A Family of Immersed Finite Element Spaces and Applications to Three-Dimensional \(\bf{H}(\operatorname{curl})\) Interface Problems

.Efficient and accurate computation of \({\bf H}(\operatorname{curl})\) interface problems is of great importance in many electromagnetic applications. Unfitted mesh methods are especially attractive in three-dimensional (3D) computation as they can circumvent generating complex 3D interface-fitted meshes. However, many unfitted mesh methods rely on nonconforming approximation spaces, which may cause a loss of accuracy for solving Maxwell-type 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 Nédélec-type immersed finite element (IFE) spaces with a Petrov–Galerkin scheme that is able to produce optimal-convergent solutions. To establish a systematic framework, we analyze all the \(H^1\), \({\bf H}(\operatorname{curl})\), and \({\bf H}(\operatorname{div})\) 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 generalized minimal residual (GMRES) and conjugate gradient (CG) methods for solving the nonsymmetric linear algebraic system. The approximation capabilities of the proposed IFE spaces will be also established.Reproducibility of computational results.This paper has been awarded the "SIAM Reproducibility Badge: code and data available", as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available in https://github.com/lyc102/ifem OR in the Supplementary Materials.Keywordsinterface problemsMaxwell equationsimmersed finite element methodsPetrov–Galerkin formulationde Rham complexNédélec elementsRaviart–Thomas elementspreconditioner\({\bf H}(\text{curl;}\,\Omega )\)-elliptic equationsMSC codes65N1265N1565N3065N55