  The close vicinity of neutron stars remains poorly constrained by
observations. Although plenty of data are available for the peculiar class of
pulsars we are still unable to deduce the underlying plasma distribution in
their magnetosphere. In the present paper, we try to unravel the magnetospheric
structure starting from basic physics principles and reasonable assumptions
about the magnetosphere. Beginning with the monopole force-free case, we
compute accurate general-relativistic solutions for the electromagnetic field
around a slowly rotating magnetized neutron star. Moreover, here we address
this problem by including the important effect of plasma screening. This is
achieved by solving the time-dependent Maxwell equations in a curved space-time
following the 3+1~formalism. We improved our previous numerical code based on
pseudo-spectral methods in order to allow for possible discontinuities in the
solution. Our algorithm based on a multi-domain decomposition of the simulation
box belongs to the discontinuous Galerkin finite element methods. We performed
several sets of simulations to look for the general-relativistic force-free
monopole and split monopole solutions. Results show that our code is extremely
powerful in handling extended domains of hundredth of light-cylinder
radii~$\rlight$. The code has been validated against known exact analytical
monopole solutions in flat space-time. We also present semi-analytical
calculations for the general-relativistic vacuum monopole.
