  We study an inexact inner-outer generalized Golub-Kahan algorithm for the
solution of saddle-point problems with a two-times-two block structure. In each
outer iteration, an inner system has to be solved which in theory has to be
done exactly. Whenever the system is getting large, an inner exact solver is,
however, no longer efficient or even feasible and iterative methods must be
used. We focus this article on a numerical study showing the influence of the
accuracy of an inner iterative solution on the accuracy of the solution of the
block system. Emphasis is further given on reducing the computational cost,
which is defined as the total number of inner iterations. We develop relaxation
techniques intended to dynamically change the inner tolerance for each outer
iteration to further minimize the total number of inner iterations. We
illustrate our findings on a Stokes problem and validate them on a mixed
formulation of the Poisson problem.
