Optimization-based numerical methods for the two-and three-dimensional time harmonic Maxwell problem with sign-changing coefficients

We study the two-and three-dimensional time-harmonic Maxwell equations in a medium composed of a classical material and a metamaterial. The problem is modeled as an interface problem with sign-changing coefficients, with the electric field as the main vector-valued unknown. To carry out the theoretical and numerical analyses, we only require that the problem has a unique solution. We then construct a convergent numerical approximation, with the help of Nédélec finite elements and prove its convergence in the strong H(curl)-norm. At the discrete level, we consider two approaches. The first directly adapts the arguments used for the continuous problem, while the second uses the extra regularity of the tangential traces of discrete fields. Finally, numerical experiments illustrate the theoretical results.