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

 

 

 

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
      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
            Diffusion of a Gaussian function
      A nonlinear Poisson equation
            PDE problem
            Variational formulation
            FEniCS implementation
      The equations of linear elasticity
            PDE problem
            Variational formulation
            FEniCS implementation
      The Navier–Stokes equations
            PDE problem
            Variational formulation
            FEniCS implementation
            Exercise 2: Simulate channel flow in a 3D geometry
            Flow past a cylinder
      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
            Exercise 5: Refactor the code for membrane deflection
      Working with linear solvers
            Controlling the solution process
            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
            Computing convergence rates
            Taking advantage of structured mesh data
      Bibliography