  In the present work, we introduce a discrete formulation of the nonlinear
Dirac equation in the form of a discretization of the Gross-Neveu model. The
motivation for this discrete model proposal is both computational (near the
continuum limit) and theoretical (using the understanding of the anti-continuum
limit of vanishing coupling). Numerous unexpected features are identified
including a staggered solitary pattern emerging from a single site excitation,
as well as two- and three-site excitations playing a role analogous to one- and
two-site, respectively, excitations of the discrete nonlinear Schr\"odinger
analogue of the model. Stability exchanges between the two- and three-site
states are identified, as well as instabilities that appear to be persistent
over the coupling strength $\epsilon$, for a subcritical value of the
propagation constant $\Lambda$. Variations of the propagation constant,
coupling parameter and nonlinearity exponent are all examined in terms of their
existence and stability implications and long dynamical simulations are used to
unravel the evolutionary phenomenology of the system (when unstable).
