This talk presents a geometric multigrid algorithm for the incompressible Navier-Stokes equations based on a discretely divergence-free variant of the well-known Q_2-P_1 finite element pair. The key idea of this discretization technique is the construction of vector-valued basis functions for the velocity field that automatically satisfy the discrete incompressibility condition. As a result, the saddle-point structure of system matrices commonly encountered in mixed finite element formulations is eliminated. Moreover, the overall problem size is reduced and the computation of the pressure variable is postponed to an inexpensive post-processing step. This allows for the design of efficient solution strategies that do not have to deal with saddle-point structures.
The talk outlines the main ideas behind discretely divergence-free finite elements and discusses possible challenges associated with this approach. In particular, we consider complex geometries in which the distribution of fluxes between different branches of the domain is not known a priori. In such cases, we argue that so-called 'global' finite element functions must be introduced to predict the correct flow behavior. Finally, a geometric multigrid solver is presented that efficiently incorporates the influence of these 'global' finite element functions. Numerical examples illustrate the performance of this solution strategy for three-dimensional flow simulations.

Are you interested in pursuing a doctorate, but can't quite picture what day-to-day life during a PhD looks like? Or are you unsure how to find a position that includes the opportunity to do a doctorate? Then feel free to stop by the PhD consultation hour of the GAMM Junior Research Group on 23/04/2026 – no strings attached!
There you can ask your questions in a relaxed atmosphere to those who should know: doctoral researchers in engineering and mathematics. We conduct research in the fields of applied mathematics and mechanics – an exciting playground between fundamental research and application!
The consultation hour will take place at 4:00 PM in room R165 in the MB I building and digitally via Zoom, and is open to all interested parties – whether Bachelor's or Master's students, whether a doctorate is already being concretely planned or you simply want to find out a little more. We warmly welcome everyone!

Hat and Spectre are recently discovered aperiodic monotiles for the Euclidean plane. Each of them tiles the plane, without gaps or overlaps, but only aperiodically. As such, each of them gives rise to a higly ordered class of tiling dynamical systems under the translation action of R^2.
Combining rather different methods from algebraic topology, dynamical systems theory and harmonic analysis, we show that the tilings have pure-point spectrum and display quasiperiodic long-range order. In fact, both tilings are reprojections of self-similar planar tilings with a 4D embedding space, and thus rather different from the limit-periodic structure of the earlier Taylor-Socolar monotile.
(This is joint work with Franz Gähler and Lorenzo Sadun).