It seems like you want Discrete Exterior Calculus. This formalizes discrete 'approximations' to continuous ideas like differential forms and such, but without depending on an underlying continuous theory: it's developed through chains and cochains on a simplicial complex from the bottom up. Here's the main original thesis, and Here's a good overview of the field with some interesting algorithms. It's a self-contained analgoue to the associated continuous ideas, rather than being an approximation. It's actually discrete, rather than just discretized, so it seems to be what you're looking for.