$$ \newcommand{\dt}{\Delta t} \newcommand{\tp}{\thinspace .} \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\x}{\boldsymbol{x}} \newcommand{\dx}{\, \mathrm{d}x} \newcommand{\ds}{\, \mathrm{d}s} \newcommand{\Real}{\mathbb{R}} \newcommand{\uI}{u_{_0}} \newcommand{\ub}{u_{_\mathrm{D}}} \newcommand{\GD}{\Gamma_{_\mathrm{D}}} \newcommand{\GN}{\Gamma_{_\mathrm{N}}} \newcommand{\GR}{\Gamma_{_\mathrm{R}}} \newcommand{\inner}[2]{\langle #1, #2 \rangle} $$

Contents
- Table of contents
- Preface
- Preliminaries
- The FEniCS Project
- What you will learn
- Working with this tutorial
- Obtaining the software
- Installation using Docker containers
- Installation using Ubuntu packages
- Testing your installation
- Obtaining the tutorial examples
- Background knowledge
- Programming in Python
- The finite element method
- Fundamentals: Solving the Poisson equation
- Mathematical problem formulation
- Finite element variational formulation
- Abstract finite element variational formulation
- Choosing a test problem
- FEniCS implementation
- The complete program
- Running the program
- Terminal window
- Spyder
- Jupyter notebooks
- Dissection of the program
- The important first line
- Generating simple meshes
- Defining the finite element function space
- Defining the trial and test functions
- Defining the boundary conditions
- Defining the source term
- Defining the variational problem
- Forming and solving the linear system
- Plotting the solution using the plot command
- Plotting the solution using ParaView
- Computing the error
- Examining degrees of freedom and vertex values
- Deflection of a membrane
- Scaling the equation
- Defining the mesh
- Defining the load
- Defining the variational problem
- Plotting the solution
- Making curve plots through the domain
- A Gallery of finite element solvers
- The heat equation
- PDE problem
- Variational formulation
- FEniCS implementation
- Test problem 1: A known analytical solution
- FEniCS implementation
- Test problem 2: Diffusion of a Gaussian function
- FEniCS implementation
- Visualization in ParaView
- A nonlinear Poisson equation
- PDE problem
- Variational formulation
- FEniCS implementation
- Test problem
- FEniCS implementation
- The equations of linear elasticity
- PDE problem
- Variational formulation
- FEniCS implementation
- Test problem
- FEniCS implementation
- Vector function spaces
- Constant vectors
- `nabla_grad`
- Stress computation
- Scaling
- The Navier--Stokes equations
- PDE problem
- Variational formulation
- FEniCS implementation
- Test problem 1: Channel flow
- FEniCS implementation
- Verification
- Test problem 2: Flow past a cylinder
- FEniCS implementation
- A system of advection--diffusion--reaction equations
- PDE problem
- Variational formulation
- FEniCS implementation
- Setting initial conditions for mixed systems
- Setting boundary conditions for mixed systems
- Accessing components of mixed systems
- Subdomains and boundary conditions
- Combining Dirichlet and Neumann conditions
- PDE problem
- Variational formulation
- FEniCS implementation
- Setting multiple Dirichlet conditions
- Defining subdomains for different materials
- Using expressions to define subdomains
- Using mesh functions to define subdomains
- Using C++ code snippets to define subdomains
- Setting multiple Dirichlet, Neumann, and Robin conditions
- Three types of boundary conditions
- PDE problem
- Variational formulation
- FEniCS implementation
- Test problem
- Debugging boundary conditions
- Generating meshes with subdomains
- PDE problem
- Variational formulation
- FEniCS implementation
- Extensions: Improving the Poisson solver
- Refactoring the Poisson solver
- A more general solver function
- Writing the solver as a Python module
- Verification and unit tests
- Parameterizing the number of space dimensions
- Working with linear solvers
- Choosing a linear solver and preconditioner
- Choosing a linear algebra backend
- Setting solver parameters
- An extended solver function
- A remark regarding unit tests
- List of linear solver methods and preconditioners
- High-level and low-level solver interfaces
- Linear variational problem and solver objects
- Explicit assembly and solve
- Examining matrix and vector values
- Degrees of freedom and function evaluation
- Examining the degrees of freedom
- Setting the degrees of freedom
- Function evaluation
- Postprocessing computations
- Test problem
- Flux computations
- Computing functionals
- Energy functional
- Error functional
- Flux Functional
- Computing convergence rates
- Computing error norms
- Computing convergence rates
- Test problem
- Experiments
- Taking advantage of structured mesh data
- Iterating over points and values
- Computing finite difference approximations
- Making surface plots
- Making contour plots
- Making curve plots through the domain
- Making curve plots of the flux
- Test problem
- Taking the next step
- Bibliography

Solving PDEs in Python -
The FEniCS Tutorial Volume I

Hans Petter Langtangen [1, 2] Anders Logg [3, 1, 4] (logg at chalmers.se)

[1] Center for Biomedical Computing, Simula Research Laboratory [2] Department of Informatics, University of Oslo [3] Department of Mathematical Sciences, Chalmers University of Technology [4] Computational Engineering and Design, Fraunhofer-Chalmers Centre

This book gives a concise and gentle introduction to finite element programming in Python based on the popular FEniCS software library. Through a series of examples, including among others the Poisson equation, the equations of linear elasticity, the incompressible Navier–Stokes equations, and systems of nonlinear advection–diffusion–reaction equations, the reader is guided through the essential steps of how to quickly solve a PDE in FEniCS, including how to define a finite variational problem, how to set boundary conditions, how to solve linear and nonlinear systems, and how to visualize solutions and structure finite element Python programs.

This document is also available in PDF and Sphinx formats.

Sep 18, 2017

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© 2017, Hans Petter Langtangen, Anders Logg. Released under CC Attribution 4.0 license