Immersed Virtual Element Methods for Maxwell Interface Problems in Three Dimensions

Finite element methods for Maxwell's equations are highly sensitive to the conformity of approximation spaces, and non-conforming methods may cause loss of convergence. This fact leads to an essential obstacle for almost all the interface-unfitted mesh methods in the literature regarding the application to Maxwell interface problems, as they are based on non-conforming spaces. In this work, a novel immersed virtual element method for solving a 3D Maxwell interface problems is developed, and the motivation is to combine the conformity of virtual element spaces and robust approximation capabilities of immersed finite element spaces. The proposed method is able to achieve optimal convergence for a 3D Maxwell interface problem. To develop a systematic framework, the $H^1$, $\mathbf{H}(\mathrm{curl})$ and $\mathbf{H}(\mathrm{div})$ interface problems and their corresponding problem-orientated immersed virtual element spaces are considered all together. In addition, the de Rham complex will be established based on which the HX preconditioner can be used to develop a fast solver for the $\mathbf{H}(\mathrm{curl})$ interface problem.