Termine Angewandte Mathematik Münster

In this talk, we investigate efficient and robust solution strategies for the incompressible Navier-Stokes equations, which are specifically designed to predict transient flow behaviors on modern hardware architectures. The main idea of the framework is to block the individual linear systems of equations at each time step into a single all-at-once saddle-point problem. To address the challenges associated with the zero block, we employ a global Schur complement technique, implicitly eliminating all velocity unknowns and leading to a problem for all pressure unknowns only. This so called pressure Schur complement (PSC) equation is then solved using a space-time multigrid algorithm with tailor-made preconditioners for smoothing purposes. However, as the Reynolds number increases, the accuracy of the PSC preconditioner deteriorates, leading to convergence issues in the overall multigrid solver. To overcome this disadvantage, we enhance the solution strategy by using the Augmented Lagrangian methodology in a global-in-time fashion. This approach guarantees a rapid convergence by drastically increasing the accuracy of the adapted PSC preconditioner by means of a strongly consistent modification of the velocity system matrix. For sufficiently large stabilization parameters, even the global-in-time Schur complement iteration by itself converges fast and makes the coarse grid acceleration of the multigrid algorithm redundant. The Augmented Lagrangian technique comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned for large stabilization parameters so that standard iterative solvers prove to be practically unusable. This calls for more sophisticated iterative solvers, like a specialized multigrid solver utilizing modified intergrid transfer operators and block diagonal preconditioners. In the limit of large stabilization parameters, the Augmented Lagrangian approach implies that all intermediate velocity solutions satisfy the discrete incompressibility condition. A straightforward consequence is therefore the use of discretely divergence-free finite elements, eliminating the pressure variable in an a priori manner. Therefore, no saddle-point structure exists and a wide range of efficient and accurate preconditioners becomes applicable. After illustrating the potential of this approach for two-dimensional global-in-time problems, we give some insight into the rarely investigated extension to three dimensions.

TBA

The space of elastic curves is an infinite-dimensional Banach space equipped with a Riemannian metric. Using a global Isometry the space can be better understood and geodesics can be computed.

Spectral graph theory is the discipline studying what properties of a graph can be deduced from the eigenvalues of associated matrices. One may wonder how far one can take this perspective. Is it typical for a random graph to be characterized up to isomorphism by spectral information? In this talk, I will discuss some of the history of this question as well as more recent progress. My main focus will be on a new line-of-attack that reduces the problem to studying random abstract-algebraic objects associated to the graph, such as groups and modules.

TBA