\documentclass[reqno]{amsart} \usepackage{mathrsfs} % for \mathscr %\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \usepackage{hyperref} \AtBeginDocument{{\noindent\small 2004-Fez conference on Differential Equations and Mechanics \newline {\em Electronic Journal of Differential Equations}, Conference 11, 2004, pp. 53--59.\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}{53} \begin{document} \title[\hfilneg EJDE/Conf/11 \hfil Error estimate for the characteristic method] {Error estimate for the characteristic method involving Oldroyd derivative in a tensorial transport problem} \author[M. Bensaada, D. Esselaoui, P. Saramito\hfil EJDE/Conf/11 \hfilneg] {Mohammed Bensaada, Driss Esselaoui, Pierre Saramito} % 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} \email{m\_bensaada@hotmail.com} \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} \address{Pierre Saramito \hfill\break LMC-IMAG \\ B.P. 53, 38041 Grenoble cedex 9, France} \email{Pierre.Saramito@imag.fr \; Fax: 33 4 76 63 12 63} \date{} \thanks{Published October 15, 2004.} \thanks{Supported by grant STCI01/03 from the CNRST-CNRS} \subjclass[2000]{65M60, 65M25, 65M15, 76A10} \keywords{Finite element method; method of characteristics; error bound; \hfill\break\indent viscoelastic fluids} \begin{abstract} An optimal {\em a priori} error estimate $O(h^{k+1}+\Delta t)$, result is presented for a tensor problem involving Oldroyd derivative when using a suitable characteristic method and a finite element method. To conclude, we present results of numerical tests which confirm the previous estimates. Our long time goal is to deal with the viscoelastic fluid flow problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} The Oldroyd derivative of a symmetric tensor $\sigma$ is defined by \begin{equation} \frac{\mathscr{D}_a \sigma}{\mathscr{D}t} = \frac{\partial\sigma}{\partial t} + ({\bf u}.\nabla)\sigma + \sigma M_a({\bf u}) + M_a^T({\bf u}) \sigma, \label{eq-oldroyd} \end{equation} where ${\bf u}$ is a given velocity field, $ M_a({\bf u}) = \left( (1-a) \boldsymbol{\nabla} {\bf u} - (1+a) \boldsymbol{\nabla} {\bf u}^T \right)/2 $ and $a\in [-1,1]$ is the parameter of the Oldroyd derivative. In this paper, we suppose also, for simplicity, that ${\bf u} = 0$ on $\partial \Omega$. Problems involving the Oldroyd derivative appear in viscoelasticity (non-Newtonian polymer melt flow problems, see e.g.~\cite{saramito-crasp-94}), in turbulence modelling ($R_{ij}-\epsilon$ models) or in liquid crystals modelling. Let $\Omega$ be a bounded polygonal subset of $\mathbb{R}^d$, $d=1,2$ or $3$, $T>0$ be a time constant, $\gamma$ a given symmetric tensor defined in $\Omega \times ]0,T[$, $\sigma_0$ given in $\Omega$. and $\lambda>0$. The aim of this paper is to study, as a preliminary result, the approximation of the linear transport equation involving the Oldroyd derivative: {\em find $\sigma$ defined in $\Omega \times ]0,T[$ such that } \begin{equation} \begin{gathered} \lambda \frac{\mathscr{D}_a \sigma}{\mathscr{D}t} + \sigma = \gamma \quad \mbox{in } \Omega \times ]0,T[, \\ \sigma(0) = \sigma_0 \quad \mbox{in } \Omega. \end{gathered} \label{eq-pb} \end{equation} and its discrete counterpart: {\em find $\sigma_h^{(n)} \in T_h$, ${1\leq n\leq N}$, such that, for all $\tau_h \in T_h$, } \begin{equation} \Big( \lambda \frac{ \sigma_h^{(n+1)} - R_{\Delta t}^{(n)} \times \sigma_h^{(n)} \circ X_{\Delta t}^{(n)} \times \left( R_{\Delta t}^{(n)} \right)^T } {\Delta t} + \sigma_h^{(n+1)}, \tau_h \Big) = \left( \gamma(t_{n+1}), \tau_h \right), \label{eq-pb-h} \end{equation} and $\sigma_h^{(0)} = \pi_h\sigma_0$, where $N\geq 1$, $\Delta t=T/N$, $t_n=n\Delta t$, $X_{\Delta t}^{(n)}(x)=x-\Delta t \, {\bf u}(x,t_{n+1})$, $R_{\Delta t}^{(n)}(x)=I-\Delta t \, M_a^T({\bf u})(x,t_{n+1})$, and $T_h$ denotes the space of continuous piecewise polynomial symmetric tensors $ T_h = \{ \tau \in \left( C^0 (\overline{\Omega}) \cap L^2(\Omega) \right)^{d \times d}; \; \tau_{|K} \in P_k, \; \forall K \in {\mathscr{T}_h} \} $. Here, $({\mathscr{T}_h})_{h>0}$ denotes a suitable family of regular triangulations of $\overline{\Omega}$. The Lagrange interpolation operator from $L^2$ tensors into $T_h$ is denoted by $\pi_h$, $k\geq 1$. The characteristic method~\cite{pironneau-82} has been proposed for the numerical treatment of convected-dominated flows and transport equations. It is based on an approximation of the material derivative $\frac{D}{D t} = \frac{\partial}{\partial t} + ({\bf u}.\nabla)$: \begin{equation} \frac{D\varphi}{Dt}(x,t) \approx \frac{\varphi(x,t) - \varphi(X(x,t;s),s)}{t-s} \label{eq-schema} \end{equation} The trajectory $X(x,s;.)$ is defined for all $(x,s)\in \overline{\Omega}\times [0,T]$ by \begin{equation} \begin{gathered} \frac{\partial X}{\partial t}(x,s;t) = {\bf u}(X(x,s;t),t), \quad t \in ]0,T[, \\ X(x,s;s) = x. \end{gathered} \label{eq-trajectory} \end{equation} In 1987, D. Esselaoui and M. Fortin~\cite{esselaoui-87} extended the characteristic method for the approximation of the Oldroyd derivative of a symmetric tensor. For all fixed $s\in [0,T]$, these authors considered the following transformed problem: {\em find $\widehat{\sigma}$(.,.;s) defined in $\Omega \times ]0,T[$ such that } \begin{equation} \begin{gathered} \lambda \frac{D\widehat{\sigma}}{D t} (x,t;s) + \widehat{\sigma}(x,t;s) = R(x,t;s) \gamma(x,t) R^T(x,t;s), \quad (x,t) \in \Omega \times ]0,T[, \\ \widehat{\sigma}(x,0;s) = R(x,0;s) \sigma_0(x) R^T(x,0;s), \quad x \in \Omega, \end{gathered} \label{eq-pb-transf} \end{equation} where $ \widehat{\sigma}(x,t;s) = R(x,t;s) \sigma(x,t) R(x,t;s)^T $ and the tensor flow $R(x,.;s)$ is defined for all $(x,s)\in \overline{\Omega}\times [0,T]$ by \begin{equation} \begin{gathered} \frac{D R}{D t} (x,t;s) = R(x,t;s) M_a^T({\bf u})(x,t), \quad t \in ]0,T[, \\ R(x,s;s) = I. \end{gathered} \label{eq-transf-tensor-hat} \end{equation} The Oldroyd derivative has been replaced by a material derivative of a tensor, suitable for the characteristic method, as~\eqref{eq-pb-h}. A short computation shows that \begin{equation} \frac{D^m \widehat{\sigma}}{D t^m} (x,t;s) = R^m(x,t;s) {\frac{\mathscr{D}^m_a \sigma}{\mathscr{D}t^m}} (x,t) (R^m)^T(x,t;s), \quad \forall m \in \mathbb{N}, \label{eq-prop-derivee} \end{equation} and thus problems~\eqref{eq-pb} and~\eqref{eq-pb-transf} are equivalent. Numerical computations involving the scheme~\eqref{eq-pb-transf} has been already performed in~\cite{esselaoui-87} while the corresponding numerical analysis was not yet available. We show a ${\mathscr{O}}\left(h^{k+1} + \Delta t\right)$ optimal estimate. Moreover, since the trajectories and the transformation are both approximated, the scheme~\eqref{eq-pb-transf} is of practical interest. To conclude, we present results of numerical tests that confirm the previous estimates. \section{Error estimate} Let $\|\cdot\|$ the $L^2(\Omega)$ norm, $\|\cdot \|_\infty$ the $L^\infty(\Omega)$ one and $|\cdot|_{m,p,\Omega}$ the $W^{m,p}(\Omega)$ semi-norm, for $m\geq 0$ et $p\in [1,\infty]$. Also, $C^{0,1}(\overline{\Omega})$ denotes the space of lipschitzian functions $\overline{\Omega}$. For a Banach space $Y$, let us denote $C(Y)$ the space $C([0,T],Y)$. We suppose also that the data ${\bf u} \in C(C^{0,1}(\overline{\Omega}))$: the existence and the continuity of $x\rightarrow X(x,s;t)$ follow then from the Cauchy-Lipschitz theorem. Let us denote finally $X^{(n)}(x) = X (x,t_{n+1};t_n)$ and $R^{(n)}(x) = R(X^{(n)}(x),t_n;t_{n+1})$. In this paper, $C_i$, $i\in\mathbb{N}$ is a positive constant, independent of $h$ and $\Delta t$. For a Banach space $Y$, with norm $\|.\|_Y$ and $1\leq p < \infty$ we introduce \begin{gather*} l^p(0,T;Y) = \Big\{ \varphi : (t_1,\ldots,t_N) \rightarrow Y; \|\varphi\|_{l^p(0,T;Y)} = \Big(\sum_{n=1}^N \| \varphi(t_i) \|^p_Y \Delta t \Big)^{1/p} < \infty \Big\}, \\ l^\infty(0,T;Y) = \Big\{ \varphi : (t_1,\ldots,t_N) \rightarrow Y; \|\varphi\|_{l^\infty(0,T;Y)} = \max_{1\leq i \leq N} \| \varphi(t_i) \|_Y < \infty \Big\}. \end{gather*} % \begin{theorem}[Error estimate] \label{th-estim-approx} Let $\sigma$ and $\sigma_h$ the solutions of \eqref{eq-pb} and \eqref{eq-pb-h}, respectively. Suppose that ${\bf u}$ is in $ \left( C(C^{0,1}(\overline{\Omega})) \cap W^{1,\infty}(W^{1,\infty}(\Omega) \right)^d$ and $\sigma$ is in $ \left( W^{1,\infty}(H^{m+1}(\Omega)) \cap W^{2,\infty}(L^{2}(\Omega)) \right)^{d\times d}$, with $m\geq 0$. Then, there exist three positive constants $\Delta t_0$, $h_0$ and $c$, independent of $h$ and $\Delta t$, such that, if $\Delta t \leq \Delta t_0$ and $h\leq h_0$, we have \begin{gather} \| \sigma - \sigma_h \|_{l^\infty(0,T;L^2(\Omega))} \leq c \left( h^{r+1}+\Delta t \right), \label{eq-estim-linf-1} \\ \| \sigma - \sigma_h \|_{l^2(0,T;L^2(\Omega))} \leq c \left( h^{r+1}+\Delta t \right), \label{eq-estim-l2-1} \end{gather} where $r=\min(k,m)$. \end{theorem} \begin{proof} Let us introduce $\tilde{\sigma}_h(t)=\pi_h \sigma (t)$, $t\in [0,T]$, $\tilde{\sigma}_h^{(n)}=\tilde{\sigma}_h(t_n)$ and $\varepsilon_h^{(n)}=\sigma_h^{(n)}-\tilde{\sigma}_h^{(n)}$. By a development and using~\eqref{eq-pb-h}, we get \begin{equation} \Bigg( \lambda \frac{\varepsilon_h^{(n+1)} - R_{\Delta t}^{(n)} \times \varepsilon_h^{(n)} \circ X_{\Delta t}^{(n)} \times \left(R_{\Delta t}^{(n)}\right)^T} {\Delta t} + \varepsilon_h^{(n+1)}, \, \varepsilon_h^{(n+1)} \Bigg) = (\rho_h^{(n+1)}, \varepsilon_h^{(n+1)}), \label{eq-dem-h-1} \end{equation} where \begin{equation} \rho_h^{(n+1)} = \gamma(t_{n+1}) - \tilde{\sigma}_h^{(n+1)} - \lambda \frac{\tilde{\sigma}_h^{(n+1)} - R_{\Delta t}^{(n)} \times \tilde{\sigma}_h^{(n)} \circ X_{\Delta t}^{(n)} \times \left(R_{\Delta t}^{(n)}\right)^T} {\Delta t}. \label{eq-rho} \end{equation} From the Cauchy-Schwartz inequality and using the identity $ab\leq (a^2+b^2)/2$, \begin{align*} &\Big( R_{\Delta t}^{(n)} \varepsilon_h^{(n)} \circ X_{\Delta t}^{(n)} \big( R_{\Delta t}^{(n)} \big)^T, \varepsilon_h^{(n+1)} \Big) \\ &\leq \frac{1}{2} \Big( \big\| R_{\Delta t}^{(n)} \varepsilon_h^{(n)} \circ X_{\Delta t}^{(n)} \left( R_{\Delta t}^{(n)} \right)^T \big\|^2 + \big\| \varepsilon_h^{(n+1)} \big\|^2 \Big) \end{align*} and \begin{equation*} \left( \rho_h^{(n+1)}, \varepsilon_h^{(n+1)} \right) \leq \frac{1}{2} \left( \frac{1}{2}\big\| \rho_h^{(n+1)} \big\|^2 + 2 \big\| \varepsilon_h^{(n+1)} \big\|^2 \right). \end{equation*} Then, \eqref{eq-dem-h-1} becomes \[ \left\| \varepsilon_h^{(n+1)} \right\|^2 \leq \left\| R_{\Delta t}^{(n)} \times \varepsilon_h^{(n)} \circ X_{\Delta t}^{(n)} \times (R_{\Delta t}^{(n)})^T \right\|^2 + \frac{\Delta t}{2\lambda} \left\| \rho_h^{(n+1)} \right\|^2. \] From lemma~\ref{eq-rhs} (paragraph~2), for $\Delta t$ small enough, we get \[ \left\| \varepsilon_h^{(n+1)} \right\|^2 \leq (1+C_0\Delta t) \left\| \varepsilon_h^{(n)} \right\|^2 + \frac{\Delta t}{2\lambda} \left\| \rho_h^{(n+1)} \right\|^2. \] From the discrete Gronwall lemma, and using $n\leq N=T/\Delta t$, \begin{equation} \left\| \varepsilon_h^{(n)} \right\|^2 \leq \frac{e^{C_0 T}}{2\lambda} \| \rho_h \|_{l^2(L^2)}^2 \label{eq-appli-gronwall} \end{equation} where $\rho_h=(\rho_h^{(1)},\ldots,\rho_h^{(N)})$. It still remains to bound the right-hand side. From \eqref{eq-pb-transf} with $s=t=t_{n+1}$, we get \begin{equation} \lambda \frac{D\widehat{\sigma}}{D t} \left(x,t_{n+1}; t_{n+1} \right) + \sigma (x,t_{n+1}) - \gamma (x,t_{n+1}) = 0. \label{eq-pb-transf-bis} \end{equation} Adding~\eqref{eq-pb-transf-bis} to the expression~\eqref{eq-rho} of $\rho_h^{(n+1)}$, \begin{align*} \rho_h^{(n+1)} &= (\sigma-\tilde{\sigma}_h)(t_{n+1})\\ &+ \lambda \Big\{ \frac{D\widehat{\sigma}}{Dt}(x,t_{n+1};t_{n+1}) - \frac{\tilde{\sigma}_h^{(n+1)} - R_{\Delta t}^{(n)} \times \tilde{\sigma}_h^{(n)} \circ X_{\Delta t}^{(n)} \times \left(R_{\Delta t}^{(n)}\right)^T} {\Delta t} \Big\}. \end{align*} Let us introduce the splitting $ \rho_h^{(n+1)} = \zeta_h^{(n+1)} + \eta_h^{(n+1)}, $ where \begin{align*} &\zeta_h^{(n+1)}\\ &= (\sigma-\tilde{\sigma}_h)(t_{n+1}) + \frac{\lambda}{\Delta t} \left\{ (\sigma-\tilde{\sigma}_h)(t_{n+1}) - R^{(n)} \times (\sigma-\tilde{\sigma}_h) (X^{(n)},t_n) \times (R^{(n)})^T \right\}, \end{align*} and \begin{align*} &\eta_h^{(n+1)}\\ &= \lambda \Big\{ \frac{D\widehat{\sigma}}{Dt} \left(x,t_{n+1};t_{n+1}\right) - \frac{\sigma(t_{n+1}) - R^{(n)} \times \sigma \left(X^{(n)},t_n\right) \times \left(R^{(n)}\right)^T} {\Delta t} \Big\} \\ &\quad - \frac{\lambda}{\Delta t} \Big\{ R^{(n)} \times \tilde{\sigma}_h \left(X^{(n)},t_n\right) \times \left(R^{(n)}\right)^T - R_{\Delta t}^{(n)} \times \tilde{\sigma}_h \left(X_{\Delta t}^{(n)},t_n\right) \times \left(R_{\Delta t}^{(n)}\right)^T \Big\}. \end{align*} On the one hand, using a classical interpolation result~\cite{brenner-scott-2002}, we have $ \left\| \varphi-\pi_h\varphi \right\| \leq C_4 h^{r+1} | \varphi |_{r+1,2,\Omega}$, for all $\varphi \in H^{r+1}(\Omega)$. Then, lemma~\ref{lemma-schema-temps} (paragraph~2) yields $ \|\zeta_h^{(n+1)}\| = {\mathscr{O}}(h^{r+1})$. On the other hand, the lemma~\ref{lemma-schema-temps} and~\ref{lemma-approx-r} (paragraph~2) and the continuity of $\pi_h$ in $H^1(\Omega)$ give $ \|\eta_h^{(n+1)}\| = {\mathscr{O}}(\Delta t)$. Thus, the result yields by reporting $\|\rho_h^{(n+1)}\| = {\mathscr{O}}(h^{r+1}+\Delta t)$ in~\eqref{eq-appli-gronwall}. \end{proof} \section{Auxiliary lemma} \begin{lemma}[Estimate on the approximate transformation] \label{eq-rhs} There exist two positive constants $C_0$ and $\Delta t_0$ such that, if $\Delta t \leq \Delta t_0$, then \[ \big\| R_{\Delta t}^{(n)} \times \tau \circ X_{\Delta t}^{(n)} \times \left( R_{\Delta t}^{(n)} \right)^T \big\|^2 \leq (1+C_0 \Delta t) \| \tau \|^2, \quad \forall \tau\in L^2(\Omega). \] \end{lemma} \begin{proof} Let $d(x)=\mathop{\rm dist}(x,\partial \Omega)$ for $x\in\Omega$. Since ${\bf u}=0$ on $\partial \Omega$, we have \[ |X_{\Delta t}^{(n)}(x)-x| = |{\bf u}(x,t_{n+1})| \, \Delta t \leq \|{\bf u}\|_{L^\infty(W^{1,\infty})} \, d(x) \Delta t. \] This leads to $X_{\Delta t}^{(n)}(x) \in \Omega$ for $\Delta t < 1/|{\bf u}|_{L^\infty(W^{1,\infty})}$. Since $X_{\Delta t}^{(n)}(\partial\Omega) = \partial\Omega$, and from the continuity of ${\bf u}(.,t_{n+1})$, we get $X_{\Delta t}^{(n)}(\Omega) = \Omega$. Let $J_{\Delta t}^{(n)} = \mbox{det} \left( R_{\Delta t}^{(n)} \right)$. Let $|.|$ denotes the matrix norm in $\mathbb{R}^{d\times d}$. We get \begin{align*} &\int_\Omega \big| R_{\Delta t}^{(n)} \times \tau \circ X_{\Delta t}^{(n)} \times \big( R_{\Delta t}^{(n)} \big)^T \big|^2 \, {\rm d}x \\ &\leq \| R_{\Delta t}^{(n)} \|_\infty^4 \int_\Omega \big| \tau \circ X_{\Delta t}^{(n)} \big|^2 \, {\rm d}x \\ &\leq \| R_{\Delta t}^{(n)} \|_\infty^4 \int_{X_{\Delta t}^{(n)}(\Omega)} \left| \tau (y) \right|^2 J_{\Delta t}^{(n)}(y) \, {\rm d}y \leq \| R_{\Delta t}^{(n)} \|_{\infty}^4 \| J_{\Delta t}^{(n)} \|_{\infty}^2 \| \tau \|^2. \end{align*} From the definition of $R_{\Delta t}^{(n)}$, we get $\| R_{\Delta t}^{(n)} \|_{\infty} \leq 1+\Delta t \| {\bf u} \|_{L^\infty(W^{1,\infty})}$ and, for $\Delta t$ small enough, there exists a positive constant $c$ depending only of ${\bf u}$ such that $\| J_{\Delta t}^{(n)} \|_{\infty} \leq 1+c \Delta t$. Then we obtain the result. \end{proof} \begin{lemma}[Time approximation] \label{lemma-schema-temps} There exists $K_1$, $C_1$ and $C_2$ such that, if $\Delta t \leq K_1 / \|{\bf u}\|_{L^\infty(W^{1,\infty})}$, then \\ (i) for all $\tau\in W^{1,\infty}(L^2(\Omega))^{d\times d}$ \[ \big\| \tau(.,t_{n+1}) - R^{(n)} \times \tau(X^{(n)},t_{n}) \times \left( R^{(n)} \right)^T \big\| \leq C_1 \Delta t \big\| \frac{\mathscr{D}_a \tau}{\mathscr{D}t} \big\|_{L^{\infty}(L^2)} \] (ii) for all $\tau\in W^{2,\infty}(L^2(\Omega))^{d\times d}$ we have \begin{align*} &\Big\| \frac{D \widehat{\tau}}{Dt} (.,t_{n+1};t_{n+1}) - \frac{ \tau(.,t_{n+1}) - R^{(n)} \times \tau(X^{(n)},t_{n}) \times \left( R^{(n)} \right)^T }{\Delta t} \Big\| \\ &\leq C_2 \Delta t \left\| {\frac{\mathscr{D}^2_a \tau}{\mathscr{D}t^2}} \right\|_{L^{\infty}(L^2)} \end{align*} \end{lemma} \begin{proof} Let $f(t) = \widehat{\tau}(X(x,t_{n+1}, t),t;t_{n+1})$. From one hand, remark that $f(t_{n+1}) = \tau(x,t_{n+1})$ and $f(t_{n}) = R^{(n)} \times \tau(X^{(n)},t_{n}) \times \left( R^{(n)} \right)^T$. From other hand, by the property of the material derivative $ f^{(m)}(t) = \frac{D^m \widehat{\tau}}{Dt^m} (X(x,t_{n+1}, t),t;t_{n+1})$, $m\geq 0$. Then, the result yields from the error estimate for the Taylor interpolation polynoms, from~\eqref{eq-prop-derivee} and from the properties of the solution $R(x,.;t_{n+1})$ of the linear differential equation~\eqref{eq-transf-tensor-hat}. \end{proof} \begin{lemma}[Trajectories and transformation approximations] \label{lemma-approx-r} If ${\bf u}$ is in the space $W^{1,\infty}(W^{1,\infty})^d$ and $\tau$ in $H^1(\Omega)^{d\times d}$ then there exists $C_3$ such that \begin{align*} &\Big\| R^{(n)} \times \tau \circ X^{(n)} \times \left( R^{(n)} \right)^T - R_{\Delta t}^{(n)} \times \tau \circ X_{\Delta t}^{(n)} \times \left( R_{\Delta t}^{(n)} \right)^T \Big\| \\ &\leq C_3 \Delta t^2 \| {\bf u} \|_{W^{1,\infty}(W^{1,\infty})} \| \tau \|_{1,2,\Omega} \end{align*} \end{lemma} \begin{proof} Let us consider the following splitting \begin{align*} &R^{(n)} \times \tau \circ X^{(n)} \times \left( R^{(n)} \right)^T - R_{\Delta t}^{(n)} \times \tau \circ X_{\Delta t}^{(n)} \times \left( R_{\Delta t}^{(n)} \right)^T \\ &= \left( R^{(n)} - R_{\Delta t}^{(n)} \right) \times \tau \circ X^{(n)} \times \left( R^{(n)} \right)^T \\ &\quad + R_{\Delta t}^{(n)} \times \tau \circ X^{(n)} \times \left( R^{(n)} - R_{\Delta t}^{(n)} \right)^T + R_{\Delta t}^{(n)} \times \left( \tau \circ X^{(n)} - \tau \circ X_{\Delta t}^{(n)} \right) \times \left( R_{\Delta t}^{(n)} \right)^T. \end{align*} From the Taylor expansion of $f_1(t) = R(X(x,t_{n+1},t),t,t_{n+1})$, we get $ R^{(n)} - R_{\Delta t}^{(n)} = {\mathscr{O}}( \Delta t^2) $. Then, from the expansion of $f_2(t) = X(x,t_{n+1},t)$, we get $ X^{(n)} - X_{\Delta t}^{(n)} = {\mathscr{O}}( \Delta t^2) $. Finally, from the development of \[ g(\theta)=\tau\left(\theta X^{(n)}(x) + (1-\theta)X_{\Delta t}^{(n)}(x) \right), \] we have $ \tau \circ X^{(n)} - \tau \circ X_{\Delta t}^{(n)} = {\mathscr{O}}( \Delta t^2) $. \end{proof} \section{Numerical experiments} In this paragraph, we present numerical results for $d=2$ and $k=1$. We choose ${\bf u}=(-x_2,x_1)$ $\Omega = ]-1/2,\,1/2[^2$, $T=2\pi$, $\gamma=0$ and the initial condition $\sigma_0$ is chosen such that the solution of~\eqref{eq-pb} is given by \begin{align*} \sigma(x,t) &= \frac{1}{2} \exp\left\{ -\frac{t}{\lambda} -\frac{(x_1-x_{1,c}(t))^2 + (x_2-x_{2,c}(t))^2} {r_0^2} \right\}\\ &\quad \times \begin{pmatrix} 1+\cos(2t) & \sin(2t) \\ \sin(2t) & 1-\cos(2t) \end{pmatrix} \end{align*} where $ x_{1,c}(t) = \bar{x}_{1,c} \cos(t) - \bar{x}_{2,c} \sin(t) $ et $ x_{2,c}(t) = \bar{x}_{1,c} \sin(t) + \bar{x}_{2,c} \cos(t) $ with $r_0>0$ and $(\bar{x}_{1,c},\,\bar{x}_{2,c})\in\mathbb{R}^2$. The numerical tests are performed for $\lambda=1$, $r_0=1/10$ and $(\bar{x}_{1,c},\,\bar{x}_{2,c})=(1/4,\,0)$. The scheme~\eqref{eq-pb-h} is based on the free software {\tt rheolef}~\cite{saramito-roquet-rheolef-2003}. The computation of the scalar product $\big( R_{\Delta}^{(n)} \sigma_h^{(n)} \circ X_{\Delta}^{(n)} \big( R_{\Delta}^{(n)} \big)^T, \tau_h \big) $ in the approximate problem~\eqref{eq-pb-h} is not an obvious task (see also~\cite{suli-88,priestley-88}). In each triangle, we use the six point fourth order Gauss quadrature formulae. Since our domain is a square, we split each edge in $M$ segments, and obtain $M^2$ small squares of length $1/M$. Then we split each of them in two triangles: the step of the mesh is then $h=\sqrt{2}/M$. Fig. 1.a represents $\|\sigma_h-\pi_h\sigma\|_{l^2(L^2)}$ as a function of $\Delta t$ for four triangulations given by $h=\sqrt{2}/2^{i}$, $3\leq i \leq 6$. We observe that when $\Delta t \rightarrow 0$, the error starts to decrease and then tends to a constant that behaves as $h^{2}$. A simultaneous choice of $(h,\Delta t)$ such that $\Delta t={\mathscr{O}}(h^2)$ is convenient~: let us choose $(h,\Delta t)=(\sqrt{2}/2^i,\,2\pi/4^i)$, $3\leq i\leq 6$. Fig. 1.b and 1.c presents the corresponding error for the $l^2(L^2)$ and $l^\infty(L^2)$ norms as a function of $h$ and $\Delta t$, respectively. These tests confirm the optimality of our estimates~\eqref{eq-estim-linf-1}-\eqref{eq-estim-l2-1}. % \input{transport-1-riesz-P1-q4-h-dt.tex} % \input{transport-1-riesz-P1-q4-dt.tex} % \input{transport-1-riesz-P1-q4-h.tex} \begin{figure}[htb] \begin{center} \input{fig1a.tex} \quad \input{fig1b.tex} \\ \input{fig1c.tex} \end{center} \caption{Method convergence versus $h$ and $\Delta t$.} \end{figure} \begin{thebibliography}{0} \bibitem{pironneau-82} M.~Bercovier and O.~Pironneau. \newblock Characteristics and the finite element method. \newblock In T.~Kawai, editor, {\em \em Proc. 4th Int. Symp. on finite element methods in flow problems}, pages 67--73. 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