  We obtain the solutions for the tunneling zone of a one-dimensional
electrostatic potential in the relativistic (Dirac to Klein-Gordon) wave
equation regime when the incoming wave packet exhibits the possibility of being
almost totally transmitted through the potential barrier. The conditions for
the occurrence of accelerated and, eventually, superluminal tunneling
transmission probabilities are all quantified and the problematic superluminal
interpretation originated from the study based on non-relativistic dynamics of
tunneling is overcome. The treatment of the problem suggests revealing insights
into condensed-matter experiments using electrostatic barriers in single- and
bi-layer graphene, for which the accelerated tunneling effect deserves a more
careful investigation.
