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Maxwell interface problems are of great importance in many electromagneticapplications. Unfitted mesh methods are especially attractive in 3D computationas they can circumvent generating complex 3D interface-fitted meshes. However,many unfitted mesh methods rely on non-conforming approximation spaces, whichmay cause a loss of accuracy for solving Maxwell equations, and the widely-usedpenalty techniques in the literature may not help in recovering the optimalconvergence. In this article, we provide a remedy by developingN\'ed\'elec-type immersed finite element spaces with a Petrov-Galerkin schemethat is able to produce optimal-convergent solutions. To establish a systematicframework, we analyze all the $H^1$, $\mathbf{H}(\text{curl})$ and$\mathbf{H}(\text{div})$ IFE spaces and form a discrete de Rham complex. Basedon these fundamental results, we further develop a fast solver using a modifiedHiptmair-Xu preconditioner which works for both the GMRES and CG methods.

A family of immersed finite element spaces and applications to three dimensional $\mathbf{H}(\text{curl})$ interface problems