The FEniCS Project is a research and software project aimed at
creating mathematical methods and software for automated computational
mathematical modeling. This means creating easy, intuitive, efficient,
and flexible software for solving partial differential equations
(PDEs) using finite element methods. FEniCS was initially created in
2003 and is developed in collaboration between researchers from a
number of universities and research institutes around the world. For
more information about FEniCS and the latest updates of the FEniCS
software and this tutorial, visit the FEniCS web page at
`https://fenicsproject.org`

.

FEniCS consists of a number of building blocks (software components) that together form the FEniCS software: DOLFIN [2], FFC [3], FIAT [4], UFL [5], mshr, and a few others. For an overview, see [1]. FEniCS users rarely need to think about this internal organization of FEniCS, but since even casual users may sometimes encounter the names of various FEniCS components, we briefly list the components and their main roles in FEniCS. DOLFIN is the computational high-performance C++ backend of FEniCS. DOLFIN implements data structures such as meshes, function spaces and functions, compute-intensive algorithms such as finite element assembly and mesh refinement, and interfaces to linear algebra solvers and data structures such as PETSc. DOLFIN also implements the FEniCS problem-solving environment in both C++ and Python. FFC is the code generation engine of FEniCS (the form compiler), responsible for generating efficient C++ code from high-level mathematical abstractions. FIAT is the finite element backend of FEniCS, responsible for generating finite element basis functions, UFL implements the abstract mathematical language by which users may express variational problems, and mshr provides FEniCS with mesh generation capabilities.

The goal of this tutorial is to demonstrate how to apply the finite element to solve PDEs in FEniCS. Through a series of examples, we demonstrate how to:

- solve linear PDEs (such as the Poisson equation),
- solve time-dependent PDEs (such as the heat equation),
- solve nonlinear PDEs,
- solve systems of time-dependent nonlinear PDEs.

We will also discuss how to best structure the Python code for a PDE solver, how to debug programs, and how to take advantage of testing frameworks.

The mathematics of the illustrations is kept simple to better focus on FEniCS functionality and syntax. This means that we mostly use the Poisson equation and the time-dependent diffusion equation as model problems, often with input data adjusted such that we get a very simple solution that can be exactly reproduced by any standard finite element method over a uniform, structured mesh. This latter property greatly simplifies the verification of the implementations. Occasionally we insert a physically more relevant example to remind the reader that the step from solving a simple model problem to a challenging real-world problem is often quite short and easy with FEniCS.

Using FEniCS to solve PDEs may seem to require a thorough
understanding of the abstract mathematical framework of the finite
element method as well as expertise in Python programming.
Nevertheless, it turns out that many users are able to pick up the
fundamentals of finite elements *and* Python programming as they go
along with this tutorial. Simply keep on reading and try out the
examples. You will be amazed at how easy it is to solve PDEs with
FEniCS!

Working with this tutorial obviously requires access to the FEniCS software. FEniCS is a complex software library, both in itself and due to its many dependencies to state-of-the-art open-source scientific software libraries. Manually building FEniCS and all its dependencies from source can thus be a daunting task. Even for an expert who knows exactly how to configure and build each component, a full build can literally take hours! In addition to the complexity of the software itself, there is an additional layer of complexity in how many different kinds of operating systems (Linux, Mac, Windows) may be running on a user's laptop or compute server, with different requirements for how to configure and build software.

For this reason, the FEniCS Project provides prebuilt packages to make the installation easy, fast, and foolproof.

FEniCS may also be installed using other methods, including Conda
packages and building from source. For more installation options and
the latest information on the simplest and best options for installing
FEniCS, check out the official FEniCS installation instructions. These
can be found at
`https://fenicsproject.org/download`.

A modern solution to the challenge of software installation on diverse
software platforms is to use so-called *containers*. The FEniCS
Project provides custom-made containers that are controlled,
consistent, and high-performance software environments for FEniCS
programming. FEniCS containers work equally well
on all operating systems, including Linux, Mac, and Windows.

**1:** Running Docker containers on Mac and Windows
involves a small performance overhead compared to running Docker
containers on Linux. However, this performance penalty is typically
small and is often compensated for by using the highly tuned and
optimized version of FEniCS that comes with the official FEniCS
containers, compared to building FEniCS and its dependencies from
source on Mac or Windows.

To use FEniCS containers, you must first install the Docker platform. Docker installation is simple and instructions are available on the Docker web page. Once you have installed Docker, just copy the following line into a terminal window:

```
Terminal> curl -s https://get.fenicsproject.org | bash
```

The command above will install the program `fenicsproject`

on your
system. This program lets you easily create FEniCS sessions
(containers) on your system:

```
Terminal> fenicsproject run
```

This command has several useful options, such as easily switching
between the latest release of FEniCS, the latest development version
and many more. To learn more, type `fenicsproject help`

. FEniCS can
also be used directly with Docker, but this typically requires
typing a relatively complex Docker command, for example:

```
docker run --rm -ti -v `pwd`:/home/fenics/shared -w
/home/fenics/shared quay.io/fenicsproject/stable:current '/bin/bash -l
-c "export TERM=xterm; bash -i"'
```

`fenicsproject run`

, it will
automatically share your current working directory (the directory
from which you run the `fenicsproject`

command) with the FEniCS
session. When the FEniCS session starts, it will automatically
enter into a directory named `shared`

which will be identical with
your current working directory on your host system. This means that
you can easily edit files and write data inside the FEniCS session, and
the files will be directly accessible on your host system. It is
recommended that you edit your programs using your favorite editor
(such as Emacs or Vim) on your host system and use the FEniCS session
only to run your program(s).
For users of Ubuntu GNU/Linux, FEniCS can also be installed easily via
the standard Ubuntu package manager `apt-get`

. Just copy the following
lines into a terminal window:

```
Terminal> sudo add-apt-repository ppa:fenics-packages/fenics
Terminal> sudo apt-get update
Terminal> sudo apt-get install fenics
Terminal> sudo apt-get dist-upgrade
```

This will add the FEniCS package archive (PPA) to your Ubuntu computer's list of software sources and then install FEniCS. It will will also automatically install packages for dependencies of FEniCS.

Once you have installed FEniCS, you should make a quick test to see that your installation works properly. To do this, type the following command in a FEniCS-enabled terminal:

**2:** For users of FEniCS containers, this means first
running the command `fenicsproject run`

.

```
Terminal> python -c 'import fenics'
```

If all goes well, you should be able to run this command without any error message (or any other output).

In this tutorial, you will learn finite element and FEniCS programming
through a number of example programs that demonstrate both how to
solve particular PDEs using the finite element method, how to program
solvers in FEniCS, and how to create well-designed Python code that
can later be extended to solve more complex problems. All
example programs are available from the web page of this book at
`https://fenicsproject.org/tutorial`

. The programs as well as the
source code for this text can also be accessed directly from the Git
repository for this
book.

While you can likely pick up basic Python programming by working
through the examples in this tutorial, you may want to study
additional material on the Python language. A natural starting point
for beginners is the classic *Python Tutorial* [6],
or a tutorial geared towards scientific computing
[7]. In the latter, you will also find
pointers to other tutorials for scientific computing in Python. Among
ordinary books we recommend the general introduction *Dive into
Python* [8] as well as texts that focus on scientific
computing with Python
[9] [10] [11] [12] [13].

```
from __future__ import print_function
```

to enable the `print`

function from Python 3 in Python 2. All
use of `print`

in the programs in this tutorial consists of function
calls, like `print('a:', a)`

. Almost all other constructions are of
a form that looks the same in Python 2 and 3.

Many good books have been written on the finite element method. The books typically fall in either of two categories: the abstract mathematical version of the method or the engineering "structural analysis" formulation. FEniCS builds heavily on concepts from the abstract mathematical exposition. The first author has a book [14] in development that explains all details of the finite element method in an intuitive way, using the abstract mathematical formulations that FEniCS employs.

The finite element text by Larson and Bengzon [15] is our recommended introduction to the finite element method, with a mathematical notation that goes well with FEniCS. An easy-to-read book, which also provides a good general background for using FEniCS, is Gockenbach [16]. The book by Donea and Huerta [17] has a similar style, but aims at readers with an interest in fluid flow problems. Hughes [18] is also recommended, especially for readers interested in solid mechanics and heat transfer applications.

Readers with a background in the engineering "structural analysis"
version of the finite element method may find Bickford
[19] an attractive bridge over to the abstract
mathematical formulation that FEniCS builds upon. Those who have a
weak background in differential equations in general should consult a
more fundamental book, and Eriksson *et al*
[20] is a very good choice. On the
other hand, FEniCS users with a strong background in mathematics
will appreciate the texts by Brenner and Scott [21],
Braess [22], Ern and Guermond [23],
Quarteroni and Valli [24], or Ciarlet
[25].