This paper describes the unerlying mathematical model and
the Balancing Neumann-Neumann methods are
introduced and studied for incompressible Stokes equations discretized
with mixed finite or spectral elements with discontinuous pressures. After
decomposing the original domain of the problem into nonoverlapping
subdomains, the interior unknowns, which are the interior velocity
component and all except the constant pressure component, of each
subdomain problem are implicitly eliminated. The resulting saddle point
Schur complement is solved with a Krylov space method with a balancing
Neumann-Neumann preconditioner based on the solution of a coarse Stokes
problem with a few degrees of freedom per subdomain and on the solution of
local Stokes problems with natural %Neumann velocity and essential
boundary conditions on the subdomains. This preconditioner is of hybrid
form in which the coarse problem is treated multiplicatively while the
local problems are treated additively. The condition number of the
preconditioned operator is independent of the number of subdomains and is
bounded from above by the product of the square of the logarithm of the
local number of unknowns in each subdomain and the inverse of the inf-sup
constants of the discrete problem and of the coarse subproblem. Numerical
results show that the method is quite fast; they are also fully consistent
with the theory.