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Solving PDEs in Python : The FEniCS Tutorial I.

Bibliographic Details
Main Author: Langtangen, Hans Petter.
Other Authors: Logg, Anders.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2017.
Edition:1st ed.
Series:Simula SpringerBriefs on Computing Series
Subjects:
Electronic books.
Online Access:Click to View
Table of Contents:
  • Intro
  • Foreword
  • Contents
  • Preface
  • 1 Preliminaries
  • 1.1 The FEniCS Project
  • 1.2 What you will learn
  • 1.3 Working with this tutorial
  • 1.4 Obtaining the software
  • 1.4.1 Installation using Docker containers
  • 1.4.2 Installation using Ubuntu packages
  • 1.4.3 Testing your installation
  • 1.5 Obtaining the tutorial examples
  • 1.6 Background knowledge
  • 1.6.1 Programming in Python
  • 1.6.2 The finite element method
  • 2 Fundamentals: Solving the Poisson equation
  • 2.1 Mathematical problem formulation
  • 2.1.1 Finite element variational formulation
  • 2.1.2 Abstract finite element variational formulation
  • 2.1.3 Choosing a test problem
  • 2.2 FEniCS implementation
  • 2.2.1 The complete program
  • 2.2.2 Running the program
  • 2.3 Dissection of the program
  • 2.3.1 The important first line
  • 2.3.2 Generating simple meshes
  • 2.3.3 Defining the finite element function space
  • 2.3.4 Defining the trial and test functions
  • 2.3.5 Defining the boundary conditions
  • 2.3.6 Defining the source term
  • 2.3.7 Defining the variational problem
  • 2.3.8 Forming and solving the linear system
  • 2.3.9 Plotting the solution using the plot command
  • 2.3.10 Plotting the solution using ParaView
  • 2.3.11 Computing the error
  • 2.3.12 Examining degrees of freedom and vertex values
  • 2.4 Deflection of a membrane
  • 2.4.1 Scaling the equation
  • 2.4.2 Defining the mesh
  • 2.4.3 Defining the load
  • 2.4.4 Defining the variational problem
  • 2.4.5 Plotting the solution
  • 2.4.6 Making curve plots through the domain
  • 3 A Gallery of finite element solvers
  • 3.1 The heat equation
  • 3.1.1 PDE problem
  • 3.1.2 Variational formulation
  • 3.1.3 FEniCS implementation
  • 3.2 A nonlinear Poisson equation
  • 3.2.1 PDE problem
  • 3.2.2 Variational formulation
  • 3.2.3 FEniCS implementation
  • 3.3 The equations of linear elasticity
  • 3.3.1 PDE problem.
  • 3.3.2 Variational formulation
  • 3.3.3 FEniCS implementation
  • 3.4 The Navier-Stokes equations
  • 3.4.1 PDE problem
  • 3.4.2 Variational formulation
  • 3.4.3 FEniCS implementation
  • 3.5 A system of advection-diffusion-reaction equations
  • 3.5.1 PDE problem
  • 3.5.2 Variational formulation
  • 3.5.3 FEniCS implementation
  • 4 Subdomains and boundary conditions
  • 4.1 Combining Dirichlet and Neumann conditions
  • 4.1.1 PDE problem
  • 4.1.2 Variational formulation
  • 4.1.3 FEniCS implementation
  • 4.2 Setting multiple Dirichlet conditions
  • 4.3 Defining subdomains for different materials
  • 4.3.1 Using expressions to define subdomains
  • 4.3.2 Using mesh functions to define subdomains
  • 4.3.3 Using C++ code snippets to define subdomains
  • 4.4 Setting multiple Dirichlet, Neumann, and Robin conditions
  • 4.4.1 Three types of boundary conditions
  • 4.4.2 PDE problem
  • 4.4.3 Variational formulation
  • 4.4.4 FEniCS implementation
  • 4.4.5 Test problem
  • 4.4.6 Debugging boundary conditions
  • 4.5 Generating meshes with subdomains
  • 4.5.1 PDE problem
  • 4.5.2 Variational formulation
  • 4.5.3 FEniCS implementation
  • 5 Extensions: Improving the Poisson solver
  • 5.1 Refactoring the Poisson solver
  • 5.1.1 A more general solver function
  • 5.1.2 Writing the solver as a Python module
  • 5.1.3 Verification and unit tests
  • 5.1.4 Parameterizing the number of space dimensions
  • 5.2 Working with linear solvers
  • 5.2.1 Choosing a linear solver and preconditioner
  • 5.2.2 Choosing a linear algebra backend
  • 5.2.3 Setting solver parameters
  • 5.2.4 An extended solver function
  • 5.2.5 A remark regarding unit tests
  • 5.2.6 List of linear solver methods and preconditioners
  • 5.3 High-level and low-level solver interfaces
  • 5.3.1 Linear variational problem and solver objects
  • 5.3.2 Explicit assembly and solve
  • 5.3.3 Examining matrix and vector values.
  • 5.4 Degrees of freedom and function evaluation
  • 5.4.1 Examining the degrees of freedom
  • 5.4.2 Setting the degrees of freedom
  • 5.4.3 Function evaluation
  • 5.5 Postprocessing computations
  • 5.5.1 Test problem
  • 5.5.2 Flux computations
  • 5.5.3 Computing functionals
  • 5.5.4 Computing convergence rates
  • 5.5.5 Taking advantage of structured mesh data
  • 5.6 Taking the next step
  • References
  • Index.

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