# Mixed Isogeometric Discretizations for Planar Linear Elasticity

Jeremias Arf and Bernd Simeon E-mail: {arf<EMAIL_ADDRESS>
 _2020 Mathematics Subject Classification. 65N30; 65N12_ ,
_Keywords._ Linear Elasticity, B-splines, Mixed Formulation.  TU
Kaiserslautern, Germany

###### Abstract

In this article we suggest two discretization methods based on isogeometric
analysis (IGA) for planar linear elasticity. On the one hand, we apply the
well-known ansatz of weakly imposed symmetry for the stress tensor and obtain
a well-posed mixed formulation. Such modified mixed problems have been already
studied by different authors. But we concentrate on the exploitation of IGA
results to handle also curved boundary geometries. On the other hand, we
consider the more complicated situation of strong symmetry, i.e. we discretize
the mixed weak form determined by the so-called Hellinger-Reissner variational
principle. We show the existence of suitable approximate fields leading to an
inf-sup stable saddle-point problem. For both discretization approaches we
prove convergence statements and in case of weak symmetry we illustrate the
approximation behavior by means of several numerical experiments.

##  1 Introduction

In this article we identify a bounded Lipschitz domain
$\Omega\subset\mathbb{R}^{2}$ with some elastic body that is loaded according
to some body force $\boldsymbol{f}$. The deformation of the body can be
expressed through the displacement field
$\boldsymbol{u}\colon\Omega\rightarrow\mathbb{R}^{2}$ that measures the
difference between current configuration $\Omega+\boldsymbol{u}$ and initial
configuration $\Omega$. Governing equations guaranteeing a physically
reasonable description of the deformation process are given by the theory of
linear elasticity and read:

$\displaystyle\boldsymbol{A}\boldsymbol{\sigma}$
$\displaystyle=\boldsymbol{\varepsilon}(\boldsymbol{u})\hskip
5.69046pt\textup{and}$ (1) $\displaystyle\nabla\cdot\boldsymbol{\sigma}$
$\displaystyle=\boldsymbol{f}\hskip 2.84544pt\textup{in}\ \ \Omega,$ (2)

where
$\boldsymbol{\varepsilon}(\boldsymbol{u})=\frac{1}{2}(\nabla\boldsymbol{u}+\nabla\boldsymbol{u}^{T})$
denotes the symmetric gradient that corresponds to the mechanical strain
within the infinitesimal strain theory. Further,
$\boldsymbol{A}=\boldsymbol{A}(\boldsymbol{x})\colon\mathbb{S}\rightarrow\mathbb{S},\
\boldsymbol{x}\in\Omega$ is the compliance tensor which comprises the material
properties and $\boldsymbol{\sigma}\colon\Omega\rightarrow\mathbb{S}$ is the
symmetric stress tensor, where $\mathbb{S}$ stands for the space of symmetric
matrices in $\mathbb{R}^{2\times 2}$. We note that the linear relation between
strain and stress as given in equation (1) is the key assumption in the
linearized framework. A classical conservation of momentum argument relates
the stress tensor and the force $\boldsymbol{f}$ and as a result one obtains
equation (2). An important special case are homogeneous and isotropic
materials. Then the compliance tensor simplifies to

$\boldsymbol{A}\boldsymbol{\sigma}=\frac{1}{2\mu}\Big{(}\boldsymbol{\sigma}-\frac{\lambda}{2\lambda+2\mu}\textup{tr}(\boldsymbol{\sigma})\boldsymbol{I}\Big{)},$

and the material properties are condensed in form of so-called Lamé parameters
$\lambda\geq 0,\ \mu>0$. In all sections below we assume $\boldsymbol{A}$ to
be symmetric, positive definite and bounded w.r.t. the $L^{2}$-norm. Actually,
we have to specify some boundary conditions to complete the model, where we
first consider the easy case of pure displacement conditions, i.e. we have
$\boldsymbol{u}=\boldsymbol{u}_{D}$ on $\Gamma\coloneqq\partial\Omega$ for
some given $\boldsymbol{u}_{D}$. Supposing the invertibility of
$\boldsymbol{A}$ one can obtain a formulation which depends only on the
displacement variable. A combination of (1) and (2) yields

$\nabla\cdot(\boldsymbol{C}\boldsymbol{\varepsilon}(\boldsymbol{u}))=\boldsymbol{f},\
\ \ \boldsymbol{C}\coloneqq\boldsymbol{{A}}^{-1},$ (3)

with $\boldsymbol{C}$ as the elasticity tensor. The standard weak form of the
mentioned equation in case of homogeneous boundary conditions is the primal
formulation:

$\textup{Find}\ \ \boldsymbol{u}\in\boldsymbol{H}_{0}^{1}(\Omega)\ \
\textup{s.t.}\ \
\langle\boldsymbol{C}\boldsymbol{\varepsilon}(\boldsymbol{u}),\boldsymbol{\varepsilon}(\boldsymbol{v})\rangle=\langle\boldsymbol{f},\boldsymbol{v}\rangle,\
\ \forall\boldsymbol{v}\in\boldsymbol{H}_{0}^{1}(\Omega),$ (4)

with $\boldsymbol{H}_{0}^{1}(\Omega)$ denoting the Sobolev space with square-
integrable weak derivatives, square-
integrable component functions and zero boundary values. Further
$\langle\cdot,\cdot\rangle$ denotes the $L^{2}$ inner product. Unfortunately,
the primal formulation becomes unstable for incompressible problems, i.e. if
$\lambda\rightarrow\infty$. Another point is the non-locality of stress-strain
relations in more general material situations. Consequently, the primal weak
formulation is not optimal also in view of generalizability. This is why we
find in literature mixed formulations that seek for stress and displacement
simultaneously. A standard mixed formulation reads:

Find
$\boldsymbol{\sigma}\in\boldsymbol{\mathcal{{H}}}(\Omega,\textup{div},\mathbb{S}),$
$\boldsymbol{u}\in\boldsymbol{L}^{2}(\Omega)$ s.t.

$\displaystyle\langle\textbf{A}\boldsymbol{\sigma},\boldsymbol{\tau}\rangle\
+\ $ $\displaystyle\langle\boldsymbol{u},\nabla\cdot\boldsymbol{\tau}\rangle\
\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle=\langle\boldsymbol{\tau}\cdot\textbf{n};\boldsymbol{u}_{D}\rangle_{\Gamma},\
\hskip 11.38092pt$
$\displaystyle\forall\boldsymbol{\tau}\in\boldsymbol{\mathcal{{H}}}(\Omega,\textup{div},\mathbb{S}),$
(5) $\displaystyle\langle\nabla\cdot\boldsymbol{\sigma},\boldsymbol{v}\rangle$
$\displaystyle=\langle\boldsymbol{f},\boldsymbol{v}\rangle,\ \hskip
11.38092pt$ $\displaystyle\forall\boldsymbol{v}\in\boldsymbol{L}^{2}(\Omega).$

Above
$\langle\boldsymbol{\tau}\cdot\textbf{n};\boldsymbol{u}_{D}\rangle_{\Gamma}$
denotes the duality pairing determined by the normal trace which corresponds
to the $L^{2}(\Gamma)$ inner product on the boundary if
$\boldsymbol{\mathcal{\tau}}$ is regular enough. Furthermore, we wrote
$\boldsymbol{\mathcal{{H}}}(\Omega,\textup{div},\mathbb{S})$ for the space of
square-integrable symmetric matrix fields which have a square-integrable
divergence and $\boldsymbol{L}^{2}(\Omega)$ denotes the space of square-
integrable vector fields. As explained in [6], background of this saddle-point
problem is the _Hellinger-Reissner_ functional where
$(\boldsymbol{u},\boldsymbol{\sigma})$ is given as the corresponding unique
critical point. Regarding the discretization of the mixed weak system there
are two main ways which are frequently used. On the one hand, one can try to
define discrete spaces
$\boldsymbol{\mathcal{V}}_{h}\subset\boldsymbol{\mathcal{H}}(\textup{div},\Omega,\mathbb{S}),\
\boldsymbol{V}_{h}\subset\boldsymbol{L}^{2}(\Omega)$ to approximate the stress
tensor and the displacement. However, the choice of suitable subspaces is not
trivial. To obtain a well-posed numerical method the discretized saddle-point
problem has to satisfy _Brezzi_ ’s stability conditions (see [9]). It becomes
apparent that the symmetry condition and divergence compatibility for
$\boldsymbol{\mathcal{V}}_{h}$ is hard to achieve simultaneously, especially
in case of more complex computational domains $\Omega$. Methods that follow
the strong symmetry ansatz can be found e.g. in [4] and [7]. However, the
mentioned discretizations are quite complicated. For example in [7] one needs
at least $24$ degrees of freedom per triangle element. Therefore, on the other
hand authors in several articles (e.g. [6, 23, 5, 15]) study a modified mixed
problem which guarantees symmetry only in a variational sense. Such
formulations with weakly imposed symmetry are related to the de Rham complex
and the elegant theory of Finite Element Exterior Calculus, see [6] and [15],
but require an additional Lagrange multiplier.

In this article we shall consider both ways of discretization, namely strong
and weak symmetry conservation. Our aim is to show that there are discrete
spaces satisfying the Brezzi conditions and thus lead to well-posedness. In
contrast to classical Finite Elements (FEM) based on simplicial decompositions
or quadrilateral elements we use B-splines and results from Isogeometric
Analysis (IGA) to define test and trial fields. Advantages of IGA are the
possibility to consider geometries with curved boundaries and the fact that
the global regularity of the B-splines can be increased efficiently. Such
special features are useful if the method should be applicable in more complex
scenarios. Although our ansatz for strongly imposed symmetry is only valid in
the so-called single-patch case, it is capable to handle domains with curved
boundaries. Nevertheless, for the weak symmetry approach the generalization to
multi-patch situations is possible.

The paper is structured as follows. In Section 2 we introduce briefly some
mathematical notation and basic notions in the context of IGA. Next, in
Section 3 we explain the mixed formulation of linear elasticity with weakly
imposed symmetry and consider spline spaces that are suitable for
discretization in planar domains. In particular, we establish a corresponding
error estimate. Afterwards, in Section 4 we face the $2D$ case with strong
symmetry, where we define likewise proper discrete spaces. Section 5 is
dedicated to numerical examples to validate the theoretical statements, where
we restrict ourselves to the weak symmetry case. A detailed numerical study of
the strong symmetry ansatz should be the issue of a further article. We close
with a short conclusion in Section 6.

##  2 Mathematical preliminaries and notation

###  2.1 Mathematical notation

In the section we introduce some notation and define several spaces.
For some bounded Lipschitz domain $D\subset\mathbb{R}^{d}$ we write for the
standard Sobolev spaces $H^{0}(D)=L^{2}(D)$ and $\ H^{k}(D),\ k\in\mathbb{N}$,
where $L^{2}(D)$ stands for the Hilbert space of square-integrable functions
endowed with the inner product $\langle\cdot,\cdot\rangle$. The norms
$\left\lVert\cdot\right\rVert_{H^{k}(D)}$ denote the classical Sobolev norms
in $H^{k}(D)$. In case of vector or matrix fields we can define Sobolev spaces
by requiring the component functions to be in suitable Sobolev spaces. To
distinguish latter case from the scalar-valued one, we use a bold-type
notation. For example we have for $\boldsymbol{v}\coloneqq(v_{1},v_{2})^{T},\
\boldsymbol{v}\in\boldsymbol{H}^{k}(D)\colon\Leftrightarrow\ v_{i}\in
H^{k}(\Omega),\,\forall i$ and $\boldsymbol{M}\coloneqq\big{(}M_{ij}\big{)},\
\boldsymbol{M}\in\boldsymbol{\mathcal{H}}^{k}(D)\colon\Leftrightarrow\
M_{ij}\in H^{k}(\Omega),\,\forall i,j$ and we define the norms

$\displaystyle\left\lVert\boldsymbol{v}\right\rVert_{\boldsymbol{H}^{k}(D)}^{2}\coloneqq\sum_{i}\left\lVert
v_{i}\right\rVert_{H^{k}(D)}^{2},\ \ \ \ \ \ \
\left\lVert\boldsymbol{M}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}(D)}^{2}\coloneqq\sum_{i,j}\left\lVert
M_{ij}\right\rVert_{{H}^{k}(D)}^{2}.$

We note that the inner product $\langle\cdot,\cdot\rangle$ introduces
straightforwardly an inner product on
$\boldsymbol{L}^{2}(D)=\boldsymbol{H}^{0}(D)$ and $\
\boldsymbol{\mathcal{L}}^{2}(D)=\boldsymbol{\mathcal{H}}^{0}(\Omega)$ which
will we denote with $\langle\cdot,\cdot\rangle$, too. Now let us consider
vector-valued mappings and set

$\displaystyle\boldsymbol{H}(D,\textup{curl})$
$\displaystyle\coloneqq\\{\boldsymbol{v}\in\boldsymbol{L}^{2}(D)\ |\
\nabla\times\boldsymbol{v}\in\boldsymbol{L}^{2}(D)\\},\ \hskip 8.5359pt$
$\displaystyle\left\lVert\boldsymbol{v}\right\rVert_{\boldsymbol{H}(D,\textup{curl})}^{2}\coloneqq\left\lVert\boldsymbol{v}\right\rVert_{\boldsymbol{L}^{2}(D)}^{2}+\left\lVert\nabla\times\boldsymbol{v}\right\rVert_{\boldsymbol{L}^{2}(D)}^{2},$
$\displaystyle\boldsymbol{H}(D,\textup{div})$
$\displaystyle\coloneqq\\{\boldsymbol{v}\in\boldsymbol{L}^{2}(D)\ |\
\nabla\cdot\boldsymbol{v}\in{L}^{2}(D)\\},\ $
$\displaystyle\left\lVert\boldsymbol{v}\right\rVert_{\boldsymbol{H}(D,\textup{div})}^{2}\coloneqq\left\lVert\boldsymbol{v}\right\rVert_{\boldsymbol{L}^{2}(D)}^{2}+\left\lVert\nabla\cdot\boldsymbol{v}\right\rVert_{{L}^{2}(D)}^{2}.$

Above we wrote $\nabla$ for the classical nabla operator. The definition for
$\boldsymbol{H}(D,\textup{div})$ and the corresponding norm can be generalized
to the matrix setting, namely $\ \boldsymbol{\mathcal{H}}(D,\textup{div})$, by
a row-wise definition. In particular, the divergence $\nabla\cdot$ acts row-
wise, too. On $\boldsymbol{H}(D,\textup{div})$ and thus on
$\boldsymbol{\mathcal{H}}(D,\textup{div})$ one can introduce a _normal trace
operator_ $\gamma_{n}$; see Lemma 7. If
$\boldsymbol{\sigma}\in\boldsymbol{\mathcal{H}}(D,\textup{div})$, we write
$\langle\boldsymbol{\sigma}\cdot\boldsymbol{n};\cdot\rangle_{\Gamma}$ for the
underlying duality pairing , where $\boldsymbol{n}$ stands for the outer unit
normal to the domain boundary. We denote the space of square-integrable
symmetric and skew-symmetric matrix fields with
$\boldsymbol{\mathcal{L}}^{2}(D,\mathbb{S})$ and
$\boldsymbol{\mathcal{L}}^{2}(D,\mathbb{K})$ and later the next operators
become useful.

$\displaystyle\textup{Skew}(q)$ $\displaystyle\coloneqq\begin{pmatrix}0&-q\\\
q&0\end{pmatrix}\in\boldsymbol{\mathcal{L}}^{2}(D,\mathbb{K}),\ \ \ q\in
L^{2}(D),$ $\displaystyle\textup{SYM}(s_{1},s_{2},s_{3})$ $\displaystyle\colon
D\rightarrow\mathbb{R}^{2\times 2},\ \
\boldsymbol{x}\mapsto\begin{pmatrix}s_{1}(\boldsymbol{x})&s_{2}(\boldsymbol{x})\\\
s_{2}(\boldsymbol{x})&s_{3}(\boldsymbol{x})\end{pmatrix},$

if $s_{i}\colon D\rightarrow\mathbb{R}$. For vector spaces we likewise use the
formal notation $\textup{SYM}()$, e.g.
$\textup{SYM}(L^{2}(D),\dots,L^{2}(D))=\boldsymbol{\mathcal{L}}^{2}(D,\mathbb{S})$.
Besides we define

$\displaystyle\boldsymbol{\mathcal{H}}(D,\textup{div},\mathbb{S})$
$\displaystyle\coloneqq\\{\boldsymbol{S}\in\boldsymbol{\mathcal{L}}^{2}(D,\mathbb{S})\
|\ \nabla\cdot\boldsymbol{S}\in\boldsymbol{L}^{2}(D)\\}.$

Further, for a matrix $\boldsymbol{M}=(M_{ij})$ we use an upper index
$\boldsymbol{M}^{j}$ to denote the $j$-th column and a lower index
$\boldsymbol{M}_{i}$ for the $i$-th row. With $C^{k}(D)$ we denote the space
of continuous functions which have continuous derivatives up to order $k$. If
we have vector or matrix fields we utilize again the notation $C^{k}(D)$ and
mean that all components are $C^{k}$-regular. Finally, we introduce the $2D$
curl operator through
$\textup{curl}(v)\coloneqq(\partial_{2}v,-\partial_{1}v)^{T},\ v\in H^{1}(D)$.

After stating some basic notation we proceed with the consideration of
B-splines.

###  2.2 B-splines

Here, we state a short overview of B-spline functions, spaces respective, and
some related basic results.
Following [12, 8] for a brief exposition, we call an increasing sequence of
real numbers $\Xi\coloneqq\\{\xi_{1}\leq\xi_{2}\leq\dots\leq\xi_{n+p+1}\\}$
for some $p\in\mathbb{N}$ _knot vector_ , where we assume
$0=\xi_{1}=\xi_{2}=\dots=\xi_{p+1},\ \xi_{n+1}=\xi_{n+2}=\dots=\xi_{n+p+1}=1$,
and call such knot vectors $p$-open. Further, the multiplicity of the $j$-th
knot is denoted by $m(\xi_{j})$ and we require $m(\xi_{j})\leq p+1$. Then the
univariate B-spline functions $\widehat{B}_{j,p}(\cdot)$ of degree $p$
corresponding to a given knot vector $\Xi$ are defined recursively by the
_Cox-DeBoor formula_ :

$\displaystyle\widehat{B}_{j,0}(\zeta)\coloneqq\begin{cases}1,\ \ \textup{if}\
\zeta\in[\xi_{j},\xi_{j+1})\\\ 0,\ \ \textup{else},\end{cases}$ (6)

and if $p\in\mathbb{N}_{\geq 1}\ \textup{we set}$

$\displaystyle\widehat{B}_{j,p}(\zeta)\coloneqq\frac{\zeta-\xi_{j}}{\xi_{j+p}-\xi_{j}}\widehat{B}_{j,p-1}(\zeta)+\frac{\xi_{j+p+1}-\zeta}{\xi_{j+p+1}-\xi_{j+1}}\widehat{B}_{j+1,p-1}(\zeta),$
(7)

where one puts $0/0=0$ to obtain well-definedness. The knot vector $\Xi$
without knot repetitions is denoted by $\\{\psi_{1},\dots,\psi_{m}\\}$.
The multivariate extension of the last spline definition is achieved by a
tensor product construction. In other words, we set for a given tensor knot
vector $\boldsymbol{\Xi}\coloneqq\Xi_{1}\times\dots\times\Xi_{d}$, where the
$\Xi_{l}=\\{\xi_{1}^{l},\dots,\xi_{n_{l}+p_{l}+1}^{l}\\},\ l=1,\dots,d$ are
$p_{l}$-open, and a given _degree vector_
$\boldsymbol{p}\coloneqq(p_{1},\dots,p_{d})$ for the multivariate case

$\displaystyle\widehat{B}_{\boldsymbol{i},\boldsymbol{p}}(\boldsymbol{\zeta})\coloneqq\prod_{l=1}^{d}\widehat{B}_{i_{l},p_{l}}(\zeta_{l}),\
\ \ \ \forall\,\boldsymbol{i}\in\mathit{\mathbf{I}},\ \
\boldsymbol{\zeta}\coloneqq(\zeta_{1},\dots,\zeta_{d}),$ (8)

with $d$ as the underlying dimension of the parametric domain
$\widehat{\Omega}\coloneqq(0,1)^{d}$ and I the multi-index set
$\textup{{I}}\coloneqq\\{(i_{1},\dots,i_{d})\ |\ 1\leq i_{l}\leq n_{l},\
l=1,\dots,d\\}$.
B-splines fulfill several properties and for our purposes the most important
ones are:

  * •

If for all internal knots the multiplicity satisfies $1\leq m(\xi_{j})\leq
m\leq p,$ then the B-spline basis functions $\widehat{B}_{i,p}({\xi})$ are
globally $C^{p-m}$-continuous. Therefore we define in this case the regularity
integer $r\coloneqq p-m$. Obviously, by the product structure, we get splines
$\widehat{B}_{\boldsymbol{i},\boldsymbol{p}}(\boldsymbol{\zeta})$ which are
$C^{r_{l}}$-smooth w.r.t. the $l$-th coordinate direction if the internal
multiplicities fulfill $1\leq m(\xi_{j}^{l})\leq m_{l}\leq p_{l},\
r_{l}\coloneqq p_{l}-r_{l},\ \forall l\in 1,\dots,d$ in the multivariate case.
In case of $r_{i}=-1$ we have discontinuous splines w.r.t. the $i$-th
coordinate direction.

  * •

The B-splines $\\{\widehat{B}_{\boldsymbol{i},\boldsymbol{p}}\ |\ \
\boldsymbol{i}\in\textbf{I}\\}$ are linearly independent.

  * •

For continuous univariate splines $\widehat{B}_{i,p},\ p\geq 1$ we have

$\displaystyle\partial_{\zeta}\widehat{B}_{i,p}(\zeta)=\frac{p}{\xi_{i+p}-\xi_{i}}\widehat{B}_{i,p-1}(\zeta)+\frac{p}{\xi_{i+p+1}-\xi_{i}}\widehat{B}_{i+1,p-1}(\zeta),$
(9)

with $\widehat{B}_{1,p-1}(\zeta)\coloneqq\widehat{B}_{n+1,p-1}(\zeta)\coloneqq
0$.

  * •

The support of the spline $\widehat{B}_{i,p}$ is the interval
$[\xi_{i},\xi_{i+p+1}]$. Moreover, the knots $\psi_{j}$ define a subdivision
of the interval $(0,1)$ and for each element $I=(\psi_{j},\psi_{j+1})$ we find
a $i$ with $(\psi_{j},\psi_{j+1})=(\xi_{i},\xi_{i+1})$ and write
$\tilde{I}\coloneqq(\xi_{i-p},\xi_{i+p+1})$ for the so-called _support
extension_.

The space spanned by all univariate splines $\widehat{B}_{i,p}$ corresponding
to given knot vector, degree $p$ and global regularity $r$ is denoted by

$S_{p}^{r}\coloneqq\textup{span}\\{\widehat{B}_{i,p}\ |\ i=1,\dots,n\\}.$

For the multivariate case we just define the spline space as the product space

$S_{p_{1},\dots,p_{d}}^{r_{1},\dots,r_{d}}\coloneqq
S_{p_{1}}^{r_{1}}\otimes\dots\otimes
S_{p_{d}}^{r_{d}}=\textup{span}\\{\widehat{B}_{\boldsymbol{i},\boldsymbol{p}}\
|\ \boldsymbol{i}\in\mathit{\mathbf{I}}\\}$

of proper univariate spline spaces.
To define discrete spaces based on splines we require a mapping
$\boldsymbol{F}\colon\widehat{\Omega}\rightarrow\mathbb{R}^{d}$ which
parametrizes the computational domain
$\Omega\coloneqq\boldsymbol{F}(\widehat{\Omega})$. The knots stored in the
knot vector $\boldsymbol{\Xi}$, corresponding to the underlying splines,
determine a mesh in the parametric domain $\widehat{\Omega}$, namely
$\widehat{M}\coloneqq\\{K_{\boldsymbol{j}}\coloneqq(\psi_{j_{1}}^{1},\psi_{j_{1}+1}^{1})\times\dots\times(\psi_{j_{d}}^{d},\psi_{j_{d}+1}^{d})\
|\ \boldsymbol{j}=(j_{1},\dots,j_{d}),\ \textup{with}\ 1\leq j_{i}<N_{i}\\},$
and with
${\boldsymbol{\Psi}}=\\{\psi_{1}^{1},\dots,\psi_{N_{1}}^{1}\\}\times\dots\times\\{\psi_{1}^{d},\dots,\psi_{N_{d}}^{d}\\}$
as the knot vector ${\boldsymbol{\Xi}}$ without knot repetitions. The image of
this mesh under the mapping $\boldsymbol{F}$, i.e.
$\mathcal{M}\coloneqq\\{{\boldsymbol{F}}(K)\ |\ K\in\widehat{M}\\}$, gives us
a mesh structure in the physical domain. By inserting knots without changing
the parametrization we can refine the mesh, which is the concept of
$h$-refinement; see [18, 12]. For a mesh $\mathcal{M}$ we define the global
mesh size $h\coloneqq\max\\{h_{\mathcal{K}}\ |\ \mathcal{K}\in\mathcal{M}\\}$,
where for $\mathcal{K}\in\mathcal{M}$ we denote with
$h_{\mathcal{K}}\coloneqq\textup{diam}(\mathcal{K})$ the _element size_.

######  Assumption 1.

(Regular mesh)
There exists a constant $c_{u}$ independent from the mesh size such that
$h_{\mathcal{K}}\leq h\leq c_{u}\,h_{\mathcal{K}}$ for all mesh elements
$\mathcal{K}\in\mathcal{M}$. Let the parametrization mapping $\boldsymbol{F}$
be a diffeomorphism and let the restrictions of $\boldsymbol{F}$ to the
closure of each mesh element be smooth.

###  2.3 Isogeometric de Rham spaces

Here and in the following we have $\Omega=\boldsymbol{F}(\widehat{\Omega})$,
where $\boldsymbol{F}$ meets the conditions in Assumption 1 and
$\boldsymbol{J}\coloneqq D\boldsymbol{F}$ stands for the Jacobian of
$\boldsymbol{F}$.

Basis for the design of stable FEM discretizations of the mixed linear
elasticity formulation with weakly imposed symmetry is the connection between
the de Rham complexes

$\mathbb{R}$$H^{1}(\Omega)$$\boldsymbol{H}(\Omega,\textup{curl})$$\boldsymbol{H}(\Omega,\textup{div})$$L^{2}(\Omega)$$\nabla$$\iota$$\nabla\times$$\nabla\cdot$$0$$\\{0\\},$
in
$3D$,$H^{1}(\Omega)$$\boldsymbol{H}(\Omega,\textup{div})$$L^{2}(\Omega)$$\\{0\\},\
$ in $2D$,curl$\nabla\cdot$$0$$\iota$$\mathbb{R}$

and the so-called elasticity complex with weak symmetry. For more information
we refer to [6] and [15]. Therefore, a structure-preserving discretization of
the de Rham chain utilizing proper spline spaces might be a reasonable
starting point for the derivation of numerical methods. A detailed discussion
on underlying isogeometric discrete differential forms in $3D$ can be found in
[11] and below we state very shortly some important results from the last
reference.

Define the pullbacks

$\displaystyle\mathcal{Y}_{1}(q)$ $\displaystyle\coloneqq
q\circ\boldsymbol{F},$ $\displaystyle q\in L^{2}(\Omega),$ (10)
$\displaystyle\mathcal{Y}_{2}(\boldsymbol{v})$
$\displaystyle\coloneqq\boldsymbol{J}^{T}\big{(}\boldsymbol{v}\circ\boldsymbol{F}),$
$\displaystyle\boldsymbol{v}\in\boldsymbol{L}^{2}(\Omega),$ (11)
$\displaystyle\mathcal{Y}_{3}(\boldsymbol{v})$
$\displaystyle\coloneqq\textup{det}(\boldsymbol{J})\boldsymbol{J}^{-1}\big{(}\boldsymbol{v}\circ\boldsymbol{F}\big{)},\
\ \ \ $ $\displaystyle\boldsymbol{v}\in\boldsymbol{L}^{2}(\Omega),\ \ $ (12)
$\displaystyle\mathcal{Y}_{4}(q)$
$\displaystyle\coloneqq\textup{det}(\boldsymbol{J})\big{(}q\circ\boldsymbol{F}\big{)},$
$\displaystyle q\in L^{2}(\Omega).$ (13)

Then, for $\Omega\subset\mathbb{R}^{3},\ p_{i}>r_{i}\geq 0$, the spaces

$\displaystyle V_{h,1}^{3}$
$\displaystyle\coloneqq\mathcal{Y}_{1}^{-1}(\widehat{V}^{3}_{h,1})\ \ \
\textup{with}\hskip 8.5359pt$ $\displaystyle\widehat{V}^{3}_{h,1}\coloneqq
S_{p_{1},p_{2},p_{3}}^{r_{1},r_{2},r_{3}},$
$\displaystyle\boldsymbol{V}_{h,2}^{3}$
$\displaystyle\coloneqq\mathcal{Y}_{2}^{-1}(\widehat{\boldsymbol{V}}^{3}_{h,2})\
\ \ \textup{with}\hskip 8.5359pt$
$\displaystyle\widehat{\boldsymbol{V}}^{3}_{h,2}\coloneqq\big{(}S_{p_{1}-1,p_{2},p_{3}}^{r_{1}-1,r_{2},r_{3}}\times
S_{p_{1},p_{2}-1,p_{3}}^{r_{1},r_{2}-1,r_{3}}\times
S_{p_{1},p_{2},p_{3}-1}^{r_{1},r_{2},r_{3}-1}\big{)}^{T},$
$\displaystyle\boldsymbol{V}_{h,3}^{3}$
$\displaystyle\coloneqq\mathcal{Y}_{3}^{-1}(\widehat{\boldsymbol{V}}^{3}_{h,3})\
\ \ \textup{with}\hskip 8.5359pt$
$\displaystyle\widehat{\boldsymbol{V}}^{3}_{h,3}\coloneqq\big{(}S_{p_{1},p_{2}-1,p_{3}-1}^{r_{1},r_{2}-1,r_{3}-1}\times
S_{p_{1}-1,p_{2}-2,p_{3}-1}^{r_{1}-1,r_{2},r_{3}-1}\times
S_{p_{1}-1,p_{2}-1,p_{3}}^{r_{1}-1,r_{2}-1,r_{3}}\big{)}^{T}$ , $\displaystyle
V_{h,4}^{3}$
$\displaystyle\coloneqq\mathcal{Y}_{4}^{-1}(\widehat{V}^{3}_{h,4})\ \ \
\textup{with}\hskip 8.5359pt$ $\displaystyle\widehat{V}^{3}_{h,4}\coloneqq
S_{p_{1}-1,p_{2}-1,p_{3}-1}^{r_{1}-1,r_{2}-1,r_{3}-1},$

form an exact subcomplex of the de Rham chain, i.e. there are continuous
projections

$\displaystyle{\Pi}^{3}_{h,i}\colon L^{2}(\Omega)\rightarrow{V}_{h,i}^{3},\
i\in\\{1,4\\}\ \ \ \textup{and}\ \ \
{\Pi}^{3}_{h,i}\colon\boldsymbol{L}^{2}(\Omega)\rightarrow{\boldsymbol{V}}_{h,i}^{3},\
i\in\\{2,3\\},$

such that we obtain the commutative diagram

$H^{1}(\Omega)$$\boldsymbol{H}(\Omega,\textup{curl})$$\boldsymbol{H}(\Omega,\textup{div})$${L}^{2}(\Omega)$$\nabla$$\nabla\times$$\nabla\cdot$$0$$\iota$$\\{0\\}$$\mathbb{R}$${V}_{h,1}^{3}$$\boldsymbol{V}_{h,2}^{3}$$\boldsymbol{V}_{h,3}^{3}$${V}_{h,4}^{3}$${\nabla}$${\nabla}\times$${\nabla}\cdot$$0$$\iota$$\\{0\\}\
.$$\mathbb{R}$id$\Pi_{h,1}^{3}$$\Pi_{h,2}^{3}$$\Pi_{h,3}^{3}$$\Pi_{h,4}^{3}$id

Due to the smoothness properties of the splines the discrete spaces are indeed
subspaces of the continuous pendants. For reasons of simplification let us
assume in the following $p\coloneqq p_{1}=p_{2}=p_{3}$ and $r\coloneqq
r_{1}=r_{2}=r_{3}$. Then, as shown in [11], we can define the above
projections in such a way that the next approximation estimates are valid.

######  Lemma 1.

Let $0\leq k\leq p,\ 0\leq r\leq p-1$. Then, assuming sufficient regularity
for $\phi$ and $\boldsymbol{v}$, it is

$\displaystyle\left\lVert{\phi}-\Pi^{3}_{h,1}{\phi}\right\rVert_{H^{l}}\leq
Ch^{k-l+1}\ \left\lVert{\phi}\right\rVert_{H^{k+1}},\ l\in\\{0,1\\},$
$\displaystyle\left\lVert{\boldsymbol{v}}-\Pi^{3}_{h,2}{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}(\textup{curl})}\leq
Ch^{k}\
\left\lVert{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}^{k}(\textup{curl})},\hskip
14.22636pt$
$\displaystyle\left\lVert{\boldsymbol{v}}-\Pi^{3}_{h,3}{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}(\textup{div})}\leq
Ch^{k}\
\left\lVert{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}^{k}(\textup{div})},$
$\displaystyle\left\lVert{\phi}-\Pi^{3}_{h,4}{\phi}\right\rVert_{{L}^{2}}\leq
Ch^{k}\ \left\lVert{\phi}\right\rVert_{{H}^{k}}.$

We use the notation
$\left\lVert{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}^{k}(\textup{curl})}^{2}\coloneqq\left\lVert{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}^{k}}^{2}+\left\lVert\nabla\times{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}^{k}}^{2}$
and
$\left\lVert{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}^{k}(\textup{div})}^{2}\coloneqq\left\lVert{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}^{k}}^{2}+\left\lVert\nabla\cdot{\boldsymbol{v}}\right\rVert_{{H}^{k}}^{2}$.

###### Proof.

See [11, Theorem 5.3,Theorem 5.4, Remark 5.1]. ∎

Here and in the rest of the article $C<\infty$ denotes a constant only
depending on the domain $\Omega,\ \boldsymbol{F}$ and the polynomial degrees
as well as on the regularity parameter $r$ and may vary at different
occurrences.

In fact, we shall study the $2D$ elasticity equation for which the planar de
Rham chain will be useful to handle the weak symmetry case. Nevertheless, we
mentioned the classical isogeometric de Rham spaces in $\mathbb{R}^{3}$ above
in order to justify the next statements without giving a detailed proof.
Namely, it is easy to check that the procedure in [11] can be adapted to the
$2D$ setting. With similar steps and proofs one can show the existence of
another commutative diagram

$H^{1}(\Omega)$$\boldsymbol{H}(\Omega,\textup{div})$$L^{2}(\Omega)$$\\{0\\}$curl$\nabla\cdot$$0$$\iota$$\mathbb{R}$${V}_{h,1}^{2}$$\boldsymbol{V}_{h,2}^{2}$${V}_{h,3}^{2}$$\\{0\\}\
,$curl${\nabla}\cdot$$0$$\iota$$\mathbb{R}$id$\Pi_{h,1}^{2}$$\Pi_{h,2}^{2}$$\Pi_{h,3}^{2}$id
Figure 1: There is a IGA discretization of the planar de Rham complex
preserving the chain structure.

where we have the spaces

$\displaystyle V_{h,1}^{2}$
$\displaystyle\coloneqq\mathcal{Y}_{1}^{-1}(\widehat{V}^{2}_{h,1})\ \ \
\textup{with}\hskip 8.5359pt$ $\displaystyle\widehat{V}^{2}_{h,1}\coloneqq
S_{p,p}^{r,r},$ $\displaystyle\boldsymbol{V}_{h,2}^{2}$
$\displaystyle\coloneqq\mathcal{Y}_{3}^{-1}(\widehat{\boldsymbol{V}}^{2}_{h,2})\
\ \ \textup{with}\hskip 8.5359pt$
$\displaystyle\widehat{\boldsymbol{V}}^{2}_{h,2}\coloneqq\big{(}S_{p,p-1}^{r,r-1}\times
S_{p-1,p}^{r-1,r}\big{)},$ $\displaystyle V_{h,3}^{2}$
$\displaystyle\coloneqq\mathcal{Y}_{4}^{-1}(\widehat{V}^{2}_{h,3})\ \ \
\textup{with}\hskip 8.5359pt$ $\displaystyle\widehat{V}^{2}_{h,3}\coloneqq
S_{p-1,p-1}^{r-1,r-1},\ \ \ $

and there are compatible $L^{2}$-continuous projections $\Pi_{h,i}^{2}$ onto
the respective spaces. One notes the same notation for the pullbacks in $2D$
and mean the straight-forward adaption of (10), (12) and (13) to the two-
dimensional setting. An important aspect showing the appropriateness of the
$\mathcal{Y}_{i}$ are the properties

$\displaystyle\widehat{\nabla}\cdot\mathcal{Y}_{3}(\boldsymbol{v})=\mathcal{Y}_{4}(\nabla\cdot\boldsymbol{v}),\
\ \ \
\widehat{\textup{curl}}(\mathcal{Y}_{1}(\phi))=\mathcal{Y}_{3}(\textup{curl}(\phi)),$
(14)

which can be proven easily. The notation " $\widehat{\ \cdot\ }$ " indicates
that the differentiation is w.r.t. the parametric coordinates. Analogously to
the $3D$ case one can derive approximation estimates for the mentioned spaces.

######  Lemma 2.

Let $0\leq k\leq p,\ 0\leq r\leq p-1$. The compatible projections
$\Pi_{h,i}^{2}$ satisfy the estimates

$\displaystyle\left\lVert{\phi}-\Pi^{2}_{h,1}{\phi}\right\rVert_{H^{l}}\leq
Ch^{k-l+1}\ \left\lVert{\phi}\right\rVert_{H^{k+1}},\ l\in\\{0,1\\},$
$\displaystyle\left\lVert{\boldsymbol{v}}-\Pi^{2}_{h,2}{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}(\textup{div})}\leq
Ch^{k}\
\left\lVert\boldsymbol{v}\right\rVert_{\boldsymbol{H}^{k}(\textup{div})},\hskip
14.22636pt$
$\displaystyle\left\lVert\phi-\Pi^{2}_{h,3}{\phi}\right\rVert_{{L}^{2}}\leq
Ch^{k}\ \left\lVert\phi\right\rVert_{{H}^{k}},$

assuming sufficient regularity.

###### Proof.

With similar proofs like for the $3D$ de Rham chain this can be shown with the
results from [11]. ∎

At the end of this short section we want to mention the Taylor-Hood type
isogeometric spaces; see e.g. [10]. They are defined through

$\displaystyle\boldsymbol{W}_{h}^{TH}(p,r)\coloneqq\big{(}\boldsymbol{V}_{h}^{TH},V_{h}^{TH}\big{)}\coloneqq\big{(}(S_{p,p}^{r,r}\times
S_{p,p}^{r,r})\circ\boldsymbol{F}^{-1},S_{p-1,p-1}^{r,r}\circ\boldsymbol{F}^{-1}\big{)},\
\ p-1>r>=0.$

Actually, latter spaces fit not directly to discrete differential forms.
Nevertheless, the space pair will prove useful later and utilizing [8, Theorem
3.2] or Lemma 2 we see the existence of a projection $\Pi_{h}^{TH}\colon
L^{2}(\Omega)\rightarrow V_{h}^{TH}$ with approximation result

$\left\lVert q-\Pi_{h}^{TH}q\right\rVert_{L^{2}}\leq Ch^{k}\left\lVert
q\right\rVert_{H^{k}},\ \ p\geq k\geq 0.$ (15)

After the introduction of elementary notions and spline spaces we look at the
mixed weak formulation for linear elasticity with weak symmetry below.

##  3 Mixed formulation with weakly imposed symmetry

To obtain a mixed weak formulation for the elasticity system that does not
require test spaces with strong symmetry, i.e. subspaces of
$\boldsymbol{\mathcal{H}}(\Omega,\textup{div},\mathbb{S})$, one can use the
weak symmetry ansatz utilizing a Lagrange multiplier as introduced in [16] and
applied several times e.g. in [1] or [6]: _Find_
$\boldsymbol{\sigma}\in\boldsymbol{\mathcal{{H}}}(\Omega,\textup{div}),\
\boldsymbol{u}\in\boldsymbol{L}^{2}(\Omega)$ and
$\boldsymbol{p}\in\boldsymbol{\mathcal{{L}}}^{2}(\Omega,\mathbb{K})$ s.t.

$\displaystyle\langle\boldsymbol{A}\boldsymbol{\sigma},\boldsymbol{\tau}\rangle\
+\ $ $\displaystyle\langle\boldsymbol{u},\nabla\cdot\boldsymbol{\tau}\rangle\
+\ $
$\displaystyle\langle\boldsymbol{p},\boldsymbol{\tau}\rangle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle=\langle\boldsymbol{\tau}\cdot\boldsymbol{n};\boldsymbol{u}_{D}\rangle_{\Gamma},\
\hskip 11.38092pt$
$\displaystyle\forall\boldsymbol{\tau}\in\boldsymbol{\mathcal{{H}}}(\Omega,\textup{div}),$
$\displaystyle\langle\nabla\cdot\boldsymbol{\sigma},\boldsymbol{v}\rangle$
$\displaystyle=\langle\boldsymbol{f},\boldsymbol{v}\rangle,\ \hskip
11.38092pt$ $\displaystyle\forall\boldsymbol{v}\in\boldsymbol{L}^{2}(\Omega),$
(16) $\displaystyle\langle\boldsymbol{\sigma},\boldsymbol{q}\rangle$
$\displaystyle=0,\ \hskip 11.38092pt$
$\displaystyle\forall\boldsymbol{q}\in\boldsymbol{\mathcal{{L}}}^{2}(\Omega,\mathbb{K}),$

where $\boldsymbol{u}_{D}\in\boldsymbol{H}^{1/2}(\Gamma),\
\Gamma\coloneqq\partial\Omega,$ determines the displacement boundary data and
$\boldsymbol{f}\in\boldsymbol{L}^{2}(\Omega).$ For an explanation of boundary
Sobolev spaces we refer the reader to [26, Chapter 2]. Starting point for the
derivation of latter system is the standard mixed weak formulation (5). Then
the symmetry condition is dropped and replaced by the requirement

$\langle\boldsymbol{\mathcal{\sigma}},\boldsymbol{s}\rangle=0,\ \
\forall\boldsymbol{s}\in\boldsymbol{\mathcal{L}}^{2}(\Omega,\mathbb{K}),$

which is equivalent to

$\langle s_{i},\sigma_{ij}-\sigma_{ji}\rangle=0,\ \ \forall s_{i}\in
L^{2}(\Omega),\ \ j\neq i.$

Latter constraint is incorporated by means of a Lagrange multiplier and this
explains the additional variable $\boldsymbol{p}$ as well as the last line in
system (3). It is easy to see the equivalence of the two formulations (5) and
(3) in the continuous setting, where the Lagrange multiplier is related to the
solution $(\boldsymbol{\sigma},\boldsymbol{u})$ by
$\boldsymbol{p}=-\textup{Skew}\big{(}\frac{1}{2}(\partial_{2}u_{1}-\partial_{1}u_{2})\big{)}$.
The well-posedness of (3) in $2D$ is shown in [5] and the consideration of the
weak symmetry formulation is in some sense justified. Main advantage of the
new form is the obtained flexibility for discretization approaches compared to
(5), but the price to pay is the increased number of degrees of freedom. At
least in $3D$ and for fine meshes this might become problematic within
numerical computations.

Nevertheless, in the following we want to face the conforming discretization
of (3) in the two-dimensional setting and thus we have to define proper test
function spaces
$\boldsymbol{\mathcal{W}}_{h}\coloneqq\boldsymbol{\mathcal{V}}_{h,2}^{w}\times\boldsymbol{V}_{h,3}^{w}\times\boldsymbol{\mathcal{V}}_{h,3}^{w}\subset\boldsymbol{\mathcal{H}}(\Omega,\textup{div})\times\boldsymbol{L}^{2}(\Omega)\times\boldsymbol{\mathcal{L}}^{2}(\Omega,\mathbb{K})$
and then we solve the discrete problem:
_Find_
$[\boldsymbol{\sigma}_{h},\boldsymbol{u}_{h},\boldsymbol{p}_{h}]\in\boldsymbol{\mathcal{V}}_{h,2}^{w}\times\boldsymbol{V}_{h,3}^{w}\times\boldsymbol{\mathcal{V}}_{h,3}^{w}$
such that

$\displaystyle\langle\boldsymbol{A}\boldsymbol{\sigma}_{h},\boldsymbol{\tau}_{h}\rangle\
+\ $
$\displaystyle\langle\boldsymbol{u}_{h},\nabla\cdot\boldsymbol{\tau}_{h}\rangle\
+\ $
$\displaystyle\langle\boldsymbol{p}_{h},\boldsymbol{\tau}_{h}\rangle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle=\langle\boldsymbol{\tau}_{h}\cdot\boldsymbol{n};\boldsymbol{u}_{D}\rangle_{\Gamma},\
\hskip 11.38092pt$
$\displaystyle\forall\boldsymbol{\tau}_{h}\in\boldsymbol{\mathcal{V}}_{h,2}^{w},$
$\displaystyle\langle\nabla\cdot\boldsymbol{\sigma}_{h},\boldsymbol{v}_{h}\rangle$
$\displaystyle=\langle\boldsymbol{f},\boldsymbol{v}_{h}\rangle,\ \hskip
11.38092pt$
$\displaystyle\forall\boldsymbol{v}_{h}\in\boldsymbol{V}_{h,3}^{w},$ (17)
$\displaystyle\langle\boldsymbol{\sigma}_{h},\boldsymbol{q}_{h}\rangle$
$\displaystyle=0,\ \hskip 11.38092pt$
$\displaystyle\forall\boldsymbol{q}_{h}\in\boldsymbol{\mathcal{V}}_{h,3}^{w}.$

The index $w$ should indicate that we are in the framework of weak symmetry.
Then the actual test spaces will be constructed through the spline based
spaces defined in Section 2.3. Before we specify the discretization method we
state the fundamental conditions that have to be satisfied in order to get a
stable saddle-point problem (3). The application of Brezzi’s stability
criterion for saddle-point problems to the mixed formulation (cf. [6, Section
2] and [9]) implies the stability requirements:

  * (S1)

It holds
$\left\lVert\boldsymbol{\tau}_{h}\right\rVert_{\boldsymbol{\mathcal{H}}(\textup{div})}^{2}\leq
C_{S1}\langle\boldsymbol{A}\boldsymbol{\tau}_{h},\boldsymbol{\tau}_{h}\rangle\
$ for all $\boldsymbol{\tau}_{h}\in\boldsymbol{\mathcal{V}}_{h,2}^{w}$
satisfying
$0=\langle\nabla\cdot\boldsymbol{\tau}_{h},\boldsymbol{v}_{h}\rangle$ and
$0=\langle\boldsymbol{\tau}_{h},\boldsymbol{q}_{h}\rangle,$ $\
\forall(\boldsymbol{v}_{h},\boldsymbol{q}_{h})\in\
\boldsymbol{V}_{h,3}^{w}\times\boldsymbol{\mathcal{V}}_{h,3}^{w}$ and some
constant $C_{S1}$.

  * (S2)

For all $(\boldsymbol{v}_{h},\boldsymbol{q}_{h})\in\
\boldsymbol{V}_{h,3}^{w}\times\boldsymbol{\mathcal{V}}_{h,3}^{w}$, there
exists a
$\boldsymbol{0}\neq\boldsymbol{\tau}_{h}\in\boldsymbol{\mathcal{V}}_{h,2}^{w}$
s.t.
$\langle\boldsymbol{\tau}_{h},\boldsymbol{q}_{h}\rangle+\langle\nabla\cdot\boldsymbol{\tau}_{h},\boldsymbol{v}_{h}\rangle\geq
C_{S2}\left\lVert\boldsymbol{\tau}_{h}\right\rVert_{\boldsymbol{\mathcal{H}}(\textup{div})}\big{(}\left\lVert\boldsymbol{v}_{h}\right\rVert_{\boldsymbol{L}^{2}}+\left\lVert\boldsymbol{q}_{h}\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}\big{)}$
for a constant $C_{S2}$.

If the conditions (S1) and (S2) are fulfilled we get automatically well-
posedness and moreover the next theorem with results from [5] and [23]
applies. More precisely, we refer to Theorem 3.1 in [23], [5, Section 2] and
[5, Remark 1].

######  Theorem 1.

Let the test spaces fulfill (S1) and (S2). Then the problem (3) has a unique
solution. Further, we get the quasi-optimal estimate

$\left\lVert[\boldsymbol{\sigma}-\boldsymbol{\sigma}_{h},\boldsymbol{u}-\boldsymbol{u}_{h},\boldsymbol{p}-\boldsymbol{p}_{h}]\right\rVert_{\mathcal{B}}\leq
C\left\lVert[\boldsymbol{\sigma}-\boldsymbol{\tau}_{h},\boldsymbol{u}-\boldsymbol{v}_{h},\boldsymbol{p}-\boldsymbol{q}_{h}]\right\rVert_{\mathcal{B}}\
\
\forall[\boldsymbol{\tau}_{h},\boldsymbol{v}_{h},\boldsymbol{q}_{h}]\in\boldsymbol{\mathcal{W}}_{h}$
(18)

between the exact solution
$(\boldsymbol{\sigma},\boldsymbol{u},\boldsymbol{p})$ of (3) and the solution
to (3),
where
$\left\lVert[\boldsymbol{\sigma},\boldsymbol{u},\boldsymbol{p}]\right\rVert_{\mathcal{B}}^{2}\coloneqq\left\lVert\boldsymbol{\sigma}\right\rVert_{\boldsymbol{\mathcal{H}}(\textup{div})}^{2}+\left\lVert\boldsymbol{u}\right\rVert_{\boldsymbol{L}^{2}}^{2}+\left\lVert\boldsymbol{p}\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}^{2}$
and $C$ depends only on $C_{S1},C_{S2}$ and $\boldsymbol{A}$.
The discretization is inf-sup stable, meaning we have a constant
$C_{\mathcal{B}}>0$ independent from $h$ s.t.

$\underset{[\boldsymbol{\sigma}_{h},\boldsymbol{u}_{h},\boldsymbol{p}_{h}]\in\boldsymbol{\mathcal{W}}_{h}\textbackslash\\{\boldsymbol{0}\\}}{\inf}\hskip
5.69046pt\underset{[\boldsymbol{\tau}_{h},\boldsymbol{v}_{h},\boldsymbol{q}_{h}]\in\boldsymbol{\mathcal{W}}_{h}}{\sup}\frac{\mathcal{B}([\boldsymbol{\tau}_{h},\boldsymbol{v}_{h},\boldsymbol{q}_{h}],[\boldsymbol{\sigma}_{h},\boldsymbol{u}_{h},\boldsymbol{p}_{h}])}{\left\lVert[\boldsymbol{\tau}_{h},\boldsymbol{v}_{h},\boldsymbol{q}_{h}]\right\rVert_{\mathcal{B}}\
\left\lVert[\boldsymbol{\sigma}_{h},\boldsymbol{u}_{h},\boldsymbol{p}_{h}]\right\rVert_{\mathcal{B}}}\geq
C_{\mathcal{B}},$

with
$\mathcal{B}([\boldsymbol{\sigma},\boldsymbol{u},\boldsymbol{p}],[\boldsymbol{\tau},\boldsymbol{v},\boldsymbol{q}])\coloneqq\langle\boldsymbol{A}\boldsymbol{\sigma},\boldsymbol{\tau}\rangle\
+\langle\boldsymbol{u},\nabla\cdot\boldsymbol{\tau}\rangle\
+\langle\boldsymbol{p},\boldsymbol{\tau}\rangle+\langle\nabla\cdot\boldsymbol{\sigma},\boldsymbol{v}\rangle+\langle\boldsymbol{\sigma},\boldsymbol{q}\rangle$
denoting the underlying bilinear form of (3). Besides, if the constant
identity matrix $\boldsymbol{I}$ is an element of
$\boldsymbol{\mathcal{V}}_{h,2}^{w}$, then the constant $C_{\mathcal{B}}$ and
the constant $C$ in (18) are independent of the Lamé coefficient $\lambda$.

After our problem model is clarified we define spaces for the two-dimensional
setting yielding a stable discretization.

###  3.1 Choice of discrete spaces

The choice of suitable test spaces for $\Omega\subset\mathbb{R}^{2}$
satisfying the discrete inf-sup condition is determined by the following lemma
that corresponds to Lemma 3.2 in [23]. It gives us a helpful guideline how to
satisfy the conditions (S1) and (S2).

######  Lemma 3.

Let $\boldsymbol{W}_{h}^{1}\subset\boldsymbol{H}(\Omega,\textup{div}),\
W_{h}^{2}\subset L^{2}(\Omega)$ be a pair of spaces such that

$\displaystyle\underset{p_{h}\in W_{h}^{2}\textbackslash\\{0\\}}{\inf}\
\underset{\boldsymbol{v}_{h}\in\boldsymbol{W}_{h}^{1}}{\sup}\
\frac{\langle\nabla\cdot\boldsymbol{v}_{h},p_{h}\rangle}{\left\lVert\boldsymbol{v}_{h}\right\rVert_{\boldsymbol{H}(\textup{div})}\left\lVert
p_{h}\right\rVert_{L^{2}}}\geq C_{P}>0,$

and let $\boldsymbol{W}^{0}_{h}\subset\boldsymbol{H}^{1}(\Omega),\
W_{h}^{3}\subset L^{2}(\Omega)$ be a second pair of discrete spaces with

$\displaystyle\underset{q_{h}\in W_{h}^{3}\textbackslash\\{0\\}}{\inf}\
\underset{\boldsymbol{w}_{h}\in\boldsymbol{W}_{h}^{0}}{\sup}\
\frac{\langle\nabla\cdot\boldsymbol{w}_{h},q_{h}\rangle}{\left\lVert\boldsymbol{w}_{h}\right\rVert_{\boldsymbol{H}^{1}}\left\lVert
q_{h}\right\rVert_{L^{2}}}\geq C_{S}>0,$

and assume the constants $0<C_{P},C_{S}$ to be independent of the underlying
mesh size $h$.
If
$\textup{curl}(w_{h,i})\in\boldsymbol{W}_{h}^{1},\forall\boldsymbol{w}_{h}=(w_{h,1},w_{h,2})^{T}\in\boldsymbol{W}^{0}_{h}$
and $\nabla\cdot\boldsymbol{W}_{h}^{1}\subset W_{h}^{2}$ then

$\boldsymbol{\mathcal{V}}^{w}_{h,2}\coloneqq\big{(}\boldsymbol{W}_{h}^{1}\times\boldsymbol{W}_{h}^{1}\big{)}^{T},\
\ \boldsymbol{{V}}_{h,3}^{w}\coloneqq\big{(}W_{h}^{2}\times
W_{h}^{2}\big{)}^{T}\ \ and\ \
\boldsymbol{\mathcal{V}}_{h,3}^{w}\coloneqq\\{\textup{Skew}(q)\ |\ q\in
W_{h}^{3}\\}$ (19)

fulfill the conditions (S1) and (S2).
111$\big{(}\boldsymbol{W}_{h}^{1}\times\boldsymbol{W}_{h}^{1}\big{)}^{T}$
denotes the space of matrix fields whose rows are elements of
$\boldsymbol{W}_{h}^{1}$.

###### Proof.

Compare Lemma 3.2 in [23]. ∎

Now we define the isogeometric discrete spaces meeting the requirements of
Lemma 3 for the two-dimensional setting. In view of the mentioned lemma and
the definitions in Section 2.3 we choose:

$\displaystyle\boldsymbol{W}_{h}^{1}$
$\displaystyle\coloneqq\boldsymbol{V}_{h,2}^{2},$ (20) $\displaystyle
W_{h}^{2}$ $\displaystyle\coloneqq V_{h,3}^{2}.$ (21)

And we use a Taylor-Hood pair, namely

$\displaystyle\boldsymbol{W}^{0}_{h}\times W^{3}_{h}$
$\displaystyle\coloneqq\boldsymbol{W}_{h}^{TH}(p,r)=\big{(}\boldsymbol{V}_{h}^{TH},V_{h}^{TH}\big{)}.$
(22)

For the above choices of the discrete spaces the following two lemmas
resulting from [8] and [12] apply.

######  Lemma 4.

There exists a constant $0<C_{P}$ independent of $h$ s.t.

$\displaystyle\underset{p_{h}\in V_{h,3}^{2}\textbackslash\\{0\\}}{\inf}\ \
\underset{\boldsymbol{v}_{h}\in\boldsymbol{V}_{h,2}^{2}}{\sup}\frac{\langle\nabla\cdot\boldsymbol{v}_{h},p_{h}\rangle}{\left\lVert\boldsymbol{v}_{h}\right\rVert_{\boldsymbol{H}(\textup{div})}\
\left\lVert p_{h}\right\rVert_{L^{2}}}\geq C_{P}>0$ (23)

###### Proof.

First one notes the commutativity of the diagram in Fig. 1 and the
$L^{2}$-boundedness of the corresponding projections $\Pi_{h,i}^{2}$, meaning
there is a structure-preserving discretization of the planar de Rham complex.
As already mentioned, this can be seen by an adaption of the discretization
approach in [11] to the planar de Rham complex. Consequently, we can easily
adapt the elegant proof of Theorem 2 in [12, pp. 155-193] to our two-
dimensional de Rham chain. ∎

######  Lemma 5.

Let $p>=2,\ r=0$ and $\Gamma_{B}\subsetneq\partial\Omega$, where $\Gamma_{B}$
is the union of full mesh edges. Then there exists $h_{max}>0$, which depends
only on $\boldsymbol{F}$, and a constant $0<C_{S}$ independent of $h$ s.t.

$\displaystyle\underset{q_{h}\in V_{h}^{TH}\textbackslash\\{0\\}}{\inf}\ \
\underset{\boldsymbol{w}_{h}\in\boldsymbol{V}_{h}^{TH}\cap\boldsymbol{H}_{\Gamma_{B}}^{1}(\Omega)}{\sup}\
\frac{\langle\nabla\cdot\boldsymbol{w}_{h},q_{h}\rangle}{\left\lVert\boldsymbol{w}_{h}\right\rVert_{\boldsymbol{H}^{1}}\
\left\lVert q_{h}\right\rVert_{L^{2}}}\geq C_{S}>0,\ \ \forall h\leq h_{max}.$
(24)

We use the abbreviation
$\boldsymbol{H}_{\Gamma_{B}}^{1}(\Omega)\coloneqq\\{\boldsymbol{v}\in\boldsymbol{H}^{1}(\Omega)\
|\ \boldsymbol{v}=\boldsymbol{0}\ \textup{on}\ \Gamma_{B}\\}$

###### Proof.

This follows from Theorem 5.2 in [8] and its proof for the case of a constant
weight function, where one notes the equivalence of the Sobolev semi-norm
$|\cdot|_{H^{1}}$ and the standard Sobolev norm
$\left\lVert\cdot\right\rVert_{H^{1}}$ in case of zero Dirichlet boundary
conditions on a part of the boundary. Further one observes Assumption 1. ∎

And we have the next auxiliary result.

######  Lemma 6.

Let $p>r+1\geq 1$.
It holds $\boldsymbol{V}_{h,2}^{2}\subset\boldsymbol{H}(\Omega,\textup{div}),\
V_{h,3}^{2}\subset L^{2}(\Omega),\
\boldsymbol{V}_{h}^{TH}\subset\boldsymbol{H}^{1}(\Omega),\ V_{h}^{TH}\subset
L^{2}(\Omega)$. Further, if $(w_{h,1},w_{h,2})^{T}\in\boldsymbol{V}_{h}^{TH}$,
then $\textup{curl}(w_{h,i})\subset\boldsymbol{V}_{h,2}^{2}$ as well as
$\nabla\cdot\boldsymbol{V}_{h,2}^{2}\subset V_{h,3}^{2}$.

###### Proof.

This is clear by the regularity properties of B-splines and (14); see Fig. 1.
∎

Hence, using the last three lemmas and applying Lemma 3, we see directly the
next result:

######  Theorem 2 (Discrete spaces for weak symmetry).

If $p\geq 2,r=0$ and if we define the test spaces according to (19) with (20)
- (22). Then, the corresponding discretized weak form with weakly imposed
symmetry is well-posed for $h$ small enough.

######  Remark 1.

An interesting point of the mixed formulation is the possibility to reach
stability w.r.t. $\lambda$, which gets important in case of nearly
incompressible materials. In view of Lemma 3 we need to guarantee
$\boldsymbol{I}\in\boldsymbol{\mathcal{V}}_{h,2}^{w}$ which is equivalent to
require
$\textup{det}(\boldsymbol{J})\boldsymbol{J}^{-1}\in\big{(}\widehat{\boldsymbol{V}}_{h,2}^{2}\times\widehat{\boldsymbol{V}}_{h,2}^{2}\big{)}$.
This implies the conditions $(\boldsymbol{J}_{22},-\boldsymbol{J}_{21})\in
S_{p,p-1}^{r,r-1}\times S_{p-1,p}^{r-1,r},\
(-\boldsymbol{J}_{12},\boldsymbol{J}_{11})\in S_{p,p-1}^{r,r-1}\times
S_{p-1,p}^{r-1,r}$. Thus, if we use B-spline parametrizations of the form
$\boldsymbol{F}\in(S_{p,p}^{r,r})^{2}$ we obtain
$\boldsymbol{I}\in\boldsymbol{\mathcal{V}}_{h,2}^{w}$. In particular, the
stability condition fits to the isoparametric paradigm within IGA.

###  3.2 Error estimation

Because of Theorem 1 and in view of the approximation properties of spline
spaces, see Lemma 2, we can estimate easily the discretization error for the
method 3. One obtains the following result.

######  Theorem 3 (Discretization error for weak symmetry).

Let $p\geq 2,r=0$ and choose discrete spaces for (3) as determined by
(20)-(22) and (19).
Further let the exact solution of (3) satisfy the regularity conditions
$\boldsymbol{\sigma}\in\boldsymbol{\mathcal{H}}^{k}(\Omega,\textup{div}),\boldsymbol{u}\in\boldsymbol{H}^{k}(\Omega),\
\boldsymbol{p}\in\boldsymbol{\mathcal{H}}^{k}(\Omega)\cap\boldsymbol{\mathcal{L}}^{2}(\Omega,\mathbb{K})$
with $p\geq k\geq 0$.
222$\boldsymbol{\tau}\in\boldsymbol{\mathcal{H}}^{k}(\Omega,\textup{div})$ iff
$\boldsymbol{\tau}\in\boldsymbol{\mathcal{H}}^{k}(\Omega)$ and
$\nabla\cdot\boldsymbol{\tau}\in\boldsymbol{H}^{k}(\Omega)$. Further,
$\left\lVert{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}(\textup{div})}^{2}\coloneqq\left\lVert{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}}^{2}+\left\lVert\nabla\cdot{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{H}^{k}}^{2}$.
Then the estimates

$\displaystyle\left\lVert[\boldsymbol{\sigma}-\boldsymbol{\sigma}_{h},\boldsymbol{u}-\boldsymbol{u}_{h},\boldsymbol{p}-\boldsymbol{p}_{h}]\right\rVert_{\mathcal{B}}$
$\displaystyle\leq Ch^{k}\
\big{(}\left\lVert\boldsymbol{\sigma}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}(\textup{div})}+\left\lVert\boldsymbol{u}\right\rVert_{\boldsymbol{H}^{k}}+\left\lVert\boldsymbol{p}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}}\big{)},$
$\displaystyle\left\lVert\nabla\cdot\boldsymbol{\sigma}-\nabla\cdot\boldsymbol{\sigma}_{h}\right\rVert_{\boldsymbol{L}^{2}}$
$\displaystyle\leq
Ch^{k}\left\lVert\nabla\cdot\boldsymbol{\sigma}\right\rVert_{\boldsymbol{H}^{k}}$

are valid for $h$ small enough (compare Lemma 5).

###### Proof.

First let us generalize the projections $\Pi_{h,i}^{2}$ from Section 2.3 to
the matrix setting, vector setting respectively, by a row-wise definition,
i.e.
$\Pi_{h,3}^{2}\boldsymbol{u}\coloneqq(\Pi_{h,3}^{2}u_{1},\Pi_{h,3}^{2}u_{2})^{T}$
if $\boldsymbol{u}=(u_{1},u_{2})^{T}$ and
$\Pi_{h,2}^{2}\boldsymbol{\sigma}\coloneqq\begin{bmatrix}\Pi_{h,2}^{2}\boldsymbol{\sigma}_{1}^{T}\
\Pi_{h,2}^{2}\boldsymbol{\sigma}_{2}^{T}\end{bmatrix}^{T},$ if
$\boldsymbol{\sigma}=\begin{bmatrix}\boldsymbol{\sigma}_{1}\
\boldsymbol{\sigma}_{2}\end{bmatrix}^{T}.$ Moreover, for
$\boldsymbol{p}=\textup{Skew}(p)$ we define
$\Pi_{h}^{TH}\boldsymbol{p}\coloneqq\textup{Skew}(\Pi_{h}^{TH}p)$. Then, the
first estimate is a consequence of Theorem 1, Lemma 2 and (15). More
precisely, it is

$\displaystyle\left\lVert[\boldsymbol{\sigma}-\boldsymbol{\sigma}_{h},\boldsymbol{u}-\boldsymbol{u}_{h},\boldsymbol{p}-\boldsymbol{p}_{h}]\right\rVert_{\mathcal{B}}$
$\displaystyle\leq
C\left\lVert[\boldsymbol{\sigma}-\Pi_{h,2}^{2}\boldsymbol{\sigma},\boldsymbol{u}-\Pi_{h,3}^{2}\boldsymbol{u},\boldsymbol{p}-\Pi_{h}^{TH}\boldsymbol{p}]\right\rVert_{\mathcal{B}}$
$\displaystyle\leq
C\Big{(}\left\lVert\boldsymbol{\sigma}-\Pi_{h,2}^{2}\boldsymbol{\sigma}\right\rVert_{\boldsymbol{\mathcal{H}}(\textup{div})}+\left\lVert\boldsymbol{u}-\Pi_{h,3}^{2}\boldsymbol{u}\right\rVert_{\boldsymbol{L}^{2}}+\left\lVert\boldsymbol{p}-\Pi_{h}^{TH}\boldsymbol{p}\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}\Big{)}$
$\displaystyle\leq Ch^{k}\
\big{(}\left\lVert\boldsymbol{\sigma}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}}+\left\lVert\boldsymbol{u}\right\rVert_{\boldsymbol{H}^{k}}+\left\lVert\boldsymbol{p}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}}\big{)}.$

The second estimate of the assertion can be obtained by a standard approach.
Due to the second lines of (3) and (3) together with
$\Pi_{h,3}^{2}\nabla\cdot\boldsymbol{\sigma}-\nabla\cdot\boldsymbol{\sigma}_{h}\in\boldsymbol{V}_{h,3}^{w}$,
we have

$\displaystyle\left\lVert\nabla\cdot\boldsymbol{\sigma}-\nabla\cdot\boldsymbol{\sigma}_{h}\right\rVert^{2}_{\boldsymbol{L}^{2}}$
$\displaystyle=\langle\nabla\cdot\boldsymbol{\sigma}-\nabla\cdot\boldsymbol{\sigma}_{h},\nabla\cdot\boldsymbol{\sigma}-\nabla\cdot\boldsymbol{\sigma}_{h}\rangle$
$\displaystyle=\langle\nabla\cdot\boldsymbol{\sigma}-\Pi_{h,3}^{2}\nabla\cdot\boldsymbol{\sigma},\nabla\cdot\boldsymbol{\sigma}-\nabla\cdot\boldsymbol{\sigma}_{h}\rangle$
$\displaystyle\ \ \ \
+\underbrace{\langle\Pi_{h,3}^{2}\nabla\cdot\boldsymbol{\sigma}-\nabla\cdot\boldsymbol{\sigma}_{h},\nabla\cdot\boldsymbol{\sigma}-\nabla\cdot\boldsymbol{\sigma}_{h}\rangle}_{=0}$
$\displaystyle\leq\left\lVert\nabla\cdot\boldsymbol{\sigma}-\Pi_{h,3}^{2}\nabla\cdot\boldsymbol{\sigma}\right\rVert_{\boldsymbol{L}^{2}}\
\left\lVert\nabla\cdot\boldsymbol{\sigma}-\nabla\cdot\boldsymbol{\sigma}_{h}\right\rVert_{\boldsymbol{L}^{2}}.$

Thus, again in view of Lemma 2, we get
$\hskip
28.45274pt\left\lVert\nabla\cdot\boldsymbol{\sigma}-\nabla\cdot\boldsymbol{\sigma}_{h}\right\rVert_{\boldsymbol{L}^{2}}\leq\left\lVert\nabla\cdot\boldsymbol{\sigma}-\Pi_{h,3}^{2}\nabla\cdot\boldsymbol{\sigma}\right\rVert_{\boldsymbol{L}^{2}}\leq
Ch^{k}\left\lVert\nabla\cdot\boldsymbol{\sigma}\right\rVert_{\boldsymbol{H}^{k}}.$
∎

In the next section we try to generalize the discretization approach from
above with weakly imposed symmetry to more general settings including multi-
patch parametrizations and the case with traction boundary conditions.

###  3.3 Remarks concerning generalizations

####  3.3.1 Higher regularity splines

A major advantage of the B-spline spaces is the possibility to increase the
global smoothness without worsening the approximation capabilities, where the
polynomial degrees bound the regularity parameter. To be more precise, one can
choose for example $r\leq p-1$ in the definition of the isogeometric de Rham
spaces. Therefore, a generalization of the discretization method for $r>0$
seems natural. In fact, for most of the steps above we do not need to set
$r=0$ and a lot of properties like the approximation estimates in Lemma 2 and
(15) are still valid if the regularity parameter is increased considering the
underlying polynomial degrees. Only the inf-sup stability result from Lemma 5
uses $r=0$. But in the IGA community there is numerical evidence indicating
the uniform inf-sup stability of the Taylor-Hood pair also for $r>0$; see
[10]. Hence, if we make the reasonable assumption of inf-sup stability of the
Taylor-Hood space pair for increased regularity parameter we obtain error
estimates analogous to the ones in Theorem 3 also for $1\leq r+1<p$. The
numerical experiments we carried out in Section 5 utilizing splines with
higher regularity support the assumption of stability.

####  3.3.2 Mixed boundary conditions

Up to now, we supposed Dirichlet data for the displacement $\boldsymbol{u}$
which are incorporated in a weak sense. Consequently, also for
$\boldsymbol{u}_{D}=0$ the discrete solution might not be exactly zero on
$\partial\Omega$ in contrast to the standard approximations via the primal
weak form where zero boundary conditions (BCs) are fulfilled exactly; see (4).
Another central boundary condition within linear elasticity is the traction
boundary condition corresponding to an application of a boundary force. In the
mentioned primal formulation with unknown $\boldsymbol{u}$ such force
conditions can be incorporated in the sense of Neumann type BCs. From the
mixed method point of view this changes and traction forces are now
implemented explicitly, in a strong sense, respectively. First of all, we
consider the model problem

$\displaystyle\boldsymbol{A}\boldsymbol{\sigma}$
$\displaystyle=\boldsymbol{\varepsilon}(\boldsymbol{u}),\hskip
73.97733pt\nabla\cdot\boldsymbol{\sigma}$ $\displaystyle=\boldsymbol{f}\ \ \ \
\textup{in}\ \ \Omega,$ (25) $\displaystyle\boldsymbol{u}$
$\displaystyle=\boldsymbol{u}_{D}\ \ \ \textup{on}\ \Gamma_{D},\hskip
42.67912pt\boldsymbol{\sigma}\cdot\boldsymbol{n}$
$\displaystyle=\boldsymbol{t}_{n}\ \ \ \textup{on}\ \Gamma_{t},$ (26)

with $\Gamma_{D}\cap\Gamma_{t}=\emptyset,\
\overline{\Gamma_{D}}\cup\overline{\Gamma_{t}}=\Gamma\coloneqq\partial\Omega$
and $\boldsymbol{n}$ denoting the outer unit normal to the domain boundary.
Further we require that $\Gamma_{D},\ \Gamma_{t}$ are composed of full mesh
edges of the initial mesh and $\emptyset\neq\Gamma_{D}$. One notes the well-
definedness of the boundary condition in case of
$\boldsymbol{\sigma}\in\boldsymbol{\mathcal{H}}(\textup{div},\Omega)$ and
$\boldsymbol{t}_{n}\in\boldsymbol{H}^{-1/2}(\Omega)$ due to the next lemma;
see e.g. [3]. 333$\boldsymbol{H}^{-1/2}(\Omega)$ is the dual space of
$\boldsymbol{H}^{1/2}(\Omega)$ which is in turn the image of the classical
(component-wise) trace operator $\gamma\colon H^{1}(\Omega)\rightarrow
H^{1/2}(\Gamma)$; see [26, Section 2].

######  Lemma 7 (Normal trace space of $\boldsymbol{H}(\textup{div})$).

There is a bounded trace operator

$\gamma_{n}\colon\boldsymbol{H}(\Omega,\textup{div})\rightarrow
H^{-1/2}(\partial\Omega),$

s.t.
$\gamma_{n}(\boldsymbol{v})=(\boldsymbol{v}\cdot\boldsymbol{n})_{|\partial\Omega}$
for all $\boldsymbol{v}\in C^{1}(\overline{\Omega})$.

######  Remark 2.

The last result can be generalized to matrix fields through a row-wise
definition and we refer to [3, Section 3.4] for more information about spaces
like $H^{-1/2}(\partial\Omega)$. Although the mentioned reference concentrates
on the three-dimensional setting, the $2D$ adaption is straight-forward.

For $\boldsymbol{t}_{n}=\boldsymbol{0}$ the traction condition can be
fulfilled exactly in the numerical computations by dropping all basis
functions violating it. Let now $\boldsymbol{t}_{n}$ be determined by the
restriction
$\boldsymbol{t}_{n}=\tilde{\boldsymbol{\sigma}}\cdot\boldsymbol{n}$ of some
given
$\tilde{\boldsymbol{\sigma}}\in\boldsymbol{\mathcal{H}}(\Omega,\textup{div})$
to the boundary, then we can easily adapt the problem to the homogeneous case
by seeking $\boldsymbol{\sigma}-\tilde{\boldsymbol{\sigma}}$. If such an
auxiliary mapping is not known or if it does not exist one can try to satisfy
the traction BC approximately by computing a suitable approximate mapping

$\tilde{\boldsymbol{t}}_{n}\approx\boldsymbol{t}_{n},\ \
\tilde{\boldsymbol{t}}_{n}=\textstyle\sum_{i}c_{i}\boldsymbol{\tau}_{h,i},\ \
\boldsymbol{\tau}_{h,i}\in\boldsymbol{\mathcal{V}}_{h,2}^{w},\
c_{i}\in\mathbb{R},$

utilizing the underlying basis functions. However, in the rest of this section
we assume $\boldsymbol{t}_{n}=\boldsymbol{0}$. Then the discrete spaces have
to be changed according to

$\displaystyle\boldsymbol{\mathcal{V}}_{h,2}^{w}\rightarrow\boldsymbol{\mathcal{V}}_{0,h,2}^{w}\coloneqq\big{(}\boldsymbol{V}_{0,h,2}^{2}\times\boldsymbol{V}_{0,h,2}^{2}\big{)}^{T},\
\
\boldsymbol{V}_{0,h,2}^{2}\coloneqq\boldsymbol{V}_{h,2}^{2}\cap\\{\boldsymbol{v}|\boldsymbol{v}\cdot\boldsymbol{n}=0\
\textup{on}\ \Gamma_{t}\\},$ (27) $\displaystyle\textup{and in
\eqref{choice_3} we replace
}\boldsymbol{V}_{h}^{TH}\rightarrow\boldsymbol{V}_{0,h}^{TH}\coloneqq\boldsymbol{V}_{h}^{TH}\cap\\{\boldsymbol{v}|\boldsymbol{v}=\boldsymbol{0}\
\textup{on}\ \Gamma_{t}\\}.$ (28)

Obviously, it is possible to apply Lemma 3 for the new function spaces to
check for inf-sup stability. The modification of the auxiliary test space
$\boldsymbol{V}_{h}^{TH}$ above is explained by the next lemma.

$\hat{\Gamma}_{1}$$\hat{\Gamma}_{2}$$\hat{\Gamma}_{3}$$\hat{\Gamma}_{4}$$\hat{\Gamma}_{t}$$\hat{\Gamma}_{D}$$\zeta_{1}$$\zeta_{2}$

$\boldsymbol{F}$

${\Gamma}_{1}$${\Gamma}_{2}$${\Gamma}_{3}$${\Gamma}_{4}$

Figure 2: Notation for the boundary edges. We assume
$\Gamma_{i}=\boldsymbol{F}(\hat{\Gamma}_{i})$. (Figure generated utilizing
[27].)

######  Lemma 8.

Assume $\boldsymbol{v}\coloneqq(v_{1},v_{2})^{T}\in\boldsymbol{V}_{0,h}^{TH}$.
Then we have $\textup{curl}(v_{i})\in\boldsymbol{V}_{0,h,2}^{2}$.

###### Proof.

Since it is $\textup{curl}(v_{i})\in\boldsymbol{V}_{h,2}^{2}$ and due to the
fact that $\mathcal{Y}_{1}$ and $\mathcal{Y}_{3}$ preserve the Dirichlet BC,
the zero normal trace, respectively, it is enough to verify the assertion for
$\boldsymbol{F}=\textup{id}$. We check it for the case
$\Gamma_{t}\subset\hat{\Gamma}_{1}\coloneqq\\{(0,\zeta_{2})\
|\zeta_{2}\in[0,1]\\}$ of $\widehat{\Omega}$. And indeed, if $v_{i}=0$ on
${\Gamma}_{t}$, then it is $\hat{\partial}_{2}v_{i}=0$ on ${\Gamma}_{t}$,
where $\hat{\partial}_{j}$ denotes the derivative w.r.t. the $j$-th parametric
coordinate. Thus,
$\widehat{\textup{curl}}(v_{i})\cdot(-1,0)^{T}=-\hat{\partial}_{2}v_{i}=0$ on
${\Gamma}_{t}$. Other cases follow by a similar reasoning. ∎

In view of latter lemma it remains to check whether the both inf-sup
conditions of Lemma 3 are still valid for the spaces with BCs. Currently, we
are only able to prove the inf-stability if we make the next assumption.

######  Assumption 2.

Let us assume that there is a boundary edge which is not part of $\Gamma_{t}$,
i.e. we have an edge $\Gamma_{i}$ s.t. $\Gamma_{i}\subset\Gamma_{D}$; see Fig.
2.

Nevertheless, as long as $\Gamma_{t}\subsetneq\partial\Omega$, we think that
stability is still valid.

######  Lemma 9.

Let $\Gamma_{D}$ fulfill Assumption 2 and $p\geq 2,\ r=0$. Then, for the
choice (21),
$\boldsymbol{W}_{h}^{1}\coloneqq\boldsymbol{V}_{0,h,2}^{2}$ and
$\boldsymbol{W}^{0}_{h}\times
W^{3}_{h}\coloneqq\big{(}\boldsymbol{V}_{0,h}^{TH},V_{h}^{TH}\big{)}$ both
inf-sup conditions of Lemma 3 are satisfied.

###### Proof.

W.l.o.g. we suppose
$\Gamma_{t}=\boldsymbol{F}(\hat{\Gamma}_{t})\subset\boldsymbol{F}(\hat{\Gamma}_{1})\cup\boldsymbol{F}(\hat{\Gamma}_{2})\cup\boldsymbol{F}(\hat{\Gamma}_{3})$;
see Fig. 2.
Let $\hat{p}\in\widehat{V}_{h,3}^{2}$ arbitrary but fixed.
Define $\hat{\boldsymbol{v}}\coloneqq(\hat{v}_{1},\hat{v}_{2})^{T}$ by

$\displaystyle\hat{v}_{1}(\zeta_{1},\zeta_{2})$
$\displaystyle\coloneqq\frac{1}{2}\Big{(}\int_{0}^{\zeta_{1}}\hat{p}(\tau,\zeta_{2})-\int_{0}^{1}\hat{p}(\eta,\zeta_{2})d\eta\
d\tau\Big{)},$ $\displaystyle\hat{v}_{2}(\zeta_{1},\zeta_{2})$
$\displaystyle\coloneqq\frac{1}{2}\Big{(}\int_{0}^{\zeta_{2}}\hat{p}(\zeta_{1},\tau)+\int_{0}^{1}\hat{p}(\eta,\tau)d\eta\
d\tau\Big{)}.$

Since $\hat{p}$ is the linear combination of product spline functions of the
form
$\widehat{B}_{\boldsymbol{i},\boldsymbol{p}}(\zeta_{1},\zeta_{2})=\widehat{B}_{i_{1},p-1}(\zeta_{1})\cdot\widehat{B}_{i_{2},p-1}(\zeta_{2})\in
S_{p-1,p-1}^{-1,-1}$. And since we have for such a spline

$\displaystyle\int_{0}^{\zeta_{1}}\widehat{B}_{i_{1},p}(\tau)\cdot\widehat{B}_{i_{2},p}(\zeta_{2})d\tau$
$\displaystyle=\int_{0}^{\zeta_{1}}\widehat{B}_{i_{1},p}(\tau)d\tau\cdot\widehat{B}_{i_{2},p}(\zeta_{2})\in
S_{p,p-1}^{0,-1},$
$\displaystyle\int_{0}^{\zeta_{1}}\int_{0}^{1}\widehat{B}_{i_{1},p}(\eta)\cdot\widehat{B}_{i_{2},p}(\zeta_{2})d\eta
d\tau$ $\displaystyle=\int_{0}^{1}\widehat{B}_{i_{1},p}(\eta)\
d\eta\cdot\int_{0}^{\zeta_{1}}\widehat{B}_{i_{2},p}(\zeta_{2})d\tau,$
$\displaystyle=\zeta_{1}\Big{(}\int_{0}^{1}\widehat{B}_{i_{1},p}(\eta)\
d\eta\Big{)}\cdot\widehat{B}_{i_{2},p}(\zeta_{2})\in S_{p,p-1}^{0,-1},$
$\displaystyle\int_{0}^{\zeta_{2}}\widehat{B}_{i_{1},p}(\zeta_{1})\cdot\widehat{B}_{i_{2},p}(\tau)d\tau$
$\displaystyle=\widehat{B}_{i_{1},p}(\zeta_{1})\cdot\int_{0}^{\zeta_{2}}\widehat{B}_{i_{2},p}(\tau)d\tau\in
S_{p-1,p}^{-1,0},$
$\displaystyle\int_{0}^{\zeta_{2}}\int_{0}^{1}\widehat{B}_{i_{1},p}(\eta)\cdot\widehat{B}_{i_{2},p}(\tau)d\eta
d\tau$
$\displaystyle=\int_{0}^{1}\widehat{B}_{i_{1},p}(\eta)d\eta\cdot\int_{0}^{\zeta_{2}}\widehat{B}_{i_{2},p}(\tau)d\tau\in
S_{p-1,p}^{-1,0},$

we obtain $\hat{\boldsymbol{v}}\in\widehat{\boldsymbol{V}}_{h,2}^{2}$. Using
the Cauchy-Schwarz inequality one can check that
$\left\lVert\hat{\boldsymbol{v}}\right\rVert_{\boldsymbol{L}^{2}}\leq
2\left\lVert\hat{p}\right\rVert_{L^{2}}.$ Further, we have

$\widehat{\nabla}\cdot\hat{\boldsymbol{v}}=\frac{1}{2}\Big{(}\hat{p}-\int_{0}^{1}\hat{p}(\eta,\zeta_{2})d\eta+\hat{p}+\int_{0}^{1}\hat{p}(\eta,\zeta_{2})d\eta\Big{)}=\hat{p}.$

Consequently,
$\left\lVert\hat{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}(\widehat{\textup{div}})}\leq\sqrt{5}\left\lVert\hat{p}\right\rVert_{L^{2}}$
and it is

$\displaystyle{\langle\nabla\cdot\hat{\boldsymbol{v}},\hat{p}\rangle}\geq\frac{1}{\sqrt{5}}\left\lVert\hat{p}\right\rVert_{L^{2}}\
\left\lVert\hat{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}(\widehat{\textup{div}})}.$

By the arbitrariness of $\hat{p}$ and the fact that

$\displaystyle\hat{v}_{1}(0,\zeta_{2})=\hat{v}_{1}(1,\zeta_{2})=0\ \ \
\textup{and}\ \ \ \hat{v}_{2}(\zeta_{1},0)=0,$

we get with
$\widehat{\boldsymbol{V}}_{0,h,2}^{2}\coloneqq\mathcal{Y}_{3}({\boldsymbol{V}}_{0,h,2}^{2})$:

$\displaystyle\underset{\hat{\boldsymbol{w}}\in\widehat{\boldsymbol{V}}_{0,h,2}^{2}}{\sup}\frac{\langle\nabla\cdot\hat{\boldsymbol{w}},\hat{q}\rangle}{\left\lVert\hat{\boldsymbol{w}}\right\rVert_{\boldsymbol{H}(\widehat{\textup{div}})}}\geq\frac{1}{\sqrt{5}}\left\lVert\hat{q}\right\rVert_{L^{2}},\hskip
8.5359pt\forall\hat{q}\in\widehat{V}_{h,3}^{2}.$

Next we use the commutativity property (14) of the pullbacks
$\mathcal{Y}_{i}$. We note that for
$\hat{\boldsymbol{v}}=\mathcal{Y}_{3}(\boldsymbol{v})\in\widehat{\boldsymbol{V}}_{0,h,2}^{2}$
and $\hat{p}=\mathcal{Y}_{4}(p)\in\widehat{V}_{h,3}^{2}$ we have

$\left\lVert\boldsymbol{v}\right\rVert_{\boldsymbol{H}(\textup{div})}\leq
C\left\lVert\hat{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}(\widehat{\textup{div}})},\hskip
28.45274ptC_{1}\left\lVert\hat{p}\right\rVert_{L^{2}}\geq\left\lVert
p\right\rVert_{L^{2}}\geq C_{2}\left\lVert\hat{p}\right\rVert_{L^{2}},\ $

where $C,C_{i}$ are suitable constants depending only on $\boldsymbol{F}$ and
$\Omega$. Hence, one obtains for a $q=\mathcal{Y}_{4}(\hat{q})\in V_{h,3}^{2}$
arbitrary

$\displaystyle\underset{{\boldsymbol{w}}\in{\boldsymbol{V}}_{0,h,2}^{2}}{\sup}\frac{\langle\nabla\cdot{\boldsymbol{w}},{q}\rangle}{\left\lVert{\boldsymbol{w}}\right\rVert_{\boldsymbol{H}(\textup{div})}}\geq
C\underset{\hat{\boldsymbol{w}}\in\widehat{\boldsymbol{V}}_{0,h,2}^{2}}{\sup}\frac{\langle\nabla\cdot\hat{\boldsymbol{w}},\hat{q}\rangle}{\left\lVert\hat{\boldsymbol{w}}\right\rVert_{\boldsymbol{H}(\widehat{\textup{div}})}}\geq
C\left\lVert\hat{q}\right\rVert_{L^{2}}\geq C\left\lVert
q\right\rVert_{L^{2}}.$

And thus the first inf-sup condition follows.
Now we face the second inf-sup condition. By assumption there is an edge
${\Gamma}_{i}$ which is not part of the traction boundary, i.e.
$\Gamma_{t}\neq\partial\Omega$. But then the application of Lemma 5 yields the
wanted estimate if we set $\Gamma_{B}=\Gamma_{t}$.
This finishes the proof. ∎

In view of the Lemma above, the mixed formulation with mixed boundary
condition is well-posed, at least if
$\boldsymbol{t}_{n}=\tilde{\boldsymbol{\sigma}}\cdot\boldsymbol{n}$ for some
given
$\tilde{\boldsymbol{\sigma}}\in\boldsymbol{\mathcal{H}}(\Omega,\textup{div})$
and if Assumption 2 is fulfilled. We have the subsequent result:

######  Corollary 1.

Let the Assumption 2 be valid and let the boundary data regular enough, i.e.
the exact solution exists and the problem can be reduced to the case of
$\boldsymbol{t}_{n}=\boldsymbol{0}$ on $\Gamma_{t}$. Then, the discrete
problem with mixed boundary conditions is well-posed, at least for $h$ small
enough.

####  3.3.3 Multi-patch parametrizations

Up to now we considered parametrizations of the form
$\boldsymbol{F}\colon[0,1]^{2}\rightarrow\mathbb{R}^{2}$ meaning we have a so-
called single-patch parametrization. Frequently in applications computational
domains have to be considered which can not be parameterized accurately by
means of a single patch. To achieve more flexibility one decomposes $\Omega$
into several simpler shapes $\Omega_{i}$ and defines a parametrization for
each patch separately. In other words, in the multi-patch framework one has

$\displaystyle\overline{\Omega}=\bigcup_{i=1}^{m}\overline{\Omega}_{i},\ \ \ \
\ \ \hskip
2.84544pt\boldsymbol{F}_{i}\colon[0,1]^{2}\rightarrow\overline{\Omega}_{i},$

with $m\in\mathbb{N}$ denoting the number of patches. For reasons of
simplification we restrict ourselves to a special case of multi-patch
discretizations, i.e. we make the next assumptions.

######  Assumption 3.

Each patch parametrization $\boldsymbol{F}_{i}$ fulfills Assumption 1.
Moreover, for $i\neq j$ it is either
$\overline{\Omega_{i}}\cap\overline{\Omega_{j}}=\emptyset$ or the two patches
share a boundary edge or a single vertex. Further, we assume for each patch
$\Omega_{i}$ the existence of an edge $S^{(i)}\subset\partial\Omega_{i}$ s.t.
$S^{(i)}\subset\partial\Omega$.

For the discretization we further assume:

######  Assumption 4.

For each patch we have the same underlying polynomial degree $p$, regularity
parameter $r=0$ and the same parametric mesh.

The multi-patch spaces are then defined by means of the single-patch spaces.
For example, we have the multi-patch Taylor-Hood pair

$\displaystyle\boldsymbol{V}_{h}^{TH,M}$
$\displaystyle\coloneqq\\{\boldsymbol{v}\in\boldsymbol{H}^{1}(\Omega)\ |\
\boldsymbol{v}_{|\Omega_{i}}\in\boldsymbol{V}_{h}^{TH,i},\ \forall i\\},$
$\displaystyle{V}_{h}^{TH,M}$ $\displaystyle\coloneqq\\{{v}\in{H}^{1}(\Omega)\
|\ {v}_{|\Omega_{i}}\in{V}_{h}^{TH,i},\ \forall i\\},$

where the upper index $i$ is used to identify the IGA spaces in the $i$-th
patch, i.e. $\boldsymbol{V}_{h}^{TH,i}\coloneqq(S_{p,p}^{r,r}\times
S_{p,p}^{r,r})\circ\boldsymbol{F}_{i}^{-1},\ $
$V_{h,3}^{2,i}\coloneqq\widehat{V}_{h,3}^{2}\circ\boldsymbol{F}_{i}^{-1}$ and
so on. The rest of the multi-patch spaces

$\boldsymbol{\mathcal{V}}_{h,2}^{w,M}\coloneqq\\{\boldsymbol{\tau}\in\boldsymbol{\mathcal{H}}(\Omega,\textup{div})\
|\ \boldsymbol{\tau}_{|\Omega_{i}}\in\boldsymbol{\mathcal{V}}_{h,2}^{w,i},\
\forall i\\},\ {V}_{h,3}^{2,M}\coloneqq\\{{q}\in{L}^{2}(\Omega)\ |\
q_{|\Omega_{i}}\in{V}_{h,3}^{2,i},\ \forall i\\},\ \ \dots$

are defined in an analogous manner.

In this subsection we are again in the context of pure displacement boundary
conditions. A combination of mixed boundary conditions with multi-patch
parametrizations seems possible, but is not considered here; see Remark 3.

To obtain a well-posed method in the multi-patch framework, we have to check
again the requirements in Lemma 3, where the appearing spaces have to be
replaced by the multi-patch pendants. We do not prove the multi-patch inf-sup
stability in general, but with the additional restrictions determined by the
above assumptions we succeed. On the one hand we have the result:

######  Lemma 10.

Let the multi-patch discretization be regular as described in Assumption 3 and
Assumption 4.
Then there exist mesh size independent constants $h_{\textup{max}}>0$,
$C_{S}>0$ s.t.

$\displaystyle\underset{q_{h}\in V_{h}^{TH,M}\textbackslash\\{0\\}}{\inf}\
\underset{\boldsymbol{w}_{h}\in\boldsymbol{V}_{h}^{TH,M}}{\sup}\frac{\langle\nabla\cdot\boldsymbol{w}_{h},q_{h}\rangle}{\left\lVert\boldsymbol{w}_{h}\right\rVert_{\boldsymbol{H}^{1}}\
\left\lVert q_{h}\right\rVert_{L^{2}}}\geq C_{S},\ \ \forall h\leq h_{max}.$
(29)

###### Proof.

Unfortunately, we can not apply Lemma 5 directly, since the $H^{1}$-condition
at the interfaces has to be taken into account. Therefore, we first consider
the auxiliary space

$\tilde{\boldsymbol{V}}_{h}\coloneqq\\{\boldsymbol{v}\in\boldsymbol{L}^{2}(\Omega)\
|\
\boldsymbol{v}_{|\Omega_{i}}\in\boldsymbol{V}_{h}^{TH,i}\cap\boldsymbol{H}^{1}_{\Gamma_{D}^{i}}(\Omega_{i})\\},$

where $\boldsymbol{V}_{h}^{TH,i}$ is just the space corresponding to the
parametrization $\boldsymbol{F}_{i}$ and with
$\Gamma_{D}^{i}\coloneqq\partial\Omega_{i}\textbackslash S^{(i)}$. Then, the
global $H^{1}$-regularity is implied, i.e.
$\tilde{\boldsymbol{V}}_{h}\subset\boldsymbol{V}_{h}^{TH,M}$. Hence, we obtain
by means of Lemma 5 constants $C_{i}$ and mesh sizes $h_{\textup{max},i}$ such
that

$\displaystyle\underset{\boldsymbol{w}\in\tilde{\boldsymbol{V}}_{h}}{\sup}\frac{\langle\nabla\cdot\boldsymbol{w},q\rangle_{\Omega_{i}}}{\left\lVert\boldsymbol{w}\right\rVert_{\boldsymbol{H}^{1}(\Omega_{i})}}\geq\left\lVert
q\right\rVert_{L^{2}(\Omega_{i})}\geq C_{i}>0,\ \ \forall q\in V_{h}^{TH,i},\
\ h\leq h_{max,i}.$

Let now $q\in V_{h}^{TH,M}\textbackslash\\{0\\}$ arbitrary but fixed and
$J\subset\\{1,\dots,m\\}$ the set of all indices $j$ with $q_{|\Omega_{j}}\neq
0$. For $h\leq\min\\{h_{\textup{max},i}\\}$ we obtain:

$\displaystyle\underset{\boldsymbol{w}\in\boldsymbol{V}_{h}^{TH,M}}{\sup}\frac{\langle\nabla\cdot\boldsymbol{w},q\rangle_{\Omega}}{\left\lVert\boldsymbol{w}\right\rVert_{\boldsymbol{H}^{1}(\Omega)}}$
$\displaystyle\geq\sum_{j\in J}\
\underset{\boldsymbol{w}_{j}\in\boldsymbol{V}_{h}^{TH,j}\cap\boldsymbol{H}^{1}_{\Gamma_{D}^{j}}(\Omega_{j})}{\sup}\frac{\langle\nabla\cdot\boldsymbol{w}_{j},q\rangle_{\Omega_{j}}}{\left\lVert\boldsymbol{w}_{j}\right\rVert_{\boldsymbol{H}^{1}(\Omega_{j})}\
}$ $\displaystyle\geq\sum_{j\in J}C_{j}\ \left\lVert
q\right\rVert_{L^{2}(\Omega_{j})}\geq\underset{i}{\textup{min}}\\{C_{i}\\}\left\lVert
q\right\rVert_{L^{2}(\Omega)}.$

Consequently, the inequality in the assertion is true for
$C_{S}=\underset{i}{\textup{min}}\\{C_{i}\\}>0$ and
$h_{\max}=\underset{i}{\textup{min}}\\{h_{\max,i}\\}$. ∎

On the other hand one sees:

######  Lemma 11.

Under the same conditions as the previous Lemma 10, the first inf-sup
condition from Lemma 3 is satisfied in the multi-patch setting, i.e. there is
a constant $0<C_{P}$ s.t.

$\underset{p_{h}\in V_{h,3}^{2,M}\textbackslash\\{0\\}}{\inf}\
\underset{\boldsymbol{v}_{h}\in\boldsymbol{V}_{h,2}^{2,M}}{\sup}\frac{\langle\nabla\cdot\boldsymbol{v}_{h},p_{h}\rangle}{\left\lVert\boldsymbol{v}_{h}\right\rVert_{\boldsymbol{H}(\textup{div})}\
\left\lVert p_{h}\right\rVert_{L^{2}}}\geq C_{P}$

###### Proof.

We only give the proof idea since the approach is very similar to the steps in
proof of Lemma 10.
By assumption, we have for each patch an edge which is free, i.e. which is not
part of an interface. Then the patch-wise application of Lemma 9 implies the
wanted inf-sup condition. ∎

With a row-wise curl application we further see easily
$\textup{curl}(\boldsymbol{V}_{h}^{TH,M})\subset\boldsymbol{\mathcal{V}}_{h,2}^{w,M}$.
Thus, using Lemmas 10 and 11, we see with Lemma 3:

######  Corollary 2.

Let Assumptions 3 and 4 hold. Then the straight-forward adaption of the
discrete mixed system (3) to the multi-patch setting leads to a well-posed
problem provided that $h$ is small enough; see Lemma 10.

######  Remark 3.

Obviously, it makes sense to combine the different generalization steps,
meaning we have for example multi-patch parametrizations endowed with mixed
boundary conditions. However, we restricted ourselves to special situations;
see Assumption 2 and Assumptions 3 and 4. Although a strict proof is missing,
we think that stability and convergence of the discretization method can be
achieved also for more general parametrizations and mixed boundary conditions.

After we studied the mixed system with weakly imposed symmetry we face the
more complicated situation of test spaces with strong symmetry.

##  4 Mixed formulation with strong symmetry

Here we want to show the existence of suitable spline spaces for the case of
planar linear elasticity with strong symmetry. Later, the actual proofs are
within the scope of pure Dirichlet problems, meaning
$\Gamma_{D}=\partial\Omega$. But before we are able to define proper discrete
spaces we first look at an auxiliary problem endowed with a particular mixed
boundary condition. Besides, in the whole section we consider only single-
patch parametrizations.

###  4.1 An auxiliary mixed BC problem

To begin with, we assume for the underlying problem:

######  Assumption 5 (Mixed boundary conditions).

We have the mixed boundary conditions

$\displaystyle\boldsymbol{\sigma}\cdot\boldsymbol{n}=\boldsymbol{0}\ \
\textup{on}\ \ \Gamma_{t}=\boldsymbol{F}(\hat{\Gamma}_{1}),\ \
\boldsymbol{u}=\boldsymbol{u}_{D}\in\boldsymbol{H}^{1/2}(\partial\Omega)\ \
\textup{on}\ \Gamma_{D}=\partial\Omega\textbackslash\Gamma_{t}.$ (30)

For the labeling of the boundary edges we refer to Fig. 2 and we require
$\boldsymbol{F}^{-1}\in C^{3}(\overline{\Omega}),\ \boldsymbol{F}\in
C^{3}(\overline{\widehat{\Omega}})$.

In other words we consider the weak formulation: _Find_
$(\boldsymbol{\sigma},\boldsymbol{u})\in\boldsymbol{\mathcal{H}}_{\Gamma_{1}}(\Omega,\textup{div},\mathbb{S})\times\boldsymbol{L}^{2}(\Omega)$
such that

$\displaystyle\langle\boldsymbol{A}\boldsymbol{\sigma},\boldsymbol{\tau}\rangle\
+\ $ $\displaystyle\langle\boldsymbol{u},\nabla\cdot\boldsymbol{\tau}\rangle\
\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle=\langle\boldsymbol{\tau}\cdot\boldsymbol{n};\boldsymbol{u}_{D}\rangle_{\Gamma},\
\hskip 11.38092pt$
$\displaystyle\forall\boldsymbol{\tau}\in\boldsymbol{\mathcal{{H}}}_{\Gamma_{1}}(\Omega,\textup{div},\mathbb{S}),$
$\displaystyle\langle\nabla\cdot\boldsymbol{\sigma},\boldsymbol{v}\rangle$
$\displaystyle=\langle\boldsymbol{f},\boldsymbol{v}\rangle,\ \hskip
11.38092pt$ $\displaystyle\forall\boldsymbol{v}\in\boldsymbol{L}^{2}(\Omega),$
(31)

with

$\displaystyle\boldsymbol{\mathcal{{H}}}_{\Gamma_{1}}(\Omega,\textup{div},\mathbb{S})\coloneqq\overline{\\{\boldsymbol{\tau}\in\boldsymbol{\mathcal{{H}}}(\Omega,\textup{div},\mathbb{S})\cap
C^{1}(\overline{\Omega})\ |\
\boldsymbol{\tau}\cdot\boldsymbol{n}=\boldsymbol{0}\ \textup{on}\
\Gamma_{1}\\}}^{\boldsymbol{\mathcal{H}}(\textup{div})},$ (32)

where on right-hand side of the last line we have the closure of the space in
brackets w.r.t. the norm
$\left\lVert\cdot\right\rVert_{\boldsymbol{\mathcal{H}}(\textup{div})}$. For
the sake of clarity we write
$\boldsymbol{\mathcal{H}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$
and
$\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$
for the corresponding spaces on the parametric domain. For more information
about such spaces with boundary conditions we refer to the articles [20, 21].
In latter references the explanations are for the three-dimensional case, but
they can be adapted easily to planar domains. To simplify the expressions we
will use the _Einstein summation convention_ , where the different indices
take the values $1$ and $2$. Further, we use several times the dot "$\cdot$"
to indicate a matrix multiplication or scalar multiplication. The usage of the
dot sign should clarify the appearing longer terms.

Basic idea of our approach is to work in the parametric domain and to use in
some sense proper pullback and push-forward operations to handle the situation
in $\Omega$. Therefore, we define the next two transformations

$\displaystyle\mathcal{Y}_{2,\Gamma_{1}}^{s}$
$\displaystyle\colon\boldsymbol{\mathcal{L}}^{2}(\Omega,\mathbb{S})\rightarrow\boldsymbol{\mathcal{L}}^{2}(\widehat{\Omega},\mathbb{S}),$
(33)
$\displaystyle\boldsymbol{S}\mapsto\textup{det}(\boldsymbol{J})^{2}\boldsymbol{J}^{-1}(\boldsymbol{S}\circ\boldsymbol{F})\boldsymbol{J}^{-T}+\widehat{\textup{Airy}}({F}_{n})\cdot\Big{(}\int_{0}^{\zeta_{1}}(-1)^{n+1}\
\textup{det}(\boldsymbol{J})\ J_{2l}^{-1}\
(S_{\sigma(n)l}\circ\boldsymbol{F})\ d1\Big{)},$

with $\sigma(n)\coloneqq 3-n,\ \boldsymbol{F}=(F_{1},F_{2})^{T}$ and

$\widehat{\textup{Airy}}(\hat{\phi})\coloneqq\begin{pmatrix}\hat{\partial}_{22}\hat{\phi}&-\hat{\partial}_{12}\hat{\phi}\\\
-\hat{\partial}_{21}\hat{\phi}&\hat{\partial}_{11}\hat{\phi}\end{pmatrix},\ \
\ \
\textup{Airy}({\phi})\coloneqq\begin{pmatrix}{\partial}_{22}{\phi}&-{\partial}_{12}{\phi}\\\
-{\partial}_{21}{\phi}&{\partial}_{11}{\phi}\end{pmatrix},\ \
\textup{respectively}.$

Above "$\ d1$" means that we integrate w.r.t. to the first parametric
coordinate, i.e. for some $g=g(\zeta_{1},\zeta_{2})$ we have
$\int_{0}^{\zeta_{1}}g\
d1\coloneqq\int_{0}^{\zeta_{1}}g(\tau,\zeta_{2})d\tau$. Besides,
$\hat{\partial}_{i},\ \partial_{i}$ denote the derivatives w.r.t. to the
$i$-th parametric coordinate, $i$-th physical coordinate respectively, and we
write $\hat{\partial}_{ij},\ \partial_{ij}$ for
$\hat{\partial}_{i}\hat{\partial}_{j}$ and $\partial_{i}\partial_{j}$. We
remark that $J_{ij}^{-1}=J_{ij}^{-1}(\boldsymbol{x})$ stands for the entry
$(i,j)$ of the Jacobian $\boldsymbol{J}^{-1}$ of $\boldsymbol{F}^{-1}$. With
the relation $\boldsymbol{F}(\zeta_{1},\zeta_{2})=\boldsymbol{x}$ we interpret
the integrand functions in the whole section as mappings defined on
$\widehat{\Omega}$. For example $\int_{0}^{\zeta_{1}}J_{ij}^{-1}\
d1\coloneqq\int_{0}^{\zeta_{1}}J_{ij}^{-1}\circ\boldsymbol{F}(\tau,\zeta_{2})d\tau$.
With the relation for the adjugate matrix
$\textup{adj}(\boldsymbol{J})=\textup{det}(\boldsymbol{J})(\boldsymbol{J}^{-1}\circ\boldsymbol{F})$
and using $\tilde{\boldsymbol{J}}\coloneqq\textup{adj}(\boldsymbol{J})$ we can
write

$\mathcal{Y}_{2,\Gamma_{1}}^{s}(\boldsymbol{S})=\tilde{\boldsymbol{J}}(\boldsymbol{S}\circ\boldsymbol{F})\tilde{\boldsymbol{J}}^{T}+\widehat{\textup{Airy}}({F}_{n})\cdot\Big{(}\int_{0}^{\zeta_{1}}(-1)^{n+1}\
\tilde{{J}}_{2l}\ (S_{\sigma(n)l}\circ\boldsymbol{F})\ d1\Big{)}.$ (34)

One notes the simple structure of $\tilde{\boldsymbol{J}}$, namely

$\tilde{\boldsymbol{J}}\coloneqq\textup{adj}(\boldsymbol{J})=\begin{bmatrix}J_{22}&-J_{12}\\\
-J_{21}&J_{11}\end{bmatrix}.$

Secondly, we introduce the mapping

$\mathcal{Y}_{3}^{s}\colon\boldsymbol{L}^{2}(\Omega)\rightarrow\boldsymbol{L}^{2}(\widehat{\Omega})\
,\ \boldsymbol{v}\mapsto\textup{det}(\boldsymbol{J})\
\tilde{\boldsymbol{J}}\cdot(\boldsymbol{v}\circ\boldsymbol{F})+\hat{\partial}_{1}\tilde{\boldsymbol{J}}\cdot\int_{0}^{\zeta_{1}}\textup{det}(\boldsymbol{J})\
(\boldsymbol{v}\circ\boldsymbol{F})\ d1,$ (35)

where $\hat{\partial}_{1}\tilde{\boldsymbol{J}}$ means that each entry of
$\tilde{\boldsymbol{J}}$ is differentiated w.r.t. the first parametric
coordinate. Later we will use the equivalent formulation

$\mathcal{Y}_{3}^{s}(\boldsymbol{v})=\hat{\partial}_{1}\Big{[}\tilde{\boldsymbol{J}}\cdot\int_{0}^{\zeta_{1}}\textup{det}(\boldsymbol{J})\
(\boldsymbol{v}\circ\boldsymbol{F})\ d1\Big{]}.$ (36)

Next we face several lemmas involving the $\mathcal{Y}_{2,\Gamma_{1}}^{s}$ and
$\mathcal{Y}_{3}^{s}$. Basically, these auxiliary results are needed to
justify the appropriateness of the mentioned transformations for the
discretization.

We start with the the $L^{2}$-boundedness.

######  Lemma 12 (Boundedness of the transformations).

The mappings $\mathcal{Y}_{2,\Gamma_{1}}^{s}$ and $\mathcal{Y}_{3}^{s}$ are
bounded with respect to the $L^{2}$-norm.

###### Proof.

Since we assumed that $\boldsymbol{F}\in C^{3}(\overline{\widehat{\Omega}})$
with $\boldsymbol{F}^{-1}\in C^{3}(\overline{\widehat{\Omega}})$ and in view
of the Cauchy-Schwarz inequality which yields

$|\int_{0}^{\zeta_{1}}\hat{\phi}(\tau,\zeta_{2})d\tau|^{2}\leq\int_{0}^{\zeta_{1}}|\hat{\phi}(\tau,\zeta_{2})|^{2}d\tau\cdot\int_{0}^{\zeta_{1}}1d\tau\leq\int_{0}^{1}|\hat{\phi}(\tau,\zeta_{2})|^{2}d\tau\
\ \textup{for}\ \ \hat{\phi}\in L^{2}(\widehat{\Omega}),$

one sees easily the boundedness w.r.t. the $L^{2}$-norm. ∎

######  Lemma 13 (Divergence compatibility).

For
$\boldsymbol{S}\in\boldsymbol{\mathcal{H}}_{\Gamma_{1}}(\Omega,\textup{div},\mathbb{S})$
we have the relation

$\widehat{\nabla}\cdot\mathcal{Y}^{s}_{2,\Gamma_{1}}(\boldsymbol{S})=\mathcal{Y}^{s}_{3}(\nabla\cdot\boldsymbol{S}).$

In particular, with Lemma 12 we get indeed
$\mathcal{Y}^{s}_{2,\Gamma_{1}}\big{(}\boldsymbol{\mathcal{H}}_{\Gamma_{1}}(\Omega,\textup{div},\mathbb{S})\big{)}\subset\boldsymbol{\mathcal{H}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$.

###### Proof.

For the sake of clarity we moved the proof of the lemma to the appendix; see
Section 7.1. ∎

######  Lemma 14 (Invertibility of $\mathcal{Y}_{2,\Gamma_{1}}^{s}$).

The mapping $\mathcal{Y}_{2,\Gamma_{1}}^{s}$ is invertible with

$(\mathcal{Y}_{2,\Gamma_{1}}^{s})^{-1}(\tilde{\boldsymbol{S}})=\tilde{\boldsymbol{J}}^{-1}(\tilde{\boldsymbol{S}}\circ\boldsymbol{F}^{-1})\tilde{\boldsymbol{J}}^{-T}+\textup{Airy}(F_{k}^{-1})\cdot\Big{(}\int_{0}^{\zeta_{1}}(-1)^{k+1}\
\ \tilde{S}_{2\sigma(k)}\ d1\Big{)}\circ\boldsymbol{F}^{-1}.$ (37)

###### Proof.

We refer to Section 7.2 in the appendix for the detailed proof steps. Since
the proof is technical we omit it here. ∎

######  Lemma 15 (Invertibility of $\mathcal{Y}_{3}^{s}$).

The mapping $\mathcal{Y}_{3}^{s}$ is invertible, where

$(\mathcal{Y}_{3}^{s})^{-1}(\tilde{\boldsymbol{v}})=\textup{det}(\boldsymbol{J}^{-1})\
\tilde{\boldsymbol{J}}^{-1}\cdot(\tilde{\boldsymbol{v}}\circ\boldsymbol{F}^{-1})+\Big{(}\frac{\hat{\partial}_{1}\big{[}\tilde{\boldsymbol{J}}^{-1}\circ\boldsymbol{F}\big{]}}{\textup{det}(\boldsymbol{J})}\cdot\int_{0}^{\zeta_{1}}\tilde{\boldsymbol{v}}\
d1\Big{)}\circ\boldsymbol{F}^{-1}.$ (38)

###### Proof.

In this proof we write
$\tilde{\mathcal{Y}}=\tilde{\mathcal{Y}}(\tilde{\boldsymbol{v}})$ for the
mapping determined by the right-hand side of (38). Further, we use the
notation $\hat{\boldsymbol{v}}\coloneqq\boldsymbol{v}\circ\boldsymbol{F}$. To
begin with, we look at $\tilde{\mathcal{Y}}\circ{\mathcal{Y}}_{3}^{s}$.

$\displaystyle\tilde{\mathcal{Y}}\circ{\mathcal{Y}}_{3}^{s}(\boldsymbol{v})$
$\displaystyle=\boldsymbol{v}+\Big{(}\underbrace{(\tilde{\boldsymbol{J}}^{-1}\circ\boldsymbol{F})\cdot\frac{\hat{\partial}_{1}\tilde{\boldsymbol{J}}}{\textup{det}(\boldsymbol{J})}\cdot\int_{0}^{\zeta_{1}}\tilde{g}\
d1}_{\eqqcolon(L3,1)}\Big{)}\circ\boldsymbol{F}^{-1}$
$\displaystyle+\Big{(}\underbrace{\frac{\hat{\partial}_{1}\big{[}\tilde{\boldsymbol{J}}^{-1}\circ\boldsymbol{F}\big{]}}{\textup{det}(\boldsymbol{J})}\cdot\int_{0}^{\zeta_{1}}{\mathcal{Y}}_{3}^{s}(\boldsymbol{v})\
d1}_{\eqqcolon(L3,2)}\Big{)}\circ\boldsymbol{F}^{-1},$

where we set $\tilde{g}\coloneqq\textup{det}(\boldsymbol{J})\
(\boldsymbol{v}\circ\boldsymbol{F})$. Using the relations

$\displaystyle\big{(}{\mathcal{Y}}_{3}^{s}(\boldsymbol{v})\big{)}_{j}$
$\displaystyle=\textup{det}(\boldsymbol{J})\cdot\tilde{{J}}_{jk}\cdot\hat{v}_{k}+\hat{\partial}_{1}\tilde{{J}}_{jl}\cdot\int_{0}^{\zeta_{1}}\tilde{g}_{l}\
d1,$ $\displaystyle 0$
$\displaystyle=\hat{\partial}_{1}\big{[}(\tilde{{J}}_{mk}^{-1}\circ\boldsymbol{F})\cdot\tilde{{J}}_{kl}\big{]}=\hat{\partial}_{1}[\tilde{{J}}_{mk}^{-1}\circ\boldsymbol{F}]\cdot\tilde{{J}}_{kl}+[\tilde{{J}}_{mk}^{-1}\circ\boldsymbol{F}]\cdot\hat{\partial}_{1}\tilde{{J}}_{kl},$

we obtain for the $m$-th entry of $(L3,1)+(L3,2)$:

$\displaystyle\big{[}(L3,1)+(L3,2)\big{]}_{m}$
$\displaystyle=\frac{\tilde{{J}}_{mj}^{-1}\circ\boldsymbol{F}}{\textup{det}(\boldsymbol{J})}\cdot\hat{\partial}_{1}\tilde{{J}}_{jl}\cdot\int_{0}^{\zeta_{1}}\tilde{g}_{l}\
d1+\frac{\hat{\partial}_{1}[\tilde{{J}}_{mj}^{-1}\circ\boldsymbol{F}]}{\textup{det}(\boldsymbol{J})}\cdot\int_{0}^{\zeta_{1}}({\mathcal{Y}}_{3}^{s}(\boldsymbol{v}))_{j}\
d1$
$\displaystyle=-\frac{\hat{\partial}_{1}[\tilde{{J}}_{mj}^{-1}\circ\boldsymbol{F}]}{\textup{det}(\boldsymbol{J})}\cdot\tilde{{J}}_{jl}\cdot\int_{0}^{\zeta_{1}}\tilde{g}_{l}\
d1$ $\displaystyle\hskip
14.22636pt+\frac{\hat{\partial}_{1}[\tilde{{J}}_{mj}^{-1}\circ\boldsymbol{F}]}{\textup{det}(\boldsymbol{J})}\cdot\int_{0}^{\zeta_{1}}\textup{det}(\boldsymbol{J})\cdot\tilde{{J}}_{jk}\cdot\hat{v}_{k}+\hat{\partial}_{1}\tilde{{J}}_{jl}\cdot\tilde{\tilde{g}}\
d1,$ $\displaystyle\ \ \ \textup{with}\
\tilde{\tilde{g}}\coloneqq\int_{0}^{\zeta_{1}}\tilde{g}_{l}\ d1.\ \textup{This
implies }$ $\displaystyle\big{[}(L3,1)+(L3,2)\big{]}_{m}$
$\displaystyle=-\frac{\hat{\partial}_{1}[\tilde{{J}}_{mj}^{-1}\circ\boldsymbol{F}]}{\textup{det}(\boldsymbol{J})}\cdot\tilde{{J}}_{jl}\cdot\int_{0}^{\zeta_{1}}\tilde{g}_{l}\
d1+\frac{\hat{\partial}_{1}[\tilde{{J}}_{mj}^{-1}\circ\boldsymbol{F}]}{\textup{det}(\boldsymbol{J})}\cdot\tilde{{J}}_{jl}\cdot\int_{0}^{\zeta_{1}}\tilde{g}_{l}\
d1$ $\displaystyle\hskip
14.22636pt+\frac{\hat{\partial}_{1}[\tilde{{J}}_{mj}^{-1}\circ\boldsymbol{F}]}{\textup{det}(\boldsymbol{J})}\cdot\int_{0}^{\zeta_{1}}\textup{det}(\boldsymbol{J})\cdot\tilde{{J}}_{jk}\cdot\hat{v}_{k}-\tilde{{J}}_{jl}\cdot\tilde{g}_{l}\
d1$
$\displaystyle=0+\frac{\hat{\partial}_{1}[\tilde{{J}}_{mj}^{-1}\circ\boldsymbol{F}]}{\textup{det}(\boldsymbol{J})}\cdot\int_{0}^{\zeta_{1}}\underbrace{\textup{det}(\boldsymbol{J})\cdot\tilde{{J}}_{jk}\cdot\hat{v}_{k}-\tilde{{J}}_{jl}\cdot\textup{det}(\boldsymbol{J})\cdot\hat{v}_{l}}_{=0}\
d1=0.$

Thus we showed that $\tilde{\mathcal{Y}}$ is the left inverse of
$\mathcal{Y}_{3}^{s}$. In particular we showed injectivity. Hence it remains
to show the surjectivity of $\mathcal{Y}_{3}^{s}$.
Therefore consider

$\displaystyle{\mathcal{Y}}_{3}^{s}\circ\tilde{\mathcal{Y}}(\tilde{\boldsymbol{v}})$
$\displaystyle=\tilde{\boldsymbol{v}}+\Big{(}\underbrace{\hat{\partial}_{1}\tilde{\boldsymbol{J}}\cdot\int_{0}^{\zeta_{1}}\textup{det}(\boldsymbol{J})\cdot\big{(}\tilde{\mathcal{Y}}(\boldsymbol{v})\circ\boldsymbol{F}\big{)}\
d1}_{\eqqcolon(L3,3)}\Big{)}$ $\displaystyle+\ \ \
\underbrace{\textup{det}(\boldsymbol{J})\cdot\tilde{\boldsymbol{J}}\cdot\frac{\hat{\partial}_{1}\big{[}\tilde{\boldsymbol{J}}^{-1}\circ\boldsymbol{F}\big{]}}{\textup{det}(\boldsymbol{J})}\cdot\int_{0}^{\zeta_{1}}\tilde{\boldsymbol{v}}\
d1}_{\eqqcolon(L3,4)}.$

We can simplify

$\displaystyle[(L3,4)]_{m}$
$\displaystyle=-\hat{\partial}_{1}\tilde{{J}}_{ml}\cdot[\tilde{{J}}_{lk}^{-1}\circ\boldsymbol{F}]\cdot\int_{0}^{\zeta_{1}}\tilde{{v}}_{k}\
d1,\ \ \textup{since}\ \
\hat{\partial}_{1}\tilde{{J}}_{ml}\cdot[\tilde{{J}}_{lk}^{-1}\circ\boldsymbol{F}]=-\tilde{{J}}_{ml}\cdot\hat{\partial}_{1}[\tilde{{J}}_{lk}^{-1}\circ\boldsymbol{F}],$
and expl oiting the product rule we obtain $\displaystyle[(L3,3)]_{m}$
$\displaystyle=\hat{\partial}_{1}\tilde{{J}}_{ml}\cdot\int_{0}^{\zeta_{1}}\textup{det}(\boldsymbol{J})\cdot\big{(}\tilde{\mathcal{Y}}(\boldsymbol{v})\circ\boldsymbol{F}\big{)}_{l}\
d1$
$\displaystyle=\hat{\partial}_{1}\tilde{{J}}_{ml}\cdot\int_{0}^{\zeta_{1}}{[\tilde{{J}}^{-1}_{lk}\circ\boldsymbol{F}]\cdot\tilde{v}_{k}}\
d1-\hat{\partial}_{1}\tilde{{J}}_{ml}\cdot\int_{0}^{\zeta_{1}}[\tilde{{J}}^{-1}_{lk}\circ\boldsymbol{F}]\cdot\tilde{v}_{k}\
d1$ $\displaystyle\ \ \
+\hat{\partial}_{1}\tilde{{J}}_{ml}\cdot[\tilde{{J}}_{lk}^{-1}\circ\boldsymbol{F}]\cdot\int_{0}^{\zeta_{1}}\tilde{{v}}_{k}\
d1$ $\displaystyle=-[(L3,4)].$

Consequently, $\mathcal{Y}_{3}^{s}$ is bijective with inverse mapping
$\tilde{\mathcal{Y}}$. ∎

######  Lemma 16 (Boundary condition compatibility).

$\mathcal{Y}_{2,\Gamma_{1}}^{s}$ preserves the traction boundary condition for
our choice in Assumption 5. More precisely, we have

$\mathcal{Y}_{2,\Gamma_{1}}^{s}(\boldsymbol{\mathcal{H}}_{\Gamma_{1}}(\Omega,\textup{div},\mathbb{S}))=\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S}).$

###### Proof.

Let us assume
$\boldsymbol{S}\in\boldsymbol{\mathcal{H}}_{\Gamma_{1}}(\Omega,\textup{div},\mathbb{S})\cap
C^{1}(\overline{\Omega})$. Then we obtain
$\mathcal{Y}_{2,\Gamma_{1}}^{s}(\boldsymbol{S})\in\boldsymbol{\mathcal{H}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})\cap
C^{1}(\overline{\widehat{\Omega}})$. Writing $\hat{\boldsymbol{n}}_{1}$ for
the outer unit normal to the edge $\hat{\Gamma}_{1}$, it is due to the
divergence preserving transform
$\boldsymbol{v}\mapsto\textup{det}(\boldsymbol{J})\boldsymbol{J}^{-1}(\boldsymbol{v}\circ\boldsymbol{F})$:
$(\textup{det}(\boldsymbol{J})\
\boldsymbol{J}^{-1}\cdot(\boldsymbol{S}_{i}\circ\boldsymbol{F}))\cdot\hat{\boldsymbol{n}}_{1}=-\tilde{{J}}_{1j}\cdot\hat{S}_{ij}=0$
on $\hat{\Gamma}_{1}$. Note
$\hat{\boldsymbol{S}}\coloneqq\boldsymbol{S}\circ\boldsymbol{F}$.
Consequently,

$\big{(}\mathcal{Y}_{2,\Gamma_{1}}^{s}(\boldsymbol{S})\big{)}_{m}\cdot\hat{\boldsymbol{n}}_{1}=-\tilde{{J}}_{1j}\cdot\hat{S}_{ij}\cdot\tilde{{J}}_{mi}=0\cdot\tilde{{J}}_{mi}=0\
\ \textup{on}\ \hat{\Gamma}_{1}.$

One observes that the integral terms in the definition of
$\mathcal{Y}_{2,\Gamma_{1}}^{s}$ (see 33) vanish on $\hat{\Gamma}_{1}$. Hence
$\mathcal{Y}_{2,\Gamma_{1}}^{s}(\boldsymbol{S})\in\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$.
Due to Lemma 13, Lemma 12 and a density argument we have indeed
$\mathcal{Y}_{2,\Gamma_{1}}^{s}(\boldsymbol{\mathcal{H}}_{\Gamma_{1}}(\Omega,\textup{div},\mathbb{S}))\subset\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$.
On the other hand let us assume
$\tilde{\boldsymbol{S}}\in\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})\cap
C^{1}(\overline{\widehat{\Omega}})$. One shows

$\displaystyle\big{(}(\mathcal{Y}_{2,\Gamma_{1}}^{s})^{-1}(\tilde{\boldsymbol{S}})\big{)}_{m}\cdot{\boldsymbol{n}}_{1}$
$\displaystyle=\big{[}\tilde{{J}}_{mi}^{-1}\cdot\big{(}\tilde{\boldsymbol{J}}^{-1}\cdot(\tilde{\boldsymbol{S}}^{i}\circ\boldsymbol{F}^{-1})\big{)}^{T}\big{]}\cdot{\boldsymbol{n}}_{1}$
$\displaystyle=\tilde{{J}}_{mi}^{-1}\cdot\big{[}\big{(}\tilde{\boldsymbol{J}}^{-1}\cdot(\tilde{\boldsymbol{S}}^{i}\circ\boldsymbol{F}^{-1})\big{)}^{T}\cdot{\boldsymbol{n}}_{1}\big{]}=\tilde{{J}}_{mi}^{-1}\cdot
0=0\ \ \textup{on}\ \ \Gamma_{1}.$

In the last lines we wrote ${\boldsymbol{n}}_{1}$ for the outer unit normal to
the edge $\Gamma_{1}=\boldsymbol{F}(\hat{\Gamma}_{1})$ and we use the fact
$\big{(}\tilde{\boldsymbol{J}}^{-1}\cdot(\tilde{\boldsymbol{S}}^{i}\circ\boldsymbol{F}^{-1})\big{)}^{T}\cdot{\boldsymbol{n}}_{1}=0$
which is a consequence of the transformation rule for normal vectors. This
implies
$(\mathcal{Y}_{2,\Gamma_{1}}^{s})^{-1}(\tilde{\boldsymbol{S}})\in\boldsymbol{\mathcal{H}}_{\Gamma_{1}}(\Omega,\textup{div},\mathbb{S})\cap
C^{1}(\overline{\Omega})$. The general case of
$\tilde{\boldsymbol{S}}\in\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$
follows by a density argument. ∎

If we use the Lemma 16 we see that we have the commuting diagram

$\boldsymbol{L}^{2}(\Omega)$$\nabla\cdot$$\boldsymbol{\mathcal{H}}_{\Gamma_{1}}({\Omega},\textup{div},\mathbb{S})$$\boldsymbol{L}^{2}(\widehat{\Omega})$$\widehat{\nabla}\cdot$$\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$$\mathcal{Y}_{3}^{s}$$\mathcal{Y}_{2,\Gamma_{1}}^{s}$
Figure 3: The diagram commutes.

Next we look at the classical planar Dirichlet problem and use the results
derived in Section 4.1.

######  Remark 4.

The results of this subsection imply the well-posedness of the weak form
(4.1). Namely, due to the commutativity of the diagram in Fig. 3, Lemma (13) -
(15) it is enough to prove the well-posedness for
$\boldsymbol{F}=\textup{id}$. In fact, the latter can be shown with an ansatz
similar to the one in the proof of Lemma 21. To be more precise, in view of
the choice (41)-(43) for a general
$\tilde{\boldsymbol{v}}\in\boldsymbol{L}^{2}(\widehat{\Omega})$ it is easy to
check that for all such $\tilde{\boldsymbol{v}}$ there is a
$\tilde{\boldsymbol{\tau}}\in\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$
s.t. $\widehat{\nabla}\cdot\tilde{\boldsymbol{\tau}}=\tilde{\boldsymbol{v}}$
and
$\left\lVert\tilde{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}\leq\left\lVert\tilde{\boldsymbol{v}}\right\rVert_{\boldsymbol{L}^{2}}$.
This leads to the classical Brezzi stability conditions for the saddle-point
problem $\eqref{weak_form_strong_symmetry_contiuous_BC}$. Since we want to
study the case with pure displacement boundary conditions, we omit here
further details.

###  4.2 The Dirichlet problem

In this section we suppose $\Gamma_{D}=\partial\Omega$. Further, for the
moment we assume always $\boldsymbol{S}\in
C^{1}(\overline{\Omega},\mathbb{S})\coloneqq\boldsymbol{\mathcal{L}}^{2}(\Omega,\mathbb{S})\cap
C^{1}(\overline{\Omega})$ and $\tilde{\boldsymbol{S}}\in
C^{1}(\overline{\widehat{\Omega}},\mathbb{S})\coloneqq\boldsymbol{\mathcal{L}}^{2}(\widehat{\Omega},\mathbb{S})\cap
C^{1}(\overline{\widehat{\Omega}})$. The smoothness assumption is made to
guarantee the well-definedness of subsequent calculations. Analyzing the proof
of Lemma 13 we see that there is only one step where the underlying boundary
conditions are needed, namely the rearrangement from line (60) to (61). Hence,
if we do not require the boundary conditions in Assumption 5, we get an
additional term. More precisely, with
$\hat{\boldsymbol{{S}}}\coloneqq\boldsymbol{{S}}\circ\boldsymbol{F}$ it is

$\displaystyle\widehat{\nabla}\cdot\mathcal{Y}^{s}_{2,\Gamma_{1}}(\boldsymbol{S})=\mathcal{Y}^{s}_{3}(\nabla\cdot\boldsymbol{S})+\begin{bmatrix}\hat{\partial}_{1}\tilde{{J}}_{1i}\cdot\tilde{{J}}_{1l}(0,\cdot)\cdot\hat{S}_{il}(0,\cdot)\\\
\hat{\partial}_{1}\tilde{{J}}_{2i}\cdot\tilde{{J}}_{1l}(0,\cdot)\cdot\hat{S}_{il}(0,\cdot)\end{bmatrix}.$

Above and in the following we write $g(0,\cdot)$ to set the first parametric
coordinate of a mapping $g$ to zero, i.e. for example
$\int_{0}^{\zeta_{1}}g(0,\cdot)d1\coloneqq\int_{0}^{\zeta_{1}}g(0,\zeta_{2})d\tau$.
Thus we define

$\displaystyle\mathcal{Y}^{s}_{2}(\boldsymbol{S})$
$\displaystyle\coloneqq\mathcal{Y}^{s}_{2,\Gamma_{1}}(\boldsymbol{S})-\mathcal{Y}^{s}_{2,A}(\boldsymbol{S}),\
\ \forall\boldsymbol{S}\in C^{1}(\overline{\Omega},\mathbb{S}),\
\textup{where}$ (39) $\displaystyle\mathcal{Y}^{s}_{2,A}(\boldsymbol{S})$
$\displaystyle\coloneqq\begin{bmatrix}\int_{0}^{\zeta_{1}}\hat{\partial}_{1}\tilde{{J}}_{1i}\cdot\tilde{{J}}_{1l}(0,\cdot)\cdot\hat{S}_{il}(0,\cdot)\
d1&0\\\
0&\int_{0}^{\zeta_{2}}\hat{\partial}_{1}\tilde{{J}}_{2i}\cdot\tilde{{J}}_{1l}(0,\cdot)\cdot\hat{S}_{il}(0,\cdot)\
d2\end{bmatrix}.$

Obviously, we have the next Lemma.

######  Lemma 17 (Divergence compatibility).

It holds
$\widehat{\nabla}\cdot\mathcal{Y}_{2}^{s}(\boldsymbol{S})=\mathcal{Y}^{s}_{3}(\nabla\cdot\boldsymbol{S}),\
\ \forall\boldsymbol{{S}}\in C^{1}(\overline{\Omega},\mathbb{S})$.

Another point we want to mention is the fact that
$\mathcal{Y}_{2}^{s}(\boldsymbol{S})\in
C^{1}(\overline{\widehat{\Omega}},\mathbb{S})$ if $\boldsymbol{S}\in
C^{1}(\overline{\Omega},\mathbb{S})$. This follows easily by the defintion of
the mappings. Furthermore we set

$\displaystyle\tilde{\mathcal{Y}}_{2}^{s}(\tilde{\boldsymbol{S}})$
$\displaystyle\coloneqq(\mathcal{Y}_{2,\Gamma_{1}}^{s})^{-1}(\tilde{\boldsymbol{S}})-\tilde{\mathcal{Y}}_{2,A}^{s}(\tilde{\boldsymbol{S}}),\
\tilde{\boldsymbol{S}}\in C^{1}(\overline{\widehat{\Omega}},\mathbb{S}),$

with

$\displaystyle\tilde{\mathcal{Y}}_{2,A}^{s}(\tilde{\boldsymbol{S}})$
$\displaystyle\coloneqq(\mathcal{Y}^{s}_{2,\Gamma_{1}})^{-1}\Big{(}\begin{bmatrix}\int^{\zeta_{1}}_{0}\big{[}\mathcal{Y}^{s}_{3}(T(\tilde{\boldsymbol{S}}))\big{]}_{1}\
d1&0\\\
0&\int^{\zeta_{2}}_{0}\big{[}\mathcal{Y}^{s}_{3}(T(\tilde{\boldsymbol{S}}))\big{]}_{2}\
d2\end{bmatrix}\Big{)},$ $\displaystyle T(\tilde{\boldsymbol{S}})$
$\displaystyle\coloneqq\begin{bmatrix}\partial_{k}\tilde{J}^{-1}_{1i}\cdot\tilde{{J}}^{-1}_{k1}\cdot(\tilde{S}_{i1}(0,\cdot)\circ\boldsymbol{F}^{-1})\\\
\partial_{k}\tilde{J}^{-1}_{2i}\cdot\tilde{{J}}^{-1}_{k1}\cdot(\tilde{S}_{i1}(0,\cdot)\circ\boldsymbol{F}^{-1})\end{bmatrix}.$

Again one can verify the well-definedness of
$\tilde{\mathcal{Y}}_{2}^{s}(\tilde{\boldsymbol{S}})$ and that
$\tilde{\mathcal{Y}}_{2}^{s}(\tilde{\boldsymbol{S}})\in
C^{1}(\overline{\Omega},\mathbb{S})$ in case of $\tilde{\boldsymbol{S}}\in
C^{1}(\overline{\widehat{\Omega}},\mathbb{S})$. Next we show the invertibility
of the introduced map $\mathcal{Y}_{2}^{s}$.

######  Lemma 18 (Invertibility of $\mathcal{Y}_{2}^{s}$).

The mapping $\mathcal{Y}_{2}^{s}\colon
C^{1}(\overline{\Omega},\mathbb{S})\rightarrow
C^{1}(\overline{\widehat{\Omega}},\mathbb{S})$ determined by (39) is
invertible with

$(\mathcal{Y}_{2}^{s})^{-1}=\tilde{\mathcal{Y}}_{2}^{s}.$

###### Proof.

A detailed proof for the assertion can be found in the appendix; see Section
7.3. ∎

######  Lemma 19 (Boundedness w.r.t. $H^{1}$-norm).

The mappings ${\mathcal{Y}}_{2}^{s}\colon
C^{1}(\overline{\Omega},\mathbb{S})\rightarrow
C^{1}(\overline{\widehat{\Omega}},\mathbb{S})$ and
$\tilde{\mathcal{Y}}_{2}^{s}\colon
C^{1}(\overline{\widehat{\Omega}},\mathbb{S})\rightarrow
C^{1}(\overline{{\Omega}},\mathbb{S})$ are bounded in the sense

$\left\lVert{\mathcal{Y}}_{2}^{s}(\boldsymbol{{S}})\right\rVert_{\boldsymbol{\mathcal{H}}(\widehat{\textup{div}})}\leq
C\left\lVert\boldsymbol{{S}}\right\rVert_{\boldsymbol{\mathcal{H}}^{1}}$ and
$\left\lVert\tilde{\mathcal{Y}}_{2}^{s}(\tilde{\boldsymbol{S}})\right\rVert_{\boldsymbol{\mathcal{H}}({\textup{div}})}\leq
C\left\lVert\tilde{\boldsymbol{S}}\right\rVert_{\boldsymbol{\mathcal{H}}^{1}}$.

In particular, ${\mathcal{Y}}_{2}^{s}$ and $\tilde{\mathcal{Y}}_{2}^{s}$ have
unique bounded extensions on
$\boldsymbol{\mathcal{H}}^{1}(\Omega)\cap\boldsymbol{\mathcal{L}}^{2}(\Omega,\mathbb{S})$,
$\boldsymbol{\mathcal{H}}^{1}(\widehat{\Omega})\cap\boldsymbol{\mathcal{L}}^{2}(\widehat{\Omega},\mathbb{S})$
respectively. Hence, it is justified to write also

$\displaystyle{\mathcal{Y}}_{2}^{s}$
$\displaystyle\colon\boldsymbol{\mathcal{H}}^{1}(\Omega)\cap\boldsymbol{\mathcal{L}}^{2}(\Omega,\mathbb{S})\rightarrow\boldsymbol{\mathcal{H}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S}),$
$\displaystyle\tilde{\mathcal{Y}}_{2}^{s}$
$\displaystyle\colon\boldsymbol{\mathcal{H}}^{1}(\widehat{\Omega})\cap\boldsymbol{\mathcal{L}}^{2}(\widehat{\Omega},\mathbb{S})\rightarrow\boldsymbol{\mathcal{H}}({\Omega},{\textup{div}},\mathbb{S}).$

###### Proof.

We show the assertion only for ${\mathcal{Y}}_{2}^{s}$ since the second part
can be seen with a similar reasoning.
Due to Lemma 12 and Lemma 17 it is enough to check
$\left\lVert{\mathcal{Y}}_{2,A}^{s}(\boldsymbol{{S}})\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}\leq
C\left\lVert\boldsymbol{{S}}\right\rVert_{\boldsymbol{\mathcal{H}}^{1}}$.
Since we assume $\boldsymbol{F},\boldsymbol{F}^{-1}$ to be three times
continuously differentiable we have $|\hat{\partial}_{1}\tilde{{J}}_{mi}|\in
C^{0}(\overline{\widehat{\Omega}})$ and in view of the Cauchy-Schwarz
inequality it is for some constants $C<\infty$, i.e. $C$ may vary at different
occurrences,

$\displaystyle\Big{|}\int_{0}^{\zeta_{1}}\hat{\partial}_{1}\tilde{{J}}_{1i}\cdot\tilde{{J}}_{1l}(0,\cdot)\cdot\hat{S}_{il}(0,\cdot)\
d1\Big{|}^{2}\leq
C\int_{0}^{\zeta_{1}}\big{|}\hat{\boldsymbol{S}}(0,\cdot)\big{|}^{2}\ d1,\ \ \
\hat{\boldsymbol{{S}}}\coloneqq\boldsymbol{{S}}\circ\boldsymbol{F}.$

Then applying a standard trace estimate one sees

$\displaystyle\int_{0}^{1}\Big{|}\int_{0}^{\zeta_{1}}\hat{\partial}_{1}\tilde{{J}}_{1i}\cdot\tilde{{J}}_{1l}(0,\cdot)\cdot\hat{S}_{il}(0,\cdot)\
d1\Big{|}^{2}d2$ $\displaystyle\leq
C\int_{0}^{1}\big{|}\hat{\boldsymbol{S}}(0,\cdot)\big{|}^{2}\ d2\ $
$\displaystyle\leq
C\left\lVert\hat{\boldsymbol{{S}}}\right\rVert_{\boldsymbol{\mathcal{H}}^{1}}^{2}\leq
C\left\lVert\boldsymbol{{S}}\right\rVert_{\boldsymbol{\mathcal{H}}^{1}}^{2}.$

And in an analogous manner we get

$\displaystyle\Big{|}\int_{0}^{\zeta_{2}}\hat{\partial}_{1}\tilde{{J}}_{2i}\cdot\tilde{{J}}_{1l}(0,\cdot)\cdot\hat{S}_{il}(0,\cdot)\
d2\Big{|}^{2}\leq
C\left\lVert\boldsymbol{{S}}\right\rVert_{\boldsymbol{\mathcal{H}}^{1}}^{2}.$

Latter two estimates yield the boundedness result for
${\mathcal{Y}}_{2,A}^{s}$. ∎

###  4.3 Discretization

Let us apply the $\mathcal{Y}_{i}^{s}$ for the discretization of the mixed
elasticity system with strong symmetry. Our starting point is the next
definition of spline spaces:

$\displaystyle\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}$
$\displaystyle=\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}(p,r)\coloneqq\textup{SYM}(S_{p+1,p-1}^{r+1,r-1},S_{p,p}^{r,r},S_{p-1,p+1}^{r-1,r+1}),$
$\displaystyle\ \ \ \widehat{\boldsymbol{V}}_{h,3}^{s}$
$\displaystyle=\widehat{\boldsymbol{V}}_{h,3}^{s}(p,r)\coloneqq\big{(}S_{p,p-1}^{r,r-1}\times
S_{p-1,p}^{r-1,r}\big{)}^{T}\textup{with}\ \ p>r\geq 1.$

We note that with $r\geq 1$ we get directly
$\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}\subset\boldsymbol{\mathcal{H}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})\cap\boldsymbol{\mathcal{H}}^{1}(\widehat{\Omega})$.
And the isogeometric test spaces are defined by means of
$\tilde{\mathcal{Y}}^{s}_{2},\ (\mathcal{Y}^{s}_{3})^{-1}$, namely

$\displaystyle{\boldsymbol{\mathcal{V}}}_{h,2}^{s}\coloneqq\tilde{\mathcal{Y}}_{2}^{s}\big{(}\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}\big{)},\
\ \ \
{\boldsymbol{V}}_{h,3}^{s}\coloneqq(\mathcal{Y}_{3}^{s})^{-1}\big{(}\widehat{\boldsymbol{V}}_{h,3}^{s}\big{)}.$
(40)

Exploiting the piecewise smoothness of splines one obtains the next
properties.

######  Lemma 20.

It holds
${\mathcal{Y}}_{2}^{s}\circ\tilde{\mathcal{Y}}_{2}^{s}(\hat{\boldsymbol{\tau}}_{h})=\hat{\boldsymbol{\tau}}_{h},\
\forall\hat{\boldsymbol{\tau}}_{h}\in\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}$
and
$(\widehat{\nabla}\cdot)\circ{\mathcal{Y}}_{2}^{s}={\mathcal{Y}}_{3}^{s}\circ{\nabla}\cdot\
$ on ${\boldsymbol{\mathcal{V}}}_{h,2}^{s}$.

###### Proof.

Above in Section 4.2 we assumed several times the mappings $\boldsymbol{{S}}$
and $\tilde{\boldsymbol{S}}$ to be smooth enough, namely $C^{1}$-regular. By
the piecewise smoothness of spline functions and the fact that
$\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}\subset\boldsymbol{\mathcal{H}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})\cap\boldsymbol{\mathcal{H}}^{1}(\widehat{\Omega})$
and
${\boldsymbol{\mathcal{V}}}_{h,2}^{s}\subset\boldsymbol{\mathcal{H}}({\Omega},{\textup{div}},\mathbb{S})\cap\boldsymbol{\mathcal{H}}^{1}({\Omega})$
the proof steps for Lemma 18 (see Section 7.3) and the divergence relation
from Lemma 17 are still valid for our choice of test functions. ∎

We get the next useful result:

######  Lemma 21.

The diagram in Fig. 4 commutes and it is exact meaning the right column
corresponds to the image of the left column w.r.t. the divergence operator.
Moreover we have
$\widehat{\nabla}\cdot\big{(}\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}\cap\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})\big{)}=\widehat{\boldsymbol{V}}_{h,3}^{s}$.

###### Proof.

We show the last equation of the assertion and
$\widehat{\nabla}\cdot\big{(}\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}\big{)}\subset\widehat{\boldsymbol{V}}_{h,3}^{s}$.
Then the commutativity and exactness of the diagram in Fig. 4 follows by Lemma
20.
Since

$\displaystyle\widehat{\nabla}\cdot\big{[}S_{p+1,p-1}^{r+1,r-1}\times
S_{p,p}^{r,r}\big{]}$
$\displaystyle=\hat{\partial}_{1}\big{(}S_{p+1,p-1}^{r+1,r-1}\big{)}+\hat{\partial}_{2}\big{(}S_{p,p}^{r,r}\big{)}\subset
S_{p,p-1}^{r,r-1},$
$\displaystyle\widehat{\nabla}\cdot\big{[}S_{p,p}^{r,r}\times
S_{p-1,p+1}^{r-1,r+1}\big{]}$
$\displaystyle=\hat{\partial}_{1}\big{(}S_{p,p}^{r,r}\big{)}+\hat{\partial}_{2}\big{(}S_{p-1,p+1}^{r-1,r+1}\big{)}\subset
S_{p-1,p}^{r-1,r},$

we obtain indeed
$\widehat{\nabla}\cdot\big{(}\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}\big{)}\subset\widehat{\boldsymbol{V}}_{h,3}^{s}$.
Now let
$\tilde{\boldsymbol{v}}\coloneqq(\tilde{v}_{1},\tilde{v}_{2})^{T}\in\widehat{\boldsymbol{V}}_{h,3}^{s}$
arbitrary but fixed. Then define
$\tilde{\boldsymbol{\tau}}=(\tilde{\tau}_{ij})$ through

$\displaystyle\tilde{\tau}_{11}$
$\displaystyle\coloneqq\int_{0}^{\zeta_{1}}\tilde{v}_{1}\
d1\in\int_{0}^{\zeta_{1}}S_{p,p-1}^{r,r-1}\ d1\subset S_{p+1,p-1}^{r+1,r-1},$
(41) $\displaystyle\ \tilde{\tau}_{22}$
$\displaystyle\coloneqq\int_{0}^{\zeta_{2}}\tilde{v}_{2}\
d2\in\int_{0}^{\zeta_{2}}S_{p-1,p}^{r-1,r}\ d2\subset S_{p-1,p+1}^{r-1,r+1},$
(42) $\displaystyle\ \tilde{\tau}_{12}$
$\displaystyle=\tilde{\tau}_{21}\coloneqq 0.$ (43)

Clearly, we have
$\tilde{\boldsymbol{\tau}}\in\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}\cap\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$
and $\widehat{\nabla}\cdot\tilde{\boldsymbol{\tau}}=\tilde{\boldsymbol{v}}$.
The arbitrariness of $\tilde{\boldsymbol{v}}$ finishes the proof. ∎

${\boldsymbol{V}}_{h,3}^{s}$$\nabla\cdot$${\boldsymbol{\mathcal{V}}}_{h,2}^{s}$$\widehat{\boldsymbol{V}}_{h,3}^{s}$$\widehat{\nabla}\cdot$$\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}$$\mathcal{Y}_{2}^{s}$$\mathcal{Y}_{3}^{s}$
Figure 4: The diagram commutes and it is exact.

######  Theorem 4 (Well-posedness of the mixed formulation with strong
symmetry).

The weak formulation (5) with strong symmetry is well-posed. In other words,
there exists a unique solution and the underlying inf-sup condition is
satisfied.

###### Proof.

Follows directly by the explanations in [3, Section 8.8] and [3, Section
4.4.2]. ∎

Finally we can prove now that the weak formulation with Dirichlet BC and
strong symmetry can be discretized properly by means of the spaces
${\boldsymbol{\mathcal{V}}}_{h,2}^{s},\ {\boldsymbol{V}}_{h,3}^{s}$.

######  Theorem 5 (Discrete method in the strong symmetry case).

Let $(\boldsymbol{\sigma},\boldsymbol{u})$ be the exact solution of the weak
form with strong symmetry (5) equipped with Dirichlet data
$\boldsymbol{u}_{D}\in\boldsymbol{H}^{1/2}(\Gamma)$. The discretized weak form
that seeks for
$\boldsymbol{\sigma}_{h}\in{\boldsymbol{\mathcal{V}}}_{h,2}^{s},\
\boldsymbol{u}_{h}\in{\boldsymbol{V}}_{h,3}^{s}$ s.t.

$\displaystyle\langle\textbf{A}\boldsymbol{\sigma}_{h},\boldsymbol{\tau}_{h}\rangle\
+\ $
$\displaystyle\langle\boldsymbol{u}_{h},\nabla\cdot\boldsymbol{\tau}_{h}\rangle\
\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle=\langle\boldsymbol{\tau}_{h}\cdot\textbf{n};\boldsymbol{u}_{D}\rangle_{\Gamma},\
\hskip 11.38092pt$
$\displaystyle\forall\boldsymbol{\tau}_{h}\in{\boldsymbol{\mathcal{V}}}_{h,2}^{s},$
$\displaystyle\langle\nabla\cdot\boldsymbol{\sigma}_{h},\boldsymbol{v}_{h}\rangle$
$\displaystyle=\langle\boldsymbol{f},\boldsymbol{v}_{h}\rangle,\ \hskip
11.38092pt$
$\displaystyle\forall\boldsymbol{v}_{h}\in{\boldsymbol{V}}_{h,3}^{s},$ (44)

is inf-sup stable. More precisely, there are constants
$0<\gamma_{1},\gamma_{2}$ independent from $h$ satisfying

$\displaystyle\underset{\boldsymbol{v}_{h}\in{\boldsymbol{V}}_{h,3}^{s}\textbackslash\\{0\\}}{\inf}\
\
\underset{\boldsymbol{\tau}_{h}\in{\boldsymbol{\mathcal{V}}}_{h,2}^{s}}{\sup}\
\frac{\langle\nabla\cdot\boldsymbol{\tau}_{h},\boldsymbol{v}_{h}\rangle}{\left\lVert\boldsymbol{\tau}_{h}\right\rVert_{\boldsymbol{\mathcal{H}}(\textup{div})}\left\lVert\boldsymbol{v}_{h}\right\rVert_{\boldsymbol{L}^{2}}}\geq\gamma_{1}>0$
(45)

and

$\displaystyle\langle\textbf{A}\boldsymbol{\tau}_{h},\boldsymbol{\tau}_{h}\rangle$
$\displaystyle\geq\gamma_{2}\left\lVert\boldsymbol{\tau}_{h}\right\rVert_{\boldsymbol{\mathcal{H}}(\textup{div})}^{2},\
\ \forall\boldsymbol{\tau}_{h}\in B_{h},\ \textup{with}\ $ $\displaystyle
B_{h}$
$\displaystyle\coloneqq\\{\boldsymbol{\tau}_{h}\in{\boldsymbol{\mathcal{V}}}_{h,2}^{s}\
|\ \langle\nabla\cdot\boldsymbol{\tau}_{h},\boldsymbol{v}_{h}\rangle=0,\
\forall\boldsymbol{v}_{h}\in{\boldsymbol{V}}_{h,3}^{s}\\}.$

###### Proof.

Due to Lemma 21 it is
$\left\lVert{\boldsymbol{s}}\right\rVert_{\boldsymbol{\mathcal{H}}({\textup{div}})}=\left\lVert{\boldsymbol{s}}\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}$
for all ${\boldsymbol{s}}\in B_{h}$. Then it follows easily

$\langle\textbf{A}{\boldsymbol{s}},{\boldsymbol{s}}\rangle\geq\gamma_{2}\left\lVert\boldsymbol{s}\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}^{2}=\gamma_{2}\left\lVert\boldsymbol{s}\right\rVert_{\boldsymbol{\mathcal{H}}({\textup{div}})}^{2},\
\ \forall\boldsymbol{s}\in B_{h},$

for some A-dependent constant $\gamma_{2}$.
Now we show the inf-sup condition first in the parametric domain, i.e. if
$\boldsymbol{F}=\textup{id}$.
Let
$\tilde{\boldsymbol{v}}\in\widehat{\boldsymbol{V}}_{h,3}^{s}\textbackslash\\{0\\}$
arbitrary but fixed. With the choice (41) - (43) in proof of the mentioned
lemma we see that there is a
$\tilde{\boldsymbol{\tau}}=\tilde{\boldsymbol{\tau}}(\tilde{\boldsymbol{v}})\in\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}\cap\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$
s.t. $\widehat{\nabla}\cdot\tilde{\boldsymbol{\tau}}=\tilde{\boldsymbol{v}}$
and
$\left\lVert\tilde{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}\leq\left\lVert\tilde{\boldsymbol{v}}\right\rVert_{\boldsymbol{L}^{2}}$.
And one obtains

$\displaystyle\underset{\tilde{\boldsymbol{s}}\in\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}}{\sup}\frac{\langle\nabla\cdot\tilde{\boldsymbol{s}},\tilde{\boldsymbol{v}}\rangle}{\left\lVert\tilde{\boldsymbol{s}}\right\rVert_{\boldsymbol{\mathcal{H}}(\widehat{\textup{div}})}}\geq\frac{\langle\nabla\cdot\tilde{\boldsymbol{\tau}},\tilde{\boldsymbol{v}}\rangle}{\left\lVert\tilde{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}(\widehat{\textup{div}})}}\geq\frac{1}{\sqrt{2}}\frac{\left\lVert\tilde{\boldsymbol{v}}\right\rVert_{\boldsymbol{L}^{2}}^{2}}{\left\lVert\tilde{\boldsymbol{v}}\right\rVert_{\boldsymbol{L}^{2}}}\geq\frac{1}{\sqrt{2}}\left\lVert\tilde{\boldsymbol{v}}\right\rVert_{\boldsymbol{L}^{2}}.$

The arbitrariness of $\tilde{\boldsymbol{v}}$ shows us the validity of the
inf-sup stability result in the parametric case.

The general case is now shown utilizing Lemma 16 and Lemma 13. Let
$\boldsymbol{v}\in{\boldsymbol{V}}_{h,3}^{s}\textbackslash\\{0\\}$ and
$\tilde{\boldsymbol{v}}\coloneqq\mathcal{Y}^{s}_{3}(\boldsymbol{v})$ arbitrary
but fixed. Then if $\tilde{\boldsymbol{\tau}}$ is chosen as above and setting
$\boldsymbol{\tau}\coloneqq(\mathcal{Y}_{2,\Gamma_{1}}^{s})^{-1}\tilde{\boldsymbol{\tau}}$
we get $\nabla\cdot\boldsymbol{\tau}={\boldsymbol{v}}$. One observes
$\tilde{\boldsymbol{\tau}}\in\boldsymbol{\mathcal{H}}_{\hat{\Gamma}_{1}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$.
By Lemma 12 and the invertibility of $\mathcal{Y}_{2,\Gamma_{1}}^{s}$ there
are constants $C>0$ s.t.

$\left\lVert\boldsymbol{\tau}\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}\leq
C\left\lVert\tilde{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}\leq
C\left\lVert\tilde{\boldsymbol{v}}\right\rVert_{\boldsymbol{L}^{2}}\leq
C\left\lVert{\boldsymbol{v}}\right\rVert_{\boldsymbol{L}^{2}}.$

Consequently,

$\displaystyle\underset{{\boldsymbol{s}}\in{\boldsymbol{\mathcal{V}}}_{h,2}^{s}}{\sup}\frac{\langle\nabla\cdot{\boldsymbol{s}},{\boldsymbol{v}}\rangle}{\left\lVert{\boldsymbol{s}}\right\rVert_{\boldsymbol{\mathcal{H}}(\textup{div})}}\geq\frac{\langle\nabla\cdot{\boldsymbol{\tau}},{\boldsymbol{v}}\rangle}{\left\lVert{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}(\textup{div})}}\geq\frac{\left\lVert{\boldsymbol{v}}\right\rVert_{\boldsymbol{L}^{2}}^{2}}{C\
\sqrt{2}\
\left\lVert{\boldsymbol{v}}\right\rVert_{\boldsymbol{L}^{2}}}\geq\frac{1}{C\
\sqrt{2}}\left\lVert{\boldsymbol{v}}\right\rVert_{\boldsymbol{L}^{2}}\ \ .$

Latter estimate yields the inf-sup stability result. ∎

###  4.4 A first error estimate

In this last theory section we look at an error estimate for the Dirichlet
problem with strong symmetry. We use a standard approach involving spline
projections and orient ourselves towards [11].

######  Definition 1.

For $p_{1}-1\geq r_{1}\geq-1,\ p_{2}-1\geq r_{2}\geq 0$ and $p_{3}-1\geq
r_{3}\geq 1$ we set

$\displaystyle\widehat{\Pi}_{p_{1}}$ $\displaystyle\colon
L^{2}((0,1))\rightarrow S_{p_{1}}^{r_{1}},\
v\mapsto\sum_{i}(\lambda_{i}^{p_{1}}v)\widehat{B}_{i,p_{1}},\ \
\textup{analogous to }(3.6)\ \textup{in
\cite[cite]{[\@@bibref{}{Buffa2011IsogeometricDD}{}{}]}},$
$\displaystyle\widehat{\Pi}_{p_{2}-1}^{c,1}$ $\displaystyle\colon
L^{2}((0,1))\rightarrow S_{p_{2}-1}^{r_{2}-1},\
v\mapsto\frac{d}{d\zeta}\widehat{\Pi}_{p_{2}}\int_{0}^{\zeta_{1}}v(\tau)\
\textup{d}\tau,\ \ \textup{analogous to }(3.9)\ \textup{in
\cite[cite]{[\@@bibref{}{Buffa2011IsogeometricDD}{}{}]}},$
$\displaystyle\widehat{\Pi}_{p_{3}-2}^{c,2}$ $\displaystyle\colon
L^{2}((0,1))\rightarrow S_{p_{3}-2}^{r_{3}-2},\
v\mapsto\frac{d}{d\zeta}\widehat{\Pi}_{p_{3}-1}^{c,1}\int_{0}^{\zeta_{1}}v(\tau)\
\textup{d}\tau=\frac{d^{2}}{d\zeta^{2}}\widehat{\Pi}_{p_{3}}\int_{0}^{\zeta_{1}}\int_{0}^{\tau}v(r)\
\textup{d}r\,\textup{d}\tau.$

Above $\lambda_{i}^{p}$ denote the canonical dual basis functionals
corresponding to the spline basis functions, i.e.
$\lambda_{i}^{p}(\widehat{B}_{j,p})=\delta_{ij}$. For more information we
refer to [24, Section 4.6].

Below, we assume for the underlying regularity parameter always $r_{i}=r$.
Useful for our purposes are the subsequent properties.

######  Lemma 22.

The above projections of the univariate case satisfy the following properties,
where we assume $r\geq-1,\ p\geq 0$ for the first two lines and $p-1\geq r\geq
0$ otherwise and let $I\coloneqq(\psi_{i},\psi_{i+1})$ denote an arbitrary
sub-interval induced by the discretization.

$\displaystyle\widehat{\Pi}_{p}s$ $\displaystyle=s$ $\displaystyle\forall s\in
S^{r}_{p},$ (46) $\displaystyle|\widehat{\Pi}_{p}v|_{H^{l}(I)}$
$\displaystyle\leq C\ |v|_{H^{l}(\tilde{I})}$ $\displaystyle\forall v\in
H^{l}((0,1)),\ 0\leq l\leq p+1,$ $\displaystyle\widehat{\Pi}_{p-1}^{c,1}s$
$\displaystyle=s$ $\displaystyle\forall s\in S^{r-1}_{p-1},$ (47)
$\displaystyle\widehat{\Pi}_{p-1}^{c,1}\partial_{\zeta}v$
$\displaystyle=\partial_{\zeta}\widehat{\Pi}_{p}v$ $\displaystyle\forall v\in
H^{1}((0,1)),$ (48) $\displaystyle|\widehat{\Pi}_{p-1}^{c,1}v|_{H^{l}(I)}$
$\displaystyle\leq C\ |v|_{H^{l}(\tilde{I})}$ $\displaystyle\forall v\in
H^{l}((0,1)),\ 0\leq l\leq p,$ (49)

for some constant $C$ independent of mesh refinement.
And if it is $r\geq 1,\ p\geq 2$ we further have

$\displaystyle\widehat{\Pi}_{p-2}^{c,2}s$ $\displaystyle=s$
$\displaystyle\forall s\in S^{r-2}_{p-2},$ (50)
$\displaystyle\widehat{\Pi}_{p-2}^{c,2}\partial_{\zeta}v$
$\displaystyle=\partial_{\zeta}\widehat{\Pi}_{p-1}^{c,1}v$
$\displaystyle\forall v\in H^{1}((0,1)),$ (51)
$\displaystyle|\widehat{\Pi}_{p-2}^{c,2}v|_{H^{l}(I)}$ $\displaystyle\leq C\
|v|_{H^{l}(\tilde{I})}$ $\displaystyle\forall v\in H^{l}((0,1)),\ 0\leq l\leq
p-1.$ (52)

Above $|\cdot|_{H^{l}}$ denotes the standard Sobolev seminorm.

###### Proof.

The first five statements correspond to the properties (3.7), (3.8) and
(3.10)-(3.12) in [11] if we keep the regular mesh assumption in mind. Hence we
only check the points (50)-(52).
In the following we use (3.4) of [11], which gives immediately that
$\partial_{\zeta}^{2}S_{p}^{r}=S_{p-2}^{r-2}$. Firstly, with property (47) we
have

$\displaystyle\widehat{\Pi}_{p-2}^{c,2}s=\frac{d}{d\zeta}\widehat{\Pi}_{p-1}^{c,1}\underbrace{\int_{0}^{\zeta_{1}}s(\tau)\
\textup{d}\tau}_{\in
S_{p-1}^{r-1}}=\frac{d}{d\zeta}{\int_{0}^{\zeta_{1}}s(\tau)\
\textup{d}\tau}=s.$

In view of (46) and (48) one gets

$\displaystyle\widehat{\Pi}_{p-2}^{c,2}\partial_{\zeta}v$
$\displaystyle=\frac{d^{2}}{d\zeta^{2}}\widehat{\Pi}_{p}\int_{0}^{\zeta_{1}}v(\tau)+a\
\textup{d}\tau\,=\frac{d^{2}}{d\zeta^{2}}\widehat{\Pi}_{p}\int_{0}^{\zeta_{1}}v(\tau)\
\textup{d}\tau=\frac{d}{d\zeta}\widehat{\Pi}_{p-1}^{c,1}v.$

In the last line we wrote $a\in\mathbb{R}$ for a suitable constant of
integration. Hence (51) follows.
The last point can be proven in a similar fashion like equation 3.12 in [11]
using the fact that if $v\in H^{l}((0,1))$ then
$w(\zeta)\coloneqq\int_{0}^{\zeta_{1}}v\textup{d}\tau\in H^{l+1}((0,1))$ and
using estimates (49), (51). It is

$|\widehat{\Pi}_{p-2}^{c,2}v|_{H^{l}(I)}\leq|\widehat{\Pi}_{p-1}^{c,1}w|_{H^{l+1}(I)}\leq
C\,|w|_{H^{l+1}(\tilde{I})}\leq C\,|v|_{H^{l}(\tilde{I})}.$

∎

######  Lemma 23 (Projections on the parametric domain).

We define the next projections on the parametric domain

$\displaystyle\widehat{\Pi}_{h,2}^{s}=\widehat{\Pi}_{h,2}^{s}(p,r)$
$\displaystyle\coloneqq\textup{SYM}(\widehat{\Pi}_{p+1}\otimes\widehat{\Pi}_{p-1}^{c,2},\
\widehat{\Pi}_{p}^{c,1}\otimes\widehat{\Pi}_{p}^{c,1},\
\widehat{\Pi}_{p-1}^{c,2}\otimes\widehat{\Pi}_{p+1}),$ (53)
$\displaystyle\widehat{\Pi}_{h,3}^{s}=\widehat{\Pi}_{h,3}^{s}(p,r)$
$\displaystyle\coloneqq\big{(}\widehat{\Pi}_{p}^{c,1}\otimes\widehat{\Pi}_{p-1}^{c,2},\
\widehat{\Pi}_{p-1}^{c,2}\otimes\widehat{\Pi}_{p}^{c,1}\big{)}^{T}.$ (54)

It holds the commutativity relation

$(\widehat{\nabla}\cdot)\circ\widehat{\Pi}_{h,2}^{s}=\widehat{\Pi}_{h,3}^{s}\circ(\widehat{\nabla}\cdot)\
\ \textup{on}\
\boldsymbol{\mathcal{H}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S}).$

If
$\hat{\boldsymbol{\tau}}\coloneqq\textup{SYM}(\hat{\tau}_{11},\hat{\tau}_{12},\hat{\tau}_{22})\in{\boldsymbol{\mathcal{H}}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})\cap\boldsymbol{\mathcal{H}}^{k}(\widehat{\Omega})$
and $\hat{\boldsymbol{v}}\in\boldsymbol{H}^{k}(\widehat{\Omega})$ we have for
some constant $C$ independent of $h$ the estimates

$\displaystyle\left\lVert\hat{\boldsymbol{\tau}}-\widehat{\Pi}_{h,2}^{s}\hat{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}^{1}}$
$\displaystyle\leq
Ch^{k-1}\left\lVert\hat{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}},p\geq
k\geq 1,\ p>r\geq 1,$ (55)
$\displaystyle\left\lVert\hat{\boldsymbol{v}}-\widehat{\Pi}_{h,3}^{s}\hat{\boldsymbol{v}}\right\rVert_{\boldsymbol{L^{2}}}$
$\displaystyle\leq
Ch^{k}\left\lVert\hat{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}^{k}},\ p\geq
k\geq 0.$ (56)

###### Proof.

We begin with the commutativity relation. Basically, the proof steps are the
same as of Lemma 4.3 in [11], but for reasons of completeness, we sketch the
proof steps. Therefore, we first assume that we have a smooth mapping
$\hat{\boldsymbol{\tau}}\in\boldsymbol{\mathcal{H}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})$.
We exploit now Lemma 22 and the tensor product structure of the multivariate
spline projections. For the latter we refer to the explanations in Section
4.1. of the mentioned paper [11]. This makes the next equality chain
reasonable.

$\displaystyle\Big{[}(\widehat{\nabla}\cdot)\circ\widehat{\Pi}_{h,2}^{s}\hat{\boldsymbol{\tau}}\Big{]}_{1}$
$\displaystyle=\hat{\partial}_{1}\big{(}\widehat{\Pi}_{p+1}\otimes\widehat{\Pi}_{p-1}^{c,2}\big{)}\hat{\tau}_{11}+\hat{\partial}_{2}\big{(}\widehat{\Pi}_{p}^{c,1}\otimes\widehat{\Pi}_{p}^{c,1}\big{)}\hat{\tau}_{12}$
$\displaystyle=\big{(}\hat{\partial}_{1}\widehat{\Pi}_{p+1}\otimes\widehat{\Pi}_{p-1}^{c,2}\big{)}\hat{\tau}_{11}+\big{(}\widehat{\Pi}_{p}^{c,1}\otimes\hat{\partial}_{2}\widehat{\Pi}_{p}^{c,1}\big{)}\hat{\tau}_{12}$
$\displaystyle=\big{(}\widehat{\Pi}_{p}^{c,1}\hat{\partial}_{1}\otimes\widehat{\Pi}_{p-1}^{c,2}\big{)}\hat{\tau}_{11}+\big{(}\widehat{\Pi}_{p}^{c,1}\otimes\widehat{\Pi}_{p-1}^{c,2}\hat{\partial}_{2}\big{)}\hat{\tau}_{12}$
$\displaystyle=\big{(}\widehat{\Pi}_{p}^{c,1}\otimes\widehat{\Pi}_{p-1}^{c,2}\big{)}\hat{\partial}_{1}\hat{\tau}_{11}+\big{(}\widehat{\Pi}_{p}^{c,1}\otimes\widehat{\Pi}_{p-1}^{c,2}\big{)}\hat{\partial}_{2}\hat{\tau}_{12}=\big{(}\widehat{\Pi}_{p}^{c,1}\otimes\widehat{\Pi}_{p-1}^{c,2}\big{)}(\widehat{\nabla}\cdot[\hat{\tau}_{11},\hat{\tau}_{12}]^{T}).$

A similar calculation can be applied for the second entry of
$(\widehat{\nabla}\cdot)\circ\widehat{\Pi}_{h,2}^{s}\hat{\boldsymbol{\tau}}$.
Consequently, the commutativity is clear.
The estimates for the projection are based on the boundedness and spline-
preserving properties of the univariate projections in Lemma 22. The proof is
analogous to the proof of Lemma 5.1 in [11] and we state briefly the
underlying idea. Let
$\hat{\boldsymbol{\tau}}\in{\boldsymbol{\mathcal{H}}}(\widehat{\Omega},\widehat{\textup{div}},\mathbb{S})\cap\boldsymbol{\mathcal{H}}^{k}(\widehat{\Omega}),\
p\geq k\geq 1,\ p>r\geq 1$. One uses Lemma 3.1 of [8] which gives us the
existence of a spline mapping
$\boldsymbol{s}\in\widehat{\boldsymbol{\mathcal{V}}}_{h,2}^{s}$ such that

$\left\lVert\hat{\boldsymbol{s}}-\hat{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}^{1}}\leq
Ch^{k-1}\left\lVert\hat{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}},\
\ \
\left\lVert\hat{\boldsymbol{s}}-\hat{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}\leq
Ch^{k}\left\lVert\hat{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}},$

for a suitable constant $C$. Since the univariate projections are spline
preserving and the tensor-product structure of the multivariate projections
fits to the product structure of the spline spaces we get directly the spline
preserving property of $\widehat{\Pi}_{h,2}^{s}$ and
$\widehat{\Pi}_{h,3}^{s}$. Thus, an application of the triangle inequality and
a standard inverse estimate for polynomials yields

$\displaystyle\left\lVert\hat{\boldsymbol{\tau}}-\widehat{\Pi}_{h,2}^{s}\hat{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}^{1}}$
$\displaystyle\leq\left\lVert\hat{\boldsymbol{\tau}}-\hat{\boldsymbol{s}}\right\rVert_{\boldsymbol{\mathcal{H}}^{1}}+\left\lVert\widehat{\Pi}_{h,2}^{s}\big{(}\hat{\boldsymbol{s}}-\hat{\boldsymbol{\tau}}\big{)}\right\rVert_{\boldsymbol{\mathcal{H}}^{1}}$
$\displaystyle\leq
Ch^{k-1}\left\lVert\hat{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}}+\frac{C}{h}\left\lVert\hat{\boldsymbol{s}}-\hat{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{L}}^{2}}$
$\displaystyle\leq
Ch^{k-1}\left\lVert\hat{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}^{k}}.$

Above we used the boundedness of the projection w.r.t. to the
$\boldsymbol{\mathcal{L}}^{2}$-norm following from the boundedness properties
in Lemma 22. ∎

######  Lemma 24.

If
$\boldsymbol{\tau}\in\boldsymbol{\mathcal{H}}({\Omega},{\textup{div}},\mathbb{S})\cap\boldsymbol{\mathcal{H}}^{k+1}(\Omega)$
and $\boldsymbol{v}\in\boldsymbol{H}^{k}(\Omega),\ k\geq 0$, then it holds

$\displaystyle\left\lVert\mathcal{Y}_{2}^{s}(\boldsymbol{\tau})\right\rVert_{\boldsymbol{\mathcal{H}}^{k}}$
$\displaystyle\leq
C\left\lVert{\boldsymbol{\tau}}\right\rVert_{\boldsymbol{\mathcal{H}}^{k+1}},$
$\displaystyle\left\lVert\mathcal{Y}_{3}^{s}(\boldsymbol{{v}})\right\rVert_{\boldsymbol{H}^{k}}$
$\displaystyle\leq
C\left\lVert{\boldsymbol{v}}\right\rVert_{\boldsymbol{H}^{k}},$

assuming a regular enough $\boldsymbol{F}$; e.g. $\boldsymbol{F}\in
C^{k+2}(\overline{\widehat{\Omega}})$ and $\boldsymbol{F}^{-1}\in
C^{k+2}({\overline{\Omega}})$. The constant $C$ depends on the
parametrization.

###### Proof.

Again we concentrate on the first estimate, since the second one is proven
with a similar approach. Further, we first assume $\boldsymbol{\tau}\in
C^{k+1}(\overline{\Omega})\cap\boldsymbol{\mathcal{L}}^{2}(\Omega,\mathbb{S})$
and start with the case $k=1$. Note that the assertion is clear for $k=0$; see
proof of Lemma 19. With the definition (39) of $\mathcal{Y}_{2}^{s}$, problems
may arise due to the term $\mathcal{Y}_{2,A}^{s}$ because of the evaluation on
the boundary part $\hat{\Gamma}_{1}$. Applying derivatives
$\hat{\partial}_{1},\hat{\partial}_{2}$ to
$\mathcal{Y}_{2,A}^{s}(\boldsymbol{\tau})$ we obtain expressions involving
values and first derivatives
$\partial_{m}\boldsymbol{\tau}\circ\boldsymbol{F}(0,\cdot)$ on the boundary
edge $\hat{\Gamma}_{1}$ which are multiplied by geometry dependent terms. By
the trace theorem (see [26, Chapter 2]) we obtain

$\displaystyle\left\lVert{\tau_{ij}}\circ\boldsymbol{F}\right\rVert_{{L}^{2}(\hat{\Gamma}_{1})}$
$\displaystyle\leq
C\left\lVert{\tau_{ij}}\circ\boldsymbol{F}\right\rVert_{{H}^{1}(\widehat{\Omega})}\leq
C\left\lVert{\tau_{ij}}\right\rVert_{{H}^{1}({\Omega})},$
$\displaystyle\left\lVert\partial_{m}{\tau_{ij}}\circ\boldsymbol{F}\right\rVert_{{L}^{2}(\hat{\Gamma}_{1})}$
$\displaystyle\leq
C\left\lVert\partial_{m}{\tau_{ij}}\circ\boldsymbol{F}\right\rVert_{{H}^{1}(\widehat{\Omega})}\leq
C\left\lVert{\tau_{ij}}\right\rVert_{{H}^{2}(\Omega)}.$

Consequently, if the geometry function $\boldsymbol{F}$ is regular enough, one
can see that

$\left\lVert\mathcal{Y}_{2,A}^{s}(\boldsymbol{\tau})\right\rVert_{\boldsymbol{\mathcal{H}}^{1}(\widehat{\Omega})}\leq
C\left\lVert\boldsymbol{\tau}\right\rVert_{\boldsymbol{\mathcal{H}}^{2}({\Omega})}$

and clearly
$\left\lVert\mathcal{Y}_{2}^{s}(\boldsymbol{\tau})\right\rVert_{\boldsymbol{\mathcal{H}}^{1}(\widehat{\Omega})}\leq
C\left\lVert\boldsymbol{\tau}\right\rVert_{\boldsymbol{\mathcal{H}}^{2}({\Omega})}$.
Since the smooth mappings are dense in $\boldsymbol{\mathcal{H}}^{2}(\Omega)$
the inequality is still valid for a general element in
$\boldsymbol{\mathcal{H}}^{2}(\Omega)$. A similar reasoning can be used for
the cases $k\geq 2$.

∎

######  Lemma 25 (Projections on the physical domain).

Utilizing the mappings $\mathcal{Y}_{2}^{s},\ \mathcal{Y}_{3}^{s}$ we define
the projections onto the actual test function spaces

$\displaystyle{\Pi}_{h,2}^{s}$
$\displaystyle\colon\boldsymbol{\mathcal{H}}(\Omega,\textup{div},\mathbb{S})\cap\boldsymbol{\mathcal{H}}^{1}(\Omega)\rightarrow\boldsymbol{\mathcal{V}}_{h,2}^{s}\
,\
\boldsymbol{\tau}\mapsto\tilde{\mathcal{Y}}_{2}^{s}\circ\widehat{\Pi}_{h,2}^{s}\circ\mathcal{Y}_{2}^{s}(\boldsymbol{\tau}),$
$\displaystyle{\Pi}_{h,3}^{s}$
$\displaystyle\colon\boldsymbol{L}^{2}(\Omega)\rightarrow\boldsymbol{V}_{h,3}^{s}\
,\