  The time-harmonic Maxwell equations are used to study the effect of electric
and magnetic fields on each other. Although the linear systems resulting from
solving this system using FEMs are sparse, direct solvers cannot reach the
linear complexity. In fact, due to the indefinite system matrix, iterative
solvers suffer from slow convergence. In this work, we study the effect of
using the inverse of $\CH$-matrix approximations of the Galerkin matrices
arising from N\'ed\'elec's edge FEM discretization to solve the linear system
directly. We also investigate the impact of applying an $\mathcal{H}-LU$
factorization as a preconditioner and we study the number of iterations to
solve the linear system using iterative solvers.
