Efficient preconditioning of the discrete adjoint equations for the incompressible Navier-Stokes equations

Preconditioning of the discrete adjoint equations is closely related to preconditioning the linear systems arising in the Newton linearization of the discretized flow equations. We investigate the use of an optimal preconditioner for both problems on the example of a finite element discretization of the steady state incompressible Navier-Stokes equations. It is demonstrated that complications arising from the use of a zero mean pressure condition in the problem formulation can be overcome by modifying the preconditioner suitably.