Discrete calculus with cubic cells on discrete manifolds
Leonardo De Carlo
TL;DR
The paper develops a rigorous discrete exterior calculus on cubic cell complexes, preserving core geometric-topological structures of the continuum by defining discrete k-forms on a primal cube complex and their dual, along with a coboundary operator $d$ and codifferential $oldsymbol{\delta}$ as adjoints. It establishes a discrete Hodge decomposition $\,oldsymbol{\Omega}^k({\mathcal{C}}) = d^{k-1}\boldsymbol{\Omega}^{k-1}({\mathcal{C}}) \oplus \boldsymbol{\delta}^{k+1}\boldsymbol{\Omega}^{k+1}({\mathcal{C}}) \oplus \boldsymbol{\Omega}^k_H({\mathcal{C}})$ and demonstrates the framework concretely on a 3D discrete torus, deriving explicit formulas for $d$ on 0-,1-,2-,3-forms and for $oldsymbol{\delta}$ on 3-,2-,1-forms. The work provides an implementable DEC toolkit on cubical lattices, including gradient, curl, and divergence operators, and verifies discrete Gauss and Stokes theorems via the primal-dual structure. This sets the stage for lattice-based simulations in physics and graphics where a continuous underpinning is unnecessary or unavailable.
Abstract
This work is thought as an operative guide to discrete exterior calculus (DEC), but at the same time with a rigorous exposition. We present a version of (DEC) on cubic cell, defining it for discrete manifolds. An example of how it works, it is done on the discrete torus, where usual Gauss and Stokes theorems are recovered.