On block preconditioners for saddle point problems with singular or indefinite (1,1) block. (Q2889374)

The author analyzes a class of parametrized block preconditioners for algebraic saddle point problems resulting from discretizations of partial differential equations, which covers a variety of preconditioners proposed in the literature. While in many cases the \((1,1)\) block of the saddle point matrix is assumed to be symmetric positive definite, the author extents the analysis to the case of indefinite or even singular block resulting from discretizations of, e.g., time harmonic Maxwell equations, generalized Stokes equations, or interior point methods in PDE constrained optimization. Under certain assumptions on the uniform spectral equivalence of the approximations of the mass matrices involved in the discretization of the continuous saddle point problem, the author proves that the condition number of the preconditioned is bounded independently of the discretization providing a uniform bound on the number of iterations of the preconditioned conjugate residual method.