Numerical approximation of PDEs
MATH-451
Course informationLectures : Friday from 8.00h to ...
This page is part of the content downloaded from Course informationLectures : Friday from 8.00h to ... on Monday, 30 June 2025, 16:57. Note that some content and any files larger than 50 MB are not downloaded.
Description
Course information
Lectures : Friday from 8.00h to 10.00h in MA A1 12,
Exercises sessions : Thursday from 8.00h to 10.00h in MA A1 12 and/or MA B1 486.
Lecturer: Annalisa Buffa (annalisa.buffa@epfl.ch)
Assistant: Mohamed Ben Abdelouahab (mohamed.benabdelouahab@epfl.ch)
Student Assistants: Vincenzo Scurria (vincenzo.scurria@epfl.ch)
Content
The aim of the course is to give
a theoretical and practical knowledge of the finite element method in dimension one and higher partial differential
equations.
The main topics are:
Exams
An introduction to Python in scientific computing: https://scipy-lectures.org/intro/ (you can skip 1.3)
References and books
The main topics are:
- Linear elliptic problems: weak form, well-posedness, Galerkin approximation
- Linear and quadratic finite element approximation for elliptic problems in one and two dimensions: stability, convergence, a-priori error estimates in different norms, implementation aspects
- Stabilized finite elements methods for advection-diffusion equations
- Time-discretizations for parabolic and hyperbolic problems
- A posteriori error estimation
- Linear elliptic problems: weak form, well-posedness, Galerkin approximation
Mini projects
- One mini project in Python at the end of April.
- The solution can be submitted for additional credit.
- Submission deadline at the beginning of June - exact date to be announced.
- Bonus 0.5 on final grade.
Exams
The final grade is based on a written exam.
- Written exam: the exam covers all the material treated during the course and the exercise sessions and worths 100% of the final grade.
- Bonus subject to participation in the mini projects.
Python Resources
We will provide one one hour lecture on Numpy vectorisation and the role it plays in this course's python code that we will provide for everyone to use in the mini projects.
The students' main coding objective is implementing assembly routines using Numpy vectorisation. The rest is taken care of by the provided code.
Further ressources:
An introduction to Python in scientific computing: https://scipy-lectures.org/intro/ (you can skip 1.3)
If you anticipate to program in the future, it will be helpful to get acquainted with good editors and IDEs (integrated development environments). Some that we recommend are:
- Microsoft Visual Studio Code: https://code.visualstudio.com/
- PyCharm: https://www.jetbrains.com/de-de/pycharm/
- Spyder: https://www.spyder-ide.org/
- Sublime: https://www.sublimetext.com/
References and books
- Lecture notes in class
- Additional lecture notes provided by the teacher
- A.Quarteroni, Numerical Models for Differential Problems, Springer, 2009 (can be downloaded from Springer website following this link, using a connection on campus or the EPFL VPN service)
- S.C. Brenner, L.R. Scott The Mathematical Theory of Finite Element Methods, Springer, 3rd ed, 2007 (can be downloaded from Springer website following this link, using a connection on campus or the EPFL VPN service)
- A. Ern, J-L. Guermond, Theory and Practice of Finite Elements, Springer, 2004
- Obtain a free student version of Matlab : https://www.epfl.ch/campus/services/ressources-informatiques/support-informatique/logiciels-pro/
- If you want , you can work with a Virtual Machine (VM) created for the course and available here (choose the VM SB-MATH-LINUX or SB-MATH-WINDOWS) : http://vdi.epfl.ch
- MOOC tutorial for Matlab of Prof. Simone Deparis : https://courseware.epfl.ch/courses/course-v1:EPFL+MatlabOctaveBeginners+2018/about
- Rappels et bases pour l'analyse numérique (résolution de systèmes linéaires et méthodes numériques pour équations différentielles) : J. Rappaz et M. Picasso, Introduction à l'analyse numérique, PPUR
References on finite differences (not discussed in this class)
- Griffiths, D. F., Dold, J. W., & Silvester, D. J. (2015). Essential partial differential equations. Springer.
- Larsson, S., & Thomée, V. (2008). Partial differential equations with numerical methods (Vol. 45). Springer Science & Business Media.
- Morton, K. W., & Mayers, D. F. (2005). Numerical solution of partial differential equations: an introduction. Cambridge university press.
- Tveito, Aslak, and Ragnar Winther. Introduction to partial differential equations: a computational approach. Vol. 29. Springer Science & Business Media, 2004.
- Jovanović, Boško S., and Endre Süli. Analysis of Finite Difference Schemes: For Linear Partial Differential Equations with Generalized Solutions. Vol. 46. Springer Science & Business Media, 2013.