The work to be presented in this talk focuses on an accelerated global-in-time Oseen solver, which highly exploits the augmented Lagrangian methodology 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 all-at-once 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 and customized preconditioners for smoothing purposes. However, the accuracy of these PSC preconditioners deteriorates as the Reynolds number increases and, hence, causes convergence issues of the overall multigrid scheme. To improve the robustness of the solution strategy and accelerate its convergence behavior, the augmented Lagrangian approach is exploited in a global-in-time fashion by modifying the velocity system matrix in a strongly consistent manner. While the introduced discrete grad-div stabilization does not modify the solution of the discretized Oseen equations, the accuracy of the adapted PSC preconditioners drastically improves and, hence, guarantees a rapid convergence. If the stabilization parameter is chosen sufficiently large, the coarse grid acceleration of the multigrid algorithm may not even be worth the effort. 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. The potential of the proposed overall solution strategy is discussed in several numerical studies. Benefits and drawbacks of the approach are summarized and the influence of the stabilization parameter on the PSC iteration and the involved velocity solver is illustrated.