  We study the continuum limit for Dirac-Hodge operators defined on the $n$
dimensional square lattice $h\mathbb{Z}^n$ as $h$ goes to $0$. This result
extends to a first order discrete differential operator the known convergence
of discrete Schr\"odinger operators to their continuous counterpart. To be able
to define such a discrete analog, we start by defining an alternative framework
for a higher-dimensional discrete differential calculus. We believe that this
framework, that generalize the standard one defined on simplicial complexes,
could be of independent interest. We then express our operator as a
differential operator acting on discrete forms to finally be able to show the
limit to the continuous Dirac-Hodge operator.
