2D discrete Hodge-Dirac operator on the torus
Volodymyr Sushch
TL;DR
This work develops a faithful 2D discretization of de Rham–Hodge theory using discrete exterior calculus on a combinatorial chain complex $C(2)$ and its dual cochain complex $K(2)$. It defines discrete analogues of the exterior derivative $d^c$, the codifferential $\delta^c$, the Hodge star $\ast$, and the cup product $\cup$, together with a discrete Hodge–Dirac operator $d^c+\delta^c$ and a discrete Laplacian $\Delta^c$, establishing a discrete Hodge decomposition $H^r(V)=R_{d^c}^r\oplus R_{\delta^c}^r\oplus N_{\Delta^c}^r$. The paper further specializes to a combinatorial torus, providing explicit matrix representations for these operators and showing the discrete cohomology groups satisfy $\mathcal{H}^0(T)\cong \mathbb{R}$, $\mathcal{H}^1(T)\cong \mathbb{R}^2$, and $\mathcal{H}^2(T)\cong \mathbb{R}$, matching the continuum. By delivering exact discrete analogues of Hodge theory on a torus, the work enables robust computational approaches to de Rham–Hodge problems on discrete toroidal geometries.
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.