1. Introduction
Discrete Exterior Calculus is introduced in [6]
as a discrete version of Exterior Differential Calculus. Most mathematical objects in the continuous theory and the potential applications, have been explored actively with DEC. An application of interest in this work, is the numerical approximation of Partial Differential Equations (PDE). The attractive feature is that differential forms are the natural language for physical laws modeled by PDE.
At present, the method has proven successful. For instance, in [7] it is applied to solve Darcy flow and Poisson’s equation. As illustrated in [9], the method is natural for curved meshes. Therein, they show an application to the incompressible NavierStokes equations over surface simplicial meshes.
These numerical results are proof that the DEC method is a welcomed addition to PDE numerics. But, to our knowledge, theoretical results on convergence are scarce at best. There are works exploring numerical convergence such as the recent by Mohamed et al [8].
In [3], the authors show that in simple cases (e.g. flat geometry and regular meshes), the equations resulting from DEC are equivalent to classical numerical methods, finite difference or finite volume. It is well known that the latter involves a dual mesh in close analogy to the DEC method. This leads to the earlier paper of [1], where the box method, a finite volume type method, is analyzed. It is straightforward to join the dots and see that the box and DEC methods coincide for the Poisson equation and other simple self adjoint variants. Our purpose is to provide a proof of this fact while introducing the basics of the DEC method. Consequently a convergence result is established, namely the DEC method converges comparable to the Finite Element Method with linear elements. We do not aim for generality, thus our description of the DEC and BOX methods just provides enough details as to make the connection.
The outline is as follows.
In Section 2 we review the BOX method following closely [1]. Local aspects, boxwise, are stressed. In Section 3 we introduce cell complexes in DEC theory, and interpret the meshes in the FEM and BOX methods as the primal and dual cell complexes respectively. The discrete operators to approximate the Poisson equation are presented in Section 4. The proof of DEC equals BOX method is also carried out. Next, error estimates and first order convergence are established. A case for DEC is made, when classical methods are limited by geometric issues.
2. The box method for the Poisson equation in 2D
This section is a summary of [1].
2.1. Meshes
Let be a bounded polygonal domain in and its boundary. Let a Finte Element triangulation with vertices
To each vertex associate a region consisting of those triangles with as a vertex, as in Figure 1.
A box is a basic element of a dual mesh constructed as follows, see Figure 2. For each triangle select a point , later we shall restrict the mesh in order to select the circumcenter.
The point is connected with straight line segments (, , ) to the edge midpoints of , (, , ). We obtain a partition of in three subregions (, , ). Sometimes for simplicity, denote both the set and its length (area).
We are led to a dual mesh for made out of boxes. For each vertex there is a corresponding box , consisting of the union of subregions in which have as a corner, see Figure 3.
The straight line segments from the selected points to the edge midpoints are collected in a set of edges, .
We stress the following.
(2.1) Definition. For a box , let us define its boundary as the intersection of the topological boundary with the set of edges.
Notice that only for vertices in the open set the boundary is the topological one.
2.2. Function spaces
Let and denote the usual Sobolev spaces equipped with the norms
where
If the norm is equivalent to the energy norm
In we also define bilinear form
Following the linear Finite Element Method, let be the subspace of continuous piecewise linear polynomials associated with . Also consider
The subspace of , is the space of discontinuous piecewise constants with respect to the boxes.
Let the usual nodal basis for satisfying
If , then for some unique scalars we have
Easily, the map
given by
is bijective. Moreover, and take the same value at the vertices of .
Let be the subspace of of functions that are zero in . For , the energy norm is defined by
A basic result is the following,
(2.2) Lemma. Let , and . Then
where is the outward pointing normal.
Proof. (Lemma 3 in[1])
An important observation is that the proof is local. That is, it suffices to show for , , namely
The equality is shown triangle by triangle for such that .
Let us assume that the triangle , it is proven that
Here
where is a cyclic permutation of .
2.3. The Poisson equation
Consider the Dirichlet Boundary Value problem,
where .
Let be the weak solution of the Poisson equation. This solution satisfies
The FEM solution with linear elements is such that
In light of the lemma, let us introduce the bilinear form
Define by and the subspaces of and whose elements are zero on .
The box method consists on finding such that
The method leads to the linear system
where, by the proposition
(2.3) Remark. The stiffness matrix is identical to that arising from the Finite Element Method with continuous piecewise linear approximations. The corresponding linear systems differ only in the right hand side which are close in an average sense.
3. FEM and BOX meshes and cell complexes
In this section we describe the correspondence between FEM and BOX meshes and cell complexes.
3.1. Primal Complex
A dimensional simplex is the convex hull of geometrically independent points .
Let the dimensional face with the vertex removed.
(3.1) Definition. A simplicial complex , is a collection of simplices in such that

Every face of a simplex of is in .

The intersection of any two simplices of is either a face of each of them or it is empty.
The union of all simplices of treated as a subset of is called the underlying space of and is denoted by .
(3.2) Definition. A simplicial complex of dimension is called a manifoldlike simplicial complex if and only if is a manifold, with or without boundary. More precisely

All simplices of dimension with must be a face of some simplex of dimension in .

Each point on has a neighborhood homeomorphic to or dimensional halfspace.
An oriented simplex is denoted by . Its orientation corresponds to the orientation of the corner basis at . Namely, . It is equivalent to state that two orderings of the vertices have the same orientation if they differ one from one another by an even permutation.
Let be a dimensional face. We say that the induced orientation on this face is the same of if is even. If
is odd, it is the opposite.
Assume that two oriented dimensional simplices, , share a face of dimension . They have the same orientation if the induced orientation of the shared face induced by is the opposite to that induced by .
(3.3) Definition. A manifoldlike simplicial complex of dimension is called an oriented manifoldlike simplicial complex if adjacent simplices (i.e., those that share a common face) have the same orientation (orient the shared face oppositely) and simplices of dimensions and lower are oriented individually.
2D FEM interpretation:

The triangulation may be regarded as a simplicial complex of dimension with underlying space .

For two adjacent triangles, orientation corresponds to set outward normals on the intersecting side, opposite to each other.
3.2. Dual Complex
Given the triangulation , a point is selected in the closure of each triangle in the box method. In the context of DEC, a natural choice is the circumcenter.
(3.4) Definition. The circumcenter of a simplex is given by the center of the circumsphere. We will denote the circumcenter of a simplex by . If the circumcenter of a simplex lies in its interior we call it a wellcentered simplex. A simplicial complex all of whose simplices (of all dimensions) are wellcentered will be called a wellcentered simplicial complex.
Hereafter we shall consider triangulations corresponding to a wellcentered simplicial complex.
(3.5) Definition. The circumcentric subdivision of a wellcentered simplicial complex of dimension is denoted , and it is a simplicial complex with the same underlying space as and consisting of all simplices (each of which is called a subdivision simplex) of the form for (note that the index here is not dimension since it is a subscript). Here (i.e., is a proper face of for all ) and the are in .
Each subdivision simplex in a given simplex will be called a subdivision simplex of . Of these, a simplex () will be called a subdivision simplex of .
(3.6) Circumcentric subdivision in 2D: Consider a simplicial complex with vertices , and , i.e., the complex consists of a triangle , its edges and its vertices. Then consists of the following elements:

(the simplices of ): consists of the circumcenters , and , the midpoints of the edges , and and the circumcenter of the triangle, i.e., ,

(the simplices of ): consists of edges, the two halves of each edge and edges joining the circumcenter of the triangle to the vertices and midpoints of the edges,

(the simplices of ): consists of triangles, for instance
.
(3.7) Definition. Let be a wellcentered manifoldlike simplicial complex of dimension and let be one of its simplices. The circumcentric dual operator is given by
where the coefficient ensure the the orientation of is consistent with the orientation of the primal simplex, and the ambient volumeform.
The union of the interiors of the simplices in the definition of is the (circumcentric) duall cell also denoted by . We will call each simplex an elementary dual simplex of . This is an simplex in . The collection of dual cells is called the dual cell decomposition of . This is dual cell complex and will be denoted .
2D BOX interpretation:
If is a vertex in the triangulation of , then the dual cell of the simplex is the box .
4. DEC approximation of the Poisson equation
We shall use the FEMBOX notation when appropriate.
4.1. Chains, Forms, Integrals and Derivatives
We start with an oriented manifoldlike simplicial complex of dimension.
(4.1) Definition. The space of chains of simplices is given by
where is the number of simplices. is equipped with an Abelian group structure.
(4.2) Definition. The space of discrete forms (cochains) is the dual space of chains, namely
(4.3) Definition. The integral of a cochain over a chain is defined by
This bilinear pairing of cochain with chain is also denoted using bracket notation
The same construction can be applied to the dual cell complex. We denote the Abelian group of chains on the dual complex by
Also, the space of forms on the dual complex is denoted by
(4.4) Definition. The th boundary operator denoted by is the map
given by
where measn that is omitted.
(4.5) Definition. Let be a cochain. The th discrete exterior derivative of is the transpose of the st boundary operator
We have
The discrete exterior derivative operates on primal cochains. For dual cochains we have
The negative sign comes from the orientation on the dual mesh induced by the orientation on the primal mesh. See [2].
Two more facts. It is readily seen that
and the Stokes’ Theorem holds,
4.2. The Discrete Hodge star operator and the Poisson Equation
In the exterior calculus for smooth manifolds, the Hodge star, denoted , is an isomorphism between the space of forms and forms. It is apparent that a definition of a discrete Hodge star ought to involve this relation in correspondence with the fact that the dual of a simplex is a cell. This is accomplished with the bilinear pairing.
(4.6) Definition. The discrete Hodge star operator is a map
For a cochain it is defined as
for all simplices .
For all cochains , it follows that
Also the analogue for chains is valid. Namely,
The Poisson equation.
Let us denote by the measure of a set . Point sets are assigned unit measure. We restrict our discussion to the plane, and regard as a wellcentered oriented manifoldlike simplicial complex of dimension 2.
Assume that and are primal forms. Consequently, a DEC approximation of the Poisson equation will result on nodal values for .
Since is a dual form we may write the Poisson equation in DEC formalism as follows,
Let be a vertex in the triangulation . It is also a simplex . Applying the previous equation to its dual cell, we have
By the definition of Hodge star, the right hand side becomes
But , thus
Notice that the dual cell of is exactly the box ,
Hence, in box notation we have
For the left hand side we prove the following
(4.7) Theorem.
Proof.
Discrete derivatives are the transpose of boundary operators, thus
It is apparent that
By linearity of the boundary operator and the integral, it is enough to show the equality for a triangle , . Let be one of such triangles, or the primal simplex .
Let us discuss the orientation of the dual simplices and . For the former, consider the simplex . This is related to the area form , up to a sign determined by the relative orientation of and . Thus we have that
Thus we have that the correct orientation for the simplex is given by
Similarly,
and the orientation for the simplex is given by
We are led to compute in the simplex
By the definition of the dual cell operator
Applying the definition of the discrete Hodge star operator we obtain
Now
The result follows.
Consequently, in the dual cell we have
Assembling these block submatrices we obtain , the DEC solution of this system with zero boundary conditions.
4.3. Error estimates in energy norm
We show that the stiffness matrix of de BOX and DEC methods coincide.
Let us consider the triangle in Figure 4. Using the fact that
and
we obtain by elementary geometry that
(1) 
It is apparent that this is valid for every vertex in We are led to the following result.
(4.8) Theorem. Let be piecewise constant with respecto to . Then, there is a positive constant such that
Proof.
Multiplying the Poisson equation by and integrating by parts we obtain
Recall that the box solution satisfies
4.4. General triangulations
Let us assume that the triangle . In the box method, the line integral
depends only on and , the endpoints of the integration path. So the location of the distinguished point is arbitrary.
For the DEC method, we argue that the circumcenter need not be in the interior of .
Starting with the FEM local matrix computation, it is straightforward to see that
These expressions are valid regardless of the location of the circumcenter and can, indeed, take negative values. Such signs are essential for the calculations to work for general meshes. In [4] we tested the DEC method for bad quality meshes, in the FEM jargon. As expected, The numerical performance was of the first order in these not well centered meshes.
References
 [1] Bank, Randolph E., and Donald J. Rose. Some error estimates for the box method. SIAM Journal on Numerical Analysis 24.4 (1987): 777787.
 [2] M. Desbrun, E. Kanso, and Y. Tong. Discrete differential forms for computational modeling. In SIGGRAPH ’06: ACM SIGGRAPH 2006 Courses, pages 3954, New York, NY, USA, 2006. ACM.
 [3] Griebel, M., Rieger, C., Schier, A. (2017). Upwind Schemes for Scalar AdvectionDominated Problems in the Discrete Exterior Calculus. In D. Bothe, A. Reusken (eds.), Transport processes at Fluidic Interfaces, Advances in Mathematical Fluids Mechanics, DOI 10.1007/97833195660236, Chapter 6, 145175.
 [4] R. Herrera, S. Botello, H. Esqueda and M. A. Moreles, A geometric description of Discrete Exterior Calculus for general triangulations, Rev. int. metodos numer. calc. diseño ing. (Online first) URL https://www.scipedia.com/public/Herreraetal2018b
 [5] Hirani, A. N. (2003). Discrete exterior calculus (Doctoral dissertation, California Institute of Technology).
 [6] A. N. Hirani: “Discrete exterior calculus”. Diss. California Institute of Technology, 2003.
 [7] A. N. Hirani, K. B. Nakshatrala, J. H. Chaudhry: “Numerical method for Darcy flow derived using Discrete Exterior Calculus.” International Journal for Computational Methods in Engineering Science and Mechanics 16.3 (2015): 151169.
 [8] Mohamed, Mamdouh S., Anil N. Hirani, and Ravi Samtaney. ”Numerical convergence of discrete exterior calculus on arbitrary surface meshes.” International Journal for Computational Methods in Engineering Science and Mechanics 19.3 (2018): 194206.
 [9] Mohamed, Mamdouh S., Anil N. Hirani, and Ravi Samtaney. ”Discrete exterior calculus discretization of incompressible Navier?Stokes equations over surface simplicial meshes.” Journal of Computational Physics 312 (2016): 175191.