\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 82, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/82\hfil Handling geometric singularities]
{Handling geometric singularities by the mortar spectral element method
for fourth-order problems}

\author[M. Abdelwahed, N. Chorfi, V. R\u{a}dulescu \hfil EJDE-2017/82\hfilneg]
{Mohamed Abdelwahed, Nejmeddine Chorfi, Vicen\c{t}iu D. R\u{a}dulescu}

\address{Mohamed Abdelwahed \newline
Department of Mathematics,
College of Sciences,
King Saud University, Riyadh, Saudi Arabia}
\email{mabdelwahed@ksu.edu.sa}

\address{Nejmeddine Chorfi \newline
Department of Mathematics,
College of Sciences,
King Saud University, Riyadh, Saudi Arabia}
\email{nchorfi@ksu.edu.sa}

\address{Vicen\c{t}iu  R\u{a}dulescu \newline
Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,
P.O. Box 1-764, 014700 Bucharest, Romania. \newline
Department of Mathematics, University of Craiova,
200585 Craiova, Romania}
\email{vicentiu.radulescu@imar.ro}

\dedicatory{Communicated by Giovanni Molica Bisci}

\thanks{Submitted January 7, 2017. Published March 24, 2017.}
\subjclass[2010]{78M22, 35J15}
\keywords{Biharmonic problem; mortar method; spectral discretizatio;
 \hfill\break\indent Strang and Fix algorithm}

\begin{abstract}
 This article concerns the numerical analysis and the error estimate
 of the biharmonic problem with homogeneous boundary conditions
 using the mortar spectral element method in domains with corners.
 Since the solution of this problem can be written as a sum of a regular
 part and known singular functions, we propose to use the Strang and
 Fix algorithm for improving the order of the error.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

It is well known that the solutions of an elliptic equations in polygonal
domains are not very regular despite the regularity of the second member and
boundary data \cite{G,G1,K}. More precisely, the solution of an elliptic problem
in such domains is the sum of a regular part and another one which is presented
as a linear combination of functions which the regularity gets lower as the angle
of singularity gets greater. This singular part of the solution  pollutes the
error estimate. Different numerical methods for the most part related to
finite element method have been developed to calculate the singular part of
the solution or to improve the error estimate \cite{B,BR,BS};
this is the case of the mesh refinement method near the singular angle corners.
Among these methods the Strang and Fix algorithm \cite{SF} which was extended
to the mortar method for a spectral discretization \cite{ABM,C3}.

The high precision of the spectral methods makes them well adapted to the
treatment of the singularities. In fact, the numerical analysis using this
 method in the Laplacian case \cite{ABM,C1} confirms this expectation of
sufficient precision. Furthermore, the study of the singular function
approximation by polynomials near the singular corners shows that the convergence
is better than what the general approximation theory lets to believe and
explains the appearance of super convergence \cite{BM1}.  Calculations have
also been made for the stokes system \cite{C4}.

The Strang and Fix algorithm consists on the enlargement of the test function
space and the resolution of the discrete problem in this space.
This algorithm permits us the computing of the singular coefficient which
is usually issued from the physics (case of the elastic crack) \cite{AM}.

In this work we propose to study this algorithm for the homogeneous biharmonic
problem. For that we place ourselves within the framework of the Mortar element
method with spectral discretization \cite{BM3,BMP}.  The analysis and the
implementation of the mortar element method has been done in the work of
Belhachmi et al. \cite{Z1,Z2,Z3} for a problem of order 4.
We present in this work an extension in the case of the non regular domains
in order to improve the estimation of the order of the error.


An outline of this article is as follows.
In section 2, we present the geometry aspects of the domain.
In section 3 we present the continuous problem, then we give the singular
functions and some regularity results.
In section 4, we define the discrete problem. Section 5 is devoted to the
numerical analysis and the error estimation of the mortar spectral element
 method of the Strang and Fix algorithm for the harmonic problem.

\section{Geometric aspects}

Let $\Omega$ an open polygonal, bounded, Lipschitzian and connected domain of
$\mathbb{R}^2$, decomposed on $K$ rectangles $\Omega^k$, $1 \le k \le K$ such that
$$
\overline{\Omega}=\cup_{k=1}^K \overline{\Omega}^k \quad\text{and}\quad
 \Omega^k\cup\Omega^l=\emptyset \,, \; 1 \le k \neq l \le K.
$$
We denote by $\overline{\Gamma}^{k,j}$, $1 \le j \le 4$ the sides of the
sub-domain $\overline{\Omega}^k$,   $1 \le k \le K$ and
$$
\overline{\gamma}_{kl}=\overline{\Omega}^k\cap\overline{\Omega}^l, \quad
 1 \le k \neq l \le K
$$
the interface of the decomposition.

 We define the skeleton of the decomposition
$$
\mathcal S = {\cup_{k=1}^K \cup_{j=1}^4 \overline{\Gamma}^{k,j}}.
$$
We associate to each decomposition the set of vertices of the sub-domain,
denoted by $\mathcal V$.

We choose $\mathcal M$ a set of integers $m$ such that the open segment
$\Gamma^{k(m),j(m)}$ are two by two disjoints and
$$
\mathcal S = \cup_{m\in \mathcal M} \overline{\Gamma}^{k(m),j(m)}.
$$
The sides $\Gamma^{k(m),j(m)}, \; m\in \mathcal M$ is called mortars and
denoted by $\gamma_m$.
 We suppose that the intersection of a sub-domain $\Omega^k$ with the
boundary $\partial \Omega$ can be reduced to a vertex (see Figure \ref{fig}).


\begin{figure}[ht]
 \includegraphics[angle=270,width=0.8\textwidth]{fig1}
\caption{Domain $\Omega$}\label{fig}
\end{figure}


The angles of the singular vertices are $\pi/2$,
$3\pi/2$ or $2 \pi$.
Thereafter we will be interested specially to the case $3\pi/2$
because of its applications in fluid mechanic (step case in Stokes flow) and to
 the case of ${2\pi}$ for its applications in mechanics (crack propagation).
The local influence of the singularity allows to limit the study to one vertex.
We denote $\mathbf{a}$ this vertex and $\omega$ the associated angle.
To simplify the problem analysis, the sides of the sub-domains are supposed to be
parallel to the axis of the scale of origin $\mathbf{a}$.
We introduce the polar coordinates $(r,\theta)$ with $r$ the distance from a
point to the vertex $\mathbf{a}$ and the line $\theta=0$ contains a side
of $\partial \Omega$.

Also we consider the following conformity assumption.

\begin{assumption}\label{as1} \rm
We denote $\Delta$ the union of sub-domains containing the vertex $\mathbf{a}$.
We suppose that the decomposition of the domain $\Delta$ is conforming
(see Figure \ref{fig}): If $\mathbf{a}$ is a vertex of the mortar
$\Gamma^{k(m),j(m)}$ which coincides with $\Gamma^l$ a side of a
sub-domain $\Omega^l$, $l \ne k(m)$ then $N_{k(m)}\le N_l$, such that the
restriction of a function to $\Delta$ is in $H^2(\Delta)$.
\end{assumption}

\section{Continuous problem and singular functions}

Consider the homogeneous biharmonic problem
\begin{equation}\label{1}
\begin{gathered}
\Delta^2u=f \quad \text{in } \Omega,\\
u=0 \quad\text{on } \partial\Omega,\\
\frac{\partial u}{\partial n}=0 \quad\text{on } \partial\Omega.
\end{gathered}
\end{equation}
For $f\in H^{-2}(\Omega)$ the problem \eqref{1} is equivalent to
the following variational formulation:
Find $u\in H_0^{2}(\Omega)$, such that for all $v\in H_0^{2}(\Omega)$,
\begin{equation}\label{2}
a(u,v)=\langle f,v\rangle,
\end{equation}
where $a(u,v)=\int_{\Omega}\Delta u : \Delta v \,dx$ and $\langle\cdot,\cdot\rangle$
 is the duality mapping between $H^{-2}(\Omega)$ and $H_0^{2}(\Omega)$.
Since the bilinear form $a(\cdot,\cdot)$ is continuous in
$H_0^{2}(\Omega)\times H_0^{2}(\Omega)$ and coercive in $H_0^{2}(\Omega)$,
we conclude using the Lax-Milgram theorem that for $f \in H^{-2}(\Omega)$
the problem \eqref{2} has a unique solution $u \in H_0^{2}(\Omega)$ such that
$$
\|u\|_{H^{2}(\Omega)} \le C \|f\|_{H^{-2}(\Omega)},
$$
where $C$ is a constant independent of $\Omega$.

Let $V$ a neighborhood of the singular point $\mathbf{a}$ included in
the domain $\overline{\Delta}$, let $s\ge 1$ and $f \in H^{s-2}(\Omega)$
then we know that the solution of problem \eqref{1} is written as \cite{G,G1}
\begin{equation}\label{dc}
u=u_R+u_{S},
\end{equation}
where $u_R \in H^{s+2}(\Omega)\cap H_0^{2}(\Omega)$ and $u_S$ is given by
\begin{equation}
u_S(r,\theta)=\sum_{0< \operatorname{Real}(z_k)<s+2}\lambda_k r^{1+z_k}\varphi_k(\theta)
+\sum_{0< \operatorname{Real}(\hat{z}_k)<s+2}\hat{\lambda}_k r^{1+\hat{z}_k}[\sigma_k(\theta)
+\ln(r)\eta_k(\theta)]
\end{equation}
with $\lambda_k$ and $\hat {\lambda}_k$ are real numbers, $\varphi_k$,
$\sigma_k$, $\eta_k$ are functions defined on a finite dimension sub-space of
$C^{\infty}([0,\omega])\cap H^2([0,\omega])$ (see \cite{G1} for the explicit
 expression of these functions) and $z_k$ (respectively  $\hat z_k$) are the
simple (respectively double) roots of the characteristic equation of
the bilaplacian
\begin{equation}\label{4}
\sin(\omega z)^2=z^2 \sin(\omega ^2)
\end{equation}
in the band,  $0 < \operatorname{Real}(z)<s+2$, except $1$ if $\omega \ne \tan(\omega)$,
without exception if $\omega =\tan(\omega)$ which has the unique solution
$\omega_e=1.430397 \pi$  in $]0,2\pi[$.

The study of  equation \eqref{4} shows that $z$ is a double root if and only
if $z=0$ or $z=\pm \sqrt{\frac{1}{\sin \omega^2}-\frac{1}{\omega^2}}$.
This is given by the following necessary and sufficient condition \cite{G1}
\begin{equation}\label{5}
\sin\big(\frac{\omega^2}{\sin \omega ^2}-1\big)
=\pm \sqrt{1-\frac{\sin \omega^2}{\omega^2}}.
\end{equation}
For handling the singularities we define
$$
\eta(\omega)=\inf\{ \operatorname{Real} (z),\; z\hbox{ is a solution of }\eqref{4},
\, z\neq \pm 1\}.
$$
In the case of $\omega=3\pi/2$ we have $\eta(\omega)=0.54484$ and
$s < 1.544$. We decompose  $u=u_R+\lambda S$, such that
$ u_R\in H^{s+2}(\Omega)$ and
$$
\|u_R\|_{H^{s+2}(\Omega)}+|\lambda|\le C \|f\|_{H^{s-2}(\Omega)},
$$
where
\begin{itemize}
\item
\begin{equation}\label{6}
S(r,\theta)=r^{1+\eta(\omega)} \varphi(\theta)
\end{equation}
with $\varphi(\theta)=2.093\Big(\cos(0.459 \,\theta)-\cos(1.544 \,
\theta)\Big)+1.093\Big(2.193\sin(0.459 \, \theta)-\sin(1.544\, \theta)\Big)$,

\item $\lambda$ is the first singular coefficient of the singularity $S$.
\end{itemize}
Furthermore, if $f \in H^{s-2}(\Omega)$, with $s<2.908$, we can again decompose
the singular part as follows
\begin{equation}\label{p5}
u={\tilde u}_R+\lambda S + \tilde \lambda \tilde S,
\end{equation}
where
\begin{itemize}
\item ${\tilde u}_R \in H^{s+2}(\Omega)$,
\item
\begin{equation}\label{7}
\tilde S(r,\theta)=r^{1+z_2} \psi(\theta),
\end{equation}
with $z_2$ is the second solution of equation \eqref{4} in the band
$0 < \operatorname{Real}(z) < 1$  $(z_2\simeq 0.908529)$ and
$\psi(\theta)=4.302\big(\cos(0.092 \theta)-\cos(1.908  \theta)\big) \\ -1.815
\big(10.869\sin(0.092 \theta)-0.524\sin(1.908 \theta)\big)$,

\item $\tilde \lambda$ is the coefficient of the second singularity $\tilde S$
satisfying
$$
\|{\tilde u}_R\|_{H^{s+2}(\Omega)}+|\lambda|+|\tilde \lambda|
\le C \|f\|_{H^{s-2}(\Omega)}.
$$
\end{itemize}
 when $\omega=2\pi$, we have $\eta(\omega)=0.5$ and $s <1.5$.
If ${f}$ belongs to $H^{s-2}(\Omega)$, then $u$ belongs to the space
$H^{s+2}(\Omega)$. We decompose
$$
u= u_R+\lambda S+\tilde\lambda \tilde S,
$$
where
\begin{gather*}
S(r,\theta)=r^{3/2}\big((\sin(3\theta/2)-3\sin(\theta/2))
+ (\cos(3\theta/2)-\cos(\theta/2))\big), \\
\tilde S(r,\theta)=r^{5/2}\big((\cos(5\theta/2)
-5\sin(\theta/2)) + (\cos(5\theta/2)-\cos(\theta/2))\big),
\end{gather*}
and $(\lambda, \tilde\lambda)$ is the singular coefficient associated
to the singular function
$(S,\tilde S)$.
If ${ f}$ belongs to $H^{s-2}(\Omega)$, $u_R$ belongs to $H^{s+2}(\Omega)$
for $s<2,5$. We have the following stability condition:
$$
\| { u}_R \|_{H^{s+2}(\Omega)} +| \lambda |+| \tilde \lambda |
\leq C \| { f} \|_{H^{s-2}(\Omega)}.
$$

\section{Discrete problem}

Firstly, we recall the space of mortar functions. As the considered problem
is posed in $H^2(\Omega)$, two matching conditions are necessary on each
interface; one for the trace of the function and the other for its normal
derivative.

We introduce $\delta=(N_k)_{1\le k \le K}$, a strictly positive sequence
of integers. $\delta$ is called parameter of discretization and
$(N_k)$, ${1\le k \le K}$, are the degrees of  polynomials in each sub-domain.
($\mathbb{P} _n(\Omega)$ is the space of polynomial functions of degree less than
or equal to $n$).

The mortar method requires the introduction of a space of functions,
which we call mortar functions. These are defined on the skeleton and ensure
the matching of the locally approximation functions.
The space of mortar functions is then defined by
$$
W^\delta=\Big\{(\varphi_0,\varphi_1); \varphi_0/_{\gamma^m}
=v_{\delta}/_{\Gamma^{k(m),j(m)}} \text{ and }
 \varphi_1/_{\gamma^m}=(\frac{\partial v_{\delta}}{\partial n})/_{\Gamma^{k(m),j(m)}}
\; \forall m \in \mathcal M\Big\}.
$$
where $v_{\delta}$ is a test function.

We propose a discretization by the Galerkin method with numerical integration.
In the case of the problem of order four, it is more appropriate to use
a quadrature formula which takes into account the values of the function
on the boundary. The following lemma defines this quadrature formula
(see \cite{BM3} for a proof).

\begin{lemma} \label{lem4.1}
Let $N\ge 2$ be an integer. Then there exists a unique set of points $\xi_j$,
$1\le j \le N-1$, a unique set of positive reals $\rho_j$, $1\le j \le N-1$,
$\rho_+$, $\rho_-$ such that for all polynomials $\varphi$ in
$\mathbb{P}_{2N-1}(]-1,1[)$
\begin{equation}\label{lem41}
\int_{-1}^1 \varphi(x)\,dx
= \sum_{j=1}^{N-1}\varphi(\xi_j)\rho_j+\varphi(-1)\rho_- + \varphi(1)\rho_+\,.
\end{equation}
\end{lemma}

\begin{remark} \label{rmk4.1} \rm
The nodes $\xi_j$; $1\le j \le N-1$, are the zeros of the derivative of the
Legendre polynomial $L_N$. We refer to \cite{BM3} for the calculation of
$\xi_j$ and $\rho_j$,  $1\le j \le N-1$.
\end{remark}

Given two functions $u$, $v$ continuous on $\overline \Omega =[-1,1]\times[-1,1]$
and vanishing on its boundary, we define the discrete scalar product
$$
(u,v)_N=\sum_{i=1}^{N-1}\sum_{j=1}^{N-1}u(\xi_i,\xi_j) v(\xi_i,\xi_j) \rho_i \rho_j.
$$
If $T^k$ is the bijection from $]-1,1[^2$ in $\Omega_k$, we define
$$
(u,v)_{N_k}=\frac{|\Omega_k|}{4}\sum_{i=1}^{N_k-1}\sum_{j=1}^{N_k-1}
(u\circ T^k)(\xi_i,\xi_j) (v\circ T^k)(\xi_i,\xi_j) \rho_i \rho_j.
$$
Thus, we define the space of approximation $X_{\delta}$ as the space of
functions $v_{\delta}$ such that
\begin{itemize}
\item for all $k$, $1\le k \le K$, $v_{\delta}/_{\Omega^k}\in \mathbb{P}_{N_k}
(\Omega^k)$,

\item $v_{\delta}$ and $\frac{\partial v_{\delta}}{\partial n}$ vanishes on
$\partial \Omega$,

\item there exist a couple $(\varphi_0,\varphi_1)\in W_{\delta}$ such that, for all
$1\le k \le K$, $1 \le j \le 4$, and all $\psi \in \mathbb{P}_{N_k-4(\Gamma^{k,j})}$,
$\int_{\Gamma^{k,j}}(v_{\delta}-\varphi_0)(\tau) \psi(\tau) \,d \tau=0$ and
 $\int_{\Gamma^{k,j}}(\frac{\partial v_{\delta}}{\partial n}-\varphi_1)(\tau)
\psi(\tau) \,d \tau=0$.
\end{itemize}
Finally the discrete problem is written:
For $f\in \mathcal C(\overline\Omega)$, find $u_\delta \in X_{\delta}$ such that
for all $v_{\delta} \in X_{\delta}$, 
\[
a_{\delta}(u_{\delta},v_{\delta})=(f,v)_{\delta},
\]
where $a_{\delta}(u_{\delta},v_{\delta})
=\sum_{k=1}^{K}(\Delta u_{\delta}^k,\Delta v_{\delta}^k)_{N_k}$ and
$(f,v_{\delta})_{\delta}=\sum_{k=1}^{K}(f,v_{\delta}^k)_{N_k}$.
We refer to \cite{Z2} for the a priori analysis of this problem and
its implementation by the mortar spectral element method.

\section{Strang and Fix algorithm}

The Strang and Fix algorithm \cite{SF} consists of the enlargement of the
discrete space $X_{\delta}$ as
$$
X_{\delta}^*=X_{\delta}+\mathbb{R}  S,
$$
where $S$ is the first singular function.
We have, then $u_{\delta}^*=u_{\delta}+\lambda S$ and
$v_{\delta}^*=v_{\delta}+\mu S$  in $X_{\delta}^*$,
\begin{align*}
a_{\delta}^*(u_{\delta}^*,v_{\delta}^*)
&=\sum_{k=1}^{K}\Big[(\Delta u_{\delta}^k,\Delta v_{\delta}^k)_{N_k}
+\lambda \int_{\Omega_k}\Delta v_{\delta}^k \Delta S \,dx \\
&\quad +\mu  \int_{\Omega_k}\Delta u_{\delta}^k \Delta S \,dx
+ \lambda \mu \int_{\Omega_k} (\Delta S)^2 \,dx\Big].
\end{align*}
The discrete problem becomes: Find $u_{\delta}^*\in X_{\delta}^*$ such that
\begin{equation}\label{4.1}
\forall v_{\delta}^* \in X_{\delta}^*\, ,\quad a_{\delta}(u_{\delta}^*,
v_{\delta}^*)=\sum_{k=1}^{K}\int_{\Omega_k} f v_{\delta k}^*\,dx,
\end{equation}
where $v_{\delta k}^*$ is the restriction of $v_{\delta}^*$ to sub-domain
$\Omega_k$.

For the analysis of this problem, we introduce the following two norms on
$X_{\delta}^*$,
$$
\|u_{\delta}^*\|_{*1}
=\sum_{k=1}^K \Big(\|u_{\delta}^k \|_{H^2(\Omega_k)}^2
 +|\lambda|^2 \| S/_{\Omega_k}\|_{H^2(\Omega_k)}^2 \Big)^{1/2}
$$
and
$$
\|u_{\delta}^*\|_{*2}
=\Big(\sum_{k=1}^K \|u_{\delta}^* \|_{H^2(\Omega_k)}^2\Big)^{1/2}.
$$

\begin{proposition} \label{prop5.1}
Let $\eta(\delta)=\sup_{u_{\delta}^*\in X_{\delta}^*}
\frac{\|u_{\delta}^*\|_{*2}}{\|u_{\delta}^*\|_{*1}}$.
Then $\lim_{\delta \to +\infty} \eta(\delta)=0$.
\end{proposition}

\begin{proof}
Because of the conformity of the decomposition, we consider the proof in
the domain $\Delta$. Let $N_{\Delta}=\min_{\Omega_k\subset \Delta}(N_k)$,
$\eta(N_{\Delta})$ is the sine of the angle between the space $X_{\delta}$
and the singular function $S$. Then
$$
\eta(N_{\Delta})^2=1-\Big(\sup_{u_{N_{\Delta}}
\in \mathbb{P}_{N_{\Delta}}(\Delta)}
\frac{(v_{N_{\Delta}},S)}{\|u_{N_{\Delta}}\|_{H^2(\Delta)}\|S\|_{H^2(\Delta)}}
 \Big)^2;
$$
$(\cdot,\cdot)$ is the scalar product on $H^2(\Delta)$.
If we consider $\Pi_{N_{\Delta}}:L^2(\Delta) \to \mathbb{P}_{N_{\Delta}}(\Delta)$,
we conclude that
$$
\eta(N_{\Delta})^2=1-\Big(\frac{(\Pi_{N_{\Delta}}S,S)}{\|\Pi_{N_{\Delta}}
 S\|_{H^2(\Delta)}\|S\|_{H^2(\Delta)}} \Big)^2.
$$
Let
$$
\frac{(\Pi_{N_{\Delta}}S,S)}{\|\Pi_{N_{\Delta}} S\|_{H^2(\Delta)}\|S\|_{H^2(\Delta)}}
 = \frac{(\Pi_{N_{\Delta}}S-S,S)}{\|\Pi_{N_{\Delta}} S\|_{H^2(\Delta)}
 \|S\|_{H^2(\Delta)}}+\frac{(S,S)}{\|\Pi_{N_{\Delta}} S\|_{H^2(\Delta)}
 \|S\|_{H^2(\Delta)}}.
$$
We conclude that $\eta(N_{\Delta})$ has the same order as
 $\big(\|\Pi_{N_{\Delta}} S - S\|_{H^2(\Delta)}\big)^{1/2}$ which is
$N^{-\pi/w}$ \cite{BM1}. This completes the proof.
\end{proof}

To study problem \eqref{4.1}, we begin by giving the properties of the bilinear
form $a_{\delta}^*(\cdot,\cdot)$ in the following proposition.

\begin{proposition} \label{prop5.2}
There exist two positive functions $C_1$ and $C_2$ independent of $\delta$
such that for all $u_{\delta}^*$, $v_{\delta}^*$ in $X_{\delta}^*$,
\begin{gather}\label{prop521}
|a_{\delta}^*(u_{\delta}^*,v_{\delta}^*)|
\le C_1 \| u_{\delta}^*\|_{*1}\| v_{\delta}^*\|_{*1}, \\
\label{prop521b}
a_{\delta}^*(u_{\delta}^*,v_{\delta}^*)\ge C_2 \| u_{\delta}^*\|_{*2}^2.
\end{gather}
\end{proposition}

\begin{proof}
Consider $\Omega_k=]a_k,b_k[\times]c_k,d_k[$. For $u_{\delta}^*$ and
$v_{\delta}^*$ in $X_{\delta}^*$, we have
\begin{align*}
a_{\delta}^*(u_{\delta}^*,v_{\delta}^*)
&=\sum_{k=1}^{K}\Big[(\Delta u_\delta^k,\Delta v_\delta^k)_{N_k}
 + \lambda \int_{k}\Delta S\, \Delta v_{\delta}^k\,dx\,dy
 + \mu \int_{\Omega_k} \Delta u_{\delta}^k \Delta S \,dx\,dy\\
&\quad +\lambda \mu \int_{\Omega_k} (\Delta S)^2 \,dx\,dy  \Big].
\end{align*}
where
\begin{align*}
&(\Delta u_\delta^k,\Delta v_\delta^k)_{N_k}\\
&=\sum_{i=1}^{N_K-1}\sum_{j=1}^{N_K-1}
 \Big[\frac{\partial^2 u_{\delta}^k}{\partial x^2}
 \frac{\partial^2 v_{\delta}^k}{\partial x^2}
 +\frac{\partial^2 u_{\delta}^k}{\partial x^2}
 \frac{\partial^2 v_{\delta}^k}{\partial y^2}
 +\frac{\partial^2 u_{\delta}^k}{\partial y^2}
 \frac{\partial^2 v_{\delta}^k}{\partial x^2}
 +\frac{\partial^2 u_{\delta}^k}{\partial y^2}
 \frac{\partial^2 v_{\delta}^k}{\partial y^2}\Big](\xi_i^k,\xi_j^k)\rho_i\rho_j.
\end{align*}
The terms $\frac{\partial^2 u_{\delta}^k}{\partial x^2}
\frac{\partial^2 v_{\delta}^k}{\partial x^2}$,
$\frac{\partial^2 u_{\delta}^k}{\partial x^2}
\frac{\partial^2 v_{\delta}^k}{\partial y^2}$,
$\frac{\partial^2 u_{\delta}^k}{\partial y^2}
\frac{\partial^2 v_{\delta}^k}{\partial x^2}$ and
$\frac{\partial^2 u_{\delta}^k}{\partial y^2}
\frac{\partial^2 v_{\delta}^k}{\partial y^2}$
are the polynomials of degree less or equal to $2 N_k -1$ with respect to
$x$ and $y$ respectively.

Using the exactness of the quadrature formula, the Cauchy-Schwartz inequality
and $a\cdot b\le \frac{a^2+b^2}{2}$ we obtain
\begin{align*}
(\Delta u_\delta^k,\Delta v_\delta^k)N_k
&\le \int_{a_k}^{b_k} \Big(\sum_{j=1}^{N_k-1}
 \frac{\partial^2 u_{\delta}^k}{\partial x^2}(x,\xi_j)^2\rho_j \Big)^{1/2}
 \Big(\sum_{j=1}^{N_k-1}\frac{\partial^2 v_{\delta}^k}{\partial x^2}(x,\xi_j)^2
 \rho_j \Big)^{1/2}\,dx\\
&\quad + \int_{c_k}^{d_k} \Big(\sum_{i=1}^{N_k-1}
 \frac{\partial^2 u_{\delta}^k}{\partial y^2}(\xi_i,y)^2\rho_i \Big)^{1/2}
 \Big(\sum_{i=1}^{N_k-1}\frac{\partial^2 v_{\delta}^k}{\partial y^2}(\xi_i,y)^2
 \rho_i \Big)^{1/2}\,dy\\
&\quad +\frac{1}{2}\int_{a_k}^{b_k} \Big(\sum_{j=1}^{N_k-1}
 \frac{\partial^2 u_{\delta}^k}{\partial x^2}(x,\xi_j)^2\rho_j
 + \frac{\partial^2 v_{\delta}^k}{\partial x^2}(x,\xi_j)^2\rho_j\Big)\,dx\\
&\quad +\frac{1}{2}\int_{c_k}^{d_k}
 \Big(\sum_{i=1}^{N_k-1}\frac{\partial^2 u_{\delta}^k}{\partial y^2}
 (\xi_i,y)^2\rho_i+ \frac{\partial^2 v_{\delta}^k}{\partial y^2}(\xi_i,y)^2\rho_i
 \Big)\,dy.
\end{align*}
Using that for all $\varphi_N \in \mathbb{P}_N(\Lambda)$,
\[
\|\varphi_N \|_{L^2(\Lambda)} \le (\varphi_N,\varphi_N)_N
\le C \|\varphi_N \|_{L^2(\Lambda)},
\]
where $C$ is a constant independent of $N$ \cite{BCM}, we deduce \eqref{prop521}.

For the ellipticity proof, we have
\begin{align*}
a_{\delta}(u_{\delta}^*,v_{\delta}^*)
&\ge \sum_{k=1}^{K}\|\Delta u_{\delta}^k\|_{L^2(\Omega_k)}^2
 +\lambda^2\|\Delta S \|_{L^2(\Omega_k)}^2
 +2 \lambda \int_{\Omega_k}\Delta u^k \Delta S\,dx\\
&\ge \sum_{k=1}^{K}\|\Delta u_{\delta k}^*\|_{L^2(\Omega_k)}^2.
\end{align*}
We  distinguish the two cases $\Omega\backslash\overline{\Delta}$ and $\Delta$.

(1) If $\Omega_k \subset \Delta$; since the functions $X_{\delta}^*$  and
their normal derivatives vanish on $\partial \Delta$ and using the conformity
hypothesis it is therefore sufficient to show that
 \begin{equation}\label{4.2}
 \sum_{\Omega_k\subset\Delta}\|\Delta u_{\delta k}^* \|_{L^2(\Omega_k)}^2
\ge C \sum_{\Omega_k\subset\Delta}| u_{\delta k}^* |_{H^2(\Omega_k)}.
\end{equation}
    It suffices to handle the terms of the cross product, using Green formula
    \begin{equation}\label{4.3}
    \begin{aligned}
\int_{\Omega_k}\frac{\partial^2 u_{\delta k}^*}{\partial x^2}
 \frac{\partial^2 u_{\delta k}^*}{\partial y^2}\,dx
&=\int_{\Omega_k}\Big(\frac{\partial^2 u_{\delta k}^*}{\partial x \partial y}\Big)^2
   - \int_{\partial \Omega_k}\frac{\partial u_{\delta k}^*}{\partial x}
 \frac{\partial^2 u_{\delta k}^*}{\partial x \partial y} n_y^k \,d\tau \\
&\quad + \int_{\partial \Omega_k}\frac{\partial u_{\delta k}^*}{\partial x}
 \frac{\partial^2 u_{\delta k}^*}{\partial y^2} n_x^k \,d\tau.
    \end{aligned}
\end{equation}
The sum on the mortar of $\Delta$ reveals the jumps
$[\frac{\partial u_{\delta k}^*}{\partial x}
\frac{\partial^2 u_{\delta k}^*}{\partial x \partial y}] $
and $[ \frac{\partial u_{\delta k}^*}{\partial x}
\frac{\partial^2 u_{\delta k}^*}{\partial y^2}] $ on the interfaces terms.
Since $S$ is continuous as well as its normal derivative on the interfaces,
these jumps are reduced to
$[\frac{\partial u_{\delta k}^*}{\partial x}
 \frac{\partial^2 u_{\delta k}}{\partial x \partial y}] $ and
$[\frac{\partial u_{\delta k}^*}{\partial x}
\frac{\partial^2 u_{\delta k}}{\partial y^2}]$.

 These terms are then written:
$[\frac{\partial u_{\delta k}}{\partial x}
\frac{\partial^2 u_{\delta k}}{\partial x \partial y}
+\lambda \frac{\partial S}{\partial x}
\frac{\partial^2 u_{\delta k}}{\partial x \partial y}] $ and
 $[\frac{\partial u_{\delta k}}{\partial x}
\frac{\partial^2 u_{\delta k}}{ \partial y^2}
+\lambda \frac{\partial S}{\partial x}
\frac{\partial^2 u_{\delta k}}{\partial y^2}] $.

The integral of  $[\frac{\partial u_{\delta k}}{\partial x}
\frac{\partial^2 u_{\delta k}}{\partial x \partial y}]$ and
$[\frac{\partial u_{\delta k}}{\partial x}\frac{\partial^2
u_{\delta k}}{ \partial y^2}]$ vanishes (see \cite{Z2}).

We also show that the integral of these terms
$[ \frac{\partial S}{\partial x}\frac{\partial^2 u_{\delta k}}
{\partial x \partial y}]$ and
$[\frac{\partial S}{\partial x}\frac{\partial^2 u_{\delta k}}{\partial y^2} ]$
vanishes since $\frac{\partial S}{\partial x}=\sum_{n\ge 0} \alpha_n L_n(x).
$
Then, the sum on the sub-domain $\Delta$ in \eqref{4.3} no longer counts
jump terms which gives \eqref{4.2}.

(2) If $\Omega_k \subset \Omega\backslash \overline{\Delta}$, the restriction
of the functions from $X_{\delta}^*$ to $\Omega\backslash \overline{\Delta}$
coincides with that of $X_{\delta}$ and we conclude (see \cite{Z2})
     \begin{equation}\label{4.4}
    \sum_{\Omega_k\subset\Omega\backslash\overline{\Delta}}
\|\Delta u_{\delta k}^* \|_{L^2(\Omega_k)}^2
\ge C \sum_{\Omega_k\subset\Omega\backslash\overline{\Delta}}
\| u_{\delta k}^* \|_{H^2(\Omega_k)}^2.
    \end{equation}
Then from inequalities \eqref{4.2} and \eqref{4.3} we have
that for all $v_{\delta}^* \in X_{\delta}^*$,
\[
\sum_{k=1}^K \|\Delta u_{\delta k}^* \|_{L^2(\Omega_k)}^2
\ge C \|u_{\delta k}^* \|_{*2}^2.
\]
Hence we obtain the ellipticity of $a_{\delta}^*(\cdot,\cdot)$.
\end{proof}

\begin{proposition} \label{prop5.3}
For $f\in L^2(\Omega)$, the discrete problem \eqref{4.1} has a unique solution
 $u_{\delta}^*$ in $X_{\Delta}^*$ and
$$
\|u_{\delta}^* \|_{2*} \le C \|f\|_{L^2(\Omega)}.
$$
\end{proposition}

\begin{remark} \label{rmk5.1} \rm
The norms $\|\cdot \|_{1*}$ and $\|\cdot \|_{2*}$ are equivalent with a
constant depending on the discretization parameter $\delta$.
In the following we will use the norm $\|\cdot \|_{1*}$ and we will show an
inf-sup condition on the bilinear form $a_{\delta}^*(\cdot,\cdot)$ using this norm.
\end{remark}

\begin{proposition} \label{prop5.4}
There exists a constant $\alpha$ such that for all $v_{\delta}^*\in X_{\delta}^*$,
 \begin{equation}\label{4.5}
\sup_{t_{\delta}^*\in X_{\delta}^*}\frac{a_{\delta}^*(v_{\delta}^*,
t_{\delta}^*)}{\|t_{\delta}^* \|_{1*}}\ge \alpha \|v_{\delta}^* \|_{1*}.
    \end{equation}
\end{proposition}

\begin{proof}
Consider $t_{\delta}^*=v_{\delta}+\beta(\lambda S)$ and find a value for
$\beta$ which satisfies inequality \eqref{4.5},
\begin{align*}
&a_{\delta}^*(v_{\delta}^*,t_{\delta}^*)\\
&=a_{\delta}^*(v_{\delta}+\lambda S,v_{\delta}+\beta(\lambda S))\\
& \ge \sum_{k=1}^{K}\Big[\int_{\Omega_k}(\Delta v_{\delta k})^2\,dx
 +|\lambda|(1+\beta)\int_{\Omega_k}\Delta S  \Delta v_{\delta k}\,dx
 + \beta^2 \lambda^2 \int_{\Omega_k}\Delta S^2\,dx\Big]\\
&\ge \sum_{k=1}^{K}\Big[\|\Delta v_{\delta k}\|_{L^2(\Omega_k)}^2
 -|\lambda|(1+\beta)\|\Delta S\|_{L^2(\Omega_k)}\|\Delta v_{\delta k}
 \|_{L^2(\Omega_k)}+\beta^2\lambda^2 \|\Delta S \|_{L^2(\Omega_k)}^2\Big]\\
&\ge \sum_{k=1}^{K}\Big[\frac{1}{2}\|\Delta v_{\delta k}\|_{L^2(\Omega_k)}^2
 +|\lambda|\Big(\beta^2-\frac{(\beta+1)^2}{2}\Big)\|\Delta S \|_{L^2(\Omega_k)}^2
\Big]
\end{align*}
Using  Young's inequality $ab \le \frac{a^2+b^2}{2}$ and choosing $\beta=3$,
we complete the proof.
\end{proof}

Using inequality \eqref{4.5} and the Strang lemma we obtain the following result.

\begin{proposition} \label{prop5.5}
The error estimate between $u$ the solution of problem \eqref{1} and
$u_{\delta^*}$ the solution of problem \eqref{4.1} is
\begin{equation}\label{4.6}
\begin{aligned}
&\|u- u_{\delta^*}\|_{1*} \\
&\le C \Big[\inf_{v_{\delta}^*\in X_{\delta}^*}
 \Big(\|u-v_{\delta}^*\|_{1*}+\sup_{\omega_{\delta}^*\in X_{\delta}^*}
 \frac{a(v_{\delta}^*,\omega_{\delta}^*)-a_{\delta}^*(v_{\delta}^*,
 \omega_{\delta}^*)}{\|\omega_{\delta}^*\|_{1*}} \Big)  \\
&\quad\times \sup_{\omega_{\delta}^*\in X_{\delta}^*}
 \frac{\sum_{k=1}^K\sum_{l=k+1}^K\big(\int_{\gamma_{kl}}
 \frac{\partial (\Delta u)}{\partial n}[\omega_{\delta}^*]\,dx
    - \int_{\gamma_{kl}}\Delta u [\frac{\partial \omega_{\delta}^*}{\partial n}]
 \,dx\big)}{\| \omega_{\delta}^*\|_{1*}}\Big],
\end{aligned}
\end{equation}
where $n$  and $[\omega]$   are respectively the  normal and the jump of
$\omega$ on the interfaces.
\end{proposition}

To find the order of convergence, we have to estimate each term of
the inequality \eqref{4.6}. Recall that the singular function $S$ is of class
$\mathcal C^1$, then the jump terms $(\omega_{\delta k}^*-\omega_{\delta l}^*)$
and $\big(\frac{\partial \omega_{\delta k}^*}{\partial n}
-\frac{\partial \omega_{\delta l}^*}{\partial n}\Big)$ through each interface
$\gamma_{kl}$ are reduced to $(\omega_{\delta k}-\omega_{\delta l})$ and
$\big(\frac{\partial \omega_{\delta k}}{\partial n}
 -\frac{\partial \omega_{\delta l}}{\partial n}\big)$.

The conformity hypothesis on $\Delta$ implies that these quantities vanish.
 Moreover $u$ and $u_R$ coincide on $\Omega \backslash\overline{\Delta}$;
the consistence error term is then written on each interface $\gamma_{kl}$,
\begin{align*}
&\int_{\gamma_{kl}}\frac{\partial (\Delta u)}{\partial n}[\omega_{\delta}]\,dx
 +\int_{\gamma_{kl}}\Delta u [\frac{\partial \omega_{\delta}}{\partial n}]\,dx\\
&= \int_{\gamma_{kl}}\frac{\partial (\Delta u_R)}{\partial n}
 (\varphi_0-\omega_{\delta k})\,dx
 + \int_{\gamma_{kl}}\frac{\partial (\Delta u_R)}{\partial n}
 (\varphi_0-\omega_{\delta l})\,dx\\
&\quad + \int_{\gamma_{kl}}(\Delta u_R)
 \big(\varphi_1-\frac{\partial \omega_{\delta k}}{\partial n}\big)\,dx
 +\int_{\gamma_{kl}}(\Delta u_R)
 \big(\varphi_1-\frac{\partial \omega_{\delta l}}{\partial n}\big)\,dx
\end{align*}
where $\varphi_0$ and $\varphi_1$ are the mortar functions associated with
$(\omega_\delta,\frac{\partial \omega_{\delta}}{\partial n})$.
Then we obtain, \cite{Z2},
\begin{equation}\label{17}
\begin{aligned}
&\sum_{k=1}^{K}\sum_{l=k+1}^{K}\int_{\gamma_{kl}}
 \frac{\partial (\Delta u_R)}{\partial n}[\omega_{\delta}]\,dx
 +\int_{\gamma_{kl}}\Delta u_R [\frac{\partial \omega_{\delta}}{\partial n}]\,dx\\
&\le c \sum_{k=1}^{K}\sum_{j=1}^{4}\Big(\inf_{\psi_{kj}\in\mathbb{P}_{N_k-4}
(\Gamma^{kj})}\|\frac{\partial (\Delta u_R)}{\partial n}-\psi_{kj}
 \|_{(H^{3/2}(\Gamma^{kj}))'}\\
&\quad+\inf_{\psi_{kj}\in\mathbb{P}_{N_k-4}(\Gamma^{kj})}\|\Delta u_R-\psi_{kj}
\|_{(H^{1/2}(\Gamma^{kj}))'}\Big).
\end{aligned}
\end{equation}
Following the definition of $X_{\delta}^*$ and \eqref{dc}, we have
$$
\inf_{v_{\delta}^*\in X_{\delta}^*}\|u-v_{\delta}^*\|_{1*}
\le C \inf_{v_{\delta}\in X_{\delta}^{-}}\|u_R-v_{\delta} \|_{1*}
$$
where
$$
X_{\delta}^{-}=\{v_{\delta}\in X_{\delta};
 v_{\delta k}\in \mathbb{P}_{N-1}(\Omega_k) \}.
$$
Finally the term
$$
\sup_{\omega_{\delta}^*\in X_{\delta}^*}\frac{a(v_{\delta}^*,\omega_{\delta}^*)-a_{\delta}^*(v_{\delta}^*,\omega_{\delta}^*)}{\|\omega_{\delta}^*\|_{1*}}
$$
vanishes if we choose $v_{\delta}^*=v_{\delta}\in X_{\delta}^-$ following the exactness of the quadrature formula \eqref{lem41}.\\
Doing the sum of these results, we obtain
\begin{equation}\label{p12}
\begin{aligned}
&\|u-u_{\delta}^* \|_{1*} \\
&\le C \Big[\inf_{v_{\delta}\in X_{\delta}^-}
\|u-v_{\delta}\|_{1*}+\sum_{k=1}^K \sum_{j=1}^4
\Big(\inf_{\psi_{kj}\in\mathbb{P}_{N_k-4}(\Gamma^{kj})}
\|\frac{\partial \Delta u_R}{\partial n}-\psi_{kj} \|_{(H^{3/2}(\Gamma^{kj}))'}\\
&\quad +\inf_{\psi_{kj}\in\mathbb{P}_{N_k-4}(\Gamma^{kj})}
 \|\Delta u_R-\psi_{kj} \|_{(H^{1/2}(\Gamma^{kj}))'}  \Big)\Big].
\end{aligned}
\end{equation}
Suppose $f$ in $H^{s-2}(\Omega)$ for $\eta(\omega)<s<\eta(\omega)+2$,
then $u_R\in H^{s+2}(\Omega)$ and the trace (respectively the normal
derivative trace) of $u_R$ belongs to
$H^{s-\frac{1}{2}}(\partial \Omega_k)$ (respectively
$H^{s-\frac{3}{2}}(\partial \Omega_k)$); $1\le k \le K$.
Taking $\psi_{kj}$ (respectively  $\chi_{kj}$) the orthogonal projection on
$\mathbb{P}_{N_k -4}(\Gamma^{kj})$, we deduce
\begin{gather*}
\|\Delta u_R-\psi_{kj} \|_{(H^{1/2}(\Gamma^{kj}))'}
\le C \,N_k^{-s}\|u_R \|_{H^{s+2}(\Omega_k)}, \\
\|\frac{\partial \Delta u_R}{\partial n}-\chi_{kj} \|_{H^{-3/2}(\Gamma^{kj})}
\le C \,N_k^{-s}\|u_R \|_{H^{s+2}(\Omega_k)}.
\end{gather*}
Furthermore, we have
$$
\inf_{v_{\delta}\in X_{\delta}^-} \|u-v_{\delta}\|_{1*}
\le C \sum_{k=1}^{K} N_k^{-s}\|u_R \|_{H^{s+2}(\Omega_k)}.
$$
Suppose that $f\in H^{s-2}(\Omega)$ with $s<2+\eta_1(\omega)$ where
$\eta_1(\omega)$ is the second real solution of the equation \eqref{4},
in the band $ 0<\operatorname{Real}(z)<s$, then from the decomposition
\eqref{p5} and Assumption \ref{as1}, we show exactly in the same way  that
\begin{align*}
&\|u-u_{\delta}^* \|_{1*} \\
&\le C \Big[\inf_{v_{\delta}\in X_{\delta}^-} \|u_R-v_{\delta}\|_{1*}
+\sum_{k=1}^K \sum_{j=1}^4 \Big(\inf_{\psi_{kj}\in\mathbb{P}_{N_k-4}
(\Gamma^{kj})}\|\frac{\partial \Delta \tilde{u}_R}{\partial n}-\psi_{kj}
\|_{(H^{3/2}(\Gamma^{kj}))'}\\
&\quad +\inf_{\chi_{kj}\in\mathbb{P}_{N_k-4}(\Gamma^{kj})}\|\Delta \tilde{u}_R
-\chi_{kj} \|_{(H^{1/2}(\Gamma^{kj}))'}  \Big)\Big].
\end{align*}
We note that
$$
\inf_{v_{\delta}\in X_{\delta}^-} \|u_R-v_{\delta}\|_{1*}
\le C \Big\{\inf_{v_{\delta}\in X_{\delta}^-} \|\tilde{u}_R-v_{\delta}\|_{1*}
+|\tilde{\lambda}|\inf_{v_{\delta}\in X_{\delta}^-}
\|\tilde{S}-v_{\delta}\|_{1*} \Big\}.
$$
Using the approximation result of the singular functions by polynomials
\cite{BM1} we have
$$
\inf_{v_{\delta}\in X_{\delta}^-} \|\tilde{S}-v_{\delta}\|_{1*}
\le C  N^{\varepsilon-2\eta_1(\omega)}\quad \forall \varepsilon>0.
$$
Then
$$
\inf_{v_{\delta}\in X_{\delta}^-} \|{u}_R-v_{\delta}\|_{1*}
\le C  N^{2-s}(\|\tilde{u}_R \|_{H^s(\Omega)}+|\tilde{\lambda}|),
$$
hence
$$
\|{u}-u_{\delta}^*\|_{1*}\le C  N^{2-s} \|f \|_{H^{s-2}(\Omega)}\quad
\text{ for } s <2+\eta_1(\omega).
$$
Combining these results we have the following theorem.

\begin{theorem} \label{thm5.1}
If $f\in H^{s-2}(\Omega)$ for $s>0$ and $\varepsilon>0$ then
$$
\|{u}-u_{\delta}^*\|_{1*}\le C \Big(\sum_{k=1}^K  N_k^{-\sigma_k}\Big)
\| f\|_{H^{s-2}(\Omega)}
$$
where $\sigma_k$, $1\le k \le K$ satisfies
\begin{equation}\label{eq}
\sigma_k=
\begin{cases}
s-2  & \text{if $\overline{\Omega}_k$ does not contain any vertices of }\Omega,\\
\inf(s-2,2\eta_1(\pi/2)-\varepsilon)
& \text{if $\overline{\Omega}_k$ contains a vertex of $\Omega$  other than }
 \mathbf{a},\\
\inf(s-2,2\eta_1(\omega)-\varepsilon) & \text{if $\overline{\Omega}_k$ contains
}  \mathbf{a}.
\end{cases}
\end{equation}
\end{theorem}

Using the Aubin-Nische duality we have the following corollary.

\begin{corollary} \label{cor5.1}
Let ${f}$ in $H^{s-2}(\Omega)$, for $s>0$, then, for all $\epsilon > 0$,
$$
\| {u} - { u}_\delta^*\|_{L^2(\Omega)} \leq C \Big(N^{-2}(\sum_{k=1}^K
 N_k^{-\sigma_k})\Big)\|
{f}\|_{H^{s-2}(\Omega)}
$$
where $\sigma_k$ satisfies \eqref{eq} and $N={\inf_{1\leq k\leq K}}N_k$.
\end{corollary}

\subsection*{Conclusion}
We studied the biharmonic problem with homogeneous boundary conditions
in a domain of $\mathbb{R}^2$ with corners. The discrete problem 
was studied  using the mortar spectral element method.
We showed that if we consider the decomposition
of the solution in a regular part and a singular one, we improve the order of
the error. Using the Strang and Fix algorithm, which consists on adding the
singular function in the discrete space, we prove an optimal order of the error
on the solution. The numerical implementation of the obtained results will
be presented in a forthcoming work. The extension of this discretization
to the three dimension axi-symmetric domain is presently under consideration.


\subsection*{Acknowledgments}
 The authors would like to extend their sincere appreciation to the Deanship
of Scientific Research at King Saud University for funding this Research
group No (RG-1435-026).
The third author thanks the Visiting Professor Programming at King Saud
University for funding this work.


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\end{document}