Constraint Schur complement preconditioners for nonsymmetric saddle point problems

We consider constraint preconditioners for block two-by-two generalized saddle point problems, this is the general nonsymmetric, nonsingular case where the (1,2) block need not equal the transposed (2,1) block and the (2,2) block may not be zero. The constraint preconditioners are derived from splittings of the (1,1) block of the generalized saddle point matrix. We show that fast convergence of the preconditioned iterative methods depends mainly on the quality of the splittings and on the effectively solving for the Schur complement systems which arise from the implementation of the constraint preconditioners. Results concerning the eigensolution distribution of the preconditioned matrix and its minimal polynomial are given. To demonstrate the effectiveness of the constraint Schur complement preconditioners we show convergence results and spectra for two model Navier–Stokes problems.