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\proceedings{Finite volumes for complex applications V}{-}
\begin{document}
% \date{October 15}
%\maketitle % Exemple de 1ere page sans les valeurs de champs
            % pour afficher les commandes a utiliser

%\journal[Mode d'emploi]{Signature de l'article~: nom de la revue}%
%   {10}{5}{1999}{1}{15}% Exemple avec numeros de page

%\journal{\LOBJET}{8}{\LMO}{2002}{45}{57}
\journal{}{6}{}{2008}{1}{15}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% Macros for the different tests
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\ocvld{{\bf ocvl2}}
\def\ocvgradld{{\bf ocvgradl2}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%
\title[Benchmark on anisotropic problems]%
      {Benchmark on anisotropic problems  }

\subtitle{Insert title - {\small
    (December 8th)} }
\author{Author name 1, Author name 2}

 
\address{% affiliation
INSERT ADRESS\\[3pt]
email}



\abstract{Insert abstract}



\keywords{Anisotropy benchmark, finite volume....}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Presentation of the scheme}
\section{Numerical results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ {{Test 1.1} Mild anisotropy, $ u(x,y) = 16 x(1-x) y(1-y)$, $\min = 0, \,  \max = 1$}, {{regular triangular mesh,}} {\tt mesh1}} 
%subroutine solution1_1 in sources.f90
%benchmark meshes mesh1_i, i=1, 7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% in the tables please enter the results using the format: sign integer dot integer ingeger E sign integer integer. 
%For instance: 3.01E00 or -6.02E-01 or 1.10E04. 
% This format is given for instance by the FORTRAN format S9.2

\begin{center}
\begin{tabular}{|c|ccccccc|}
i&nunkw&nnmat&sumflux&erl2&ergrad&ratiol2&ratiograd\\
\hline
1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
6&&&&&&&\\
7&&&&&&&\\
\hline
\end{tabular}
\end{center}
\ocvld$=???$, \ \ocvgradld $=???$.
\medskip
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&erflx0&erflx1&erfly0&erfly1&erflm&umin&umax\\
\hline
1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
6&&&&&&&\\
7&&&&&&&\\
\hline
\end{tabular}
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ {{Test 1.1} Mild anisotropy, $u(x,y) = 16 x(1-x) y(1-y)$,
  $\min = 0, \,  \max = 1$}, {{coarse  (C) and fine (F) distorted quadrangular meshes}}, {\tt mesh4\_1}  and {\tt mesh4\_2}}
%subroutine solution1_1 in sources.f90
%benchmark meshes mesh4_1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabular}{|c|ccccc|}
grid&nunkw&nnmat&sumflux&erl2&ergrad\\
\hline
C&&&&&\\
F&&&&&\\
\hline
\end{tabular}
\medskip

\begin{tabular}{|c|ccccccc|}
grid&erflx0&erflx1&erfly0&erfly1&erflm&umin&umax\\
\hline
C&&&&&&&\\
F&&&&&&&\\
\hline
\end{tabular}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ {{Test 1.2} Mild anisotropy,  $u(x,y) = \sin((1-x)(1-y))
+(1-x)^3(1-y)^2$, $\min = 0,\, \max = 1+  \sin 1$}, {{regular triangular mesh}}, {\tt mesh1}} 
%subroutine solution1_2 in sources.f90
%benchmark meshes mesh1_i, i=1, 7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&nunkw&nnmat&sumflux&erl2&ergrad&ratiol2&ratiograd\\
\hline
1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
6&&&&&&&\\
7&&&&&&&\\
\hline
\end{tabular}

\end{center}
\ocvld$=???$, \ \ocvgradld $=???$.
\medskip
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&erflx0&erflx1&erfly0&erfly1&erflm&umin&umax\\
\hline
1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
6&&&&&&&\\
7&&&&&&&\\
\hline
\end{tabular}
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ {{Test 1.2} Mild anisotropy,  $u(x,y) = \sin((1-x)(1-y))
+(1-x)^3(1-y)^2$, $\min = 0,\, \max = 1+  \sin 1$}, {{locally refined nonconforming rectangular mesh}}, {\tt mesh3}}
%subroutine solution1_2 in sources.f90
%benchmark meshes mesh3_i, i=1, 5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&nunkw&nnmat&sumflux&erl2&ergrad&ratiol2&ratiograd\\
\hline1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
\hline
\end{tabular}
\end{center}
\ocvld$=???$, \ \ocvgradld $=???$.
\medskip
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&erflx0&erflx1&erfly0&erfly1&erflm&umin&umax\\
\hline
1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
\hline
\end{tabular}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ Test 2 Numerical locking, $u(x,y) = \sin(2\pi x)e^{-2\pi \sqrt{\frac 1 \delta} y},\,\delta=10^5$,  $\min = -1,\, \max = 1$, 
regular triangular mesh, {\tt mesh1} }
%subroutine solution2 in sources.f90
%benchmark meshes mesh1_i, i=1, 7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&nunkw&nnmat&sumflux&erl2&ergrad&ratiol2&ratiograd\\
\hline
1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
6&&&&&&&\\
7&&&&&&&\\
\hline
\end{tabular}
\end{center}
\ocvld$=???$, \ \ocvgradld $=???$.
\medskip
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&erflx0&erflx1&fluy0&fluy1&erflm&umin&umax\\
\hline
1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
6&&&&&&&\\
7&&&&&&&\\
\hline
\end{tabular}
\end{center}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ {{Test 2} Numerical locking, $u(x,y) = \sin(2\pi x)e^{-2\pi\sqrt{\frac 1 \delta} y},\,\delta=10^6$, $\min = -1,\, \max = 1$}, {{regular triangular mesh}}, {\tt mesh1}}
%subroutine solution2 in sources.f90
%benchmark meshes mesh1_i, i=1, 7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&nunkw&nnmat&sumflux&erl2&ergrad&ratiol2&ratiograd\\
\hline
1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
6&&&&&&&\\
7&&&&&&&\\
\hline
\end{tabular}
\end{center}
\ocvld$=???$, \ \ocvgradld $=???$.
\medskip
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&erflx0&erflx1&fluy0&fluy1&erflm&umin&umax\\
\hline
1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
6&&&&&&&\\
7&&&&&&&\\
\hline
\end{tabular}
\end{center}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \paragraph*{\bf$\bullet$ {{Test 3} Oblique flow, $\min = 0,\, \max = 1$}, {{uniform rectangular mesh}}, {\tt mesh2}}
%subroutine solution3 in sources.f90
%benchmark meshes mesh2_i, i= 1, 5
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Description of the user chosen reference mesh (and step size) if available.

 \begin{center}
 \begin{tabular}{|c|ccccccc|}
 i&nunkw&nnmat&sumflux&umin&umax\\
 \hline1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
ref&&&&&&&\\
 \hline
 \end{tabular}
\end{center}

\medskip
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&flux0&flux1&fluy0&fluy1&ener1&ener2&eren\\
\hline
1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
ref&&&&&&&\\
\hline
\end{tabular}

 \end{center}

INSERT approximate solutions for  mesh2\_i for i=2 , i=3 ,
  i=4 , if available.

% \begin{figure}
% \begin{center}
% \includegraphics[width=3cm]{oblique_2.eps}\hskip 0.5cm\includegraphics[width=3cm]{oblique_3.eps}\hskip 0.5cm\includegraphics[width=3cm]{oblique_4.eps}
% \end{center}\caption{Solutions for the oblique flow on mesh2\_i for i=2 (left), i=3 (center),
%   i=4 (right)}
% \end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ {{Test 4} Vertical fault, $\min = 0.,\, \max = 1.$}, {{non conforming rectangular mesh}}, {\tt mesh5}}
%subroutine solution4 in sources.f90
%benchmark meshes mesh5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Description of the user chosen reference mesh (and step size) if available.

 \begin{center}
 \begin{tabular}{|c|ccccccc|}
 i&nunkw&nnmat&sumflux&umin&umax\\
 \hline
1&&&&&&&\\
reg&&&&&&&\\
ref&&&&&&&\\
 \hline
 \end{tabular}
\end{center}

\medskip
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&flux0&flux1&fluy0&fluy1&ener1&ener2&eren\\
\hline
1&&&&&&&\\
reg&&&&&&&\\
ref&&&&&&&\\
\hline
\end{tabular}

 \end{center}

INSERT the figure of  approximate solution  if available.

% \begin{figure}
% \begin{center}
% \includegraphics[width=3cm]{verticalfault.eps}}\hskip 0.5cm\includegraphics[width=3cm]{verticalfault_reg.eps}}\hskip 0.5cm\includegraphics[width=3cm]{verticalfault_ref.eps}
% \end{center}\caption{Solution for the vertical fault on a non conforming rectangular mesh}
% 
% \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ {{Test 5} Heterogeneous rotating anisotropy, $\min = 0,\, \max = 1$ }, {{uniform rectangular mesh}}, {\tt mesh2}}
%subroutine solution5 in sources.f90
%benchmark meshes mesh2_i, i= 1, 5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&nunkw&nnmat&sumflux&erl2&ergrad&ratiol2&ratiograd\\
\hline1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
\hline
\end{tabular}
\end{center}
\ocvld$=???$, \ \ocvgradld $=???$.
\medskip
\begin{center}
\begin{tabular}{|c|ccccccc|}
i&erflx0&erflx1&erfly0&erfly1&erflm&umin&umax\\
\hline
1&&&&&&&\\
2&&&&&&&\\
3&&&&&&&\\
4&&&&&&&\\
5&&&&&&&\\
\hline
\end{tabular}
\end{center}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ {{Test 6} Oblique drain, $\min = -1.2,\, \max = 0$}, {{coarse (C) and fine (F) oblique meshes}}, {\tt mesh6} and  {\tt mesh7}}
%subroutine solution6 in sources.f90
%benchmark meshes mesh6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabular}{|c|ccccc|}
grid&nunkw&nnmat&sumflux&erl2&ergrad\\
\hline
C&&&&&\\
F&&&&&\\
\hline
\end{tabular}
\medskip

\begin{tabular}{|c|ccccccc|}
grid&erflx0&erflx1&erfly0&erfly1&erflm&umin&umax\\
\hline
C&&&&&&&\\
F&&&&&&&\\
\hline
\end{tabular}
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ {{Test 7} Oblique barrier, $\min = -5.575,\, \max = 0.575$}, {{coarse oblique mesh  {\tt mesh6}}}}

%subroutine solution7 in sources.f90
%benchmark meshes mesh6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabular}{|ccccc|}
nunkw&nnmat&sumflux&erl2&ergrad\\
\hline
 &&&&\\

\hline
\end{tabular}
\medskip

\begin{tabular}{|ccccccc|}
erflx0&erflx1&erfly0&erfly1&erflm&umin&umax\\
\hline
&&&&&&\\
\hline
\end{tabular}
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ {{Test 8} Perturbed parallelograms, $\min = 0 $}, {{perturbed parallelograms mesh  {\tt mesh8}}}}


%benchmark meshes mesh8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabular}{|ccccc|}
nunkw&nnmat&sumflux&umin&umax\\
\hline
&&&&\\

\hline
\end{tabular}

\begin{tabular}{|cccc|}
flux0&flux1&fluy0&fluy1\\
\hline
&&&\\
\hline
\end{tabular}

\end{center}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{\bf $\bullet$ {{Test 9} Anisotropy with wells, $\min =
    0,\, \max = 1.$}, {{square uniform grid   {\tt mesh9}}}}

%benchmark meshes mesh9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabular}{|ccccc|}
nunkw&nnmat&sumflux&umin&umax\\
\hline
&&&&\\

\hline
\end{tabular}

\end{center}


\section{Comments on the results} 



INSERT Bibliography
\cite{*}
%\bibliography{biblio}


\end{document}
