This talk focuses on an accelerated global-in-time Oseen solver, which highly exploits the augmented Lagrangian approach to improve the convergence behavior of the Schur complement iteration. The main idea of the solution strategy is to block the individual linear systems of equations at each time step into a single saddle point problem. By elimination of all velocity unknowns, the resulting pressure Schur complement~(PSC) equation can be solved efficiently on modern hardware architectures using a space-time multigrid algorithm [Lohmann and Turek, 2023]. However, the accuracy of the involved PSC~preconditioners deteriorates as the Reynolds number increases and, hence, causes convergence issues.

To improve the robustness of the solution strategy and accelerate its convergence behavior, the augmented Lagrangian approach is exploited by modifying the velocity system matrix in a strongly consistent manner. While the introduced discrete grad-div stabilization does not modify the solution, the accuracy of the adapted PSC~preconditioners drastically improves and, hence, guarantees a rapid convergence. This strategy comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned so that standard iterative solution strategies are no longer efficient. This calls for highly specialized multigrid solvers, which are based on modified intergrid transfer operators and block diagonal preconditioners (cf. [Benzi and Olshanskii, 2006; Wechsung, 2019]).