The FEniCS project is a research and software project aimed at creating mathematical methods and software for solving partial differential equations (PDEs). This includes creating intuitive, efficient and flexible software. The project was initiated in 2003, and is developed in a collaboration among researchers from a number of universities and research institutes around the world. For the latest updates and more information about the FEniCS project, visit the FEniCS webpage.
The latest version of the FEniCS project, FEniCSx, consists of several building blocks, namely DOLFINx, UFL, FFCx, and Basix. We will now go through the main objectives of each of these building blocks. DOLFINx is the high performance C++ backend of FEniCSx, where structures such as meshes, function spaces and functions are implemented. Additionally, DOLFINx also contains compute intensive functions such as finite element assembly and mesh refinement algorithms. It also provides an interface to linear algebra solvers and data-structures, such as PETSc. UFL is a high-level form language for describing variational formulations with a high-level mathematical syntax. FFCx is the form compiler of FEniCSx; given variational formulations written with UFL, it generates efficient C code. Basix is the finite element backend of FEniCSx, responsible for generating finite element basis functions.
The goal of this tutorial is to demonstrate how to apply the finite element to solve PDEs using FEniCS. Through a series of examples, we will demonstrate how to:
- Solve linear PDEs (such as the Poisson equation),
- Solve time-dependent PDEs (such as the heat equation),
- Solve non-linear PDEs,
- Solve systems of time-dependent non-linear PDEs.
Important topics include: how to set boundary conditions of various types (Dirichlet, Neumann, Robin), how to create meshes, how to define variable coefficients, how to interact with linear and non-linear solvers, and how to post-process and visualize solutions.
Most of the mathematical part of the examples will be kept at a simple level, such that we can keep the focus on the functionality and syntax of FEniCSx. Therefore, we will mostly use the Poisson equation and the time-dependent diffusion equation as model problems. We will use adjusted input data, such that the solution of the problem can be exactly reproduced on uniform, structured meshes with the finite element method. This greatly simplifies the verification of the implementations. Occasionally we will consider a more physically relevant example to remind the reader that there are no big leaps from solving simple model problems to challenging real-world problems when using FEniCSx.
As this book has been published as a Jupyter Book, we provide interactive notebooks that can be run in the browser. To start such a notebook click the 
-symbol in the top right corner of the relevant tutorial.
If you would like to work with DOLFINx outside of the binder-notebooks, you need to install the FEniCS software. The recommended way of installing DOLFINx for new users is by using Docker. Docker is a software that uses containers to supply software across different kinds of operating systems (Linux, Mac, Windows). The first step is to install docker, following the instructions at their webpage.
All notebooks can be converted to python files using nbconvert.
The tutorial uses several dependencies for meshing, plotting and timings. A compatible JupyterLab image is available in the Github Packages.
To use the notebooks in this tutorial with DOLFINx on your own computer, you should use the docker image obtained using the following command:
docker run --init -p 8888:8888 -v "$(pwd)":/root/shared ghcr.io/jorgensd/dolfinx-tutorial:v0.7.2This image can also be used as a normal docker container by adding:
docker run --ti -v "$(pwd)":/root/shared --entrypoint="/bin/bash" ghcr.io/jorgensd/dolfinx-tutorial:v0.7.2The tutorials can also be exported as an IPython notebook or PDF by clicking the 
-symbol in the top right corner of the relevant tutorial. The notebook can in turn be used with a Python kernel which has DOLFINx.
The FEniCS project supplies pre-built docker images at https://hub.docker.com/r/dolfinx/dolfinx. The Dockerfile provides a definitive build recipe. As the DOLFINx docker images are hosted at Docker-hub, one can directly access the image using:
docker run dolfinx/dolfinx:v0.8.0There are several ways of customizing a docker container, such as mounting volumes/sharing folder, setting a working directory, sharing graphical interfaces etc. See docker run --help for an extensive list.
Once you have installed DOLFINx, either by using docker or installing from source, you can test the installation by running python3 -c "import dolfinx".
If all goes well, no error-messages should appear.
The software is quite complex, and building the software and all the dependencies from source can be a daunting task. The list of dependencies can be found at docs.fenicsproject.org/dolfinx/main/python/installation.html.
If you are a beginner in Python, we suggest reading {cite}Langtangen2016 by Hans Petter Langtangen, which will give you a gentle introduction to the Python programming language. Note that DOLFINx, being a state of the art finite element solver, only supports Python 3, as Python 2 reached its end of life January 1st, 2020. To automatically transfer Python 2 scripts to Python 3, it is suggested to use the 2to3-package, which provides automated translation of the code.
In the last decade, several sets of lecture notes on finite element methods have been made open access. See for instance:
- Numerical methods for partial differential equations, by Hans Petter Langtangen
- Finite elements - analysis and implementation, by David A. Ham and Colin J. Cotter
- Finite element analysis for coupled problems, by David Kamensky.
- DefElement: an encyclopedia of finite element definitions, by Matthew W. Scroggs.
Many good textbooks on the finite element method have been written, and we refer to the original FEniCS tutorial for references to these, see Chapter 1.6.2 of The FEniCS tutorial {cite}FenicsTutorial.
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