In many applications in computational science and engineering the solution of a partial differential 
equation is sought for, often these applications demand a huge amount of compute power or memory, thus 
requiring the use of supercomputers.
For many problems multigrid methods are optimal solvers. By optimality we mean that the convergence 
rate is bounded from above independently from the system size and that the number of arithmetic 
operations grows linear with the system size. Multigrid methods have been developed especially for the 
solution of linear systems that arise when partial differential equations are discretized. They rely 
on a grid hierarchy that is available naturally when structured grids are used. If this is not the 
case, other, more expensive techniques like algebraic multigrid have to be used. Besides allowing for 
the use of computationally cheaper geometric multigrid methods, the presence of structure also enables 
to use more efficient implementations on modern computer architectures, including GPUs or vector units 
in general.
We work on highly scalable multigrid methods on high performance computers and accelerators. This 
includes the design and analysis of coarse grid and grid transfer operators, as well as the 
development of smoothers with a special emphasize on scalability. E.g., block smoothers that posses a 
higher arithmetic complexity and better smoothing properties, resulting in shorter time to solution. 
Additionally, for time-dependent problems parallelization in time is employed.

In the talk our work on block smoothers and analysis techniques will be presented. Further, results on 
different parallel architectures will be shown, including the solution of parabolic PDEs using 
parallelization in time.