}:\bm{v}_{T}=\bm{v}|_{T}\in\left[H^{2}(T)\right]^{3}\quad\forall
T\in\mathcal{T}\right\\}.$

The associated broken seminorms can be defined as follows

$\displaystyle\left|\bm{u}\right|_{H^{1}(\Omega,\mathcal{T})}=\left(\sum_{T\in\mathcal{T}}\left|\bm{u}\right|^{2}_{H^{1}(T)}\right)^{1/2},\qquad\left|\bm{u}\right|_{H^{2}(\Omega,\mathcal{T})}=\left(\sum_{T\in\mathcal{T}}\left|\bm{u}\right|^{2}_{H^{2}(T)}\right)^{1/2}.$

In addition, we can define piecewise derivative operators $\forall
T\in\mathcal{T}$

$\displaystyle\bm{\epsilon}_{\mathcal{T}}(\bm{u})|_{T}=\bm{\epsilon}\left(\bm{u}_{T}\right),\qquad\text{div}_{\mathcal{T}}(\bm{u})|_{T}=\text{div}\left(\bm{u}_{T}\right).$

With the above discussion in mind, we can formally state a new class of
generalized, discrete Korn’s inequalities for all
$\bm{v}\in\left[H^{2}(\Omega,\mathcal{T})\right]^{3}$

(3.2) $\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega,\mathcal{T})}^{2}\leq
C\Bigg{(}$
$\displaystyle\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}+\left(\bm{\Phi}(\bm{v})\right)^{2}$
$\displaystyle+\sum_{\sigma\in
S(\mathcal{T},\Omega)}\left(\text{diam}\,\sigma\right)^{-1}\left\|\pi_{\sigma}\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}\right\|^{2}_{L_{2}(\sigma)}\Bigg{)}.$

In accordance with Eq. (3.1), we can immediately eliminate the dependence of
the inequality on the projection operator $\pi_{\sigma}$, and obtain

(3.3) $\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega,\mathcal{T})}^{2}\leq
C\Bigg{(}$
$\displaystyle\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}+\left(\bm{\Phi}(\bm{v})\right)^{2}$
$\displaystyle+\sum_{\sigma\in
S(\mathcal{T},\Omega)}\left(\text{diam}\,\sigma\right)^{-1}\left\|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}\right\|^{2}_{L_{2}(\sigma)}\Bigg{)}.$

The Korn’s inequalities above can be expressed explicitly for each seminorm,
as follows

(3.4) $\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega,\mathcal{T})}^{2}\leq
C\Bigg{(}$
$\displaystyle\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}+\left\|G\bm{v}\right\|_{L_{2}(\Omega)}^{2}$
$\displaystyle+\sum_{\sigma\in
S(\mathcal{T},\Omega)}\left(\text{diam}\,\sigma\right)^{-1}\left\|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}\right\|^{2}_{L_{2}(\sigma)}\Bigg{)},$
(3.5) $\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega,\mathcal{T})}^{2}\leq
C\Bigg{(}$
$\displaystyle\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}+\sup_{\begin{subarray}{c}\bm{m}\in\mathbf{CK}(\Omega)\\\
\left\|\bm{m}\right\|_{L_{2}(\partial\Omega)}=1,\int_{\partial\Omega}\bm{m}\,ds=\bm{0}\end{subarray}}\left(\int_{\partial\Omega}\bm{v}\cdot\bm{m}\,ds\right)^{2},$
$\displaystyle+\sum_{\sigma\in
S(\mathcal{T},\Omega)}\left(\text{diam}\,\sigma\right)^{-1}\left\|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}\right\|^{2}_{L_{2}(\sigma)}\Bigg{)},$

or equivalently,

(3.6) $\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega,\mathcal{T})}^{2}\leq
C\Bigg{(}$
$\displaystyle\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}+\left\|G\bm{v}\right\|_{L_{2}(\Omega)}^{2}$
$\displaystyle+\sum_{\sigma\in
S(\mathcal{T},\Omega)}\left(\text{diam}\,\sigma\right)^{-1}\left\|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}\right\|^{2}_{L_{2}(\sigma)}\Bigg{)},$
(3.7) $\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega,\mathcal{T})}^{2}\leq
C\Bigg{(}$
$\displaystyle\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}+\left\|\bm{v}\right\|^{2}_{L_{2}(\partial\Omega)}$
$\displaystyle+\sum_{\sigma\in
S(\mathcal{T},\Omega)}\left(\text{diam}\,\sigma\right)^{-1}\left\|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}\right\|^{2}_{L_{2}(\sigma)}\Bigg{)}.$

The inequalities in Eqs. (3.6) and (3.7) are the most useful in practice. In
particular, they can be used for mixed finite element methods and
discontinuous Galerkin methods. They can also be used for continuous Galerkin
methods. In this latter case, all of the jump terms vanish, and one obtains
the following simplified inequalities

(3.8) $\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega,\mathcal{T})}^{2}\leq
C\Bigg{(}$
$\displaystyle\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}+\left\|G\bm{v}\right\|_{L_{2}(\Omega)}^{2}\Bigg{)},$ (3.9)
$\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega,\mathcal{T})}^{2}\leq
C\Bigg{(}$
$\displaystyle\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}+\left\|\bm{v}\right\|^{2}_{L_{2}(\partial\Omega)}\Bigg{)}.$

It remains for us to prove the key discrete result (Eq. (3.2)), from whence
all other discrete results follow. This will be the focus of the next section.

##  4\. Proofs of Discrete Results

In this section, we will construct a rigorous proof for Eq. (3.2). This will
require the introduction of two lemmas, prior to the final result. The final
result itself will be stated in a theorem at the end of the section.

Prior to the first lemma, it is necessary to introduce a mapping operator,
along with some additional notation. Let us denote the set of all vertices in
$\mathcal{T}$ as $\mathcal{V}(\mathcal{T})$, and the set of all edge midpoints
as $\mathcal{M}(\mathcal{T})$. Thereafter, one may define a mapping operator
$E$ which maps piecewise quadratic vector fields to continuous piecewise
quadratic vector fields. More precisely, we define
$E:V_{\mathcal{T}}\rightarrow W_{\mathcal{T}}$, where

$\displaystyle V_{\mathcal{T}}$
$\displaystyle=\left\\{\bm{v}\in\left[L_{2}(\Omega)\right]^{3}:\bm{v}_{T}=\bm{v}|_{T}\in\left[P_{2}(T)\right]^{3}\quad\forall
T\in\mathcal{T}\right\\},$ $\displaystyle W_{\mathcal{T}}$
$\displaystyle=\left\\{\bm{w}\in\left[H^{1}(\Omega)\right]^{3}:\bm{w}_{T}=\bm{w}|_{T}\in\left[P_{2}(T)\right]^{3}\quad\forall
T\in\mathcal{T}\right\\},$

and where

(4.1) $\displaystyle\left(E\bm{v}\right)(p)$
$\displaystyle=\frac{1}{|\mathcal{T}_{p}|}\sum_{T\in\mathcal{T}_{p}}\bm{v}_{T}(p),\qquad\forall
p\in\mathcal{V}(\mathcal{T}),$ (4.2) $\displaystyle\left(E\bm{v}\right)(m)$
$\displaystyle=\frac{1}{|\mathcal{T}_{m}|}\sum_{T\in\mathcal{T}_{m}}\bm{v}_{T}(m),\qquad\forall
m\in\mathcal{M}(\mathcal{T}).$

Here, we define $\mathcal{T}_{p}$ as the set of tetrahedra sharing a common
vertex $p$

$\displaystyle\mathcal{T}_{p}=\left\\{T\in\mathcal{T}:p\in\partial
T\right\\},$

and $\mathcal{T}_{m}$ as the set of tetrahedra sharing a common edge with
midpoint $m$

$\displaystyle\mathcal{T}_{m}=\left\\{T\in\mathcal{T}:m\in\partial
T\right\\}.$

Figure 1 provides an example of a shared vertex and a shared edge midpoint.

Figure 1. A collection of tetrahedra: $T_{1},T_{2},\ldots,T_{5}$. The
tetrahedra share two common vertices, one of which ($p$) is shown on the left.
The tetrahedra also share a single edge midpoint ($m$) which is shown on the
right.

In general, we assume that the number of tetrahedra sharing a vertex or
midpoint is relatively small, i.e.

$\displaystyle|\mathcal{T}_{p}|$ $\displaystyle\lesssim 1,\qquad\forall
p\in\mathcal{V}(\mathcal{T}),$ $\displaystyle|\mathcal{T}_{m}|$
$\displaystyle\lesssim 1,\qquad\forall m\in\mathcal{M}(\mathcal{T}).$

Here, and throughout the remainder of this section, we say that $A\lesssim B$
when $A\leq c_{0}B$, and $c_{0}$ is a constant do not depend on the mesh. In a
natural fashion, we say that $A\lesssim B$ and $B\lesssim A$ implies $A\approx
B$.

In addition, the continuous piecewise quadratic vector fields (which reside in
$W_{\mathcal{T}})$ can be represented using piecewise quadratic, continuous
Galerkin finite elements. These elements can be visualized as shown in figure
2.

Figure 2. A collection of continuous, piecewise quadratic, finite elements.
The degrees of freedom for each element are represented using solid dots.

We are now ready to introduce the first lemma of this section.

######  Lemma 4.1.

The pointwise differences between the piecewise vector fields $\bm{v}\in
V_{\mathcal{T}}$ and the continuous mapping $E\bm{v}\in W_{\mathcal{T}}$ are
given by the following inequalities

(4.3) $\displaystyle\left|\bm{v}_{T}(p)-E\bm{v}(p)\right|^{2}$
$\displaystyle\lesssim\sum_{\sigma\in\Xi_{p}}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2},\qquad\forall
T\in\mathcal{T},\;\forall p\in\mathcal{V}(\mathcal{T}),$ (4.4)
$\displaystyle\left|\bm{v}_{T}(m)-E\bm{v}(m)\right|^{2}$
$\displaystyle\lesssim\sum_{\sigma\in\Xi_{m}}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(m)\right|^{2},\qquad\forall
T\in\mathcal{T},\;\forall m\in\mathcal{M}(\mathcal{T}),$

where $\Xi_{p}$ is the set of interior faces sharing $p$ as a common vertex

$\displaystyle\Xi_{p}=\left\\{\sigma\in
S(\mathcal{T},\Omega):p\in\partial\sigma\right\\},$

and $\Xi_{m}$ is the set of interior faces sharing $m$ as a common edge
midpoint

$\displaystyle\Xi_{m}=\left\\{\sigma\in
S(\mathcal{T},\Omega):m\in\partial\sigma\right\\}.$

###### Proof.

The proof of Eq. (4.3) appears in Lemma 2.1 of [6]. It remains for us to prove
Eq. (4.4). We begin by using the definition of $E$ from Eq. (4.2) in order to
obtain the following expression

(4.5)
$\displaystyle\bm{v}_{T}(m)-E\bm{v}(m)=\frac{1}{|\mathcal{T}_{m}|}\sum_{T^{\prime}\in\mathcal{T}_{m}}\left(\bm{v}_{T}(m)-\bm{v}_{T^{\prime}}(m)\right).$

Next, we introduce a sequence of tetrahedra which share the edge midpoint $m$,
where $T=T_{1}$ and $T^{\prime}=T_{k}$, and $k>1$. An example of such a
sequence is $T_{1},T_{2},\ldots,T_{j},T_{j+1},\ldots,T_{k-1},T_{k}$, where
$k\leq|\mathcal{T}_{m}|$. Furthermore, suppose that $T_{j}$ and $T_{j+1}$
share a common face $\sigma_{j}\in\Xi_{m}$. Under these circumstances, the
following identity holds

(4.6) $\displaystyle\left|\bm{v}_{T}(m)-\bm{v}_{T^{\prime}}(m)\right|^{2}$
$\displaystyle=\left|\sum_{j=1}^{k-1}\left(\bm{v}_{T_{j}}(m)-\bm{v}_{T_{j+1}}(m)\right)\right|^{2}$
$\displaystyle\lesssim\sum_{j=1}^{k-1}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma_{j}}(m)\right|^{2},$

where we have used the root-mean-square-arithmetic-mean inequality on the last
line. Upon combining Eqs. (4.5) and (4.6), and using the root-mean-square-
arithmetic-mean inequality a second time, we obtain the desired result (Eq.
(4.4)). ∎

######  Lemma 4.2.

Suppose that a seminorm $\bm{\Phi}$ satisfies the following inequalities

(4.7) $\displaystyle\bm{\Phi}(\bm{w})$ $\displaystyle\leq
C_{\bm{\Phi}}\left\|\bm{w}\right\|_{H^{1}(\Omega,\mathcal{T})},\qquad\bm{w}\in\left[H^{1}(\Omega,\mathcal{T})\right]^{3},$
(4.8) $\displaystyle\left(\bm{\Phi}(\bm{v}-E\bm{v})\right)^{2}$
$\displaystyle\lesssim\sum_{\sigma\in
S(\mathcal{T},\Omega)}(\text{diam}\,\sigma)\left(\sum_{p\in\mathcal{V}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}+\sum_{m\in\mathcal{M}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(m)\right|^{2}\right),\quad\forall\bm{v}\in
V_{\mathcal{T}},$

in conjunction with

$\displaystyle\bm{\Phi}(\bm{m})=0\qquad\text{and}\qquad\bm{m}\in\mathbf{CK}(\Omega)\qquad\Longleftrightarrow\qquad\bm{m}=\text{const},$

where $\mathcal{V}(\sigma)$ is the set of all vertices of $\sigma$, and
$\mathcal{M}(\sigma)$ is the set of all edge midpoints of $\sigma$. Under
these circumstances, the following inequality holds for all $\bm{v}\in
V_{\mathcal{T}}$

(4.9) $\displaystyle\left|\bm{v}\right|^{2}_{H^{1}(\Omega,\mathcal{T})}$
$\displaystyle\lesssim\left\|\epsilon_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|^{2}_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left(\bm{\Phi}(\bm{v})\right)^{2}$
$\displaystyle+\sum_{\sigma\in
S(\mathcal{T},\Omega)}(\text{diam}\,\sigma)\left(\sum_{p\in\mathcal{V}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}+\sum_{m\in\mathcal{M}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(m)\right|^{2}\right).$

###### Proof.

We begin by assuming that the diameter of each triangle is comparable in size
to the diameters of the nearby faces

(4.10) $\displaystyle\text{diam}\,T\approx\text{diam}\,\sigma,\qquad$
$\displaystyle\forall
T\in\mathcal{T}_{p},\;\sigma\in\Xi_{p},\;p\in\mathcal{V}(\mathcal{T}),$
$\displaystyle\forall
T\in\mathcal{T}_{m},\;\sigma\in\Xi_{m},\;m\in\mathcal{M}(\mathcal{T}).$

Next, we develop the following piecewise $H^{1}$ estimate by using standard
$L_{\infty}$ error estimates (cf. [7], Chapter 4), in conjunction with Eqs.
(4.3), (4.4), and (4.10)

(4.11)
$\displaystyle\left|\bm{v}-E\bm{v}\right|^{2}_{H^{1}(\Omega,\mathcal{T})}$
$\displaystyle\lesssim\sum_{T\in\mathcal{T}}\left(\text{diam}\,T\right)\left(\sum_{p\in\mathcal{V}(T)}\left|\bm{v}_{T}(p)-E\bm{v}(p)\right|^{2}+\sum_{m\in\mathcal{M}(T)}\left|\bm{v}_{T}(m)-E\bm{v}(m)\right|^{2}\right)$
$\displaystyle\lesssim\sum_{\sigma\in
S(\mathcal{T},\Omega)}(\text{diam}\,\sigma)\left(\sum_{p\in\mathcal{V}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}+\sum_{m\in\mathcal{M}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(m)\right|^{2}\right).$

In addition, we develop the following $H^{1}$ estimate based on the triangle
inequality, root-mean-square-arithmetic-mean inequality, Eqs. (2.9), (4.8),
and (4.11), and Eq. (A.2) from Lemma A.1 in Appendix A

$\displaystyle\left|\bm{v}\right|^{2}_{H^{1}(\Omega,\mathcal{T})}$
$\displaystyle\leq\left|E\bm{v}\right|^{2}_{H^{1}(\Omega)}+\left|\bm{v}-E\bm{v}\right|^{2}_{H^{1}(\Omega,\mathcal{T})}$
$\displaystyle\lesssim\left\|\bm{\epsilon}(E\bm{v})-\frac{1}{3}\text{div}(E\bm{v})\mathbb{I}\right\|^{2}_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left(\bm{\Phi}(E\bm{v})\right)^{2}+\left|\bm{v}-E\bm{v}\right|^{2}_{H^{1}(\Omega,\mathcal{T})}$
$\displaystyle\lesssim\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|^{2}_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v}-E\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v}-E\bm{v})\mathbb{I}\right\|^{2}_{L_{2}(\Omega)\times
L_{2}(\Omega)}$
$\displaystyle+\left(\bm{\Phi}(\bm{v})\right)^{2}+\left(\bm{\Phi}(\bm{v}-E\bm{v})\right)^{2}+\left|\bm{v}-E\bm{v}\right|^{2}_{H^{1}(\Omega,\mathcal{T})}$
$\displaystyle\lesssim\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|^{2}_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left(\bm{\Phi}(\bm{v})\right)^{2}+\left(\bm{\Phi}(\bm{v}-E\bm{v})\right)^{2}+\left|\bm{v}-E\bm{v}\right|^{2}_{H^{1}(\Omega,\mathcal{T})}$
$\displaystyle\lesssim\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|^{2}_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left(\bm{\Phi}(\bm{v})\right)^{2}$
$\displaystyle+\sum_{\sigma\in
S(\mathcal{T},\Omega)}(\text{diam}\,\sigma)\left(\sum_{p\in\mathcal{V}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}+\sum_{m\in\mathcal{M}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(m)\right|^{2}\right).$

The last line contains the desired result (Eq. (4.9)). ∎

In the following examples, we will demonstrate that the inequality in Eq.
(4.8) holds for the seminorms in Eqs. (2.10) and (2.11).

######  Example 4.3.

For the seminorm in Eq. (2.10), we have that

$\displaystyle\left(\bm{\Phi}_{1}(\bm{v}-E\bm{v})\right)^{2}$
$\displaystyle=\left\|G(\bm{v}-E\bm{v})\right\|_{L_{2}(\Omega)}^{2}$
$\displaystyle\leq C\left\|\bm{v}-E\bm{v}\right\|_{L_{2}(\Omega)}^{2}$
$\displaystyle\lesssim\sum_{T\in\mathcal{T}}\left(\text{diam}\,T\right)^{3}\left(\sum_{p\in\mathcal{V}(T)}\left|\bm{v}_{T}(p)-E\bm{v}(p)\right|^{2}+\sum_{m\in\mathcal{M}(T)}\left|\bm{v}_{T}(m)-E\bm{v}(m)\right|^{2}\right)$
$\displaystyle\lesssim\sum_{\sigma\in
S(\mathcal{T},\Omega)}(\text{diam}\,\sigma)^{3}\left(\sum_{p\in\mathcal{V}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}+\sum_{m\in\mathcal{M}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(m)\right|^{2}\right),$

where we have used the boundedness of the projection operator (Eq. (2.12)), a
standard $L_{\infty}$ estimate [7], and Eqs. (4.3), (4.4), and (4.10).

######  Example 4.4.

For the seminorm in Eq. (2.11), we have that

$\displaystyle\left(\bm{\Phi}_{2}(\bm{v}-E\bm{v})\right)^{2}$
$\displaystyle=\sup_{\begin{subarray}{c}\bm{m}\in\mathbf{CK}(\Omega)\\\
\left\|\bm{m}\right\|_{L_{2}(\partial\Omega)}=1,\int_{\partial\Omega}\bm{m}\,ds=\bm{0}\end{subarray}}\left(\int_{\partial\Omega}\left(\bm{v}-E\bm{v}\right)\cdot\bm{m}\,ds\right)^{2}$
$\displaystyle\leq C\left\|\bm{v}-E\bm{v}\right\|_{L_{2}(\partial\Omega)}^{2}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}T\in\mathcal{T}\\\ \partial
T\cap\partial\Omega\neq\emptyset\end{subarray}}\left(\text{diam}\,T\right)^{2}\left(\sum_{p\in\mathcal{V}(T)}\left|\bm{v}_{T}(p)-E\bm{v}(p)\right|^{2}+\sum_{m\in\mathcal{M}(T)}\left|\bm{v}_{T}(m)-E\bm{v}(m)\right|^{2}\right)$
$\displaystyle\lesssim\sum_{\sigma\in
S(\mathcal{T},\Omega)}(\text{diam}\,\sigma)^{2}\left(\sum_{p\in\mathcal{V}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}+\sum_{m\in\mathcal{M}(\sigma)}\left|\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(m)\right|^{2}\right),$

where we have used Hölder’s inequality, a standard $L_{\infty}$ estimate [7],
a discrete trace inequality, a root-mean-square-arithmetic-mean inequality,
and Eqs. (4.3), (4.4), and (4.10).

In order to provide an appropriate context for our discussion of the final
theorem, we will introduce the following interpolation operator for all
$\bm{v}\in\left[H^{2}(T)\right]^{3}$

(4.12)
$\displaystyle\left|\int_{T}\left(\bm{v}-\Pi_{T}\bm{v}\right)dx\right|=0,\qquad\left|\int_{T}\text{div}\left(\bm{v}-\Pi_{T}\bm{v}\right)dx\right|=0,$
$\displaystyle\left|\int_{T}\text{curl}\left(\bm{v}-\Pi_{T}\bm{v}\right)dx\right|=0,\qquad\left|\int_{T}\text{curl}\left(\text{curl}\left(\bm{v}-\Pi_{T}\bm{v}\right)\right)dx\right|=0.$

This interpolation operator is noteworthy because it projects $\bm{v}$ on to
the kernel space $\mathbf{CK}(T)$. In fact, it can be shown that for all
$\bm{m}\in\mathbf{CK}(T)$

$\displaystyle\left|\int_{T}\bm{m}\,dx\right|=0,\qquad\left|\int_{T}\text{div}(\bm{m})\,dx\right|=0,$
$\displaystyle\left|\int_{T}\text{curl}(\bm{m})\,dx\right|=0,\qquad\left|\int_{T}\text{curl}(\text{curl}(\bm{m}))\,dx\right|=0,\qquad\Longleftrightarrow\qquad\bm{m}=\bm{0}.$

The proof of the above statement appears in Appendix B, Lemma B.1. Using the
element-wise definition of the interpolation operator above, we can construct
a global interpolation operator

$\displaystyle\Pi:\left[H^{2}(\Omega,\mathcal{T})\right]^{3}\rightarrow
V_{\mathcal{T}},$
$\displaystyle\left(\Pi\bm{v}\right)|_{T}=\Pi_{T}\bm{v}_{T},\qquad\forall
T\in\mathcal{T}.$

In addition, upon combining Eq. (4.12) with Eq. (C.20), (the latter of which
is derived in Appendix C, Theorem C.4), we obtain

(4.13) $\displaystyle\left|\bm{v}-\Pi_{T}\bm{v}\right|_{H^{1}(T)}$
$\displaystyle\lesssim\left\|\bm{\epsilon}(\bm{v}-\Pi_{T}\bm{v})-\frac{1}{3}\text{div}(\bm{v}-\Pi_{T}\bm{v})\mathbb{I}\right\|_{L_{2}(T)\times
L_{2}(T)}$
$\displaystyle=\left\|\bm{\epsilon}(\bm{v})-\frac{1}{3}\text{div}(\bm{v})\mathbb{I}\right\|_{L_{2}(T)\times
L_{2}(T)},$

for all $T\in\mathcal{T}$ and
$\bm{v}\in\left[H^{2}(\Omega,\mathcal{T})\right]^{3}$. Finally, by a standard
Poincaré-type inequality, we have that

(4.14)
$\displaystyle\left\|\bm{v}-\Pi_{T}\bm{v}\right\|_{L_{2}(T)}\lesssim\left(\text{diam}\,T\right)\left|\bm{v}-\Pi_{T}\bm{v}\right|_{H^{1}(T)},$

for all $T\in\mathcal{T}$ and
$\bm{v}\in\left[H^{1}(\Omega,\mathcal{T})\right]^{3}$. We are now ready to
state the final result.

######  Theorem 4.5.

Suppose that a seminorm
$\bm{\Phi}:\left[H^{1}(\Omega,\mathcal{T})\right]^{3}\rightarrow\mathbb{R}$
satisfies the following condition

(4.15)
$\displaystyle\bm{\Phi}\left(\bm{v}-\Pi\bm{v}\right)\lesssim\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)},\qquad\forall\bm{v}\in\left[H^{2}(\Omega,\mathcal{T})\right]^{3},$

where $\Pi:\left[H^{2}(\Omega,\mathcal{T})\right]^{3}\rightarrow
V_{\mathcal{T}}$. Under these circumstances, the following inequality holds

(4.16)
$\displaystyle\left|\bm{v}\right|^{2}_{H^{1}(\Omega,\mathcal{T})}\leq\kappa\left(\theta_{\mathcal{T}}\right)\Bigg{(}$
$\displaystyle\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}+\left(\bm{\Phi}(\bm{v})\right)^{2}$
$\displaystyle+\sum_{\sigma\in
S(\mathcal{T},\Omega)}\left(\text{diam}\,\sigma\right)^{-1}\left\|\pi_{\sigma}\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}\right\|^{2}_{L_{2}(\sigma)}\Bigg{)},\qquad\forall\bm{v}\in\left[H^{2}(\Omega,\mathcal{T})\right]^{3},$

where $\kappa:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is a continuous
function, which does not depend on $\mathcal{T}$.

###### Proof.

We start by constructing the following $H^{1}$ estimate using the triangle
inequality, the root-mean-square-arithmetic-mean inequality, Eq. (2.3), Eq.
(4.9), and Eq. (4.13)

(4.17) $\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega,\mathcal{T})}^{2}$
$\displaystyle\leq\left|\bm{v}-\Pi\bm{v}\right|^{2}_{H^{1}(\Omega,\mathcal{T})}+\left|\Pi\bm{v}\right|^{2}_{H^{1}(\Omega,\mathcal{T})}$
$\displaystyle\lesssim\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}+\left(\bm{\Phi}(\Pi\bm{v})\right)^{2}$
$\displaystyle+\sum_{\sigma\in
S(\mathcal{T},\Omega)}(\text{diam}\,\sigma)\left(\sum_{p\in\mathcal{V}(\sigma)}\left|\left[\\!\left[\Pi\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}+\sum_{m\in\mathcal{M}(\sigma)}\left|\left[\\!\left[\Pi\bm{v}\right]\\!\right]_{\sigma}(m)\right|^{2}\right).$

In addition, we can use the triangle inequality and Eq. (4.15) to obtain the
following estimate

(4.18) $\displaystyle\bm{\Phi}(\Pi\bm{v})$
$\displaystyle\leq\bm{\Phi}(\bm{v}-\Pi\bm{v})+\bm{\Phi}(\bm{v})$
$\displaystyle\lesssim\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\bm{\Phi}(\bm{v}).$

It remains for us to construct an upper bound for the jump terms on the RHS of
Eq. (4.17). Towards this end, we re-introduce the quadratic projection
operator

$\displaystyle\pi_{\sigma}:\left[L_{2}(\sigma)\right]^{3}\rightarrow\left[P_{2}(\sigma)\right]^{3},\qquad\forall\sigma\in
S(\mathcal{T},\Omega).$

Then, by the triangle inequality, the root-mean-square-arithmetic-mean
inequality, the discrete trace inequality, and standard inverse estimates (see
[7], Chapter 4), we obtain

(4.19)
$\displaystyle\left|\left[\\!\left[\Pi\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}$
$\displaystyle=\left|\left[\\!\left[\Pi\bm{v}-\bm{v}+\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}$
$\displaystyle=\left|\pi_{\sigma}\left[\\!\left[\Pi\bm{v}-\bm{v}+\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}$
$\displaystyle\lesssim\left|\pi_{\sigma}\left[\\!\left[\bm{v}-\Pi\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}+\left|\pi_{\sigma}\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}$
$\displaystyle\lesssim\left(\text{diam}\,\sigma\right)^{-2}\left(\left\|\pi_{\sigma}\left[\\!\left[\bm{v}-\Pi\bm{v}\right]\\!\right]_{\sigma}\right\|_{L_{2}(\sigma)}^{2}+\left\|\pi_{\sigma}\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}\right\|_{L_{2}(\sigma)}^{2}\right).$

Following a similar argument, we have that

(4.20)
$\displaystyle\left|\left[\\!\left[\Pi\bm{v}\right]\\!\right]_{\sigma}(m)\right|^{2}\lesssim\left(\text{diam}\,\sigma\right)^{-2}\left(\left\|\pi_{\sigma}\left[\\!\left[\bm{v}-\Pi\bm{v}\right]\\!\right]_{\sigma}\right\|_{L_{2}(\sigma)}^{2}+\left\|\pi_{\sigma}\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}\right\|_{L_{2}(\sigma)}^{2}\right).$

Next, we can construct the following $L_{2}$ estimate by using the discrete
trace inequality in conjunction with Eqs. (4.13), (4.14), and (3.1)

(4.21)
$\displaystyle\left\|\pi_{\sigma}\left[\\!\left[\bm{v}-\Pi\bm{v}\right]\\!\right]_{\sigma}\right\|_{L_{2}(\sigma)}^{2}$
$\displaystyle\leq
C\left\|\left[\\!\left[\bm{v}-\Pi\bm{v}\right]\\!\right]_{\sigma}\right\|_{L_{2}(\sigma)}^{2}$
$\displaystyle\lesssim\sum_{T\in\mathcal{T}_{\sigma}}\Bigg{(}\left(\text{diam}\,T\right)\left|\bm{v}_{T}-\Pi_{T}\bm{v}_{T}\right|_{H^{1}(T)}^{2}$
$\displaystyle+\left(\text{diam}\,T\right)^{-1}\left\|\bm{v}_{T}-\Pi_{T}\bm{v}_{T}\right\|_{L_{2}(T)}^{2}\Bigg{)}$
$\displaystyle\lesssim\sum_{T\in\mathcal{T}_{\sigma}}\left(\text{diam}\,T\right)\left\|\bm{\epsilon}(\bm{v}_{T})-\frac{1}{3}\text{div}(\bm{v}_{T})\mathbb{I}\right\|_{L_{2}(T)\times
L_{2}(T)}^{2},$

where $\mathcal{T}_{\sigma}$ is the set of tetrahedra sharing the common side
$\sigma$. Now, upon combining Eqs. (4.10), (4.19), (4.20), and (4.21), we
obtain

(4.22) $\displaystyle\sum_{\sigma\in
S(\mathcal{T},\Omega)}(\text{diam}\,\sigma)\left(\sum_{p\in\mathcal{V}(\sigma)}\left|\left[\\!\left[\Pi\bm{v}\right]\\!\right]_{\sigma}(p)\right|^{2}+\sum_{m\in\mathcal{M}(\sigma)}\left|\left[\\!\left[\Pi\bm{v}\right]\\!\right]_{\sigma}(m)\right|^{2}\right)$
$\displaystyle\lesssim\sum_{\sigma\in
S(\mathcal{T},\Omega)}(\text{diam}\,\sigma)^{-1}\left\|\pi_{\sigma}\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}\right\|_{L_{2}(\sigma)}^{2}+\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}.$

Lastly, on combining Eqs. (4.17), (4.18), and (4.22) we obtain

$\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega,\mathcal{T})}^{2}$
$\displaystyle\lesssim\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2}+\left(\bm{\Phi}(\bm{v})\right)^{2}$
$\displaystyle+\sum_{\sigma\in
S(\mathcal{T},\Omega)}(\text{diam}\,\sigma)^{-1}\left\|\pi_{\sigma}\left[\\!\left[\bm{v}\right]\\!\right]_{\sigma}\right\|_{L_{2}(\sigma)}^{2}.$

This equation is identical to the desired result (Eq. (4.16)). ∎

In the following examples, we will demonstrate that the inequality in Eq.
(4.15) holds for the seminorms in Eqs. (2.10) and (2.11).

######  Example 4.6.

For the seminorm in Eq. (2.10), we have that

$\displaystyle\left(\bm{\Phi}_{1}(\bm{v}-\Pi\bm{v})\right)^{2}$
$\displaystyle=\sum_{T\in\mathcal{T}}\left\|G(\bm{v}_{T}-\Pi_{T}\bm{v}_{T})\right\|_{L_{2}(T)}^{2}$
$\displaystyle\leq
C\sum_{T\in\mathcal{T}}\left\|\bm{v}_{T}-\Pi_{T}\bm{v}_{T}\right\|_{L_{2}(T)}^{2}$
$\displaystyle\lesssim\sum_{T\in\mathcal{T}}\left(\text{diam}\,T\right)^{2}\left\|\bm{\epsilon}(\bm{v}_{T})-\frac{1}{3}\text{div}(\bm{v}_{T})\mathbb{I}\right\|_{L_{2}(T)\times
L_{2}(T)}^{2}$
$\displaystyle\lesssim\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2},$

where we have used Eqs. (2.12), (4.13), and (4.14).

######  Example 4.7.

For the seminorm in Eq. (2.11), we have that

$\displaystyle\left(\bm{\Phi}_{2}(\bm{v}-\Pi\bm{v})\right)^{2}$
$\displaystyle=\sup_{\begin{subarray}{c}\bm{m}\in\mathbf{CK}(\Omega)\\\
\left\|\bm{m}\right\|_{L_{2}(\partial\Omega)}=1,\int_{\partial\Omega}\bm{m}\,ds=\bm{0}\end{subarray}}\left(\int_{\partial\Omega}\left(\bm{v}-\Pi\bm{v}\right)\cdot\bm{m}\,ds\right)^{2}$
$\displaystyle\leq
C\left\|\bm{v}-\Pi\bm{v}\right\|_{L_{2}(\partial\Omega)}^{2}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}T\in\mathcal{T}\\\ \partial
T\cap\partial\Omega\neq\emptyset\end{subarray}}\left(\text{diam}\,T\right)\left\|\bm{\epsilon}(\bm{v}_{T})-\frac{1}{3}\text{div}(\bm{v}_{T})\mathbb{I}\right\|_{L_{2}(T)\times
L_{2}(T)}^{2}$
$\displaystyle\lesssim\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{v})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}^{2},$

where we have used a discrete trace inequality, and Eqs. (4.13) and (4.14).

##  5\. Conclusion

In this work, we introduced a class of discrete generalized Korn’s
inequalities. These inequalities are similar to standard Korn’s inequalities,
with the key caveat that they involve the trace-free symmetric gradient
operator instead of the symmetric gradient operator. We have shown that the
trace-free symmetric gradient operator requires a careful analysis, as it
possesses a more complicated kernel space than the standard symmetric gradient
operator. As a result, it is necessary for us to introduce a new quadratic
mapping procedure, and a new projection operator in order to construct the
corresponding discrete Korn’s inequalities. To the best of our knowledge, this
endeavor has been successful in generating a new class of discrete
inequalities for piecewise $H^{2}$ vector fields in 3D.

There are several interesting possibilities for future research. In
particular, the extension of this work to 2D remains an open research
question. In particular, the kernel space becomes infinite dimensional in this
case, unless the vector field vanishes on the boundary [11]. In principle,
such an extension is possible, although it is beyond the scope of the current
work. In addition, we note that there are still opportunities to extend our
analysis to higher dimensions, where more complicated derivative operators are
present. This extension may be of particular interest for space-time finite
element methods in 4D.

##  Appendix A Seminorms and Generalized Korn’s Inequalities

In this section, our objective is to construct a pair of generalized Korn’s
inequalities which are augmented on the right hand side with seminorms. In
preparation for constructing these inequalities, we will first construct an
‘inverse generalized Korn’s inequality’ that will facilitate our subsequent
analysis.

######  Lemma A.1.

For all $\bm{u}\in\left[H^{1}(\Omega)\right]^{3}$, the following upper bound
on the trace-free symmetric gradient holds

(A.1)
$\displaystyle\left\|\bm{\epsilon}(\bm{u})-\frac{1}{3}\text{div}(\bm{u})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}\leq C\left|\bm{u}\right|_{H^{1}(\Omega)}.$

In addition, for all $\bm{u}\in\left[H^{1}(\Omega,\mathcal{T})\right]^{3}$

(A.2)
$\displaystyle\left\|\bm{\epsilon}_{\mathcal{T}}(\bm{u})-\frac{1}{3}\text{div}_{\mathcal{T}}(\bm{u})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}\leq C\left|\bm{u}\right|_{H^{1}(\Omega,\mathcal{T})}.$

###### Proof.

We begin by considering a vector field,
$\bm{v}\in\left[H^{1}(\Omega)\right]^{3}$. For this case, we have that

(A.3)
$\displaystyle\left\|\bm{\epsilon}(\bm{v})-\frac{1}{3}\text{div}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}$
$\displaystyle=\left\|\frac{1}{2}\left(\text{grad}(\bm{v})+\text{grad}(\bm{v})^{t}\right)-\frac{1}{3}\text{div}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}$ $\displaystyle\leq
C\left(\left\|\text{grad}(\bm{v})\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left\|\text{grad}(\bm{v})^{t}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left\|\text{div}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}\right)$ $\displaystyle\leq
C\left(\left\|\text{grad}(\bm{v})\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left\|\text{div}(\bm{v})\right\|_{L_{2}(\Omega)}\right),$

where we have used the triangle inequality, and the fact that
$\left\|\text{grad}(\bm{v})\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}=\left\|\text{grad}(\bm{v})^{t}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}$.

In accordance with Lemma 3.34 of [26],

(A.4) $\displaystyle\left\|\text{div}(\bm{v})\right\|_{L_{2}(\Omega)}\leq
C\left\|\text{grad}(\bm{v})\right\|_{L_{2}(\Omega)\times L_{2}(\Omega)}.$

Upon substituting this result into Eq. (A.3), we obtain

(A.5)
$\displaystyle\left\|\bm{\epsilon}(\bm{v})-\frac{1}{3}\text{div}(\bm{v})\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}\leq C\left\|\text{grad}(\bm{v})\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}.$

Now, upon replacing $\bm{v}$ with $\bm{u}$, we obtain the first desired result
(Eq. (A.1)).

Furthermore, upon replacing $\Omega$ with $T$ in Eq. (A.5), squaring both
sides, summing over all $T\in\mathcal{T}$, taking the square root, and letting

$\displaystyle\bm{u}|_{T}=\bm{v},$

we obtain the second desired result (Eq. (A.2)). ∎

We are now ready to introduce the first generalized Korn’s inequality that
applies to $H^{1}$ vector fields.

######  Lemma A.2.

Suppose that $\bm{v}\in\left[H^{1}(\Omega)\right]^{3}$ and that a seminorm
$\bm{\Phi}(\bm{v})$ satisfies the conditions in Eqs. (2.6) and (2.7). Under
these circumstances, we have that

(A.6) $\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega)}\leq
C_{\Omega}\left(\left\|\bm{\epsilon}\left(\bm{v}\right)-\frac{1}{3}\text{div}\left(\bm{v}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\bm{\Phi}\left(\bm{v}\right)\right),\qquad\forall\bm{v}\in\left[H^{1}(\Omega)\right]^{3}.$

###### Proof.

The proof follows from Lemma A.1 and the compact embedding of $H^{1}$ into
$L_{2}$, (cf. [1], Theorem 6.3). ∎

Our next generalized Korn’s inequality applies to vector fields in
$\mathbb{E}(\Omega)$, which is a complete, normed vector space which lies
between $[H^{1}(\Omega)]^{3}$ and $[H^{2}(\Omega)]^{3}$. In what follows, we
will define this space, and then present the associated generalized Korn’s
inequality.

######  Definition A.3 (The Space $\mathbb{E}(\Omega)$).

We begin by defining a new norm

$\displaystyle\left\|\bm{v}\right\|_{\mathbb{E}(\Omega)}=\left\|\bm{v}\right\|_{H^{1}(\Omega)}+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}))\,dx\right|,$

for all $\bm{v}\in\mathbb{E}(\Omega)$. Here, we define $\mathbb{E}(\Omega)$ as
a Banach space which is a subspace of $\left[H^{1}(\Omega)\right]^{3}$ such
that

$\displaystyle\mathbb{E}(\Omega)=\left\\{\bm{v}\in\left[H^{1}(\Omega)\right]^{3}:\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}))\,dx\right|<\infty\right\\}.$

We note that constructing $\mathbb{E}(\Omega)$ is always possible, as we can
simply start with an incomplete metric space $\mathbb{E}^{\prime}(\Omega)$,
and then develop its completion in order to obtain $\mathbb{E}(\Omega)$ in
accordance with [28], Theorem 1.6-2. For example, we can think of
$\mathbb{E}(\Omega)$ as the completion of
$\left[C^{\infty}(\Omega)\right]^{3}$ with respect to the norm
$\left\|\cdot\right\|_{\mathbb{E}(\Omega)}$.

By inspection, we have that the following inclusion holds

$\displaystyle\left[H^{2}(\Omega)\right]^{3}\subset\mathbb{E}(\Omega)\subset\left[H^{1}(\Omega)\right]^{3}.$

Further, we assume that the following compact embedding holds

$\displaystyle\mathbb{E}(\Omega)\subset\subset\left[L_{2}(\Omega)\right]^{3}.$

######  Definition A.4 (Div-Curl Seminorm).

Consider the seminorm $\bm{\Psi}$ which is well-defined for vectors
$\bm{v}\in\mathbb{E}(\Omega)$,

(A.7)
$\displaystyle\bm{\Psi}(\bm{v})=\left|\int_{\Omega}\text{div}(\bm{v})\,dx\right|+\left|\int_{\Omega}\text{curl}(\bm{v})\,dx\right|+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}))\,dx\right|.$

This seminorm has the following properties

(A.8) $\displaystyle\bm{\Psi}\left(\bm{v}\right)\leq
C_{\bm{\Psi}}\left\|\bm{v}\right\|_{\mathbb{E}(\Omega)},\qquad\forall\bm{v}\in\mathbb{E}(\Omega),$

and

(A.9)
$\displaystyle\bm{\Psi}\left(\bm{m}\right)=0\qquad\text{and}\qquad\bm{m}\in\mathbf{CK}(\Omega)\qquad\Longleftrightarrow\qquad\bm{m}=\text{a
constant vector}.$

It immediately follows that

(A.10) $\displaystyle\bm{\Psi}\left(\bm{v}+\bm{c}\right)=\bm{\Psi}(\bm{v}).$

######  Lemma A.5.

Suppose that $\bm{v}\in\mathbb{E}(\Omega)$ and that a particular seminorm (Eq.
(A.7)) satisfies the conditions in Eqs. (A.8) and (A.9). Under these
circumstances, we have that

(A.11)
$\displaystyle\left|\bm{v}\right|_{H^{1}(\Omega)}+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}))\,dx\right|\leq
C_{\Omega}\left(\left\|\bm{\epsilon}\left(\bm{v}\right)-\frac{1}{3}\text{div}\left(\bm{v}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\bm{\Psi}\left(\bm{v}\right)\right),$
$\displaystyle\forall\bm{v}\in\mathbb{E}(\Omega).$

###### Proof.

The proof is by contradiction. We assume that Eq. (A.11) is false. In this
case, there must exist a sequence $\bm{v}_{n}\in\mathbb{E}(\Omega)$ which
satisfies the following conditions

(A.12)
$\displaystyle\left|\bm{v}_{n}\right|_{H^{1}(\Omega)}+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}_{n}))\,dx\right|=1,\qquad\forall
n\geq 1,$

and

(A.13) $\displaystyle 1\geq
n\left(\left\|\bm{\epsilon}\left(\bm{v}_{n}\right)-\frac{1}{3}\text{div}\left(\bm{v}_{n}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\bm{\Psi}\left(\bm{v}_{n}\right)\right).$

We can further specify the sequence, by assuming that

(A.14) $\displaystyle\int_{\Omega}\bm{v}_{n}\,dx=\bm{0},\qquad\forall n\geq
1,$

in accordance with Eq. (A.10). Next, Eqs. (A.12), (A.14), and a standard
Poincaré-Friedrichs inequality imply that
$\left\\{\bm{v}_{n}\right\\}_{n=1}^{\infty}$ remains bounded in
$\mathbb{E}(\Omega)$.

Now, due to the compact embedding of $\mathbb{E}(\Omega)$ into
$\left[L_{2}(\Omega)\right]^{3}$, we can consider a convergent subsequence in
$\left[L_{2}(\Omega)\right]^{3}$. More precisely, we can assume there exists
$\bm{v}_{\ast}\in\left[L_{2}(\Omega)\right]^{3}$ such that

(A.15)
$\displaystyle\lim_{n\rightarrow\infty}\left\|\bm{v}_{n}-\bm{v}_{\ast}\right\|_{L_{2}(\Omega)}=0.$

Combining this statement along with Eq. (A.13), we have that

(A.16)
$\displaystyle\lim_{n\rightarrow\infty}\left\|\bm{\epsilon}\left(\bm{v}_{n}\right)-\frac{1}{3}\text{div}\left(\bm{v}_{n}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}=0,\qquad\lim_{n\rightarrow\infty}\bm{\Psi}(\bm{v}_{n})=0.$

The limits in Eqs. (A.15) and (A.16), in conjunction with Korn’s inequality
(Eq. (2.4)) imply that $\left\\{\bm{v}_{n}\right\\}_{n=1}^{\infty}$ is a
Cauchy sequence in $\mathbb{E}(\Omega)$. For the sake of clarity, we will
further expand this argument below. First, let Eq. (2.4) be rewritten as
follows

(A.17)
$\displaystyle\left\|\bm{v}\right\|_{H^{1}(\Omega)}+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}))\,dx\right|$
$\displaystyle\leq\left(C_{\Omega}+1\right)\left(\left\|\bm{\epsilon}\left(\bm{v}\right)-\frac{1}{3}\text{div}\left(\bm{v}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left\|\bm{v}\right\|_{L_{2}(\Omega)}+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}))\,dx\right|\right).$

Next, we can consider two sequence members $\bm{v}_{n_{j}}$ and
$\bm{v}_{n_{k}}$, where $k>j$, and both $k$ and $j$ are large. Upon
substituting these members into Eq. (A.17), we obtain

$\displaystyle\left\|\bm{v}_{n_{j}}-\bm{v}_{n_{k}}\right\|_{H^{1}(\Omega)}+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}_{n_{j}}-\bm{v}_{n_{k}}))\,dx\right|$
$\displaystyle\leq\left(C_{\Omega}+1\right)\Bigg{(}\left\|\bm{\epsilon}\left(\bm{v}_{n_{j}}-\bm{v}_{n_{k}}\right)-\frac{1}{3}\text{div}\left(\bm{v}_{n_{j}}-\bm{v}_{n_{k}}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}$
$\displaystyle+\left\|\bm{v}_{n_{j}}-\bm{v}_{n_{k}}\right\|_{L_{2}(\Omega)}+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}_{n_{j}}-\bm{v}_{n_{k}}))\,dx\right|\Bigg{)}.$

On using the triangle inequality, we obtain

$\displaystyle\left\|\bm{v}_{n_{j}}-\bm{v}_{n_{k}}\right\|_{H^{1}(\Omega)}+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}_{n_{j}}-\bm{v}_{n_{k}}))\,dx\right|$
$\displaystyle\leq\left(C_{\Omega}+1\right)\Bigg{(}\left\|\bm{\epsilon}\left(\bm{v}_{n_{j}}\right)-\frac{1}{3}\text{div}\left(\bm{v}_{n_{j}}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}+\left\|\bm{\epsilon}\left(\bm{v}_{n_{k}}\right)-\frac{1}{3}\text{div}\left(\bm{v}_{n_{k}}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}$
$\displaystyle+\left\|\bm{v}_{n_{j}}-\bm{v}_{n_{k}}\right\|_{L_{2}(\Omega)}+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}_{n_{j}}))\,dx\right|+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}_{n_{k}}))\,dx\right|\Bigg{)}.$

In accordance with Eq. (A.13), the RHS can be rewritten as follows

(A.18)
$\displaystyle\left\|\bm{v}_{n_{j}}-\bm{v}_{n_{k}}\right\|_{H^{1}(\Omega)}+\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}_{n_{j}}-\bm{v}_{n_{k}}))\,dx\right|$
$\displaystyle\leq\left(C_{\Omega}+1\right)\Bigg{(}\frac{1}{n_{j}}+\frac{1}{n_{k}}+\left\|\bm{v}_{n_{j}}-\bm{v}_{n_{k}}\right\|_{L_{2}(\Omega)}\Bigg{)}.$

Evidently, the sequence is Cauchy in $\mathbb{E}(\Omega)$, as
$n_{j}\rightarrow\infty$, $n_{k}\rightarrow\infty$, and Eq. (A.15) holds.

In addition, the limit of $\left\\{\bm{v}_{n}\right\\}$ is
$\bm{v}_{\ast}\in\mathbb{E}(\Omega)$ because $\mathbb{E}(\Omega)$ is a Banach
space, and naturally

(A.19)
$\displaystyle\lim_{n\rightarrow\infty}\left\|\bm{v}_{n}-\bm{v}_{\ast}\right\|_{\mathbb{E}(\Omega)}=0.$

Having established the properties of our sequence in $\mathbb{E}(\Omega)$, we
now turn our attention to the seminorm $\bm{\Psi}$. In accordance with Eq.
(A.8) and the reverse triangle inequality, we have

(A.20)
$\displaystyle\left|\bm{\Psi}(\bm{v}_{n})-\bm{\Psi}(\bm{v}_{\ast})\right|\leq\bm{\Psi}(\bm{v}_{n}-\bm{v}_{\ast})\leq
C_{\bm{\Psi}}\left\|\bm{v}_{n}-\bm{v}_{\ast}\right\|_{\mathbb{E}(\Omega)}.$

Upon combining Eqs. (A.16), (A.19), and (A.20), we obtain

(A.21) $\displaystyle\bm{\Psi}(\bm{v}_{\ast})=0.$

In a similar fashion, we have that

(A.22)
$\displaystyle\left|\left\|\bm{\epsilon}\left(\bm{v}_{n}\right)-\frac{1}{3}\text{div}\left(\bm{v}_{n}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}-\left\|\bm{\epsilon}\left(\bm{v}_{\ast}\right)-\frac{1}{3}\text{div}\left(\bm{v}_{\ast}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}\right|$
$\displaystyle\leq\left\|\bm{\epsilon}\left(\bm{v}_{n}-\bm{v}_{\ast}\right)-\frac{1}{3}\text{div}\left(\bm{v}_{n}-\bm{v}_{\ast}\right)\mathbb{I}\right\|_{L_{2}(\Omega)\times
L_{2}(\Omega)}\leq
C\left\|\bm{v}_{n}-\bm{v}_{\ast}\right\|_{H^{1}(\Omega)}\leq
C\left\|\bm{v}_{n}-\bm{v}_{\ast}\right\|_{\mathbb{E}(\Omega)},$

where we have used Eq. (A.1) from Lemma A.1 on the last line. Upon combining
Eqs. (A.16), (A.19), and (A.22), we obtain

(A.23)
$\displaystyle\bm{\epsilon}\left(\bm{v}_{\ast}\right)-\frac{1}{3}\text{div}\left(\bm{v}_{\ast}\right)\mathbb{I}=\bm{0}.$

Finally, based on Eqs. (2.3), (A.9), (A.21), and (A.23), we conclude that
$\bm{v}_{\ast}$ belongs to the kernel space $\mathbf{CK}(\Omega)$, and is a
constant vector. However, this immediately contradicts Eqs. (A.12) and (A.19).
∎

##  Appendix B Projection Operator

In this section, we introduce an element-wise projection operator that
projects H2 vector fields on to the kernel space $\mathbf{CK}(T)$, which
itself is contained in the space of quadratic polynomial vector fields.

######  Lemma B.1.

Consider the projection operator
$\Pi_{T}:\left[H^{2}(T)\right]^{3}\rightarrow\left[P_{2}(T)\right]^{3}$, which
is defined such that

(B.1)
$\displaystyle\left|\int_{T}\left(\bm{v}-\Pi_{T}\bm{v}\right)dx\right|=0,\qquad\left|\int_{T}\text{div}\left(\bm{v}-\Pi_{T}\bm{v}\right)dx\right|=0,$
$\displaystyle\left|\int_{T}\text{curl}\left(\bm{v}-\Pi_{T}\bm{v}\right)dx\right|=0,\qquad\left|\int_{T}\text{curl}\left(\text{curl}\left(\bm{v}-\Pi_{T}\bm{v}\right)\right)dx\right|=0.$

This operator is well-posed, and projects $\bm{v}\in\left[H^{2}(T)\right]^{3}$
on to the kernel space $\mathbf{CK}(T)$.

###### Proof.

Let us introduce the function $\bm{w}\in\mathbf{CK}(T)$, where

(B.2)
$\displaystyle\bm{w}=2\left(\bm{a}\cdot\bm{x}\right)\bm{x}-\left|\bm{x}\right|^{2}\bm{a}+\rho\bm{x}+\bm{Q}\bm{x}+\bm{b},$

and where

(B.3) $\displaystyle\bm{a}=\begin{bmatrix}a_{1}\quad a_{2}\quad
a_{3}\end{bmatrix}^{T},\qquad\bm{b}=\begin{bmatrix}b_{1}\quad b_{2}\quad
b_{3}\end{bmatrix}^{T},\qquad\bm{Q}=\begin{bmatrix}0&q_{1}&q_{2}\\\\[6.45831pt]
-q_{1}&0&q_{3}\\\\[6.45831pt] -q_{2}&-q_{3}&0\end{bmatrix}.$

In order to ensure well-posedness of $\Pi_{T}$, and that the range of its
projection is the appropriate kernel space, we require that $\bm{w}=\bm{0}$
when

(B.4)
$\displaystyle\int_{T}\bm{w}\,dx=\bm{0},\qquad\int_{T}\text{div}(\bm{w})\,dx=0,$
(B.5)
$\displaystyle\int_{T}\text{curl}(\bm{w})\,dx=\bm{0},\qquad\int_{T}\text{curl}(\text{curl}(\bm{w}))\,dx=\bm{0}.$

Fortunately, this relationship is relatively easy to demonstrate. We start by
substituting Eq. (B.2) into the rightmost part of Eq. (B.5) as follows

$\displaystyle\int_{T}\text{curl}(\text{curl}(\bm{w}))\,dx=8\int_{T}\bm{a}\,dx=\bm{0}.$

This immediately implies that $\bm{a}=\bm{0}$. Next, we substitute Eq. (B.2)
into the leftmost part of Eq. (B.5) in order to obtain

$\displaystyle\int_{T}\text{curl}(\bm{w})\,dx=2\int_{T}\begin{bmatrix}-\left(q_{3}+2a_{3}x_{2}-2a_{2}x_{3}\right)\\\\[6.45831pt]
\left(q_{2}+2a_{3}x_{1}-2a_{1}x_{3}\right)\\\\[6.45831pt]
-\left(q_{1}+2a_{2}x_{1}-2a_{1}x_{2}\right)\end{bmatrix}\,dx=2\int_{T}\begin{bmatrix}-q_{3}\\\\[6.45831pt]
q_{2}\\\\[6.45831pt] -q_{1}\end{bmatrix}\,dx=\bm{0}.$

This immediately implies that $\bm{Q}=\bm{0}$. Next, we substitute Eq. (B.2)
into the rightmost part of Eq. (B.4), such that

$\displaystyle\int_{T}\text{div}(\bm{w})\,dx=\int_{T}\left(6\left(a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}\right)+3\rho\right)\,dx=3\int_{T}\rho\,dx=0.$

This immediately implies that $\rho=0$. Finally, we substitute Eq. (B.2) into
the leftmost part of Eq. (B.4) as follows

$\displaystyle\int_{T}\bm{w}\,dx$
$\displaystyle=\int_{T}\begin{bmatrix}b_{1}+q_{1}x_{2}+q_{2}x_{3}+a_{1}\left(x_{1}^{2}-x_{2}^{2}-x_{3}^{2}\right)+x_{1}\left(2a_{2}x_{2}+2a_{3}x_{3}+\rho\right)\\\\[6.45831pt]
b_{2}-q_{1}x_{1}+q_{3}x_{3}-a_{2}\left(x_{1}^{2}-x_{2}^{2}+x_{3}^{2}\right)+x_{2}\left(2a_{1}x_{1}+2a_{3}x_{3}+\rho\right)\\\\[6.45831pt]
b_{3}-q_{2}x_{1}-q_{3}x_{2}-a_{3}\left(x_{1}^{2}+x_{2}^{2}-x_{3}^{2}\right)+x_{3}\left(2a_{1}x_{1}+2a_{2}x_{2}+\rho\right)\end{bmatrix}\,dx$
$\displaystyle=\int_{T}\bm{b}\,dx=\bm{0}.$

This immediately implies that $\bm{b}=\bm{0}$. Therefore, since $\rho=0$,
$\bm{a}=\bm{b}=\bm{0}$, and $\bm{Q}=\bm{0}$, by inspection we have that
$\bm{w}=\bm{0}$. ∎

##  Appendix C Generalized Korn’s Inequality for Tetrahedra

In this section, we derive a generalized Korn’s inequality for tetrahedral
meshes. Prior to introducing this inequality, we establish several useful
definitions along with an important lemma. Thereafter, we summarize the
principal result of the section in a theorem.

######  Definition C.1 (Reformulation of the H2 Seminorm).

Let us begin by redefining the H2 seminorm for vectors
$\bm{v}\in[H^{2}(\Omega)]^{3}$,

$\displaystyle\left|\bm{v}\right|_{H^{2}(\Omega)}=\left(\sum_{i=1}^{3}\int_{\Omega}\left|\text{hess}\left(v_{i}\right)\right|^{2}dx\right)^{1/2},$

where the ‘hess’ operator is the second-derivative Hessian operator which acts
on a generic scalar $\phi\in H^{2}(\Omega)$ and returns a $3\times 3$
symmetric matrix as follows

$\displaystyle\left[\text{hess}(\phi)\right]_{ij}=\frac{\partial^{2}\phi}{\partial
x_{i}\partial x_{j}}.$

######  Definition C.2 (Div-Curl Seminorm with Scaling).

Consider the seminorm $\bm{\Psi}_{\ell}$ for vectors
$\bm{v}\in[H^{2}(\Omega)]^{3}$,

(C.1)
$\displaystyle\bm{\Psi}_{\ell}(\bm{v})=\ell^{a}\left(\left|\int_{\Omega}\text{div}(\bm{v})\,dx\right|+\left|\int_{\Omega}\text{curl}(\bm{v})\,dx\right|\right)+\ell^{b}\left|\int_{\Omega}\text{curl}(\text{curl}(\bm{v}))\,dx\right|,$

where $\ell$ is a generic length scale, and $a$ and $b$ are generic constants.
In 3-dimensional space, it is common practice to set $a=-3/2$ and $b=-1/2$ for
scaling purposes, although this is not strictly necessary.

We are now ready to introduce an important lemma.

######  Lemma C.3.

Consider a vector field defined on the reference element $\widehat{T}$ such
that $\widehat{\bm{v}}\in[H^{2}(\widehat{T})]^{3}$ and

(C.2) $\displaystyle\left\|\widehat{\bm{v}}\right\|_{H^{2}(\widehat{T})}=1.$

Next, suppose that an invertible linear mapping exists between $\widehat{T}$
and any generic tetrahedron $T$

(C.3) $\displaystyle\bm{\alpha}(\widehat{\bm{x}})=\bm{B}\widehat{\bm{x}},$

where $\bm{B}$ belongs to the Lie group of non-singular matrices, $GL(3)$.
Under these circumstances, the following function is continuous on $GL(3)$

(C.4)
$\displaystyle\Lambda_{\widehat{\bm{v}}}(\bm{B})=\frac{\left|\widehat{\bm{v}}\circ\bm{\alpha}^{-1}\right|_{H^{1}(T)}}{\left\|\bm{\epsilon}\left(\widehat{\bm{v}}\circ\bm{\alpha}^{-1}\right)-\frac{1}{3}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}^{-1})\mathbb{I}\right\|_{L_{2}(T)\times
L_{2}(T)}+\bm{\Psi}_{\ell}\left(\widehat{\bm{v}}\circ\bm{\alpha}^{-1}\right)}.$

###### Proof.

In order to prove continuity of the function in Eq. (C.4), our goal is to
construct an upper bound for the following quantity

$\displaystyle\left|\Lambda_{\widehat{\bm{v}}}(\bm{B}_{1})-\Lambda_{\widehat{\bm{v}}}(\bm{B}_{2})\right|$
$\displaystyle=\Bigg{|}\frac{\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}\right|_{H^{1}(T_{1})}}{\left\|\bm{\epsilon}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}\right)-\frac{1}{3}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1})\mathbb{I}\right\|_{L_{2}(T_{1})\times
L_{2}(T_{1})}+\bm{\Psi}_{\ell}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}\right)}$
(C.5)
$\displaystyle-\frac{\left|\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{1}(T_{2})}}{\left\|\bm{\epsilon}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right)-\frac{1}{3}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\mathbb{I}\right\|_{L_{2}(T_{2})\times
L_{2}(T_{2})}+\bm{\Psi}_{\ell}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right)}\Bigg{|},$

in terms of $\left\|\bm{B}_{1}^{-1}-\bm{B}_{2}^{-1}\right\|_{F}$, where
$T_{1}$ and $T_{2}$ are generic tetrahedra, (figure 3 illustrates the
geometric configuration).

Figure 3. Mapping between reference tetrahedra $\widehat{T}$ and arbitrary
tetrahedra $T_{1}$ and $T_{2}$.

Towards this end, we define the following surrogates for the quotient
denominators in Eq. (C.5)

$\displaystyle\mathcal{S}_{1}$
$\displaystyle=\left\|\bm{\epsilon}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}\right)-\frac{1}{3}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1})\mathbb{I}\right\|_{L_{2}(T_{1})\times
L_{2}(T_{1})}+\bm{\Psi}_{\ell}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}\right),$
$\displaystyle\mathcal{S}_{2}$
$\displaystyle=\left\|\bm{\epsilon}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right)-\frac{1}{3}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\mathbb{I}\right\|_{L_{2}(T_{2})\times
L_{2}(T_{2})}+\bm{\Psi}_{\ell}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right).$

Upon substituting the expressions above into Eq. (C.5), and manipulating the
result, we obtain

(C.6)
$\displaystyle\left|\Lambda_{\widehat{\bm{v}}}(\bm{B}_{1})-\Lambda_{\widehat{\bm{v}}}(\bm{B}_{2})\right|$
$\displaystyle=\left|\frac{\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}\right|_{H^{1}(T_{1})}\mathcal{S}_{2}-\left|\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{1}(T_{2})}\mathcal{S}_{1}}{\mathcal{S}_{1}\mathcal{S}_{2}}\right|.$

Next, we observe that the following identity holds for generic real numbers
$c$, $d$, $e$, and $f$

$\displaystyle\left|cd-ef\right|$
$\displaystyle\leq|c||d-f|+|f||c-e|\leq\left(|c|+|f|\right)\left(|d-f|+|c-e|\right).$

On substituting this identity into the numerator of Eq. (C.6), we obtain

(C.7)
$\displaystyle\left|\Lambda_{\widehat{\bm{v}}}(\bm{B}_{1})-\Lambda_{\widehat{\bm{v}}}(\bm{B}_{2})\right|\leq
C\left(\left|\mathcal{S}_{1}-\mathcal{S}_{2}\right|+\left|\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}\right|_{H^{1}(T_{1})}-\left|\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{1}(T_{2})}\right|\right),$

where

$\displaystyle
C=\frac{\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}\right|_{H^{1}(T_{1})}+\mathcal{S}_{1}}{\mathcal{S}_{1}\mathcal{S}_{2}}.$

Now, upon expanding the RHS of Eq. (C.7), and extending the domains of
integration for each seminorm over the domain $T_{1}\cup T_{2}$, we obtain

$\displaystyle\left|\Lambda_{\widehat{\bm{v}}}(\bm{B}_{1})-\Lambda_{\widehat{\bm{v}}}(\bm{B}_{2})\right|$
$\displaystyle\leq
C\Bigg{(}\Bigg{|}\left\|\bm{\epsilon}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}\right)-\frac{1}{3}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1})\mathbb{I}\right\|_{L_{2}(T_{1}\cup
T_{2})\times L_{2}(T_{1}\cup T_{2})}$
$\displaystyle-\left\|\bm{\epsilon}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right)-\frac{1}{3}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\mathbb{I}\right\|_{L_{2}(T_{1}\cup
T_{2})\times L_{2}(T_{1}\cup T_{2})}\Bigg{|}$
$\displaystyle+\ell^{a}\left|\left|\int_{T_{1}\cup
T_{2}}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1})\,dx\right|-\left|\int_{T_{1}\cup
T_{2}}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\,dx\right|\right|$
$\displaystyle+\ell^{a}\left|\left|\int_{T_{1}\cup
T_{2}}\text{curl}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1})\,dx\right|-\left|\int_{T_{1}\cup
T_{2}}\text{curl}(\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\,dx\right|\right|$
$\displaystyle+\ell^{b}\left|\left|\int_{T_{1}\cup
T_{2}}\text{curl}(\text{curl}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}))\,dx\right|-\left|\int_{T_{1}\cup
T_{2}}\text{curl}(\text{curl}(\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}))\,dx\right|\right|$
$\displaystyle+\left|\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}\right|_{H^{1}(T_{1}\cup
T_{2})}-\left|\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{1}(T_{1}\cup
T_{2})}\right|\Bigg{)}$

and furthermore

$\displaystyle\left|\Lambda_{\widehat{\bm{v}}}(\bm{B}_{1})-\Lambda_{\widehat{\bm{v}}}(\bm{B}_{2})\right|$
$\displaystyle\leq
C\Bigg{(}\left\|\bm{\epsilon}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right)-\frac{1}{3}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\mathbb{I}\right\|_{L_{2}(T_{1}\cup
T_{2})\times L_{2}(T_{1}\cup T_{2})}$
$\displaystyle+\ell^{a}\left(\left|\int_{T_{1}\cup
T_{2}}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\,dx\right|+\left|\int_{T_{1}\cup
T_{2}}\text{curl}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\,dx\right|\right)$
(C.8) $\displaystyle+\ell^{b}\left|\int_{T_{1}\cup
T_{2}}\text{curl}(\text{curl}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}))\,dx\right|+\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{1}(T_{1}\cup
T_{2})}\Bigg{)}.$

Here, we have used the reverse triangle inequality on the last three lines.
Next, the first term on the RHS of Eq. (C.8) can be treated as follows

$\displaystyle\left\|\bm{\epsilon}\left(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right)-\frac{1}{3}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\mathbb{I}\right\|_{L_{2}(T_{1}\cup
T_{2})\times L_{2}(T_{1}\cup T_{2})}$ (C.9) $\displaystyle\leq
C\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{1}(T_{1}\cup
T_{2})},$

in accordance with Lemma A.1. In addition, we have that

$\displaystyle\left|\int_{T_{1}\cup
T_{2}}\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\,dx\right|$
$\displaystyle\leq\int_{T_{1}\cup
T_{2}}\left|\text{div}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\right|\,dx$
(C.10) $\displaystyle\leq C\sqrt{\left|T_{1}\cup
T_{2}\right|}\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{1}(T_{1}\cup
T_{2})},$

where we have used Hölder’s inequality. Next, following similar arguments, we
obtain

(C.11) $\displaystyle\left|\int_{T_{1}\cup
T_{2}}\text{curl}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\,dx\right|$
$\displaystyle\leq C\sqrt{\left|T_{1}\cup
T_{2}\right|}\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{1}(T_{1}\cup
T_{2})},$ (C.12) $\displaystyle\left|\int_{T_{1}\cup
T_{2}}\text{curl}(\text{curl}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}))\,dx\right|$
$\displaystyle\leq C\sqrt{\left|T_{1}\cup
T_{2}\right|}\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{2}(T_{1}\cup
T_{2})}.$

Now, on substituting Eqs. (C.9)–(C.12) into Eq. (C.8), we obtain

$\displaystyle\left|\Lambda_{\widehat{\bm{v}}}(\bm{B}_{1})-\Lambda_{\widehat{\bm{v}}}(\bm{B}_{2})\right|$
$\displaystyle\leq C\Bigg{(}\left(1+\ell^{a}\sqrt{\left|T_{1}\cup
T_{2}\right|}\right)\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{1}(T_{1}\cup
T_{2})}$ (C.13) $\displaystyle+\ell^{b}\sqrt{\left|T_{1}\cup
T_{2}\right|}\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{2}(T_{1}\cup
T_{2})}\Bigg{)}.$

The expression above can be further simplified by using the following chain-
rule identities

(C.14) $\displaystyle\text{grad}(\widehat{\bm{v}}\circ\bm{\alpha}^{-1})$
$\displaystyle=\left(\widehat{\text{grad}}(\widehat{\bm{v}})\circ\bm{\alpha}^{-1}\right)\bm{B}^{-1},$
(C.15) $\displaystyle\text{hess}(\widehat{v}_{i}\circ\bm{\alpha}^{-1})$
$\displaystyle=\bm{B}^{-T}\left(\widehat{\text{hess}}(\widehat{v}_{i})\circ\bm{\alpha}^{-1}\right)\bm{B}^{-1},$
(C.16) $\displaystyle\int_{\widehat{T}}\widehat{\phi}\,d\widehat{x}$
$\displaystyle=\int_{T}\left(\widehat{\phi}\circ\bm{\alpha}^{-1}\right)\text{det}(\bm{B}^{-1})\,dx,$

where $\widehat{\phi}$ is a generic scalar function of $\widehat{\bm{x}}$ and

$\displaystyle\left[\widehat{\text{grad}}(\widehat{\phi})\right]_{i}=\frac{\partial\widehat{\phi}}{\partial\widehat{x}_{i}},\qquad\left[\widehat{\text{hess}}(\widehat{\phi})\right]_{ij}=\frac{\partial^{2}\widehat{\phi}}{\partial\widehat{x}_{i}\partial\widehat{x}_{j}}.$

Based on these identities, we obtain the following upper bound for the H1 term
in Eq. (C.13)

$\displaystyle\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{1}(T_{1}\cup
T_{2})}$ $\displaystyle=\left(\int_{T_{1}\cup
T_{2}}\left|\text{grad}(\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1})\right|^{2}dx\right)^{1/2}$
$\displaystyle=\left(\int_{T_{1}\cup
T_{2}}\left|\left(\widehat{\text{grad}}(\widehat{\bm{v}})\circ\bm{\alpha}_{1}^{-1}\right)\bm{B}_{1}^{-1}-\left(\widehat{\text{grad}}(\widehat{\bm{v}})\circ\bm{\alpha}_{2}^{-1}\right)\bm{B}_{2}^{-1}\right|^{2}dx\right)^{1/2}$
$\displaystyle\leq
C\Bigg{(}\int_{T_{1}}\left|\left(\widehat{\text{grad}}(\widehat{\bm{v}})\circ\bm{\alpha}_{1}^{-1}\right)\left(\bm{B}_{1}^{-1}-\bm{B}_{2}^{-1}\right)\right|^{2}dx+\int_{T_{2}\setminus
T_{1}}\left|\left(\widehat{\text{grad}}(\widehat{\bm{v}})\circ\bm{\alpha}_{2}^{-1}\right)\left(\bm{B}_{1}^{-1}-\bm{B}_{2}^{-1}\right)\right|^{2}dx\Bigg{)}^{1/2}$
$\displaystyle\leq
C\Bigg{(}\frac{1}{\text{det}(\bm{B}_{1}^{-1})}\int_{\widehat{T}}\left|\widehat{\text{grad}}(\widehat{\bm{v}})\left(\bm{B}_{1}^{-1}-\bm{B}_{2}^{-1}\right)\right|^{2}d\widehat{x}+\frac{1}{\text{det}(\bm{B}_{2}^{-1})}\int_{\widehat{T}}\left|\widehat{\text{grad}}(\widehat{\bm{v}})\left(\bm{B}_{1}^{-1}-\bm{B}_{2}^{-1}\right)\right|^{2}d\widehat{x}\Bigg{)}^{1/2}$
$\displaystyle\leq
C\left(\frac{1}{\text{det}(\bm{B}_{1}^{-1})}+\frac{1}{\text{det}(\bm{B}_{2}^{-1})}\right)^{1/2}\left(\int_{\widehat{T}}\left|\widehat{\text{grad}}(\widehat{\bm{v}})\left(\bm{B}_{1}^{-1}-\bm{B}_{2}^{-1}\right)\right|^{2}d\widehat{x}\right)^{1/2}$
$\displaystyle\leq
C\left(\frac{1}{\text{det}(\bm{B}_{1}^{-1})}+\frac{1}{\text{det}(\bm{B}_{2}^{-1})}\right)^{1/2}\left|\widehat{\bm{v}}\right|_{H^{1}(\widehat{T})}\left\|\bm{B}_{1}^{-1}-\bm{B}_{2}^{-1}\right\|_{\infty}$
(C.17) $\displaystyle\leq
C\left(\frac{1}{\text{det}(\bm{B}_{1}^{-1})}+\frac{1}{\text{det}(\bm{B}_{2}^{-1})}\right)^{1/2}\left\|\bm{B}_{1}^{-1}-\bm{B}_{2}^{-1}\right\|_{F}.$

In the above equations, we have used the triangle inequality, root-mean-
square-arithmetic-mean inequality, Hölder’s inequality, and the equivalence of
norms in a finite-dimensional space.

In a similar fashion, for the H2 term in Eq. (C.13) we obtain the following

(C.18)
$\displaystyle\left|\widehat{\bm{v}}\circ\bm{\alpha}_{1}^{-1}-\widehat{\bm{v}}\circ\bm{\alpha}_{2}^{-1}\right|_{H^{2}(T_{1}\cup
T_{2})}\leq
C\left(\frac{1}{\text{det}(\bm{B}_{1}^{-1})}+\frac{1}{\text{det}(\bm{B}_{2}^{-1})}\right)^{1/2}\left\|\bm{B}_{1}^{-1}-\bm{B}_{2}^{-1}\right\|_{F}^{2}.$

Upon substituting Eqs. (C.17) and (C.18) into Eq. (C.13), we obtain

$\displaystyle\left|\Lambda_{\widehat{\bm{v}}}(\bm{B}_{1})-\Lambda_{\widehat{\bm{v}}}(\bm{B}_{2})\right|$
$\displaystyle\leq
C\left(\frac{1}{\text{det}(\bm{B}_{1}^{-1})}+\frac{1}{\text{det}(\bm{B}_{2}^{-1})}\right)^{1/2}$
(C.19) $\displaystyle\times\Bigg{(}1+\ell^{a}\sqrt{\left|T_{1}\cup
T_{2}\right|}+\ell^{b}\sqrt{\left|T_{1}\cup
T_{2}\right|}\left\|\bm{B}_{1}^{-1}-\bm{B}_{2}^{-1}\right\|_{F}\Bigg{)}\left\|\bm{B}_{1}^{-1}-\bm{B}_{2}^{-1}\right\|_{F}.$

This completes the proof. ∎

We are now ready to present the final result of this section.

######  Theorem C.4.

Suppose that $\mathcal{T}$ is a valid mesh with non-degenerate elements. Then,
for all $\bm{v}\in\left[H^{2}(T)\right]^{3}$ and $T\in\mathcal{T}$, the
following Korn’s inequality holds

(C.20)
$\displaystyle\left|\bm{v}\right|_{H^{1}(T)}\leq\kappa(\theta_{T})\Bigg{[}$
$\displaystyle\left\|\bm{\epsilon}(\bm{v})-\frac{1}{3}\text{div}(\bm{v})\mathbb{I}\right\|_{L_{2}(T)\times
L_{2}(T)}+\ell^{a}\left|\int_{T}\text{div}(\bm{v})\,dx\right|$
$\displaystyle+\ell^{a}\left|\int_{T}\text{curl}(\bm{v})\,dx\right|+\ell^{b}\left|\int_{T}\text{curl}(\text{curl}(\bm{v}))\,dx\right|\Bigg{]},$

where $\ell$ is a characteristic length scale associated with $\mathcal{T}$,
$a$ and $b$ are constants that depend on the number of dimensions (3), and
$\kappa:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a continuous, decreasing
function of the minimum angle of each tetrahedron, $\theta_{T}$.

###### Proof.

We begin by introducing the following inequality

(C.21) $\displaystyle\left|\bm{v}\right|_{H^{1}(T)}\leq
k(T)\left(\left\|\bm{\epsilon}\left(\bm{v}\right)-\frac{1}{3}\text{div}\left(\bm{v}\right)\mathbb{I}\right\|_{L_{2}(T)\times
L_{2}(T)}+\bm{\Psi}_{\ell}\left(\bm{v}\right)\right),\qquad\forall\bm{v}\in\left[H^{2}(T)\right]^{3},$

where $k(T)$ is the smallest possible number such that the inequality above
holds. We note that this inequality is guaranteed to hold for a suitable
family of constants, in accordance with Lemma A.5.

Following the arguments in [6], we introduce a mapping between the physical
domain $T$ and a reference domain $\widehat{T}$, such that
$\bm{\alpha}:\widehat{T}\rightarrow T$, where

$\displaystyle\bm{\alpha}(\widehat{\bm{x}})=\bm{B}\widehat{\bm{x}}+\bm{b}.$

Here, $\bm{B}$ is a matrix which belongs to the Lie group of non-singular
matrices $GL(3)$, (as we discussed previously), and $\bm{b}$ is a constant
vector. If we assume without loss of generality that $\bm{b}=\bm{0}$, then we
have that $k(T)=k(\bm{B})$. It then follows that

$\displaystyle
k(\bm{B})=\sup_{\begin{subarray}{c}\widehat{\bm{v}}\in\left[H^{2}(\widehat{T})\right]^{3}\\\
\left\|\widehat{\bm{v}}\right\|_{H^{2}(\widehat{T})}=1\end{subarray}}\left(\frac{\left|\widehat{\bm{v}}\circ\bm{\alpha}^{-1}\right|_{H^{1}(T)}}{\left\|\bm{\epsilon}\left(\widehat{\bm{v}}\circ\bm{\alpha}^{-1}\right)-\frac{1}{3}\text{div}\left(\widehat{\bm{v}}\circ\bm{\alpha}^{-1}\right)\mathbb{I}\right\|_{L_{2}(T)\times
L_{2}(T)}+\bm{\Psi}_{\ell}\left(\widehat{\bm{v}}\circ\bm{\alpha}^{-1}\right)}\right).$

We note that Lemma C.3 in conjunction with the fact that
$\left\|\widehat{\bm{v}}\right\|_{H^{2}(\widehat{T})}=1$ imply that the RHS of
the equation above represents a family of equicontinuous functions on $GL(3)$.
It immediately follows that the supremum of this family $k(\cdot)$ is
continuous on $GL(3)$, (see [15] for details).

The argument above holds for a generic element $T\in\mathcal{T}$. Of course,
the mesh is usually composed of many such elements, and it remains for us to
generalize our result to the entire mesh. Therefore, we consider the set
$\left\\{T_{i}:i\in I\right\\}$, the family of tetrahedral elements which are
affine homeomorphic to the reference element $\widehat{T}$, and we assume that
all elements in the mesh belong to this set. Furthermore, we assume that the
aspect ratios of all elements in the set are uniformly bounded, and (without
loss of generality) that $\text{diam}(T_{i})=1$ for all $i$. Under these
circumstances, the following inequalities hold in accordance with Theorem
3.1.3 of [10]

$\displaystyle\left\|\bm{B}_{i}\right\|\leq
C,\qquad\left\|\bm{B}^{-1}_{i}\right\|\leq C,$

where $\left\|\cdot\right\|$ is the matrix 2-norm for the element-specific
transformation matrix $\bm{B}_{i}$. The inequalities above imply that the set
of matrices $\left\\{\bm{B}_{i}:i\in I\right\\}$ is a precompact subset of
$GL(3)$. In turn, the continuity of $k(\cdot)$ on $GL(3)$ ensures the
boundedness of $\left\\{k(\bm{B}_{i}):i\in I\right\\}$. As a result, the
following inequality holds

$\displaystyle\left|\bm{v}\right|_{H^{1}(T_{i})}\leq
C\left(\left\|\bm{\epsilon}\left(\bm{v}\right)-\frac{1}{3}\text{div}\left(\bm{v}\right)\mathbb{I}\right\|_{L_{2}(T_{i})\times
L_{2}(T_{i})}+\bm{\Psi}_{\ell}\left(\bm{v}\right)\right),\qquad\forall\bm{v}\in\left[H^{2}(T_{i})\right]^{3},$

where $C$ is a constant independent of the particular tetrahedron $T_{i}$.

Finally, the constant $k(\bm{B})$ can be expressed in terms of the minimum
angle, $\theta_{T}$ of $T$. In a natural fashion, we can define the following
set

$\displaystyle S_{\theta}=\left\\{k(T):T\,\text{is a tetrahedron and the
minimum angle of}\;T\,\text{is}\geq\theta>0\right\\}.$

This set is bounded because $k(\bm{B})$ is bounded (as discussed above). We
can define $\eta(\theta)=\sup S_{\theta}$, which is a non-negative, decreasing
function of $\theta$, where $\eta:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$.
Now, in accordance with Eq. (C.21), we have that

(C.22)
$\displaystyle\left|\bm{v}\right|_{H^{1}(T)}\leq\eta(\theta_{T})\left(\left\|\bm{\epsilon}\left(\bm{v}\right)-\frac{1}{3}\text{div}\left(\bm{v}\right)\mathbb{I}\right\|_{L_{2}(T)\times
L_{2}(T)}+\bm{\Psi}_{\ell}\left(\bm{v}\right)\right),\qquad\forall\bm{v}\in\left[H^{2}(T)\right]^{3}.$

We complete the proof by defining a continuous, decreasing function
$\kappa(\theta_{T})\geq\eta(\theta_{T})$, and substituting this function into
Eq. (C.22) in order to obtain the desired result (Eq. (C.20)). ∎

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