On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems

The development of preconditioners for PDE-constrained optimization problems is a field of numerical

analysis which has recently generated much interest. One class of problems which has been investigated in particular

is that of Stokes control problems, that is, the problem of minimizing a functional with the Stokes (or Navier-Stokes)

equations as constraints. In this manuscript, we present an approach for preconditioning Stokes control problems

using preconditioners for the Poisson control problem and, crucially, the application of a commutator argument. This

methodology leads to two block diagonal preconditioners for the problem, one of which was previously derived by

W. Zulehner in 2011 [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 536-560] using a nonstandard norm argument for

this saddle point problem, and the other of which we believe to be new. We also derive two related block triangular

preconditioners using the same methodology and present numerical results to demonstrate the performance of the four

preconditioners in practice.