Additional block triangular preconditioners for coupled block two-by-two linear systems arising from Stokes control problems

For the coupled block two-by-two linear systems arising from the finite element discretization of the velocity tracking problem of Stokes flow, we construct a class of additive block triangular preconditioners in an inner-outer implementation framework. The related splitting iterative solution methods are proved to be convergent with sharp parameter independent convergence rates. The corresponding preconditioned matrices have tight and parameter free eigenvalue distributions. Moreover, we derive inexact variants to further simplify the proposed preconditioners to avoid their inner-outer implementation process. Numerical experiments indicate that the new preconditioners may lead to satisfactory performance compared with some existing solution methods.