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Efficient and accurate computation of H(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 H¹, 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 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. [ABSTRACT FROM AUTHOR]

This paper is the second of two articles, in which we aim to prove the convergence of the Hiptmair--Xu (HX) preconditioner (originally proposed by [R. Hiptmair and J. Xu, SIAM J. Numer. Anal., 45 (2007), pp. 2483--2509]) for H(curl)-elliptic boundary value problems with jump coefficients. In this paper, based on the auxiliary results obtained in our first article [Q. Hu, SIAM J. Numer. Anal., 59 (2021), pp. 2500--2535], we establish two new regular decompositions for lowdimensional edge finite element functions in three dimensions, which, under suitable assumptions on the distribution of the coefficients, are stable up to a polylogarithmic factor of the meshwidth with respect to weighted norms defined by the coefficients. Using these regular decompositions, we analyze the convergence of the HX preconditioner for the case of strongly discontinuous coefficients. We show that the HX preconditioner is asymptotically optimal up to a polylogarithmic factor in the meshwidth and is not severely affected by large jumps of the coefficients across the interface between two neighboring subdomains. [ABSTRACT FROM AUTHOR]

A mixed finite element method is proposed for the Maxwell eigenproblem under the general setting. The method is based on a modification of the Kikuchi mixed formulation in terms of the electric field and the multiplier, with a mesh-dependent Gauss law of the electric field enforced in the formulation. The electric field is discretized by discontinuous elements, and the multiplier is always discretized by the lowest-order continuous nodal element (e.g., the linear element). The method renders four key features: the discrete de Rham complex exact sequence is not required and is replaced by a gradient inclusion condition of a low-order scalar element; i.e., the finite element space of the electric field includes the gradient of an auxiliary scalar H¹-conforming finite element space of low order; the discrete compactness property holds; the strong convergence of the Gauss law is ensured globally for the finite element solution; the method converges nearly optimally for both singular and smooth solutions. With these features, we develop a general analysis to prove that whether or not the discrete eigenmodes are spurious-free and spectral-correct attributes essentially to the first-order approximation property in the H(curl ; Ω) norm. As a direct application, except three lowest-order elements that do not have the first-order approximation property on nonaffine meshes, the first-kind Nédélec elements on nonaffine quadrilateral and hexahedral meshes and the second-kind Nédélec elements on affine and nonaffine quadrilateral and hexahedral meshes, including their discontinuous versions, are spurious-free and spectral-correct in the new mixed method, while these Nédélec elements generate spurious and incorrect discrete eigenmodes in the classical methods. [ABSTRACT FROM AUTHOR]

This paper is the first in a series of two articles, aiming to prove the convergence of the HX preconditioner originally proposed by Hiptmair and Xu [SIAM J. Numer. Anal., 45 (2007), pp. 2483-2509] for Maxwell's equations with jump coefficients. In this paper, we establish several extensions of the discrete regular decomposition for edge finite element functions defined in threedimensional domains. The functions defined by the new discrete regular decompositions can inherit zero degrees of freedom of the considered edge finite element function on some faces and edges of polyhedral domains as well as of some non-Lipschitz domains and possess nearly optimal stability with only a logarithmic factor. [ABSTRACT FROM AUTHOR]

A main issue in nonstationary, compressible magnetohydrodynamic (MHD) simulations is controlling the divergence of the magnetic flux. This paper presents a general procedure showing how to modify the intercell fluxes in a conservative MHD finite volume code such that the scheme becomes locally divergence preserving. That is, a certain discrete divergence operator vanishes exactly during the entire simulation, which results in the suppression of any divergence error. The procedure applies to arbitrary finite volume schemes provided they are based on intercell fluxes. We deduce the necessary modifications for numerical methods based on rectangles and triangles and present numerical experiments with the new schemes. The theoretical justifition of the schemes is given in two independent ways. One way starts with the discrete divergence operator that has to be preserved and modifies the fluxes accordingly. The second way uses a finite element reconstruction via Nedelec elements. Both methods lead to equivalent numerical methods. [ABSTRACT FROM AUTHOR]