A Hierarchical Space-Time Solver for Optimal Distributed Control of Fluid Flow

We present a space-time hierarchical solution concept for optimization problems governed by the time-dependent Stokes– and Navier–Stokes system. Discretisation is carried out with finite elements in space and a one-step-θ-scheme in time. By combining a Newton solver for the treatment of the nonlinearity with a space-time multigrid solver for linear subproblems, we obtain a robust solver whose convergence behaviour is independent of the number of unknowns of the discrete problem and robust with regard to the considered flow configuration. A set of numerical examples analyses the solver behaviour for various problem settings with respect to the efficiency of this approach.