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\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)~~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}
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