We present a local formulation for 2D Discrete Exterior Calculus (DEC) similar to that of the Finite Element Method (FEM), which allows a natural treatment of material heterogeneity (assigning material properties element by element). It also allows us to deduce, in a principled manner, anisotropic fluxes and the DEC discretization of the pullback of 1forms by the anisotropy tensor, i.e. we deduce the discrete action of the anisotropy tensor on primal 1forms. Due to the local formulation, the computational cost of DEC is similar to that of the Finite Element Method with Linear interpolating functions (FEML). The numerical DEC solutions to the anisotropic Poisson equation show numerical convergence, are very close to those of FEML on fine meshes and are slightly better than those of FEML on coarse meshes.
The theory of Discrete Exterior Calculus (DEC) is a relatively recent discretization [8] of the classical theory of Exterior Differential Calculus developed by E. Cartan [3], which is a fundamental tool in Differential Geometry and Topology. The aim of DEC is to solve partial differential equations preserving their geometrical and physical features as much as possible. There are only a few papers about implementions of DEC to solve certain PDEs, such as the Darcy flow and Poisson's equation [9], the NavierStokes equations [10], the simulation of elasticity, plasticity and failure of isotropic materials [5], some comparisons with the finite differences and finite volume methods on regular flat meshes [7], as well as applications in digital geometry processing [4].
In this paper, we describe a local formulation of DEC which is reminiscent of that of the Finite Element Method (FEM). Indeed, once the local systems of equations have been established, they can be assembled into a global linear system. This local formulation is also efficient and helpful in understanding various features of DEC that can otherwise remain unclear if one is dealing dealing with an entire mesh. Besides, we believe the local description to DEC will be accesible to a wide readership. We will, therefore, take a local approach when recalling all the objects required by 2D DEC [6]. Our main results are the following:
The paper is organized as follows. In Section 2, we describe the local versions of the discrete derivative operator, the dual mesh, the discrete Hodge star operator and the meaning of a continuous 1form on the plane. In Section 3, we deduce the discretization of the pullback operator on primal 1forms. In Section 4, we deduce the natural way of computing flux vectors in DEC (which turns out to be equivalent to the FEML result), as well as the anisotropic flux vectors. In Section 5, we present the local DEC formulation of the 2D anisotropic Poisson equation and compare it with the local system of FEML, proving that the diffusion terms are identical in both schemes, while the source terms are differentl due to a different areaweight assignment for the nodes. In Section 6, we reexamine the geometry of some of the local DEC quantities. In Section 7, we present and compare numerical examples of DEC and FEML approximate solutions to the 2D anisotropic Poisson equation on different domains with meshes of various resolutions. In Section 8, we summarize the contributions of this paper.
In this section, we will recall the basic operators of Dicrete Exterior Calculus restricting ourselves to a mesh made up of one simplex/triangle. The local results we derive in the paper can be assembled, just as in the Finite elemnt Method, due to the additivity of both differentiation and integration.
Let us consider a primal mesh made up of a single (positively oriented) triangle with vertices (Figure 1).
Figure 1. Triangle 
Such a mesh has



There is a well known boundary operator

(1) 
which describes the boundary of the triangle as an alternated sum of its ordered oriented edges , and .
Similarly, one can compute the boundary of each edge

If we consider
then the map (1), which sends the oriented triangle to a sum of its oriented edges, is represented by the matrix

while the map (2), which sends the oriented edges to sums of their oriented vertices, is represented by the matrix

It has been argued that the DEC discretization of the differential of a function is given by the transpose of the matrix of the boundary operator on edges [6,8]. More precisely, suppose we have a function discretized by its values at the vertices

Its discrete derivative, according to DEC, is

Indeed, such differences are rough approximations of the directional derivatives of along the oriented edges. For instance, is a rough approximation of the directional derivative of at in the direction of the vector , i.e.,

It is precisely in this sense that, according to DEC,
Let

The dual mesh of the primal mesh consisting of a single triangle is constructured as follows:
Figure 2. Circumcenter of the triangle 
Figure 3. Dual segment of the edge 
Figure 4. Dual quadrilateral of the vertex 
For the Poisson equation in 2D, we need two matrices: one relating original edges to dual edges, and another relating vertices to dual cells.

assigned to the segment ;

assigned to the edge .

assigned to the segment ;
In other words,


assigned to the quadrilateral ;

assigned to the quadrilateral ;

assigned to the quadrilateral .
In other words,

A differential 1form on can be thought of as a function whose arguments are a point in and a vector in , i.e.

and which is linear on the vector argument.
The dot product of allows us to see continuous 1forms in a more familiar form as vector fields in the following way: If is a vector field on , at each point we can consider the 1form

where . On the other hand, from a continuos 1form we can build a vector field as follows

where

Given a tensor (a linear transformation) and the vector field associated to a 1form as above, consider the vector field . For any vector , the value of the corresponding continuous 1form is given by the product

which is the value of the 1form called the pullback of by the tensor .
In order to produce a primal 1form from such a continuous 1form we need to inegrate it along the edges of the mesh (the de Rham map). In our local description, this produces a collection of 3 numbers associated to the 3 oriented edges of the triangle, i.e. for the oriented edge joining to we have

This is equivalent to calculating the line integral (circulation) of the corresponding vector field along the path , , parametrizing the edge

In Section 2, we dealt with the primal discretization of a continuous 1form. Now we wish to understand the pullback of a primal 1form. In order to do this, we will interpolate a primal 1form using the Whitney interpolation forms to produce a continuous 1form (Whitney map) to which we can apply the pullback operator by a tensor (linear transformation), and then intergrate along the edges (de Rham map).
The Whitney interpolation forms were introduced by Whitney in 1957 [12], and Bossavit [1] explained their relevance to “mixed methods” of finite elements. For the sake of simplicity, we will only define the Whitney 0 and 1forms.
First consider the Finite Element linear basis functions (or Whitney interpolation functions)

and their differentials

The Whitney interpolation 1forms are

These differential 1forms are such that, along the edge

i.e. or .
Now, suppose we have a primal 1form given on our triangle by the collection of numbers

We form a continuous 1form as follows

which is such that, for

Now we will calculate

for an arbitrary linear transformation

Let

and

As it turns out,

where

Anyhow, these quantities are areas of the parallelograms formed by one fixed edge of the original triangle and one edge transformed by .
Thus, we have that the induced transformation on primal 1forms is

(6) 
When the material is isotropic, i.e. is a constant multiple of the identity matrix, we get a constant multiple of the identity matrix above as discrete pullback operator.
In this section, we deduce the DEC formulae for the local flux, the local anisotropic flux and the local anisotropy operator for primal 1forms.
We wish to find a natural construction for the discrete flux (discrete gradient vector) of a discrete function. Recall from Vector Calculus that the directional derivative of a differentiable function at a point in the direction of is defined by

Thus, we have three Vector Calculus identities

As in Subsection 2.2, the rough approximations to directional derivatives of a function in the directions of the (oriented) edges are given as follows

Thus, if we want to find a discrete gradient vector of at the point , we need to solve the equations

If

then

Now, if we were to find a discrete gradient vector of at the point , we need to solve the equations

The vectors solving these equations is actually equal to . Indeed, consider

so that

(10) 
Thus, adding up (7) and (9) we get

(11) 
Subtracting (8) from (10) we get

(12) 
Since and are linearly independent and the two inner products in (11) and (12) vanish,

Analogously, the corresponding gradient vector of at the vertex is equal to . This means that the three approximate gradient vectors at the three vertices coincide, i.e. we have a unique discrete flux vector on the given element. Note that the discrete flux satisfies

This means that the primal 1form discretizing can be obtained by performing the dot products of the discrete flux vector with the vectors given by the triangle's oriented edges.
The local flux (gradient) of in FEML is given by

where and are the basis functions given in (3). Explicitly

where

is the area of the triangle, so that the FEML flux is given by

and we can see that it coincides with the DEC flux.
We will now discuss how to discretize anisotropy in 2D DEC. Let denote the symmetric anisotropy tensor

and recall the anisotropic Poisson equation

First recall that, in Exterior Differential Calculus, for any ,

where is the pullback of by .
Since we already have a discrete candidate for on the given element, the product of the matrix and gives us a candidate for the anisptropic flux vector on such an element

In order to produce a primal 1form from , all we have to do is assume that is constant on the given triangle and take dot products with the edges of the triangle, i.e.

Nevertheless, in order to better understand the structure of the local system (as in FEML), it is desirable to have a factorization of the result as a matrix product

where the matrix is a real matrix depending on the geometry of the triangle and the matrix , and such that it is a multiple of the identity matrix when the domain is isotropic. In order to do this, we will use the discretization of the pullback opertator of Exterior Differential Calculus for arbitrary 1forms from Section 3. Thus, using (6), is the following matrix where we have set , i.e.

where

Consider the Figure 5, where denotes the anticlockwise rotation

Figure 5. Geometric interpretation of the entries of the anisotropy tensor discretization 
We have

where is the area of the red triangle. Thus, the numbers are quotients of areas of transformed triangles divided by the area of the original triangle.
In this section, we describe the local DEC discretization of the 2D anisotropic Poisson equation and compare it to that of FEML.
The anisotropic Poisson equation reads as follows

where and are two functions on a certain domain in and is the anisotropy tensor. In terms of the exterior derivative and the Hodge star operator it reads as follows

where and . Following the discretization of the discretized divergence operator [6], the corresponding local DEC discretization of the anisotropic Poisson equation is

or equivalently

(13) 
In order to simplify the notation, consider the lengths and areas defined in Figure 6.
Figure 6. Triangle 
Now, the discretized equation (13) looks as follows:

The diffusive term matrix is actually symmetric (see Subsection 5.3.1).
The diffusive elemental matrix in FEM (frequently called “stiffness matrix”) on an element is given by

where is the matrix representing the anisotropic diffusion tensor, and the matrix is given explicitly by

Since the matrix is constant on an element of the mesh, the integral is easy to compute. Thus, the difussive matrix for a linear triangular element (FEML) is given by

Now, let us consider the first diagonal entry of the local FEML anisotropic Poisson diffusive matrix ,

where is the anticlockwise rotation,

In this notation, the diffusive term in local FEML is given as follows

For the sake of brevity, we are only going to compare the entries of the first row and first column of each formulation. Consider the various lengths, areas and angles labeled in Figures 6 and 7.
Figure 7. Circumscribed triangle. 
We have the following identities:

For instance, we claim that

Indeed,

Thus,

All we have to do now is show that

Note that, since is orthogonal to , must be parallel to . Thus,

(14) 
Now we are going to express in terms of . Let us consider

where are coefficients to be determined. Taking inner products with and we get the two equations

Solving for and

Substituting all the relevant quantities in (14) we have, for instance, that the coefficient of is

and similarly for the coefficient of . The calculations for the remaining entries are similar.
Thus, the local DEC and FEML diffusive terms of the 2D anisotropic Poisson equation coincide.
As already observed in [6], the right hand sides of the local DEC and FEML systems are different

While FEML uses a barycentric subdivision to calculate the areas associated to each node/vertex, DEC uses a circumcentric subdivision. Eventually, this leads the DEC discretization to a better approximation of the solution (on coarse meshes).
The numbers appearing in the local DEC matrices can be expressed both in terms of determinants and in terms of trigonometric functions. More precisely,

These expressions are valid regardless of the location of the circumcenter and can, indeed, take negative values. The angles that are measured in the scheme can be negative as in the obtuse triangle of Figure 8.
Figure 8. Negative (exterior) angles measured in an obtuse triangle 
and some quantities can even be zero. For instance, if

then

In order to understand how local DEC assigns area weights to vertices differently from FEML, let us consider the obtuse triangle shown in Figure 8. Let be the middle points of the segments respectively. As shown in Figure 9, the triangle lies completely outside of the triangle . Geometrically, this implies that its area must be assigned a negative sign, which is confirmed by the determinant formulas of Subsection 6.1. On the other hand, the triangle will have positive area. Thus, their sum gives us the area in Figure 9.
Figure 9. Area weight assigned to 
The area is computed similarly, where the triangle is assigned negative area (Figure 10).
Figure 10. Area weight assigned to 
Note that for , the two triangles and both have positive areas (Figure 11).
Figure 11. Area weight assigned to 
In this section, we present three numerical examples in order to illustrate the performance of DEC resulting from the local formulation and its implementation. In all cases, we solve the anisotropic Poisson equation. The FEML methodology that we have used in the comparison can be consulted [2,11,13].
This example is intended to highlight how Local DEC deals effectively with heterogeneous materials. Consider the region in the plane given in Figure 12.
Figure 12. Square and inner circle with different conditions 
The meshes used in this example are shown in Figure 13 and vary from coarse to very fine.
(a)  (b)  (c) 
(d)  (e)  (f) 
Figure 13. Six of the meshes used in the first example 
The numerical results for the maximum temperature value are exemplified in Table 1.
Mesh  nodes  elements  Max. Temp. Value  Max. Flux Magnitude  
DEC  FEML  DEC  FEML  
Figure 13(a)  49  80  5.51836  5.53345  13.837  13.453 
Figure 13(b)  98  162  5.65826  5.66648  14.137  14.024 
Figure 13(c)  258  466  5.70585  5.71709  14.858  14.770 
Figure 13(d)  1,010  1,914  5.72103  5.72280  15.008  15.006 
Figure 13(e)  3,813  7,424  5.72725  5.72725  15.229  15.228 
Figure 13(f)  13,911  27,420  5.72821  5.72826  15.342  15.337 
50,950  101,098  5.72841  5.72842  15.395  15.396  
135,519  269,700  5.72845  5.72845  15.420  15.417  
298,299  594,596  5.72848  5.72848  15.430  15.429  
600,594  1,198,330  5.72848  5.72848  15.433  15.433  
1,175,238  2,346,474  5.72849  5.72849  15.43724  15.43724 
The temperature and fluxmagnitude distribution fields are shown in Figure 14.
(a) Contour fill of temperatures  (b) Contour fill of flux vectors on Elems 
Figure 14. Temperature and fluxmagnitude distribution fields of the first example 
Figure 15 shows the graphs of the temperature and the fluxmagnitude along a horizontal line crossing the inner circle for the first two meshes.
(a)  (b) 
(c)  (d) 
(e)  (f) 
Figure 15. Temperature and Flux magnitude graphs of the first example along a crosssection of the domain for different meshes 
Table 2 shows some global error metrics for different meshes. Figure 16 shows the error evolution in norm for this example.
Mesh  Nodes  norm  
1  49  1.0537e02  1.4438e01 
2  98  4.2447e03  4.8558e02 
3  258  6.9781e04  3.0390e03 
4  1,010  8.8386e05  1.4877e04 
5  3,813  1.0736e05  7.7369e06 
6  13,911  1.4422e06  4.9791e07 
7  50,950  1.7582e07  2.9608e08 
8  135,518  3.2621e08  2.9233e09 
9  298,299  7.3566e09  3.3610e10 
10  603,440  1.8577e09  4.9496e11 
Figure 16. DEC error in the first example 
Let us solve the Poisson equation in a circle of radius one centered at the origin under the following conditions (Figure 17):
Figure 17. Disk of radius one 
The meshes used in this example are shown in Figure 18 and vary from very coarse to very fine.
(a)  (b)  (c) 
(d)  (e)  (f) 
Figure 18. First six meshes used for unit disk 
The numerical results for the maximum temperature value () are exemplified in Table 3 where a comparison with the Finite Element Method with linear interpolation functions (FEML) is also shown.
Mesh  Nodes  Elements  Temp. Value at  Flux Magnitude at  
DEC  FEML  DEC  FEML  
Figure 18(a)  17  20  10.20014  10.19002  0.42133  0.43865 
Figure 18(b)  41  56  10.20007  10.19678  0.48544  0.49387 
Figure 18(c)  201  344  10.20012  10.20158  0.52470  0.52428 
Figure 18(d)  713  1304  10.20000  10.19969  0.54143  0.54224 
Figure 18(e)  2455  4660  10.20000  10.19990  0.54971  0.55138 
Figure 18(f)  8180  15862  10.20000  10.20002  0.55326  0.55409 
20016  39198  10.20000  10.19999  0.55470  0.55520  
42306  83362  10.20000  10.20000  0.55540  0.55572 
The temperature distribution and Flux magnitude fields for the finest mesh are shown in Figure 19.
(a) Contour Fill of Temperatures  (b) Contour Fill of Flux vectors on Elems 
Figure 19. Temperature distribution and Flux magnitude fields for the finest mesh of the second example 
Figures 20(a), (b) and (c) show the graphs of the temperature and flux magnitude values along a diameter of the circle for the different meshes of Figures 18(a), (b) and (c), respectively.
(a)  (b) 
(c)  (d) 
(e)  (f) 
Figure 20. Temperature and Flux magnitude graphs of the second example along a diameter of the circle for different meshes: mesh in Figure Figure 18(a), aTemperature, bFlux; mesh in Figure 18(b), cTemperature, dFlux; mesh in Figure 18(c), eTemperature, fFlux 
Table 4 shows some global error metrics for different meshes. Figure 21 shows the error evolution in norm for this example.
Mesh  Nodes  norm  
1  17  1.5818e04  2.3555e06 
2  51  2.5395e05  1.5639e07 
3  201  3.3643e06  1.0517e08 
4  713  5.1563e07  8.3543e10 
5  2,455  8.9235e08  7.6073e11 
6  8,180  3.1731e08  2.9858e11 
7  20,016  2.0217e08  2.6580e11 
8  42,306  1.4421e08  2.7062e11 
9  82,722  9.8533e09  2.6164e11 
10  156,274  7.1352e09  2.5954e11 
11  420,013  4.4277e09  2.6151e11 
12  935,016  2.9635e09  2.6003e11 
Figure 21. DEC error evolution in the second example 
Let us solve the Poisson equation in a circle of radius on the following domain (Figure 22) with various material properties. The geometry of the domain is defined by segments of ellipses passing through the given points which have centers at the origin .
Figure 22. Egglike domain with different materials 
Point  Point  
a  5  0  A  0  4 
b  4  0  B  0  3 
c  3  0  C  0  2 
d  1  0  D  0  1 
e  1  0  E  0  1 
f  6  0  F  0  2 
g  7  0  G  0  3 
h  8  0  H  0  4 
Figure 23. Dirichlet condition 
Angle  
Domain mat1  5  25  30  15 
Domain mat2  25  5  0  5 
Domain mat3  50  12  45  5 
Domain mat4  10  35  0  5 
The meshes used in this example are shown in Figure 24.
(a)  (b)  
(c)  (d)  
Figure 24. Meshes for layered egglike figure 
The numerical results for the maximum temperature value () are exemplified in Table 5 where a comparison with the Finite Element Method with linear interpolation functions (FEML) is also shown.
Mesh  Nodes  Elements  Max. Temp. Value  Max. Flux Magnitude  
DEC  FEML  DEC  FEML  
Figure 24(a)  342  616  2.79221  2.79854  18.41066  18.40573 
Figure 24(b)  1,259  2,384  2.83929  2.84727  18.93838  18.91532 
Figure 24(c)  4,467  8,668  2.85608  2.85717  19.13297  19.13193 
Figure 24(d)  14,250  28,506  2.85994  2.86056  19.20982  19.20909 
20,493  40,316  2.86120  2.86177  19.23120  19.23457  
60,380  119,418  2.86219  2.86231  19.26655  19.26628  
142,702  283,162  2.86249  2.86256  19.28045  19.28028  
291,363  579,360  2.86263  2.86267  19.28727  19.28755  
495,607  986,724  2.86275  2.86269  19.29057  19.29081  
1,064,447  2,122,160  2.86272  2.86273  19.29385  19.29389  
2,106,077  4,202,536  2.86274  2.86274  19.29618  19.29615  
4,031,557  8,049,644  2.86275  2.86275  19.29763  19.29765 
The temperature distribution and Flux magnitude fields for the finest mesh are shown in Figure 25.
(a) Contour fill of temperatures  (b) Contour till of flux vectors on elems 
Figure 25. Temperature distribution and Flux magnitude fields for the finest mesh of the third example 
Figure 26 shows the graphs of the temperature and flux magnitude values along a diameter of the circle for different meshes of Figure 24.
(a)  (b) 
(c)  (d) 
(e)  (f) 
Figure 26. Temperature and Flux magnitude graphs of the third example along a crosssection of the domain
for different meshes: Mesh in Figure 24(a), aTemperature, bFlux; Mesh in Figure 24(b), cTemperature, dFlux; Mesh in Figure 24(c), eTemperature, fFlux 
Table 6 shows some global error metrics for different meshes. Figure 27 shows the error evolution in norm for this example.
Mesh  Nodes  norm  
1  342  9.3638e04  2.7786e02 
2  1,259  1.5173e04  3.2731e03 
3  4,467  2.2110e05  2.2937e04 
4  14,250  3.7233e06  1.9151e05 
5  20,492  2.2311e06  9.5930e06 
6  60,380  4.3769e07  8.7330e07 
7  142,702  1.1664e07  1.3926e07 
8  291,369  3.9764e08  3.3314e08 
9  497,378  1.6680e08  1.0275e08 
10  1,067,171  3.9594e09  1.2190e09 
11  2,106,248  9.6949e10  1.4415e10 
Figure 27. DEC evolution in the third example 
Remark. As can be seen from the previous examples, DEC behaves well on coarse meshes. As expected, the results of DEC and FEML are identical for fine meshes. We would also like to point out the the computational costs of DEC and FEML are very similar since the local matrices are very similar in profile, or even identical in some cases.
DEC is a relatively recent discretization scheme for PDE's which takes into account the geometric and analytic features of the operators and the domains involved. The main contributions of this paper are the following:
On the other hand we would like to point the following features:
Our future work will include the DEC discretization of convective terms, and DEC on 2dimensional simplicial surfaces in 3D. Preliminary results on both problems are promising and competitive with FEML.
The second named author was partially supported by a CONACyT grant, and would like to thank the International Centre for Numerical Methods in Engineering (CIMNE) and the University of Swansea for their hospitality. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan X Pascal GPU used for this research.
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Published on 19/06/20
Accepted on 24/05/20
Submitted on 22/04/20
Volume 36, Issue 2, 2020
DOI: 10.23967/j.rimni.2020.05.003
Licence: CC BYNCSA license