2D Discrete Hodge–Dirac Operator on the Torus

Abstract

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.

1. Introduction

The choice of technique to approximate the solution of partial differential equations depends on the discretization scheme. In recent years, there has been a growing interest within the computing community in discrete models that preserve the geometric structure of their continuum counterparts [1,2,3,4,5]. Ideally, a geometric discretization scheme should share the same properties with the continuum, but practically it is difficult to achieve. Computational methods often fail to preserve some fundamental geometric structures of underlying continuum models. Usually, there are difficulties with definitions of discrete counterparts of the Hodge star and of the exterior product of differential forms. Various approaches to discretizing exterior calculus have been proposed in the literature (see, e.g., [6,7,8,9,10,11]). Most of them use simplicial chains and cochains as basic constructs for the discrete exterior calculus. An approach based on the use of the Whitney map, which maps chains to differential forms, was developed in [6,7,8,11]. Another approach to discrete exterior calculus was presented in [5,9,10]. These authors consider discrete forms as real-valued functions on the space of simplicial chains. Recently, also, a quite general framework of discrete calculus based on a new type of discrete geometry (script geometry) was proposed in [12].

In this article, we follow the approach that was initially introduced by Dezin [13] and later further developed in the author’s previous papers [14,15,16,17]. We present a discretization scheme using cochains over rectangular meshes as the discrete representation of differential forms. Our approach is also different from those available in the literature in terms of definitions of the Hodge star and a discrete analogue of the exterior product. The discrete analogue of the exterior product is defined in a way that allows the Leibniz-type rule to be valid for a discrete analogue of the differential when it acts on this product of discrete forms.

Let $\mathsf{\Lambda }\left({\mathbb{R}}^2\right)={\mathsf{\Lambda }}^0\left({\mathbb{R}}^2\right)\oplus {\mathsf{\Lambda }}^1\left({\mathbb{R}}^2\right)\oplus {\mathsf{\Lambda }}^2\left({\mathbb{R}}^2\right)$ denote the graded vector space of smooth differential forms on ${\mathbb{R}}^2$, where ${\mathsf{\Lambda }}^r\left({\mathbb{R}}^2\right)$ denotes the subspace of r-forms, $r=0,1,2$. Let $d:{\mathsf{\Lambda }}^r\left({\mathbb{R}}^2\right)\to {\mathsf{\Lambda }}^{r+1}\left({\mathbb{R}}^2\right)$ be the exterior derivative. The codifferential $\delta :{\mathsf{\Lambda }}^r\left({\mathbb{R}}^2\right)\to {\mathsf{\Lambda }}^{r-1}\left({\mathbb{R}}^2\right)$ is defined by $\delta =-\ast d\ast$, where ∗ is the Hodge star operator such that $\ast :{\mathsf{\Lambda }}^r\left({\mathbb{R}}^2\right)\to {\mathsf{\Lambda }}^{2-r}\left({\mathbb{R}}^2\right)$ and ${\ast }^2={(-1)}^r$. The operator

$$d+\delta :\mathsf{\Lambda }\left({\mathbb{R}}^2\right)\to \mathsf{\Lambda }\left({\mathbb{R}}^2\right)$$

(1)

is called the Hodge–Dirac operator on ${\mathbb{R}}^2$. The Laplacian $\Delta :{\mathsf{\Lambda }}^r\left({\mathbb{R}}^2\right)\to {\mathsf{\Lambda }}^r\left({\mathbb{R}}^2\right)$ is defined by

$$\Delta ={(d+\delta )}^2=d\delta +\delta d.$$

(2)

Our goal is to develop a satisfactory discrete model of the de Rham–Hodge theory on manifolds which are homeomorphic to the torus. We consider a chain complex as a combinatorial model of ${\mathbb{R}}^2$. Supplemented by discrete analogues of the exterior derivative, the Hodge star operator, and the exterior product acting on cochains, it provides all the basic ingredients for the calculus of discrete counterparts of differential forms. We show that discrete analogues of the operators (1) and (2) have the same properties as those in the continual case. We formulate and prove a discrete version of the Hodge decomposition theorem and provide an example which illustrates how cohomology groups are calculated for our discrete model. Note that our construction of discrete versions of the Hodge–Dirac and Laplace operators is very similar to one proposed in Section 5 of [12]. Matrix forms of these discrete operators on the torus are the same in both cases. The difference between the two approaches is in the definitions of discrete Hodge–Dirac and Laplace operators. In [12], the definitions are given in terms of the exterior derivative and boundary operators, while we define these operators in terms of the exterior derivative and its adjoint, as in the continual theory. We believe our approach is also simpler than previous ones.

2. Discrete Model

In this section, we briefly review the construction of a discrete exterior calculus framework, which was initiated in [13] and developed in, e.g., [14,15].

The starting point of consideration is a two-dimensional chain complex (a combinatorial model of ${\mathbb{R}}^2$). Let the sets $\left\{x_k\right\}$ and $\left\{e_k\right\}$, $k\in \mathbb{Z}$, be the generators of free abelian groups of zero-dimensional and one-dimensional chains of the one-dimensional complex $C=C_0\oplus C_1$. The free abelian group is understood as the direct sum of infinity cyclic groups generated by $\left\{x_k\right\}$, $\left\{e_k\right\}$. The boundary operator $\partial :C_1\to C_0$ is the homomorphism defined by $\partial e_k=x_{k+1}-x_k$ and the boundary of every zero chain is defined to be zero. Geometrically, we can interpret the zero-dimensional basis elements $x_k$ as points of the real line and the one-dimensional basis elements $e_k$ as open intervals between points. We call the complex C a combinatorial real line. Let the tensor product

$$C\left(2\right)=C\otimes C$$

be a combinatorial model of the two-dimensional Euclidean space ${\mathbb{R}}^2$. The two-dimensional complex $C\left(2\right)=C_0\left(2\right)\oplus C_1\left(2\right)\oplus C_2\left(2\right)$ contains the free abelian groups of r-chains, $r=0,1,2$, generated by the basic elements

$$\begin{array}{c}\hfill x_{k,s}=x_k\otimes x_s,e_{k,s}^1=e_k\otimes x_s,e_{k,s}^2=x_k\otimes e_s,V_{k,s}=e_k\otimes e_s,\end{array}$$

where $k,s\in \mathbb{Z}$. It is convenient to introduce the shift operators $\tau ,\sigma$ in the set of indices by

$$\tau k=k+1\sigma k=k-1.$$

(3)

The boundary operator $\partial :C_r\left(2\right)\to C_{r-1}\left(2\right)$ is given by

$$\begin{array}{c}\hfill \partial x_{k,s}=0,\partial e_{k,s}^1=x_{\tau k,s}-x_{k,s}\partial e_{k,s}^2=x_{k,\tau s}-x_{k,s},\\ \hfill \partial V_{k,s}=e_{k,s}^1+e_{\tau k,s}^2-e_{k,\tau s}^1-e_{k,s}^2.\end{array}$$

(4)

The definition (4) is extended to arbitrary chains by linearity.

Let $K\left(2\right)$ be a complex of cochains with real coefficients. The cochain complex $K\left(2\right)$ is the dual object to the chain complex $C\left(2\right)$. It has a similar structure to $C\left(2\right)$ and consists of cochains of dimension 0, 1 and 2. Then, $K\left(2\right)$ can be expressed by

$$K\left(2\right)=K^0\left(2\right)\oplus K^1\left(2\right)\oplus K^2\left(2\right),$$

where $K^r\left(2\right)$ is the set of all r-cochains. We will call cochains forms (or discrete forms), emphasizing their relationship with differential forms. Then, the complex $K\left(2\right)$ is a discrete analogue of the grade algebra of differential forms $\mathsf{\Lambda }\left({\mathbb{R}}^2\right)$. Denote by $\left\{x^{k,s}\right\}$, $\{e_1^{k,s},e_2^{k,s}\}$ and $\left\{V^{k,s}\right\}$ are the basis elements of $K^0\left(2\right)$, $K^1\left(2\right)$ and $K^2\left(2\right)$, respectively. The pairing is defined with the basis elements of $C\left(2\right)$ by the rule

$$⟨x_{k,s},x^{i,j}⟩=⟨e_{k,s}^1,e_1^{i,j}⟩=⟨e_{k,s}^2,e_2^{i,j}⟩=⟨V_{k,s},V^{i,j}⟩={\delta }_k^i{\delta }_s^j,$$

(5)

where ${\delta }_k^i$ is the Kronecker delta. The operation (5) is linearly extended to arbitrary chains and cochains. Let $\stackrel{r}{\omega }\in K^r\left(2\right)$, then we have

$$\stackrel{0}{\omega }=\sum _{k,s}{\stackrel{0}{\omega }}_{k,s}{x}^{k,s},\stackrel{1}{\omega }=\sum _{k,s}({\omega }_{k,s}^1{e}_1^{k,s}+{\omega }_{k,s}^2{e}_2^{k,s}),\stackrel{2}{\omega }=\sum _{k,s}{\stackrel{2}{\omega }}_{k,s}{V}^{k,s},$$

(6)

where ${\stackrel{0}{\omega }}_{k,s}$, ${\omega }_{k,s}^1$, ${\omega }_{k,s}^2$ and ${\stackrel{2}{\omega }}_{k,s}$ are real numbers for any $k,s\in \mathbb{Z}$.

The coboundary operator $d^c:K^r\left(2\right)\to K^{r+1}\left(2\right)$ is defined by

$$⟨\partial a_{r+1},\stackrel{r}{\omega }⟩=⟨a_{r+1},d^c\stackrel{r}{\omega }⟩,$$

(7)

where $a_{r+1}\in C_{r+1}\left(2\right)$. The operator $d^c$ is an analog of the exterior differential. From the above it follows that

$$d^c\stackrel{2}{\omega }=0\mathrm{and}{d}^c{d}^c\stackrel{r}{\omega }=0\mathrm{for}\mathrm{any}r=0,1.$$

By (4) and (5) we can calculate

$$d^c\stackrel{0}{\omega }=\sum _{k,s}\left({\Delta }_k{\stackrel{0}{\omega }}_{k,s}\right)e_1^{k,s}+\left({\Delta }_s{\stackrel{0}{\omega }}_{k,s}\right)e_2^{k,s},$$

(8)

$$d^c\stackrel{1}{\omega }=\sum _{k,s}({\Delta }_k{\omega }_{k,s}^2-{\Delta }_s{\omega }_{k,s}^1)V^{k,s},$$

(9)

where ${\Delta }_k$ and ${\Delta }_s$ are the difference operators defined by

$${\Delta }_k{\stackrel{r}{\omega }}_{k,s}={\stackrel{r}{\omega }}_{\tau k,s}-{\stackrel{r}{\omega }}_{k,s},{\Delta }_s{\stackrel{r}{\omega }}_{k,s}={\stackrel{r}{\omega }}_{k,\tau s}-{\stackrel{r}{\omega }}_{k,s}.$$

(10)

Here, ${\stackrel{r}{\omega }}_{k,s}$ is a component of $\stackrel{r}{\omega }\in K^r\left(2\right)$ and $\tau$ is given by (3). Note that ${\stackrel{1}{\omega }}_{k,s}=\{{\omega }_{k,s}^1,{\omega }_{k,s}^2\}$.

We now consider a multiplication of discrete forms, which is an analogue of the exterior multiplication for differential forms. Denote this multiplication by ∪. For the basis elements of $K\left(2\right)$, the ∪-multiplication is defined as follows

$$x^{k,s}\cup x^{k,s}=x^{k,s},x^{k,s}\cup e_1^{k,s}=e_1^{k,s},x^{k,s}\cup e_2^{k,s}=e_2^{k,s},$$

$$x^{k,s}\cup V^{k,s}=V^{k,s},e_1^{k,s}\cup x^{\tau k,s}=e_1^{k,s},e_2^{k,s}\cup x^{k,\tau s}=e_2^{k,s},$$

$$V^{k,s}\cup x^{\tau k,\tau s}=V^{k,s},e_1^{k,s}\cup e_2^{\tau k,s}=V^{k,s},e_2^{k,s}\cup e_1^{k,\tau s}=-V^{k,s},$$

supposing the product to be zero in all other cases. The operation is extended to arbitrary forms by linearity. It is important to note that this definition leads to the following discrete counterpart of the Leibniz rule for differential forms.

Proposition 1.

Let $\stackrel{r}{\omega }\in K^r\left(2\right)$ and $\stackrel{p}{\phi }\in K^p\left(2\right)$. Then

$$d^c(\stackrel{r}{\omega }\cup \stackrel{p}{\phi })=d^c\stackrel{r}{\omega }\cup \stackrel{p}{\phi }+{(-1)}^r\stackrel{r}{\omega }\cup d^c\stackrel{p}{\phi }.$$

(11)

This was proved by Dezin [13].

Define the operation $\ast :K^r\left(2\right)\to K^{2-r}\left(2\right)$ by the rule

$$\ast x^{k,s}=V^{k,s},\ast e_1^{k,s}=e_2^{\tau k,s},\ast e_2^{k,s}=-e_1^{k,\tau s},\ast V^{k,s}=x^{\tau k,\tau s}.$$

(12)

Again, the operation is extended to arbitrary forms by linearity. This operation is a discrete analogue of the Hodge star operator. It is true that for any $\stackrel{r}{\omega }\in K^r\left(2\right)$, we have

$$\stackrel{r}{\omega }\cup \ast \stackrel{r}{\omega }=\sum _{k,s}{\left({\stackrel{r}{\omega }}_{k,s}\right)}^2{V}^{k,s}.$$

(13)

Consider the two-dimensional finite chain $V\in C_2\left(2\right)$ with unit coefficients of the form

$$V=\sum _{k,s}{V}_{k,s},k=1,2,...,N,s=1,2,..,M.$$

(14)

This chain imitates a rectangle. Using (4), we have

$$\partial V=\sum _{k=1}^N{e}_{k,1}^1+\sum _{s=1}^M{e}_{N,s}^2-\sum _{k=1}^N{e}_{k,M}^1-\sum _{s=1}^M{e}_{1,s}^2.$$

(15)

Then, for forms $\stackrel{r}{\phi },\stackrel{r}{\omega }\in K^r\left(2\right)$ of the same degree r, the inner product over the set V is defined by the rule

$${(\stackrel{r}{\phi },\stackrel{r}{\omega })}_V=⟨V,\stackrel{r}{\phi }\cup \ast \stackrel{r}{\omega }⟩.$$

(16)

For forms of different degrees, the product (16) is set equal to zero. From (13) and (5), we have

$${(\stackrel{r}{\phi },\stackrel{r}{\omega })}_V=\sum _{k=1}^N\sum _{s=1}^M{\stackrel{r}{\phi }}_{k,s}{\stackrel{r}{\omega }}_{k,s}.$$

Proposition 2.

Let $\stackrel{r}{\phi }\in K^r\left(2\right)$ and $\stackrel{r+1}{\omega }\in K^{r+1}\left(2\right)$, $r=0,1$. Then, we have

$${(d^c\stackrel{r}{\phi },\stackrel{r+1}{\omega })}_V=⟨\partial V,\stackrel{r}{\phi }\cup \ast \stackrel{r+1}{\omega }⟩+{(\stackrel{r}{\phi },{\delta }^c\stackrel{r+1}{\omega })}_V,$$

(17)

where

$${\delta }^c\stackrel{r+1}{\omega }={(-1)}^{r+1}{\ast }^{-1}{d}^c\ast \stackrel{r+1}{\omega }$$

(18)

is the operator formally adjoint of $d^c$.

Here, ${\ast }^{-1}$ is the inverse of ∗, i.e., $\ast {\ast }^{-1}=1$. By (12) for the basic elements, we have

$${\ast }^{-1}{x}^{k,s}=V^{\sigma k,\sigma s},{\ast }^{-1}{e}_1^{k,s}=-e_2^{k,\sigma s},{\ast }^{-1}{e}_2^{k,s}=e_1^{\sigma k,s},{\ast }^{-1}{V}^{k,s}=x^{k,s},$$

where $\sigma$ is given by (3).

Proof. 

By (7), (16) and Formula (11), we obtain

$$\begin{array}{c}\hfill {(d^c\stackrel{r}{\phi },\stackrel{r+1}{\omega })}_V=⟨V,d^c\stackrel{r}{\phi }\cup \ast \stackrel{r+1}{\omega }⟩=⟨V,d^c(\stackrel{r}{\phi }\cup \ast \stackrel{r+1}{\omega })-{(-1)}^r\stackrel{r}{\phi }\cup d^c\ast \stackrel{r+1}{\omega }⟩\\ \hfill =⟨\partial V,\stackrel{r}{\phi }\cup \ast \stackrel{r+1}{\omega }⟩+{(-1)}^{r+1}⟨V,\stackrel{r}{\phi }\cup \ast {\ast }^{-1}{d}^c\ast \stackrel{r+1}{\omega }⟩\\ \hfill =⟨\partial V,\stackrel{r}{\phi }\cup \ast \stackrel{r+1}{\omega }⟩+⟨V,\stackrel{r}{\phi }\cup \ast \left({\delta }^c\stackrel{r+1}{\omega }\right)⟩.\end{array}$$

□

The operator ${\delta }^c:K^{r+1}\left(2\right)\to K^r\left(2\right)$ given by (18) is a discrete analogue of the codifferential $\delta$. For the 0-form $\stackrel{0}{\omega }\in K^0\left(2\right)$, we have ${\delta }^c\stackrel{0}{\omega }=0$. It is obvious from (18) that ${\delta }^c{\delta }^c\stackrel{r}{\omega }=0$ for any $r=1,2.$ Using (8)–(10), (12) and (18), we can calculate

$${\delta }^c\stackrel{1}{\omega }=\sum _{k,s}(-{\Delta }_k{\omega }_{\sigma k,s}^1-{\Delta }_s{\omega }_{k,\sigma s}^2)x^{k,s},$$

(19)

$${\delta }^c\stackrel{2}{\omega }=\sum _{k,s}\left({\Delta }_s{\stackrel{2}{\omega }}_{k,\sigma s}\right)e_1^{k,s}-\left({\Delta }_k{\stackrel{2}{\omega }}_{\sigma k,s}\right)e_2^{k,s}.$$

(20)

In the particular case $r=0$, the Equality (17) can be expressed as

$$\begin{array}{c}\hfill \sum _{k=1}^N\sum _{s=1}^M(\left({\Delta }_k{\stackrel{0}{\phi }}_{k,s}\right){\omega }_{k,s}^1+\left({\Delta }_s{\stackrel{0}{\phi }}_{k,s}\right){\omega }_{k,s}^2)\\ \hfill =\sum _{k=1}^N({\stackrel{0}{\phi }}_{k,\tau M}{\omega }_{k,M}^2-{\stackrel{0}{\phi }}_{k,1}{\omega }_{k,0}^2)+\sum _{s=1}^M({\stackrel{0}{\phi }}_{\tau N,s}{\omega }_{N,s}^1-{\stackrel{0}{\phi }}_{1,s}{\omega }_{0,s}^1)\\ \hfill +\sum _{k=1}^N\sum _{s=1}^M{\stackrel{0}{\phi }}_{k,s}(-{\Delta }_k{\omega }_{\sigma k,s}^1-{\Delta }_s{\omega }_{k,\sigma s}^2),\end{array}$$

where ${\stackrel{1}{\omega }}_{k,s}=\{{\omega }_{k,s}^1,{\omega }_{k,s}^2\}$. The similar equality holds in the case $r=1$.

It should be noted that the relation (17) includes not only the forms with the components ${\stackrel{r}{\phi }}_{k,s}$ and ${\stackrel{r+1}{\omega }}_{k,s}$, where the subscripts $k,s$ would run only over the values from (14), but also the components ${\stackrel{r}{\phi }}_{0,s}$, ${\stackrel{r}{\phi }}_{\tau N,s}$, ${\stackrel{r+1}{\omega }}_{0,s}$, ${\stackrel{r+1}{\omega }}_{\tau N,s}$, ${\stackrel{r}{\phi }}_{k,0}$, ${\stackrel{r}{\phi }}_{k,\tau M}$, ${\stackrel{r+1}{\omega }}_{k,0}$ and ${\stackrel{r+1}{\omega }}_{k,\tau M}$.

Let us set

$$\begin{array}{c}\hfill {\stackrel{r}{\omega }}_{0,s}={\stackrel{r}{\omega }}_{N,s},{\stackrel{r}{\omega }}_{\tau N,s}={\stackrel{r}{\omega }}_{1,s},s=1,2,..,M,\\ \hfill {\stackrel{r}{\omega }}_{k,0}={\stackrel{r}{\omega }}_{k,M},{\stackrel{r}{\omega }}_{k,\tau M}={\stackrel{r}{\omega }}_{k,1},k=1,2,...,N.\end{array}$$

(21)

For r-forms satisfying conditions (21), the inner product (16) generates the finite-dimensional Hilbert spaces $H^r\left(V\right)$. Now, we consider the operators

$$d^c:H^r\left(V\right)\to H^{r+1}\left(V\right),{\delta }^c:H^{r+1}\left(V\right)\to H^r\left(V\right).$$

Proposition 3.

Let $\stackrel{r}{\phi }\in H^r\left(V\right)$ and $\stackrel{r+1}{\omega }\in H^{r+1}\left(V\right)$, $r=0,1$. Then, we have

$${(d^c\stackrel{r}{\phi },\stackrel{r+1}{\omega })}_V={(\stackrel{r}{\phi },{\delta }^c\stackrel{r+1}{\omega })}_V.$$

(22)

Proof. 

In fact, by use of conditions (21) one has

$$⟨\partial V,\stackrel{0}{\phi }\cup \ast \stackrel{1}{\omega }⟩=\sum _{k=1}^N({\stackrel{0}{\phi }}_{k,\tau M}{\omega }_{k,M}^2-{\stackrel{0}{\phi }}_{k,1}{\omega }_{k,0}^2)+\sum _{s=1}^M({\stackrel{0}{\phi }}_{\tau N,s}{\omega }_{N,s}^1-{\stackrel{0}{\phi }}_{1,s}{\omega }_{0,s}^1)=0,$$

$$⟨\partial V,\stackrel{1}{\phi }\cup \ast \stackrel{2}{\omega }⟩=\sum _{k=1}^N({\phi }_{k,1}^1{\stackrel{2}{\omega }}_{k,0}-{\phi }_{k,\tau M}^1{\stackrel{2}{\omega }}_{k,M})+\sum _{s=1}^M({\phi }_{\tau N,s}^2{\stackrel{2}{\omega }}_{N,s}-{\phi }_{1,s}^2{\stackrel{2}{\omega }}_{0,s})=0,$$

where ${\stackrel{1}{\omega }}_{k,s}=\{{\omega }_{k,s}^1,{\omega }_{k,s}^2\}$ and ${\stackrel{1}{\phi }}_{k,s}=\{{\phi }_{k,s}^1,{\phi }_{k,s}^2\}$. Hence, by Proposition 2, it follows (22). □

3. Discrete Hodge Decomposition

In this section, we discuss properties of discrete analogues of the Laplacian and Hodge–Dirac operators using concepts introduced in the previous section. We also present a discrete version of the Hodge decomposition theorem, emphasizing that it provides an exact counterpart to the continuum theory.

Let us consider the operator

$${\Delta }^c=d^c{\delta }^c+{\delta }^c{d}^c:H^r\left(V\right)\to H^r\left(V\right).$$

This is a discrete analogue of the Laplacian (2).

Proposition 4.

For any r-form $\phi \in H^r\left(V\right)$, we have

$${({\Delta }^c\phi ,\phi )}_V\ge 0$$

and ${({\Delta }^c\phi ,\phi )}_V=0$ if and only if ${\Delta }^c\phi =0$.

Proof. 

By Proposition 3, one has

$$\begin{array}{c}\hfill {({\Delta }^c\phi ,\phi )}_V={(d^c{\delta }^c\phi ,\phi )}_V+{({\delta }^c{d}^c\phi ,\phi )}_V\\ \hfill ={({\delta }^c\phi ,{\delta }^c\phi )}_V+{(d^c\phi ,d^c\phi )}_V=\parallel {\delta }^c{\phi \parallel }^2+{\parallel d^c\phi \parallel }^2,\end{array}$$

where $\parallel ·\parallel$ denotes the norm and

$${\parallel \phi \parallel }^2={(\phi ,\phi )}_V=\sum _{k=1}^N\sum _{s=1}^M{\left({\phi }_{k,s}\right)}^2.$$

From this, if ${({\Delta }^c\phi ,\phi )}_V=0$ then $\parallel {\delta }^c{\phi \parallel }^2=0$ and $\parallel d^c{\phi \parallel }^2=0$. It gives ${\delta }^c\phi =0$ and $d^c\phi =0$. Hence,

$${\Delta }^c\phi =d^c{\delta }^c\phi +{\delta }^c{d}^c\phi =0.$$

□

Corollary 1.

${\Delta }^c\phi =0$ if and only if ${\delta }^c\phi =0$ and $d^c\phi =0$.

Proposition 5.

The operator ${\Delta }^c:H^r\left(V\right)\to H^r\left(V\right)$ is self-adjoint, i.e.,

$${({\Delta }^c\phi ,\omega )}_V={(\phi ,{\Delta }^c\omega )}_V.$$

Proof. 

By (22), it is obvious. □

Consider the spaces

$$R_{d^c}^r=\{d^c\phi \in H^r\left(V\right):\phi \in H^{r-1}\left(V\right)\},$$

$$R_{{\delta }^c}^r=\{{\delta }^c\omega \in H^r\left(V\right):\omega \in H^{r+1}\left(V\right)\},$$

and

$$N_{{\Delta }^c}^r=\{\psi \in H^r\left(V\right):{\Delta }^c\psi =0\}.$$

By analogy with the continuum case, the discrete r-form $\omega$ is called closed if $d^c\omega =0$ and exact if $\omega \in R_{d^c}^r$.

Proposition 6.

For each $r=0,1,2$, we have the direct sum decomposition

$$H^r\left(V\right)=R_{d^c}^r\oplus R_{{\delta }^c}^r\oplus N_{{\Delta }^c}^r.$$

Proof. 

The space $H^1\left(V\right)$ decomposes into

$$H^1\left(V\right)=R_{d^c}^1\oplus N_{{\delta }^c}^1,H^1\left(V\right)=R_{{\delta }^c}^1\oplus N_{d^c}^1,$$

(23)

where $N_{{\delta }^c}^1$ and $N_{d^c}^1$ are the orthogonal complements of the corresponding spaces. For any $\omega \in R_{{\delta }^c}^1$, we have $\omega ={\delta }^c\psi$ and

$${(d^c\phi ,\omega )}_V={(d^c\phi ,{\delta }^c\psi )}_V={(d^c{d}^c\phi ,\psi )}_V=0$$

for any $\phi \in H^0\left(V\right)$. Therefore, $\omega$ is orthogonal to $R_{d^c}^1$. It follows that

$$R_{{\delta }^c}^1\subset N_{{\delta }^c}^1.$$

Similarly, we find $R_{d^c}^1\subset N_{{\delta }^c}^1$. Hence, (23) becomes

$$H^1\left(V\right)=R_{d^c}^1\oplus R_{{\delta }^c}^1\oplus N^1,$$

where

$$N^1=N_{{\delta }^c}^1\cap N_{d^c}^1=\{\phi \in H^1\left(V\right):d^c\phi =0,{\delta }^c\phi =0\}.$$

By Corollary 1, we have $N^1=N_{{\Delta }^c}^1$.

Similar reasonings apply to the spaces $H^0\left(V\right)$ and $H^2\left(V\right)$. Thus, we have

$$H^0\left(V\right)=R_{{\delta }^c}^0\oplus N_{{\Delta }^c}^0,H^2\left(V\right)=R_{d^c}^2\oplus N_{{\Delta }^c}^2,$$

since $R_{d^c}^0=\left\{0\right\}$ and $R_{{\delta }^c}^2=\left\{0\right\}$. □

The Proposition 6 is a discrete version of the well-known Hodge decomposition theorem (see, e.g., [18]).

Let $\mathsf{\Omega }$ be an inhomogeneous discrete form, i.e., $\mathsf{\Omega }=\stackrel{0}{\omega }+\stackrel{1}{\omega }+\stackrel{2}{\omega }$, where $\stackrel{r}{\omega }\in H^r\left(V\right)$. The inner product (16) can be extended to an inner product of inhomogeneous discrete forms by the rule

$${(\mathsf{\Omega },\mathsf{\Phi })}_V=\sum _{r=0}^2{(\stackrel{r}{\omega },\stackrel{r}{\phi })}_V,$$

(24)

where $\mathsf{\Phi }=\stackrel{0}{\phi }+\stackrel{1}{\phi }+\stackrel{2}{\phi }$. The inner product (24) generates the finite-dimensional Hilbert space $H\left(V\right)$. It is true that

$$H\left(V\right)=H^0\left(V\right)\oplus H^1\left(V\right)\oplus H^2\left(V\right).$$

By Proposition 6, the following holds

$$H\left(V\right)=R_{d^c}\oplus R_{{\delta }^c}\oplus N_{{\Delta }^c},$$

(25)

where

$$R_{d^c}=R_{d^c}^1\oplus R_{d^c}^2,R_{{\delta }^c}=R_{{\delta }^c}^0\oplus R_{{\delta }^c}^1,$$

and

$$N_{{\Delta }^c}=N_{{\Delta }^c}^0\oplus N_{{\Delta }^c}^1\oplus N_{{\Delta }^c}^2.$$

The discrete Hodge–Dirac operator is defined as

$$d^c+{\delta }^c:H\left(V\right)\to H\left(V\right).$$

(26)

Proposition 7.

Operator (26) is self-adjoint with respect to the inner product (24), i.e.,

$${((d^c+{\delta }^c)\mathsf{\Phi },\mathsf{\Omega })}_V={(\mathsf{\Phi },(d^c+{\delta }^c)\mathsf{\Omega })}_V.$$

Proof. 

By (22), it is obvious. □

Proposition 8.

$$(d^c+{\delta }^c)\mathsf{\Omega }=0ifandonlyif\mathsf{\Omega }\in N_{{\Delta }^c}.$$

Proof. 

The equation $(d^c+{\delta }^c)\mathsf{\Omega }=0$ can be written as

$$\begin{array}{c}\hfill {\delta }^c\stackrel{1}{\omega }=0,d^c\stackrel{1}{\omega }=0,\\ \hfill d^c\stackrel{0}{\omega }+{\delta }^c\stackrel{2}{\omega }=0.\end{array}$$

Let $\mathsf{\Omega }\in N_{{\Delta }^c}$. This means that $\stackrel{0}{\omega }\in N_{{\Delta }^c}^0$, $\stackrel{1}{\omega }\in N_{{\Delta }^c}^1$ and $\stackrel{2}{\omega }\in N_{{\Delta }^c}^2$. By Corollary 1, $\stackrel{1}{\omega }\in N_{{\Delta }^c}^1$ if and only if ${\delta }^c\stackrel{1}{\omega }=0$ and $d^c\stackrel{1}{\omega }=0$. It is easy to show that

$$\parallel d^c\stackrel{0}{\omega }+{\delta }^c\stackrel{2}{\omega }{\parallel }^2={(d^c\stackrel{0}{\omega }+{\delta }^c\stackrel{2}{\omega },d^c\stackrel{0}{\omega }+{\delta }^c\stackrel{2}{\omega })}_V=\parallel d^c\stackrel{0}{\omega }{\parallel }^2+{\parallel {\delta }^c\stackrel{2}{\omega }\parallel }^2$$

(27)

for any $\stackrel{0}{\omega }\in H^0\left(V\right)$ and $\stackrel{2}{\omega }\in H^2\left(V\right)$. For $\stackrel{0}{\omega }\in N_{{\Delta }^c}^0$ and $\stackrel{2}{\omega }\in N_{{\Delta }^c}^2$, by (22) and (27), we have

$$0={({\delta }^c{d}^c\stackrel{0}{\omega },\stackrel{0}{\omega })}_V+{(d^c{\delta }^c\stackrel{2}{\omega },\stackrel{2}{\omega })}_V={(d^c\stackrel{0}{\omega },d^c\stackrel{0}{\omega })}_V+{({\delta }^c\stackrel{2}{\omega },{\delta }^c\stackrel{2}{\omega })}_V={\parallel d^c\stackrel{0}{\omega }+{\delta }^c\stackrel{2}{\omega }\parallel }^2$$

and, thus, $d^c\stackrel{0}{\omega }+{\delta }^c\stackrel{2}{\omega }=0$. The converse is trivially true. □

Proposition 9.

For any inhomogeneous form $F\in R_{d^c}\oplus R_{{\delta }^c}$, there exists a unique form $\mathsf{\Omega }\in R_{d^c}\oplus R_{{\delta }^c}$ which is a solution to the equation

$$(d^c+{\delta }^c)\mathsf{\Omega }=F.$$

(28)

Proof. 

Since the operator (26) is self-adjoint and $H\left(V\right)$ is a finite-dimensional Hilbert space, the existence of the solution is a consequence of the uniqueness of the solution and vice versa. By Proposition 8, $(d^c+{\delta }^c)\mathsf{\Omega }=0$ implies $\mathsf{\Omega }\in N_{{\Delta }^c}$. From this and by (25), the uniqueness of the solution $\mathsf{\Omega }\in R_{d^c}\oplus R_{{\delta }^c}$ of Equation (28) follows immediately. □

Corollary 2.

If $\mathsf{\Omega }\in H\left(V\right)$ is a solution of Equation (28), then the following holds

$${\parallel \mathsf{\Omega }\parallel }^2\le c(\parallel d^c{\mathsf{\Omega }\parallel }^2+\parallel {\delta }^c{\mathsf{\Omega }\parallel }^2)+\parallel {\mathsf{\Omega }}_{{\Delta }^c}{\parallel }^2,$$

where c is a constant, $d^c\mathsf{\Omega }=d^c\stackrel{0}{\omega }+d^c\stackrel{1}{\omega }$, ${\delta }^c\mathsf{\Omega }={\delta }^c\stackrel{1}{\omega }+{\delta }^c\stackrel{2}{\omega }$ and ${\mathsf{\Omega }}_{{\Delta }^c}$ is the projection of Ω onto $N_{{\Delta }^c}$.

4. Combinatorial Torus

In this section, we consider an example of the discrete model of the torus in detail. We recall that the torus can be regarded as the topological space obtained by taking a rectangle and identifying each pair of opposite sides with the same orientation. Let us consider the partitioning of the plane ${\mathbb{R}}^2$ by the straight lines $x=k$ and $y=s$, where $k,s\in \mathbb{Z}$. Denote by $V_{k,s}$, an open square bounded by the lines $x=k,x=\tau k,y=s$ and $y=\tau s$, where $\tau$ is given by (3). Denote the vertices of $V_{k,s}$ by $x_{k,s},x_{\tau k,s},x_{k,\tau s}$, $x_{\tau k,\tau s}$. Let $e_{k,s}^1$ and $e_{k,s}^2$ be the open intervals $(x_{k,s},x_{\tau k,s})$ and $(x_{k,s},x_{k,\tau s})$, respectively. These geometric objects can be identified with the combinatorial objects considered in previous sections. We identify the collection $V_{k,s}$ with V given by (14) and let $N=M=2$. In this case, the conditions (21) take the form

$${\stackrel{r}{\omega }}_{0,s}={\stackrel{r}{\omega }}_{2,s},{\stackrel{r}{\omega }}_{3,s}={\stackrel{r}{\omega }}_{1,s},{\stackrel{r}{\omega }}_{k,0}={\stackrel{r}{\omega }}_{k,2},{\stackrel{r}{\omega }}_{k,3}={\stackrel{r}{\omega }}_{k,1},$$

(29)

where $k=1,2$ and $s=1,2$. If we identify the points and the intervals on the boundary of V in the following way

$$\begin{array}{c}\hfill x_{1,1}=x_{3,1}=x_{1,3}=x_{3,3},x_{1,2}=x_{3,2},x_{2,1}=x_{2,3},\\ \hfill e_{1,1}^1=e_{1,3}^1,e_{2,1}^1=e_{2,3}^1,e_{1,1}^2=e_{3,1}^2,e_{1,2}^2=e_{3,2}^2,\end{array}$$

(30)

we obtain the geometric object which is homomorphic to the torus (see Figure 1). Denote by $C\left(T\right)$ the complex $C\left(2\right)$, which corresponds to the introduced geometric object. We call $C\left(T\right)$ a combinatorial torus.

Figure 1. Combinatorial torus.

It is clear that by (30), the conditions (29) hold for any r-form on the combinatorial torus. The Hilbert space $H\left(V\right)$ considered in previous sections can be regarded as the space of cochains of the complex $K\left(T\right)$ dual to $C\left(T\right)$.

Let us now consider the forms $\phi \in K^0\left(T\right)$, $\omega \in K^1\left(T\right)$ and $\psi \in K^2\left(T\right)$, that is

$$\phi =\sum _{k=1}^2\sum _{s=1}^2{\phi }_{k,s}{x}^{k,s},\omega =\sum _{k=1}^2\sum _{s=1}^2(u_{k,s}{e}_1^{k,s}+v_{k,s}{e}_2^{k,s}),\psi =\sum _{k=1}^2\sum _{s=1}^2{\psi }_{k,s}{V}^{k,s}.$$

By (30), for these forms, the formulas (8), (9), (19) and (20) become

$$\begin{array}{c}\hfill d^c\phi =({\phi }_{2,1}-{\phi }_{1,1})e_1^{1,1}+({\phi }_{1,1}-{\phi }_{2,1})e_1^{2,1}+({\phi }_{2,2}-{\phi }_{1,2})e_1^{1,2}\\ \hfill +({\phi }_{1,2}-{\phi }_{2,2})e_1^{2,2}+({\phi }_{1,2}-{\phi }_{1,1})e_2^{1,1}+({\phi }_{1,1}-{\phi }_{1,2})e_2^{1,2}\\ \hfill +({\phi }_{2,2}-{\phi }_{2,1})e_2^{2,1}+({\phi }_{2,1}-{\phi }_{2,2})e_2^{2,2},\end{array}$$

(31)

$$\begin{array}{c}\hfill d^c\omega =(u_{1,1}-u_{1,2}+v_{2,1}-v_{1,1})V^{1,1}+(u_{2,1}-u_{2,2}-v_{2,1}+v_{1,1})V^{2,1}\\ \hfill +(u_{1,2}-u_{1,1}+v_{2,2}-v_{1,2})V^{1,2}+(u_{2,2}-u_{2,1}+v_{1,2}-v_{2,2})V^{2,2},\end{array}$$

(32)

$$\begin{array}{c}\hfill {\delta }^c\omega =(u_{2,1}-u_{1,1}+v_{1,2}-v_{1,1})x^{1,1}+(u_{1,1}-u_{2,1}-v_{2,1}+v_{2,2})x^{2,1}\\ \hfill +(u_{2,2}-u_{1,2}+v_{1,1}-v_{1,2})x^{1,2}+(u_{1,2}-u_{2,2}+v_{2,1}-v_{2,2})x^{2,2},\end{array}$$

(33)

$$\begin{array}{c}\hfill {\delta }^c\psi =({\psi }_{1,1}-{\psi }_{1,2})e_1^{1,1}+({\psi }_{2,1}-{\psi }_{2,2})e_1^{2,1}+({\psi }_{2,2}-{\psi }_{1,2})e_2^{1,2}\\ \hfill +({\psi }_{2,1}-{\psi }_{1,1})e_2^{1,1}+({\psi }_{1,2}-{\psi }_{1,1})e_1^{1,2}+({\psi }_{2,2}-{\psi }_{2,1})e_1^{2,2}\\ \hfill +({\psi }_{1,2}-{\psi }_{2,2})e_2^{2,2}+({\psi }_{1,1}-{\psi }_{2,1})e_2^{2,1}.\end{array}$$

(34)

It should be noted that the formulas above can be expressed in the matrix form. Let us introduce the following row vectors

$$\left[\phi \right]=\left[{\phi }_{1,1}{\phi }_{2,1}{\phi }_{1,2}{\phi }_{2,2}\right],\left[\omega \right]=\left[u_{1,1}{u}_{2,1}{v}_{1,2}{v}_{1,1}{u}_{1,2}{u}_{2,2}{v}_{2,2}{v}_{2,1}\right],$$

$$\left[x\right]=\left[x^{1,1}{x}^{2,1}{x}^{1,2}{x}^{2,2}\right],\left[e\right]=\left[e_1^{1,1}{e}_1^{2,1}{e}_2^{1,2}{e}_2^{1,1}{e}_1^{1,2}{e}_1^{2,2}{e}_2^{2,2}{e}_2^{2,1}\right],$$

$$\left[\psi \right]=\left[{\psi }_{1,2}{\psi }_{2,2}{\psi }_{1,1}{\psi }_{2,1}\right],\left[V\right]=\left[V^{1,2}{V}^{2,2}{V}^{1,1}{V}^{2,1}\right].$$

Denote by ${[·]}^T$, a corresponding column vector. Then, we have

$$d^c\phi =A{\left[\phi \right]}^T\left[e\right],d^c\omega =B{\left[\omega \right]}^T\left[V\right],{\delta }^c\omega =A^T{\left[\omega \right]}^T\left[x\right],{\delta }^c\psi =B^T{\left[\psi \right]}^T\left[e\right],$$

where

$$A=\left[\begin{array}{cccc}-1& \phantom{-}1& \phantom{-}0& \phantom{-}0\\ \phantom{-}1& -1& \phantom{-}0& \phantom{-}0\\ \phantom{-}1& \phantom{-}0& -1& \phantom{-}0\\ -1& \phantom{-}0& \phantom{-}1& \phantom{-}0\\ \phantom{-}0& \phantom{-}0& -1& \phantom{-}1\\ \phantom{-}0& \phantom{-}0& \phantom{-}1& -1\\ \phantom{-}0& \phantom{-}1& \phantom{-}0& -1\\ \phantom{-}0& -1& \phantom{-}0& \phantom{-}1\end{array}\right],B=\left[\begin{array}{cccccccc}-1& \phantom{-}0& -1& \phantom{-}0& \phantom{-}1& \phantom{-}0& \phantom{-}1& \phantom{-}0\\ \phantom{-}0& -1& \phantom{-}1& \phantom{-}0& \phantom{-}0& \phantom{-}1& -1& \phantom{-}0\\ \phantom{-}1& \phantom{-}0& \phantom{-}0& -1& -1& \phantom{-}0& \phantom{-}0& \phantom{-}1\\ \phantom{-}0& \phantom{-}1& \phantom{-}0& \phantom{-}1& \phantom{-}0& -1& \phantom{-}0& -1\end{array}\right]$$

and $A^T$, $B^T$ are the transpose of A, B.

The discrete Hodge–Dirac operator (26) on the combinatorial torus can be represented by the following block matrix

$$\left[\begin{array}{ccc}0& A^T& 0\\ A& 0& B^T\\ 0& B& 0\end{array}\right],$$

where 0 is a zero square matrix of the corresponding size.

In the same way, the discrete Laplacian on the combinatorial torus can be written as

$${\Delta }^c\phi =D{\left[\phi \right]}^T\left[x\right],{\Delta }^c\omega =D_1{\left[\omega \right]}^T\left[e\right],{\Delta }^c\psi =D{\left[\psi \right]}^T\left[V\right],$$

where

$$D=\left[\begin{array}{cccc}\phantom{-}4& -2& -2& \phantom{-}0\\ -2& \phantom{-}4& \phantom{-}0& -2\\ -2& \phantom{-}0& \phantom{-}4& -2\\ \phantom{-}0& -2& -2& \phantom{-}4\end{array}\right],D_1=\left[\begin{array}{cccccccc}\phantom{-}4& -2& \phantom{-}0& \phantom{-}0& -2& \phantom{-}0& \phantom{-}0& \phantom{-}0\\ -2& \phantom{-}4& \phantom{-}0& \phantom{-}0& \phantom{-}0& -2& \phantom{-}0& \phantom{-}0\\ \phantom{-}0& \phantom{-}0& \phantom{-}4& -2& \phantom{-}0& \phantom{-}0& -2& \phantom{-}0\\ \phantom{-}0& \phantom{-}0& -2& \phantom{-}4& \phantom{-}0& \phantom{-}0& \phantom{-}0& -2\\ -2& \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}4& -2& \phantom{-}0& \phantom{-}0\\ \phantom{-}0& -2& \phantom{-}0& \phantom{-}0& -2& \phantom{-}4& \phantom{-}0& \phantom{-}0\\ \phantom{-}0& \phantom{-}0& -2& \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}4& -2\\ \phantom{-}0& \phantom{-}0& \phantom{-}0& -2& \phantom{-}0& \phantom{-}0& -2& \phantom{-}4\end{array}\right].$$

Let us define analogues of the cohomology groups ${\mathcal{H}}^r\left(T\right)$ of the combinatorial torus $C\left(T\right)$. The quotient space of the linear space of closed r-forms

$$N_{d^c}^r\left(T\right)=\{\omega \in K^r\left(T\right):d^c\omega =0\}$$

modulo the subspace of exact r-forms

$$R_{d^c}^r\left(T\right)=\{\omega \in K^r\left(T\right):\exists \phi \in K^{r-1}\left(T\right)\omega =d^c\phi \}$$

is called the r-th cohomology group of $C\left(T\right)$, that is,

$${\mathcal{H}}^r\left(T\right)=N_{d^c}^r\left(T\right)/R_{d^c}^r\left(T\right).$$

Two closed r-forms ${\omega }^1$ and ${\omega }^2$ are cohomologous, ${\omega }^1\sim {\omega }^2$, if, and only if, they differ by an exact form, i.e.,

$${\omega }^1\sim {\omega }^2\iff {\omega }^1-{\omega }^2\in R_{d^c}^r\left(T\right).$$

An element of ${\mathcal{H}}^r\left(T\right)$ is, thus, an equivalence class $\left[\omega \right]$ of closed r-forms $\omega +d^c\phi$, defined by the equivalence relation ∼. These equivalence classes endow ${\mathcal{H}}^r\left(T\right)$ with a group structure.

Calculation of ${\mathcal{H}}^0\left(T\right)$. Since there are no $-1$-forms, a 0-form $\phi$ can never be exact, i.e., $R_{d^c}^0\left(T\right)=\left\{0\right\}$. If $\phi \in N_{d^c}^0\left(T\right)$, then $d^c\phi =0$. By (31), it follows immediately that ${\phi }_{1,1}={\phi }_{2,1}={\phi }_{1,2}={\phi }_{2,2}$. Hence,

$$\phi =c(x^{1,1}+x^{2,1}+x^{1,2}+x^{2,2}),$$

where $c\in \mathbb{R}$ is a constant. Thus, ${\mathcal{H}}^0\left(T\right)$ is isomorphic to the group generated by one independent generator which is isomorphic to $\mathbb{R}$, and we write ${\mathcal{H}}^0\left(T\right)\cong \mathbb{R}$.

Calculation of ${\mathcal{H}}^1\left(T\right)$. Let ${\omega }^1=\{u^1,v^1\}\in R_{d^c}^1\left(T\right)$. Then, ${\omega }^1=d^c\phi$ for some $\phi \in K^0\left(T\right)$. From (31), we have

$$\begin{array}{c}\hfill u_{1,1}^1={\phi }_{2,1}-{\phi }_{1,1},u_{1,2}^1={\phi }_{2,2}-{\phi }_{1,2},\\ \hfill u_{2,1}^1={\phi }_{1,1}-{\phi }_{2,1},u_{2,2}^1={\phi }_{1,2}-{\phi }_{2,2},\\ \hfill v_{1,1}^1={\phi }_{1,2}-{\phi }_{1,1},v_{2,1}^1={\phi }_{2,2}-{\phi }_{2,1},\\ \hfill v_{1,2}^1={\phi }_{1,1}-{\phi }_{1,2},v_{2,2}^1={\phi }_{2,1}-{\phi }_{2,2}.\end{array}$$

It follows that

$$u_{2,1}^1=-u_{1,1}^1,u_{1,2}^1=-u_{2,2}^1,v_{1,2}^1=-v_{1,1}^1,v_{2,1}^1=-v_{2,2}^1.$$

Hence, any form ${\omega }^1\in R_{d^c}^1\left(T\right)$ can be written as

$$\begin{array}{c}\hfill {\omega }^1=u_{1,1}^1{e}_1^{1,1}-u_{1,1}^1{e}_1^{2,1}-u_{2,2}^1{e}_1^{1,2}+u_{2,2}^1{e}_1^{2,2}\\ \hfill +v_{1,1}^1{e}_2^{1,1}-v_{1,1}^1{e}_2^{1,2}-v_{2,2}^1{e}_2^{2,1}+v_{2,2}^1{e}_2^{2,2}\\ \hfill =u_{1,1}^1(e_1^{1,1}-e_1^{2,1})+u_{2,2}^1(e_1^{2,2}-e_1^{1,2})+v_{1,1}^1(e_2^{1,1}-e_2^{1,2})+v_{2,2}^1(e_2^{2,2}-e_2^{2,1}).\end{array}$$

Let $\omega =\{u,v\}$ now be a closed 1-form, i.e., $\omega \in N_{d^c}^1\left(T\right)$ and $d^c\omega =0$. By (32), the requirement $d^c\omega =0$ means that

$$\begin{array}{c}\hfill u_{1,1}-u_{1,2}+v_{2,1}-v_{1,1}=0,\\ \hfill u_{2,1}-u_{2,2}-v_{2,1}+v_{1,1}=0,\\ \hfill u_{1,2}-u_{1,1}+v_{2,2}-v_{1,2}=0,\\ \hfill u_{2,2}-u_{2,1}+v_{1,2}-v_{2,2}=0.\end{array}$$

From this, we obtain

$$\begin{array}{c}\hfill u_{1,1}-u_{1,2}+u_{2,1}-u_{2,2}=0,\\ \hfill v_{1,1}-v_{2,1}+v_{1,2}-v_{2,2}=0.\end{array}$$

Hence, any form $\omega \in N_{d^c}^1\left(T\right)$ can be written as

$$\begin{array}{c}\hfill \omega =u_{1,1}{e}_1^{1,1}+(u_{1,2}-u_{1,1}+u_{2,2})e_1^{2,1}+u_{1,2}{e}_1^{1,2}+u_{2,2}{e}_1^{2,2}\\ \hfill +v_{1,1}{e}_2^{1,1}+(v_{1,1}+v_{1,2}-v_{2,2})e_2^{2,1}+v_{1,2}{e}_2^{1,2}+v_{2,2}{e}_2^{2,2}.\end{array}$$

This yields

$$\begin{array}{c}\hfill \omega =u_{1,1}(e_1^{1,1}-e_1^{2,1})+u_{1,2}(e_1^{2,1}+e_1^{1,2})+u_{2,2}(e_1^{2,2}+e_1^{2,1})\\ \hfill +v_{1,1}(e_2^{1,1}+e_2^{2,1})+v_{1,2}(e_2^{2,1}+e_2^{1,2})+v_{2,2}(e_2^{2,2}-e_2^{2,1})\\ \hfill =(u_{1,2}+u_{2,2})(e_1^{2,1}+e_1^{1,2})+(v_{1,2}+v_{1,1})(e_2^{2,1}+e_2^{1,2})+{\omega }^0,\end{array}$$

where

$${\omega }^0=u_{1,1}(e_1^{1,1}-e_1^{2,1})+u_{2,2}(e_1^{2,2}-e_1^{1,2})+v_{1,1}(e_2^{1,1}-e_2^{1,2})+v_{2,2}(e_2^{2,2}-e_2^{2,1})$$

and note that ${\omega }^0\in R_{d^c}^1\left(T\right)$. Hence,

$$\omega \sim (u_{1,2}+u_{2,2})(e_1^{2,1}+e_1^{1,2})+(v_{1,2}+v_{1,1})(e_2^{2,1}+e_2^{1,2})\in {\mathcal{H}}^1\left(T\right).$$

This means that ${\mathcal{H}}^1\left(T\right)$ has two independent generators. Thus, ${\mathcal{H}}^1\left(T\right)\cong {\mathbb{R}}^2$.

Calculation of ${\mathcal{H}}^2\left(T\right)$. A 2-form

$${\psi }^1={\psi }_{1,1}^1{V}^{1,1}+{\psi }_{2,1}^1{V}^{2,1}+{\psi }_{1,2}^1{V}^{1,2}+{\psi }_{2,2}^1{V}^{2,2}$$

is an element of $R_{d^c}^2\left(T\right)$ if ${\psi }^1=d^c\omega$ for some 1-form $\omega =\{u,v\}$. Using (32), ${\psi }^1=d^c\omega$ gives rise to

$$\begin{array}{c}\hfill {\psi }_{1,1}^1=u_{1,1}-u_{1,2}+v_{2,1}-v_{1,1},{\psi }_{2,1}^1=u_{2,1}-u_{2,2}-v_{2,1}+v_{1,1},\\ \hfill {\psi }_{1,2}^1=u_{1,2}-u_{1,1}+v_{2,2}-v_{1,2},{\psi }_{2,2}^1=u_{2,2}-u_{2,1}+v_{1,2}-v_{2,2}.\end{array}$$

Adding these equations, we obtain

$${\psi }_{1,1}^1+{\psi }_{2,1}^1+{\psi }_{1,2}^1+{\psi }_{2,2}^1=0.$$

Hence, any element ${\psi }^1\in R_{d^c}^2\left(T\right)$ can be written as

$${\psi }^1={\psi }_{1,1}^1{V}^{1,1}+{\psi }_{2,1}^1{V}^{2,1}+{\psi }_{1,2}^1{V}^{1,2}+(-{\psi }_{1,1}^1-{\psi }_{2,1}^1-{\psi }_{1,2}^1)V^{2,2}.$$

Since $d^c\psi =0$ for any 2-form $\psi$, $N_{d^c}^2\left(T\right)=K^2\left(T\right)$. Any element of $N_{d^c}^2\left(T\right)$ can be expressed as

$$\psi =({\psi }_{1,1}+{\psi }_{2,1}+{\psi }_{1,2}+{\psi }_{2,2})V^{2,2}+{\psi }^0,$$

where

$${\psi }^0={\psi }_{1,1}{V}^{1,1}+{\psi }_{2,1}{V}^{2,1}+{\psi }_{1,2}{V}^{1,2}+(-{\psi }_{1,1}-{\psi }_{2,1}-{\psi }_{1,2})V^{2,2}.$$

Thus,

$$\psi \sim ({\psi }_{1,1}+{\psi }_{2,1}+{\psi }_{1,2}+{\psi }_{2,2})V^{2,2},$$

since ${\psi }^0\in R_{d^c}^2\left(T\right)$. This means that ${\mathcal{H}}^2\left(T\right)$ has only one independent generator; so ${\mathcal{H}}^2\left(T\right)\cong \mathbb{R}$.

5. Conclusions

We proposed a discretization scheme based on the use of the differential forms calculus. This scheme was applied to the Hodge–Dirac operator in the two-dimensional case and the properties of the discrete Hodge–Dirac operator were discussed. A discrete version of the Hodge decomposition theorem was proved. The proposed discrete model was applied to the combinatorial torus and the cohomology groups were calculated. Moreover, it was also shown that the cohomology groups in the discrete case are exactly the same as those in the continuum one. Issues of convergence and numerical implementation remain to be studied in detail. We finish by remarking that the content of this work can be developed for the n-dimensional torus. These problems are the subject of current work in progress.