Isogeometric multi-patch C^1-mortar coupling for the biharmonic equation

A. Benvenuti^* G. LoliUniversità di Pavia, Dipartimento di Matematica “F. Casorati”, Via A. Ferrata 1, 27100 Pavia, Italy. G. Sangalli^*IMATI-CNR “Enrico Magenes”,  Pavia, Italy.
T. TakacsJohann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Str. 69, 4040 Linz, Austria
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We propose an isogeometric mortar method to fourth order elliptic problems. In particular we are interested in the discretization of the biharmonic equation on $C^0$-conforming multi-patch domains and we exploit the mortar technique to weakly enforce $C^1$-continuity across interfaces.
In order to obtain discrete inf-sup stability, a particular choice for the Lagrange multiplier space is needed. Actually, we use as multipliers splines of degree reduced by two, w.r.t. the primal spline space, and with merged elements in the neighbourhood of the corners. In this framework, we are able to show optimal a priori error estimates.
We also perform numerical tests that reflect theoretical results.
Keywords:  Isogeometric analysis, splines, multi-patch, mortar method, fourth-order PDEs, biharmonic equation.

§ INTRODUCTION

Isogeometric Analysis (IgA) is a computational approach for the discretization of Partial Differential Equations (PDE), introduced in <cit.>, with the aim of integrating Finite Element Analysis (FEA) and Computer Aided Design (CAD). Indeed, in IgA, the same mathematical objects, such as B-splines and Non-Uniform Rational B-splines (NURBS), are used for both the geometrical representation of the computational domain and the numerical approximation of the solution of the PDE that model the problem of interest. The use of smooth splines for the discrete solution of PDEs bring several advantages. Indeed, the possibility of choosing high-continuity splines improves the accuracy per degree of freedom, when compared to $C^0$ piecewise polynomials, see <cit.>, and it also allows direct approximation of PDEs of order higher than two, for example Kirchhoff-Love plates/shells <cit.>,the Cahn-Hilliard equation <cit.> and the Navier-Stokes-Korteweg equation <cit.>.

In problems of practical relevance, it is often too difficult to represent the whole geometry by a single patch parametrization. Although high-order continuity is attained naturally within a patch, in the multi-patch framework a direct strong coupling between patches is not straightforward to achieve, cf. <cit.>. One possible technique to couple subdomains is the mortar method <cit.>, originally introduced as a domain decomposition method in the Finite Element context. The mortar method weakly enforces continuity constraints across interfaces by means of Lagrange multipliers. The choice of the Lagrange multipliers space is crucial and must satisfy two abstract requirements: to provide a sufficient approximation order and to fulfil an inf-sup stability condition.
In the isogeometric context, applications of the mortar method can be found in <cit.>.
In <cit.> the implementation of $C^0$-continuity constraints along the interfaces of a multi-patch domain is investigated theoretically and numerically in an
isogeometric mortar framework.
In particular, given an isogeometric test space of degree $p$, multipliers spaces of degree $p,p-1$ and $p-2$ are considered. The pairing $p/p-1$ is numerically proved to be unstable while for the equal order pairing $p/p$ a boundary modification on the Lagrange multipliers is required in order to avoid instabilities at the cross-points. This modification is performed by reducing the dimension of the dual space, such that a counting argument for the degrees of freedom still holds and this is done by a degree reduction in the neighbourhood of cross-points. Finally, the pairing $p/p-2$ is proved to be stable and, even if the expected order of convergence is suboptimal, for some parametrizations a better order is obtained.

In this paper, given a $C^0$-conforming multi-patch domain, for weakly imposing $C^1$-continuity constraints along patch interfaces, we adopt a $p/p-2$ pairing and reduce the dimension of the Lagrange multiplier spaces by coarsening the mesh in the neighbourhood of the vertices (i.e. merging the first and last two elements).

The structure of the paper is as follows. Section <ref> proposes a brief overview on B-splines, introducing some
notations and basic concepts. In Section <ref> we describe the model problem and state the well-posedness of its mortar formulation.
Section <ref> analyses the isogeometric discretization of the model problem and provides an a priori error estimate.
In Section <ref> we study the validity of the discrete inf-sup condition.
Finally, Section <ref> collects several numerical results assessing the effectiveness of the proposed method and its optimal approximation properties. Finally, conclusions are drawn in Section <ref>.

§ PRELIMINARIES ABOUT ISOGEOMETRIC ANALYSIS

In this section we propose a brief overview on B-splines, introducing some notations and basic concepts. For a complete overview we refer to cottrell,bazilevs,schumaker.
In order to define the univariate B-spline basis functions we need to introduce knot vectors.

A knot vector in one dimension is a non-decreasing vector of real numbers

\begin{equation*}
\Xi = \left(\xi_1, \xi_2, \dots, \xi_{n+p+1}\right),
\end{equation*}

where $p$ is the polynomial degree and $n$ is the number of basis functions. Knot values may be repeated, $\xi_1 \leq \xi_2 \leq \dots \leq \xi_{n+p+1}$. Without loss of generality, we can assume that $\xi_1 = 0$ and $\xi_{n+p+1} = 1$.  In the following we always make use of open knot vectors, that is, the first and the last knots appear exactly $p+1$ times:

\begin{equation*}
0 = \xi_1 = \xi_{p+1} < \xi_{p+2} \leq \dots \leq \xi_{n} < \xi_{n+1} = \xi_{n+p+1} = 1.
\end{equation*}

We also introduce the vector

\begin{equation*}
Z = (\zeta_1,\ldots,\zeta_{n_{\textsc{bp}}}),\, \mbox{with}\, 0=\zeta_1<\cdots<\zeta_{n_{\textsc{bp}}} = 1.
\end{equation*}

of knots without repetitions, also called breakpoint vector. For any breakpoint $\zeta_j \in Z$, we define $m_j$ as its multiplicity in $\Xi$. The breakpoint vector forms a partition of the parametric domain $\widehat{\Omega} = \left(0,1\right)$.

B-spline basis functions of degree $p$ are defined recursively by means of Cox-de Boor formula: for all $1 \leq i \leq n$ we set

\begin{equation}\label{eq:coxdeboor}
\begin{split}
B^0_i(x) &=
\begin{cases}
1 & \text{if $\xi_i \leq x < \xi_{i+1}$,} \\
0 & \text{otherwise,}
\end{cases}\\
B^p_i(x) &= \dfrac{x - \xi_i}{\xi_{i+p} - \xi_i}B^{p-1}_{i}(x) + \dfrac{\xi_{i+p+1} - x}{\xi_{i+p+1} - \xi_{i+1}}B^{p-1}_{i+1}(x), \, p \geq 1,
\end{split}
\end{equation}

where we assume that if a denominator is zero, the corresponding coefficient is defined to be zero.
These $n$ B-spline functions are linear independent non-negative piecewise polynomials of degree $p$, with $p - m_j$ continuous derivatives at the breakpoint $\zeta_j$, $j \in \Set{ 2, \dots, n_{\textsc{bp}} -1 }$. They form a partition of unity, namely

\begin{equation*}
\sum_{i = 1}^{n} B^p_i(x) = 1, \quad \forall x \in [0,1],
\end{equation*}

and, since the knot vector is open, they are interpolatory at the first and last knots.
The space of splines of degree $p$ is denoted by

\begin{equation*}
\Sp^p \equiv \Sp^p(\Xi) := \sspan\Set{ B^p_i }_{i=1}^n.
\end{equation*}

If $p - m_j = r$ for all $j \in \Set{ 2, \dots, n_{\textsc{bp}} -1 }$, we indicate by $\Sp^p_r$ the space of splines of degree $p$ that are globally $C^r$-continuous.

In the following we will make use of spline spaces $\Sp^p_r$ that are at least globally $C^1$-continuous, i.e., with $p \geq 2$ and $r \geq 1$.

The definition of spline spaces can be extended to the two-dimensional case by means of tensor product structure.
Let $\widehat{\Omega} = (0,1)^2 \subset \R^2$ be an open parametric domain and $\left( \Xi_{\delta}\right)_{\delta=1}^2$ a set of open knot vectors. This then defines a mesh $\Q_h$ on $(0,1)^2$, namely, a partition of the parametric domain into two-dimensional elements:

\begin{equation*}
\begin{split}
\Q_h &\equiv \Q_h(\Xi_1, \Xi_2) \\
&:= \Set{ Q= \left( \zeta_{i}^1, \zeta_{i+1}^1 \right) \times \left( \zeta_{j}^2, \zeta_{j+1}^2 \right) \mid  1 \leq i < n_{\textsc{bp}_1}, \,  1 \leq j < n_{\textsc{bp}_2}}.
\end{split}
\end{equation*}

The parameter $h$ will be referred to as global mesh size and it is defined as:

\begin{equation*}
h:= \max_{\substack{1 \leq k \leq 2 \\ 1 \leq i \leq m+p+1 }}\{ h_{k,i} \},
\end{equation*}

where we have defined $h_{k,i}:= | \zeta_{k,i+1} - \zeta_{k,i} |$ and $\zeta_{k,i}$ denotes the $i$-th break point in the break point vector $Z_k$.

We assume that the knot vectors are quasi-uniform, that is, there exists  $\alpha > 0$, independent
of $h$, such that each nonempty knot span $( \xi_{k,i} ,
\xi_{k,i+1})$ fulfils $ \alpha h \leq \xi_{k,i+1} - \xi_{k,i}$, for $1\leq k \leq 2$.

For each multi-index

\begin{equation*}
\textbf{i} \in \mathcal{I} := \Set{ (i_1, i_2) | 1 \leq i_1 \leq n_1, \, 1 \leq i_2 \leq n_2},
\end{equation*}

given a vector of polynomial degrees $\textbf{p} = (p_1, p_2)$, we define the bivariate B-spline basis functions as:

\begin{equation*}
B_{\textbf{i}}^{\textbf{p}} = B_{i,1}^{p_1} \otimes B_{j,2}^{p_2},
\end{equation*}

where any $B_{i_\delta,\delta}^{p_\delta}$ is a univariate B-spline basis function of the form (<ref>).

The tensor-product spline space $\H{\Sp}^{\textbf{p}}$ is the space spanned by the bivariate B-spline basis functions:

\begin{equation*}
\H{\Sp}^{\textbf{p}} = \H{\Sp}^{\textbf{p}}(\Xi_1, \Xi_2) = \Sp^{p_1}(\Xi_1) \otimes \Sp^{p_2}(\Xi_2) := \sspan \left\lbrace B_{\textbf{i}}^{\textbf{p}}\right\rbrace _{\textbf{i} \in I}.
\end{equation*}

Given the vector $\textbf{r} = (r_1, r_2)$, with $0 \leq r_\delta \leq p_\delta$, we indicate by $\H{\Sp}^{\textbf{p}}_{\textbf{r}}$ the space of splines of degree $p_\delta$ that are globally $C^{r_\delta}$-continuous along the $\delta$-th parametric direction, for $\delta = 1, 2$.
A spline function $\widehat{s} \in \H{\Sp}^{\textbf{p}}_{\textbf{r}}$ is written as

\begin{equation*}
\widehat{s}(\widehat{x},\widehat{y}) = \sum_{\textbf{i} \in \mathcal{I}} \textbf{C}_{\textbf{i}} B_{\textbf{i}}^{\textbf{p}} (\widehat{x}, \widehat{y}),
\end{equation*}

where the $\textbf{C}_{\textbf{i}}$ are the degrees-of-freedom or control points of $\widehat{s}$. Throughout the paper we assume that all meshes are quasi-uniform.
For the sake of simplicity, in the following we assume that the spline degree is the same for both the parametric directions, i.e. $p_1 = p_2 = p$. Moreover we assume that $n_1 = n_2 = n$, $n_{\textsc{bp}_1} = n_{\textsc{bp}_2} = n_{\textsc{bp}}$ and $r_1 = r_2 = r$.

§ A MORTAR $C^1$ MULTI-PATCH MODEL PROBLEM

Let $\Omega \subset \R ^2$ be a bounded connected domain with a piecewise polynomial boundary.  We consider the following fourth order elliptic problem with homogeneous boundary conditions

\begin{equation}\label{eq:bilaplace_strong}
\begin{cases}
\begin{aligned}
\Delta^2 u &= f & &\text{in $\Omega$} \\
u &= 0 & &\text{on $\partial\Omega$} \\
\partial_n  u &= 0 & &\text{on $\partial\Omega$.}
\end{aligned}
\end{cases}
\end{equation}

We recall that a possible variational formulation of (<ref>) is: Find $u \in H^2_0(\Omega)$ such that

\begin{align}\label{eq:bilaplace_var}
\int_{\Omega} \nabla (\nabla u ) : \nabla (\nabla v ) \, dx = \int_{\Omega} fv \, dx, & & \forall v \in H^2_0(\Omega).
\end{align}

In (<ref>), $f$ is required to be a functional of $H^{-2}(\Omega) = (H^2_0(\Omega))'$.
The mortar method applies to a decomposition of the domain $\Omega$ into non-overlapping subdomains, i.e., we introduce a family of open sets $\left\lbrace \Omega_k\right\rbrace _{k \in \io}$ such that

\begin{equation*}
\overline{\Omega} = \bigcup_{k \in \io} \overline{\Omega}_k, \quad \Omega_k \cap \Omega_\ell = \emptyset \quad \forall \, k,\ell \in \io, \, k \neq \ell,
\end{equation*}

where $\io$ is the finite set of patch indices.
The skeleton $\Sigma$ of the decomposition is defined by

\begin{equation*}
\Sigma := \bigcup_{k \in \io} \partial \Omega_k \setminus \partial \Omega,
\end{equation*}

and admits a partition. Indeed, for every pair $k, k' \in \io$ such that $\partial \Omega_k \cap \partial \Omega_{k'} \neq \emptyset$, we can define

\begin{equation*}
\Gamma_{k,k'} = \partial \Omega_k \cap \partial \Omega_{k'},
\end{equation*}

and then, enumerating all the interfaces $\left\lbrace  \Gamma_{k,k'} \right\rbrace  \leftrightarrow \left\lbrace  \Gamma_{\ell} \right\rbrace _{\ell \in \ig} $, we can write

\begin{equation*}
\Sigma = \bigcup_{\ell \in \ig} \Gamma_{\ell},
\end{equation*}

where $\ig$ is the finite set of interface indices.
Moreover, each interface $\Gamma_\ell$ can be associated to a pair of indices,  $m(\ell)$ and $s(\ell)$, such that $\overline{\Gamma}_\ell = \overline{\Omega}_{m(\ell)} \cap \overline{\Omega}_{s(\ell)}$. We will refer to $\Omega_{m(\ell)}$ as the primary subdomain of $\Gamma_\ell$ and to $\Omega_{s(\ell)}$ as the secondary subdomain, (this choice is arbitrary).
We denote by $\textbf{n}_\ell$ the outer normal to $\Gamma_\ell$ from the primary side, while $\partial_n (\cdot) := \nicefrac{\partial (\cdot)}{\partial \textbf{n}_\ell}$ is the outer normal derivative on $\Gamma_\ell$ along $\textbf{n}_\ell$.

In what follows, we will denote by $C$ a positive, dimensionless constant, possibly different at each occurrence and independent from $h$.

Each subdomain $\Omega_k$ is the image of the parametric space $\widehat{\Omega} = (0,1)^2$ by a spline geometry mapping $\bm{F}_k \colon \widehat{\Omega} \to \Omega_k$, with $\bm{F}_k \in \H{\Sp}^{\textbf{p}}_{\textbf{r}} \times \H{\Sp}^{\textbf{p}}_{\textbf{r}}$, satisfying $\det \nabla \bm{F}_k >0$.

We assume that the meshes are $C^0$-conforming, that is for all $\ell \in \ig$, the mesh that $\gl$ inherits from $\Omega_{m(\ell)}$ coincides with the one that it inherits from $\Omega_{s(\ell)}$, and hence the mesh size on $\gl$ can be simply denoted, with abuse of notation, as

\begin{equation*}
h_\ell := h_{m(\ell)} \simeq h_{s(\ell)} \simeq h.
\end{equation*}

In the following we will indicate with $\iv$ the set of indices that enumerate the vertices $\left\lbrace \vv_i\right\rbrace _{i \in \iv}$ of the decomposition, i.e. the vertices of $\Sigma$, while we will denote by $\mathcal{I}_{\vv,k}$ the set of indices of the four corners of the patch $\Omega_k$.
The four corners on the reference domain $\widehat{\Omega}$ are denoted by $\widehat{\vv}_{i,k} := \bm{F}_k^{-1}(\vv_i)$, or with the more compact notation $\widehat{\vv}_{i} \equiv \widehat{\vv}_{i,k}$, when $k$ is fixed.

For every $k \in \io$ we define the space $H_*^2(\Omega_k)$ as

\begin{equation*}\label{eq:defH2star}
H_*^2(\Omega_k) := \Set{ v_k \in H^2(\Omega_k) | v_k = \partial_n v_k = 0, \quad \text{on $\partial \Omega \cap \partial \Omega_k$}}.
\end{equation*}

In the sequel we make use of finite dimensional spaces of functions that satisfy the following requirements:

* their restriction on $\Omega_k$ belongs to $H^2_*(\Omega_k)$, for all $k \in \io$;

* they are $C^0$ on the whole domain $\Omega$;

* they are $C^2$ at the vertices;

hence we introduce the space

\begin{equation*}\label{eq:spaceX}
X:=\Set{v \in \prod_{k \in \io}H^2_*(\Omega_k)| \jump{\dn v}_{\gl} \in H^{\nicefrac{1}{2}}_{00}(\gl), \, \forall \ell \in \ig} \cap H^1(\Omega),
\end{equation*}

where $\jump{\cdot}_{\gl}$ denotes the jump defined as the trace from the secondary side $s(\ell)$ minus the trace from the primary side $m(\ell)$, equipped with the norm

\begin{equation}\label{eq:normX}
\norm{v}_X^2 := \sum_{k \in \io} \norm{v}_{H^2(\Omega_k)}^2 + \sum_{\ell \in \ig} \norm{\jump{\dn v}_{\gl}}_{H^{\nicefrac{1}{2}}_{00}(\gl)}^2.
\end{equation}

We define the continuous Lagrange multipliers space as

\begin{equation*}
M := \prod_{\ell \in \ig} H^{\nicefrac{-1}{2}}(\Gamma_\ell).
\end{equation*}

Let us introduce the continuous bilinear form on $X \times X$

\begin{equation*}
a(u,v) := \sum_{k \in \io} \int_{\Omega_k} \nabla (\nabla u ) : \nabla (\nabla v ) \, dx,
\end{equation*}

and the continuous bilinear form on $X \times M$

\begin{equation*}
b(v,\mu) := \sum_{\ell \in \ig} \int_{\Gamma_\ell} \mu \jump{\partial_n v}_{\gl} \, d\sigma
\end{equation*}

Let us consider the linear continuous operator $B:X \rightarrow M'$, defined as

\begin{align*}
\left\langle Bv,\mu \right\rangle_{M' \times M}= b(v,\mu), & & \forall v \in X,\quad \forall \mu \in M.
\end{align*}

The following characterization holds:

\begin{equation}\label{eq:char_ker}
H^2_0(\Omega) = \Set{v \in X | b(v,\mu) = 0, \quad \forall \mu \in M}=\ker B.
\end{equation}

For all $u \in H^2_0(\Omega)$ there exists a constant $C > 0$ such that

\begin{equation*}
\norm{u}_{H^2(\Omega)} \leq C \abs{u}_{H^2(\Omega)}
\end{equation*}

For the proof  we refer to <cit.>.

The norms $\abs{u}_{H^2(\Omega)}$ and $\norm{u}_{H^2(\Omega)}$ are equivalent for all $u \in H^2_0(\Omega)$.

We introduce the following saddle point problem:
Find $(u,\mu)\in X \times M$ such that

\begin{equation}\label{eq:q}
\begin{cases}
\begin{aligned}
a(u, v) + b(v, \mu) & = f(v) & & \forall v \in X \\
b(u, \psi)          & = 0    & & \forall \psi \in M.
\end{aligned}
\end{cases}
\end{equation}

The saddle point problem (<ref>) has a unique solution.

From (<ref>) and Lemma <ref> it follows that $a$ is coercive on $\ker B$. Further, by construction it holds $\mathrm{Im} B=M'$. Hence, from <cit.>, the existence and uniqueness for the solution of (<ref>) follows.

If $(u, \mu) \in X \times M$ is the solution of problem (<ref>), then $u \in H^2_0(\Omega)$ and is the solution of problem (<ref>). Furthermore if $u \in H^3(\Omega)$, for all $\ell \in \ig$, then it holds

\begin{equation}\label{eq:lm}
\mu|_{\Gamma_\ell}  = \left( \partial_{tt}u - \Delta u \right)|_{\gl}.
\end{equation}

Let $(u, \mu) \in X \times M$ be the solution of (<ref>). Choosing $\psi|_{\gl}=\jump{\dn u}_{\gl}$, from the second equation of (<ref>) we obtain $u \in H^2_0(\Omega)$. Now, for all $v \in H^2_0(\Omega)$, the first equation of (<ref>) becomes

\begin{align*}
\int_{\Omega} \nabla (\nabla u ) : \nabla (\nabla v ) \,dx= \int_{\Omega} fv \, dx,
\end{align*}

which coincides with (<ref>).

Let us assume the solution $u$ being also in $H^3(\Omega)$. For all $v \in X$, it holds

\begin{align*}
\int_{\Omega}fv \, dx&= \int_{\Omega} \Delta^2 uv \, dx= \sum_{k \in \io} \int_{\Omega_k} \Delta u \Delta v \, dx + \sum_{\ell \in \ig} \int_{\gamma_{\ell}}
%\left( \cancel{\partial_n \Delta u \jump{v}}
- \Delta u \jump{\partial_n v}_{\gl} \, d\sigma
\\
&=\sum_{k \in \io} \int_{\Omega_k}  \, \nabla (\nabla u ) : \nabla (\nabla v ) \, dx + \sum_{\ell \in \ig} \int_{\gamma_{\ell}} \left(
%\cancel{\partial_n \Delta u \jump{v}}
- \Delta u \jump{\partial_n v}_{\gl} + \partial_{tt}u \jump{\partial_n v}_{\gl} - \partial_{nt}u \jump{\partial_t v}_{\gl} \right) \, d\sigma\\
&=\sum_{k \in \io} \int_{\Omega_k} \nabla (\nabla u ) : \nabla (\nabla v ) \, dx+ \sum_{\ell \in \ig} \int_{\gamma_{\ell}} \left(
%\cancel{\partial_n \varDelta u \jump{v}}
- \Delta u \jump{\partial_n v}_{\gl} + \partial_{tt}u \jump{\partial_n v}_{\gl} %+ \cancel{\partial_{ntt}u \jump{v}}
\right) \, d\sigma\\
\end{align*}

\begin{align*}
\mu|_{\Gamma_\ell} := \left( \partial_{tt}u - \Delta u  \right)|_{\gl}.
\end{align*}

The statement follows from the uniqueness of the solution of (<ref>).

§ THE ISOGEOMETRIC MORTAR FORMULATION

Let us define the isogeometric space on each $\Omega_k$ by

\begin{equation*}
\Sp^{p}_{r}(\Omega_k):=\Set{ v_k = \widehat{v}_k \circ F_{k}^{-1} | \widehat{v}_k \in \H{\Sp}^{p}_{r}(\Xi_k) },
\end{equation*}

recalling that $\Sp^{p}_{r}(\Xi_k)$ is the spline space of degree $p \geq 2$ and regularity $r \geq 1$ on the parametric domain $\widehat{\Omega}$ and $\bm{F}_k$ is the spline parametrization from $\widehat{\Omega}$ to $\Omega_k$, introduced in Assumption <ref>. Let us now introduce the local primal space

\begin{equation}\label{eq:defXkh}
X_{k,h} := \Sp^{p}_{r}(\Omega_k) \cap H_*^2(\Omega_k).
\end{equation}

On $\Omega$ we define the space of isogeometric functions with $C^0$-continuity constraints along the skeleton $\Sigma$ and $C^2$-continuity at the vertices of the decomposition

\begin{equation}\label{eq:spaceXh}
X_h := \left\{v \in C^0(\overline{\Omega}) : \,   v|_{\Omega_{k}} \in X_{k,h},\, \forall k \in \io \text{ and } v \in C^2(\vv_i), \, \forall i \in \iv \right\}.
\end{equation}

We have $X_h \subset X$.

Since $\jump{\dn v}_{\gl} \in C^0(\gl)$ and $v \in C^2(\vv_i), \, \forall i \in \iv$, $\jump{\dn v}_{\gl}$ is zero at the endpoints of $\gl$, it holds $\jump{\dn v}_{\gl} \in H^{1/2}_{00}(\gl)$, which yields the thesis.

Furthermore we denote the mortar kernel as

\begin{equation}\label{eq:mortarkernel}
V_h := \Set{v_h \in X_h | b(v_h, \mu_h) = 0, \quad \forall \mu_h \in M_h}.
\end{equation}

$V_h$ is the isogeometric space with weak $C^1$-continuity that depends on $M_h := \prod_{\ell \in \ig} M_\ell$.
$M_h$ is a finite dimensional subspace of $M$, that will be defined later.
Now we are able to define the weak formulation of the model problem as a saddle point problem with $C^1$-mortar constraints along $\Sigma$:

Find $(u_h, \tau_h) \in X_h \times M_h$ such that

\begin{equation}\label{eq:qh}
\begin{cases}
\begin{aligned}
a(u_h, v_h) + b(v_h, \tau_h)	 & = f(v_h) & \forall   v_h \in X_h \\
b(u_h, \mu_h)	                 & = 0      & \forall \mu_h \in M_h.
\end{aligned}
\end{cases}
\end{equation}

The choice of the discrete Lagrange multipliers space $M_h$ plays an essential role for the stability and the optimality of the discretization.

§.§ Approximation properties of $X_h$

Theoretical aspectsApproximation properties of $X_h$

We consider the operator defined in <cit.> $\widetilde{\Pi}_h: H^1(0,1) \rightarrow \Sp^p$, such that

\begin{equation}\label{eq:cappi}
\widetilde{\Pi}_h(\widehat{v})(\widehat{x}) = \sum_{j=1}^{n} \widetilde{\lambda}_j^p(\widehat{v}) B^p_j(\widehat{x}),
\end{equation}

where $\widehat{v}$ is a continuous function defined on the parametric interval $(0,1)$, $\widetilde{\lambda}_j^p$ is a dual basis with $\widetilde{\lambda}_1^p(\widehat{v}) = \widehat{v}(0)$, $\widetilde{\lambda}_n^p(\widehat{v}) = \widehat{v}(1)$ and $\widetilde{\lambda}_j^p(\widehat{v})$ for $2 \leq j \leq n-1$ is defined, e.g., as in <cit.>. By tensorization we can generalize (<ref>) to the bivariate setting, obtaining $\widetilde{\Pi}_h: H^1(\widehat{\Omega}) \rightarrow \Sp^p \times \Sp^p$

\begin{equation}\label{eq:bivcappi}
\widetilde{\Pi}_h(\widehat{v})(\widehat{x}, \widehat{y}) = \sum_{i,j=1}^{n} \widetilde{\lambda}_{i,j}^p(\widehat{v}) B^p_{(i,j)}(\widehat{x}, \widehat{y}),
\end{equation}

where $\widetilde{\lambda}_{i,j}^p = \widetilde{\lambda}_{i}^p \otimes \widetilde{\lambda}_{j}^p$. See Figure <ref> for a visualization of the dual basis.

We recall that, from <cit.> the following bounds hold:

\begin{align}\label{eq:lambdaprop}
\abs{\widetilde{\lambda}_{i,j}^p (\widehat{v})} &\leq Ch^{-1} \norm{\widehat{v}}_{L^2\left( \supp{B^p_{(i,j)}} \right) }, & 2 \leq i,j \leq n-1, \notag \\
\abs{\widetilde{\lambda}_{i,1}^p (\widehat{v})} &\leq Ch^{-\nicefrac{1}{2}} \norm{\widehat{v}}_{L^2\left( (\supp{B^p_{i}})\times\{0\} \right) }, & 2 \leq i \leq n-1, \notag \\
\abs{\widetilde{\lambda}_{i,n}^p (\widehat{v})} &\leq Ch^{-\nicefrac{1}{2}} \norm{\widehat{v}}_{L^2\left( (\supp{B^p_{i}})\times\{1\} \right) }, & 2 \leq i \leq n-1, \\
\abs{\widetilde{\lambda}_{1,j}^p (\widehat{v})} &\leq Ch^{-\nicefrac{1}{2}} \norm{\widehat{v}}_{L^2\left( \{0\}\times(\supp{B^p_{j}}) \right) }, & 2 \leq j \leq n-1, \notag \\
\abs{\widetilde{\lambda}_{n,j}^p (\widehat{v})} &\leq Ch^{-\nicefrac{1}{2}} \norm{\widehat{v}}_{L^2\left( \{1\}\times(\supp{B^p_{j}}) \right) }, & 2 \leq j \leq n-1. \notag
\end{align}

Furthermore $\widetilde{\Pi}_h(\widehat{v})$ on each edge of $\partial \widehat{\Omega}$ only depends on $\widehat{v}$ on the given edge.

As $\widehat{\Omega} \subset \R^2$ is a bounded connected domain with a Lipschitz boundary, if $s > 3$ the following Sobolev compact embedding holds, see <cit.>,

\begin{equation*}
H^s(\widehat{\Omega}) \subset C^2(\overline{\widehat{\Omega}}).
\end{equation*}

Given an integer $s \geq 0$, the bent Sobolev space of order $s$, $\hh^s(\widehat{\Omega})$, is the space of functions that are in $H^s(Q)$ for all elements $Q$ of the mesh and have matching traces, up to $\min\{r, s-1 \}$ order derivatives, at the interelement interfaces. The seminorms and norm of $\hh^s$ are defined as

\begin{equation*}
\begin{split}
\abs{v}^2_{\hh^i(\widehat{\Omega})}  &:= \sum_{Q  \in \Q_h} \abs{v}^2_{H^i(Q)}, \quad 0 \leq i \leq s,\\
\norm{v}^2_{\hh^s(\widehat{\Omega})} &:= \sum_{i = 0}^{s} \abs{v}^2_{\hh^i(\widehat{\Omega})}.
\end{split}
\end{equation*}

Bent Sobolev spaces on subdomains of $\widehat{\Omega}$, on $\Omega$, and its subspaces, are constructed in a similar way, see <cit.> for more details.

Let $m,s$ and $t$ be integers such that $s \geq 4$ and $0 \leq t \leq m \leq \min\{ p+1, s\}$.
Then, there exists an operator $\widehat{\Pi}_{k,h} \colon H^s(\widehat{\Omega}) \to \Sp_{r}^{p}$ such that

\begin{equation*}\label{eq:defproj}
\widehat{\Pi}_{k,h} \widehat{v} = \widehat{v}, \quad \forall \, \widehat{v} \in \Sp_{r}^{p}, \\
\end{equation*}

for all $\iota \in \mathcal{I}_{\vv,k}$,

\begin{equation}\label{eq:projc2}
\begin{split}
(\widehat{\Pi}_{k,h} \widehat{v})(\widehat{\vv}_\iota) 		&=       \widehat{v}(\widehat{\vv}_\iota) \\
\nabla (\widehat{\Pi}_{k,h} \widehat{v})(\widehat{\vv}_\iota)	&=\nabla \widehat{v}(\widehat{\vv}_\iota)\\
\hess (\widehat{\Pi}_{k,h} \widehat{v})(\widehat{\vv}_\iota)	&= \hess \widehat{v}(\widehat{\vv}_\iota),
\end{split}
\end{equation}

where $\hess (\cdot)(\cdot)$ is the Hessian matrix, and, for all $\widehat{v} \in H^s(\widehat{\Omega})$

\begin{equation}\label{eq:appproj}
\norm*{\widehat{v} - \widehat{\Pi}_{k,h} \widehat{v}}_{\hh^t(\widehat{\Omega})} \leq C \, h^{m-t} \norm*{\widehat{v}}_{H^r(\widehat{\Omega})},
\end{equation}

and, for each edge $\hgamma \subset \partial \widehat{\Omega}$, we have

\begin{equation}\label{eq:appprojbis}
\norm{\nabla(\widehat{v} - \widehat{\Pi}_{k,h} \widehat{v})}_{L^2(\hgamma)} + h\, \abs{\nabla(\widehat{v} - \widehat{\Pi}_{k,h} \widehat{v})}_{H^1(\hgamma)} \leq C\,h^{m - \nicefrac{3}{2}} \norm{\widehat{v}}_{H^r(\widehat{\Omega})},
\end{equation}

with $r=\max \Set{4,m}$.

We modify the operator $\widetilde{\Pi}_h$ defined in (<ref>) in order to fulfill (<ref>).
It is convenient to distinguish the degrees of freedom that are involved in the $C^2$ interpolation at the corners from the others on the boundary and the remaining ones in the interior of the patch. Hence we define

\begin{equation*}
\iC := \Set{\textbf{i}=(i,j) \in \mathcal{I} \ | \ \exists \iota \in \mathcal{I}_{\vv,k}: \ \widehat{\vv}_\iota \in \supp(B_{(i,j)})}.
\end{equation*}

For all $(i,j) \in \mathcal{I} \setminus \iC$, $\lambda_{i,j}$ can be defined as

\begin{equation*}
\lambda_{i,j} := \widetilde{\lambda}_{i,j}.
\end{equation*}

Otherwise, for all $(i,j) \in \iC$, the functionals $\lambda_{i,j}$ have to be defined to fit the following constraints

\begin{align}
\widehat{\Pi}_{k,h} \widehat{v}(\widehat{\vv}_\iota)			  &= \widehat{v}(\widehat{\vv}_\iota),			 & \partial_{\widehat{x}\widehat{x}} \widehat{\Pi}_{k,h} \widehat{v}(\widehat{\vv}_\iota) &= \partial_{\widehat{x}\widehat{x}} \widehat{v}(\widehat{\vv}_\iota) \notag\\
\partial_{\widehat{x}} \widehat{\Pi}_{k,h} \widehat{v}(\widehat{\vv}_\iota) &= \partial_{\widehat{x}} \widehat{v}(\widehat{\vv}_\iota)	 & \partial_{\widehat{x}\widehat{y}} \widehat{\Pi}_{k,h} \widehat{v}(\widehat{\vv}_\iota) &= \partial_{\widehat{x}\widehat{y}} \widehat{v}(\widehat{\vv}_\iota) \label{eq:const}\\
\partial_{\widehat{y}} \widehat{\Pi}_{k,h} \widehat{v}(\widehat{\vv}_\iota) &= \partial_{\widehat{y}} \widehat{v}(\widehat{\vv}_\iota)	 & \partial_{\widehat{y}\widehat{y}} \widehat{\Pi}_{k,h} \widehat{v}(\widehat{\vv}_\iota) &= \partial_{\widehat{y}\widehat{y}} \widehat{v}(\widehat{\vv}_\iota), \notag
\end{align}

for all $\iota \in \mathcal{I}_{\vv,k}$.
It is easy to see that (<ref>) is a well posed interpolation problem that determines $\lambda_{i,j}$ for all $(i,j) \in \iC$. In particular,
for $\widehat{\vv}_\iota = (0,0)$ and $\bigl\{ \lambda_{i,j} \bigr\}_{i+j \leq 4}$, we obtain

\begin{equation*}
\begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
-1 & 1 & 0 & 0 & 0 & 0 \\
-1 & 0 & 1 & 0 & 0 & 0 \\
\frac{h_{1,1}+h_{1,2}}{h_{1,1}} & -\frac{2h_{1,1}+h_{1,2}}{h_{1,1}} & 0 & 1 & 0 & 0 \\
1 & -1 & -1 & 0 & 1 & 0 \\
\frac{h_{2,1}+h_{2,2}}{h_{2,1}} & 0 & -\frac{2h_{2,1}+h_{2,2}}{h_{2,1}} & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
\lambda_{1,1}\\
\lambda_{2,1}\\
\lambda_{1,2}\\
\lambda_{3,1}\\
\lambda_{2,2}\\
\lambda_{1,3}
\end{bmatrix}
\begin{bmatrix}
\widehat{v}(0,0)\\
\frac{h_{1,1}}{p_1} \, \partial_{\widehat{x}} \widehat{v}(0,0)\\
\frac{h_{2,1}}{p_2} \, \partial_{\widehat{y}} \widehat{v}(0,0)\\
\frac{h_{1,1}(h_{1,1}+h_{1,2})}{p_1(p_1-1)} \, \partial_{\widehat{x}\widehat{x}} \widehat{v}(0,0)\\
\frac{h_{1,1} h_{2,1}}{p_1 p_2} \, \partial_{\widehat{x}\widehat{y}} \widehat{v}(0,0)\\
\frac{h_{2,1}(h_{2,1}+h_{2,2})}{p_2(p_2-1)} \, \partial_{\widehat{y}\widehat{y}} \widehat{v}(0,0)
\end{bmatrix},
\end{equation*}

where the coefficient matrix is non-singular, since it is lower-triangular. Assuming shape-regularity and thanks to Assumption <ref>, if $\widehat{v} \in \hh^4(\widehat{\Omega})$, we obtain

\begin{equation}\label{eq:1}
\sum_{i+j \leq 4} \abs{\lambda_{i,j}\widehat{v}} \leq C \left( \abs{\widehat{v}(0,0)} + h \norm{\nabla \widehat{v}(0,0)} +  h^2 \norm{H \widehat{v}(0,0)}\right),
\end{equation}

where $\norm{\cdot}$ denotes the vector or matrix norm and $C$ is independent of $h$.

A similar bound holds for all $| \lambda_{i,j} |$ with $(i,j) \in \iC$. Assume now that $Q \in \Q_h$ satisfies  $Q \subseteq \cup_{i + j \leq 4} \supp(B^p_{(i,j)})$,
then we have

\begin{equation}\label{eq:11}
\begin{split}
\norm{\widehat{\Pi}_{k,h}(\widehat{v})}_{L^2(Q)} &\leq \sum_{(i,j):\ Q \subseteq \supp(B^p_{(i,j)})} \abs{\lambda_{i,j}\widehat{v}} \norm{B^p_{(i,j)}}_{L^2(Q)} \\
&\leq C \sum_{\substack{(i,j):\ Q \subseteq \supp(B^p_{(i,j)}) \\ i+j > 4}} h \abs{\lambda_{i,j}\widehat{v}} + C \sum_{ i+j \leq 4} h \abs{\lambda_{i,j}\widehat{v}}.
\end{split}
\end{equation}

Then, from (<ref>), (<ref>) and (<ref>), we conclude that

\begin{equation}\label{eq:12}
\begin{split}
\norm{\widehat{\Pi}_{k,h} \widehat{v}}_{L^2(Q)} &\leq C
\left( \norm{\widehat{v}}_{L^2(\widetilde{Q})} + h^{\nicefrac{1}{2}} \norm{\widehat{v}}_{L^2\left( \partial \widetilde{Q} \cap \partial \widehat{\Omega} \right) } + h \abs{\widehat{v}(0,0)} + h^2 \norm{\nabla \widehat{v}(0,0)} + h^3 \norm{H \widehat{v}(0,0)}\right) \\
&= C \opnorm{\widehat{v}}_{\widetilde{Q}},
\end{split}
\end{equation}

where $\widetilde{Q}$ is the support extension of $Q$, that is the union of the supports of basis functions whose support intersects $Q$, and $\opnorm{\cdot}_{\widetilde{Q}}$ is a short notation for the sum of norms appearing on the right-hand-side of (<ref>).

Let us consider a linear parametrization $T:[0,1]^2 \rightarrow \widetilde{Q}$ of $\widetilde{Q}$, i.e. $T([0,1]^2)=\widetilde{Q}$. Thanks to a scaling argument, it holds

\begin{equation}\label{eq:trace_scaling}
\begin{aligned}
\norm{\widehat{v}}_{L^2\left( \partial \widetilde{Q} \cap \partial \widehat{\Omega} \right)} &\leq C h^{\nicefrac{1}{2}} \norm{\widehat{v} \circ T}_{L^2(T^{-1}(\partial \widetilde{Q} \cap \partial \widehat{\Omega}))} \leq C h^{\nicefrac{1}{2}} \norm{\widehat{v} \circ T}_{\hh^1([0,1]^2)} \\
&= C h^{\nicefrac{1}{2}} \left( \norm{\widehat{v} \circ T}_{L^2([0,1]^2)} + \abs{\widehat{v} \circ T}_{H^1([0,1]^2)} \right) \leq C h^{\nicefrac{1}{2}} \left( h^{-1} \norm{\widehat{v}}_{L^2(\widetilde{Q})} +  \abs{\widehat{v}}_{\hh^1(\widetilde{Q})} \right) \\
&= C \left( h^{-\nicefrac{1}{2}} \norm{\widehat{v}}_{L^2(\widetilde{Q})} + h^{\nicefrac{1}{2}} \abs{\widehat{v}}_{\hh^1(\widetilde{Q})} \right)%, \quad \forall \, \widehat{v} \in H^1(\widetilde{Q})
\end{aligned}
\end{equation}

\begin{equation*}
\begin{aligned}
\abs{\widehat{v}(0,0)} &\leq \abs{\widehat{v}(T (0,0))} \leq C \left( \norm{\widehat{v} \circ T}_{L^2(T^{-1}(\partial \widetilde{Q} \cap \partial \widehat{\Omega}))} + \abs{\widehat{v} \circ T}_{\hh^1(T^{-1}(\partial \widetilde{Q} \cap \partial \widehat{\Omega}))} \right) \\
&\leq C \left( \norm{\widehat{v} \circ T}_{L^2(T^{-1}(\widetilde{Q}))} + \abs{\widehat{v} \circ T}_{\hh^1(T^{-1}(\widetilde{Q}))} + \abs{\widehat{v} \circ T}_{\hh^2(T^{-1}(\widetilde{Q}))} \right) \\
&\leq C \left( h^{-1}\norm{\widehat{v}}_{L^2(\widetilde{Q})} + \abs{\widehat{v}}_{\hh^1(\widetilde{Q})} + h \abs{\widehat{v}}_{\hh^2(\widetilde{Q})} \right)%, \quad \forall \, \widehat{v} \in H^2(\widetilde{Q})
\end{aligned}
\end{equation*}

Similarly, it also holds

\begin{equation*}
\abs{\nabla \widehat{v}(0,0)} \leq C \left( h^{-1}\abs{\widehat{v}}_{\hh^1(\partial \widetilde{Q})} + \abs{\widehat{v}}_{\hh^2(\partial \widetilde{Q})} + h \abs{\widehat{v}}_{\hh^3(\partial \widetilde{Q})} \right)%, \quad \forall \, \widehat{v} \in H^3(\widetilde{Q})
\end{equation*}

\begin{equation*}
\abs{H \widehat{v}(0,0)} \leq C \left( h^{-1}\abs{\widehat{v}}_{\hh^2(\partial \widetilde{Q})} + \abs{\widehat{v}}_{\hh^3(\partial \widetilde{Q})} + h \abs{\widehat{v}}_{\hh^4(\partial \widetilde{Q})} \right)%, \quad \forall \, \widehat{v} \in H^4(\widetilde{Q})
\end{equation*}

Following an argument like in <cit.>, we can prove that for any $\widehat{v} \in \hh^s (\widetilde{Q})$ with $s \geq 4$, there exists a spline $\widehat{s} \in \Sp^p_r$ such that

\begin{equation*}
\opnorm{\widehat{v} - \widehat{s}}_{\widetilde{Q}} + h^t \abs{\widehat{v} - \widehat{s}}_{\hh^t (\widetilde{Q})} \leq C\,h^s \norm{\widehat{v}}_{\hh^s (\widetilde{Q})}%, \quad \forall \, \widehat{v} \in H^4(\widetilde{Q})
\end{equation*}

Combining the previous results and a standard inverse inequality, when $Q \subseteq \cup_{i + j \leq 4} \supp(B^p_{i,j})$

\begin{equation}\label{eq:hnorm}
\begin{split}
\abs{\widehat{v} - \widehat{\Pi}_{k,h} \widehat{v}}_{\hh^t(Q)} &= \abs{\widehat{v} - \widehat{s} - \widehat{\Pi}_{k,h} (\widehat{v} - \widehat{s})}_{\hh^t(Q)} \\
&\leq \abs{\widehat{v} - \widehat{s}}_{H^t(Q)} + C\,h^{-t} \norm{\widehat{\Pi}_{k,h} (\widehat{v} - \widehat{s})}_{L^2(Q)} \\
&\leq \abs{\widehat{v} - \widehat{s}}_{\hh^t(Q)} + C\,h^{-t} \opnorm{\widehat{v} - \widehat{s}}_{\widetilde{Q}} \\
&\leq C\,h^{m-t} \norm{\widehat{v}}_{\hh^r (\widetilde{Q})}.
\end{split}
\end{equation}

We remark that the same construction can be repeated for all $\widehat{\vv}_\iota$, with $\iota \in \mathcal{I}_{\vv,k}$, and (<ref>) holds for $Q \nsubseteq \cup_{i + j \leq 4} \supp(B^p_{(i,j)})$, as proved in <cit.>. Finally (<ref>) follows by summation over all mesh elements.

In order to prove (<ref>) we proceed similarly. Given a mesh element $Q$ adjacent to the boundary $\partial \widehat{\Omega}$, let $E$ be one edge of $\partial Q$ on $\partial \widehat{\Omega}$.  Reasoning as in (<ref>) and thanks to (<ref>), if we set $\eta := \widehat{v} - \widehat{\Pi}_{k,h} \widehat{v}$, we obtain

\begin{equation*}
\begin{aligned}
\norm{\nabla \eta}_{L^2(E)} &\leq C \left( h^{-\nicefrac{1}{2}} \norm{\nabla \eta}_{L^2(Q)} + h^{\nicefrac{1}{2}} \abs{\nabla \eta}_{H^1(Q)} \right) \\
&\leq C \left( h^{-\nicefrac{1}{2}} \abs{\eta}_{H^1(Q)} + h^{\nicefrac{1}{2}} \abs{\eta}_{H^2(Q)} \right) \\
&\leq C h^{m-\nicefrac{3}{2}} \norm{\widehat{v}}_{H^r(Q)}
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\norm{\nabla \eta}_{H^1(E)} &\leq C \left( h^{-\nicefrac{1}{2}} \abs{\nabla \eta}_{H^1(Q)} + h^{\nicefrac{1}{2}} \abs{\nabla \eta}_{H^2(Q)} \right) \\
&\leq C \left( h^{-\nicefrac{1}{2}} \abs{\eta}_{H^2(Q)} + h^{\nicefrac{1}{2}} \abs{\eta}_{H^3(Q)} \right) \\
&\leq C h^{m-\nicefrac{5}{2}} \norm{\widehat{v}}_{H^r(Q)}.
\end{aligned}
\end{equation*}

Then (<ref>) follows. Note also that $\nabla \widehat{v} -  \nabla \widehat{\Pi}_{k,h} \widehat{v} \in H^1_0(\hgamma)$ for each edge $\hgamma$ of $\partial \widehat{\Omega}$. Indeed $\nabla \widehat{v}$ is interpolated at the vertices, that is $\nabla \widehat{v} -  \nabla \widehat{\Pi}_{k,h} \widehat{v}$ vanishes at the endpoints of $\hgamma$.

Let $m,s$ and $t$ be integers such that $s \geq 4$ and $0 \leq t \leq m \leq \min\{ p+1, s\}$.
Then, there exists an operator $\Pi_{k,h} \colon H^s(\Omega_k) \cap H^2_*(\Omega_k) \to X_{k,h}$ such that

\begin{equation*}
\Pi_{k,h} v = v, \quad \forall v \in X_{k,h}, \\
\end{equation*}

\begin{equation}\label{eq:c2phys}
\begin{split}
(\Pi_{k,h} v)(\vv_\iota) 		&=  v(\vv_\iota) \\
\nabla (\Pi_{k,h} v)(\vv_\iota)	&=\nabla v(\vv_\iota)\\
\hess (\Pi_{k,h} v)(\vv_\iota)	&= \hess v(\vv_\iota),
\end{split}
\end{equation}

for all $\iota \in \mathcal{I}_{\vv,k}$, and

\begin{equation}\label{eq:physapp}
\norm*{v - \Pi_{k,h} v}_{\hh^t(\Omega_k)} \leq C \, h^{m-t} \norm*{v}_{\hh^r(\Omega_k)}.
\end{equation}

Moreover, for each edge $\Gamma$ of $\partial \Omega$, we have

\begin{equation}\label{eq:physappbis}
\norm{\nabla(v - \Pi_{k,h} v)}_{L^2(\Gamma)} + h\, \abs{\nabla(v - \Pi_{k,h} v)}_{H^1(\Gamma)} \leq C\, h^{m - \nicefrac{3}{2}} \norm{v}_{H^r(\Omega)},
\end{equation}

with $r=\max \Set{4,m}$.

We define the projector $\Pi_{k,h} \colon H^s(\Omega_k)\cap H^2_*(\Omega_k) \to X_{k,h}$ as

\begin{equation*}
\Pi_{k,h} v := (\widehat{\Pi}_{k,h}(v \circ \bm{F}_k)) \circ \bm{F}_k^{-1}, \quad \forall \, v \in H^s(\Omega_k)\cap H^2_*(\Omega_k),
\end{equation*}

where $\widehat{\Pi}_{k,h}$ is defined in Proposition <ref>.
Then, (<ref>) follows from (<ref>) and the chain rule. Also (<ref>)-(<ref>) follow from the chain rule, see <cit.> for more details.

§.§ Approximation properties of $V_h$

Theoretical aspectsApproximation properties of $V_h$
The aim of this section is to prove the estimate

\begin{equation*}
\inf_{v_h \in V_h} \norm{u - v_h}_X \leq C h^{p - 1} \norm{u}_{H^{p+1}(\Omega)},
\end{equation*}

where $\norm{\cdot}_X$ is the norm defined in (<ref>). Since $V_h$ depends on the choice of $M_h$, the estimate also depends on $M_h$. Postponing specific constructions, we ask in general the following.

For every $\gl \in \ig$ and for any $\tau \in H^{p - \nicefrac{3}{2}}(\gl)$, it holds

\begin{equation*}
\inf_{\mu_h \in M_\ell} \norm{\tau - \mu_h}_{L^2(\gl)} \leq C h^{p - \nicefrac{3}{2}} \norm{\tau}_{H^{p - \nicefrac{3}{2}}(\gl)}.
\end{equation*}

Let $m,s$ and $t$ be integers such that $s \geq 4$ and $0 \leq t \leq m \leq \min\{ p+1, s\}$.
There exists a multi-patch operator $\Pi_h \colon H^s(\Omega) \cap H^2_*(\Omega) \to X_h$ such that

\begin{equation*}
\norm{v - \Pi_h v}_X \leq C\,h^{m-2}\,\norm{v}_{H^r(\Omega)},
\end{equation*}

with $r=\max \Set{4,m}$.

Let $v$ be in $H^s(\Omega) \cap H^2_*(\Omega)$. We define $\Pi_h$ as

\begin{equation*}
(\Pi_h v)|_{\Omega_k} = \Pi_{k,h} (v|_{\Omega_k}), \quad \forall k \in \io.
\end{equation*}

We only need to prove that, for all $\ell \in \ig$,

\begin{equation*}
\norm{\jump{\dn v - \dn \Pi_h v}}_{H^{\nicefrac{1}{2}}(\gl)} \leq C\,h^{m-2}\,\norm{v}_{H^r\left( \Omega_{m(\ell)} \cup \Omega_{s(\ell)}\right) }.
\end{equation*}

From (<ref>), it holds

\begin{align*}
\norm{(\nabla v - \nabla \Pi_h v)|_{\Omega_{s(\ell)}}}_{L^2(\gl)} &\leq C\,h^{m-\nicefrac{3}{2}}  \norm{v}_{H^s\left( \Omega_{s(\ell)}\right) }, \\
\norm{(\nabla v - \nabla \Pi_h v)|_{\Omega_{s(\ell)}}}_{H^1(\gl)} &\leq C\,h^{m-\nicefrac{5}{2}} \norm{v}_{H^r\left( \Omega_{s(\ell)}\right)}.
\end{align*}

Interpolating between $L^2(\gl)$ and $H^1(\gl)$, we get

\begin{equation*}
\norm{(\nabla v - \nabla \Pi_h v)|_{\Omega_{s(\ell)}}}_{H^{\nicefrac{1}{2}}(\gl)} \leq C\,h^{m-2} \abs{v}_{H^r\left( \Omega_{s(\ell)}\right) }.
\end{equation*}

Similarly, we also obtain

\begin{equation*}
\norm{(\nabla v - \nabla \Pi_h v)|_{\Omega_{m(\ell)}}}_{H^{\nicefrac{1}{2}}(\gl)} \leq C\,h^{m-2} \abs{v}_{H^r\left( \Omega_{m(\ell)}\right) }.
\end{equation*}

Finally, it holds

\begin{align*}
\norm{\jump{\dn v - \dn \Pi_h v}}_{H^{\nicefrac{1}{2}}(\gl)} &\leq \norm{(\nabla v - \nabla \Pi_h v)|_{\Omega_{s(\ell)}}}_{H^{\nicefrac{1}{2}}(\gl)} + \norm{(\nabla v - \nabla \Pi_h v)|_{\Omega_{m(\ell)}}}_{H^{\nicefrac{1}{2}}(\gl)}\\
& \leq  C\,h^{m-2}\,\norm{v}_{H^r\left( \Omega_{m(\ell)} \cup \Omega_{s(\ell)}\right) }.
\end{align*}

We introduce the mortar projection $\mortar_h \colon L^2(\Sigma) \to W_h$. Here the discrete trace derivative space $W_h$ is defined as $W_h := \prod_{\ell \in \ig} W_{\ell}$, where

\begin{equation}\label{eq:defWl}
W_{\ell} := \Set{w \in L^2(\gl)|
\begin{aligned}
w &= \dn v_h|_{\gl}, \\
v_h &\in X_{s(\ell),h} \cap H^1_0\left( \Omega_{s(\ell)}\right), \, { \dn v_h|_{\partial \Omega_{s(\ell)} \setminus \mathring{\Gamma}_{\ell}}} = 0
\end{aligned}}.
\end{equation}

Functions in $W_{\ell}$ are zero with zero derivative at the endpoints of the interface $\gl$.

Observe that the degrees-of-freedom (control points) associated to the function $v_h$ appearing in (<ref>) that are relevant for $W_\ell$ are not involved in $W_{\ell'}$, $\ell' \neq \ell$. In Figure <ref> the degrees of freedom associated to the $C^2$-constraints on the vertices are marked in red, the ones associated with the $C^0$-constraints along the interfaces are marked in blue, while the degrees of freedom of $v_h$ such that $\partial_n v_h$ generates $W_\ell$ are depicted in green.

For each $w \in H_{00}^{\nicefrac{1}{2}}(\gl)$, it holds

\begin{equation}\label{eq:orderW-H12}
\inf_{w_h \in W_{\ell}} \norm{w - w_h}_{L^2(\gl)} \leq C \, h^{\nicefrac{1}{2}} \, \norm{w}_{H^{\nicefrac{1}{2}}(\gl)}.
\end{equation}

We observe that, given $w \in H_{00}^{\nicefrac{1}{2}}(\gl)$, there exists $v \in H^2(\Omega_{s(\ell)}) \cap H^1_0(\Omega_{s(\ell)})$ such that $\dn v|_{\gl}=w$, $\dn v_h|_{\partial \Omega_{s(\ell)} \setminus \mathring{\Gamma}_{\ell}} = 0$ and $\norm{v}_{H^2(\Omega_{s(\ell)})} \leq \norm{w}_{H^{\nicefrac{1}{2}}(\gl)}$.

Let us introduce the following space with boundary conditions

\begin{equation*}
X^{00}_{\ell}:=\Set{v \in H^2(\Omega_{s(\ell)})\cap H^1_0(\Omega_{s(\ell)}) \ | \ \partial_{n}v|_{\partial \Omega_{s(\ell)} \setminus \mathring{\Gamma}_{\ell}} = 0}
\end{equation*}

and its discrete counterpart

\begin{equation*}
X^{00}_{\ell,h}:=\Set{v_h \in X_{s(\ell),h} \cap H^1_0(\Omega_{s(\ell)}) \ | \ \partial_{n}v_h|_{\partial \Omega_{s(\ell)} \setminus \mathring{\Gamma}_{\ell}} = 0}.
\end{equation*}

Reasoning as in <cit.>, we can define a projector $\Pi^{00}_{s(\ell),h}: X^{00}_{\ell} \rightarrow X^{00}_{\ell,h}$ such that, thanks to an argument similar to the proof of <ref>, it satisfies

\begin{equation*}\label{eq:est_trace1}
\norm{\nabla(v - \Pi^{00}_{s(\ell),h} v)}_{L^2(\gl)} %+ h \abs{\nabla(v - \Pi^{00}_{s(\ell),h} v)}_{H^1(\gl)}
\leq C\, h^{\nicefrac{1}{2}} \norm{v}_{H^2(\Omega_{s(\ell)})}.
\end{equation*}

Thus, we can define a projector $P_{h,\ell}:H_{00}^{\nicefrac{1}{2}}(\gl) \rightarrow W_{\ell}$ as $P_{h,\ell}(w):= (\dn \Pi^{00}_{s(\ell),h} v)|_{\gl}$.

Finally, it holds

\begin{align*}
\norm{w - P_{h,\ell}w}_{L^2(\gl)} &= \norm{\dn v - \dn \Pi_{s(\ell),h} v}_{L^2(\gl)} \leq \norm{\nabla (v - \Pi^{00}_{s(\ell),h} v)}_{L^2(\gl)} \leq Ch^{\nicefrac{1}{2}} \abs{v}_{H^2(\Omega_{s(\ell)})} \\
&\leq Ch^{\nicefrac{1}{2}} \norm{w}_{H^{\nicefrac{1}{2}}(\gl)}.
\end{align*}

We define the mortar projection $\mortar_h$ by

\begin{equation*}
(\mortar_h v)|_{\gl} := \mortar_{\ell}(v|_{\gl}), \quad \forall v \in L^2(\Sigma),
\end{equation*}

where, for every $\ell \in \ig$, the local projection $\mortar_\ell \colon L^2(\gl) \to W_\ell$ is defined as

\begin{equation*}\label{eq:mortarproj}
\int_{\gl} \mortar_\ell v \, \mu \, d\sigma = \int_{\gl} v \, \mu \, d\sigma, \quad \forall \mu \in M_{\ell}.
\end{equation*}

In order to guarantee the well-posedness of the mortar projection the following assumption is introduced.

For every $\ell \in \ig$ the discrete trace space $W_\ell$ and the discrete multiplier space $M_\ell$ have the same dimension, i.e., $\dim(W_\ell) = \dim(M_\ell)$, and there exists a constant $\beta_\ell > 0$ independent of $h$ such that

\begin{equation*}
\sup_{w_h \in W_\ell} \dfrac{\int_{\gl} w_h \, \mu_h d\sigma}{\norm{w_h}_{L^2(\gl)}} \geq \beta_\ell \norm{\mu_h}_{L^2(\gl)}, \quad \forall \mu_h \in M_\ell.
\end{equation*}

Given Assumption <ref>, it follows from <cit.> that, for every $\ell \in \ig$, it holds

\begin{equation*}
\sup_{\mu_h \in M_\ell} \dfrac{\int_{\gl} w_h \, \mu_h d\sigma}{\norm{\mu_h}_{L^2(\gl)}} \geq \beta_\ell \norm{w_h}_{L^2(\gl)}, \quad \forall w_h \in W_\ell .
\end{equation*}

Given Assumption <ref>, for every $\ell \in \ig$ the mortar projection $\mortar_\ell \colon L^2(\gl) \to W_\ell$ is well defined and $L^2$-stable.

From Assumptions <ref> and from <cit.> the matrix $B$ associated with the scalar product $(w,\mu)_{L^2}$ on $W_{\ell} \times M_{\ell}$ is square and invertible, so $\mortar_\ell$ is weel defined. Moreover, from <cit.> it follows that $\mortar_{\ell}$ is $L^2$-stable.

Given Assumption <ref>, for every $\ell \in \ig$ and for every $v \in H^{\nicefrac{1}{2}}_{00}(\gl)$, there exists a constant $C > 0$ such that

\begin{equation*}
\norm{v - \mortar_\ell \, v}_{L^2(\gl)} \leq C \,h^{\nicefrac{1}{2}}\norm{v}_{H^{\nicefrac{1}{2}}(\gl)}.
\end{equation*}

Furthermore, the mortar projection $\mortar_\ell$ is $H^{\nicefrac{1}{2}}_{00}$-stable.

From Corollary <ref>, the mortar projection $\mortar_\ell$ is well-defined and $L^2$-stable. Let $w \in H^1_{00}(\gl)$ and let $P_{h,\ell}$ be the spline approximation operator as in the proof of Proposition <ref>. It holds

\begin{equation}\label{eq:mortar_est}
\begin{split}
\norm{w - \mortar_\ell \, w}_{L^2(\gl)} &\leq \norm{w - P_{h,\ell} w - \mortar_\ell (w - P_{h,\ell} w)}_{L^2(\gl)} \\
&\leq C \norm{w - P_{h,\ell} w}_{L^2(\gl)} \leq C \, h^{\nicefrac{1}{2}} \norm{w}_{H^{\nicefrac{1}{2}}(\gl)}.
\end{split}
\end{equation}

For what concern the $H^{\nicefrac{1}{2}}_{00}$-stability, using the inverse inequality for splines and (<ref>), we have

\begin{align*}
\norm{\mortar_\ell \, w}_{H^{\nicefrac{1}{2}}(\gl)} &\leq \norm{w}_{H^{\nicefrac{1}{2}}(\gl)} + \norm{w - \mortar_\ell \, w}_{H^{\nicefrac{1}{2}}(\gl)} \leq \norm{w}_{H^{\nicefrac{1}{2}}(\gl)} + C h^{-\nicefrac{1}{2}}\norm{w - \mortar_\ell \, w}_{L^2(\gl)} \\
&\leq C \norm{w}_{H^{\nicefrac{1}{2}}(\gl)}.
\end{align*}

In addition to the local projector $\mortar_\ell$, for every $\ell \in \ig$, we introduce the dual operator $\mortar_{\ell}^{*} \colon L^2(\gl) \to M_\ell$ by

\begin{equation}\label{eq:mortarprojdual}
\int_{\gl} \mortar_{\ell}^{*} \mu \, v \, d\sigma = \int_{\gl} \mu \, v \, d\sigma, \quad \forall v \in W_{\ell}.
\end{equation}

Given Assumptions <ref>, <ref>, for every $\ell \in \ig$, the dual operator $\mortar_{\ell}^{*} \colon L^2(\gl) \to M_\ell$ is well-defined and $L^2$-stable. Furthermore, there exists a constant $C > 0$ such that, for all $\mu \in H^{p - \nicefrac{3}{2}}(\gl)$, it holds

\begin{equation}
\norm{\mu - \mortar_{\ell}^{*} \, \mu}_{L^2(\gl)} \leq C\,h^{p - \nicefrac{3}{2}}\, \norm{\mu}_{H^{p - \nicefrac{3}{2}}(\gl)}.
\end{equation}

As in the proof of Corollary <ref>, Assumption <ref> guarantees that $\mortar_{\ell}^{*}$ is well-defined. Combining Assumption <ref> with (<ref>), we get

\begin{equation*}
\begin{split}
\norm{\mortar_{\ell}^{*}\,\mu}_{L^2(\gl)} &\leq \frac{1}{\beta_\ell} \sup_{w_h \in W_{\ell}}\frac{\int_{\gl}\mortar_{\ell}^{*}\,\mu \, w_h\,d\sigma}{\norm{w_h}_{L^2(\gl)}} \\
&\leq \frac{1}{\beta_\ell} \sup_{w_h \in W_{\ell}}\frac{\int_{\gl}\mu \, w_h\,d\sigma}{\norm{w_h}_{L^2(\gl)}} \\
&\leq \frac{1}{\beta_\ell} \sup_{w_h \in W_{\ell}} \frac{\norm{\mu}_{L^2(\gl)} \norm{w_h}_{L^2(\gl)}}{\norm{w_h}_{L^2(\gl)}} = \frac{1}{\beta_\ell} \norm{\mu}_{L^2(\gl)}.
\end{split}
\end{equation*}

The approximation property of $\mortar_{\ell}^{*}$ follows from Assumption <ref>. Indeed, for  $\mu \in H^{p - \nicefrac{3}{2}}(\gl)$, let $\mu_h \in M_\ell$ be such that

\begin{equation*}
\norm{\mu - \mu_h}_{L^2(\gl)} \leq C h^{p - \nicefrac{3}{2}} \norm{\mu}_{H^{p - \nicefrac{3}{2}}(\gl)},
\end{equation*}

\begin{equation*}
\begin{split}
\norm{\mu - \mortar_{\ell}^{*} \mu}_{L^2(\gl)} &= \norm{\mu - \mu_h - \mortar_{\ell}^{*}(\mu - \mu_h)}_{L^2(\gl)} \\
&\leq C' \norm{\mu - \mu_h}_{L^2(\gl)} \leq C\,h^{p - \nicefrac{3}{2}}\, \norm{\mu}_{H^{p - \nicefrac{3}{2}}(\gl)}.
\end{split}
\end{equation*}

We can now address the study of the properties of $V_h$.

There exists a constant $C > 0$, independent of $h$ and $|\io|$, such that, for every $v_h \in V_h$, it holds

\begin{equation*}
\sum_{k \in \io} \norm{v_h}^2_{H^2(\Omega_k)} \leq C\sum_{k \in \io} \abs{v_h}^2_{H^2(\Omega_k)}.
\end{equation*}

The proof follows the lines of <cit.>. We first demonstrate that for all $v_h \in V_h$ it holds

\begin{equation}\label{eq:est1}
\sum_{k \in \io} \norm{\partial_x v_h}^2_{L^2(\Omega_k)} \leq C\sum_{k \in \io} \abs{v_h}^2_{H^2(\Omega_k)}.
\end{equation}

For any $(x,y) \in \Omega$, let $(a,y),(b,y) \in \partial \Omega$ be the endpoints of a segment that contains $(x,y)$ and let us define

\begin{equation*}
I\jump{a,x}:= \Set{t \in \jump{a,x} | (t, y) \in \Gamma},
\end{equation*}

\begin{equation}\label{eq:est2}
\partial_x v_h(x,y) = \int_a^x \partial_{xx} v_h(t,y) \, dt + \sum_{t_{\ell} \in I\jump{a,x}}\jump{\partial_x v_h}_{\gl}(t_{\ell},y).
\end{equation}

The first term in the right-hand side of (<ref>) is bounded by

\begin{equation*}
\abs*{\int_a^x \partial_{xx} v_h(t,y) \, dt} \leq \int_a^b \abs*{\partial_{xx} v_h(t,y)}\, dt
\leq \left( b-a\right)^{\nicefrac{1}{2}} \left( \int_a^b \abs*{\partial_{xx} v_h(t,y)}^2dt\right)^{\nicefrac{1}{2}},
\end{equation*}

while the second term is bounded by

\begin{equation*}
\begin{split}
\abs*{\sum_{t_{\ell} \in I\jump{a,x}}\jump{\partial_x v_h}_{\gl}(t_{\ell},y)} &\leq \sum_{t_{\ell} \in I\jump{a,b}}\abs*{\jump{\partial_x v_h}_{\gl}(t_{\ell},y)}\\
&\leq \left( \sum_{t_{\ell} \in I\jump{a,x}} \abs*{\gl} \right)^{\nicefrac{1}{2}} \left( \sum_{t_\ell \in I\jump{a,b}} \abs*{\gl}^{-1}\abs*{\jump{\partial_x v_h}(t_\ell,y)}^2 \right)^{\nicefrac{1}{2}}\\
&\leq C \abs*{b-a}^{\nicefrac{1}{2}} \left( \sum_{t_\ell \in I\jump{a,b}} \abs*{\gl}^{-1}\abs*{\jump{\partial_x v_h}(t_\ell,y)}^2 \right)^{\nicefrac{1}{2}}.
\end{split}
\end{equation*}

Hence we obtain

\begin{equation}\label{eq:aux01}
\abs*{\partial_x v_h(x,y)}^2 \leq C \int_a^b \abs*{\partial_{xx} v_h(t,y)}^2 \, dt + C \sum_{t_\ell \in I\jump{a,b}} \abs*{\gl}^{-1}\abs*{\jump{\partial_x v_h}_{\gl}(t_\ell,y)}^2.
\end{equation}

Assume that $\Omega \subset \left[ -M,  M \right] ^2$ and that $\abs*{\partial_{xx} v_h(t,y)}^2$ is extended to zero in $\left[ -M,  M \right] ^2 \setminus \Omega$. Integrating over $\Omega$ the first term of the sum in the right-hand side of (<ref>) we get

\begin{equation*}
\begin{split}
\int_\Omega \left( \int_a^b \abs*{\partial_{xx} v_h(t,y)}^2 \,dt \right) \, dx \, dy &\leq \int_{-M}^{M} \int_{-M}^{M} \int_{-M}^{M} \abs*{\partial_{xx} v_h(t,y)}^2 \, dt \, dx \, dy \\
&= 2M \int_{-M}^{M} \int_{-M}^{M} \abs*{\partial_{xx} v_h(t,y)}^2 \, dt \, dy \\
&= 2M \int_{\Omega} \abs*{\partial_{xx} v_h(t,y)}^2 \, dt \, dy.
\end{split}
\end{equation*}

In a similar way, integrating over $\Omega$ the last term of (<ref>), we obtain

\begin{equation*}
\int_\Omega \sum_{t_\ell \in I\jump{a,b}} \abs*{\gl}^{-1}\abs*{\jump{\partial_x v_h}_{\gl}(t_\ell,y)}^2 \, dx \, dy \leq C \sum_{\ell \in \ig} \abs*{\gl}^{-1} \int_{\gl} \jump{\partial_n v_h}_{\gl}^2 + \jump{\partial_s v_h}_{\gl}^2 \, ds,
\end{equation*}

where $\partial_n$ and $\partial_s$ indicate the normal and tangential derivative, respectively. Hence it holds

\begin{equation*}
\int_{\Omega} \abs*{\partial_x v_h(x,y)}^2 \, dx \, dy \leq C \int_{\Omega} \abs*{\partial_{xx} v_h}^2 \, dx \, dy + C \sum_{\ell \in \ig} \abs*{\gl}^{-1} \int_{\gl} \jump{\partial_n v_h}_{\gl}^2 + \jump{\partial_s v_h}_{\gl}^2 \, ds.
\end{equation*}

We focus on the estimate of the last integral. By the mortar condition (<ref>) the average value of $\jump{\partial_n v_h}_{\gl}$ over $\gl$ is zero. Thus the standard trace theorem and Poincaré inequality yield

\begin{equation*}
\int_{\gl} \jump{\partial_n v_h}_{\gl}^2 ds \leq C_{m(\ell)} \abs*{v_h}^2_{H^2(\Omega_{m(\ell)})} + C_{s(\ell)} \abs*{v_h}^2_{H^2(\Omega_{s(\ell)})}.
\end{equation*}

For what concerns the term with tangential derivative, from the $C^0$-continuity of the functions of $V_h$, we get $\int_{\gl} \jump{\partial_s v_h}_{\gl}^2 ds = 0$, for all $\ell \in \ig$.
Summing over all the interfaces we obtain (<ref>).
With similar computations, we can find a constant $C>0$ such that for every $v_h \in V_h$, it holds

\begin{equation}\label{eq:est_h1}
\sum_{k \in \io} \abs*{v_h}^2_{H^1(\Omega_k)} \leq C\sum_{k \in \io} \abs{v_h}^2_{H^2(\Omega_k)}.
\end{equation}

Furthermore, observing that $V_h \subset H^1_0(\Omega)$, standard Poincaré inequality yields

\begin{equation}\label{eq:est_l2}
\sum_{k \in \io} \norm{v_h}^2_{L^2(\Omega_k)} \leq C\sum_{k \in \io} \abs{v_h}^2_{H^1(\Omega_k)}.
\end{equation}

Finally, the thesis follows by combining (<ref>) and (<ref>).

We recall that $\norm{\cdot}_X$ is defined by

\begin{equation*}
\norm{v}_X^2 := \sum_{k \in \io} \norm{v}_{H^2(\Omega_k)}^2 + \sum_{\ell \in \ig} \norm{\jump{\dn v}_{\gl}}_{H^{\nicefrac{1}{2}}(\gl)}^2.
\end{equation*}

Let $p \geq 2$. There exists a constant $C > 0$ such that, for all $v \in H^{p+1}(\Omega)$,

\begin{equation*}
\inf_{v_h \in V_h} \norm{v - v_h}_X \leq C h^{p - 1} \norm{w}_{H^{p+1}(\Omega)}.
\end{equation*}

As $W_{\ell} \subset H^{\nicefrac{1}{2}}_{00}(\gl)$, every $w_h \in W_{\ell}$ can be extended to a function
$\tilde{w}_h \in H^{\nicefrac{1}{2}}(\partial\Omega_{s(\ell)})$.
Let $\hh_{h,\ell} \tilde{w}_h \in X_{k,h}$ be

such that $\abs{\hh_{h,\ell} \tilde{w}_h}_{H^2(\Omega_{s(\ell)})}=0$ and
fulfilling $\hh_{h,\ell} \tilde{w}_h \in H^1_0(\Omega_{s(\ell)})$ and $\partial_n \left( \hh_{h,\ell} \tilde{w}_h \right) = \tilde{w}_h$ on $\gl$, where we recall that $\textbf{n}$ on $\gl$ is the outer normal with respect to $\partial \Omega_{m(\ell)}$.

\begin{equation*}
\norm{\hh_{h,\ell} \tilde{w}_h}_{H^2(\Omega_{s(\ell)})} \leq C \norm{\tilde{w}_h}_{H^{\nicefrac{1}{2}}(\partial \Omega_{s(\ell)})} \leq C \norm{w_h}_{H^{\nicefrac{1}{2}}(\gl)}.
\end{equation*}

We further extend $\hh_{h,\ell} \tilde{w}_h$ to $0$ outside $\Omega_{s(\ell)}$. Let $v \in H^{p+1}(\Omega)$ and let $\Pi_h v \in X_h$ be the global spline projection of $v$ in $X_h$, defined in Proposition <ref>. We define an element $v_h \in V_h$ such that

\begin{equation*}
v_h := \Pi_h v - \sum_{\ell \in \ig} \hh_{h,\ell} \mortar_\ell \jump{\partial_n \Pi_h v}_{\gl}.
\end{equation*}

From the definition of $\Pi_h$ we have that, for all $\tau_h \in M_h$,

\begin{equation*}
\int_{\gl}\jump{\dn w_h}\,\tau_h \, d\sigma = \int_{\gl} \left( \jump{\dn \Pi_h v} - \mortar_\ell \jump{\dn \Pi_h v} \right) \tau_h \, d\sigma = 0.
\end{equation*}

That is $v_h \in V_h$. Then, we can write

\begin{equation*}
\norm{v - v_h}_X \leq \norm{v - \Pi_h v}_X + \norm*{\sum_{\ell \in \ig} \hh_{h,\ell} \mortar_\ell \jump{\dn \Pi_h v}_{\gl}}_X.
\end{equation*}

The first term is estimated in Proposition <ref>.
On the other hand, using the definition of $\hh_{h,\ell}$, we get

\begin{equation*}
\begin{split}
\norm*{\sum_{\ell \in \ig} \hh_{h,\ell} \mortar_\ell \jump{\dn \Pi_h v}_{\gl}}_X^2 &\leq \sum_{\ell \in \ig} \norm*{\hh_{h,\ell} \mortar_\ell \jump{\dn \Pi_h v}_{\gl}}_{H^2\left( \Omega_{s(\ell)}\right) }^2 + \sum_{\ell \in \ig} \norm*{\mortar_\ell \jump{\dn \Pi_h v}_{\gl}}_{H^{\nicefrac{1}{2}}(\gl)}^2 \\ &\leq C \sum_{\ell \in \ig} \norm*{\mortar_\ell \jump{\dn \Pi_h v}_{\gl}}_{H^{\nicefrac{1}{2}}(\gl)}^2.
\end{split}
\end{equation*}

The $H^{\nicefrac{1}{2}}_{00}$-stability of the mortar projector $\mortar_\ell$, stated in Lemma <ref>, and Proposition <ref> yield

\begin{equation*}
\begin{split}
\norm*{\mortar_\ell \jump{\dn \Pi_h v}_{\gl}}_{H^{\nicefrac{1}{2}}(\gl)}^2 &\leq \norm*{\jump{\dn \Pi_h v}_{\gl}}_{H^{\nicefrac{1}{2}}(\gl)}^2 \\
&\leq C h^{2(p-1)} \left(\norm*{v}^2_{H^{p+1}\left( \Omega_{m(\ell)}\right) } + \norm*{v}^2_{H^{p+1}\left( \Omega_{s(\ell)}\right) } \right).
\end{split}
\end{equation*}

Thanks to Assumption <ref>, summing over the interfaces we obtain the thesis.

§.§ Error analysis for the model problem

Since $V_h \nsubseteq H^2(\Omega)$ (the space $V_h$ is only weakly $C^1$) we need to use a Strang lemma for bounding the numerical error of our formulation.

Let $u$ be the solution of Problem (<ref>) and $u_h$ the solution of the discrete Problem (<ref>). Then, there exists a constant $C > 0$ such that

\begin{equation*}\label{eq:strang}
\norm{u - u_h}_X \leq C \left( \inf_{v_h \in V_h} \norm{u - v_h}_X + \sup_{v_h \in V_h} \dfrac{\abs*{\int_{\Sigma} (\partial_{tt}u - \Delta u) \jump{\partial_n v_h} \, d\sigma}}{\norm{v_h}_X}\right) .
\end{equation*}

From Lemma <ref>, the bilinear form $a(\cdot, \cdot)$ is continuous on $X$ and coercive on $V_h$ with respect to the broken norm $\sum_{k \in \io}\norm{\cdot}^2_{H^2(\Omega_k)}$. We want to prove the coercivity on $V_h$ with respect to $\norm{\cdot}_X$. Using the inverse inequality and function space interpolation we obtain

\begin{equation*}
\norm{\jump{\dn v_h}_{\gl}}_{H^{\nicefrac{1}{2}}}^2 \leq C h^{-1} \norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)}^2. \\
\end{equation*}

Now, following <cit.> and Assumptions <ref> and <ref>, if $v_h \in V_h$ and $\mortar_{\ell}^{*} \colon L^2(\gl) \to M_\ell$ is the projector defined by equation (<ref>), we have

\begin{equation*}
\norm{(\id - \mortar_{\ell}^{*})\jump{\dn v_h}_{\gl}}_{L^2(\gl)}^2 =\norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)}^2 + \norm{\jump{\mortar_{\ell}^{*}\dn v_h}_{\gl}}_{L^2(\gl)}^2,
\end{equation*}

where the equality follows from the fact that the jump $\jump{\dn v_h}_{\gl}$ is orthogonal to $M_\ell$, for all $v_h \in V_h$ and for every $\ell \in \ig$. Now, applying the approximation properties of $\mortar_{\ell}^{*}$, as stated in Lemma <ref>, we obtain

\begin{equation*}
\begin{split}
\norm{\jump{\dn v_h}_{\gl}}_{H^{\nicefrac{1}{2}}(\gl)}^2 &\leq C h^{-1} \norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)}^2 \leq C h^{-1} \norm{(\id - \mortar_{\ell}^{*})\jump{\dn v_h}_{\gl}}_{L^2(\gl)}^2\\
&\leq C h^{-1} \left( \norm{(\id - \mortar_{\ell}^{*}) \dn v_h|_{\Omega_{m(\ell)}}}_{L^2(\gl)}^2 + \norm{(\id - \mortar_{\ell}^{*})\dn v_h|_{\Omega_{s(\ell)}}}_{L^2(\gl)}^2 \right) \\
&\leq C h^{-1} \left( h \norm{\dn v_h|_{\Omega_{m(\ell)}}}_{H^{\nicefrac{1}{2}}(\gl)}^2 + h \norm{\dn v_h|_{\Omega_{s(\ell)}}}_{H^{\nicefrac{1}{2}}(\gl)}^2 \right) \\
&\leq C \left(\norm{v_h}_{H^2\left( \Omega_{m(\ell)}\right) }^2 + \norm{v_h}_{H^2\left( \Omega_{s(\ell)}\right) }^2 \right).
\end{split}
\end{equation*}

Then, $a(\cdot, \cdot)$ is coercive on $(V_h, \norm{\cdot}_X)$ and the second Strang's Lemma can be applied, see <cit.>. Indeed, if $\alpha^* > 0$ denotes the coercivity constant of $a(\cdot, \cdot)$ on $(V_h, \norm{\cdot}_X)$, for any $v_h \in V_h$ it holds

\begin{align*}
\alpha^*\,\norm{u_h - v_h}^2_X &\leq a(u_h - v_h, u_h-v_h)\\
&=  a(u-v_h,u_h-v_h)+a(u_h-u,u_h-v_h) \\
&= a(u-v_h,u_h-v_h)-a(u,u_h-v_h)+f(u_h-v_h)+b(u_h-v_h,\tau_h)
\end{align*}

and, recalling (<ref>), we get

\begin{equation*}
\alpha^*\,\norm{u_h - v_h}^2_X \leq a(u-v_h,u_h-v_h)-a(u,u_h-v_h)+f(u_h-v_h).
\end{equation*}

Then, if $M^*>0$ denotes the continuity constant of $a(\cdot, \cdot)$, we deduce

\begin{equation*}
\begin{split}
\alpha^*\,\norm{u_h - v_h}_X &\leq M^* \norm{u - v_h}_X + \dfrac{f(u_h - v_h) - a(u, u_h - v_h)}{\norm{u_h - v_h}_X} \\
&\leq M^* \norm{u - v_h}_X + \sup_{w_h \in V_h} \dfrac{f(w_h) - a(u, w_h)}{\norm{w_h}_X}.
\end{split}
\end{equation*}

Finally, from the triangular inequality and equation (<ref>), we obtain

\begin{equation*}
\begin{split}
\norm{u - u_h}_X &\leq \norm{u - v_h}_X + \norm{v_h - u_h}_X \\
&\leq \left(1 + \frac{M^*}{\alpha^*} \right) \norm{u - v_h}_X + \frac{1}{\alpha^*} \sup_{w_h \in V_h} \dfrac{\int_{\Sigma}(\partial_{tt}u - \Delta u) \jump{\partial_n w_h} \, d\sigma}{\norm{w_h}_X},
\end{split}
\end{equation*}

which yields the thesis.

The first term of the estimate of Lemma <ref> denotes the best approximation error in $V_h$ and was studied in Lemma <ref>. The second term stands for the consistency error, which is related to the nonconformity of the method. The next result provides an optimal estimate for the consistency error.

Let $p \geq 3$. There exists a constant $C > 0$ such that

\begin{equation*}
\sup_{v_h \in V_h} \dfrac{\abs*{\int_{\Sigma} (\partial_{tt}u - \Delta u) \jump{\partial_n v_h} \, d\sigma}}{\norm{v_h}_X} \leq C \sum_{k \in \io} h^{p -1} \norm{u}_{H^{p+1}(\Omega_k)}.
\end{equation*}

For the definition of $V_h$ and Lemma <ref>, it holds

\begin{equation*}
\begin{split}
\int_{\Sigma} (\partial_{tt}u - \Delta u) \jump{\partial_n v_h} \, d\sigma &= \int_{\Sigma} (\partial_{tt}u - \Delta u - \mortar_{\ell}^{*}(\partial_{tt}u - \Delta u)) \jump{\partial_n v_h} \, d\sigma \\
&\leq \sum_{\ell \in \ig} \norm{\partial_{tt}u - \Delta u - \mortar_{\ell}^{*}(\partial_{tt}u - \Delta u)}_{L^2(\gl)} \norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)} \\
&\leq C h^{p-\nicefrac{3}{2}} \sum_{\ell \in \ig} \norm{(\partial_{tt}u - \Delta u)}_{H^{p-\nicefrac{3}{2}}(\gl)} \norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)},
\end{split}
\end{equation*}

From the definition of $V_h$ and Lemma <ref>, we get

\begin{equation*}
\begin{split}
\norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)} &\leq \norm{(\id - \mortar_{\ell})\jump{\dn v_h}_{\gl}}_{L^2(\gl)} \\
&\leq C h^{\nicefrac{1}{2}} \norm{\jump{\dn v_h}_{\gl}}_{H^{\nicefrac{1}{2}}(\gl)} \leq C h^{\nicefrac{1}{2}} \norm{v_h}_X.
\end{split}
\end{equation*}

Trace theorem yields

\begin{equation*}
\norm{\partial_{tt}u - \Delta u}_{H^{p-\nicefrac{3}{2}}(\gl)} \leq C \norm{u}_{H^{p+\nicefrac{1}{2}}(\gl)}  \leq C \norm{u}_{H^{p+1}(\Omega_{s(\ell)})},
\end{equation*}

and thus we obtain

\begin{equation*}
\begin{split}
\int_{\Sigma} (\partial_{tt}u - \Delta u) \jump{\partial_n v_h} \,d\sigma \leq C h^{p-1} \sum_{k \in \io} \norm{u}_{H^{p+1}\left( \Omega_{k}\right) }\norm{v_h}_X.
\end{split}
\end{equation*}

We combine the previous results to get the optimal a priori error estimate for the problem (<ref>).

Given Assumptions <ref>, <ref> and $p \geq 3$, for $u \in H^{p+1}(\Omega)$ it holds

\begin{equation*}
\norm{u - u_h}_X \leq C\,h^{p-1} \norm{u}_{H^{p+1}(\Omega)}.
\end{equation*}

Since $V_h \nsubseteq H^2(\Omega)$ (the space $V_h$ is only weakly $C^1$) we need to use a Strang lemma for bounding the numerical error of our formulation.

Let $u$ be the solution of Problem (<ref>) and $u_h$ the solution of the discrete Problem (<ref>). Then, there exists a constant $C > 0$ such that

\begin{equation}\label{eq:strang}
\norm{u - u_h}_X \leq C \left( \inf_{v_h \in V_h} \norm{u - v_h}_X + \sup_{v_h \in V_h} \dfrac{\abs*{\int_{\Sigma} (\partial_{tt}u - \Delta u) \jump{\partial_n v_h} \, d\sigma}}{\norm{v_h}_X}\right) .
\end{equation}

From Lemma <ref>, the bilinear form $a(\cdot, \cdot)$ is continuous on $X$ and coercive on $V_h$ with respect to the broken norm $\sum_{k \in \io}\norm{\cdot}^2_{H^2(\Omega_k)}$. We want to prove the coercivity on $V_h$ with respect to $\norm{\cdot}_X$. Using the inverse inequality and function space interpolation we obtain

\begin{equation*}
\norm{\jump{\dn v_h}_{\gl}}_{H^{\nicefrac{1}{2}}}^2 \leq C h^{-1} \norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)}^2. \\
\end{equation*}

Now, following <cit.> and Assumptions <ref> and <ref>, if $v_h \in V_h$ and $\mortar_{\ell}^{*} \colon L^2(\gl) \to M_\ell$ is the projector defined by equation (<ref>), we have

\begin{equation*}
\norm{(\id - \mortar_{\ell}^{*})\jump{\dn v_h}_{\gl}}_{L^2(\gl)}^2 =\norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)}^2 + \norm{\jump{\mortar_{\ell}^{*}\dn v_h}_{\gl}}_{L^2(\gl)}^2,
\end{equation*}

where the equality follows from the fact that the jump $\jump{\dn v_h}_{\gl}$ is orthogonal to $M_\ell$, for all $v_h \in V_h$ and for every $\ell \in \ig$. Now, applying the approximation properties of $\mortar_{\ell}^{*}$, as stated in Lemma <ref>, we obtain

\begin{equation*}
\begin{split}
\norm{\jump{\dn v_h}_{\gl}}_{H^{\nicefrac{1}{2}}(\gl)}^2 &\leq C h^{-1} \norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)}^2 \leq C h^{-1} \norm{(\id - \mortar_{\ell}^{*})\jump{\dn v_h}_{\gl}}_{L^2(\gl)}^2\\
&\leq C h^{-1} \left( \norm{(\id - \mortar_{\ell}^{*}) \dn v_h|_{\Omega_{m(\ell)}}}_{L^2(\gl)}^2 + \norm{(\id - \mortar_{\ell}^{*})\dn v_h|_{\Omega_{s(\ell)}}}_{L^2(\gl)}^2 \right) \\
&\leq C h^{-1} \left( h \norm{\dn v_h|_{\Omega_{m(\ell)}}}_{H^{\nicefrac{1}{2}}(\gl)}^2 + h \norm{\dn v_h|_{\Omega_{s(\ell)}}}_{H^{\nicefrac{1}{2}}(\gl)}^2 \right) \\
&\leq C \left(\norm{v_h}_{H^2\left( \Omega_{m(\ell)}\right) }^2 + \norm{v_h}_{H^2\left( \Omega_{s(\ell)}\right) }^2 \right).
\end{split}
\end{equation*}

Then, $a(\cdot, \cdot)$ is coercive on $(V_h, \norm{\cdot}_X)$ and the second Strang's Lemma can be applied, see <cit.>. Indeed, if $\alpha^* > 0$ denotes the coercivity constant of $a(\cdot, \cdot)$ on $(V_h, \norm{\cdot}_X)$, for any $v_h \in V_h$ it holds

\begin{align*}
\alpha^*\,\norm{u_h - v_h}^2_X &\leq a(u_h - v_h, u_h-v_h)\\
&=  a(u-v_h,u_h-v_h)+a(u_h-u,u_h-v_h) \\
&= a(u-v_h,u_h-v_h)-a(u,u_h-v_h)+f(u_h-v_h)+b(u_h-v_h,\tau_h)
\end{align*}

and, recalling (<ref>), we get

\begin{equation*}
\alpha^*\,\norm{u_h - v_h}^2_X \leq a(u-v_h,u_h-v_h)-a(u,u_h-v_h)+f(u_h-v_h).
\end{equation*}

Then, if $M^*>0$ denotes the continuity constant of $a(\cdot, \cdot)$, we deduce

\begin{equation*}
\begin{split}
\alpha^*\,\norm{u_h - v_h}_X &\leq M^* \norm{u - v_h}_X + \dfrac{f(u_h - v_h) - a(u, u_h - v_h)}{\norm{u_h - v_h}_X} \\
&\leq M^* \norm{u - v_h}_X + \sup_{w_h \in V_h} \dfrac{f(w_h) - a(u, w_h)}{\norm{w_h}_X}.
\end{split}
\end{equation*}

Finally, from the triangular inequality and equation (<ref>), we obtain

\begin{equation*}
\begin{split}
\norm{u - u_h}_X &\leq \norm{u - v_h}_X + \norm{v_h - u_h}_X \\
&\leq \left(1 + \frac{M^*}{\alpha^*} \right) \norm{u - v_h}_X + \frac{1}{\alpha^*} \sup_{w_h \in V_h} \dfrac{\int_{\Sigma}(\partial_{tt}u - \Delta u) \jump{\partial_n w_h} \, d\sigma}{\norm{w_h}_X},
\end{split}
\end{equation*}

which yields the thesis.

The first term of the estimate of Lemma <ref> denotes the best approximation error in $V_h$ and was studied in Lemma <ref>. The second term stands for the consistency error, which is related to the nonconformity of the method. The next result provides an optimal estimate for the consistency error.

Let $p \geq 3$. There exists a constant $C > 0$ such that

\begin{equation*}
\sup_{v_h \in V_h} \dfrac{\abs*{\int_{\Sigma} (\partial_{tt}u - \Delta u) \jump{\partial_n v_h} \, d\sigma}}{\norm{v_h}_X} \leq C \sum_{k \in \io} h^{p -1} \norm{u}_{H^{p+1}(\Omega_k)}.
\end{equation*}

For the definition of $V_h$ and Lemma <ref>, it holds

\begin{equation}
\begin{split}
\int_{\Sigma} (\partial_{tt}u - \Delta u) \jump{\partial_n v_h} \, d\sigma &= \int_{\Sigma} (\partial_{tt}u - \Delta u - \mortar_{\ell}^{*}(\partial_{tt}u - \Delta u)) \jump{\partial_n v_h} \, d\sigma \\
&\leq \sum_{\ell \in \ig} \norm{\partial_{tt}u - \Delta u - \mortar_{\ell}^{*}(\partial_{tt}u - \Delta u)}_{L^2(\gl)} \norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)} \\
&\leq C h^{p-\nicefrac{3}{2}} \sum_{\ell \in \ig} \norm{(\partial_{tt}u - \Delta u)}_{H^{p-\nicefrac{3}{2}}(\gl)} \norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)},
\end{split}
\end{equation}

From the definition of $V_h$ and Lemma <ref>, we get

\begin{equation}
\begin{split}
\norm{\jump{\dn v_h}_{\gl}}_{L^2(\gl)} &\leq \norm{(\id - \mortar_{\ell})\jump{\dn v_h}_{\gl}}_{L^2(\gl)} \\
&\leq C h^{\nicefrac{1}{2}} \norm{\jump{\dn v_h}_{\gl}}_{H^{\nicefrac{1}{2}}(\gl)} \leq C h^{\nicefrac{1}{2}} \norm{v_h}_X.
\end{split}
\end{equation}

Trace theorem yields

\begin{equation*}
\norm{\partial_{tt}u - \Delta u}_{H^{p-\nicefrac{3}{2}}(\gl)} \leq C \norm{u}_{H^{p+\nicefrac{1}{2}}(\gl)}  \leq C \norm{u}_{H^{p+1}(\Omega_{s(\ell)})},
\end{equation*}

and thus we obtain

\begin{equation*}
\begin{split}
\int_{\Sigma} (\partial_{tt}u - \Delta u) \jump{\partial_n v_h} \,d\sigma \leq C h^{p-1} \sum_{k \in \io} \norm{u}_{H^{p+1}\left( \Omega_{k}\right) }\norm{v_h}_X.
\end{split}
\end{equation*}

We combine the previous results to get the optimal a priori error estimate for the problem (<ref>).

Given Assumptions <ref>, <ref> and $p \geq 3$, for $u \in H^{p+1}(\Omega)$ it holds

\begin{equation}
\norm{u - u_h}_X \leq C\,h^{p-1} \norm{u}_{H^{p+1}(\Omega)}.
\end{equation}

§ DISCRETE INF-SUP CONDITION

This section is devoted to prove the discrete inf-sup condition for a Lagrange multipliers space $\Sp^{p-2}_{\textsc{m}}$, i.e. a spline space of degree $p-2$ and merged elements in the neighbourhood of corners, see (<ref>) below. The choice of the degree $p-2$ is motivated by <cit.>.

Let $\Xi$ be an open knot vector on the reference interface $\hgamma := \left(0,1\right)$ with internal knots of multiplicity $1$:  setting $\xi_1=\dots=\xi_{p+1}=0$ and $\xi_{n+1}=\dots=\xi_{n+p+1}=1$, we have

\begin{equation*}
\Xi \equiv \Xi^p = \Set{\underbrace{0 = \dots = 0}_{p+1} < \xi_{p+2} < \dots < \xi_{n} < \underbrace{1 = \dots = 1}_{p+1}}.
\end{equation*}

As before, $\Sp^p(\Xi)$ is the spline space on $\hgamma$ generated by the knot vector $\Xi$. The global regularity of this spline space is always $C^{p-1}$.
Now we can introduce the modified knot vector

\begin{equation*}
\Xi_{\textsc{m}} \equiv \Xi^p_{\textsc{m}} = \Set{\underbrace{0 = \dots = 0}_{p+1} <  \xi_{p+3} < \dots < \xi_{n-1}  < \underbrace{1 = \dots =  1}_{p+1}}
\end{equation*}

by removing $\xi_{p+2}$ and $\xi_{n}$, the first and last internal knots, respectively.
In this way, $\Sp^p(\Xi_{\textsc{m}})$ is the spline space of degree $p$ on $\hgamma$ with the first and the last elements  of $\Xi$ merged.
Similarly, the knot vectors of $\Sp^{p-1}_{\textsc{m}}$ on the same mesh subdivision and $\Sp^{p-2}_{\textsc{m}}$ are, respectively,

\begin{equation*}
\begin{split}
\Xi_{\textsc{m}}^{p-1} &= \Set{\underbrace{0 = \dots = 0}_{p} < \xi_{p+2} < \dots < \xi_{n-1} <  \underbrace{1 = \dots = 1}_{p}},\\
\Xi_{\textsc{m}}^{p-2} &= \Set{\underbrace{0 = \dots = 0}_{p-1} <  \xi_{p+2} < \dots < \xi_{n-1} < \underbrace{1 = \dots = 1}_{p-1}}.
\end{split}
\end{equation*}

We introduce the notation $\partial\hgamma := \Set{0,1}$ to indicate the endpoints of $\hgamma$.
In this section we make use of the following spline spaces:

\begin{equation}\label{eq:spaces}
\begin{split}
\Sp^p_{0,\partial}	 &:= \Set{v \in \Sp^p(\Xi) | v = v' = 0, \text{in} \; \partial\hgamma} \\
\Sp^{p-1}_{0,\int} &:= \Set{v \in \Sp^{p-1}(\Xi) | v = 0 \; \text{in} \; \partial\hgamma \; \text{and} \; \int_{\hgamma} v d\sigma = 0} \\
\Sp^{p-1}_{\textsc{m},\int} &:= \Set{v \in \Sp^{p-1}(\Xi^{p-1}_{\textsc{m}}) | \int_{\hgamma} v d\sigma = 0}\\
\Sp^{p-2}_{\textsc{m}}     &:= \Sp^{p-2}(\Xi^{p-2}_{\textsc{m}}).
\end{split}
\end{equation}

Note that the derivative operator is a bijection between $\Sp^p_{0,\partial}$ and $\Sp^{p-1}_{0,\int}$ and between $\Sp^{p-1}_{\textsc{m},\int}$ and $\Sp^{p-2}_{\textsc{m}}$.

For any $v \in H^1_0(\hgamma)$ it holds

\begin{equation*}
\norm{v}_{L^2(\hgamma)} \leq C \norm{v'}_{[H^{1}(\hgamma)]'},
\end{equation*}

where $[H^{1}(\hgamma)]'$ is the dual of $H^1(\hgamma)$.

For all $w \in L^2(\hgamma)$ there exists a function $\bar{w} \in H^1_{\int}(\hgamma)$, with

\begin{equation*}
H^1_{\int}(\hgamma):=\Set{\bar{w} \in H^1(\hgamma) | \int_{\hgamma} \bar{w} \, d\sigma = 0},
\end{equation*}

such that $\bar{w}' = w$. For all $v \in H^1_0(\hgamma)$ it holds

\begin{equation*}
\begin{split}
\norm{v}_{L^2(\hgamma)} &= \sup_{w \in L^2(\hgamma)} \dfrac{\int_{\hgamma} v \, w \, d\sigma}{\norm{w}_{L^2(\hgamma)}} = \sup_{\bar{w} \in H^1_{\int}(\hgamma)} \dfrac{\int_{\hgamma} v \bar{w}' \, d\sigma}{\abs{\bar{w}}_{H^1(\hgamma)}} \\
&\leq C \sup_{\bar{w} \in H^1_{\int}(\hgamma)} \dfrac{\int_{\hgamma} \bar{w} \, v' \, d\sigma}{\norm{\bar{w}}_{H^1(\hgamma)}} \leq C \norm{v'}_{H^1(\hgamma)'},
\end{split}
\end{equation*}

where $C$ is the Poincaré constant, i.e. $\norm{\bar{w}}_{H^1(\hgamma)} \leq C \abs{\bar{w}}_{H^1(\hgamma)}$ for all $\bar{w} \in H^1_{\int}(\hgamma)$.

Let $p \leq 9$. For a quasi-uniform mesh defined on $\hgamma$, there exists a $\beta > 0$, independent on $h$, such that, for any $\phi \in \Sp^{p-1}_0$ it holds

\begin{equation*}
\sup_{\tau \in \Sp^{p-1}_{\textsc{m}}} \dfrac{\int_{\hgamma} \phi \tau d\sigma}{\norm{\tau}_{L^2(\hgamma)}} \geq \beta \norm{\phi}_{L^2(\hgamma)}.
\end{equation*}

If we prove that, for all $\phi \in \Sp^{p-1}_0$, there exists $\tau \in \Sp^{p-1}_{\textsc{m}}$ such that

\begin{align}
\norm{\tau}_{L^2(\hgamma)} &\leq C_1 \norm{\phi}_{L^2(\hgamma)}      \label{eq:norm1} \\
\norm{\phi}_{L^2(\hgamma)}^2  &\leq C_2 \int_{\hgamma} \phi \, \tau \, d\sigma \label{eq:norm2}
\end{align}

then we have the thesis with $\beta = C_1^{-1} C_2^{-2}$.
Let $(\widehat{B}_i)_{i=1}^n$ and $(\widetilde{B}_i)_{i=1}^n$ be the B-spline basis functions of $\Sp^{p-1}_0$ and $\Sp^{p-1}_{\textsc{m}}$, respectively. We denote by $\widehat{I}_i$ and $\widetilde{I}_i$ the supports of $\widehat{B}_i$ and $\widetilde{B}_i$, respectively, and by $\abs{\widehat{I}_i}$ and $\abs{\widetilde{I}_i}$ their respective lengths. As $\phi \in \Sp^{p-1}_0$ we can express it by $\phi = \sum_{i=1}^{n} c_i \widehat{B}_i$, and use the coefficients $(c_i)_{i=1}^n$ to define $\tau = \sum_{i=1}^{n} c_i \widetilde{B}_i$.
Let $I \subset (0,1)$ be an element of the mesh. The restriction of $\tau$ on the element $I$ can be expressed by $\tau|_{I} = \sum_{i: I \subset \widetilde{I}_i} c_i \widetilde{B}_i$, and then we have that

\begin{equation*}
\norm{\tau}_{L^2(I)}^2 \leq \int_I \left( \sum_{i: I \subset \widetilde{I}_i} c_i \widetilde{B}_i \right)^2 d\sigma \leq \int_I \left( \sum_{i: I \subset \widetilde{I}_i} c_i^2 \cdot \sum_{i: I \subset \widetilde{I}_i} \widetilde{B}_i^2 \right) d\sigma .
\end{equation*}

Since every function of the basis $(\widetilde{B}_i)_{i=1}^n$ is such that $0 \leq \widetilde{B}_i \leq 1$, we can write $\sum_{i: I \subset \widetilde{I}_i}  \widetilde{B}_i^2 \leq \sum_{i: I \subset \widetilde{I}_i}  \widetilde{B}_i \leq 1$, and then we obtain

\begin{equation*}
\norm{\tau}_{L^2(I)}^2 \leq \int_I \left( \sum_{i: I \subset \widetilde{I}_i} c_i^2 \cdot \sum_{i: I \subset \widetilde{I}_i} \widetilde{B}_i^2 \right) d\sigma \leq \sum_{i: I \subset \widetilde{I}_i} c_i^2 \cdot \abs{I}.
\end{equation*}

From <cit.>, there exists a constant $C > 0$ independent on $h$ such that

\begin{equation*}
\norm{\tau}_{L^2(I)}^2 \leq \sum_{i: I \subset \widetilde{I}_i} c_i^2 \cdot \abs{I} \leq C\,\abs{I} \sum_{i: I \subset \widetilde{I}_i} \abs{\widehat{I}_i}^{-1} \norm{\phi}^2_{L^2(\widehat{I}_i)} \leq C \sum_{i: I \subset \widetilde{I}_i} \norm{\phi}^2_{L^2(\widehat{I}_i)},
\end{equation*}

where the last inequality follows from the hypothesis of quasi-uniform mesh. Finally, summing over $I \subset (0,1)$, we get (<ref>).

In order to demonstrate (<ref>), we recall that $\Xi_0^{p-1}$ is the knot vector of $\Sp^{p-1}_0$ and $\Xi_{\textsc{m}}^{p-1}$ the knot vector of $\Sp^{p-1}_{\textsc{m}}$. Hence

\begin{equation*}
\begin{split}
\Xi_0^{p-1} &= \Set{\underbrace{0 = \dots = 0}_{p-1} < \xi_{p} < \dots < \xi_{n+1} < \underbrace{1 = \dots = 1}_{p-1}}\\
\Xi_{\textsc{m}}^{p-1} &= \Set{\underbrace{0 = \dots = 0}_{p} < \xi_{p+1} < \dots < \xi_{n} < \underbrace{1 = \dots = 1}_{p}}.
\end{split}
\end{equation*}

Let $n$ be such that $\xi_{2p} < \xi_{n-p+1}$.
We note that, for all $p+1 \leq i \leq n - p$, it holds that $\widehat{B}_i = \widetilde{B}_i$, and then

\begin{equation}\label{eq:aux03}
\int_{\hgamma} \phi\,\tau \, d\sigma = \int_{0}^{\xi_{2p}} \phi\,\tau \, d\sigma + \norm*{\phi}^2_{L^2 \left( \xi_{2p}, \xi_{n-p+1}\right)} + \int_{\xi_{n-p+1}}^{1} \phi\,\tau \, d\sigma.
\end{equation}

We focus on the first term of (<ref>). We want to prove that there exists a $\gamma > 0$ independent of $h$ such that, for all $\phi \in \Sp^{p-1}_0$, it holds

\begin{equation}\label{eq:aux04}
\int_{0}^{\xi_{2p}} \phi\,\tau \, d\sigma \geq \gamma \, \norm*{\phi}^2_{L^2 \left(0, \xi_{2p}\right) }.
\end{equation}

Since, from the previous construction, $\tau$ depends linearly from $\phi$, we have that $\int_{0}^{\xi_{2p}} \phi\,\tau \, d\sigma$ is a quadratic form. Then (<ref>) can be tested numerically by solving a generalized eigenvalue problem of the form

\begin{equation*}
\gamma= \inf_{\bm{\phi} \neq 0} \frac{\bm{\phi}^t M_1 \bm{\phi}}{\bm{\phi}^t M_2 \bm{\phi}}
\end{equation*}

where $\bm{\phi} \in \R^n$ is the vector associated to the function $\phi \in \Sp^{p-1}_0$, and $M_1$, $M_2$ are the matrices associated to the forms $\int_{0}^{\xi_{2p}} \phi\,\tau \, d\sigma$ and $\norm*{\phi}^2_{L^2 \left(0, \xi_{2p}\right) }$, respectively.  Thus, $\gamma$ is the minimum generalized eigenvalue $\mu_{\textsc{min}}$ of

\begin{equation}\label{eq:aux05}
M_1^{\textsc{s}} \, \textbf{v} = \mu \, M_2 \, \textbf{v},
\end{equation}

where $M_1^{\textsc{s}}$ is the symmetric part of the matrix $M_1$.
Thanks to a scaling argument, for uniform meshes the value of  $\mu_\textsc{min}$ does not depend of $h$.  A good behaviour of $\mu_\textsc{min}$ is observed for different values of $p$: see Figure <ref> for the $p$-dependence of $\mu_\textsc{min}$ on a uniform mesh. Numerical experiments show that $\mu_\textsc{min}$ is stable also for non-uniform meshes, see Figure <ref>.
This gives a numerical proof of condition (<ref>). The same argument can be used to show that exists a $\gamma' > 0$ independent of $h$ such that, for all $\phi \in \Sp^{p-1}_0$, it holds

\begin{equation}\label{eq:aux06}
\int_{\xi_{n-p+1}}^{1} \phi\,\tau \, d\sigma \geq \gamma' \, \norm*{\phi}^2_{L^2 \left(\xi_{n-p+1},1\right) }.
\end{equation}

Combining equations (<ref>), (<ref>) and (<ref>), we get (<ref>).

The operator $\widehat{\mortar}^{*}_h \colon L^2(\hgamma) \to \Sp^{p-1}_{\textsc{m}}$ given by

\begin{equation*}\label{eq:defprojd}
\int_{\hgamma} \widehat{\mortar}^{*}_h \phi \, \psi \, d\sigma = \int_{\hgamma} \phi \, \psi \, d\sigma, \quad \forall \psi \in \Sp^{p-1}_0,
\end{equation*}

is well defined and $\widehat{\mortar}^{*}_h \colon \Sp^{p-1}_0 \to \Sp^{p-1}_{\textsc{m}}$ is bijective. This follows from the fact that $\dim(\Sp^{p-1}_0) = \dim(\Sp^{p-1}_{\textsc{m}})$ and from Proposition <ref>.

The operator $\widehat{\mortar}^{*}_h \colon \Sp^{p-1}_0 \to \Sp^{p-1}_{\textsc{m}}$ is bijective.

Let us assume that there exists a $\phi \in \Sp^{p-1}_0$ such that $\widehat{\mortar}^{*}_h \phi = 0$. For all $\psi \in \Sp^{p-1}_0$ it holds that

\begin{equation*}
\int_{\hgamma} \phi \, \psi \, d\sigma = \int_{\hgamma} \left( \phi - \widehat{\mortar}^{*}_h \phi \right)  \, \psi \, d\sigma = 0,
\end{equation*}

and then $\phi = 0$ on $\hgamma$.
By the fact that $\dim(\Sp^{p-1}_0) = \dim(\Sp^{p-1}_{\textsc{m}})$, surjectivity follows.

For any $\phi \in \Sp^{p-1}_0$ it holds

\begin{equation}\label{eq:infsuph1}
\sup_{\tau \in \Sp^{p-1}_{\textsc{m}}} \dfrac{\int_{\hgamma} \phi \tau d\sigma}{\norm{\tau}_{H^1(\hgamma)}} \geq \beta \norm{\phi}_{H^1(\hgamma)'},
\end{equation}

with $\beta > 0$ independent from $h$.

From an argument similar to Lemma <ref>, the operator $\widehat{\mortar}^{*}_h$ is $L^2$-stable. We check that it is also $H^1$-stable. Let $\tau \in H^1(\hgamma)$ and let $\widetilde{\Pi}_h:H^1(\hgamma) \rightarrow \Sp^{p-1}_{\textsc{m}}$  be the quasi-interpolant operator defined in (<ref>). We recall that, for $t=0,1$, it holds

\begin{equation}\label{eq:qiest}
\norm{ \tau - \widetilde{\Pi}_h \tau}_{H^t(\hgamma)} \leq C h^{1-t} \norm{ \tau}_{H^t(\hgamma)}.
\end{equation}

Using the inverse inequality for splines, we get

\begin{equation*}
\begin{split}
\norm{\widehat{\mortar}^{*}_h \, \tau}_{H^1(\hgamma)} &\leq \norm{\widehat{\mortar}^{*}_h \, \tau - \widetilde{\Pi}_h \tau}_{H^1(\hgamma)} + \norm{\widetilde{\Pi}_h \tau}_{H^1(\hgamma)} \\
&= \norm{\widehat{\mortar}^{*}_h (\tau - \widetilde{\Pi}_h \tau)}_{H^1(\hgamma)} + \norm{\widetilde{\Pi}_h \tau}_{H^1(\hgamma)} \\
&\leq C h^{-1} \norm{\widehat{\mortar}^{*}_h (\tau - \widetilde{\Pi}_h \tau)}_{L^2(\hgamma)} + \norm{\widetilde{\Pi}_h \tau}_{H^1(\hgamma)},
\end{split}
\end{equation*}

but for the $L^2$-stability and (<ref>) we have

\begin{equation*}
\begin{split}
\norm{\widehat{\mortar}^{*}_h \, \tau}_{H^1(\hgamma)} &\leq C h^{-1} \norm{\tau - \widetilde{\Pi}_h \tau}_{L^2(\hgamma)} + \norm{\widetilde{\Pi}_h \tau}_{H^1(\hgamma)} \\
&\leq C \norm{\tau}_{H^1(\hgamma)}.
\end{split}
\end{equation*}

For any $\phi \in H^1(\hgamma)'$ it holds

\begin{equation}\label{eq:aux02}
\norm{\phi}_{H^1(\hgamma)'}=\sup_{\eta \in H^1(\hgamma)} \dfrac{\int_{\hgamma} \eta \, \phi \, d\sigma}{\norm{\eta}_{H^1(\hgamma)}},
\end{equation}

and, in particular, it holds for any $\phi \in \Sp^{p-1}_0$. Now, from Lemma <ref>, for any $\psi \in \Sp^{p-1}_{\textsc{m}}$ there exists $\eta \in \Sp^{p-1}_0$ such that $\psi = \widehat{\mortar}^{*}_h \eta$, that is

\begin{equation*}
\int_{\hgamma} \psi \, \phi \, d\sigma = \int_{\hgamma} \eta \, \phi \, d\sigma, \quad \forall \phi \in \Sp^{p-1}_0
\end{equation*}

and $\norm{\psi}_{H^1(\hgamma)} \leq C^{-1} \norm{\eta}_{H^1(\hgamma)}$.
Hence, for all $\phi \in \Sp^{p-1}_0$ there exists $\psi = \widehat{\mortar}^{*}_h \eta \in \Sp^{p-1}_{\textsc{m}}$ such that

\begin{equation*}
\dfrac{\int_{\hgamma} \psi \, \phi \, d\sigma}{\norm{\psi}_{H^1(\hgamma)}} \geq C \dfrac{\int_{\hgamma} \eta \, \phi \, d\sigma}{\norm{\eta}_{H^1(\hgamma)}};
\end{equation*}

taking the supremum over $\eta \in H^1(\hgamma)$ and using (<ref>) yields (<ref>).

For any $\tau \in \Sp^{p-1}_{\textsc{m},\int}$ it holds

\begin{equation*}
\sup_{\phi \in \Sp^{p-1}_{0,\int}} \dfrac{\int_{\hgamma} \phi \tau d\sigma}{\norm{\phi}_{H^1(\hgamma)'}} \geq \beta \norm{\tau}_{H^1(\hgamma)},
\end{equation*}

with $\beta > 0$ independent from $h$.

From Proposition <ref> we have that for all $\phi \in \Sp^{p-1}_{0}$ there exists $\tau \in \Sp^{p-1}_{\textsc{m}}$ such that

\begin{equation*}
\int_{\hgamma} \phi \tau d\sigma  \geq \beta \norm{\phi}_{H^1(\hgamma)'} \norm{\tau}_{H^1(\hgamma)},
\end{equation*}

and in particular it holds for all $\phi \in \Sp^{p-1}_{0,\int}$. Now we can choose $\tau^* := \tau - \bar{\tau}$, where $\bar{\tau}$ is the mean value of $\tau$ on $\hgamma$. Then we obtain

\begin{equation*}
\int_{\hgamma} \phi \tau^* d\sigma =  \int_{\hgamma} \phi \tau d\sigma \geq \beta \norm{\phi}_{H^1(\hgamma)'} \norm{\tau}_{H^1(\hgamma)}.
\end{equation*}

It is easy to see that $\norm{\tau^*}_{H^1(\hgamma)} \leq \norm{\tau}_{H^1(\hgamma)}$, and so, for all $\phi \in \Sp^{p-1}_{0,\int}$ it holds

\begin{equation*}
\sup_{\tau^* \in \Sp^{p-1}_{\textsc{m},\int}} \dfrac{\int_{\hgamma} \phi \tau^* d\sigma}{ \norm{\tau^*}_{H^1(\hgamma)}} \geq \beta \norm{\phi}_{H^1(\hgamma)'} .
\end{equation*}

As in Remark <ref>, we can interchange the inf-sup spaces and obtain the thesis.

Let $p \geq 2$. For any $\mu \in \Sp^{p-2}_{\textsc{m}}$ it holds

\begin{equation}\label{eq:sup2}
\sup_{\psi \in \Sp^p_{0,\partial}} \dfrac{\int_{\hgamma} \psi \mu d\sigma}{\norm{\psi}_{L^2(\hgamma)}} \geq \beta \norm{\mu}_{L^2(\hgamma)},
\end{equation}

with $\beta > 0$ independent from $h$.

We first recall Proposition <ref>: for all $\tau \in \Sp^{p-1}_{\textsc{m},\int}$ it holds

\begin{equation}\label{eq:sup1}
\sup_{\psi \in \Sp^p_{0,\partial}} \frac{\int_{\widehat{\Gamma}}\psi' \tau}{\norm{\psi'}_{H^1(\widehat{\Gamma})'}} \geq \widetilde{\beta}\norm{\tau}_{H^1(\widehat{\Gamma})},
\end{equation}

where we have used the bijectivity between $\Sp^p_{0,\partial}$ and $\Sp^{p-1}_{0,\int}$ of the derivative operator, setting $\psi'=\phi$. For the quotient in (<ref>) we have, integrating by parts and using Poposition <ref>

\begin{equation*}
\frac{\int_{\widehat{\Gamma}}\psi' \tau}{\norm{\psi'}_{H^1(\widehat{\Gamma})'}} \leq C \frac{\int_{\widehat{\Gamma}}\psi \tau'}{\norm{\psi}_{L^2(\widehat{\Gamma})}}= \frac{\int_{\widehat{\Gamma}}\psi \mu}{\norm{\psi}_{L^2(\widehat{\Gamma})}},
\end{equation*}

where $\mu=\tau'$ is then an arbitrary function of $\Sp^{p-2}_{\textsc{m}}$, and it holds

\begin{equation*}
\norm{\mu}_{L^2(\widehat{\Gamma})}= \abs*{\tau}_{H^1(\widehat{\Gamma})} \leq \norm{\tau}_{H^1(\widehat{\Gamma})}.
\end{equation*}

Therefore (<ref>) follows.

We need now to transfer the infsup stability from $\widehat{\Gamma}$ to $\gl=\partial\Omega_{m(\ell)}\cap \partial\Omega_{s(\ell)}$. For the sake of simplicity, in the following we omit the index $\ell$. Let $\F_{m} \colon \widehat{\Omega}_{m} \to \Omega_{m}$ and $\F_{s} \colon \widehat{\Omega}_{s} \to \Omega_{s}$ be given such that they agree on the parametric interface $\hgamma := \Set{(0, \hy) | \hy \in \left[0,1\right]}$,
\begin{equation*}
\Gamma = \Set{\F_{m}(0,\hy) = \F_{s}(0,\hy) | \hy \in \left[0,1\right] },
\end{equation*}

where $\widehat{\Omega}_{m} := \left[ -1,0\right]  \times \left[ 0,1\right]$ and $\widehat{\Omega}_{s} := \left[ 0,1\right] \times \left[ 0,1\right]$. Then, we can indicate the restriction of $\F_{m}$ and $\F_{s}$ to $\hgamma$ as

\begin{equation*}
\F_0(\hy) := \F_{m}(0,\hy) = \F_{s}(0,\hy), \quad \forall \hy \in \left[0,1\right].
\end{equation*}

On $\Gamma$ we define the normal vector $\textbf{n}$ such that

\begin{equation*}
\textbf{n} \circ \F_0(\hy) :=  \frac{1}{\norm{\partial_{\hy} \F_0(\hy)}^2} \begin{bmatrix}
\partial_{\hy} \F_0(\hy) \cdot \textbf{e}_2\\
-\partial_{\hy} \F_0(\hy) \cdot \textbf{e}_1
\end{bmatrix}.
\end{equation*}

Let $\varphi = \hphi \circ \F_{s}^{-1}$ with $\hphi \in \Sp^p(\homega_{s})$, then the outer normal derivative from the secondary side $\Omega_{s}$ can be expressed as

\begin{equation*}
\left(\nabla \varphi\cdot \textbf{n}\right)\circ \F_0(\hy) =
\left[
\partial_{\hx} \hphi(0,\hy) \, \partial_{\hy} \hphi(0,\hy)
\right]
\left[
\partial_{\hx} \F_{s}(0,\hy) \, \partial_{\hy} \F_0(\hy)
\right]^{-1}
\cdot \left(\textbf{n} \circ \F_0(\hy)\right),
\end{equation*}

for all $(x,y) = \F_0(\hy) \in \Gamma$. From the Cramer's rule it follows that

\begin{equation*}
\begin{split}
\partial_{\hx} \F_{s}(0,\hy) \, \partial_{\hy} \F_0(\hy) \right]^{-1}  \cdot \left(\textbf{n} \circ \F_0(\hy)\right) \\
\left( \det \left[
\partial_{\hx} \F_{s}(0,\hy) \, \partial_{\hy} \F_0(\hy)
\right] \right) ^{-1}
\begin{bmatrix}
\partial_{\hy} \F_0(\hy) \cdot \textbf{e}_2   & -\partial_{\hy} \F_0(\hy) \cdot \textbf{e}_1\\
-\partial_{\hx} \F_{s}(0,\hy) \cdot \textbf{e}_2 &  \partial_{\hx} \F_{s}(0,\hy) \cdot \textbf{e}_1
\end{bmatrix} \cdot \left(\textbf{n} \circ \F_0(\hy)\right) \\
\left( \det \left[
\partial_{\hx} \F_{s}(0,\hy) \, \partial_{\hy} \F_0(\hy)
\right] \right)^{-1}
\begin{bmatrix}
1 \\ -\dfrac{\partial_{\hx} \F_{s}(0,\hy) \cdot \partial_{\hy} \F_0(\hy)}{\norm{\partial_{\hy} \F_0(\hy)}^2}
\end{bmatrix} ,
\end{split}
\end{equation*}

and hence we obtain

\begin{equation}\label{eq:normalder}
\left(\nabla \varphi\cdot \textbf{n}\right)\circ \F_0(\hy)  = \alpha_s(\hy) \partial_{\hx} \hphi(0,\hy) + \beta_s(\hy) \partial_{\hy} \hphi(0,\hy),
\end{equation}

\begin{equation}\label{eq:coefs}
\begin{split}
\alpha_s(\hy) &:= \det \left[
\partial_{\hx} \F_{s}(0,\hy) \, \partial_{\hy} \F_0(\hy)
\right] ^{-1}\\
\beta_s(\hy)  &:= - \dfrac{\partial_{\hx} \F_{s}(0,\hy) \cdot \partial_{\hy} \F_0(\hy)}{\norm{\partial_{\hy} \F_0(\hy)}^2 \det \left[
\partial_{\hx} \F_s(0,\hy) \, \partial_{\hy} \F_0(\hy)
\right] }.
\end{split}
\end{equation}

We can now state the discrete inf-sup stability for the mapped (isogeometric) pairing $\Sp^p_{0,\partial}$/$\Sp^{p-2}_{\textsc{m}}$ on the physical domain.

For $h$ sufficiently small, for each $\mu \in M_{\ell}$ it holds

\begin{equation*}
\sup_{\psi \in W_{\ell}} \dfrac{\int_{\gl} \psi \mu d\sigma}{\norm{\psi}_{L^2(\gl)}} \geq \beta \norm{\mu}_{L^2(\gl)},
\end{equation*}

with $\beta > 0$ independent of $h$.

Let $\mu \in M_\ell$ be such that $\mu = \widehat{\mu} \circ \F_{0}^{-1}$, with $\widehat{\mu} \in \Sp^{p-2}_{\textsc{m}}(\hgamma)$. For (<ref>) there exists $\varphi \in \widetilde{X}_{s(\ell),h}$ such that $\psi = \dn \varphi|_{\gl}$, where

\begin{equation*}
\widetilde{X}_{s(\ell),h} := \Set{ \varphi \in X_{s(\ell),h}\cap H^1_0(\Omega_{s(\ell)}) | \dn \varphi|_{\partial \Omega_{s(\ell)} \setminus \gl} = 0}.
\end{equation*}

For (<ref>) there exists $\varphi \in X_{s(\ell),h}$ be such that $\varphi = \widehat{\varphi} \circ \F_{s(\ell)}^{-1}$, with $\widehat{\varphi} \in \widehat{X}_{h} := \Sp^{p}(\homega) \cap H^2_*(\homega)$. Thanks to (<ref>) we have

\begin{equation*}
\begin{split}
\sup_{\psi \in W_\ell} \dfrac{\int_{\gl} \psi \, \mu \, d\sigma}{\norm{\psi}_{L^2(\gl)}} &= \sup_{\varphi \in \widetilde{X}_{s(\ell),h}} \dfrac{\int_{\gl} \dn \varphi \, \mu \, d\sigma}{\norm{\dn \varphi}_{L^2(\gl)}} \\
&\geq \norm{\partial_{\widehat{y}} \F_{0}}_{L^{\infty}(\hgamma)}^{-\nicefrac{1}{2}} \sup_{\widehat{\varphi} \in \widehat{X}_{h}} \dfrac{\int_{\hgamma} \left( \alpha_{s(\ell)} \partial_{\hx} \hphi + \beta_{s(\ell)} \partial_{\hy} \hphi \right) \widehat{\mu} \, \widehat{\rho} \, d\hy}{\norm{\alpha_{s(\ell)} \partial_{\hx} \hphi + \beta_{s(\ell)} \partial_{\hy} \hphi}_{L^2(\hgamma)}},
\end{split}
\end{equation*}

where $\widehat{\rho} = \norm{\partial_{\widehat{y}} \F_{0}}$ and $\alpha_{s(\ell)}, \beta_{s(\ell)}$ are as in (<ref>). Now, if we compute the supremum on the subspace

\begin{equation*}
\widehat{Y} := \Set{\hphi \in \widehat{X}_{h}\cap H^1_0(\homega) | \partial_{\widehat{n}} \hphi|_{\partial \homega \setminus \hgamma} = 0},
\end{equation*}

then $\hpsi := \partial_{\hx} \hphi \in \Sp^p_{0,\partial}(\hgamma)$ for all $\hphi \in \widehat{Y}$, and hence

\begin{equation*}
\begin{split}
\sup_{\psi \in W_\ell} \dfrac{\int_{\gl} \psi \mu d\sigma}{\norm{\psi}_{L^2(\gl)}} &\geq C \sup_{\widehat{\varphi} \in \widehat{Y}} \dfrac{\int_{\hgamma} \alpha_{s(\ell)} \partial_{\hx} \hphi \, \widehat{\mu} \, \widehat{\rho} \, d\hy}{\norm{\alpha_{s(\ell)} \partial_{\hx} \hphi}_{L^2(\hgamma)}} \\
&\geq C \sup_{\hpsi \in \Sp^p_{0,\partial}(\hgamma)} \dfrac{\int_{\hgamma} \hpsi \, \widehat{\mu} \, d\hy}{\norm{\hpsi}_{L^2(\hgamma)}},
\end{split}
\end{equation*}

where we have used that $\alpha_{s(\ell)}$ and $\widehat{\rho}$ are uniformly bounded from above and bounded away from $0$, due to Assumption <ref>. Moreover, it holds the norm equivalence

\begin{equation}\label{eq:normequivalence}
C^{-1} \norm{\widehat{\mu}}_{L^2(\hgamma)} \leq C \norm{\alpha_{s(\ell)}\widehat{\rho} \widehat{\mu}}_{L^2(\hgamma)} \leq \norm{\widehat{\mu}}_{L^2(\hgamma)}.
\end{equation}

Let $\widehat{\pi}^{*} \colon L^2(\hgamma) \to \Sp^{p-2}_{\textsc{m}}(\hgamma)$ be a local projection with optimal approximation properties.
The integration weight $\rho$ is smooth except at the mesh lines, where is only $C^{r-1}$ and then the following super-approximation holds

\begin{equation}\label{eq:superapp}
\norm{\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho} - \widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho})}_{L^2(\hgamma)} \leq C h \norm{\widehat{\mu}}_{L^2(\hgamma)},
\end{equation}

see <cit.>. Thanks to Theorem <ref>, we can choose $\widehat{\psi}^* \in \Sp^{p-1}_0(\hgamma)$ such that

\begin{equation*}
\dfrac{\int_{\hgamma} \widehat{\psi}^* \widehat{\pi}^{*} (\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho}) d\widehat{y}}{\norm{\widehat{\psi}^*}_{L^2(\hgamma)}} \geq C \norm{\widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho})}_{L^2(\hgamma)}.
\end{equation*}

Therefore, using the bounds above,

\begin{equation*}
\begin{split}
\sup_{\psi \in W_\ell} \dfrac{\int_{\gl} \psi \mu d\sigma}{\norm{\psi}_{L^2(\gl)}} &\geq C \dfrac{\int_{\hgamma} \widehat{\psi}^* \, \widehat{\mu} \, \alpha_{s(\ell)}\widehat{\rho} \, d\hy}{\norm{\hpsi}_{L^2(\hgamma)}} \\
&= C \dfrac{\int_{\hgamma} \hpsi^* \widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho}) d\hy}{\norm{\widehat{\psi}^*}_{L^2(\hgamma)}} + C \dfrac{\int_{\hgamma} \hpsi^* (\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho} - \widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho})) d\hy}{\norm{\widehat{\psi}^*}_{L^2(\hgamma)}}\\
&\geq C \norm{\widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho})}_{L^2(\hgamma)} - C \norm{\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho} - \widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho})}_{L^2(\hgamma)}\\
&\geq C \norm{\widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho})}_{L^2(\hgamma)} - C' h \norm{\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho}}_{L^2(\hgamma)}.
\end{split}
\end{equation*}

Now, we use (<ref>) and (<ref>) to bound $\norm{\widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho})}_{L^2(\hgamma)}$:

\begin{equation*}
\norm{\widehat{\mu}}_{L^2(\hgamma)} \leq \norm{\widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho})}_{L^2(\hgamma)} + \norm{\widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho}) - \widehat{\mu} \alpha_{s(\ell)}\widehat{\rho}}_{L^2(\hgamma)} \leq \norm{\widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho})}_{L^2(\hgamma)} + C'' h \norm{\widehat{\mu}}_{L^2(\hgamma)},
\end{equation*}

which shows that $\norm{\widehat{\pi}^{*}(\widehat{\mu} \alpha_{s(\ell)}\widehat{\rho})}_{L^2(\hgamma)} \geq C \norm{\widehat{\mu}}_{L^2(\hgamma)}$ for sufficiently small $h$.
The thesis follows.

For $h$ sufficiently small, for each $\mu \in M_{h}$ it holds

\begin{equation*}
\sup_{\psi \in W_{h}} \dfrac{\int_{\Sigma} \psi \mu d\sigma}{\norm{\psi}_{L^2(\Sigma)}} \geq \beta \norm{\mu}_{L^2(\Sigma)},
\end{equation*}

with $\beta > 0$ independent of $h$.

§ NUMERICAL RESULTS

In this section, we present numerical tests with different multipliers spaces.  All the tests have been obtained with the GeoPDEs software suite for Matlab, see <cit.>. We consider several different geometries, depicted in Figures <ref>: 12-patches square (fig:Multisquare_mp), 3-patches quarter of a circle (fig:Quartercircle_mp) and 10-patches airfoil (fig:Airfoil_mp). For every simulation, the boundary conditions and the source term are set such that the exact solution is $u_{\mathrm{ex}}(x,y)=\cos(x) \cos(y)$.

Multi-patch square.

Physical domains.

For the sake of simplicity, throughout the previous sections we have assumed that the physical domain is described by a spline geometrical mapping, see Assumption <ref>. This hypothesis is not used in any crucial steps of the theory. Numerical experiments (see e.g. the tests on the quartercircle domain) show that the stability is preserved in the more general case of NURBS geometries.

§.§ Tests with $C^2$-continuity at the vertices

Numerical resultsTests with $C^2$-continuity at the vertices
We start with a primal space $X_h$ defined as in equations (<ref>) and (<ref>). This means that $X_h$ is generated from spline spaces of degree $p$ and strong $C^2$-continuity enforced at the corners.
The dual space on the interfaces is generated from spline spaces of degree $p-2$ and merged elements in the neighbourhood of the corners, i.e. $M_h := \prod_{\ell \in \ig} M_\ell$ with

\begin{equation*}
M_\ell = \Set{ \mu \circ F_\ell^{-1} | \mu \in \Sp^{p-2}_{\textsc{m}} },
\end{equation*}

and $\Sp^{p-2}_{\textsc{m}}$ as in equation (<ref>).

From a theoretical point of view, this coupling fulfills the discrete inf-sup stability, as proved in Section <ref>.
Figures <ref> and <ref> show the convergence plots in patch-wise $H^2$ broken norms, denoted by $\mathcal{H}^2$, $H^1$, $L^2$ and $L^{\infty}$ norms, for primal spaces of degrees $p = 2, \dots, 5$. For the case $p = 2$ the rates of convergence are suboptimal in $L^2$  and this is expected and occurs in the single patch case too (refercences?). Note that the Aubin-Nitsche technique can not be used when $p=2$ (see <cit.>).

In all the other examples the computed rates of convergence are optimal, as stated in Corollary <ref>.

§.§ Tests without $C^2$-continuity at the vertices

Numerical resultsTests without $C^2$-continuity at the vertices
Additionally, we have tested the pairing $\Sp^p/\Sp^{p-2}_{\textsc{m}}$ for the bilaplacian problem on the quartercircle domain, without imposing strong $C^2$-continuity at the corner. As before, Figure <ref> shows the convergence plots in $\mathcal{H}^2$, $H^1$, $L^2$ and $L^{\infty}$ norms for primal spaces of degrees $p = 2, \dots, 5$. Numerical tests reveal that the pairing $\Sp^p/\Sp^{p-2}_{\textsc{m}}$ retains optimal rate of convergence also in this $C^2$-unconstrained case. Figure <ref> shows that the difference between the numerical solutions obtained with or without $C^2$ conditions is located in a neighbourhood of the corner.

Difference between exact and numerical solutions for $h \approx 0.155$ on the quartercircle domain  with: $C^2$ continuity at the vertices and $\Sp^3/\Sp^{1}_{\textsc{m}}$ pairing (fig:Quartercircle_Diff_uex_uc2), $C^0$ continuity at the vertices and $\Sp^3/\Sp^{1}_{\textsc{m}}$ pairing (fig:Quartercircle_Diff_uex_uc0) and $C^0$ continuity at the vertices and $\Sp^3/\Sp^{1}$ pairing (fig:Quartercircle_Diff_uex_uc0_nme).

§.§ Other tests

Numerical resultsOther tests
In the context of numerical experiments without imposing strong $C^2$-continuity at the corners, we have tested the bilaplacian problem for the pairing $\Sp^p/\Sp^{p-2}$, without using merged elements. Although these tests are beyond the theory, the numerical orders of convergence are optimal, see Figures <ref>. This confirms that Isogeometric mortar method is a flexible technique for the numerical approximation of PDEs that guarantee globally $C^1$-continuous solutions on multi-patch geometries.

§ CONCLUSIONS

In this work, we propose an isogeometric method for solving the biharmonic equation on $C^0$-conforming multi-patch domains. The required $C^1$-continuity across patch interfaces is enforced weakly through a mortar formulation. Considering a spline space of degree $p$ as the primal space and choosing, as Lagrange multipliers space, a spline space of degree $p-2$ built on the same underlying mesh of the primal space, but with merged elements in the neighbourhood of patch vertices, the method satisfies a discrete inf-sup stability condition. Consequently, we have established well-posedness for the method and proved an optimal order a priori error estimates in a patch-wise $H^2$ broken norm.
To conclude, numerical examples perfectly reflect theoretical a priori error estimates and also show optimal order convergence in $L^2$, $H^1$ and $L^{\infty}$ norms.

§ ACKNOWLEDGEMENTS

G. Loli and G. Sangalli were partially supported by the European Research Council through the FP7 Ideas Consolidator Grant HIGEOM n.616563, and by the Italian Ministry of Education, University and Research (MIUR) through the "Dipartimenti di Eccellenza Program (2018-2022) - Dept. of Mathematics, University of Pavia". They are also members of the Gruppo Nazionale Calcolo Scientifico - Istituto Nazionale di Alta Matematica (GNCS-INDAM). These supports are gratefully acknowledged.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

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title$n$-widths, sup-infs, and optimality ratios for the
$k$-version of the isogeometic finite element method
journaltitleComputer Methods in Applied Mechanics and Engineering

titleApproximation in FEM, DG and IGA: a theoretical
journaltitleNumerische Mathematik

pages117 133
titleMatrix-free weighted quadrature for a computationally
efficient isogeometric $k$-method
journaltitleComputer Methods in Applied Mechanics and Engineering

titleIsogeometric analysis for sixth-order boundary value problems
of gradient-elastic Kirchhoff plates
journaltitleComputer Methods in Applied Mechanics and Engineering

automatic meshing, Bézier triangle, feature-preserving, isogeometric
analysis, Kirchhoff plate, Lagrange multiplier
titleA geometrically exact isogeometric Kirchhoff plate:
Feature-preserving automatic meshing and C1 rational triangular Bézier
spline discretizations
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titleAn isogeometric method for the Reissner-Mindlin plate bending
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journaltitleComput. Methods Appl. Mech. Eng.

titleIsogeometric analysis of the Cahn–Hilliard phase-field
journaltitleComputer methods in applied mechanics and engineering

titleIsogeometric analysis of the isothermal
Navier-Stokes-Korteweg equations
volume199 (25)
journaltitleComput. Methods Appl. Mech. Eng.

titleAnalysis-suitable $G^1$ multi-patch parametrizations for
$C^1$ isogeometric spaces
journaltitleComputer Aided Geometric Design

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journaltitleAsymptotic and Numerical Methods for Partial
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titleA new nonconforming approach to domain decomposition: the
mortar element method
journaltitleNonlinear Partial Differrential Equations and their

titleBasics and some applications of the mortar element method
journaltitleGesellschaft für Angewandte Mathematik und

domain decomposition methods, isogeometric analysis, nonconforming
NURBS multi-patches, Nitsche-type formulation
titleA Nitsche-type formulation and comparison of the most common
domain decomposition methods in isogeometric analysis
journaltitleInternational Journal for Numerical Methods in

titleIsogeometric analysis and domain decomposition methods
journaltitleComputer Methods in Applied Mechanics and Engineering

nonlinear isogeometric analysis, NURBS, domain decomposition, weak
substitution method, multi-patch connection, mortar methods
titleThe weak substitution method – an application of the mortar
method for patch coupling in NURBS-based isogeometric analysis
journaltitleInternational Journal for Numerical Methods in

titleA projected super-penalty method for the $C^1$-coupling of
multi-patch isogeometric Kirchhoff plates
journaltitleComputational Mechanics

titleIsogeometric mortar methods
journaltitleComputer Methods in Applied Mechanics and Engineering

titleIsogeometric Analysis: Approximation, stability and error
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Cambridge University Press

titleSpline Functions: Basic Theory. Third Edition

seriesMono. Univ. Math.
titleEquations aux dérivées partielles de type elliptique

titleMixed Finite Element Methods and Applications

Cambridge University Press

titleMathematical analysis of variational isogeometric methods
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Academic Press

titleSobolev Spaces

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North-Holland Publishing Company

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volume42 (12)
journaltitleAdvances in Engineering Software

seriesPrentice-Hall Series in Automatic Computation
titleAn Analysis of the Finite Element Method

[font=, trim axis left]
cycle list name=Lista1,

legend columns=1,
legend pos=south west,
legend style=at=(0.01,0.01),anchor=south west,
legend entries=$\beta$,$O(1/p)$,$O(1/\sqrt{p})$]
table [x=p, y=Beta, col sep=comma] ;
table [x=p, y=pInv, col sep=comma] ;
table [x=p, y=pInvSqrt, col sep=comma] ;

Discrete inf-sup constant on $[0,1]$ with $n_{\mathrm{sub}}=32$ on uniform mesh.