\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 258, pp. 1--8.\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-2015/258\hfil
 $L^p$ estimates for Dirichlet-to-Neumann operator]
{$L^p$ estimates for Dirichlet-to-Neumann operator and applications}

\author[T. El Arwadi, T. Sayah \hfil EJDE-2015/258\hfilneg]
{Toufic El Arwadi, Toni Sayah}

\address{Toufic El Arwadi \newline
Department of Mathematics and computer science,
Faculty of Science, Beirut Arab university,
P.O. Box: 11-5020, Beirut, Lebanon}
\email{t.elarwadi@bau.edu.lb}

\address{Toni Sayah \newline
Research unit "EGFEM", Faculty of sciences,
Saint-Joseph University, B.P. 11-514 Riad El Solh,
Beirut 1107 2050, Lebanon}
\email{toni.sayah@usj.edu.lb}

\thanks{Submitted September 5, 2015. Published October 2, 2015.}
\subjclass[2010]{47D06, 47A99, 35J15, 35M13, 65M60}
\keywords{Dynamic boundary condition; Dirichlet-to-Neumann operator; 
\hfill\break\indent $L^p$ estimation; finite element method}

\begin{abstract}
 In this article, we consider the time dependent linear
 elliptic problem with dynamic boundary condition. We recall the
 corresponding Dirichlet-to-Neumann operator on $\Gamma$ denoted by
 $-\Lambda_\gamma$. Then we show that when $\gamma=1$ near the
 boundary,  $\Lambda_\gamma-\Lambda_1$ is bounded by $\gamma-1$ in
 $L^p(\Omega)$ norm. This result is a generalization of the bound
 with the $L^\infty(\Omega)$ norm  and is applicable for comparing the
 Dirichlet to Neumann semigroup and the Lax semigroup. Finally, we
 present numerical experiments for validation of our results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

 Let $\Omega\subset\mathbb{R}^2$ be a bounded open set of class
$C^2$,  with boundary $\Gamma$, and let $]0, T[$ to denote an
interval in $\mathbb{R}$ where $T\in (0,+\infty)$ is a fixed final time.
We denote by $n(x)$ the unit outward normal vector at $x\in
\Gamma$. We intend to work with  the following time dependent
linear elliptic problem with dynamic boundary condition:
\begin{equation}\label{P}
\begin{gathered}
-\operatorname{div} \gamma(x)  \nabla u(t,x) =  0 \quad \text{in }
 ]0,T[ \times \Omega,\\
\frac{\partial u}{\partial t}(t,x)+\gamma(x)   n (x)\cdot\nabla u(t,x)
 =  0 \quad \text{on } ]0,T[ \times   \Gamma,\\
  u(0,x)  =  u_0  \quad \text{on }\Gamma,
\end{gathered}
\end{equation}
where $\gamma \in L^\infty_+ (\Omega)$ and
$u_0 \in H^{1/2}(\Gamma)$, and we suppose that there exists a real
positive number $\beta$ such that
\[
\beta^{-1}\leq \gamma(x) \leq \beta \quad \forall x \in \overline{\Omega}.
\]
The unknown is $u$ while $u_0$ is the initial condition at time $t=0$.

 The trace value of the solution $u(t,x)$ on $\Gamma$ is
directly related to the elliptic Dirichlet-to-Neumann map. In
fact, for a given $f$, $u^\gamma$ solves the  Dirichlet
problem
\begin{equation}
\begin{gathered}
\operatorname{div}(\gamma\nabla u^\gamma)  =  0\quad\text{in } \Omega, \\
u^\gamma  =  f\quad\text{on }\Gamma.
\end{gathered} \label{DP}
\end{equation}
For any $f\in H^{1/2}(\Gamma)$, it is well known that the
Dirichlet problem \eqref{DP} is uniquely solvable in $H^1(\Omega)$. We
denote by $u^\gamma=L_\gamma f$ where the function $u^\gamma$ is
called the $\gamma$-harmonic lifting of $f$ and the operator
$L_\gamma$ is called the $\gamma$-harmonic lifting operator. If
$u^\gamma$ and $\gamma$ are smooth, the Dirichlet-to-Neumann
operator is defined by
\begin{equation}
\Lambda_\gamma f=(n.\gamma\nabla u^\gamma)|_{\Gamma}\,.
\end{equation}
In another words  $\Lambda_\gamma=n\cdot\gamma\nabla L_\gamma$
(see for instance \cite{EmSh}).

 We can extend $\Lambda_\gamma$ uniquely to an operator
$\Lambda_\gamma\in \mathcal{L}(H^{1/2}(\Gamma),H^{-\frac{1}{2}}(\Gamma))$.
If we denote its part in $L^2(\Gamma)$ again by $\Lambda_\gamma$, we
define the Dirichlet-to-Neumann operator as an unbounded operator
with domain
\begin{equation}
D(\Lambda_\gamma)=\{f\in H^{1/2}(\Gamma); \Lambda_\gamma f\in L^2(\Gamma)\}.
\end{equation}
The Dirichlet-to-Neumann operator $\Lambda_\gamma$ is positive,
self adjoint and a first order pseudo-analytic operator (see for
instance \cite{TaylorI:98} and \cite{TaylorII:98}).
 By Lummer-Phillips theorem, $-\Lambda_\gamma$ generates a $C_0$
semigroup denoted by $e^{-t\Lambda_\gamma}$ in $L^2(\Gamma)$ (see \cite{s9}).

 For the existence and the uniqueness of the solution of
problem \eqref{P}, we refer to \cite[Theorem 1.1,  page 169]{s9}.


\begin{theorem} \label{thm1.2}
If $\Gamma$ is of class $C^2$,
$\gamma$ is of class $C^\alpha$ ($\alpha >2$), and for each $u_0
\in L^2(\Gamma)$, problem \eqref{P} has a unique solution
$u:[0,+\infty)\to H^1(\Omega)$ satisfying:
\begin{enumerate}
    \item $u \in C([0,+\infty);H^1(\Omega))\cap
    L^2([0,+\infty);H^1(\Omega))$;
    \item $u|_\Gamma \in C([0,+\infty);L^2(\Gamma))\cap
    C^1([0,+\infty);L^2(\Gamma))$;
    \item $n.\nabla u \in C([0,+\infty);L^2(\Gamma))$.
\end{enumerate}
\end{theorem}

 By taking the trace of the solution to
\eqref{P} and denoting it by $u(t,.)|_{\Gamma}$, the
Dirichlet-to-Neumann semigroup $e^{-t\Lambda_\gamma}u_0$ is
defined by
\begin{equation}
(e^{-t\Lambda_\gamma}u_0)(x)=u(t,x)|_{\Gamma}, \quad x\in\Gamma\,.
\end{equation}

\begin{remark} \label{rmk1.2} \rm
Lax introduced an explicit representation for the
Dirichlet-to-Neu\-mann semigroup for $\gamma=1$ and $\Omega=B(0,1)$.
The Lax semigroup is defined by
\begin{equation}
(e^{-t\Lambda_1}u_0)(x)=u^1(e^{-t}x) \quad \text{for }
x\in \partial B(0,1),
\end{equation}
where $u^1=L_1f$ is the harmonic lifting of $f$ (see \cite{Lax}).

 For $\Omega\neq B(0,1)$ there is no explicit
representation of the Dirichlet to Neumann semigroup (see
\cite{EmSh}). This motivate several authors to construct families
of approximation via Chernoff's theorem (see \cite{EmSh,CEES}).
Here an important question arises: what is the
effect of the support of $\gamma$ on the comparison of the general
Dirichlet-to-Neumann semigroup $e^{-t\Lambda_\gamma}$ and the Lax
semigroup?
\end{remark}

 In \cite{CoKnSi}, the authors showed that for $\gamma=1$
near the boundary, the distance
$\|\Lambda_\gamma-\Lambda_1\|_{\mathcal{L}(H^{1/2}(\Gamma),
H^s(\Gamma))}$
is bounded by $\|\gamma-1\|_{L^\infty(\Omega)}$ for
any $s\in \mathbb{R}$. The assumption $\gamma=1$ near the boundary has
multiple physical applications, in particular it is usually used
in the EIT (electrical Impedance Tomography) community (see
\cite{MSS}).

 In this article, we compare the general
Dirichlet-to-Neumann semigroup $e^{-t\Lambda_\gamma}$ to the Lax
semigroup. We start by comparing $\Lambda_\gamma$ to $\Lambda_1$
for $\gamma=1$ near the boundary. In particular we show that
$\|\Lambda_\gamma-\Lambda_1\|_{\mathcal{L}(H^{1/2}(\Gamma),
H^s(\Gamma))}$ is bounded by $\|\gamma-1\|_{L^p(\Omega)}$ for all
$s\in \mathbb{R}$ and $p>2$.  As a straightforward consequence, we
show that for the particular case where $\Omega =B(0,1)$,
$\|e^{-t\Lambda_\gamma}u_0-e^{-t\Lambda_1}u_0\|_{L^2(\Gamma)}$ is
also bounded by $\|\gamma-1\|_{L^p(\Omega)}$. At the end we give a
numerical example which justify our theoretical results.

We suppose that $u_0 \in H^{1/2}(\Gamma)$ and introduce the
following variational problem  in the sense of distributions on
$]0, T [$: Find $u(t,.) \in H^1(\Omega)$ such that,
\begin{equation}
\label{V}
\begin{gathered}
u(0)=u_0 \quad \text{on }\Gamma, \\
 \int _{\Omega} \gamma(x) \nabla u(t,x) \nabla v(x)\,dx
+ \frac{d}{d t} \big( \int_{\Gamma} u(t,s)v(s)\,ds
\big)=0, \quad \forall  v \in H^1(\Omega).
\end{gathered}
\end{equation}


\begin{theorem}[\cite{ElDiSa}] \label{thm1.3}
If $u \in L^2(0,T;H^1(\Omega))$ and
$u|_\Gamma \in L^\infty(0,T;L^2(\Gamma))$, then problem \eqref{P} is
equivalent to the variational problem  \eqref{V}. Furthermore, we have
the  bound
\[
 \|\nabla u\|_{L^2(0,\tau,L^2(\Omega)^2)}^2 +  \|u(\tau,.)
\|_{L^2(\Gamma)}^2 \le  c \|u_0\|_{L^2(\Gamma)}^2,
\]
where $c$ is a positive constant and $\tau \in ]0,T]$.
\end{theorem}

\section{Main result}

To avoid the complexity of notations, we denote by
$\|\cdot\|_{1/2,s}:=\|\cdot\|_{\mathcal{L}(H^{1/2}(\Gamma),H^s(\Gamma))}$.
As it was proved in \cite{CEES}, the distance between the General
Dirichlet-to-Neumann semigroup $e^{-t\Lambda_\gamma}$ and the Lax
semigroup $e^{-t\Lambda_1}$ with respect to the $L^2(\Gamma)$
topology depends directly on the distance $\gamma$ to 1 with
respect to the $L^\infty(\Omega)$ topology. However, as it was
proved in \cite{Toufic}, the support of $\gamma-1$ plays an
important role in the comparison of the Dirichlet-to-Neumann
maps.

  In this section, we show that when
$\|\gamma-1\|_{L^p(\Omega)}$, $p>2$, tends to zero and $\gamma=1$
near $\Gamma$, the general Dirichlet-to-Neumann semigroup
$e^{-t\Lambda_\gamma}$ tends to the Lax semigroup
$e^{-t\Lambda_1}$. In particular for $t\in ]0,T]$, the following estimate holds,
\begin{equation}\label{semi}
\|e^{-t\Lambda_\gamma}u_0-e^{-t\Lambda_1}u_0\|_{L^2(\Gamma)}
\leq C(T)\|\gamma-1\|_{L^p(\Omega)}\|u_0\|_{H^{1/2}(\Gamma)}.
\end{equation}
Like the $L^\infty$ estimate (see \cite{CEES}), it is clear that this
estimate is a straightforward consequence of the following lemma.

\begin{lemma} \label{lem2.1}
Let $\gamma\in L_+^\infty(\Omega)$ be a positive conductivity satisfying
 $\gamma=1$ near $\Gamma$. Then for $p>2$ and for all $s\in \mathbb{R}$,
the following estimate holds:
\begin{equation}\label{dtn}
\|\Lambda_\gamma-\Lambda_1\|_{1/2,s}\leq C_2\|\gamma-1\|_{L^p(\Omega)}
\end{equation}
where the constant $C_2$ depends on $s, \Omega$ and $\beta$.
\end{lemma}

\begin{proof}
For $\gamma=1$ near the boundary, the operator
$\Lambda_\gamma-\Lambda_1$ is a smoothing operator, i.e. it acts
from $H^{1/2}(\Gamma)$ to $H^s(\Gamma)$ for all values of $s\in \mathbb{R}$.
Depending on the values of $s$, the proof is divided into three steps.
\smallskip

\noindent\textbf{Step 1:} $s\leq -\frac{1}{2}$.
Since $H^{-1/2}(\Gamma)$ is continuously embedded in $H^s(\Gamma)$,
\begin{equation}\label{lone}
\|(\Lambda_\gamma-\Lambda_1)f\|_{H^s(\Gamma)}\leq
C\|(\Lambda_\gamma-\Lambda_1)f\|_{H^{-1/2}(\Gamma)}.
\end{equation}
As shown in \cite{Toufic}, the following estimate holds for $p>1$,
\begin{equation}\label{ltwo}
\|(\Lambda_\gamma-\Lambda_1)f\|_{H^{-1/2}(\Gamma)}\leq
C\|\gamma-1\|_{L^{2p}(\Omega)}\|f\|_{H^{1/2}(\Gamma)}.
\end{equation}
The estimate \eqref{dtn} follows by combining \eqref{lone} and
\eqref{ltwo}.
\smallskip

\noindent\textbf{Step 2:} $ s\geq \frac{3}{2}$.
First, we recall the following estimate (proved in
\cite{CoKnSi} for $m= \frac{1}{2})$:
\begin{equation}\label{phim}
\|(\Lambda_\gamma-\Lambda_1)f\|_{H^{3/2}(\Gamma)}
\leq C\|u^\gamma-u^1\|_{H^1(\Omega)}.
\end{equation}
Since
\begin{gather*}
\operatorname{div} (\gamma\nabla u^\gamma)=0 \quad \text{in } \Omega, \\
\Delta u^1=0 \quad \text{in }  \Omega, \\
u^\gamma=u^1=f \quad \text{on } \Gamma.
\end{gather*}
It is clear that
$(u^\gamma-u^1)\in H_0^1(\Omega)$ solves the  homogenous Dirichlet problem
\begin{gather*}
\operatorname{div}(\gamma\nabla( u^\gamma-u^1))
 =  -\operatorname{div}((\gamma-1)\nabla u^1)\quad\text{in }\Omega, \\
u^\gamma-u^1 =  0\quad\text{on }\Gamma.
\end{gather*}
Since $u^1\in H^1(\Omega)$ and $(\gamma-1)\in L_+^\infty(\Omega)$,
it follows that $\operatorname{div}((\gamma-1)\nabla u^1)\in H^{-1}(\Omega)$.
From standard estimates for linear elliptic boundary-value problems,
the following estimate holds
\begin{equation}\label{estim}
\|u^\gamma-u^1\|_{H^1(\Omega)}\leq C\|\operatorname{div}((\gamma-1)\nabla
u^1)\|_{H^{-1}(\Omega)}.
\end{equation}
 By denoting $\rho=\operatorname{supp}(\gamma-1)$ and using the
divergence theorem, one gets
\begin{align*}
&\|\operatorname{div}((\gamma-1)\nabla u^1)\|_{H^{-1}(\Omega)}\\
&= \sup_{v\in H_0^1;\|v\|_{H^1(\Omega)}\leq 1}|
\langle \operatorname{div}((\gamma-1)\nabla u^1),v\rangle |\\
&= \sup_{v\in H_0^1;\|v\|_{H^1(\Omega)}\leq 1}
\big|\int_{\rho}(\gamma-1)\nabla u^1\nabla v dx\big|\\
&\leq  \sup_{v\in H_0^1;\|v\|_{H^1(\Omega)}\leq 1}
\Big(\int_{\rho}(\gamma-1)^2|\nabla
u^1|^2dx\Big)^{1/2}\Big(\int_\rho |\nabla v|^2\Big)^{1/2}.
\end{align*}
 Since $\|v\|_{H^1(\Omega)}\leq 1$ we get (see \cite{Toufic})
\begin{gather*}
\int_\rho |\nabla v|^2dx \leq 1\,,\\
\int_\rho |\nabla u^1|^{2q'}dx <\infty \quad \text{for } q'> 1.
\end{gather*}
Now we are able to apply the Holder inequality and we
deduce that for $(p',q')\in ]1,\infty[^2$ such that $1/p'+1/q'=1$,
\begin{equation}\label{hmone}
\|\operatorname{div}((\gamma-1)\nabla u^1)\|_{H^{-1}(\Omega)}
\leq \Big(\int_\rho(\gamma-1)^{2p'}\Big)^{\frac{1}{2p'}}
\Big(\int_\rho|\nabla u^1|^{2q'}\Big)^{\frac{1}{2q'}}.
\end{equation}
In \cite{Toufic}, the following estimate was proved,
\begin{equation}\label{normlp}
\Big(\int_\rho |\nabla u^1|^{2q'}\Big)^{\frac{1}{2q'}}\leq
C\|u^1\|_{H^1(\Omega)}.
\end{equation}
By denoting $p=2p'$, combining the energy estimate
$\|u^1\|_{H^1(\Omega)}\leq C\|f\|_{H^{1/2}(\Gamma)}$ and \eqref{normlp}, we
deduce
$$
\|\operatorname{div}((\gamma-1)\nabla u^1)\|_{H^{-1}(\Omega)}\leq
C\|\gamma-1\|_{L^{p}(\Omega)}\|f\|_{H^{1/2}(\Gamma)}.
$$
Finally
\[
\|(\Lambda_\gamma-\Lambda_1)f\|_{\frac{3}{2}}\leq
C\|\gamma-1\|_{L^{p}(\Omega)}\|f\|_{H^{1/2}(\Gamma)}.
\]

\noindent\textbf{Step 3:} $ -1/2< s\leq 3/2$.
In this case we have
$s=(1-\theta)(-\frac{1}{2})+\theta(3/2)$ for $\theta \in ]0,1]$;
so the space $H^s(\Gamma)$ is an interpolation space
of $H^{-1/2}(\Gamma)$ and $H^{3/2}(\Gamma)$. In other words,
$H^s(\Gamma)=[H^{-1/2}(\Gamma),H^{3/2}(\Gamma)]_\theta$
(See \cite{Lions}).
By applying the interpolation inequality we deduce
$$
\|(\Lambda_\gamma-\Lambda_1)f\|_{H^s(\Gamma)}\leq
C\|(\Lambda_\gamma-\Lambda_1)f\|_{H^{-\frac{1}{2}}(\Gamma)}
^\theta\|(\Lambda_\gamma-\Lambda_1)f\|_{H^{3/2}(\Gamma)}^{1-\theta}
$$
Finally, by using the estimates of step 1 and step 2, we deduce
\eqref{dtn} for $ -1/2< s\leq 3/2$.
\end{proof}

\begin{theorem} \label{thm2.2}
For $\gamma=1$ near $\Gamma$ such that $\gamma\in C^2(\Omega)$,
and $u_0 \in H^{1/2}(\Gamma)$, there exists a constant $C(T)$ depending on
$\beta$, $u_0$, and $T$ such that :
\begin{equation}\label{semi2}
\|e^{-t\Lambda_\gamma}u_0-e^{-t\Lambda_1}u_0\|_{L^2(\Gamma)}\leq
C(T)\|\gamma-1\|_{L^p(\Omega)}.
\end{equation}
\end{theorem}
The estimate in the above theorem follows directly from \eqref{dtn},
see \cite{CEES}. We omit its proof.


\section{The discrete problem}

For the rest of this article, we assume that $\partial \Omega$ is a polyhedron.
To describe the time discretization with an  adaptive choice
of local time steps, we introduce a partition of the interval
$[0,T]$ into  equal subintervals $I_n=[t_{n-1},t_{n}]$,
$1\le n \le N$, such that $0=t_0 \le t_1 \le \dots \le t_N=T$. We denote
by $\tau$ the length of the subintervals $I_n$.

 Now, we describe the space discretization. Let
$(\mathcal{T}_{h})_h$ be a regular triangulation of $\Omega$.
$(\mathcal{T}_{h})_h$ is a set of non degenerate elements which
satisfies:
\begin{itemize}
\item for each $h$, $\bar{\Omega}$ is the union of all elements of
$\mathcal{T}_{h}$;

\item the intersection of two distinct elements of
$\mathcal{T}_{h}$, is either empty, a common vertex, or an entire
common edge;

\item the ratio of the diameter of an element $\kappa$ in
$\mathcal{T}_{h}$ to the diameter of its inscribed circle is
bounded by a constant independent of $n$ and $h$.
\end{itemize}
As usual, $h$ denotes the maximal diameter of the elements of all
$\mathcal{T}_{h}$. For each $\kappa$ in $\mathcal{T}_{h}$, we
denote by $P_1(\kappa)$ the space of restrictions to $\kappa$ of
polynomials with two variables and total degree at most one.

For a given triangulation $\mathcal{T}_{h}$, we define by $X_{h}$
a finite dimensional space of functions such that their
restrictions to any element $\kappa$ of $\mathcal{T}_h$ belong to
a space of polynomials of degree one. In other words,
\[
X_{h}=\{v_n^h\in C^0(\overline{\Omega}), \,\, v_hh|_{\kappa}\,
\text{is affine for all }\kappa \in \mathcal{T}_{h} \}.
\]
 We note that for each  $h$, $X_{h}\subset H^1(\Omega)$.

The full discrete implicit scheme associated with the problem
\eqref{V} is as follows:

\noindent Given ${u}^{n-1}_h \in X_{h}$, find $u^n_h$ with
values in $X_{h}$ such that for all $v_h \in X_h$ we have:
\begin{equation}\label{Vh}
 \int_\Omega \gamma(x) \nabla  u^n_h \nabla v_h dx +
\int_\Gamma  \frac{u_h^{n} - { u}_h^{n-1}}{\tau_{n}} \, v_h
d\sigma = 0.
\end{equation}
by assuming that $u_h^0$ is an approximation of $u(0)$ in $X_{h}$.

\begin{remark} \label{rmk3.1} \rm
 It is a simple exercise to prove existence and
uniqueness of the solution of problem \eqref{Vh} as a consequence
of discrete problem of Poisson's equation with a Robin condition.
\end{remark}

\begin{theorem} \label{thm3.2}
For each $m=1,\dots ,N$, the solution $u_{h}^m$ of
the problem \eqref{Vh} satisfies
\begin{equation}\label{numbound}
\|u_{h}^{m}\|_{0,\Gamma}^2+\sum_{n=1}^{m}\tau_{n}|u_{h}^{n}|_{1,\Omega}^2\leq
 c \|u_h^0\|_{0,\Gamma}^2,
\end{equation}
\end{theorem}

\begin{remark} \label{rmk3.3} \rm
In \cite{ElDiSa}, we establish optimal \emph{a priori} and
\emph{a posteriori} error estimates for the problem \eqref{Vh} an
shown numerical results of validation.
\end{remark}

\section{Numerical results}


\begin{figure}[ht]
\begin{center}
\subfigure[$\operatorname{Err}_u^n$ with respect to the iteration numbers for 
$\gamma^1_{5,3/4}$, ($\operatorname{Err}^4_\gamma=0.86$)]
{\includegraphics[width=0.45\textwidth]{fig1a} } % F1_erru_alpha_5_rho_3s4_p_4.pdf
\quad
\subfigure[$\operatorname{Err}_u^n$ with respect to the iteration numbers for
 $\gamma^2_{10,3/4}$, ($\operatorname{Err}^4_\gamma=1.14$)]
{\includegraphics[width=0.45\textwidth]{fig1b}} % F2_erru_alpha_10_rho_3s4_p_4.pdf
\\
\subfigure[$\operatorname{Err}_u^n$ with respect to the
iteration numbers for $\gamma^3_{e^8,1/2}$,
($\operatorname{Err}^4_\gamma=0.84$)]
{\includegraphics[width=0.45\textwidth]{fig1c}}  % F3_erru_alpha_e8_rho_1s2_p_4.pdf
\end{center}
\caption{$\operatorname{Err}_u^n$ with respect to the iteration
numbers for different functions $\gamma^i_{\alpha,\rho}$, $i=1,2,3$.}
\label{fig1}
\end{figure}

To validate the theoretical results, we present several numerical
simulations using the FreeFem++ software (see \cite{hecht}). We
choose $T=3$,
\[
u(0,x,y) =  \frac{x^2-y^2}{2} + y + \frac{1}{2},
\]
and the function $\gamma$ as (see \cite{JMSS})
%
\begin{equation}\label{gamma}
\gamma^i_{\alpha,\rho}(x) = (\alpha F_{i,\rho}(|x|) + 1)^2,
i=1,2,3,
\end{equation}
where the function $F_{i,\rho} \in C^4(\mathbb{R})$ satisfies
$F_{i,\rho}(x) = 0$ for $|x| > \rho$ and for $|x| \le \rho$ takes one of the
following three forms:
\begin{gather}\label{exemp1}
 F_{1,\rho}(x) = (x^2 - \rho^2)^4 (1.5 - \cos \frac{3 \pi x}{2 \rho}), \\
\label{exemp2}
 F_{2,\rho}(x) = (x^2 - \rho^2)^4  \cos \frac{3 \pi x}{2 \rho}, \\
\label{exemp3}
 F_{3,\rho}(x) =  e^{ -\frac{2(x^2+\rho^2)}{(x+\rho)^2(x-\rho)^2} }.
\end{gather}

We consider the two-dimensional unit circle. In fact,
the mesh corresponding to $\Omega$ is a polygon and we introduce
here a geometrical approximation. Nevertheless, the numerical
results given in the end of this section show that this
approximation has not a major influence.
The considered mesh contains $15542$ triangles with
$m=300$ segments on the boundary $\Gamma$. Thus, the mesh step
size is $ h=\frac{2\pi }{m}$. We choose a time step $\tau=h$
and we consider the numerical scheme \eqref{Vh}.

 We denote by $u_{h,\gamma}^n$ the solution of 
problem \eqref{Vh} for a given $\gamma$ and $u_{h,1}^n$ the
solution of the same problem for $\gamma=1$. We define the
errors
\begin{gather*}
\operatorname{Err}_u^n = \|u_{h,\gamma}^n - u_{h,1}^n \|_{L^2(\Gamma)}, \\
\operatorname{Err}_u = \max_{1\le i \le N} \operatorname{Err}_u^i, \\
\operatorname{Err}^p_\gamma = \|\gamma - 1 \|_{L^p(\Omega)}.
\end{gather*}
We choose $p=4$ and followed \cite{JMSS} for the choice
of $\rho$ and $\alpha$. Figures \ref{fig1}(a)-(c) show the evolution
of $\operatorname{Err}_u^n$ with respect to the iteration numbers for the three
cases of $\gamma$. It is easy to check that all this curves are
bounded and smaller than the corresponding $\operatorname{Err}_\gamma^4$. For
example, Figure \ref{fig1}(b) represents the error $\operatorname{Err}_\gamma^u$
for the second function $\gamma^2_{10,3/4}$ with a maximum of
$0.0309$ which is smaller the corresponding $\operatorname{Err}_u^4 = 1.14$.


 To show the dependency of this errors with $\rho$, in an
other word where it equals to $1$ in a neighborhood of $\Gamma$
(the neighborhood depends on $\rho$), table \ref{fgh} shows
$\operatorname{Err}_u$ and $\operatorname{Err}^4_\gamma$ with respect
 to $\rho$ for the functions $\gamma^1_{5,\rho}$ and $\gamma^2_{10,\rho}$, and for
$T=1$ and $p=4$. We remark that $\operatorname{Err}_u$ is always smaller than
$\operatorname{Err}^4_\gamma$ in all the considered cases.

\begin{table}[htb] \scriptsize
 \caption{$\operatorname{Err}_u$ and $\operatorname{Err}^4_\gamma$ with respect to
$\rho$ for the three cases of $\gamma$: $\gamma^1_{5,\rho}$ and
$\gamma^2_{10,\rho}$.} \label{fgh}
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular} {|l|*{10}{c|}}
\hline 
$\gamma^1_{5,\rho}$ &\multicolumn{9}{r}{}& \\
 \hline 
$\rho$ &  0.5 &  0.55 &  0.6 & 0.65 & 0.7 & 0.75 &  0.8 &  0.85 & 0.9 & 0.95 \\
 \hline 
$\operatorname{Err}_u$ & 0.002 & 0.005 & 0.012 & 0.026 & 0.053 & 0.098  
 & 0.169  & 0.267  & 0.391  & 0.537 \\
 \hline 
$\operatorname{Err}_\gamma$ & 0.022 & 0.051 & 0.109 & 0.2232 & 0.44 & 0.855 
& 1.65 & 3.23 & 6.37 & 12.72\\
\hline 
\end{tabular}
\\
\medskip

\begin{tabular} {|l|*{10}{c|}}
\hline 
$\gamma^2_{10,\rho}$  &\multicolumn{9}{r}{}& \\
 \hline 
$\rho$ &  0.5 &  0.55 &  0.6 & 0.65 & 0.7 & 0.75 &  0.8 &  0.85 & 0.9 & 0.95 \\
 \hline 
$\operatorname{Err}_u$ & 0.0002 & 0.0007 & 0.0018 & 0.0047 
& 0.01182 & 0.02999 & 0.077 & 0.200 & 0.4833 & 0.5680\\
 \hline 
$\operatorname{Err}_\gamma$ & 0.0263 & 0.0601 & 0.1301 & 0.2716 
 & 0.5574 & 1.1440 & 2.3757 & 5.0105 & 10.6903 & 22.8861\\
 \hline
\end{tabular}
\end{center}
\end{table}


To show the dependency with $p$, we consider for example
the functions $\gamma^1_{5,3/4}$ and $\gamma^3_{e^8,1/2}$ and we
study the errors for different values of $p>2$. Figures \ref{fig2}(a)
and \ref{fig2}(b) show $\operatorname{Err}^p_\gamma$ with respect to $p$.
 We remark that the corresponding curves increase with $p$ starting from
$0.75$ for Figure \ref{fig2}(a) and from $0.34$ for the Figure \ref{fig2}(b),
whereas the values of $\operatorname{Err}_u$ are $0,03$ for the first case
$\gamma^1_{5,3/4}$ and $0.08$ for the third one
$\gamma^3_{e^8,1/2}$.

\begin{figure}[htb]
\begin{center}
\subfigure[$\operatorname{Err}^p_\gamma$ with respect to $p$ 
for  $\gamma^1_{5,3/4}$]
{\includegraphics[width=0.45\textwidth]{fig2a}}
% Errgamma_F1_alpha_10_rho_3s4_different_p.pdf}
\quad
\subfigure[$\operatorname{Err}^p_\gamma$ with respect to $p$  
$\gamma^3_{e^8,1/2}$]
{\includegraphics[width=0.45\textwidth]{fig2b}} 
%Errgamma_F3_alpha_e8_rho_1s2_different_p.pdf}
\end{center}
\caption{$\operatorname{Err}^p_\gamma$ with respect to $p$ for the
first and the third function $\gamma^i_{\alpha,\rho},i=1,3$.}
\label{fig2}
\end{figure}

 We remark that all the numerical results validate
the theoretical estimates. 

\subsection*{Acknowledgments}
 The authors want to thank the anonymous referees for their careful reading 
of the orignal manuscript and for their suggestions.

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\end{document}