An unfitted finite element method with direct extension stabilization for time-harmonic Maxwell problems on smooth domains

TL;DR

This paper introduces an unfitted finite element method with direct extension stabilization for solving time-harmonic Maxwell equations on smooth domains, allowing flexible boundary handling and ensuring stability and optimal convergence.

Contribution

The paper presents a novel unfitted finite element approach with direct extension stabilization for Maxwell problems, incorporating a mixed formulation and boundary enforcement via Nitsche's method.

Findings

Proves inf-sup stability of the method.

Achieves optimal convergence rates in energy and L2 norms.

Numerical experiments confirm the method's accuracy in 2D and 3D.

Abstract

We propose an unfitted finite element method for numerically solving the time-harmonic Maxwell equations on a smooth domain. The model problem involves a Lagrangian multiplier to relax the divergence constraint of the vector unknown. The embedded boundary of the domain is allowed to cut through the background mesh arbitrarily. The unfitted scheme is based on a mixed interior penalty formulation, where Nitsche penalty method is applied to enforce the boundary condition in a weak sense, and a penalty stabilization technique is adopted based on a local direct extension operator to ensure the stability for cut elements. We prove the inf-sup stability and obtain optimal convergence rates under the energy norm and the $L^2$ norm for both the vector unknown and the Lagrangian multiplier. Numerical examples in both two and three dimensions are presented to illustrate the accuracy of the method.