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

. Maxwell interface problems are of great importance in many electromagnetic applications. Unfitted mesh methods are especially attractive in 3D computation as they can circumvent generating complex 3D interface-fitted meshes. However, many unfitted mesh methods rely on non-conforming approximation spaces, which may cause a loss of accuracy for solving Maxwell equations, and the widely-used penalty techniques in the literature may not help in recovering the optimal convergence. In this article, we provide a remedy by developing N´ed´elec-type immersed finite element spaces with a Petrov-Galerkin scheme that is able to produce optimal-convergent solutions. To establish a systematic framework, we analyze all the H 1 , H (curl) and H (div) IFE spaces and form a discrete de Rham complex. Based on these fundamental results, we further develop a fast solver using a modified Hiptmair-Xu preconditioner which works for both the GMRES and CG methods.

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