2D Discrete Hodge-Dirac Operator on the Torus

We discuss a discretization of the de Rham-Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge-Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide and prove a discrete version of the Hodge decomposition theorem. The goal of this work is to develop a satisfactory discrete model of the de Rham-Hodge theory on manifolds that are homeomorphic to the torus. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups.

Automated summary

In this study, we discuss the discretization of the two-dimensional de Rham-Hodge theory using a discrete exterior calculus framework. We present discrete analogues of the Hodge-Dirac and Laplace operators that capture key geometric aspects of the continuous counterpart. We also provide and prove a discrete version of the Hodge decomposition theorem, and define and compute the cohomology groups in the case of combinatorial torus.