# Graduate seminar WS 24/25 Graduate Seminar on Numerical Analysis

## Space-time methods

Dates will be fixed at the beginning of the semester depending on availability of the students.

Time-dependent partial differential equations (PDEs) are typical models for many scientific and engineering applications. The standard approach for the numerical solution of such problems are time-stepping methods, which either discretize first in space and then in time or vice versa. Concerning adaptive refinement, these suffer from the fact that the time increment is independent of the spatial location. However, since the PDE solution can have singularities in space and time, flexible adaptive refinement is mandatory to regain optimal convergence rates of the error with respect to the computational cost. Another major drawback of time-stepping methods is that they are inherently sequential in time, and therefore not well suited for parallelization.

Due to the rapid development of parallel computers, space-time methods, which aim to solve the problem as a whole and treat time as yet another dimension, have become a promising alternative. Indeed, the space-time approach has none of the aforementioned shortcomings. It has the potential to construct meshes that are optimally adapted to singularities, and it allows for massive parallelization.

In the seminar, we will study recent space-time methods from the literature. Potential topics include:

- Space-time finite element methods for parabolic problems (see TU Graz for a publicly available preprint)
- Stability of Galerkin discretizations of a mixed space–time variational formulation of parabolic evolution equations
- Space–time least-squares finite elements for parabolic equations
- Space-time finite element discretization of parabolic optimal control problems with energy regularization
- Space-time least-squares finite element methods for parabolic distributed optimal control problems (see arXiv for a publicly available preprint)

Knowledge from numerical analysis (finite element method) and functional analysis (basics on Sobolev spaces) as taught in Scientific Computing I are required to follow the course.

In case you are interested in the seminar, please contact me via ed* tod *nnob-inu* tod *sni* ta *rentnaga* tod *b@foo* tod *de.