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

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 and the boundary conditions
- Defining the source term
- Defining the variational problem
- Forming and solving the linear system
- Plotting the solution
- Exporting and post-processing the solution
- 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
- Visualizing the solution in ParaView
- Using the built-in visualization tool
- Exercise 1: Visualize a solution in a cube
- A Gallery of finite element solvers
- The heat equation
- PDE problem
- Variational formulation
- FEniCS implementation
- Test problem
- FEniCS implementation
- Diffusion of a Gaussian function
- The mathematical problem
- 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
- The code
- 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
- Exercise 2: Simulate channel flow in a 3D geometry
- 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
- Vectorized version of subdomain definitions
- Using C++ code snippets to define subdomains
- Exercise 3: Efficiency of Python vs C++ expressions
- 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
- Exercise 4: Solve a Poisson problem
- Remarks
- Exercise 5: Refactor the code for membrane deflection
- Working with linear solvers
- Controlling the solution process
- 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
- Linear variational problem and solver objects
- Explicit assembly and solve
- Examining matrix and vector values
- Examining the degrees of freedom
- Postprocessing computations
- A variable-coefficient Poisson 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
- Bibliography

Solving PDEs in Minutes -
The FEniCS Tutorial Volume I

Hans Petter Langtangen [1, 2] (hpl at simula.no) Anders Logg [3, 1] (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

This book gives a concise and gentle introduction to finite element programming in Python based on the popular FEniCS software library. The library delivers high performance since FEniCS automatically delegates compute-intensive tasks to C++ by help of code generation. We show in detail how to write finite element solvers for the Poisson equation, the time-dependent diffusion equation, the equations of elasticity, and the Navier–Stokes equations, in heterogeneous media and with different types of boundary conditions.

This document is also available in PDF and Sphinx web format.

Sep 18, 2016

Read »

Solving PDEs in Minutes - The FEniCS Tutorial Volume I

Sep 18, 2016

Solving PDEs in Minutes -
The FEniCS Tutorial Volume I