Analysis of variational formulations and low-regularity solutions for time-harmonic electromagnetic problems in complex anisotropic media

We consider the time-harmonic Maxwell's equations with physical parameters, namely, the electric permittivity and the magnetic permeability, that are complex, possibly non-Hermitian, tensor fields. Both tensor fields verify a general ellipticity condition. In this work, the well-posedness of formulations for the Dirichlet and Neumann problems (i.e., with a boundary condition on the electric field or its curl, respectively) is proven using well-suited function spaces and Helmholtz decompositions. For both problems, the a priori regularity of the solution and the solution's curl is analyzed. The regularity results are obtained by splitting the fields and using shift theorems for second-order divergence elliptic operators. Finally, the discretization of the formulations with a H (curl)-conforming approximation based on edge finite elements is considered. An a priori error estimate is derived and verified thanks to numerical results with an elementary benchmark.