Lecture “Numerical Methods for PDEs – Galerkin Methods”
In this course I will give an introduction to the finite element Method (FEM). The FEM is – in contrast to, e.g., finite difference methods – based on the variational form a PDE and approximates the solution (which lives in an infinite dimensional space) in finite dimensional subspaces of the infinite-dimensional solution space (this makes it a Galerkin method). The approximation space is spanned by a suitable basis constructed on a decomposition of the domain into simple building blocks (finite elements) such as triangles. This makes the method accessible by methods from functional analysis and approximation theory.
FEM theory is rich ob both theory and (real world) applications and I will have to make a compromise between the two here. I thought about the following topics (but may change it during the course):
- Recapturing basic notions for elliptic PDEs (integration by parts, Sobolev spaces, variational form, Lax-Milgram, etc)
- General insights for Galerkin Methods
- Introduction to FEM the classical mathematical toy example (Laplace equation)
- Common finite elements
- Some approximation theorems and error estimates for elliptic problems
- Practical aspects:
- Implementation aspect. How could you do it and how shouldn’t you? Ingredients and tools. Large scale FEM-simulations with C++.
- Practical implementation of model problems
- Applications (probably focusing on linear elasticity)
- If time permits we will either talk about Petrov-Galerkin methods, saddle-point problems or more exotic finite elements (Nedelec, Raviart-Thomas, Brezzi-Douglas-Marini, PEERS, …) for systems of PDEs such as Maxwell, flow or mechanical applications
!!! NOTE !!! Due to the exceptional situation that we currently experience I will do my best to make the teaching as effective as possible for you. I thought about the following format:
- I will provide lecture notes as PDF for you (they will grow during the course)
- Before each video lecture (recorded, downloadable, not live) which I will provide according to our needs (but at least once a week) I will ask you to read a certain part of the script. The video lecture will mostly be presentation that elaborates on the things you will have read.
- Exercises can be theoretical or practical (programming)
- Exercise classes I will try to give in an interactive format. This way I can try to answer your questions and maybe demonstrate thing on the computer to guide you
The success of the course needs your active participation.
Seminar “Finite Element Exterior Calculus”
This seminar is organized jointly with Jörn Behrens.
The seminar will be postponed and is planned to be given in a block format by the end of the semester.