Immersed Virtual Element Methods for Electromagnetic Interface Problems in Three Dimensions

TL;DR

This paper introduces a novel immersed virtual element method for 3D electromagnetic interface problems that ensures optimal convergence by combining virtual element space conformity with immersed finite element robustness.

Contribution

It develops a new immersed virtual element method for 3D Maxwell interface problems, unifying various interface problems and establishing a de Rham complex for efficient solvers.

Findings

Achieves optimal convergence for 3D electromagnetic interface problems.

Unifies $H^1$, $oldsymbol{H}( ext{curl})$, and $oldsymbol{H}( ext{div})$ interface problems.

Enables fast solvers using the Hiptmair-Xu preconditioner based on the de Rham complex.

Abstract

Finite element methods for electromagnetic problems modeled by Maxwell-type 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 electromagnetic interface problems, as they are based on non-conforming spaces. In this work, a novel immersed virtual element method for solving a 3D $\mathbf{H}(\mathrm{curl})$ interface problem 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. 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 Hiptmair-Xu (HX) preconditioner can be used to develop a fast solver for the $\mathbf{H}(\mathrm{curl})$ interface problem.