Solving PDEs in Minutes - <br> The FEniCS Tutorial Volume I

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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