2D discrete Hodge-Dirac operator on the torus

TL;DR

This paper develops a discrete exterior calculus framework for the 2D de Rham-Hodge theory on a torus, introducing discrete Hodge-Dirac and Laplace operators, and proving a discrete Hodge decomposition theorem.

Contribution

It presents a novel discretisation of the Hodge-Dirac operator on a torus that preserves key geometric properties of the continuum theory.

Findings

Discrete Hodge-Dirac operator captures geometric features

Hodge decomposition theorem is valid in the discrete setting

Cohomology groups are explicitly computed for the torus

Abstract

We discuss a discretisation 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. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups.