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

TL;DR

This paper introduces a new family of immersed finite element spaces for 3D Maxwell interface problems, achieving optimal convergence and efficient solving through a Petrov-Galerkin scheme and specialized preconditioners.

Contribution

It develops Nédélec-type immersed finite element spaces with a Petrov-Galerkin scheme, ensuring optimal convergence for 3D Maxwell interface problems, and constructs a fast solver with a modified Hiptmair-Xu preconditioner.

Findings

Achieved optimal convergence in Maxwell interface problems using new IFE spaces.

Established a systematic framework including a discrete de Rham complex.

Developed a fast solver effective for GMRES and CG methods.

Abstract

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 $H^1$, $\mathbf{H}(\text{curl})$ and $\mathbf{H}(\text{div})$ 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.