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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.

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).