Continuum Limit for a Discrete Hodge-Dirac Operator on Square Lattices

We study the continuum limit for Dirac–Hodge operators defined on the n dimensional square lattice hZⁿ as h goes to 0. This result extends to a first order discrete differential operator the known convergence of discrete Schrödinger 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.