2D discrete Hodge-Dirac operator on the torus

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.


Introduction
The choice of technique to approximate the solution of partial differential equations depends on a discretisation scheme. The interest for discrete models which preserve a geometric structure of continuum counterparts has grown in the computing community [1,2,3,10,12]. It is known that a geometric discretisation scheme would be ideal if this had the same properties as the continuum. However, there are difficulties usually in definitions of discrete counterparts of the Hodge star and the wedge product between forms. Various approaches with geometric discretisation have been proposed in the literature. See, for example, [4,5,8,9,11,19,20,22]. Recently, in [6], it was described a quite general framework of discrete calculus based on a new type of discrete geometry called script geometry. The proposed approach in this article was introduced by Dezin [7] and later developed in the author's previous papers [13,14,15,16,17,18].
Our purpose is to develop a satisfactory discrete model of de Rham-Hodge theory on manifolds which are homeomorphic to the torus. We consider a chain complex as a combinatorial model of R 2 . When we add to this discrete analogues of the exterior derivative, the Hodge star operator, and the exterior product acting on cochains, we have all of the basic ingredients for the calculus of discrete counterparts of differential forms. We show that discrete analogues of the operators (1.1) and (1.2) have properties like those in the continual case. We formulate and prove a discrete version of the Hodge decomposition theorem. We give an example illustrating 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 close to the construction in Section 5 of [6]. Matrix forms of these discrete operators on the torus are the same in both of case. The difference between our approach and one in [6] is in the definitions of discrete Hodge-Dirac and Laplace operators. In [6], the definitions are given in terms of the exterior derivative and boundary operators. Meanwhile, as in the continual theory, we define these operators in terms of the exterior derivative and its adjoint.

Discrete model
In this section, we briefly review the construction of a discrete exterior calculus framework, which was initiated in [7] and developed in e.g., [14,15]. The starting point for consideration is a two-dimensional chain complex (a combinatorial model of R 2 ). Let the sets {x k } and {e k }, k ∈ Z, be the generators of free abelian groups of zero-dimensional and one-dimensional chains of the onedimensional complex C = C 0 ⊕ C 1 . The free abelian group is understood as the direct sum of infinity cyclic groups generated by {x k }, {e k }. The boundary operator ∂ : C 1 → C 0 is the homomorphism defined by ∂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(2) = C ⊗ C be a combinatorial model of the two-dimensional Euclidean space R 2 . The twodimensional complex C(2) = C 0 (2) ⊕ C 1 (2) ⊕ C 2 (2) contains the free abelian groups of r-chains, r = 0, 1, 2, generated by the basic elements where k, s ∈ Z. It is convenient to introduce the shift operators τ, σ in the set of indices by The boundary operator ∂ : C r (2) → C r−1 (2) is given by The definition (2.2) is extended to arbitrary chains by linearity. Let K(2) be a complex of cochains with real coefficients. The cochain complex K(2) with a coboundary operator defined in it is the dual object to the chain complex C(2). It has a similar structure to C(2) and consists of cochains of dimension 0, 1 and 2. Then K(2) can be expressed by where K r (2) 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(2) is a discrete analogue of the grade algebra of differential forms Λ(R 2 ). Denote by {x k,s }, {e k,s 1 , e k,s 2 } and {V k,s } the basis elements of K 0 (2), K 1 (2) and K 2 (2) respectively. The pairing is defined with the basis elements of C(2) by the rule where δ i k is the Kronecker delta. The operation (2.3) is linearly extended to arbitrary chains and cochains. Let r ω ∈ K r (2), then we have where 0 ω k,s , ω 1 k,s , ω 2 k,s and 2 ω k,s are real numbers for any k, s ∈ Z. The coboundary operator d c : where a r+1 ∈ C r+1 (2). The operator d c is an analog of the exterior differential. From the above it follows that d c 2 ω = 0 and d c d c r ω = 0 for any r = 0, 1.

By (2.2) and (2.3) we can calculate
where ∆ k and ∆ s are the difference operators defined by Here r ω k,s is a component of r ω ∈ K r (2) and τ is given by (2.1). Note that 1 ω k,s = {ω 1 k,s , ω 2 k,s }. We now consider a multiplication of discrete forms which is an analogue of the exterior multiplication for differential forms. Denote by ∪ this multiplication. For the basis elements of K(2) the ∪-multiplication is defined as follows 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.
This was proved by Dezin [7]. Define the operation * : 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 This chain imitates a rectangle. Using (2.2) we have Then for forms r ϕ, r ω ∈ K r (2) of the same degree r the inner product over the set V is defined by the rule (2.14) For forms of different degrees the product (2.14) is set equal to zero. From (2.11) and (2.3) we have is the operator formally adjoint of d c .
Proof. By Definitions (2.5), (2.14) and Formula (2.9) we obtain The operator δ c : K r+1 (2) → K r (2) given by (2.16) is a discrete analogue of the codifferential δ. For the 0-form 0 ω ∈ K 0 (2) we have δ c 0 ω = 0. It is obvious from (2.16) that δ c δ c r ω = 0 for any r = 1, 2. Using (2.6)-(2.8), (2.10) and (2.16) we can calculate In the particular case r = 0, the equality (2.15) can be expressed as The similar equality holds in the case r = 1. It should be noted that the relation (2.15) includes not only the forms with the components r ϕ k,s and r+1 ω k,s , where the subscripts k, s would run only over the values from (2.12), but also the components For r-forms satisfying conditions (2.19) the inner product (2.14) generates the finite-dimensional Hilbert spaces H r (V ). Now we consider the operators

Discrete Hodge decomposition
In this section, we discuss the properties of discrete analogues of the Laplacian and Hodge-Dirac operators using the concepts of 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 This is a discrete analogue of the Laplacian (1.2). Proof. By Proposition 2.3 one has where · denotes the norm and Proof. By (2.20) it is obvious.

Consider the spaces
: ∆ c ψ = 0}. By analogy with the continuum case the discrete r-form ω is called closed if d c ω = 0 and exact if ω ∈ R r d c .
Proposition 3.4. For each r = 0, 1, 2 we have the direct sum decomposition where N 1 δ c and N 1 d c are the orthogonal complements of the corresponding spaces. For any ω ∈ R 1 δ c we have ω = δ c ψ and Similar reasonings apply to the spaces H 0 (V ) and H 2 (V ). Thus we have The Proposition 3.4 is a discrete version of the well-known Hodge decomposition theorem (see, e.g., [21]).
Let Ω be an inhomogeneous discrete form, i.e. Ω = 0 ω + 1 ω + 2 ω, where r ω ∈ H r (V ). The inner product (2.14) can be extend to an inner product of inhomogeneous discrete forms by the rule The inner product (3.2) generates the finite-dimensional Hilbert space H(V ). It is true that

By Proposition 3.4 the following holds
where Proof. The equation (d c + δ c )Ω = 0 can be written as Let Ω ∈ N ∆ c . This means that ∆ c if and only if δ c 1 ω = 0 and d c 1 ω = 0. It is easy to show that for any 0 ω ∈ H 0 (V ) and and thus d c 0 ω + δ c 2 ω = 0. The converse is trivially true.

Combinatorial torus
In this section, we consider an example of a 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 consider the partitioning of the plane R 2 by the straight lines x = k and y = s, where k, s ∈ Z. Denote by V k,s an open square bounded by the lines x = k, x = τ k, y = s and y = τ s, where τ is given by (2.1). Denote the vertices of V k,s by x k,s , x τ k,s , x k,τ s , x τ k,τ s . Let e 1 k,s and e 2 k,s be the open intervals (x k,s , x τ k,s ) and (x k,s , x k,τ s ), respectively. Such introduced geometric objects can be identified with the combinatorial objects we have considered in the previous sections. We identify the collection V k,s with V given by (2.12) and let N = M = 2. In this case the conditions (2.19) take the form 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 we obtain the geometric object which is homomorphic to the torus (see Figure 1). Denote by C(T ) the complex C(2) which corresponds to the introduced geometric object. We call C(T ) a combinatorial torus. It is clear that by (4.2) the conditions (4.1) hold for any r-form on the combinatorial torus. The Hilbert space H(V ) considered in the previous sections can be regarded as a space of cochains of the complex K(T ) dual to C(T ). Let now consider the forms ϕ ∈ K 0 (T ), ω ∈ K 1 (T ) and ψ ∈ K 2 (T ), that is By x 1,2
In the same way the discrete Laplacian on the combinatorial torus can be written as Let us define analogues of the cohomology groups H r (T ) of the combinatorial torus C(T ). The quotient space of the linear space of closed r-forms modulo the subspace of exact r-forms is called the r-th cohomology group of C(T ), that is, Two closed r-forms ω 1 and ω 2 are cohomologous, ω 1 ∼ ω 2 , if and only if they differ by an exact form, i.e., An element of H r (T ) is thus an equivalence class [ω] of closed r-forms ω + d c ϕ, defined by the equivalence relation ∼. These equivalence classes endow H r (T ) with a group structure.
Thus the cohomology groups are exactly the same as in the continuum case.