On block diagonal and block triangular iterative schemes and preconditioners for stabilized saddle point problems

We review the use of block diagonal and block lower/upper triangular splittings for constructing iterative methods and preconditioners for solving stabilized saddle point problems. We introduce new variants of these splittings and obtain new results on the convergence of the associated stationary iterations and new bounds on the eigenvalues of the corresponding preconditioned matrices. We further consider inexact versions as preconditioners for flexible Krylov subspace methods, and show experimentally that our techniques can be highly effective for solving linear systems of saddle point type arising from stabilized finite element discretizations of two model problems, one from incompressible fluid mechanics and the other from magnetostatics.