Block triangular Schur complement preconditioners for saddle point problems and application to the Oseen equations

We study block triangular Schur complement preconditioners for two by two block linear systems. Two block triangular Schur complement preconditioners are derived from a splitting of the (1,1)-block of the two by two block matrix. The two block triangular Schur complement preconditioners are different only in taking the opposite sign in the (2,2)-block (i.e. the Schur complement) of the preconditioners. We analyze the properties of the corresponding preconditioned matrices, in particular their spectra and discuss the computational performances of the preconditioned iterative methods. We show that fast convergence depends mainly on the quality of the splitting of the (1,1)-block. Moreover, we discuss some strategies of implementation of our preconditioners based on purely algebraic considerations. Thus, for applying our preconditioners to the Oseen equations we obtain preconditioning iterative methods in “black box” fashion.