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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 157, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2015/157\hfil Approximation of the singularity coefficients]
{Approximation of the singularity coefficients of an
elliptic equation by mortar spectral element method}
\author[N. Chorfi, M. Jleli \hfil EJDE-2015/157\hfilneg]
{Nejmeddine Chorfi, Mohamed Jleli}
\address{Nejmeddine Chorfi \newline
Department of Mathematics,
College of Science, King Saud University,
P.O. Box 2455, Riyadh 11451, Saudi Arabia}
\email{nchorfi@ksu.edu.sa}
\address{Mohamed Jleli \newline
Department of Mathematics,
College of Science, King Saud University,
P.O. Box 2455, Riyadh 11451, Saudi Arabia}
\email{jleli@ksu.edu.sa}
\thanks{Submitted April 5, 2015. Published June 12, 2015.}
\subjclass[2010]{35J15, 78M22}
\keywords{Mortar spectral method; singularity; crack}
\begin{abstract}
In a polygonal domain, the solution of a linear elliptic problem is written
as a sum of a regular part and a linear combination of singular functions
multiplied by appropriate coefficients. For computing the leading singularity
coefficient we use the dual method which based on the first singular dual function.
Our aim in this paper is the approximation of this leading singularity coefficient
by spectral element method which relies on the mortar decomposition domain technics.
We prove an optimal error estimate between the continuous and the discrete
singularity coefficient. We present numerical experiments which are in perfect
coherence with the analysis.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
If the data are smooth, the solution of an elliptic partial differential
equation is not regular when the domain is polygonal. For a Dirichlet problem
of the Laplace operator, we define some singular functions depending only on
the geometry of the domain. The solution is written as the sum of a regular
part and singular functions multiplied by appropriate coefficients \cite{K,G}.
For approximating the leading singularity coefficients we use two algorithms.
The first one is Strang and Fix algorithm \cite{SF}, which consists to add the
leading singularity function to the discrete space see \cite{C}.
For the second algorithm we apply the dual method. The numerical computation
of the leading singularity coefficient has been performed by finite elements,
see Amara and Moussaoui \cite{AA1,AA2}. This coefficient is physically
meaningful in solid mechanics (crack propagation). We use the mortar spectral
element method: the domain is decomposed in a union of finite number of
disjoint rectangles, the discrete functions are polynomials of high degree
on each rectangle and are enforced to satisfy a matching condition on the interfaces.
This technique is nonconforming because the discrete functions are not continuous.
We refer to Bernardi, Maday and Patera \cite{BMP} for the introduction of the
mortar spectral element method.
An outline of this article is as follows. In the second section, we give
the dual singular function and the formula for finding the leading coefficient
of the singularity. This coefficient does not depend on the data function
or the geometry of the domain but it just depends on the solution.
In section $3$, we present the discrete problem and the discrete leading
singularity coefficient. The section $4$ is devoted to the estimation of the
error and we prove the optimality. Finally, a results of a numerical test
are given in Section $5$.
\section{Dual singular function and the coefficient of the singularity}
In the rest of the paper, we assume that our domain ${\Omega}$ is a polygon
of $\mathbb{R}^2$ such that there exists a finite number of open rectangles
$\Omega_k,1 \leq k \leq K$, for which
\begin{equation}
\bar{\Omega}= \cup_{k=1}^{K}\bar \Omega_k\quad\text{and}\quad
\Omega_k\cap \Omega_l=\emptyset \quad \text{for } k\not = l.\label{2.1}
\end{equation}
We suppose also that the intersection of each $\bar \Omega_k$
(for $1 \leq k \leq K$) with the boundary $\partial \Omega$ is either
empty or a corner or one or several entire edges of $\Omega_k$.
We choose the coordinate axis parallel to the edge of the $\Omega_k$.
Handling the singularity is a local process. Therefore, it is not restricted
to suppose that $\Omega$ has a unique non-convex corner $\mathbf{a}$ with an
angle $\omega$ equal either to $3\pi/ 2$ or to $2\pi$ (case of the crack).
We choose the origin of the coordinate axis at the point $\mathbf{a}$,
we introduce a system of polar coordinates $(r,\theta)$ where $r$ stands for
the distance from $\mathbf{a}$ and $\theta$ is such that the line $\theta=0$
contains an edge of $\partial\Omega$. For more technical proof that will
appear later we need to make the following conformity assumption: if the
intersection of $\bar \Omega_k$ and $\bar \Omega_l, k \not= l$ contains
$\mathbf{a}$, then it contains either $\mathbf{a}$ or both an edge of $\Omega_k$ and $\Omega_l$.
Let $\Sigma$ be the open domain in $\Omega$ such that $\bar\Sigma$ is the union
of the $\bar\Omega_k$ which contain $\mathbf{a}$.
The model equation under consideration is the following Dirichlet problem
for the Laplace operator:
\begin{equation}
\begin{gathered}
-\Delta u=f \quad \text{in }\Omega\\
u=0 \quad \text{on }\partial\Omega.
\end{gathered} \label{2.2}
\end{equation}
If the data $f$ belongs to $H^{s-2}(\Omega)$, then the above problem admits
a unique solution $u$ belongs to $H^{s}(\Omega)$. This solution is decomposed as:
\begin{equation}
u=u_r+\lambda S_1,\label{2.3}
\end{equation}
where the function $u_r$ belongs to $ H^{s}(\Omega)$ for $s<1+{2\pi\over{\omega}}$
such that
$$
\| u_r\|_{H^{s}(\Omega)}+\mid\lambda\mid 0$
$$
|\lambda-\lambda^*_{\delta}|
\leq C N^{-1}\Big(\sum^{K}_{k=1}N_k^{-\sigma_k}\Big)\| f\|_{H^{s-2}(\Omega)},
$$
where $N=\inf_{1\leq k\leq K}^{} N_k$ and
$$
\sigma_k =\begin{cases}
s-1 &\text{if $\bar{\Omega}^k$ contains no corners of $\bar{\Omega}$},\\
\inf\{s-1,8-\varepsilon\} &\text{if $\bar{\Omega}^k$ contains corners
different from $\mathbf{a}$},\\
\inf\{s-1,{4\pi\over\omega}-\varepsilon\} &\text{if $\bar{\Omega}^k$ contains
$\mathbf{a}$}.
\end{cases}
$$
\end{theorem}
\begin{proof}
To estimate $| \lambda- \lambda^*_{\delta}|$, we have to estimate each term
of the inequality \eqref{3.4}. For the first term, using Cauchy-Schwarz
and Poincar{\'e}-Friedrichs inequalities we deduce that
\begin{equation}
\big| \sum^{K}_{k=1}\int_{\Omega_k}\nabla(u-u^*_{\delta})
\nabla(\varphi^*-\varphi^*_{\delta})\,dx \big|
\leq C\| u-u^*_{\delta}\|_*\|\varphi^*-\varphi^*_{\delta}\|_*
\label{4.1}.
\end{equation}
Since $u$ (respectively $u^*_{\delta}$) is the solution of the continuous
problem \eqref{2.2} (respectively the discrete problem \eqref{3.1}).
As the same for $\varphi$ and $\varphi^*_{\delta}$, are respectively the
solutions of the problems \eqref{2.4} and \eqref{3.2} with second member equal
to $\Delta S^*_1$ in $L^2(\Omega)$, we conclude by
\cite[result (5.16)]{C}.
From the continuity of $S_1$, we deduce that for any function
$\varphi^*_{\delta}$ in $X^*_{\delta}$ the jump
$\varphi^*_{\delta_{/\Omega_k}}-\varphi^*_{\delta_{/\Omega_l}}$ is equal to
$\varphi_{\delta_{/\Omega_k}}-\varphi_{\delta_{/\Omega_l}}$ which vanishes on
$\Sigma$ due to the conformity assumption.
We note also that $u=u_r$ on $\Omega\setminus\bar{\Sigma}$. Hence
$$
\int_{\gamma^{kl}}({\partial u\over\partial
n_k})(\varphi^*_{\delta_{/\Omega_k}}-\varphi^*_{\delta_{/\Omega_l}})\,d\tau
= \int_{\gamma^{kl}}({\partial u_r\over\partial
n_k})(\varphi^*_{\delta_{/\Omega_k}}-\phi)\,d\tau
-\int_{\gamma^{kl}}({\partial u_r\over\partial
n_k})(\varphi^*_{\delta_{/\Omega_l}}-\phi)\,d\tau,
$$
where $\phi$ is the mortar function associated to $\varphi_{\delta}$.
So, the estimation of this quantity can be evaluated as in
\cite[result (5.24)]{C}. If $\Gamma^k$ is not a mortar then
\begin{equation}
\begin{aligned}
&\big|\sum^{}_{1\leq k