A brick-tetrahedron finite-element interface with stable hybrid explicit–implicit time-stepping for Maxwell’s equations

A new brick-tetrahedron finite-element interface with stable hybrid explicit–implicit time-stepping for Maxwell’s equations is described and tested. The tetrahedrons are connected directly to the bricks, as opposed to previous curl-conforming formulations that use an intermediate layer of pyramids. The electric field is expanded in linear edge elements, which yields a discontinuous tangential electric field at the brick-tetrahedron interface and this discontinuity is treated by Nitsche’s method. In addition, tangential continuity for an arbitrary constant electric field is imposed in the strong sense at the interface, which makes it possible to avoid penalization that perturbs the frequency spectrum. This hybridization preserves the null-space of the curl–curl operator and is free from non-physical spurious modes, which is confirmed by numerical tests. The implicit Newmark time-stepping scheme is employed for the tetrahedrons, which allows for local mesh refinement without reduced time-step. For the brick elements, spatial lumping and explicit time-stepping is employed, which yields the standard finite-difference time-domain scheme. Furthermore, we prove that the explicit–implicit time-stepping employed at the hybrid interface is stable for time-steps up to the Courant limit of the explicitly time-stepped bricks. Second order of convergence is achieved for field solutions without singular behavior. The reflection from the brick-tetrahedron interface is small and scattering from a thin layer of tetrahedrons indicates levels at approximately −49 dB for a resolution of 14 cells per wavelength.