  Convergence analysis of a nested iterative scheme proposed by Bank,Welfert
and Yserentant (BWY) ([Numer. Math., 666: 645-666, 1990]) for solving saddle
point system is presented. It is shown that this scheme converges under weaker
conditions: the contraction rate for solving the $(1,1)$ block matrix is bound
by $(\sqrt{5}-1)/2$. Similar convergence result is also obtained for a class of
inexact Uzawa method with even weaker contraction bound $\sqrt{2}/2$.
Preconditioned generalized minimal residual method using BWY method as a
preconditioner is shown to converge with realistic assumptions.
