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.

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