The authors construct and analyze fast solution methods for the algebraic saddle-point problems arising from the finite element discretization of the mixed variational formulation of the boundary value problem
in the computational domain
with vanishing tangential component
of the vector-valued function
. The main result consists in the construction of an almost optimal substructuring (non-overlapping domain decomposition) preconditioner for the eventually regularized finite element matrix arising from the first equation above. Finally, using this result and a similar result for the Schur-complement preconditioner, one can solve the algebraic saddle saddle-point problems very efficiently by a Uzawa-like iteration.