Seminar
Numerical Methods for Parabolic Problems and Eigenvalue Problems

Tutors:	D. Jando, G. Kanschat
Class data:	LSF,
Preliminary meeting:	Tue, August 2, 2016, 11h, Mathematikon (INF 205), Seminarraum 10
Seminar:	Tue 11-13h, Mathematikon (INF 205), Seminarraum 11 (Start: Oct 18, 2016)

Summary

Parabolic partial differential equations are differential equations which depend on space and time. The heat equation and advection-diffusion-reaction equations are important members of that class. In this seminar we will analyse these problems and will treat numerical methods to solve such problems. Our focus lies on the combination of space and time discretization. Moreover, we will also consider eigenvalue problems and numerical methods to solve this kind of equations.

Target group

Students of mathematics (BSc and MSc) as well as students of scientific computing (MSc).
The seminar is in particular recommended for those students simultaneously taking the programming course deal.II or the lecture Numerical analysis of partial differential equations.

Prior knowledge

Numerical analysis of ordinary differential equations. Optionally: Numerics for PDEs and/or Analysis of PDEs.

Participation

For participation in the seminar you have to register via Müsli (binding registration).

Additionally, please write us a mail with your subject of study, semester, BSc / MSc, ... . Please also mention related lectures you have already attended, courses you will take in the winter term and, if you already have ideas, the area where you would like to write your thesis.

Topics

The seminar covers the following topics:

1. Finite Differences and Method of Lines (Grossmann, Roos: p. 23-118) Proseminar
2. Weak formulation and functional analytic setting of parabolic PDEs (Evans: 349-377; Dautray, Lions: p. 467-523 ) > F. Parzer
3. Adaptive Finite Element methods for parabolic problems (exemplarily with a linear model problem) (Eriksson, Johnson, 1991, SINUM 28) > B. Cakir
4. Adaptive Finite Element methods for parabolic problems: Optimal Error Estimates in L∞ L2 and L∞ L∞ (Eriksson, Johnson, 1995, SINUM 32) > S. Mehta
5. Semigroup theory of parabolic PDEs (Evans: p. 412- 424; Dautray, Lions: p. 297- )
6. Finite element approximation for eigenvalue problems (Fix 1973) > S. Mueller
7. Finite element approximation for higher dimensional eigenspaces (Babuska, Osborn 1987)
8. Eigensolver LOBPCG (Knyazev 2001)

Literature

• Grossmann, Roos, Stynes: Numerical Treatment of partial Differential Equations, Springer, 2007.
• Quarteroni, Valli: Numerical approximation of partial differential equations. Springer, 2008.
• Johnson: Numerical solution of partial differential equations by the finite element method. Dover, 2009.
• Evans: Partial Differential Equations, AMS, 1991.
• Dautray, Lions: Mathematical Analysis and Numerical Methods for Science and Technology, Vol 5: Evolution Problems I, Springer 2000.
• Strang, Fix: An Analysis of the Finite Element Method.