Numerical Analysis for Nonlinear PDE
Lecture: Dr. Alexei Gazca, Thursday 15:40-17:10, Room K4.
Tutorial: Dr. Alexei Gazca, Thursday 17:20-18:50, Room K4.
The lecture notes (still preliminary!) for the course can be found here
Contents
1. Compactness methods
- Strongly monotone and Lipschitz problems: Convergence of Galerkin discretisations to minimal regularity solutions.
- $\Gamma$-convergence: Convergence for discrete variational problems.
- Quasilinear PDE: Nonlinearities induced by a maximal monotone graph.
2. Error estimates via convex duality
- (Quick) Recap of convex duality: Fenchel duality and the concept of a dual problem.
- Basics of a posteriori error estimation: Deriving error identities from duality principles.
- Applications: The p-Laplacian, the obstacle problem, etc.
- A priori error estimates via duality: Applying duality at the discrete level using the Crouzeix–Raviart and Raviart–Thomas finite elements.
3. Nonlinear solvers
- Relaxed Kačanov iterations for the p-Laplacian.
- Non-smooth problems I: The augmented Lagrangian method (or ADMM) for viscoplastic flow.
- Non-smooth problems II: The proximal Galerkin method for the obstacle problem.
4. Other interesting PDE (if time allows…)
Problem Sheets
Every two weeks a problem sheet will be uploaded here.
In order to obtain the credits from the problem sheets, you should present the solution of one problem during the tutorials at least once during the semester.
| Sheet | Date | Remarks |
|---|---|---|
| Sheet 1 | 19.2.26 | |
| Sheet 2 | 26.2.26 |
Literature
- S. Bartels: Numerical Methods for Nonlinear Partial Differential Equations, Springer Series in Computational Mathematics 47, 2015.
- H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, Second Edition, MOS-SIAM Series on Optimization, 2014.
- T. Roubíček: Nonlinear Partial Differential Equations with Applications, Second Edition, Birkhäuser Verlag, 2015.
- Various research papers to be provided during the semester.