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

$\text{curl}\left(\alpha \phantom{\rule{0.166667em}{0ex}}\text{curl}\left(u\right)\right)+{\gamma}_{0}\beta u=f$ and

$\text{div}\left(\beta u\right)=g$ in the computational domain

${\Omega}$ with vanishing tangential component

$u\times n$ of the vector-valued function

$u$. 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.