This is the webpage for the University of Luxembourg Master in Mathematics course Numerical Methods for Variational Problems.

This webpage is adapted with permission from the original authored by Prof. Colin Cotter and Prof. David Ham at Imperial College London. The section on Modern Finite Element practice was authored by Dr. Jack S. Hale at the University of Luxembourg.

The authors welcome feedback and would particularly appreciate an email if this material is used to teach anywhere.

• Practical matters
  • Instructor
  • Teaching Units
  • Attendance policy
  • Accreditation
  • Assumed knowledge
  • Assessment
    • Submitting code for feedback
  • Moodle

Weekly material

The numerical analysis and implementation parts of the module run in parallel. Each part has videos and exercises embedded in its notes.

A PDF version of the course notes is available here.

Part 1: Numerical analysis

The material is presented by means of a set of lecture notes, with lecture videos interspersed in the notes.

The text for this part of the module is Brenner and Scott The Mathematical Theory of Finite Element Methods. The University of Luxembourg has paid for PDF access to this book, so it is accessible from the University of Luxembourg network at Springer Link.

However, this course has the material rearranged to synchronise the content better between the lecture notes and the implementation exercise.

Lecture notes:

• 1. Introduction
  • 1.1. Poisson’s equation in the unit square
  • 1.2. Triangulations
  • 1.3. Our first finite element space
  • 1.4. Integral formulations and \(L_2\)
  • 1.5. Finite element derivative
  • 1.6. Towards the finite element discretisation
  • 1.7. Practical implementation
• 2. Finite element spaces: local to global
  • 2.1. Ciarlet’s finite element
  • 2.2. Vandermonde matrix and unisolvence
  • 2.3. 2D and 3D finite elements
  • 2.4. Some more exotic elements
  • 2.5. Global continuity
• 3. Interpolation operators
  • 3.1. Local and global interpolation operators
  • 3.2. Measuring interpolation errors
  • 3.3. Approximation by averaged Taylor polynomials
  • 3.4. Local and global interpolation errors
• 4. Finite element problems: solvability and stability
  • 4.1. Finite element spaces and other Hilbert spaces
  • 4.2. Linear forms on Hilbert spaces
  • 4.3. Variational problems on Hilbert spaces
  • 4.4. Solvability and stability of some finite element discretisations
• 5. Convergence of finite element approximations
  • 5.1. Weak derivatives
  • 5.2. Sobolev spaces
  • 5.3. Variational formulations of PDEs
  • 5.4. Galerkin approximations of linear variational problems
  • 5.5. Interpolation error in \(H^k\) spaces
  • 5.6. Convergence of the finite element approximation to the Helmholtz problem
  • 5.7. Epilogue
• 6. Stokes equation
  • 6.1. Strong form of the equations
  • 6.2. Variational form of the equations
  • 6.3. The inf-sup condition
  • 6.4. Solveability of mixed problems
  • 6.5. Solveability of Stokes equation
  • 6.6. Discretisation of Stokes equations
  • 6.7. The MINI element

Past exam papers

Past examination papers from the Imperial College London course are available here:

• 2018 exam paper and solutions

• 2019 exam paper and solutions

• 2020 exam paper and solutions

• 2021 exam paper and solutions

• 2022 exam paper and solutions

• 2022 exam paper and solutions

• 2023 exam paper and solutions

The examination at the University of Luxembourg will be adapted to the specific needs of the students attending the course.

Part 2: Implementation

The implementation part of the module aims to give the students a deeper understanding of the finite element method through writing software to solve finite element problems in one and two dimensions.

The key to this part of the module is to build up a working finite element implementation over the course of the term, and thereby to gain a practical understanding of the method.

The starting point for the implementation method is the skeleton code, an outline of a simple finite element library written in Python. Many of the critical algorithms in the skeleton code are left unimplemented, it will be your task to implement them over the course of the term.

Each chapter of the implementation describes how a part of the finite element method is implemented. Along the way are videos which explain the text, or walk through the algorithms presented. Also interspersed in the text are exercises, typically requiring you to implement the algorithm just presented. We will look at how to do this in more detail presently.

• Software tools
  • Python
  • Git
  • The command line
• The implementation exercise
  • Formalities and marking scheme
  • Obtaining the skeleton code
    • Set up a folder to hold the repository and virtual environment
    • Setting up your venv
    • Activating your venv
    • Setting up your repository
    • Cloning a local copy
    • Installing the course Python package
  • Skeleton code documentation
  • How to do the implementation exercises
  • Testing your work
  • Coding style and commenting
  • Getting help
    • Using Ed
    • Formulating a good question
  • Tips and tricks for the implementation exercise

Implementation exercise contents:

• 1. Numerical quadrature
  • 1.1. Exact and incomplete quadrature
  • 1.2. Examples in one dimension
  • 1.3. Reference cells
  • 1.4. Quadrature rules on reference elements
  • 1.5. Legendre-Gauß quadrature in one dimension
  • 1.6. Extending Legendre-Gauß quadrature to two dimensions
  • 1.7. Implementing quadrature rules in Python
• 2. Constructing finite elements
  • 2.1. A worked example
  • 2.2. Types of node
  • 2.3. The Lagrange element nodes
  • 2.4. Solving for basis functions
  • 2.5. Implementing finite elements in Python
  • 2.6. Implementing the Lagrange Elements
  • 2.7. Tabulating basis functions
  • 2.8. Gradients of basis functions
  • 2.9. Interpolating functions to the finite element nodes
• 3. Meshes
  • 3.1. Mesh entities
  • 3.2. Adjacency
  • 3.3. Mesh geometry
  • 3.4. A mesh implementation in Python
• 4. Function spaces: associating data with meshes
  • 4.1. Local numbering and continuity
  • 4.2. Implementing local numbering
  • 4.3. Global numbering
  • 4.4. The cell-node map
  • 4.5. Implementing function spaces in Python
• 5. Functions in finite element spaces
  • 5.1. A python implementation of functions in finite element spaces
  • 5.2. Interpolating values into finite element spaces
  • 5.3. Integration
• 6. Assembling and solving finite element problems
  • 6.1. Assembling the right hand side
  • 6.2. Assembling the left hand side matrix
  • 6.3. The method of manufactured solutions
  • 6.4. Errors and convergence
  • 6.5. Implementing finite element problems
• 7. Dirichlet boundary conditions
  • 7.1. An algorithm for homogeneous Dirichlet conditions
  • 7.2. Implementing boundary conditions
  • 7.3. Inhomogeneous Dirichlet conditions
• 8. Nonlinear problems
  • 8.1. A model problem
  • 8.2. Residual form
  • 8.3. Linearisation and Gâteaux Derivatives
  • 8.4. A Taylor expansion and Newton’s method
  • 8.5. Implementing a nonlinear problem
• 9. Mixed problems
  • 9.1. Vector-valued finite elements
  • 9.2. Vector-valued function spaces
  • 9.3. Functions in vector-valued spaces
  • 9.4. Mixed function spaces
  • 9.5. Manufacturing a solution to the Stokes equations
  • 9.6. Implementing the Stokes problem
• 10. References
• 11. fe_utils package
  • 11.1. Subpackages
  • 11.2. Submodules
  • 11.3. fe_utils.finite_elements module
  • 11.4. fe_utils.function_spaces module
  • 11.5. fe_utils.mesh module
  • 11.6. fe_utils.quadrature module
  • 11.7. fe_utils.reference_elements module
  • 11.8. fe_utils.utils module
  • 11.9. Module contents