  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.
