Numerical solution of nonlocal hydrodynamic Drude model for arbitrary shaped nano-plasmonic structures using N\'ed\'elec finite elements

Nonlocal material response distinctively changes the optical properties of nano-plasmonic scatterers and waveguides. It is described by the nonlocal hydrodynamic Drude model, which -- in frequency domain -- is given by a coupled system of equations for the electric field and an additional polarization current of the electron gas modeled analogous to a hydrodynamic flow. Recent works encountered difficulties in dealing with the grad-div operator appearing in the governing equation of the hydrodynamic current. Therefore, in these studies the model has been simplified with the curl-free hydrodynamic current approximation; but this causes spurious resonances. In this paper we present a rigorous weak formulation in the Sobolev spaces $H(\mathrm{curl})$ for the electric field and $H(\mathrm{div})$ for the hydrodynamic current, which directly leads to a consistent discretization based on N\'ed\'elec's finite element spaces. Comparisons with the Mie theory results agree well. We also demonstrate the capability of the method to handle any arbitrary shaped scatterer.