\documentclass[reqno]{amsart}


\AtBeginDocument{{\noindent\small 2004-Fez conference on
Differential Equations and Mechanics \newline
 {\em Electronic Journal of Differential Equations},
Conference 11, 2004, pp. 41--51.\newline ISSN: 1072-6691. URL:
http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{41}

\begin{document}

\title[\hfilneg EJDE/Conf/11 \hfil Numerical analysis of Euler-SUPG method] 
{Numerical analysis of Euler-SUPG modified method for
transient viscoelastic flow}

\author[M. Bensaada, D. Esselaoui\hfil EJDE/Conf/11 \hfilneg]
{Mohammed Bensaada, Driss Esselaoui}  % in alphabetical order

\address{Mohammed Bensaada \hfill\break
Laboratoire des Sciences de l'Ing\'enieur\\
 Analyse Num\'erique et Optimisation (SIANO)\\
 Facult\'e des Sciences, Universit\'e Ibn Tofail \\
 B.P.133, 14000-K\'enitra, Maroc}
 

\address{Driss Esselaoui \hfill\break
Laboratoire des Sciences de l'Ing\'enieur \\
 Analyse Num\'erique et Optimisation (SIANO)\\
 Facult\'e des Sciences, Universit\'e Ibn Tofail \\
 B.P.133, 14000-K\'enitra, Maroc}
 \email{desselaoui@yahoo.fr \; Fax: 212 37 37 27 70}

\date{}
\thanks{Published October 15, 2004.}
\thanks{Supported  by CNRST - CNRS, Program STIC01/03, and
by CNRST- GRICES Portugal} 
\subjclass[2000]{65N30, 34K25, 74S05}
\keywords{Stabilized method; finite elements; modified SUPG method;
 \hfill\break\indent
transient viscoelastic flow}

\begin{abstract}
 We study a new approximation scheme
 of transient viscoelastic fluid flow obeying an Oldroyd-B type
 constitutive law. The approximation stress, velocity,
 pressure are respectively $P_{1}$-continuous,
 $P_{2}$-continuous, $P_{1}$-continuous. We use the modified
 streamline upwinding Petrov-Galerkin method induced by the
 modified Euler method. We assume that the continuous problem
 admits a sufficiently smooth and sufficiently small solution. We
 show that the approximate problem has a solution and we give an
 error bound.
\end{abstract}

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

\section{Introduction and Presentation of the problem}
In the numerical simulation of the viscoelastic fluid flows, the
hyperbolic character of the constitutive equation (when using
differential models) has to be taken into account (see
\cite{EsF,MM}). This hyperbolic character implies that some
upwinding is needed to avoid oscillations as in the method of
characteristics \cite{EsF,B}, the Lesaint-Raviart discontinuous
finite element method \cite{BBS,EsR}, the streamline-upwind method
(SU) and the streamline-upwind-Petrov-Galerkin method (SUPG)
\cite{EsRZ,MM}. The numerical analysis of the steady case of the
viscoelastics fluids flows is abundant. Although the list is not
exhaustive, one may see for example \cite{BBS,EsRZ,S}. Moreover
the numerical analysis for transient viscoelastic flow remain
quite few \cite{K,WB}.  For example, some difficulties appear,
when we use continuous finite element approximation for
$(\sigma,u,p)$ and the standard SUPG method for the convection of
the extra stress tensor. To give some response to this
difficulties, we develop in this paper the study of continuous
finite element(F.E) approximation of  a transient viscoelastic
fluid flow obeying an Oldroyd-B model. For the convective term of
the constitutive equation we use some modified SUPG method linked
to a variant of implicit Euler method (see \cite{BEs}). Under this
condition we are able to show that the approximate problem is
stabilized and has a solution and we give an error bound.

The transient viscoelastic fluid flow obeying an Oldroyd-B type
constitutive law is considered flowing in a bounded, connected
open set $\Omega$ in $IR^{2}$ with lipshitzian boundary $\Gamma$;
$n$  is the outward unit normal to $\Gamma$. The basic set of
equations of the Oldroyd-B model with a single relaxation time is
given by
\begin{equation} \label{O}
\begin{gathered}
\lambda (\frac{\partial\sigma}{\partial
t}+(u.\nabla)\sigma+\beta(\sigma,\nabla u) )
+\sigma-2\alpha D(u)=0 \quad \mbox{in } \Omega \times ] 0,T [,\\
Re\;\frac{\partial u}{\partial
t}-\nabla.\sigma-2(1-\alpha)\nabla.D(u)+\nabla p
=f\quad \mbox{in } \Omega \times] 0,T[, \\
 \nabla.u =0 \quad\mbox{in } \Omega\times] 0,T[,\\
u=0  \quad \mbox{on } \Gamma\times] 0,T [,\\
u=u_{0} ,\quad  \sigma=\sigma_{0}\quad\mbox{in } \Omega \;  t=0.
\end{gathered}
\end{equation}
Where  $\lambda > 0$,  $Re$  and $0 <\alpha < 1$  are respectively
the Weissenberg number,  the Reynolds number and the viscosity
ratio constant. $f$  is a density of forces.
$D(u)=\frac{1}{2}(\nabla u+\nabla u^{\top})$ the rate of strain
tensor, and  $\beta(\sigma,\nabla u)=-\nabla u\sigma-\sigma\nabla
u^{\top}$.

\begin{remark} \label{rmk1} \rm
The boundary condition $u=0$  on $\Gamma$ can be replaced by $u=
u_d$  on $\Gamma$. Regarding  $\sigma$ and the hyperbolic
character of the constitutive equation, we have to impose $\sigma
= \sigma_d$  on $\Gamma^{-}= \{x \in\Gamma;\;u_d.n(x) <0 \}$.
\end{remark}

\begin{remark} \label{rmk2} \rm
The inertia term $(u.\nabla)u$  is neglected in the momentum
equation in order to  make the analysis simpler.
\end{remark}

Let us define the following spaces:
\begin{gather*}
T=\{\tau=(\tau_{ij})_{1\leq i,j\leq 2} :
\tau_{ij}=\tau_{ji};\,\tau_{ij} \in L^{2}(\Omega);\,i,j=1,2\},
\quad
X=(H_{0}^{1}(\Omega))^{2} \\
Q=\{ q\in L^{2}(\Omega) / {\int_{\Omega}}q\,dx=0\},\quad V=\{v\in
X / (q,\nabla.v)=0;\forall q\in Q \}.
\end{gather*}
The norm and scalar product in $L^{2}(\Omega)$ of functions,\
vectors and tensors are denoted respectively by $\vert \cdot\vert$
and $(\cdot,\cdot)$; ($\vert\cdot\vert_{\Gamma}$ and
$(\cdot,\cdot)_{\Gamma}$ in $L^{2}(\Gamma)$); $\langle f,v\rangle
$ will denote the duality between $f\in (H^{-1}(\Omega))^{2}$ and
$v\in X$.

\begin{remark} \label{rmk3} \rm
Existence results for problem \eqref{O} are proved in \cite{GS}.
In order to make some theoretical analysis of approximate problem
of \eqref{O}  we use the regularity imposed in \cite{GS}.
\end{remark}

\section{Description of the approximation scheme}

\subsection*{FE approximation}
We suppose $\Omega$ polygonal and we consider a triangulation
$\Im_{h}$ on $\Omega$ made of triangles $K$ such that
$\overline{\Omega}=\{\bigcup K;K\in \Im_{h}\}$ uniformly regular,
$\exists \nu_{0},\nu_{1}:  \nu_{0}h\leq h_{K}\leq
\nu_{1}\varrho_{K}$ where $\varrho_{K}$ is the diameter of the
greatest ball included in $K$ and $h_{\rm
max}=\max_{K\in\Im_{h}}h_{K}$.

We use the  Taylor-Hood finite element method for approximations
in space of $(u,p)$: $P_{2}$-continuous in velocity,
$P_{1}$-continuous in pressure and we consider $P_{1}$-continuous
approximation of the stresses. The corresponding $FE$ space are:
\begin{gather*}
 X_{h}=\{v\in X \cap C^{0}(\Omega)^{2}: v_{|K} \in P_{2}(K)^{2},
\forall K\in \Im_{h}\}\\
 Q_{h}=\{q\in Q \cap C^{0}(\Omega): q_{|K} \in P_{1}(K), \forall K\in
\Im_{h}\}\\
 V_{h}=\{ v\in X_{h}: (q,\nabla . v)=0, \,\forall q\in Q_{h}\}. \\
 T_{h}=\{\tau\in T\cap C^{0}(\Omega):\tau|_{K}\in P_{1}(K), \forall
K\in  \Im_{h}\}\,,
\end{gather*}
where $P_{m}(K)$ denotes the space of polynomials of degrees less
or equal to $m$ on $K\in T_{h}$. The term $((u.\nabla)\sigma,
\tau)$ is approximated by means of an operator $B$  on
$X_{h}\times T_{h}\times T_{h}$ defined by
\begin{align*}
B(u_{h},\sigma_{h};\tau_{h})
&=((u_{h}(t).\nabla)\sigma_{h}(t),\tau_{h})
+\delta_{0}(h,t)((u_{h}(t)\nabla)\sigma_{h}(t),(u_{h}(t).\nabla)\tau)\\
&\quad +(1/2)((\nabla.u_{h})(t) \sigma,\tau).
\end{align*}
For the steady case you can see \cite{S}.

\subsection*{Numerical method}
We propose Euler-SUPG modified scheme, implicit in time, based on
the scheme proposed for the transport equation  (see \cite{BEs}).
We construct an approximation of the solution at each time step
$nk, n=0,\dots,N$ in the following way. We start with
$u_{h}^{0}=\tilde{u}_{0}$: elliptic projection of $u_{0}$ into
$V_{h},\sigma_{h}^{0}=\tilde{\sigma}_{0}$: orthogonal projection
of $\sigma_{0}$ into $T_{h}$. Given $u_{h}^{0},\dots,u_{h}^{n}$;
 $\sigma_{h}^{0},\dots,\sigma_{h}^{n}$, because
$(X_{h},Q_{h})$ satisfies the inf sup condition, we look for the
solution of the following problem, find
$(u_{h}^{n+1},\sigma_{h}^{n+1})\in V_{h}\times T_{h}$, such that
\begin{gather}
\begin{aligned}
&\lambda(\frac{\sigma_{hu_{h}^{n},\delta}^{n+1}-\sigma_{hu_{h}^{n},\delta}^{n}}{k},
\tau_{u_{h}^{n},\lambda})+
(\sigma_{h}^{n+1},\tau_{u_{h}^{n},\lambda})+B(\lambda
u_{h}^{n},\sigma_{h}^{n+1};\tau)  \\
&-2\alpha (D(u_{h}^{n+1}),\tau_{u_{h}^{n},\lambda})+
\lambda(\beta(\sigma_{h}^{n+},\nabla
u_{h}^{n+1}),\tau_{u_{h}^{n},\lambda})=0\quad \forall\tau\in
T_{h};
\end{aligned} \label{eq:sc1} \\
\begin{aligned}
&Re(\frac{u_{h}^{n+1}-u_{h}^{n}}{k},v)+(\sigma_{h}^{n+1},D(v))+
2(1-\alpha))(D(u_{h}^{n+1}),D(v)) \\
&=\langle f(t_{n+1},x),v\rangle\quad  \forall v\in V_{h},
\end{aligned} \label{eq:sc2}
\end{gather}
where
$\sigma_{hu_{h}^{n},\delta}^{i}=\sigma_{h}^{i}+\delta(h,k)(u_{h}^{n}.
\nabla)\sigma_{h}^{i}$ $(i=n,n+1)$;
$\tau_{u_{h}^{n},\lambda}=\tau+\lambda
\delta_{0}(h,k)(u_{h}^{n}.\nabla)\tau$ for all $\tau\in T_{h}$,
and $ \delta (,)$ (resp. $\delta_{0}(,)$) will be specified later.
In order to show that equation $(\ref{eq:sc1})-(\ref{eq:sc2})$
defined uniquely $(u_{h}^{n+1},\sigma_{h}^{n+1})$, we multiply
equation \eqref{eq:sc2} by $2\alpha$ and add the equation obtained
to equation  \eqref{eq:sc1}, we get
\begin{align*}
&\lambda(\frac{\sigma_{hu_{h}^{n},\delta}^{n+1}
-\sigma_{hu_{h}^{n},\delta}^{n}}{k},\tau_{u_{h}^{n},\lambda})+2\alpha
Re(\frac{u_{h}^{n+1}-u_{h}^{n}}{k},v_{h})+ B(\lambda u_{h}^{n},
\sigma_{h}^{n+1};\tau_{h})\\
&+A(u_{h}^{n};(\sigma_{h}^{n+1},u_{h}^{n+1}),(\tau,v)) +\lambda
(\beta(\sigma_{h}^{n+1},\nabla
u_{h}^{n+1}),\tau_{u_{h}^{n},\lambda})\\
&=2\alpha\langle f(t_{n+1}),v_{h}\rangle\,,\quad \forall
(\tau_{h},v_{h}) \in T_{h} \times V_{h}.
\end{align*}
where $A(.,.)$ is a bilinear form on $T_{h}\times V_{h}$ defined
as
$$
A(w;(\sigma,u),(\tau,v))=(\sigma,\tau_{w,\lambda})+2\alpha(\sigma,D(v))
-2\alpha(D(u),\tau_{w,\lambda})+4\alpha(1-\alpha)(D(u),D(v))
$$
 From this, we can establish some error bound and then the
following existence result

\begin{theorem} \label{thm1}
There exists $M_{0}$, and  $h_{0}$ such that if problem
\eqref{O} admits a solution $(\sigma,u,p)$ with,
\begin{align*}
M=\max\big\{&\Vert\sigma\Vert_{C^{1}([t_{n},t_{n+1}];H^{2})},
\Vert u\Vert_{C^{1}([t_{n},t_{n+1}];H^{3})},
\Vert p\Vert_{C^{0}([t_{n},t_{n+1}];H^{2})}, \\
&\Vert\sigma\Vert_{C^{2}([t_{n},t_{n+1}];L^{2})}, \Vert
u\Vert_{C^{2}([t_{n},t_{n+1}],T;L^{2})}\big\}< M_{0},
\end{align*}
then if $\delta(,)$ satisfies $\delta(,)=\lambda \delta_{0}(,)$
and $h\leq h_{0}$ there exists a unique solution in $T_{h}\times
V_{h}$ of problem  \eqref{eq:sc1}- \eqref{eq:sc2}.
\end{theorem}

\begin{proof}
For this purpose we define a mapping $\Phi: T_{h}\times
V_{h}\longmapsto T_{h}\times V_{h}$, which to $(\sigma_{1},u_{1})$
associates $(\sigma_{2},u_{2})=\Phi(\sigma_{1},u_{1})$, where
$(\sigma_{2},u_{2})\in T_{h}\times V_{h}$ satisfies
\begin{align*}
&\lambda(\frac{\sigma_{2u_{h}^{n},\delta}
-\sigma_{hu_{h}^{n},\delta}^{n}}{k},\tau_{u_{h}^{n},\lambda})
+2\alpha Re(\frac{u_{2}-u_{h}^{n}}{k},v_{h})
+ B(\lambda u_{h}^{n},\sigma_{h}^{n+1};\tau_{h})\\
&+A(u_{h}^{n};(\sigma_{2},u_{2}),(\tau,v)) \\
&=-\lambda (\beta(\sigma_{1},\nabla
u_{1}),\tau_{u_{h}^{n},\lambda})+2\alpha\langle
f(t_{n+1}),v_{h}\rangle\,,\quad \forall (\tau_{h},v_{h}) \in
T_{h}\times V_{h}.
\end{align*}
We define a ball $B_{h}^{(n+1)}$ as follows: let $C^{\ast}$ be
given. Then we define
\begin{align*}
B_{h}^{(n+1)}=\big\{&(\tau_{h},v_{h})\in T_{h}\times V_{h}:
[\frac{\alpha Re}{k}\Vert v_{h}-u(t_{n+1})\Vert_{0,\Omega}^{2}\\
&+\frac{\lambda}{2k}\Vert(\tau_{h}-\sigma(t_{n+1}))_{u_{h}^{n},\lambda}
\Vert_{0,\Omega}^{2}]^{1/2} \leq
C^{\ast}(k+\delta+h\sqrt{\delta_{0}}+\frac{h^{2}}{\sqrt{\delta_{0}}}
+h^{\frac{3}{2}}),\\
& [\frac{1}{4}\{4\alpha(1-\alpha)\Vert
D(v_{h}-u(t_{n+1})\Vert_{0,\Omega}^{2}
+\Vert\tau_{h}-\sigma(t_{n+1})\Vert_{0,\Omega}^{2}]^{1/2}\\
& \leq
C^{\ast}(k+\delta+h\sqrt{\delta_{0}}+\frac{h^{2}}{\sqrt{\delta_{0}}}
+h^{\frac{3}{2}})\big\}.
\end{align*}
The proof is decomposed in five parts:\\
(a) $\Phi$ is well defined for $h\leq h_{1}=h_{1}(h_{\rm
max},\alpha)$ and
 bounded on bounded sets,\\
(b) let $C_{0}$ be a positive constant independent of $h,k$ and
$\lambda,\alpha,Re$. If $h\leq h_{0}=\min\big\{h_{1},\sqrt{k}
\big(\frac{C^{*}}{C_{0}M\sqrt{\alpha Re}}\big)^{2/3},
\frac{1}{C_{0}M} \sqrt{k/ \lambda}\big\}$ and $\sqrt{\delta/
k}\leq \frac{1}{C_{0}M}$, we have $B_{h}^{(n+1)}$ is non\-empty. On
the other hand, if $M_{0}=C^{*} \frac{\gamma}{\lambda+\gamma}$,
$\gamma=\sqrt{\alpha(1-\alpha)}$, we can prove that
$\Phi({\overbrace{B}^{\circ}}{_{h}^{(n+1)}}) \subset
{\overbrace{B}^{\circ}}{_{h}^{(n+1)}}$,
\\
(c) $\Phi$ is continuous on $T_{h}\times V_{h}$.
$\Phi(B_{h}^{(n+1)})\subset B_{h}^{(n+1)}$, Brouwer's theorem then
gives the existence of fixed point
$(\sigma_{h}^{n+1},u_{h}^{n+1})$ of $\Phi$ solution of problem
$(1)-(2)$.
\\
(d) Furthermore, if $\lambda M$ and $\lambda M\gamma^{-1}$ is
sufficiently small, $\Phi$ is a contraction mapping on
$B_{h}^{(n+1)}$.

Then a result of existence and uniqueness follows from the fixed
theorem.
\end{proof}

\begin{remark} \label{rmk4}
When we use only the classical Euler-scheme in time and SUPG
method, we can't point out result of (a) in the above proof.
\end{remark}

\section{Main result and error estimates}

Suppose that the continuous problem admits a sufficiently smooth
and sufficiently small solution, we can show, the following
result.

\begin{theorem} \label{thm2}
There exists $M_{0}$ and $h_{0}$ such that if problem \eqref{O}
admits a solution $(\sigma,u,p)$ with $\sigma\in
C^{1}([0,T],(H^{2})^{4})\cap C^{2}([0,T],(L^{2})^{4})$; $u\in
C^{1}([0,T],(H^{3})^{2})\cap C^{2}([0,T],(L^{2})^{2})$; $p\in
L^{2}([0,T],(H^{2})\cap L_{0}^{2})\cap C^{0}([0,T],H^{2})$,
satisfying
$$
\max\{\Vert\sigma\Vert_{C^{1}(0,T;H^{2})},\Vert
u\Vert_{C^{1}(0,T;H^{3})},\Vert p\Vert_{C^{0}(0,T;H^{2})},
\Vert\sigma\Vert_{C^{2}(0,T;L^{2})},\Vert
u\Vert_{C^{2}(0,T;L^{2})}\} \leq M_{0}
$$
then for $kh^{-(1+\varepsilon)}(0<\varepsilon\leq 1/2)$ bounded,
there exists a constant $C$ independent  of $h$ and $k$ such that
$$
\max_{0\leq n\leq N}\vert
(\sigma_{h}^{n}-\sigma(t_{n}))_{u_{h}^{n-1}}\vert+(
\sum_{n=0}^{N}k\vert\sigma_{h}^{n}-\sigma(t_{n})\vert)^{2})^{1/2}
\leq C
(k+\delta+h\sqrt{\delta_{0}}+\frac{h^{2}}{\sqrt{\delta_{0}}}
+h^{1+\varepsilon})
$$
and
$$
\max_{0\leq n\leq N}\vert
u_{h}^{n}-u(t_{n})\vert+(\sum_{n=0}^{N}k\vert
D(u_{h}^{n}-u(t_{n}))\vert^{2})^{1/2}\leq C
(k+\delta+h\sqrt{\delta_{0}}+\frac{h^{2}}{\sqrt{\delta_{0}}}+h^{1+\varepsilon})
$$
where $N\in N^{\ast}:Nk=T$,  $u_{h}^{-1}=u_{h}^{0}$ and
$(\sigma_{h}^{i},u_{h}^{i})_{1\leq i\leq N}$ are solutions of
$(\eqref{eq:sc1}-\eqref{eq:sc2})_{1\leq i\leq N} $.
\end{theorem}

\begin{remark} \label{rmk5}\rm
 The discontinuous stresses approach case with
Euler semi-implicit method in time was treated by Baranger and
all. \cite{WB}. In the proof of the present result we can fund
some similar technical like in \cite{WB}.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm2}]
For $0\leq n<N$, let $u(t_{n})$, $\sigma(t_{n})$, $p(t_{n})$ be
respectively in
 $H^{3}(\Omega)$, $^{2}(\Omega)$, $H^{2}(\Omega)$ and so, there exists
$(\tilde{u}(t_{n}),\tilde{p}(t_{n}))\in V_{h}\times Q_{h}$ such
that,
\begin{gather}
\vert (u-\tilde{u})(t_{n})\vert+h\Vert
(u-\tilde{u})(t_{n})\Vert_{1,2}\leq C_{1}h^{3}\Vert
u(t_{n})\Vert_{3,2},\label{eq:er1}
\\
\vert(p-\tilde{p})(t_{n})\vert\leq C_{2}h^{2}\Vert
p(t_{n})\Vert_{2,2} ,\label{eq:er2}
\end{gather}
(see \cite{GR}) and there exists $\tilde{\sigma}(t_{n})\in T_{h}$
such that
\begin{equation}
\vert(\sigma-\tilde{\sigma})(t_{n})\vert\leq
C_{3}h^{2}\Vert\sigma(t_{n})\Vert_{2,2} \label{eq:er4}
\end{equation}
(see \cite{GR}). We remark that we can define $\tilde{u}(.)$ by
the elliptic projection of $u(.)$ on $V_{h}$ such that
$a((u-\tilde{u})(.),v_{h})=0, \forall v_{h}\in V_{h}$, where
$a(u,v)=(d(u),d(v))$; then the following properties are also
satisfied
$$
d\tilde{u}/dt=(du/dt)^{\sim}
$$
and
\begin{equation} \label{eq:er5}
\begin{aligned}
\vert(du/dt)(s)-(du/dt)^{\sim}(s)\vert
&\leq C_{4}h\Vert(du/dt)(s)-(du/dt)^{\sim}(s)\Vert_{1,2}\\
&\leq C_{5}h^{3}\Vert(du/dt)(s)\Vert_{3,2}
\end{aligned}
\end{equation}
(see(\cite{RT})), $u$ being in $C^{1}([0,T],H^{3})$; same
properties are satisfied for $\sigma$:
$$
d\tilde{\sigma}/dt=(d\sigma/dt)^{\sim}
$$
and
\begin{equation} \label{eq:er6}
\vert\frac{d\sigma}{dt}(s) -\tilde{\frac{d\sigma}{dt}}(s)\vert
\leq C_{6}h\Vert \frac{d\sigma}{dt}(s)-
\tilde{\frac{d\sigma}{dt}}(s)\Vert _{1,2} \leq
C_{7}h^{2}\Vert\frac{d\sigma}{dt}(s)\Vert_{2,2},
\end{equation}
(see(\cite{RT})).
\end{proof}

In the sequel we shall use the following inverse inequalities (see
\cite{C}).

\begin{lemma} \label{lem1}
 Let $k\geq 0$ be an integer and $W_{h}= \{v, v_{\vert_{K}} \in
P_{k}(K) \,
   \forall K\in \Im_{h}\}$.
Let $r$ and $p$ be reals with $1\leq r,p\leq \infty$ and let
$l\geq 0$ and $m\geq 0$ be integers such that $l\leq m$. Then
there exists a constant $C=C(\nu_{0},\nu_{1},l,r,m,p,k)$ such that
$\forall v\in W_{h}\cap W^{l,r}(\Omega)\cap
W^{m,p}(\Omega),\;\vert v\vert_{m,p}\leq
Ch^{l-m-2\max\{0,1/r-1/p\}}\vert v\vert_{l,r}$.
\end{lemma}

We shall also use the following Sobolev's imbedding theorems.

\begin{lemma} \label{lem2}
 Let $m\geq 0$ be an integer. The following embedding hold
algebraically and topologically:
$$
W^{m+1,2}(\Omega)\subset W^{m,q}(\Omega)\;\forall q\in
[1,\infty[,\quad \mbox{and}\quad  W^{m,p}(\Omega) \subset
C^{0}(\bar{\Omega}).
$$
\end{lemma}
Now, let us denote $e_{h}^{n}=u_{h}^{n}-\tilde{u}(t_{n})$,
$\varepsilon_{h}^{n}=\sigma_{h}^{n}-\tilde{\sigma}(t_{n})$. From
equations (\ref{eq:sc1})-(\ref{eq:sc2}) we have for $(\tau, v)\in
T_{h}\times V_{h}$
\begin{equation}
\begin{aligned}
&2\alpha Re(\frac{e_{h}^{n+1}-e_{h}^{n})}{k},v)+\lambda
(\frac{\varepsilon_{hu_{h}^{n},\delta}^{n+1}-\varepsilon_{hu_{h}^{n},\delta}^{n}}{k},
\tau_{u_{h}^{n},\lambda})+B(\lambda
u_{h}^{n},\varepsilon_{h}^{n+1};
\tau)\\
&+A(u_{h}^{n};(\varepsilon_{h}^{n+1},e_{h}^{n+1}),(\tau,v))\\
&=-\lambda(\frac{\tilde{\sigma}_{hu_{h}^{n},\delta}(t_{n+1})
-\tilde{\sigma}_{hu_{h}^{n},\delta}(t_{n})}{k},\tau_{u_{h}^{n},\lambda})
-2\alpha e(\frac{\tilde{u}(t_{n+1})-\tilde{u}(t_{n})}{k},v)\\
&\quad - B(\lambda u_{h}^{n},\tilde{\sigma}(t_{n+1});\tau)
+A(u_{h}^{n};(-\tilde{\sigma}(t_{n+1}),-\tilde{u}(t_{n+1})),(\tau,v))\\
&\quad -\lambda(\beta(\sigma_{h}^{n},\nabla
u_{h}^{n}),\tau_{u_{h}^{n},\lambda})+2\alpha\langle
f(t_{n+1},x),v\rangle,\quad \forall (\tau,v)\in T_{h}\times V_{h}.
\end{aligned} \label{eq:10}
\end{equation}
But $(\sigma,u,p)$ being the exact solution of problem \eqref{O},
it satisfies the following consistency equation
\begin{align*}
&2\alpha Re(\frac{du}{dt}(t),v)+\lambda(\frac{d\sigma}{dt}(t),
\tau_{u_{h}^{n},\lambda})+B(\lambda u(t),\lambda
u_{h}^{n},\sigma(t);\tau)\\
&+A(u_{h}^{n};(\sigma(t),u(t)),(\tau,v))\\
&=2\alpha(p(t),\nabla.v)+2\alpha\langle f(t),v\rangle
-\lambda(\beta(\sigma(t),\nabla
u(t)),\tau_{u_{h}^{n},\lambda})\quad \forall (\tau,v)\in
T_{h}\times V_{h}
\end{align*}
Inserting the value of $\langle f(t_{n+1}),.\rangle$ in equation
(\ref{eq:10}) we obtain
\begin{equation}
\begin{aligned}
&Re(\frac{e_{h}^{n+1}-e_{h}^{n})}{k},v)+\lambda
(\frac{\varepsilon_{hu_{h}^{n},\delta}^{n+1}-\varepsilon_{hu_{h}^{n},\delta}^{n}}{k},\tau_{u_{h}^{n},\lambda})+B(\lambda
u_{h}^{n},\varepsilon_{h}^{n+1};\tau)\\
&+A(u_{h}^{n};(\varepsilon_{h}^{n+1},e_{h}^{n+1}),(\tau,v))\\
&=\lambda(g_{a}(\sigma(t_{n+1}),\nabla   u(t_{n+1}))
-g_{a}(\sigma_{h}^{n},\nabla u_{h}^{n}),\tau)\\
&\quad +2\alpha Re(\frac{du}{dt}(t_{n+1})-\frac{\tilde{u}(t_{n+1})
-\tilde{u}(t_{n})}{k},v)\\
&\quad  +\lambda (\frac{d\sigma}{dt}(t_{n+1})-
\frac{\tilde{\sigma}_{u_{h}^{n}}(t_{n+1})-
\tilde{\sigma}_{u_{h}^{n}}(t_{n})}{k},\tau_{u_{h}^{n},\lambda})\\
&\quad
+A(u_{h}^{n},((\sigma-\tilde{\sigma})(t_{n+1}),(u-\tilde{u})(t_{n+1})),
(\tau,v))\\
&\quad +\lambda [B(\lambda u(t_{n+1}),\lambda
u_{h}^{n},\sigma(t_{n+1});\tau) -B(\lambda
u_{h}^{n},\tilde{\sigma}(t_{n+1});\tau)]
 +2\alpha(p(t_{n+1}),\nabla.v).
\end{aligned} \label{eq:11}
\end{equation}
Taking $v=e_{h}^{n+1}$ and $\tau=\varepsilon_{h}^{n+1}$ in
equation (\ref{eq:11}) and using the identity
$(a-b,a)=\frac{1}{2}(\vert a\vert^{2}-\vert b\vert^{2}+\vert
a-b\vert^{2})$, and coercivity, we obtain
\begin{equation}
\begin{aligned}
&\frac{\alpha Re}{k}\{\vert e_{h}^{n+1}\vert^{2}-\vert
e_{h}^{n}\vert^{2} +\vert
e_{h}^{n+1}-e_{h}^{n}\vert^{2}\}+\vert\varepsilon_{h}^{n+1}\vert^{2}\\
&+\frac{\lambda}{2k}\{\vert\varepsilon_{hu_{h}^{n},\lambda}^{n+1}\vert^{2}
-\vert\varepsilon_{hu_{h}^{n},\lambda}^{n}\vert^{2}
+\vert\varepsilon_{hu_{h}^{n},\lambda}^{n+1}-\varepsilon_{hu_{h}^{n},\lambda}^{n}
\vert^{2}\}\\
&+2\alpha(1-\alpha)\vert D(e_{h}^{n+1})\vert^{2}
+(1/2)\vert\varepsilon_{h}^{n+1}\vert^{2}
+(\delta_{0}/4)\vert\lambda
u_{h}^{n}.\nabla\varepsilon_{h}^{n+1}\vert^{2}
\end{aligned} \label{eq:12}
\end{equation}
which is less than ro equal to the right-hand side of
\eqref{eq:11}. To bound each term of the second member of
inequality (\ref{eq:12}), let us define for $C_{0}>0$, the ball
$B_{h,k}^{m}$ for $0\leq m\leq N$, by
\begin{align*}
&B_{h,k}^{m}\\
&=\big\{(\tau_{i},v_{i})_{i=0,..,m}\in(T_{h}\times V_{h})^{m+1}:
\max_{0\leq i\leq
m}\{\vert(\tau_{i}-\sigma(t_{i}))_{v_{i-1}}\vert^{2}
+\vert v_{i}-u(t_{i})\vert^{2}\}^{1/2}\\
&\quad \leq
C_{0}(k+\delta+h\sqrt{\delta_{0}}+\frac{h^{2}}{\sqrt{\delta_{0}}}
+h^{1+\varepsilon})\mbox{ and}\\
&\quad
[\sum_{n=0}^{m}k\lbrace\vert\tau_{i}-\sigma(t_{i})\vert^{2}+\vert
D(v_{i}-u(t_{i}))\vert^{2}\rbrace]^{1/2}
 \leq
C_{0}(k+\delta+h\sqrt{\delta_{0}}+\frac{h^{2}}{\sqrt{\delta_{0}}}
+h^{1+\varepsilon})\big\}.
\end{align*}
Our aim is to prove that we can choose $M_{0},h_{0},C_{0}$ such
that for $M\leq M_{0}$, $h\leq h_{0}$ if
$(\sigma_{h}^{n},u_{h}^{n})_{0\leq n\leq m-1}\in B_{h,k}^{m-1}$
for a $C_{0}=C_{0}(M_{0},h_{0},C_{i})$ then
$(\sigma_{h}^{n},u_{h}^{n})_{0\leq n\leq m}\in B_{h,k}^{m}$ for
the same $C_{0}$, thus for all $m$ such $mk\leq T$. Firstly, by
equations (\ref{eq:er1}) to (\ref{eq:er6}) we have
$$
\vert (\sigma_{h}^{0}-\sigma(0))_{u_{h}^{0}}\vert+\vert u_{h}^{0}
-u(0)\vert\leq Mh^{2}\{C_{3}(1+\lambda M\delta_{0}h^{-1})+C_{1}\}
$$
and
$$
[k\{\vert\sigma_{h}^{0}-\sigma(0)\vert^{2}+\vert D(u_{h}^{0}-u(0)
\vert^{2}\}]^{1/2}\leq \sqrt{2k}(C_{1}+C_{3})Mh^{2}.
$$
To ensure that
$(\sigma_{h}^{0},u_{h}^{0})=(\tilde{\sigma}_{0},\tilde{u}_{0})\in
B_{h,k}^{0}$, it suffices to impose, for $h<h_{1}$,
$$
Mh^{1/2}\{C_{3}(1+\lambda M\delta_{0}h^{-1})+C_{1}\}\leq C_{0}
$$
and for $h\leq h_{2}$,
$$
(C_{1}+C_{3})\sqrt{2\bar{C}}Mh^{\frac{3-\varepsilon}{2}}\leq
C_{0}.
$$
So, if we take $h_{0}\leq \min\{h_{1},h_{2}\}$ we have for $h\leq
h_{0}$:
 $B_{h}^{0}\not=\emptyset$.
Now, let us suppose that $(\sigma_{h}^{n},u_{h}^{n})_{0\leq n\leq
m-1}\in B_{h,k}^{(m-1)}$. We multiply inequality (\ref{eq:12}) by
$k$ and sum it for $n=0$ to $n=m-1$,
\begin{align}
&\alpha Re \sum_{n=0}^{m-1}[\vert e_{h}^{n+1}\vert^{2}-\vert
e_{h}^{n}\vert^{2} +\vert
e_{h}^{n+1}-e_{h}^{n}\vert^{2}]+(1/2)\sum_{n=0}^{m-1}k\vert
\varepsilon_{h}^{n+1}\vert^{2} \nonumber\\
&+\frac{\lambda}{2}\sum_{n=0}^{m-1}[
\vert\varepsilon_{hu_{h}^{n},\lambda}^{n+1}\vert^{2}-\vert\varepsilon_{hu_{h}^{n},
\lambda}^{n}\vert+\vert\varepsilon_{hu_{h}^{n},\lambda}^{n+1}
-\varepsilon_{hu_{h}^{n},\lambda}^{n}\vert^{2}]
+2\alpha(1-\alpha)\sum_{n=0}^{m-1}k\vert D(e_{h}^{n+1})\vert^{2}
\nonumber\\
&+(\delta_{0}/4)\sum_{n=0}^{m-1}k\vert\lambda u_{h}^{n}.\nabla
\varepsilon_{h}^{n+1}\vert^{2} \nonumber\\
&\leq \lambda\sum_{n=0}^{m-1}k(g_{a}(\sigma(t_{n+1}),\nabla
u(t_{n+1})) -g_{a}(\sigma_{h}^{n+1},\nabla
u_{h}^{n+1}),\varepsilon_{hu_{h}^{n},\lambda}^{n+1})
\label{eq:131}\\
&\quad +\lambda\sum_{n=0}^{m-1}k(\frac{d\sigma}{dt}(t_{n+1})-
\frac{\tilde{\sigma}_{u_{h}^{n},\delta}(t_{n+1})
-\tilde{\sigma}_{u_{h}^{n},\delta}(t_{n})}{k},\varepsilon_{hu_{h}^{n},\lambda}^{n+1})
\label{eq:132}\\
&\quad +2\alpha Re\sum_{n=0}^{m-1}k(\frac{du}{dt}(t_{n+1})
-\frac{\tilde{u}(t_{n+1})-\tilde{u}(t_{n})}{k},e_{h}^{n+1})
\label{eq:133}\\
&\quad
+\sum_{n=0}^{m-1}kA(u_{h}^{n};(\sigma-\tilde{\sigma})(t_{n+1}),
(u-\tilde{u})(t_{n+1}),(\varepsilon_{h}^{n+1},e_{h}^{n+1}))
\label{eq:134}\\
&\quad +\lambda\sum_{n=0}^{m-1}k\lbrace B(\lambda
u(t_{n+1}),\lambda u_{h}^{n},
\sigma(t_{n+1});\varepsilon_{h}^{n+1})-B(\lambda
u_{h}^{n},\tilde{\sigma}(t_{n+1});
\varepsilon_{h}^{n+1})\rbrace  \label{eq:135}\\
&\quad +2\alpha\sum_{n=0}^{m-1}k(p(t_{n+1}),\nabla.e_{h}^{n+1}).
\label{eq:136}
\end{align}
We use for the estimate of terms \eqref{eq:133}--\eqref{eq:136},
the results \eqref{eq:er1}--\eqref{eq:er6}. However, for the
estimation of the term  \eqref{eq:131} we choose the fixed point
$(\sigma_{h}^{m},u_{h}^{m})$ of the mapping $\Phi$ defined in
section 2. This choice is possible because we have,
$B_{h}^{(0)}\cap B_{h,\Delta t}^{0}\not=\emptyset$  and by
construction:
$$
(\sigma_{h}^{l},u_{h}^{l})_{l=0;..;m-1}\in B_{h,\Delta
t}^{m-1}\Rightarrow (\sigma_{h}^{m-1},u_{h}^{m-1})\in
B_{h}^{(m-1)}.
$$
On the other hand for estimate of the term \eqref{eq:132} we
prepare the following lemma:

\begin{lemma} \label{lem3}
Let \eqref{eq:er1}--\eqref{eq:er6} hold. Then
\begin{align*}
&\vert(\frac{d\sigma}{dt}(t_{n+1})
-\frac{\tilde{\sigma}_{u_{h}^{n},\delta}(t_{n+1})
-\tilde{\sigma}_{u_{h}^{n},\delta}(t_{n})}{k},\varepsilon_{hu_{h}^{n},
\lambda}^{n+1})\vert\\
&\leq \lambda M[ C_{7}h^{2}+k+\delta
MC_{10}(M+hM+C_{1}Mh^{\frac{3}{2}})] (\sum_{n=0}^{m-1}k
)^{1/2}\max_{0\leq n\leq m-1}\vert
\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert \\
&\quad +\big\{\delta M[C_{7}C_{8}kh(\sum_{n=0}^{m-1}k\vert
\nabla(u_{h}^{n}-u(t_{n}))\vert^{2})^{1/2}\\
&\quad +(C_{7}\lambda^{-1}h^{2}
+\frac{C{7}}{C_{6}}C_{9}\delta_{0}(M+\max_{0\leq n\leq m-1}\vert
u(t_{n})-u_{h}^{n}\vert))]
\big(\sum_{n=0}^{m-1}k )^{1/2}\\
&\quad +\lambda(1+\sqrt{\delta_{0}})\delta h^{-1}
(\sum_{n=0}^{m-1}k\vert
\nabla(u_{h}^{n}-u(t_{n}))\vert^{2}\big)^{1/2}\big\}\\
&\quad\times
\big(\sum_{n=0}^{m-1}k\{\vert\varepsilon_{hu_{h}^{n},\lambda}^{n+1}
\vert^{2}+
\delta_{0}\vert\lambda(u_{h}^{n}.\nabla)\varepsilon_{h}^{n+1}
\vert^{2}\}\big)^{1/2}.
\end{align*} %\label{eq:19}
\end{lemma}

\begin{proof} To proof the inequality in this lemma, we
write each term as follows:
\begin{align*}
&\frac{d\sigma}{dt}(t_{n+1})-\frac{\tilde{\sigma}_{u_{h}^{n},\delta}(t_{n+1})
-\tilde{\sigma}_{u_{h}^{n},\delta}(t_{n})}{k}\\
&=\big(\frac{d\sigma}{dt}(t_{n+1})-\frac{\sigma_{u_{h}^{n},\delta}(t_{n+1})
-\sigma_{u_{h}^{n},\delta}(t_{n})}{k}\big)\\
&\quad +\big(\frac{\sigma_{u_{h}^{n},\delta}(t_{n+1})
-\sigma_{u_{h}^{n},\delta}(t_{n})}{k}
-\frac{\tilde{\sigma}_{u_{h}^{n},\delta}(t_{n+1})
-\tilde{\sigma}_{u_{h}^{n},\delta}(t_{n})}{k}\big)
\end{align*}
we can write the second term in the form,
\[
\frac{1}{k}(\int_{t_{n}}^{t_{n+1}}(\frac{d\sigma}{dt}
-\tilde{(\frac{d\sigma}{dt})})(s)\,ds,\varepsilon_{hu_{h}^{n},\lambda}^{n+1})
+ \frac{\delta}{k}(\int_{t_{n}}^{t_{n+1}}u_{h}^{n}.
\nabla(\frac{d\sigma}{dt}-(\tilde{\frac{d\sigma}{dt})})(s)\,ds,
\varepsilon_{hu_{h}^{n},\lambda}^{n+1})
\]
so,
\begin{align*}
&\vert k(\frac{\sigma_{u_{h}^{n},\delta}(t_{n+1})
-\sigma_{u_{h}^{n},\delta}(t_{n+1})}{k}-\frac{\tilde{\sigma}_{u_{h}^{n},\delta}
(t_{n+1})-\tilde{\sigma}_{u_{h}^{n},\delta}(t_{n+1})}{k},\varepsilon_{hu_{h}^{n},
\lambda}^{n+1})\vert \\
&\leq \vert(\int_{t_{n}}^{t_{n+1}}(\frac{d\sigma}{dt}
-\tilde{(\frac{d\sigma}{dt})})(s)\,ds,\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert
+
\delta\vert(\int_{t_{n}}^{t_{n+1}}u_{h}^{n}.\nabla(\frac{d\sigma}{dt}
-\tilde{(\frac{d\sigma}{dt})})(s)\,ds,\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert
\end{align*}
we estimate the first term in this inequality using
(\ref{eq:er6}):
\begin{align*}
\vert (\int_{t_{n}}^{t_{n+1}}(\frac{d\sigma}{dt}
-\tilde{(\frac{d\sigma}{dt})})(s)\,ds,\varepsilon_{hu_{h}^{n},
\lambda}^{n+1})\vert &\leq k \max_{t_{n}\leq s\leq t_{n+1}}\vert
(\frac{d\sigma}{dt}
-\tilde{(\frac{d\sigma}{dt})})(s)\vert\vert\varepsilon_{hu_{h}^{n},
\lambda}^{n+1}\vert \\
& \leq
C_{7}Mkh^{2}\vert\varepsilon_{hu_{h}^{n},\lambda}^{n+1}\vert .
\end{align*}
To study the second term,  we write
\begin{align*}
&(\int_{t_{n}}^{t_{n+1}}u_{h}^{n}.\nabla(\frac{d\sigma}{dt}
-\tilde{(\frac{d\sigma}{dt})})(s)\,ds,\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\\
&=-(\int_{t_{n}}^{t_{n+1}}(\frac{d\sigma}{dt}-\tilde{(\frac{d\sigma}{dt})})(s)\,ds,
(u_{h}^{n}.\nabla)\varepsilon_{h}^{n+1})\\
&\quad +((\nabla.u_{h}^{n})\int_{t_{n}}^{t_{n+1}}
(\frac{d\sigma}{dt}-\tilde{(\frac{d\sigma}{dt})})(s)\,ds,\varepsilon_{h}^{n+1})
\\
&\quad
+\delta_{0}(\int_{t_{n}}^{t_{n+1}}u_{h}^{n}.\nabla(\frac{d\sigma}{dt}
-\tilde{(\frac{d\sigma}{dt})})(s)\,ds,\lambda(u_{h}^{n}.\nabla)
\varepsilon_{hu_{h}^{n},\lambda}^{n+1})
\end{align*}
and
\begin{align*}
&\delta\vert(\int_{t_{n}}^{t_{n+1}}u_{h}^{n}.\nabla(\frac{d\sigma}{dt}
-\tilde{(\frac{d\sigma}{dt})})(s)\,ds,\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert
\\
& \leq \delta\lambda^{-1}k \max_{t_{n}\leq s\leq t_{n+1}}\vert
(\frac{d\sigma}{dt}
-\tilde{(\frac{d\sigma}{dt})})(s)\vert\vert\lambda(u_{h}^{n}.\nabla)
\varepsilon_{h}^{n+1}\vert \\
&\quad +\delta k \max_{t_{n}\leq s\leq t_{n+1}}\vert
(\frac{d\sigma}{dt} -\tilde{(\frac{d\sigma}{dt})})(s)\vert\vert
\nabla(u_{h}^{n}-u(t_{n}))
\vert\vert\varepsilon_{h}^{n+1}\vert_{0,\infty} \\
&\quad +k\delta_{0}\delta\vert
u_{h}^{n}\vert_{0,\infty}\max_{t_{n}\leq s \leq t_{n+1}}\vert
(\frac{d\sigma}{dt} -\tilde{(\frac{d\sigma}{dt})})(s)
\vert_{1,2}\vert \lambda
(u_{h}^{n}.\nabla)\varepsilon_{h}^{n+1}\vert
\end{align*}
using (\ref{eq:er6}) and the result of Lemma \ref{lem1}, we obtain
the following estimate
\begin{align*}
&\delta\vert(\int_{t_{n}}^{t_{n+1}}u_{h}^{n}.\nabla(\frac{d\sigma}{dt}
-\tilde{(\frac{d\sigma}{dt})})(s)\,ds,\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert
\\
&\leq \delta C_{8}C_{7}M\delta kh\vert
\nabla(u_{h}^{n}-u(t_{n}))\vert\vert
\varepsilon_{h}^{n+1}\vert \\
&\quad
+\{\delta\lambda^{-1}C_{7}Mh^{2}+C_{9}\frac{C{7}}{C_{6}}Mh\delta_{0}
\delta h^{-1}\vert u_{h}^{n}\vert\}k\vert\lambda
(u_{h}^{n}.\nabla)
\varepsilon_{h}^{n+1}\vert \\
&\leq C_{7}C_{8}M\delta kh\vert
\nabla(u_{h}^{n}-u(t_{n}))\vert\varepsilon_{h}^{n+1}
\vert\\
&\quad
+\big\{C_{7}\delta\lambda^{-1}M\frac{h^{2}}{\sqrt{\delta_{0}}}
+C_{9}\frac{C{7}}{C_{6}}M\delta\sqrt{\delta_{0}}(M+\max_{0\leq
n\leq m-1}\vert
u_{h}^{n}-u(t_{n})\vert)\big\}\\
&\quad\times \sqrt{\delta_{0}}\vert\lambda
(u_{h}^{n}.\nabla)\varepsilon_{h}^{n+1}\vert
\end{align*}
and finally
\begin{align}
&\lambda\vert\sum_{n=0}^{m-1}
k(\frac{\sigma_{u_{h}^{n},\delta}(t_{n+1})-\sigma_{u_{h}^{n},\delta}(t_{n+1})}{k}
-\frac{\sigma_{u_{h}^{n},\delta}^{\sim}(t_{n+1})
-\sigma_{u_{h}^{n},\delta}^{\sim}(t_{n+1})}{k},
\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert \nonumber \\
&\leq \lambda C_{7}Mh^{2}(\sum_{n=0}^{m-1}k)\max_{0\leq n\leq
m-1}\vert
\varepsilon_{hu_{h}^{n},\lambda}^{n+1}\vert \nonumber\\
&\quad +\lambda C_{7}C_{8}M\delta kh(\sum_{n=0}^{m-1}k\vert
\nabla(u_{h}^{n}-u(t_{n}))\vert^{2})^{1/2}
(\sum_{n=0}^{m-1}k\vert\varepsilon_{h}^{n+1}\vert^{2})^{1/2}
\label{eq:20} \\
&\quad +\delta M\{C_{7}\lambda^{-1}\frac{h^{2}}{\sqrt{\delta_{0}}}
+\frac{C{7}}{C_{6}}C_{9}\sqrt{\delta_{0}}(M+\max_{0\leq n\leq m-1}
\vert u_{h}^{n}-u(t_{n})\vert))\} \nonumber\\
&\quad\times (\sum_{n=0}^{m-1}k)^{1/2}(\sum_{n=0}^{m-1}k\delta_{0}
\vert\lambda
u_{h}^{n}.\nabla\varepsilon_{h}^{n+1}\vert^{2})^{1/2}. \nonumber
\end{align}
\end{proof}

Now, we return  to the first part of term \eqref{eq:132}.
\begin{align*}
&\vert
k(\frac{d\sigma}{dt}(t_{n+1})-\frac{\sigma_{u_{h}^{n},\delta}(t_{n+1})
-\sigma_{u_{h}^{n},\delta}(t_{n})}{k},\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert\\
&\leq \vert k(\frac{d\sigma}{dt}(t_{n+1})-\frac{\sigma(t_{n+1})
-\sigma(t_{n})}{k},\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert
+\delta\vert (u_{h}^{n}.\nabla(\sigma(t_{n+1})-\sigma(t_{n})),
\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert \\
&\leq \vert
(\int_{t_{n}}^{t_{n+1}}(s-t)\frac{d^{2}\sigma}{dt^{2}}(s)\,ds,
\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert+ \delta\vert
(\int_{t_{n}}^{t_{n+1}} u_{h}^{n}.\nabla
\frac{d\sigma}{dt}(s)\,ds,
\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert \\
&\leq k^{2}\max_{t_{n}\leq s\leq t_{n+1}}\vert
\frac{d^{2}\sigma}{dt^{2}}(s)\vert\vert\varepsilon_{hu_{h}^{n},
\lambda}^{n+1}\vert+\delta k\vert u_{h}^{n}\vert_{0,\infty}
 \max_{t_{n}\leq s\leq t_{n+1}}\vert \frac{d\sigma}{dt}(s)\vert_{1,2}
 \vert\varepsilon_{hu_{h}^{n},\lambda}^{n+1}\vert\,.
\end{align*}
Using the regularity of $\sigma$, we have
$$
\vert
k(\frac{d\sigma}{dt}(t_{n+1})-\frac{\sigma_{u_{h}^{n},\delta}(t_{n+1})
-\sigma_{u_{h}^{n},\delta}(t_{n})}{k},\varepsilon_{hu_{h}^{n},\lambda}^{n+1})\vert
\leq M(k^{2}+k\delta\vert
u_{h}^{n}\vert_{0,\infty})\vert\varepsilon_{hu_{h}^{n},
\lambda}^{n+1}\vert.
$$
On the other hand, by the imbedding  result $W^{1,4}\subset
L^{\infty}$ (see Lemma \ref{lem1}) and the inverse inequality
result $\vert \cdot \vert_{0,\infty}\leq
C_{10}h^{\frac{-1}{2}}\vert \cdot \vert_{0,4}$ (see Lemma
\ref{lem2}) we can prove,
$$
\vert u_{h}^{n}\vert_{0,\infty}\leq C_{10}(M+h\Vert
u\Vert_{3,2}+C_{1}Mh^{\frac{3}{2}}+h^{-\frac{1}{2}}\vert
\nabla(u(t_{n})-u_{h}^{n})\vert).
$$
So we have,
\begin{align*}
&\lambda\vert\sum_{n=0}^{m-1}k(\frac{d\sigma}{dt}(t_{n+1})-\frac{\sigma_{u_{h}^{n},
\delta}(t_{n+1})-\sigma_{u_{h}^{n},\delta}(t_{n})}{k},\varepsilon_{hu_{h}^{n},
\lambda}^{n+1})\vert  \\
&\leq \lambda M[ k+\delta MC_{10}(M+hM+C_{1}Mh^{\frac{3}{2}})]
(\sum_{n=0}^{m-1}k)\times \max_{0\leq n\leq
m-1}\vert\varepsilon_{hu_{h}^{n},
\lambda}^{n+1}\vert\\
&\quad +\lambda(1+\sqrt{\delta_{0}})\delta h^{-\frac{1}{2}}
\big(\sum_{n=0}^{m-1}k\vert
\nabla(u(t_{n})-u_{h}^{n})\vert^{2}\big)^{1/2}\\
&\quad\times
\big[(\sum_{n=0}^{m-1}k\vert\varepsilon_{h}^{n+1}\vert^{2})^{1/2}
+(\sum_{n=0}^{m-1}k \delta_{0}\vert\lambda
u_{h}^{n}.\nabla\varepsilon_{h}^{n+1}\vert^{2})^{1/2}\big]\,.
\end{align*} %\label{eq:21}
Then Lemma \ref{lem3} follows from the above inequality and
(\ref{eq:20}).

\subsection*{Conclusion} We conclude this analysis with some comments.
The proof given here can be extended to the more realistic
rheological PTT model, to a quadrilateral FE approximation,
following \cite{BBS} and to the higher finite element
methods($P_k, k \geq 1$). With a judicious coefficient choice of
stabilization $\delta $ and $\delta_0$, we find the error bound
given respectively by Baranger and al. \cite{WB} and Ervin and
al.\ \cite{EM}.

The use of a decoupled fractional step scheme would be
computationally cheaper, following \cite{EsRZ}. Numerical analysis
of such method is currently in progress.

\begin{thebibliography}{00}

\bibitem{BBS}{A. Bahar, J. Baranger, D. Sandri },
\emph{Galerkin discontinuous approximation of the transport
equation and viscoelastic fluid flow on Quadrilateral}, Numer.
Meth. Partial Diff. Equations, 14(1)(1998), 97-114.

\bibitem{B}{F. G. Basambrio },
\emph{Flows in viscoelastic fluids treated by the method of
characteristics}, JNNFM 39 (1991), 17-34.

\bibitem{BEs}{M. Bensaada, D. Esselaoui},
\emph{Stabilization method for continuous approximation of
transient convection problem}, In press:  Numer. Methods for
Partial Differential Equations, (2004).

\bibitem{C}{P. G. Ciarlet}, \emph{The finite element method for
elliptic problems}, North-Holland, Amsterdam, (1978).

\bibitem{EsR}{D. Esselaoui, K. Najib, A. Ramadane, A. Zine},
\emph{Analyse num\'erique et approche d\'ecoupl\'ee d'un
probl\`eme d'\'ecoulements  de fluides non newtoniens de type
Phan-Thien-Tanner:contraintes discontinues}, C.R. Acad. Sci. Paris
s\'erie I (2000), 931-936.

\bibitem{EsRZ}{D. Esselaoui, A. Ramadane, A. Zine},
\emph{Decoupled approch for the problem of viscoelastic fluid flow
of PTT model I: continuous stresses}, Compt. Methods  Appl. Mech.
Engrg. (2000), 543-560.

\bibitem{EM}{V. J. Ervin and W. W. Miles},
\emph{Approximation of time-dependent  viscoelastic fluid flow:
SUPG Approximation},  SIAM J.  Numer.  Anal. 41 N0. 2 (2003)
457-486.

\bibitem{EsF}{M. Fortin, D. Esselaoui},
\emph{A finite element procedure for viscoelastic flows}. Int. J.
Num. Meth. Fluids, 7 (1987), 1035-1052.

\bibitem{GS}{C. Guillop\'e, J.C. Saut},
\emph{R\'esultats d'existence pour des fluides visco\'elastiques
\`a lois de comportements de type diff\'erentiel}, Nonlinear
Analysis Theory, methods and Applications, 15 (1990), 849-869.

\bibitem{GR}{V. Girault, P.A. Raviart},
\emph{Finite element method for Navier-Stokes equation  theory and
algorithms}, Springer, Berlin, (1986).

\bibitem{K}{R. A. Keiller},
\emph{Numerical instability of time dependant flows}. JNNFM 43
(1992), 229-246.

\bibitem{8}{V. Legat, M.J. Crochet},
\emph{The consistent streamline upwind/Petrov-Galerkin method for
viscoelastic flow recisited}. In Probl\`emes non-lin\'eaires
appliqu\'es. CEA. INRIA. EDF, (1992).

\bibitem{MM}{M.J. Marchal, M.J. Crochet},
\emph{A new finite element for calculating viscoelastic flows},
 J. N?. N?. F?. M?, 26 (1987), 77-114.

\bibitem{RT}{P.A. Raviart, J.M. Thomas},
\emph{Introduction \`a l'analyse num\'erique des E.D.P.},
 MASSON, Paris, (1983).

\bibitem{S}{D. Sandri}, \emph{Finite approximation of viscoelastic
fluid flow: existence of approximate solutions and error bounds,
continuous approximation of the stress}. SIAM J. Numer. Anal., 31
(1994), 362-377.

\bibitem{WB}{S. Wardi, J. Baranger}, \emph{Numerical analysis of a FEM
for a transient viscoelastic flow}. Comput. Methods Appl. Mech.
Engr. 125 (1995), 171-185.

\end{thebibliography}
\end{document}