  An algebraic multilevel iteration method for solving system of linear
algebraic equations arising in $H(\mathrm{curl})$ and $H(\mathrm{div})$ spaces
are presented. The algorithm is developed for the discrete problem obtained by
using the space of lowest order Nedelec and Raviart-Thomas-Nedelec elements.
The theoretical analysis of the method is based only on some algebraic
sequences and generalized eigenvalues of local (element-wise) problems. In the
hierarchical basis framework, explicit recursion formulae are derived to
compute the element matrices and the constant $\gamma$ (which measures the
quality of the space splitting) at any given level. It is proved that the
proposed method is robust with respect to the problem parameters, and is of
optimal order complexity. Supporting numerical results, including the case when
the parameters have jumps, are also presented.
