Efficient Preconditioning for an Optimal Control Problem with the Time-Periodic Stokes Equations

For the optimal control problem with time-periodic Stokes equations a practical robust preconditioner is presented. The discretization of the corresponding optimality system leads to a linear system with a large, sparse and complex 4-by-4 block matrix in saddle point form. We present a decoupling strategy, which reduces the system to two linear systems with a real 4-by-4 block matrix. Based on analytic results on preconditioners for time-harmonic control problems in Krendl et al. (Numer Math 124(1):183–213, 2013), a practical preconditioner is constructed, which is robust with respect to the mesh size h, the frequency ω and the control parameter ν. The result is illustrated by numerical examples with the preconditioned minimal residual method. Finally we discuss alternative stopping criteria.