\documentclass[a4paper,10pt]{book}
\usepackage[utf8x]{inputenc}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{cclicenses}
\usepackage{url}
\usepackage{listings}
\usepackage{hyperref}
\usepackage{listings}
\usepackage{color}
\usepackage[english]{babel}
\usepackage[T1]{fontenc}


\definecolor{dkgreen}{rgb}{0,0.6,0}
\definecolor{gray}{rgb}{0.5,0.5,0.5}
\definecolor{mauve}{rgb}{0.58,0,0.82}

\lstset{frame=tb,
  language=Octave,
  aboveskip=3mm,
  belowskip=3mm,
  showstringspaces=false,
  columns=flexible,
  basicstyle={\small\ttfamily},
  numbers=left,
  numberstyle=\tiny\color{gray},
  keywordstyle=\color{blue},
  commentstyle=\color{dkgreen},
  stringstyle=\color{mauve},
  breaklines=true,
  breakatwhitespace=true
  tabsize=3
}

\author{Marco Vassallo}
\title{ \textbf{Fem-fenics} \\ \bigskip \textit{General purpose Finite Element library \\ for GNU-Octave}\\ 
\bigskip
       \textsc{ Work in progress \\(help and remarks are welcome)}}


\begin{document}

\maketitle
\tableofcontents

\chapter{Introduction}
Fem-Fenics is an open-source package for the resolution of partial differential equations with Octave.
The project has been developed during the Google Summer of Code 2013 with the help and the sustain of the GNU-Octave 
community under the supervision of prof. De Falco.\\

The report is structured as follows:
\begin{itemize}
  \item in chapter \ref{intr} we provide a simple reference guide for beginners
  \item in chapter \ref{impl} is presented a detailed explanation of the relevant parts of the program. In this way, the 
  interested reader can see what there is ``behind'' and expecially anyone interested in it can learn quickly how
  it is possible to extend the code and contribute to the project.
  \item in chapter \ref{exem} more examples are provided. For a lot of them, we present the octave script 
  alongside with the code for Fenics (in C++ and/or Python) in order to provide the user with a quick reference
  guide.
\end{itemize}

If you think that going inside the report could be boring, it is available a wiki at
\begin{center}
\url{http://wiki.octave.org/Fem-fenics}
\end{center}
while if you want to see how the project has grown during the time you can give a look at
\begin{center}
\url{http://gedeone-gsoc.blogspot.com/}
\end{center}
Finally, the API is available at the following address
\begin{center}
\url{http://octave.sourceforge.net/fem-fenics/overview.html}
\end{center}

\chapter{Introduction to Fem-fenics}\label{intr} 

\section{Installation}
Fem-fenics is an external package for Octave. It means that you can install it only once that you
have successfully installed Octave on your PC. Furthermore, as Fem-fenics is based on Fenics,
you also need a running version of the latter. They can be easily installed following the guidelines provided
on the official Octave \cite{instoctave} and Fenics \cite{instfenics} websites.
Once that you have got Octave and Fenics, you can just launch Octave (which now is provided with a new
amazing GUI) and type

\begin{verbatim}
 >> pkg install fem-fenics -forge
\end{verbatim}

That's all! If you encounter any problem during the installation don't hesitate to contact us.
To be sure that everything is working fine, you can load the fem-fenics pkg and run
one of the examples provided within the package:

\begin{verbatim}
 >> pkg load fem-fenics
 >> femfenics_examples()
\end{verbatim}

For a description of the examples, you are referred to chapter \ref{exem}.
\\
\\
\textbf{NOTE} For completing the installation process successfully,
the form compiler FFC and the header file dolfin.h should also be available on your machine.
They are managed automatically by Fenics if you install it as a binary package or with Dorsal.
If you have done it manually, please be sure that they are available before starting the
installation of Fem-fenics.

\section{General layout and first example}

A generic problem has to be solved in two steps:
\begin{enumerate}
\item a \textbf{.ufl file} where the abstract problem is described: this file has to be written in Unified Form Language (UFL),
       which is a domain specific language for defining discrete variational forms and functionals in a notation 
       close to pen-and-paper formulation. UFL is easy to learn, and the User manual provides explanations
       and examples \cite{ufl}.
\item a script file \textbf{.m} where the abstract problem is imported and a specific problem is implemented and solved:
      this is the script file where the fem-fenics functions described in the following chapters are used.
\end{enumerate}

We provide immediately a simple example in order to familiarize the user with the code.

\paragraph{The Poisson equation}
In this example, we show how it is possible to solve the Poisson equation with mixed Boundary Conditions.
If we indicate with $\Omega$ the domain and with $\Gamma = \Gamma_{N} \cup \Gamma_{D}$ the
boundaries, the problem can be expressed as
\begin{align*}
  \Delta u &= f \qquad \text{on } \Omega \\
  u &= 0 \qquad \text{on } \Gamma_{D} \\
  \nabla u \cdot n &= g \qquad \text{on } \Gamma_{N}
\end{align*}
where $f, \, g$ are data which represent the source for and the flux
of the scalar variable $u$.
A possible variational formulation of the problem is: \\
find $u \in H_{0, \Gamma_{D}}^{1} :$
\begin{align*}
  a(u, v) &= L(v) \qquad \forall v \in H_{0, \Gamma_{D}}^{1} \\
  a(u, v) &= \int_{\Omega} \nabla u \cdot \nabla v \\
  L(v)    &= \int_{\Omega} f v + \int_{\Gamma_{N}} g v \\
\end{align*}

The abstract problem can thus be written in the \verb|Poisson.ufl| file immediately.
The only thing that we have to specify at this stage is the space of Finite Elements
which we want to use for the discretization of $H_{0, \Gamma_{D}}^{1}$. In our case,
we choose the space of continuous lagrangian polynomial of degree one 
\verb|FiniteElement("Lagrange", triangle, 1)|, but many more
possibilities are available.
\subparagraph{Poisson.ufl}

\begin{lstlisting}
element = FiniteElement("Lagrange", triangle, 1)

u = TrialFunction(element)
v = TestFunction(element)

f = Coefficient(element)
g = Coefficient(element)

a = inner(grad(u), grad(v))*dx
L = f*v*dx + g*v*ds
\end{lstlisting}

It is always a good idea to check if the ufl code is correctly written before importing it into Octave:
\begin{verbatim}
 >> ffc -l dolfin Poisson.ufl
\end{verbatim} 
shouldn't produce any error.
We can now implement and solve a specific instance of the Poisson problem in Octave.
We choose to set the parameters as follow
\begin{itemize}
 \item $\Omega = [0, 1]\times[0, 1]$
 \item $\Gamma_{D} = {(0, y) \cup(1, y)} \ \subset \partial\Omega$
 \item $\Gamma_{N} = {(x, 0) \cup(x, 1)} \ \subset \partial\Omega$
 \item $f = 10 \exp \dfrac{(x-0.5)^{2} + (y-0.5)^{2}}{0.02}$
 \item $g = \sin(5x)$
\end{itemize}

As a first thing we need to load into Octave the pkg that we have previously installed
\begin{verbatim}
    pkg load fem-fenics msh
\end{verbatim}
We can thus import the ufl file inside Octave. From the ufl file, we have to generate the 
corresponding functions for fem-fenics. There is a specific function which seeks 
every specific element defined inside the ufl file:
\begin{itemize}
    \item \verb$ufl_import_FunctionSpace ('Poisson')$ is a function which looks for the finite element 
    space defined inside the file called Laplace.ufl; if everything is ok, it generates a function
    which we will use later
    \item \verb$ufl_import_BilinearForm ('Poisson')$ is a function which looks for the rhs of the 
    equation, i.e. for the bilinear form defined inside Laplace.ufl;
    \item \verb$ufl_import_LinearForm ('Poisson')$ is a function which looks for the linear 
    form.
\end{itemize}
In some cases one could be interested in using these functions separately but if, 
as in our example, all the three elements are defined in the same ufl file (and only in this case), 
we can just call \verb$import_ufl_Problem ('Poisson')$ which generates at once all 
the three functions described above.

\begin{verbatim}
    ufl_import_Problem ('Poisson');
\end{verbatim}

To set the concrete elements which define our problem, 
the first things to do is to create a mesh. 
It can be managed easily using the msh pkg. In our case, we create a uniform mesh
\begin{verbatim}
    x = y = linspace (0, 1, 33);
    msho = msh2m_structured_mesh (x, y, 1, 1:4);
\end{verbatim}
Once that the mesh is available, we can thus initialize the 
fem-fenics mesh using the function \verb$fem_init_mesh()$:
\begin{verbatim}
    mesh = Mesh (msho);
\end{verbatim}
If instead of an Octave mesh you have a mesh stored in a different format,
you can try to convert it to the dolfin xml format using the program dolfin-convert.
In fact, \verb$Mesh ()$ accepts as arguments also a string with the name of 
the dolfin xml file which contains your mesh. For example, if you have a mesh saved
in the gmsh format, you can do as follows:
\begin{verbatim}
    Shell:
      dolfin-convert msh.gmsh msh.xml
\end{verbatim}
and then inside our Octave script:
\begin{verbatim}
    mshd = fem_init_mesh ('msh.xml');
\end{verbatim}
To initialize the functional space, we have to specify as argument only the fem-fenics mesh,
because the finite element type and the polynomial degree have been specified in the ufl file:
\begin{verbatim}
    V = FunctionSpace('Poisson', mesh);
\end{verbatim}
We can now apply essential BC using \verb$DirichletBC ()$. This function receives as argument the functional space,
a function handle which specifies the value that we want to set, and the label of the sides where we want
to apply the BC. In our case, we apply homogenous boundary condition on the left and right side of the square
\begin{verbatim}
    bc = DirichletBC(V, @(x, y) 0.0, [2;4]);
\end{verbatim}
The last thing that we have to do before solving the problem, is to set the coefficients specified
in the ufl file. It is important that they are called in the same way as in the ufl file.
In our case they are the source term 'f' and the normal flux 'g'. 
To set them, we can for example use the function \verb$Expression ()$ to which we have to pass as argument a string,
which specifies the name of the coefficient, and a function handle with the value required:
\begin{verbatim}
    f = Expression ('f', 
          @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));
    g = Expression ('g', @(x,y) sin (5.0 * x));
\end{verbatim}
Another possibility for dealing with the coefficients defined in the ufl file would be to use 
the function \verb$Constant ()$ or \verb$Function ()$.
The coefficient can thus be used together with the FunctionSpace to set 
the Bilinear and the Linear form
\begin{verbatim}
    a = BilinearForm ('Poisson', V, V);
    L = LinearForm ('Poisson', V, f, g);
\end{verbatim}
We can now obtain the discretized representation of our operator using the 
functions \verb$assemble ()$ or \verb$assemble_system ()$, which also allow us
to specify the BC(s) that we want to apply.
Whenever possible, it is better to use the \verb$assemble_system ()$ function
because it keeps the symmetry of the matrix while setting the entries for the BC:
\begin{verbatim}
    [A, b] = assemble_system (a, L, bc);
\end{verbatim}
Here A is a sparse matrix and b is a column vector. We can thus use all the 
functionalities available within Octave to solve the linear system. 
For the moment we use the easisest possibility, i.e. the backslash command to solve the linear system:
\begin{verbatim}
    u = A \ b;
\end{verbatim}
Once that the solution has been obtained, we can transform the vector into a 
fem-fenics function and plot it \verb$plot ()$, or save it \verb$save ()$ in the vtu
format.
\begin{verbatim}
    u = Function ('u', V, sol);
    save (u, 'poisson')
    plot (u);
\end{verbatim}

The complete code for the Poisson problem is reported below, while
in figure \ref{Poissonfig} is presented the output.
\begin{figure}
 \begin{center}
  \includegraphics[height=7 cm,keepaspectratio=true]{./Fem-fenics_poisson.png}
   \caption{The result for the Poisson equation}
   \label{Poissonfig}
  \end{center}
\end{figure}

\subparagraph{Poisson.m}
\begin{lstlisting}
#load the pkg and import the ufl problem
pkg load fem-fenics msh
import_ufl_Problem ('Poisson')
 
# Create the mesh and define function space
x = y = linspace (0, 1, 33);
mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4));
V = FunctionSpace('Poisson', mesh);

# Define boundary condition and source term
bc = DirichletBC(V, @(x, y) 0.0, [2;4]);
f = Expression ('f', @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));
g = Expression ('g', @(x,y) sin (5.0 * x));

#Create the Bilinear and the Linear form
a = BilinearForm ('Poisson', V, V);
L = LinearForm ('Poisson', V, f, g);
 
#Extract the matrix and compute the solution
[A, b] = assemble_system (a, L, bc);
sol = A \ b;
u = Function ('u', V, sol);
 
# Save solution in VTK format and plot it
save (u, 'poisson')
plot (u);

\end{lstlisting}



\chapter{Implementation}\label{impl} 
Two main ideas have guided us throughout the realization of the pkg:

\begin{itemize}
  \item keep the syntax as close as possible to the original one in Fenics
  \item make the interface as simple as possible.
\end{itemize}
\iffalse
Fem-fenics aims to fill a gap in Octave: even if there are pkgs for the creation of mesh \cite{msh},
for the postprocessing of data \cite{fpl} and for the resolution of some specific pde \cite{secs1d} \cite{bim},
no general purpose finite element library is available.
Our goal is thus to provide a pkg which can be used to solve user defined problem and which is able to
exploit the functionality provided from Octave. Instead of writing a library from scratch, we decided to
write an interface for one of the finite element library which are already available.
We considered a lot of possibilities: Dune Elmer Fenics FreeFem++ Hermes LifeV

``The FEniCS Project is a collection of free, open source, software
components with the common goal to enable automated solution of pde.''

Dolfin is a C++/Python interface of FEniCS, providing a consistent Problem
Solving Environment for ODE and PDE.
Why DOLFIN?
C++ library
Provides data structures and algorithms for computational meshes
and finite element assembly (... and also a namespace)
A lot of examples and broadly supported
The code is well written/organized


Our idea is to create wrappers in Octave for Dolfin, which is the problem-solving C++ environment of Fenics ,
in a similar way to what it has been done for Python. 

The general structure of the code is summarized in figure . The start point is a.
\fi
\section{General layout of a class}
All these classes derive from \verb$octave_base_value$. 
\iffalse
By now, the general layout of a class is very simple and looks like:
As a general rule, now all the private members are handled using a \verb$boost::shared_ptr< >$ to the corresponding DOLFIN type.  
This is done for two main reasons:
    in our program we have to refer in several places to resources which are built dynamically 
    (e.g. a FunctionSpace contain a reference to the Mesh) and we want that it is destroyed only
    when the last references is destroyed.
    they are widely used inside DOLFIN and it is thus easier to deal with it if we use the same types.
All the classes implement a constructor which takes as argument the corresponding dolfin type and a default 
constructor which is necessary to register a type in the Octave interpreter.

\paragraph{mesh}  The mesh class derives publicly from \verb$octave_base_value$ and dolfin::Mesh and it provides 
the minimum of functionalities needed for our TDD.
In addition to usual methods, we have implemented functionalities which allow us to deal with meshes 
currently available with the msh pkg. We have thus added the constructor
\newline
 \verb$mesh (Array<double>& p, Array<octave_idx_type>& e, Array<octave_idx_type>& t);$
\newline
and the method

  \verb$octave_scalar_map get_pet (void) const;$

We have thus implement the functions which allow us to initialize an object of this class from Octave
\newline
  \verb$  DEFUN_DLD (fem_init_mesh, args, , "...")$
\newline
and to get it back in the (p, e, t) format
\newline
  \verb$  DEFUN_DLD (fem_get_mesh, args, ,"... ")$
\newline
We can thus now run in Octave a script like:
\begin{verbatim} 
  msh1 = msh2m_structured_mesh(1:6,1:4,0,3:6);
  msh2 = fem_init_mesh (msh1);
  msh3 = fem_get_mesh (msh2);
\end{verbatim}
We can thus check that the msh1 and msh3 variables represent the same mesh and with the command whos we get
\begin{verbatim}
   Name       Size                     Bytes      Class
--------------------------------------------------------------------------
    msh1        1x1                       2240      struct
    msh2        0x0                          0      mesh
    msh3        1x1                       2240      struct
\end{verbatim}
 which confirms that msh2 is a variable of our mesh class.


\paragraph{functionspace}
A dolfin::FunctionSpace is defined by specifying a mesh and the type of the finite element which we want to use. 
The mesh is handled by our class, while the FE are specified inside the .ufl file. Possible choices are:

    Argyris                                  ARG
    Arnold–Winther                    AW
    Brezzi–Douglas–Marini        BDM
    Crouzeix–Raviart                  CR
    Discontinuous Lagrange      DG
    Hermite                                HER
    Lagrange                              CG
    Mardal–Tai–Winther             MTW
    Morley                                  MOR
    Nédélec 1st kind H (curl)     N1curl
    Nédélec 2nd kind H (curl)    N2curl
    Raviart–Thomas                   RT

Once that the .ufl file has been compiled, in the output file it is defined a namespace where we can find 
the corresponding FunctionSpace and which only needs a mesh to be initialized.

\paragraph{boundary condition}
This class is used for dealing with essential BC. We need three elements for the definition of an object of
type dolfin::DirichletBC :

    a dolfin::FunctionSpace where the BC is defined, which is handled by our class functionspace described above
    a dolfin::Function or a dolfin::Expression which represents the value that we want to assign.
    This is done deriving a new class from dolfin::Expression and overloading the eval function.
    a dolfin::subdomain which specifies where we want to apply the BC. 
\paragraph{coefficient}
This class is used to store a std::string where we save the name of the coefficient, 
and an object of type expression, which is a class derived from dolfin::Expression. 
The class expression would be explained in more detail in the next post.
\fi
\section{General layout of a function}
\iffalse
The functions which we describe below are used to set and get the values for the variable that are commonly 
used in a FEM problem.

The main functions which are available right now are:

    \verb$fem_init_mesh$: this function can take as input either a string which specify 
    the name of the file to read or a mesh produced with the msh pkg
    \verb$fem_bc$: take as input the functional space, a function handle which specifies 
    the value that we want to set on the boundary, and the label of the boundaries where we want to apply the bc
    coefficient: take as input the name of one of the coefficients defined in 
    the .ufl file and a function handle with the value that we want to set

The following functions work fine but at the moment are directly compiled including 
the output file obtained from the .ufl file (which should be avoided as discussed below) :

    \verb$fem_fs$: take as input the mesh where we want to define the functional space
    \verb$fem_rhs$: take as input the function space where the bilinear form is defined, 
    a list of coefficients and a list of BCs that we want to apply. It returns a 
    sparse matrix which contains the discretization of the bilinear operator
    \verb$fem_lhs$: take the same input as the function \verb$fem_rhs$. If you have essential BC 
    in the problem, you need to pass them also to this function in such a way that 
    the DOF corresponding to essential nodes are set to the right value

The general layout of a function is very simple and is composed of 4 steps which we describe using an example:
\begin{lstlisting}
DEFUN_DLD (fem_fs, args, , "initialize a fs from a mesh")
{
          // 1 read data
          const mesh & msho = static_cast<const mesh&> (args(0).get_rep ());
          // 2 convert the data from octave to dolfin
          const dolfin::Mesh & mshd = msho.get_msh ();
          // 3 build the new object using dolfin
          boost::shared_ptr <const dolfin::FunctionSpace> g (new Laplace::FunctionSpace (mshd));
          // 4 convert the new object from dolfin to Octave and return it
          octave_value retval = new functionspace(g);
           return retval;
}
\end{lstlisting}
All the functions presented above follow this general structure, and thus here we present 
in detail only functions which present some differences.

\paragraph{fem bc and fem coeff}
 These two functions take as input a function handle which cannot be directly evaluated by
 a dolfin function to set, respectively, the value on the boundary or the value of the coefficient. 
 We have thus derived from dolfin::Expression a class "expression" which has as private member 
 an octave function handle and which  overloads the function eval(). In this way, an object of 
 the class expression can be initialized throughout a function handle and can be used inside dolfin because 
 "it is"  a dolfin::Expression.
\begin{lstlisting}
class expression : public dolfin::Expression
{

 public:
  expression (octave_fcn_handle & _f) :
             dolfin::Expression (), f (new octave_fcn_handle (_f)) {}

  void eval (dolfin::Array<double>& values, const dolfin::Array<double>& x) const
  {
    octave_value_list b;
    b.resize (x.size ());
    for (int i = 0; i < x.size (); ++i)
      b(i) = x[i];
    octave_value_list res = feval (f->function_value (), b);
    values[0] = res(0).double_value ();
  }

 private:
  octave_fcn_handle * f;
}; 
\end{lstlisting}

\paragraph{fem lhs and fem rhs}
 For these functions, we need to set the possible coefficient of the (bi)linear form  and to apply the BC.

For the coefficient: 
\begin{lstlisting}
    // read the coefficient 
    const coefficient & cf = static_cast <const coefficient&> (args (i).get_rep ());
    // get the correct position of the coefficient in the declaration of the problem 
    std::size_t n = a.coefficient_number (cf.get_str ());
    // extract the information.. we use again the expression class descripted above 
    const boost::shared_ptr<const expression> & pexp = cf.get_expr (); 
    //set the coefficient for the (bi)linear form 
    a.set_coefficient (n, pexp);
\end{lstlisting}
For the BC
At the moment we are simply using the method apply() of the class DirichletBC, 
but it does not preserve the symmetry of the matrix. We have thus planned to insert 
later a function which instead of the method apply() calls the function \verb$assemble_system()$ , 
which preserves the symmetry but which builds together the lhs and the rhs.
The sparse matrix is then built from the dolfin::Matrix following the general guidelines of Octave.


\paragraph{other function}


    SubSpace allows to extract a subspace from a vectorial one. 
    For example, if our space is P2 x P0 we can extract the one or 
    the other and then apply BC only where it is necessary.
    \verb$fem_eval$ takes as input a Function and a coordinate and returns a 
    vector representing the value of the function at this point.
    for dealing with form of rank 0, i.e. with functional, we have now 
    added the functions \verb$fem_create_functional$ to create it from a .ufl file. 
    We have thus extended the function assemble which returns the corresponding double value.
    \verb$plot_2d$ and \verb$plot_3d$: these functions allow us to plot a function specifying 
    a mesh and the value of the function at every node of the mesh. 
    This is something which could be useful also outside of fem-fenics.


\section{Implementation Details}
The relevant implementation details which the user should know are:

    all the objects are managed using \verb$boost::shared_ptr <>$. 
    It means that the same resource can be shared by more objects and useless copies 
    should be avoided. For example, if we have two different functional spaces in the same problem, 
    like with Navier-Stokes for the velocity and the pressure, the mesh is shared between 
    them and no one has its own copy.
    The BC are imposed directly to the mesh setting to zero all the off diagonal elements 
    in the corresponding line. This means that we could loose the symmetry of the matrix, if any. 
    The simmetry is lost because we are using the method apply() of the class dolfin::DirichletBC. 
    We have thus planned to insert later a function which instead of the method apply() calls the 
    function \verb$assemble_system()$ , which preserves the symmetry of the system but which builds 
    together the lhs and the rhs.
    The coefficient of the variational problem can be specified using either a fem-coefficient 
    or a fem-function. They are different objects which behave in different ways: a fem-coefficient 
    object overloads the eval() method of the  dolfin::Expression class and it is evaluated at 
    run time using the octave function feval (). A fem-function instead doesn't need to be evaluated 
    because it is assembled copying element-by-element the values contained in the input vector.
    
    The relevant implementation details which the user should know are:

    all the objects are managed using \verb$boost::shared_ptr <>$. 
    It means that the same resource can be shared by more objects and useless copies should be avoided. 
    For example, if we have two different functional spaces in the same problem, like with Navier-Stokes f
    or the velocity and the pressure, the mesh is shared between them and no one has its own copy. 

    The essential BC are imposed directly to the matrix with the command assemble(), 
    which sets to zero all the off diagonal elements in the corresponding line, sets to 1 
    the diagonal element and sets to the exact value the rhs. This means that we could loose
    the symmetry of the matrix, if any. To avoid this problem and preserve the symmetry of 
    the system it is available the \verb$assemble_system()$ command which builds at once the lhs and the rhs. 

    The coefficient of the variational problem can be specified using either 
    an Expression(), a Constant() or a Function(). They are different objects 
    which behave in different ways: an Expession or a Constant object overloads 
    the eval() method of the dolfin::Expression class and it is evaluated at run 
    time using the octave function feval (). A Function instead doesn't need to 
    be evaluated because it is assembled copying element-by-element the values 
    contained in the input vector. 
    
    
     We have split the construction of the form into two steps:

        We set all the coefficients of the form using the function which we create on the fly. 
        They will be named \verb$ProblemName_BilinearForm$ or \verb$ProblemName_LinearForm$.
        Then we apply specific BC to the form using the assemble() function and we get back the matrix. 
        If we are assembling the whole system and we want to keep the symmetry of the matrix (if any), 
        we can instead use the command \verb$assemble_system$ (). Finally, if we are solving a non-linear problem 
        and we need to apply essential BC, we should provide to the function also the vector with the 
        tentative solution in order to modify the entries corresponding to the boundary values. 
        This will be illustrated below in the HyperElasticity example.
\fi

\subsection{Mesh generation and conversion}

\subsection{Sparse Matrices}

\subsection{Shared pointer}

\subsection{Polymorphism}

\subsection{Code release}

\subsection{Code on the fly}

\iffalse
For the creation on the fly of the code from the header file, 
Juan Pablo provided me a python code which makes a great job. I have spent some 
time adapting it to my problem, but when I finally got a working code, we realized 
that it was probably enough to use the functions available inside Octave because 
the problem was rather simple. The pyton code is however available here, while the 
final solution adopted can be found there.
For the problem related with \verb$fem_init_env ()$, we are still working on it. 
\fi

\subsection{Autoconf}
 In this section we want to discuss how we can write a config.ac and a Makefile.in files which:
\begin{itemize}
    \item check if a program is available and stop if it is not
    \item check if a header file is available and issue a warning if not, but go ahead with the compilation
\end{itemize}
We want to speak about it because, even if it is not strictly related to the fem-fenics library, 
I hope it could be helpful for someone else because some solutions which could seem right at a 
first sight are definitely wrong.

As stated above, if we want to generate automatically a Makefile, we need two components:

\paragraph{configure.ac} Is a file which checks whether the program/header is available or not 
and sets consequently the values of some variables.
\begin{lstlisting}
    # Checks if the program mkoctfile is available and sets the variable HAVE_MKOCTFILE consequently
    AC_CHECK_PROG([HAVE_MKOCTFILE], [mkoctfile], [yes], [no])
    # if mkoctfile is not available, it issues an error and stops the compilation
    if [test $HAVE_MKOCTFILE = "no"]; then 
      AC_MSG_ERROR([mkoctfile required to install $PACKAGE_NAME])
    fi

    #Checks if the header dolfin.h is available; if it is available, the value of the ac_dolfin_cpp_flags is substituted with -DHAVE_DOLFIN_H, otherwise it is left empty and a warning message is printed
    AC_CHECK_HEADER([dolfin.h],
      [AC_SUBST(ac_dolfin_cpp_flags,-DHAVE_DOLFIN_H)  AC_SUBST(ac_dolfin_ld_flags,-ldolfin)],
      [AC_MSG_WARN([dolfin headers could not be found, some functionalities will be disabled, don't worry your package will still be working, though.])] ).

    # It generates the Makefile, using the template described below
    AC_CONFIG_FILES([Makefile])
\end{lstlisting} 
\paragraph{Makefile.ac} This file is a template for the Makefile, which will be automatically generated when the configure.ac 
file is executed. The values of the variable \verb$ac_dolfin_cpp_flags$ and \verb$ac_dolfin_ld_flags$ are substituted with the 
results obtained above:
\begin{lstlisting}
    CPPFLAGS += @ac_dolfin_cpp_flags@
    LDFLAGS += @ac_dolfin_ld_flags@
\end{lstlisting} 
In this way, if dolfin.h is available, CPPFLAGS contains also the flag  \verb$-DHAVE_DOLFIN_H.$

\paragraph {program.cc}  Our .cc program, should thus include the header dolfin.h only if 
\verb$-DHAVE_DOLFIN_H$ is defined at compilation time.
For example

\begin{lstlisting}
    #ifdef HAVE_DOLFIN_H
    #include <dolfin.h> 
    #endif
    int main ()
    {  

    #ifndef HAVE_DOLFIN_H
        error("program: the program was built without support for dolfin");
    #else 
      /* Body of your function */
    #endif
     return 0;
    }

\end{lstlisting} 
\paragraph {Warning} If in the Makefile.in you write something like
\begin{lstlisting} 
    HAVE_DOLFIN_H = @HAVE_DOLFIN_H@  
    ifdef HAVE_DOLFIN_H   
      CPPFLAGS += -DHAVE_DOLFIN_H  
      LIBS += -ldolfin
    endif
 \end{lstlisting} 
 it doesn't work because the variable \verb$HAVE_DOLFIN_H$ seems to be always defined, even if the header is not available.

\chapter{More Advanced Examples}\label{exem}
\iffalse
 With the following examples, we can see directly in action the new features and understand how they work.

    Navier-Stokes: we learn how to deal with a vector-field problem and how we can save the solution using the 
    \verb$fem_save$ () function. We also use the fem pkg to generate a mesh using gmesh.
    Mixed-Poisson: we solve the Poisson problem presented in the previous posts using a mixed formulation, 
    and we see how we can extract a scalar field from a vector one.
    HyperElasticity: we exploit the fsolve () command to solve a non-linear problem. In particular, 
    we see how to use the assemble() function to apply BC also in this situation.
    Advection-Diffusion: we solve a time dependent problem using the lsode () command and save 
    the solution using the pkg flp.

For each problem, we refer the reader to the complete desciption on FEniCS or bim web-page, 
while here we highlight only the  implementation detail relevant for our pkg.
\fi
\section{Navier-Stokes equation with Chorin-Temam projection algorithm}

\paragraph{TentativeVelocity.ufl}
\begin{lstlisting}
# Copyright (C) 2010 Anders Logg
# Define function spaces (P2-P1)
V = VectorElement("CG", triangle, 2)
Q = FiniteElement("CG", triangle, 1)

# Define trial and test functions
u = TrialFunction(V)
v = TestFunction(V)

# Define coefficients
k  = Constant(triangle)
u0 = Coefficient(V)
f  = Coefficient(V)
nu = 0.01

# Define bilinear and linear forms
eq = (1/k)*inner(u - u0, v)*dx + inner(grad(u0)*u0, v)*dx + \
    nu*inner(grad(u), grad(v))*dx - inner(f, v)*dx
a  = lhs(eq)
L  = rhs(eq) 
 
\end{lstlisting}

\paragraph{PressureUpdate.ufl}
\begin{lstlisting}
 # Copyright (C) 2010 Anders Logg
 # Define function spaces (P2-P1)
V = VectorElement("CG", triangle, 2)
Q = FiniteElement("CG", triangle, 1)

# Define trial and test functions
p = TrialFunction(Q)
q = TestFunction(Q)

# Define coefficients
k  = Constant(triangle)
u1 = Coefficient(V)

# Define bilinear and linear forms
a = inner(grad(p), grad(q))*dx
L = -(1/k)*div(u1)*q*dx 
 
\end{lstlisting}

\paragraph{VelocityUpdate.ufl}
\begin{lstlisting}
 # Copyright (C) 2010 Anders Logg
# Define function spaces (P2-P1)
V = VectorElement("CG", triangle, 2)
Q = FiniteElement("CG", triangle, 1)

# Define trial and test functions
u = TrialFunction(V)
v = TestFunction(V)

# Define coefficients
k  = Constant(triangle)
u1 = Coefficient(V)
p1 = Coefficient(Q)

# Define bilinear and linear forms
a = inner(u, v)*dx
L = inner(u1, v)*dx - k*inner(grad(p1), v)*dx
\end{lstlisting}

\paragraph{NS.m}
\begin{lstlisting}
pkg load fem-fenics msh
import_ufl_Problem ("TentativeVelocity");
import_ufl_Problem ("VelocityUpdate");
import_ufl_Problem ("PressureUpdate");
 
# We can either load the mesh from the file as in Dolfin but 
# we can also use the msh pkg to generate the L-shape domain
L-shape-domain;
mesh = Mesh (msho);
 
# Define function spaces (P2-P1).
V = FunctionSpace ('VelocityUpdate', mesh);
Q = FunctionSpace ('PressureUpdate', mesh);
 
# Set parameter values and define coefficients
dt = 0.01;
T = 3.;
k = Constant ('k', dt);
f = Constant ('f', [0; 0]);
u0 = Expression ('u0', @(x,y) [0; 0]);

# Define boundary conditions
noslip = DirichletBC (V, @(x,y) [0; 0], [3, 4]);
outflow = DirichletBC (Q, @(x,y) 0, 2);

# Assemble matrices
a1 = BilinearForm ('TentativeVelocity', V, V, k);
a2 = BilinearForm ('PressureUpdate', Q, Q);
a3 = BilinearForm ('VelocityUpdate', V, V);
A1 = assemble (a1, noslip);
A3 = assemble (a3, noslip);

# Time-stepping
t = dt; i = 0;
while t < T
 
  # Update pressure boundary condition
  inflow = DirichletBC (Q, @(x,y) sin(3.0*t), 1);
 
  # Compute tentative velocity step
  L1 = LinearForm ('TentativeVelocity', V, k, u0, f);
  b1 = assemble (L1, noslip);
  utmp = A1 \ b1;
  u1 = Function ('u1', V, utmp);
 
  # Pressure correction
  L2 = LinearForm ('PressureUpdate', Q, u1, k);
  [A2, b2] = assemble_system (a2, L2, inflow, outflow);
  ptmp = A2 \ b2;
  p1 = Function ('p1', Q, ptmp);
 
  # Velocity correction
  L3 = LinearForm ('VelocityUpdate', V, k, u1, p1);
  b3 = assemble (L3, noslip);
  ut = A3 \ b3;
  u1 = Function ('u0', V, ut);
  
  # Save to file
  save (p1, sprintf ("p_%3.3d", ++i));
  save (u1, sprintf ("u_%3.3d", i));
 
  # Move to next time step
  u0 = u1;
  t += dt
 
end
\end{lstlisting}

\paragraph{L-shape-domain.m}
\begin{lstlisting}
name = [tmpnam ".geo"];
fid = fopen (name, "w");
fputs (fid,"Point (1)  = {0, 0, 0, 0.1};\n");
fputs (fid,"Point (2)  = {1, 0, 0, 0.1};\n");
fputs (fid,"Point (3)  = {1, 0.5, 0, 0.1};\n");
fputs (fid,"Point (4)  = {0.5, 0.5, 0, 0.1};\n");
fputs (fid,"Point (5) = {0.5, 1, 0, 0.1};\n");
fputs (fid,"Point (6) = {0, 1, 0,0.1};\n");
 
fputs (fid,"Line (1)  = {5, 6};\n");
fputs (fid,"Line (2) = {2, 3};\n");
 
fputs (fid,"Line(3) = {6,1,2};\n");
fputs (fid,"Line(4) = {5,4,3};\n");
fputs (fid,"Line Loop(7) = {3,2,-4,1};\n");
fputs (fid,"Plane Surface(8) = {7};\n");
fclose (fid);
msho = msh2m_gmsh (canonicalize_file_name (name)(1:end-4),...
                   "scale", 1,"clscale", .2);
unlink (canonicalize_file_name (name));
\end{lstlisting}

\section{A penalization method to take into account obstacles in incompressible viscous flows}


\newpage 

\bibliographystyle{unsrt} 
\bibliography{doc}
\end{document}