Electric and magnetic losses modeled by a stable hybrid with explicit–implicit time-stepping for Maxwell’s equations

A stable hybridization of the finite-element method (FEM) and the finite-difference time-domain (FDTD) scheme for Maxwell’s equations with electric and magnetic losses is presented for two-dimensional problems. The hybrid method combines the flexibility of the FEM with the efficiency of the FDTD scheme and it is based directly on Ampère’s and Faraday’s law. The electric and magnetic losses can be treated implicitly by the FEM on an unstructured mesh, which allows for local mesh refinement in order to resolve rapid variations in the material parameters and/or the electromagnetic field. It is also feasible to handle larger homogeneous regions with losses by the explicit FDTD scheme connected to an implicitly time-stepped and lossy FEM region. The hybrid method shows second-order convergence for smooth scatterers. The bistatic radar cross section (RCS) for a circular metal cylinder with a lossy coating converges to the analytical solution and an accuracy of 2% is achieved for about 20 points per wavelength. The monostatic RCS for an airfoil that features sharp corners yields a lower order of convergence and it is found to agree well with what can be expected for singular fields at the sharp corners. A careful convergence study with resolutions from 20 to 140 points per wavelength provides accurate extrapolated results for this non-trivial test case, which makes it possible to use as a reference problem for scattering codes that model both electric and magnetic losses.