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

Abstract

We propose an unfitted finite element method for numerically solving the time-harmonic Maxwell equations on a smooth domain. 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 the 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 variables. Numerical examples in both two and three dimensions are presented to illustrate the accuracy of the method.