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

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

A fourth-order unfitted characteristic finite element method for solving the advection-diffusion equation on time-varying domains

We propose a fourth-order unfitted characteristic finite element method to solve the advection-diffusion equation on time-varying domains. Based on a characteristic-Galerkin formulation, our method combines the cubic MARS method for interface tracking, the fourth-order backward differentiation formula for temporal integration, and an unfitted finite element method for spatial discretization. Our convergence analysis includes errors of discretely representing the moving boundary, tracing boundary markers, and the spatial discretization and the temporal integration of the governing equation. Numerical experiments are performed on a rotating domain and a severely deformed domain to verify our theoretical results and to demonstrate the optimal convergence of the proposed method