A geometric description of Discrete Exterior Calculus for general triangulations

<!-- metadata commented in wiki content
==A geometric description of Discrete Exterior Calculus for general triangulations==

'''Humberto EsquedaRafael Herrera Salvador BotelloMiguel Angel Moreles '''

{|
|-
|

Centro de Investigación en Matemáticas

Jalisco s/n, Valenciana

Guanajuato, GTO 36240,Mexico

''email: ''esqueda,rherrera,botello,[mailto:moreles@cimat.mx moreles@cimat.mx] 
|}
-->

==Abstract==

We revisit the theory of Discrete Exterior Calculus (DEC) in 2D for general triangulations, relying only on Vector Calculus and Matrix Algebra.  We present DEC numerical solutions of the Poisson equation and compare them against those  found using the Finite Element Method with linear elements (FEML).

==1 Introduction==

The purpose of this paper is to introduce the theory of Discrete Exterior Calculus (DEC)  to the widest possible audience and, therefore, we will  rely mainly on Vector Calculus and Matrix Algebra. Discrete Exterior Calculus is a relatively new method for solving partial differential equations <span id='citeF-8'></span>[[#cite-8|[8]]] based on the idea of discretizing the mathematical theory of Exterior Differential Calculus, a theory that goes back to E. Cartan <span id='citeF-3'></span>[[#cite-3|[3]]] and is fundamental  in the areas of Differential Geometry and Differential Topology. Although Exterior Differential Calculus is an abstract mathematical theory, it has been introduced in various fields such as in digital geometry processing <span id='citeF-4'></span>[[#cite-4|[4]]],  numerical schemes for partial differential equations <span id='citeF-8'></span><span id='citeF-1'></span>[[#cite-8|[8,1]]], etc.

In his PhD thesis <span id='citeF-8'></span>[[#cite-8|[8]]], Hirani laid down the fundamental concepts  of Discrete Exterior Calculus (DEC), using discrete combinatorial and geometric  operations on simplicial complexes (in any dimension), proposing discrete equivalents for differential forms, vector fields, differential and geometric operators, etc. Perhaps the first numerical application of DEC to PDE was given in <span id='citeF-9'></span>[[#cite-9|[9]]] in order to solve Darcy flow and Poisson's equation.  In <span id='citeF-7'></span>[[#cite-7|[7]]], the authors develop a modification of DEC and show that in simple cases (e.g. flat geometry and regular meshes), the equations resulting from DEC are equivalent to classical numerical schemes such as finite difference or finite volume discretizations. In <span id='citeF-11'></span>[[#cite-11|[11]]], the authors used DEC to solve the Navier-Stokes equations and,  in <span id='citeF-5'></span>[[#cite-5|[5]]] DEC was used with a discrete lattice model to simulate  elasticity, plasticity and failure of isotropic materials.

In this expository paper, we review the various operators of Exterior Differential Calculus in 2D in terms of ordinary vector calculus, and introduce only the geometrical ideas that are essential  to the formulation.  Among those ideas is that of duality between the differentiation operator (on vector fields) and the boundary operator (on the domain) contained in Green's theorem.  This duality is one of the key ideas of the method, which justifies taking the discretized derivative matrix as the transpose of the boundary operator matrix on the given mesh.   Another important ingredient is the Hodge star operator, which is hidden in the notation of Vector Calculus. In order to show the necessity of the Hodge star operator,  we carry out some simple calculations. In particular, we will introduce the notion of wedge product of vectors which, roughly speaking, helps us assign algebraic objects to parallelograms and carry out algebraic manipulations with them. We present  DEC in the simplest terms possible using easy examples. We also review the formulation of DEC for arbitrary meshes, which was first considered in <span id='citeF-10'></span>[[#cite-10|[10]]]. Performance of the method is tested on the Poisson equation and compared with the Finite Element Method with linear elements (FEML).

The paper is organized as follows.  In Section [[#2 2D Exterior Differential Calculus as Vector Calculus|2]], we introduce the wedge product of vectors and the ''geometric'' Hodge star operator, and rewrite Green's theorem appropriately in order to display the duality between the differentiation and the boundary operators. In Section [[#3 Discrete Exterior Calculus|3]], we present the operators of DEC (mesh, dual mesh, discrete derivation, discrete Hodge star operator), showing simple examples throughout. In Section [[#4 DEC for general triangulations|4]], we present  the formulation of DEC on arbitrary triangulations. In Section [[#5 Numerical examples|5]], we present the numerical solution of a Poisson equation with DEC and FEML, in order to compare their performance. In Section [[#6 Conclusions|6]], we present our conclusions.

==2 2D Exterior Differential Calculus as Vector Calculus==

In this section we introduce two geometric operators (the wedge product and the Hodge star) and explain how to use them together with the gradient operator in order to obtain the Laplacian.

===2.1 <span id='lb-2.1'></span>Wedge product for vectors in R²===

Let <math display="inline">a,b</math> be vectors in <math display="inline">\mathbb{R}^2</math>. We can assign to these vector the parallelogram they span, and to such a parallelogram its area. The latter is equal to the determinant of the transformation matrix  sending <math display="inline">e_1,e_2</math> to <math display="inline">a,b</math> respectively. In Exterior Calculus, such a parallelogram is regarded as an algebraic object <math display="inline">a\wedge b</math>, a bivector, and the set of bivectors is equipped with a vector space structure to form the one-dimensional vector space

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\textstyle \wedge ^2\mathbb{R}^2</math>
|}
|}

Thus, we have the following spaces

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbb{R}, \quad \quad \mathbb{R}^2, \quad \quad  \wedge^2\mathbb{R}^2</math>
|}
|}

of scalars, vectors and bivectors, respectively. For instance, the square formed by <math display="inline">e_1</math> and <math display="inline">e_2</math> is represented by the symbol

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>e_1\wedge e_2</math>
|}
|}

which is read as "<math display="inline">e_1</math> wedge <math display="inline">e_2</math>" (see Figure [[#lb-2.1|2.1]])

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

<span style="text-align: center; font-size: 75%;">'' [[Image:Review_239195285364_6178_text-Square00-eps-converted-to.png|180px|Wedge product of two vectors.]] ''</span></div>
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 2.1:''' Wedge product of two vectors.</span></div>


In <math display="inline">\mathbb{R}^2</math>, this represents an “element” of unit area.

Note that if we list the vectors in the opposite order, we have a different orientation and, therefore, the algebraic objects must satisfy <math display="inline">e_2\wedge e_1=-e_1\wedge e_2</math> (see Figure [[#lb-2.1|2.2]])

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_6648_text-Square01-eps-converted-to.png|250px|Change of orientation of the parallelogram implies anticommutativity in the wedge product of two vectors.]]''</span></div>
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 2.2:''' Change of orientation of the parallelogram implies anticommutativity in the wedge product of two vectors.</span></div>



More generally, given two vectors <math display="inline">a,b\in \mathbb{R}^2</math>, their wedge product <math display="inline">a\wedge b</math> looks as follows (see Figure [[#lb-2.1|2.3]])

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_4688_text-Parallelogram-eps-converted-to.png|220px|The wedge product of the vectors <math>a</math> and <math>b</math>.]]''</span></div>
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 2.3:''' The wedge product of the vectors <math>a</math> and <math>b</math>.</span></div>



The properties of the wedge product are

* it is anticommutative: <math display="inline">a\wedge b = - b\wedge a</math> (see Figure [[#lb-2.1|2.4]])

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

<span style="text-align: center; font-size: 75%;">[[Image:Review_239195285364_4197_text-ParalPos-eps-converted-to.png|220px|]] [[File:Review_239195285364_6087_text-ParalNeg-eps-converted-to.png|220px|]]</span></div>
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 2.4:''' Anticommutativity of the wedge product.</span></div>


* <math display="inline">a\wedge a = 0</math> since it is a parallelogram with area zero (see Figure [[#lb-2.1|2.5]])

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">[[Image:Review_239195285364_7320_text-ParalZero-eps-converted-to.png|180px|Parallelogram with very small area, depicting what happens when <math>b</math> tends to <math>a</math>.]]</span></div>
|} 

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 2.5:''' Parallelogram with very small area, depicting what happens when <math>b</math> tends to <math>a</math>.</span></div>


* it is distributive

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>(a + b )\wedge c = a\wedge c + b\wedge c</math>
|}
|}
* it is associative

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>(a \wedge b )\wedge c = a\wedge (b \wedge  c)</math>
|}
|}

For example, let

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>a= (a_1,a_2)\,\, = a_1 e_1 + a_2 e_2</math>
|-
| style="text-align: center;" | <math>  b= (b_1,b_2)\,\, \,= b_1 e_1 + b_2 e_2 </math>
|}
|}

Then

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>a\wedge b      =(a_1 e_1 + a_2 e_2)\wedge (b_1 e_1 + b_2 e_2)</math>
|-
| style="text-align: center;" | <math>     =a_1b_1\, e_1\wedge e_1 + a_1b_2\, e_1\wedge e_2 + a_2b_1\, e_2\wedge e_1 + a_2b_2\, e_2\wedge e_2 </math>
|-
| style="text-align: center;" | <math>     =a_1b_2\, e_1\wedge e_2 - a_2b_1\, e_1\wedge e_2  </math>
|-
| style="text-align: center;" | <math>     =(a_1b_2 - a_2b_1)\, e_1\wedge e_2  </math>
|}
|}

i.e. the determinant

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\det \left(\begin{array}{cc} a_1 & a_2 \\  b_1 & b_2 \end{array} \right)</math>
|}
|}

times the canonical bivector/square <math display="inline">e_1\wedge e_2</math>.

===2.2 <span id='lb-2.2'></span>Hodge star operator for vectors in R²===

Now consider the following situation: given the vector <math display="inline">e_1</math>, find another vector <math display="inline">v</math> such that the parallelogram that they form has area <math display="inline">1</math>. It is readily seen that, for instance, <math display="inline">v=e_2, -e_2,e_1+e_2</math> are all solutions. Requiring orthogonality and standard orientation, we see that <math display="inline">e_2</math> is the unique solution. This process is summarized in the Hodge star operator, which basically says that the complementary vector for <math display="inline">e_1</math> is <math display="inline">e_2</math>, and the one for <math display="inline">e_2</math> is <math display="inline">-e_1</math>,

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\star e_1 = e_2</math>
|-
| style="text-align: center;" | <math> {}\star e_2 = -e_1 </math>
|}
|}

In general, the equation that defines the Hodge star operator for any given vector <math display="inline">v\in \mathbb{R}^2</math> is the following

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>w\wedge (\star v) = (w\cdot v) \, e_1\wedge e_2</math>
|}
|}

for every <math display="inline">w\in \mathbb{R}^2</math>. In particular, if we take <math display="inline">v=w</math>,

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>v\wedge (\star v) = |v|^2 \, e_1\wedge e_2</math>
|}
|}

which means that <math display="inline">v</math> and <math display="inline">\star v</math> form a square of area <math display="inline">|v|^2</math>. Thus, the Hodge star operator on a vector <math display="inline">v\in \mathbb{R}^2</math> can be thought of as finding the vector that makes with <math display="inline">v</math> a positively oriented square of area <math display="inline">\vert v\vert ^2</math>.

It is somewhat less intuitive to work out the Hodge star of a bivector. First of all, we have to treat bivectors as vectors of a different space, namely the space of bivectors <math display="inline">\textstyle \wedge ^2\mathbb{R}^2</math>. Secondly, the length of a bivector <math display="inline">v\wedge w</math> is its area

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{length}(v\wedge w):= {Area}(v\wedge w)</math>
|}
|}

Thus, the defining equation of the Hodge star applied to <math display="inline">(v\wedge w)</math> and <math display="inline">\star (v\wedge w)</math> reads as follows

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>(v\wedge w)\wedge \star (v\wedge w)= \left\langle v\wedge w, v\wedge w\right\rangle e_1\wedge e_2 </math>
|}
|}

which means

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>(v\wedge w)\wedge \star (v\wedge w) = {Area}(v\wedge w)^2 e_1\wedge e_2 </math>
|}
|}

Since <math display="inline">v\wedge w={Area}(v\wedge w) e_1\wedge e_2</math> is already a bivector, <math display="inline">\star (v\wedge w)</math> must be a scalar, i.e.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\star (v\wedge w)={Area}(v\wedge w)</math>
|}
|}

When <math display="inline">v=e_1</math> and <math display="inline">w=e_2</math> we have

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\star (e_1\wedge e_2) = 1</math>
|}
|}

Finally, the Hodge star of a number <math display="inline">\lambda </math> is a bivector, i.e.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\star \lambda =\lambda \, e_1\wedge e_2</math>
|}
|}

===2.3 The Laplacian===

Let <math display="inline">f:\mathbb{R}^2\rightarrow \mathbb{R}</math> and consider the gradient

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\nabla f = {\partial f\over \partial x}e_1+{\partial f\over \partial y}e_2</math>
|}
|}

Apply the Hodge star operator to it

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\star \nabla f     = {\partial f\over \partial x} \star e_1+{\partial f\over \partial y} \star e_2</math>
|-
| style="text-align: center;" | <math>    = {\partial f\over \partial x}e_2-{\partial f\over \partial y}e_1</math>
|}
|}

Now take the gradient of each coefficient function together with wedge product

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\nabla \wedge \star \nabla f     := \nabla \left({\partial f\over \partial x}\right)\wedge e_2-\nabla \left({\partial f\over \partial y}\right)\wedge e_1</math>
|-
| style="text-align: center;" | <math>    = \left({\partial ^2 f\over \partial x^2}e_1+{\partial ^2 f\over \partial y\partial x}e_2\right)\wedge e_2    -\left({\partial ^2 f\over \partial x\partial y}e_1+{\partial ^2  f\over \partial ^2 y}e_2\right)\wedge e_1</math>
|-
| style="text-align: center;" | <math>    = {\partial ^2 f\over \partial x^2}e_1\wedge e_2+{\partial ^2 f\over \partial y\partial x}e_2\wedge e_2    -{\partial ^2 f\over \partial x\partial y}e_1\wedge e_1-{\partial ^2  f\over \partial ^2 y}e_2\wedge e_1</math>
|-
| style="text-align: center;" | <math>    = \left({\partial ^2 f\over \partial x^2}+{\partial ^2  f\over \partial ^2 y}\right)e_1\wedge e_2 </math>
|}
|}

By taking the Hodge star of this last expression we get the ordinary Laplacian of <math display="inline">f</math>

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\star \nabla ^\wedge \star \nabla (f)     = {\partial ^2 f\over \partial x^2}+{\partial ^2  f\over \partial ^2 y} </math>
|}
|}

'''Remarks. ''' 

(i) This rather convoluted looking way of computing the Laplacian of a function is based on Exterior Differential Calculus, a theory that generalizes  the operators of vector calculus (gradient, curl and divergence) to arbitrary dimensions, and is the basis for the differential topological theory of deRham cohomology.

(ii) We would like to emphasize the  necessity of using the Hodge star operator <math display="inline">\star</math> in order to make the combination of differentiation  and wedge product produce the correct answer.

===2.4 Duality in Green's theorem===

Green's theorem states that for a vector field <math display="inline">(L,M)</math> defined on a region <math display="inline">D\subset  \mathbb{R}^2</math>,

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\int _D\left({\partial M\over \partial x}-{\partial L\over \partial y}\right)dxdy=\int _{C=\partial D} (Ldx+Mdy) .</math>
|}
|}

In this section, we will explain how this identity encodes a duality between the operator of differentiation and that of taking the boundary of the domain of integration.

====2.4.1 Rewriting Green's theorem====

Note that if <math display="inline">F=(L,M)</math> is a vector field, we can write it as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>F= Le_1+Me_2</math>
|}
|}

Then, we can apply the gradient operator together with wedge product in the following fashion

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\nabla ^\wedge (L e_1 +  M  e_2)    :=\nabla L\wedge e_1 + \nabla M \wedge e_2</math>
|-
| style="text-align: center;" | <math>    =\left({\partial L\over \partial x}e_1+{\partial L\over \partial y}e_2\right)\wedge e_1   +\left({\partial M\over \partial x}e_1+{\partial M\over \partial y}e_2\right)\wedge e_2</math>
|-
| style="text-align: center;" | <math>    ={\partial L\over \partial x}e_1\wedge e_1+{\partial L\over \partial y}e_2\wedge e_1   +{\partial M\over \partial x}e_1\wedge e_2+{\partial M\over \partial y}e_2\wedge e_2</math>
|-
| style="text-align: center;" | <math>      ={\partial L\over \partial y}e_2\wedge e_1   +{\partial M\over \partial x}e_1\wedge e_2</math>
|-
| style="text-align: center;" | <math>      =\left({\partial M\over \partial x}   -{\partial L\over \partial y}\right)e_1\wedge e_2 </math>
|}
|}

Note that we have defined a new operator <math display="inline">\nabla ^\wedge </math> which combines differentiation and wedge product. Applying the Hodge star operator we obtain

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\star \nabla ^\wedge (L e_1 +  M e_2)    =\left({\partial M\over \partial x}   -{\partial L\over \partial y}\right) </math>
|}
|}

i.e. the integrand of the left hand side of the identity of integrals in Green's theorem. Thus, as a first step, Green's theorem can be rewritten as follows:

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\int _D\nabla ^\wedge (L, M)dxdy=\int _{\partial D} (L ,  M )\cdot (dx,dy)</math>
|}
|}

====2.4.2 The duality in Green's theorem====

Let us recall the following fact from Linear Algebra. Given a linear transformation <math display="inline">A</math> of Euclidean space <math display="inline">\mathbb{R}^n</math>, the transpose <math display="inline">A^T</math> satisfies

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\left\langle A(v), w \right\rangle= \left\langle v, A^T(w)\right\rangle</math>
|}
|}

for any two vectors <math display="inline">v,w\in \mathbb{R}^n</math>, where <math display="inline">\left\langle\cdot , \cdot \right\rangle</math> denotes the standard inner/dot product.  In fact, such an identity characterizes the transpose <math display="inline">A^T</math>.

Now, let us do the following notational trick: substitute the integration symbols in Green's theorem by <math display="inline">\ll \cdot ,\cdot \gg </math> as follows:

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\int _C (L,M)\cdot (dx,dy) = \ll (L,M), C\gg </math>
|-
| style="text-align: center;" | <math> \int _D\nabla ^\wedge (L, M)dxdy = \ll \nabla ^\wedge (L,M), D\gg  </math>
|}
|}

where <math display="inline">C=\partial D</math> is the boundary of the region. Using this notational change, Green's theorem reads as follows

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\ll \nabla ^\wedge (L,M), D\gg \,\,=\,\,\ll (L,M), \partial D\gg</math>
|}
|}

Roughly speaking, this means that the differential operator <math display="inline">\nabla ^\wedge </math> is the transpose of the boundary operator <math display="inline">\partial </math> by means of the product <math display="inline">\ll \cdot ,\cdot \gg </math>.

'''Remark'''. The previous observation is fundamental in the development of DEC, since the boundary operator is well understood and easy to calculate on meshes.

==3 Discrete Exterior Calculus==

Now we will discretize the differentiation operator <math display="inline">\nabla ^\wedge </math> presented above. We will start by describing the discrete version of the boundary operator on simplices/triangles. Afterwards, we will treat the differentiation operator as the transpose of the boundary operator.

We are interested in using certain geometric subsets of a given triangular mesh of a 2D region. Such subsets include vertices/nodes, edges/sides and faces/triangles. We will describe each one of them by means of the ordered list of vertices whose convex closure constitutes the subset of interest. For instance, consider the triangular mesh of the planar hexagonal region in Figure [[#3 Discrete Exterior Calculus|3.1]],

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'' [[Image:Review_239195285364_2873_text-Region-01-eps-converted-to.png|180px|]]''</span></div>
|}


<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 3.1:''' Triangular mesh of a planar hexagonal region.</span></div>


where the shaded triangle will be denoted by <math display="inline">[v_0,v_1,v_6]</math>, and its edge joining the vertices <math display="inline">v_0</math> and <math display="inline">v_1</math> will be denoted by <math display="inline">[v_0,v_1]</math>. For the sake of notational consistency, we will denote the vertices also enclosed in brackets, e.g. <math display="inline">[v_0]</math>.

===3.1 Boundary operator and discrete derivative===

There is a well known boundary operator <math display="inline">\partial </math> for oriented triangles, edges and points:

* For points/vertices:

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_9903_text-boundary00-eps-converted-to.png|80px|]]''</span></div>
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 3.2:''' Boundary of a vertex: <math>\partial [v_0] = 0</math>.</span></div>


* For sides/edges:

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_1507_text-boundary01-eps-converted-to.png|200px|]]''</span></div>
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 3.3:''' Boundary of an edge: <math>\partial [v_0,v_1] = [v_1] - [v_0]</math>.</span></div>


* For faces/triangles:

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_1837_text-boundary02-eps-converted-to.png|200px|]]''</span></div>
|}


<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 3.4:''' Boundary of a face: <math>\partial [v_0,v_1,v_2] = [v_1,v_2] - [v_0,v_2]+[v_0,v_1]</math>.</span></div>


'''Example'''. Let us consider again the mesh of the planar hexagonal (with oriented triangles) in Figure [[#3.1 Boundary operator|3.5]]

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_6355_text-boundary04-eps-converted-to.png|250px|]]''</span></div>
|}
 

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 3.5:''' Oriented triangular mesh of a planar hexagonal region.</span></div>


We will denote a triangle by the list of its vertices listed in order according to the orientation of the tringle. Thus, we have the following ordered lists:

* list of faces

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math> \{[v_0,v_1,v_6],[v_1,v_2,v_6],[v_2,v_3,v_6], [v_3,v_4,v_6],[v_4,v_5,v_6],[v_5,v_0,v_6]\}  </math>
|}
|}
* list of edges

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math> \{  [v_0,v_6], [v_1,v_6], [v_2,v_6], [v_3,v_6], [v_4,v_6], [v_5,v_6], [v_0,v_1], [v_1,v_2], [v_2,v_3], [v_3,v_4], [v_4,v_5], [v_5,v_0] \} </math>
|}
|}
* list of vertices

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math> \{  [v_0],[v_1],[v_2],[v_3],[v_4],[v_5],[v_6] \} </math>
|}
|}

A key idea in DEC is to consider each face as an element of a basis of a vector space. Namely,  column coordinate vectors are associated to faces as follows:

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>[v_0,v_1,v_6] \longleftrightarrow  (1,0,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {}[v_1,v_2,v_6] \longleftrightarrow  (0,1,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {}[v_2,v_3,v_6] \longleftrightarrow  (0,0,1,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {}[v_3,v_4,v_6] \longleftrightarrow  (0,0,0,1,0,0)^T</math>
|-
| style="text-align: center;" | <math> {}[v_4,v_5,v_6] \longleftrightarrow  (0,0,0,0,1,0)^T</math>
|-
| style="text-align: center;" | <math> {}[v_5,v_0,v_6] \longleftrightarrow  (0,0,0,0,0,1)^T</math>
|}
|}

Similarly,  coordinate vectors are associated to the edges

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{} [v_0,v_6] \longleftrightarrow  (1,0,0,0,0,0,0,0,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_1,v_6] \longleftrightarrow  (0,1,0,0,0,0,0,0,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_2,v_6] \longleftrightarrow  (0,0,1,0,0,0,0,0,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_3,v_6] \longleftrightarrow  (0,0,0,1,0,0,0,0,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_4,v_6] \longleftrightarrow  (0,0,0,0,1,0,0,0,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_5,v_6] \longleftrightarrow  (0,0,0,0,0,1,0,0,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_0,v_1] \longleftrightarrow  (0,0,0,0,0,0,1,0,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_1,v_2] \longleftrightarrow  (0,0,0,0,0,0,0,1,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_2,v_3] \longleftrightarrow  (0,0,0,0,0,0,0,0,1,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_3,v_4] \longleftrightarrow  (0,0,0,0,0,0,0,0,0,1,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_4,v_5] \longleftrightarrow  (0,0,0,0,0,0,0,0,0,0,1,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_5,v_0] \longleftrightarrow  (0,0,0,0,0,0,0,0,0,0,0,1)^T</math>
|}
|}

Finally, we do the same with the vertices

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{} [v_0] \longleftrightarrow  (1,0,0,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_1] \longleftrightarrow  (0,1,0,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_2] \longleftrightarrow  (0,0,1,0,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_3] \longleftrightarrow  (0,0,0,1,0,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_4] \longleftrightarrow  (0,0,0,0,1,0,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_5] \longleftrightarrow  (0,0,0,0,0,1,0)^T</math>
|-
| style="text-align: center;" | <math> {} [v_6] \longleftrightarrow  (0,0,0,0,0,0,1)^T</math>
|}
|}

Now, if we take the boundary of each face, we have

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\partial [v_0,v_1,v_6] = [v_1,v_6] - [v_0,v_6]+[v_0,v_1]</math>
|-
| style="text-align: center;" | <math> \partial [v_1,v_2,v_6] = [v_2,v_6] - [v_1,v_6]+[v_1,v_2]</math>
|-
| style="text-align: center;" | <math> \partial [v_2,v_3,v_6] = [v_3,v_6] - [v_2,v_6]+[v_2,v_3]</math>
|-
| style="text-align: center;" | <math> \partial [v_3,v_4,v_6] = [v_4,v_6] - [v_3,v_6]+[v_3,v_4]</math>
|-
| style="text-align: center;" | <math> \partial [v_4,v_5,v_6] = [v_5,v_6] - [v_4,v_6]+[v_4,v_5]</math>
|-
| style="text-align: center;" | <math> \partial [v_5,v_0,v_6] = [v_0,v_6] - [v_5,v_6]+[v_5,v_0] </math>
|}
|}

which, under the previous assignments of coordinate vectors, corresponds to the linear transformation given by the following matrix

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\partial _{2,1}=\left(\begin{array}{rrrrrr} -1 & 0 & 0 & 0 & 0 & 1\\ 1  & -1 & 0 & 0 &0 & 0\\  0  & 1 & -1 & 0 & 0 & 0\\  0  & 0 & 1 & -1 & 0 & 0\\  0  &0  &0  & 1 & -1 & 0\\  0  & 0 & 0 & 0 & 1 & -1\\ 1  & 0 & 0 & 0 & 0 & 0\\ 0   & 1 & 0 & 0 & 0 & 0\\  0  & 0 & 1 &0  &0  &0 \\   0 & 0 & 0 & 1 & 0 & 0\\  0  & 0 & 0 & 0 & 1 & 0\\  0  & 0 & 0 & 0 & 0 & 1         \end{array} \right)</math>
|}
|}

where the subindices in <math display="inline">\partial _{2,1}</math> indicate that we are taking the boundary of 2-dimensional elements and obtaining 1-dimensional ones. Similarly, taking the boundaries of all the edges gives

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\partial [v_0,v_6] = [v_6] - [v_0]</math>
|-
| style="text-align: center;" | <math> \partial [v_1,v_6] = [v_6] - [v_1]</math>
|-
| style="text-align: center;" | <math> \partial [v_2,v_6] = [v_6] - [v_2]</math>
|-
| style="text-align: center;" | <math> \partial [v_3,v_6] = [v_6] - [v_3]</math>
|-
| style="text-align: center;" | <math> \partial [v_4,v_6] = [v_6] - [v_4]</math>
|-
| style="text-align: center;" | <math> \partial [v_5,v_6] = [v_6] - [v_5]</math>
|-
| style="text-align: center;" | <math> \partial [v_0,v_1] = [v_1] - [v_0]</math>
|-
| style="text-align: center;" | <math> \partial [v_1,v_2] = [v_2] - [v_1]</math>
|-
| style="text-align: center;" | <math> \partial [v_2,v_3] = [v_3] - [v_2]</math>
|-
| style="text-align: center;" | <math> \partial [v_3,v_4] = [v_4] - [v_3]</math>
|-
| style="text-align: center;" | <math> \partial [v_4,v_5] = [v_5] - [v_4]</math>
|-
| style="text-align: center;" | <math> \partial [v_5,v_0] = [v_0] - [v_5] </math>
|}
|}

which, under the previous assignments of coordinate vectors, corresponds to the linear transformation given by the following matrix

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\partial _{1,0}=\left( \begin{array}{rrrrrr|rrrrrr} -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1\\ 0 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & -1\\ 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right)</math>
|}
|}

'''Remark'''. Note how these matrices encode different levels of connectivity with orientations, such as who are the edges of which oriented triangle, or which are the  end points of a given oriented edge.

Due to the duality between <math display="inline">\partial </math> and <math display="inline">\nabla ^\wedge </math>, we can define the discretiztion of <math display="inline">\nabla ^\wedge </math>  by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\nabla ^\wedge := (\partial )^T</math>
|}
|}

For instance, we see that the operator <math display="inline">\nabla ^\wedge _{0,1}</math> reads as follows

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\nabla ^\wedge _{0,1}=\left(\begin{array}{rrrrrrr} -1 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & -1 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & -1 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & -1 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & -1 & 1\\ -1 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1& 0 \\ 1 & 0 & 0 & 0 & 0 & -1& 0            \end{array} \right)</math>
|}
|}

===3.2 Dual mesh===

In order to discretize the Hodge star operator, we must first introduce  the notion of the dual mesh of a triangular mesh.

Consider the triangular mesh <math display="inline">K</math> in Figure [[#3.2 Dual mesh|3.6]](a). The construction of the dual mesh <math display="inline">K^*</math> is carried out as follows:

* The vertices of the dual mesh <math display="inline">K^*</math> are the circumcenters of the faces/triangles of the original mesh  (the blue dots in Figures   [[#3.2 Dual mesh|3.6]](b) and [[#3.2 Dual mesh|3.6]](c)).  For instance, the dual of the face <math display="inline">[v_0,v_1,v_2]</math> will be denoted by <math display="inline">[v_0,v_1,v_2]^*</math>.
* The edges of <math display="inline">K^*</math> are the straight line segments joining the circumcenters of two adjacent triangles (those which share an edge). Note that the resulting line segments   are orthogonal to one of the original edges (the blue straight line segments in   in Figures   [[#3.2 Dual mesh|3.6]](b) and [[#3.2 Dual mesh|3.6]](c)). For instance, the dual of the edge <math display="inline">[v_0,v_6]</math> will be denoted by <math display="inline">[v_0,v_6]^*</math>.
* The faces or cells of <math display="inline">K^*</math> are the areas enclosed by the new polygons determined by the new edges. For instance, the dual of the edge <math display="inline">[v_6]</math> is the inner blue hexagon in  in Figures   [[#3.2 Dual mesh|3.6]](b) and [[#3.2 Dual mesh|3.6]](c)  and will be denoted by <math display="inline">[v_6]^*</math>.



{| class="floating_imageSCP" style="text-align: center;"
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'' 
(a)  [[Image:Review_239195285364_6808_text-boundary04-eps-converted-to.png|150px|Dual mesh construction]]<math>\qquad</math>
(b)  [[Image:Review_239195285364_2587_text-Dual00-eps-converted-to.png|150px|Dual mesh construction]]<math>\qquad</math>
(c)  [[Image:Review_239195285364_5980_text-Dual01a-eps-converted-to.png|120px|]]
''</span></div>
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 3.6:'''Dual mesh construction: (a) triangular mesh <math>K</math>; (b) dual mesh <math>K^*</math> superimposed on the mesh <math>K</math>;      (c) dual mesh <math>K^*</math>.</span></div>


The orientation of the dual edges is given by the following recipe. If we have two adjacent triangles oriented as in  Figure [[#3.2 Dual mesh|3.7]](a), the dual edge crossing the edge of adjacency is oriented as in Figure [[#3.2 Dual mesh|3.7]](b)

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''(a)  [[Image:Review_239195285364_3618_text-TwoTriangles01-eps-converted-to.png|120px|]]<math>\qquad</math>(b) [[Image:Review_239195285364_1851_text-TwoTriangles02-eps-converted-to.png|120px|]]''</span></div>
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 3.7:''' (a) Two adjacent oriented triangles. (b) Compatibly oriented dual edge..</span></div>

===3.3 Boundary operator on the dual mesh===

Consider the dual mesh in Figure [[#3.3 Boundary operator on the dual mesh|3.8]] with the given labels and orientations

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_5980_text-Dual01a-eps-converted-to.png|160px|]]<math>\qquad</math>[[Image:Review_239195285364_7382_text-Dual01b-eps-converted-to.png|160px|n]]''</span></div>
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 3.8:''' Oriented dual mesh.</span></div>


The boundary operator is applied in a similar fashion as it was applied to triangles. In this case we have

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\partial ^{dual}_{2,1} F_1 =  E_{1}  + E_{7}  -  E_{12}</math>
|-
| style="text-align: center;" | <math>  \partial ^{dual}_{2,1} F_2 =  E_{2}  - E_{7}  +  E_{8}</math>
|-
| style="text-align: center;" | <math>  \partial ^{dual}_{2,1} F_3 =  E_{3}  - E_{8}  +  E_{9} </math>
|-
| style="text-align: center;" | <math>  \partial ^{dual}_{2,1} F_4 =  E_{4}  - E_{9}  +  E_{10}</math>
|-
| style="text-align: center;" | <math>  \partial ^{dual}_{2,1} F_5 =  E_{5}  - E_{10} +  E_{11}</math>
|-
| style="text-align: center;" | <math>  \partial ^{dual}_{2,1} F_6 =  E_{6}  - E_{11} +  E_{12}</math>
|-
| style="text-align: center;" | <math>  \partial ^{dual}_{2,1} F_7 =  -E_{1}- E_{2}- E_{3}- E_{4}- E_{5}- E_{6} </math>
|}
|}

If we assign coordinate vectors to the dual faces and the dual edges as before

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>F_1 \longleftrightarrow  (1,0,0,0,0,0,0)</math>
|-
| style="text-align: center;" | <math>  F_2 \longleftrightarrow  (0,1,0,0,0,0,0)</math>
|-
| style="text-align: center;" | <math>  F_3 \longleftrightarrow  (0,0,1,0,0,0,0)</math>
|-
| style="text-align: center;" | <math>  F_4 \longleftrightarrow  (0,0,0,1,0,0,0)</math>
|-
| style="text-align: center;" | <math>  F_5 \longleftrightarrow  (0,0,0,0,1,0,0)</math>
|-
| style="text-align: center;" | <math>  F_6 \longleftrightarrow  (0,0,0,0,0,1,0)</math>
|-
| style="text-align: center;" | <math>  F_7 \longleftrightarrow  (0,0,0,0,0,0,1) </math>
|}
|}

and

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>E_1 \longleftrightarrow  (1,0,0,0,0,0,0,0,0,0,0,0)</math>
|-
| style="text-align: center;" | <math>  E_2 \longleftrightarrow  (0,1,0,0,0,0,0,0,0,0,0,0)</math>
|-
| style="text-align: center;" | <math>  E_3 \longleftrightarrow  (0,0,1,0,0,0,0,0,0,0,0,0)</math>
|-
| style="text-align: center;" | <math>  E_4 \longleftrightarrow  (0,0,0,1,0,0,0,0,0,0,0,0)</math>
|-
| style="text-align: center;" | <math>  E_5 \longleftrightarrow  (0,0,0,0,1,0,0,0,0,0,0,0)</math>
|-
| style="text-align: center;" | <math>  E_6 \longleftrightarrow  (0,0,0,0,0,1,0,0,0,0,0,0)</math>
|-
| style="text-align: center;" | <math>  E_7 \longleftrightarrow  (0,0,0,0,0,0,1,0,0,0,0,0)</math>
|-
| style="text-align: center;" | <math>  E_8 \longleftrightarrow  (0,0,0,0,0,0,0,1,0,0,0,0)</math>
|-
| style="text-align: center;" | <math>  E_9 \longleftrightarrow  (0,0,0,0,0,0,0,0,1,0,0,0)</math>
|-
| style="text-align: center;" | <math>  E_{10} \longleftrightarrow  (0,0,0,0,0,0,0,0,0,1,0,0)</math>
|-
| style="text-align: center;" | <math>  E_{11} \longleftrightarrow  (0,0,0,0,0,0,0,0,0,0,1,0)</math>
|-
| style="text-align: center;" | <math>  E_{12} \longleftrightarrow  (0,0,0,0,0,0,0,0,0,0,0,1) </math>
|}
|}

we have the associated matrix

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\partial _{2,1}^{dual}=\left(\begin{array}{rrrrrrr} 1 & 0 & 0 & 0 & 0 & 0 & -1\\ 0 & 1 & 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 1 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 1 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 1 & -1\\ 1 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1& 0 \\ -1 & 0 & 0 & 0 & 0 & 1& 0            \end{array} \right)</math>
|}
|}

Notice that

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\partial _{2,1}^{dual}=-\left(\partial _{1,0}\right)^T</math>
|}
|}

which arises from the duality between the two meshes <span id='citeF-6'></span>[[#cite-6|[6]]]. In general, the discrete differential to be applied is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\nabla _{1,2}^{\wedge ,dual}=\left(\partial _{2,1}^{dual}\right)^T=-\partial _{1,0}=-\left(\nabla ^\wedge _{0,1}\right)^T</math>
|}
|}

===3.4 Discrete Hodge star===

The discretization of the Hodge star <math display="inline">\star </math> uses the    geometrical ideas described in Section [[#lb-2.2|2.2]] and the dual mesh. More precisely, the 2D Hodge star operator rotates a vector <math display="inline">90^\circ </math> counterclockwise.  For the sake of clarity, let us focus on the edge <math display="inline">[v_0,v_6]</math>, its mesh dual <math display="inline">[v_0,v_6]^*</math> and its Hodge star image <math display="inline">\star [v_0,v_6]</math>. They are represented in Figure [[#3.4 Discrete Hodge star|3.9]].

{| class="floating_imageSCP" style="text-align: center;"
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_9611_text-Dual00a-eps-converted-to.png|160px]]<math>\qquad</math>[[Image:Review_239195285364_4015_text-Dual00aa-eps-converted-to.png|160px]]''</span></div>
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 3.9'''</span></div>

Since

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{length}(\star [v_0,v_6])={length}([v_0,v_6])</math>
|}
|}

we see that the relationship between the dual edge <math display="inline">[v_0,v_6]^*</math> and the geometric <math display="inline">\star [v_0,v_6]</math> is the following

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{1\over {length}([v_0,v_6]^*)}[v_0,v_6]^*= {1\over {length}([v_0,v_6])}\star [v_0,v_6]</math>
|}
|}

As can be seen, when applying the geometric Hodge star to the edges of the mesh,  we do not end up in the dual mesh but in multiples of the elements of the dual mesh. Thus, <math display="inline">\star [v_0,v_6]</math>  must be scaled to match <math display="inline">[v_0,v_6]^*</math>. If we do this to all the Hodge star images, we get the discrete Hodge star matrix

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>M_{1,1}=\left(\begin{array}{ccccc} {{length}([v_0,v_6]^*)\over {length}([v_0,v_6])} & 0 & 0 & \dots & 0 \\ 0 & {{length}([v_1,v_6]^*)\over {length}([v_1,v_6])} & 0 & \dots & 0 \\ 0 & 0 & {{length}([v_2,v_6]^*)\over {length}([v_2,v_6])} & \dots & 0\\ \vdots & \vdots & \vdots &    \ddots &\vdots \\ 0 & 0 & 0 & \dots &  {{length}([v_5,v_0]^*)\over {length}([v_5,v_0])}           \end{array} \right)</math>
|}
|}

where the subindices in <math display="inline">M_{1,1}</math> indicate that we are sending 1-dimensional elements of the original mesh to 1-dimensional elements of the dual mesh.

Somewhat less intuitive is the meaning of the geometric Hodge star operator on nodes of the original mesh. As we saw in Subsection [[#lb-2.2|2.2]],

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\star 1=e_1\wedge e_2</math>
|}
|}

which geometrically means that the Hodge star <math display="inline">\star [v_6]</math> must be a polygon with area equal to 1 (classically, it is a parallelogram, but can also be a hexahedron of area 1 as in this example).  

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_3989_text-HodgeStarPoint01-eps-converted-to.png|160px|]]''</span></div>
|}

However, in the dual mesh we have the polygon <math display="inline">[v_6]^*</math>       

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_2797_text-HodgeStarPoint00-eps-converted-to.png|160px|]]''</span></div>
|}

so that we need to resize <math display="inline">\star [v_6]</math> as follows

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>[v_6]^*={Area}([v_6]^*)\, \star [v_6]</math>
|}
|}

If we do that for all the vertices, we obtain the discrete Hodge star matrix

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math> M_{0,2}=\left( \begin{array}{cccc} {Area}([v_0]^*) & 0 & \dots & 0\\ 0 & {Area}([v_1]^*) & \dots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & {Area}([v_6]^*) \end{array} \right)</math>
|}
|}

The inverse matrix will deal with the case when we take the Hodge star of the 2D polygons in the dual mesh to obtain points (with weight 1) in the original mesh.

In summary, the various matrices <math display="inline">M</math> representing the discrete Hodge star operator send elements of the original mesh to elements of the dual mesh.

===3.5 DEC applied to 2D Poisson's equation===

Consider the 2D Poisson's equation

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\kappa \Delta f= q</math>
|}
|}

As we have seen, this can be rewritten as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\kappa \,\,\star \nabla ^\wedge \star \nabla (f)     = q </math>
|}
|}

Suppose that we wish to solve the equation on the meshed domain <math display="inline">K</math>.  The equation can them be discretized as the matrix equation

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\kappa \,\,M_{2,0}\,\, \nabla _{1,2}^{\wedge ,dual} \,\, M_{1,1}\,\, \nabla ^\wedge _{0,1}\,\, [f]=[q]</math>
|}
|}

where <math display="inline">M_{0,2},\nabla _{1,2}^{\wedge ,dual}=-\nabla _{0,1}^T,M_{1,1}</math> and <math display="inline">\nabla ^\wedge _{0,1}</math>  denote the matrices corresponding to the relevant mesh, such as the ones described in the previous subsections, and <math display="inline">[f]</math> and <math display="inline">[q]</math> denote the discretizations of the functions <math display="inline">f</math> and <math display="inline">q</math> at the nodes/vertices.  Later on, it will be convenient to work with the  equivalent system

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\kappa \,\,(\nabla ^\wedge _{0,1})^T\,\,M_{1,1}\,\,\nabla ^\wedge _{0,1}[f]= M_{0,2}[q]</math>
|}
|}

==4 DEC for general triangulations==

Since the boundary operator is really concerned with the connectivity of the mesh and does not change under deformation of the mesh,  the change in the setup of DEC for a deformed mesh must be contained in the discrete Hodge star matrices. Since  such matrices are computed in terms of lengths and areas of oriented elements of the mesh, we will now examine how those ingredients  transform under deformation, a problem that was first considered in <span id='citeF-10'></span>[[#cite-10|[10]]].

===4.1 Dual mesh of an arbitrary triangle===

In order to explain how to implement DEC for general triangulations, let us consider first a mesh consisting of a single well-centered triangle,  as well as its dual mesh (see Figure [[#4.1 Dual mesh of an arbitrary triangle|4.1]]).

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_2549_NewCircumscribedTriangle00.png|160px|]]<math>\qquad</math>[[Image:Review_239195285364_2438_NewCircumscribedTriangle00b.png|180px|]]''</span></div>
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 4.1:''' Well-centered triangle and its dual mesh.</span></div>


The dual cells are given as follows:

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{} [v_1,v_2,v_3]^*= [c]</math>
|-
| style="text-align: center;" | <math> {} [v_1,v_2]^*= [p_1,c]</math>
|-
| style="text-align: center;" | <math> {} [v_2,v_3]^*= [p_2,c]</math>
|-
| style="text-align: center;" | <math> {} [v_3,v_1]^*= [p_3,c]</math>
|-
| style="text-align: center;" | <math> {} [v_1]^* = [v_1,p_1,c,p_3] </math>
|-
| style="text-align: center;" | <math> {} [v_2]^* = [v_2,p_2,c,p_1] </math>
|-
| style="text-align: center;" | <math> {} [v_3]^* = [v_3,p_3,c,p_2]  </math>
|}
|}

Now consider the cell <math display="inline">[v_3]^* = [v_3,p_3,c,p_2]</math> subdivided as in  Figure [[#4.1 Dual mesh of an arbitrary triangle|4.2]].

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_1856_NewCircumscribedTriangle00a.png|160px|]]''</span></div>
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 4.2:''' The subdivision of a 2-dimensional dual cell of a well-centered triangle.</span></div>

If we deform continuously the triangle <math display="inline">[v_1,v_2,v_3]</math> to become an obtuse triangle as in  Figure [[#4.1 Dual mesh of an arbitrary triangle|4.3]],

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_9664_NewCircumscribedTriangle01.png|150px|]](a)<math>\;</math>[[Image:Review_239195285364_6104_NewCircumscribedTriangle02.png|150px|]](b)<math>\;</math>[[Image:Review_239195285364_5608_NewCircumscribedTriangle03.png|150px|]](c)''</span></div>
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 4.3:''' The subdivision of a 2-dimensional dual cell of a well-centered triangle.</span></div>


we see in Figures [[#4.1 Dual mesh of an arbitrary triangle|4.1]](a) and [[#4.1 Dual mesh of an arbitrary triangle|4.1]](b) that the area of the blue subtriangle <math display="inline">[v_3,c,p_3]</math> decreases to 0 and in Figure [[#4.1 Dual mesh of an arbitrary triangle|4.1]](c) that it is completely outside of the triangle and, therefore, must be assigned a negative sign. The same can be said about the 1-dimensional cell <math display="inline">[p_3,c]</math>, which originally is completely contained in the triangle <math display="inline">[v_1,v_2,v_3]</math>, its size reduces to zero as the triangle is deformed (Figures [[#4.1 Dual mesh of an arbitrary triangle|4.1]](a) and [[#4.1 Dual mesh of an arbitrary triangle|4.1]](b)),  and eventually it is completely outside the triangle <math display="inline">[v_1,v_2,v_3]</math>  (Figure [[#4.1 Dual mesh of an arbitrary triangle|4.1]](c))  and a negative sign must be assigned to it. On the other hand, part of the red subtriangle <math display="inline">[v_3,c,p_2]</math> still intersects the interior of the triangle <math display="inline">[v_1,v_2,v_3]</math> and,  therefore, no assignment of sign is needed. Similarly for the segment <math display="inline">[p_2,c]</math>. In terms of numerical simulations, an implementation in terms of determinants contains intrinscally the aforementioned change of signs.

===4.2 Dual mesh of a general triangulation===

Now consider the well-centered mesh and its dual in Figure [[#4.2 Dual mesh of a general triangulation|4.4]].

{| class="floating_imageSCP" style="text-align: center;"
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[File:Review_239195285364_3465_NewHexagonalMesh00.png|160px|]]<math>\qquad</math>[[File:Review_239195285364_6131_NewHexagonalMesh01.png|160px|]]''</span></div>
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 4.4:''' (a) Well-centered triangular mesh of hexagon. (b) Dual mesh.</span></div>

Observe the deformation of the blue-colored dual cell <math display="inline">[v_6]^*</math>  in Figure [[#4.2 Dual mesh of a general triangulation|4.5]](a) as the vertex <math display="inline">v_0</math> is moved to make the triangle <math display="inline">[v_0,v_6,v_5]</math> non-well-centered in Figure [[#4.2 Dual mesh of a general triangulation|4.5]](b) and the vertex <math display="inline">v_4</math> is moved to make the triangle <math display="inline">[v_4,v_5,v_6]</math>  non-well-centered in Figure [[#4.2 Dual mesh of a general triangulation|4.5]](c).

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''(a)[[File:Review_239195285364_7165_NewHexagonalMesh02.png|150px|]]<math>\;</math>(b)[[File:Review_239195285364_6070_NewHexagonalMesh03.png|150px|]]<math>\;</math>(c)[[File:Review_239195285364_6698_NewHexagonalMesh04.png|150px|]]''</span></div>
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 4.5:''' Deformation of a 2-dimensional dual cell as the hexagonal region is deformed.</span></div>
 

We have colored in red the part of the dual cell <math display="inline">[v_6]^*</math> in Figure [[#4.2 Dual mesh of a general triangulation|4.2]](c) that must have the opposite orientation of the blue-colored part. Similar considerations apply to the edges.

==5 Numerical examples==

===5.1 First example===

Let us solve the Poisson equation in a circle of radius one under the following conditions (see Figure [[#5.1 First example|5.1]]):

* heat difussion constant <math display="inline">k=1</math>;
* source term <math display="inline">q= -1</math>;
* Dirichlet boundary condition <math display="inline">u=10</math>.

The exact solution is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math> u(x,y)=\frac{1}{4}(1-x^2-y^2)+10 </math>
|}
|}

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_5542_Fig_6_circle_prb.png|160px|]]''</span></div>
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 5.1:''' Disk of radius one with difussion constant <math>k=1</math>, subject to a heat source <math>q=-1</math>.</span></div>


The meshes used in this example are shown in Figure [[#5.1 First example|5.2]] and vary from very coarse to very fine.

{| class="floating_imageSCP" style="text-align: center;"
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''(a)[[Image:Review_239195285364_2714_Fig_7_circle_msh1.png|150px|]]<math>\;</math>(b)[[Image:Review_239195285364_3494_Fig_7_circle_msh2.png|150px|]]<math>\;</math>(c)[[Image:Review_239195285364_1920_Fig_7_circle_msh3.png|150px|]]''</span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''(d)[[Image:Review_239195285364_1099_Fig_7_circle_msh4.png|150px|]]<math>\;</math>(e)|[[Image:Review_239195285364_9301_Fig_7_circle_msh5.png|150px|]]''</span></div>
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">'''Figure 5.2:''' Meshes for unit disk.</span></div>

The numerical results for the maximum temperature value (<math display="inline">u(0,0)=10.25</math>) are exemplified in Table [[#5.1 First example|5.1]] where  a comparison with the Finite Element Method with linear interpolation functions (FEML) is also shown.  The FEML methodology that we used in the comparison can be consulted <span id='citeF-13'></span><span id='citeF-12'></span><span id='citeF-2'></span>[[#cite-13|[13,12,2]]]. For the sake of completeness in our comparison, here we compute the flux vectors in the same way as in FEML.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math> \begin{array}{|c|c|c|c|c|c|c|} {Mesh}& \# {nodes}& \# {elements} & {Max. Temp. Value} & {Max. Flux Magnitude}  \\ &&&{DEC\quad FEML}&{DEC\quad FEML}  \\  {Figure \,\,\, (a)}& 9& 8& {10.250\quad 10.285}& {0.270\quad 0.307}\\ {Figure \,\,\,(b)}& 17& 20& {10.250\quad 10.237}& {0.388\quad 0.405}\\ {Figure \,\,\,(c)}& 41& 56& {10.250\quad 10.246}& {0.449\quad 0.453}\\ {Figure \,\,\,(d)}& 713& 1304& {10.250\quad 10.250}& {0.491\quad 0.492}\\ {Figure \,\,\,(e)}& 42298& 83346& {10.250\quad 10.250}& {0.496\quad 0.496}\\ \end{array}     </math>
|}
|- style="text-align: center; font-size: 75%;"
| colspan="3" | '''Table 5.1:''' Maximum temperature and Flux magnitude values in the numerical simulation.
|}


The temperature distribution and Flux magnitude fields for the finest mesh are shown in Figure [[#5.1 First example|5.3]].

{| class="floating_imageSCP" style="text-align: center; "
|-
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''[[Image:Review_239195285364_7846_Fig_8_circle_contour_temp.png|200px|]]<math>\;</math>[[Image:Review_239195285364_1490_Fig_8_circle_contour_flux.png|200px|]]''</span></div>
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">''''Figure 5.3:''' Temperature distribution and Flux magnitude fields for the finest mesh.</span></div>


Figures [[#5.1 First example|5.4]](a), [[#5.1 First example|5.4]](b) and [[#5.1 First example|5.4]](c)  show the graphs of the temperature and flux magnitude values along a diameter of the circle for the different meshes of Figures [[#5.1 First example|5.4]](b), [[#5.1 First example|5.4]](c) and [[#5.1 First example|5.4]](d) respectively.

{| class="floating_imageSCP" style="text-align: center; "
|-
| (a)
|[[Image:Review_239195285364_4472_Fig23a_circle.png|260px|]]
|[[Image:Review_239195285364_6864_Fig23a_circle-flux.png|260px|]]
|-
| (b)
|[[Image:Review_239195285364_4411_Fig23b_circle.png|260px|]]
|[[Image:Review_239195285364_6460_Fig23b_circle-flux.png|260px|]]
|-
| (c)
|[[Image:Review_239195285364_8672_Fig23c_circle.png|260px|]]
|[[Image:Review_239195285364_5450_Fig23c_circle-flux.png|260px|]]
|- style="text-align: center; font-size: 75%;"
| colspan="3" | '''Figure 5.4:''' Temperature and Flux magnitude graphs along a diameter of the circle for different meshes:      (a) Graphs for the Mesh in Figure [[#5.1 First example|5.4]](b);      (b) Graphs for the Mesh in Figure [[#5.1 First example|5.4]](c);      (c) Graphs for the Mesh in Figure [[#5.1 First example|5.4]](d);
|}
 

As can be seen from Table [[#5.1 First example|5.1]] and Figure [[#5.1 First example|5.1]], DEC behaves very well on coarse meshes. Note that the maximum temperature values calculated with DEC matches the exact  theoretical value even on coarse meshes. As expected, the results of DEC and FEML are similar for fine meshes.  We would also like to point out the the computational costs of DEC and FEML are very similar.

===5.2 Second example===

In this example, we consider a region in the plane whose boundary consists of segments of a straight line, a circle, a parabola, a cubic and an ellipse (see Figure [[#5.2 Second example|5.5]]).


{| class="floating_imageSCP" style="text-align: center; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_239195285364_6878_Fig_10a_patata_prb.png|260px|]]
|[[Image:Review_239195285364_1159_Fig_10b_patata_geom.png|260px|]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 5.5:''' Region with linear, quadratic and cubic boundary segments, together with boundary conditions.
|}


The meshes used in this example are shown in Figure [[#5.2 Second example|5.6]] and vary from coarse to very fine.

{| class="floating_imageSCP" style="text-align: center; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_239195285364_6629_Fig_11c_patata_m2.png|260px|]]
|[[Image:Review_239195285364_5402_Fig_11d_patata_m3.png|260px|]]
|[[Image:Review_239195285364_3774_Fig_11e_patata_m4.png|260px|]]
|-
| (a)
| (b)
| (c)
|- style="text-align: center; font-size: 75%;"
| colspan="3" | '''Figure 5.6:''' Three of the meshes used in the first example.
|}
        

We set

* the source term <math display="inline">q=20.2</math>;
* the heat diffusion constant <math display="inline">k=80.2</math>;
* Newmann boundary condition <math display="inline">h=100</math> along the inner elliptical boundary;
* Dirichlet boundary condition <math display="inline">u=10</math> along the external boundary.

The numerical results for the maximum temperature and flux magnitude values are exemplified in Table [[#5.2 Second example|5.2]].

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math> \begin{array}{|c|c|c|c|c|c|c|} &  & & &  \\ {Mesh} & \# { nodes} &\# {elements} & {Max. Temp. Value} & {Max. Flux Magnitude}  \\ &&&{DEC \quad FEML}&{DEC \quad FEML} \\
\mbox{ Figure 5.2(a)} &162 &268  & {129.07 \quad 128.92} & {467.22  \quad 467.25}\\
\mbox{ Figure 5.2(b)} &678 &1223 & {129.71 \quad 129.70} & {547.87  \quad 548.42} \\
\mbox{ Figure 5.2(c)} &9489 &18284  & {130.00  \quad 130.00} & {591.58  \quad 591.63} \\
&27453 &53532  & {130.01 \quad 130.01} & {597.37  \quad 597.38}  \\
&651406 &1295960 &{130.02  \quad 130.02} & {602.19  \quad 602.19}  \\ \end{array} </math>
|}
|- style="text-align: center; font-size: 75%;"
| colspan="3" | '''Table 5.2:''' Numerical simulation results.
|}

The temperature and flux-magnitude distribution fields are shown in Figure [[#5.2 Second example|5.7]].

{| class="floating_imageSCP" style="text-align: center; margin: 1em auto; width: 100%;max-width: 100%; "
|-
|[[Image:Review_239195285364_1649_Fig_14_patata2_contour_temp.png|260px]]
|[[Image:Review_239195285364_8633_Fig_14_patata2_contour_flux.png|260px]]
|-
| (a)
| (b)
|- style="text-align: center; font-size: 75%;"
| colspan="3" | '''Figure 5.7:''' Temperature and Flux magnitude distribution fields.
|}


Figure [[#5.2 Second example|5.8]] shows the graphs of the temperature and the flux magnitude along a horizontal line crossing the elliptical boundary for the first two meshes.

{| class="floating_imageSCP" style="text-align: center; margin: 1em auto; width: 100%;max-width: 100%;"
|-
| (a)
|[[Image:Review_239195285364_4170_Fig_15b_patata2_m2.png|260px]]
|[[Image:Review_239195285364_1673_Fig_15b_patata2_m2_flux.png|260px]]
|-
| (b)
|[[Image:Review_239195285364_8948_Fig_15c_patata2_m3.png|260px]]
|[[Image:Review_239195285364_8624_Fig_15c_patata2_m3_flux.png|260px]]
|- style="text-align: center; font-size: 75%;"
| colspan="3" | '''Figure 5.8:''' Temperature and Flux magnitude graphs along a horizontal line crossing the elliptical boundary for the first two meshes.
|}


As can be seen from Table [[#5.2 Second example|5.2]] and Figure [[#5.2 Second example|5.8]],  the performance of DEC is very similar to that of FEML in this example.  As expected, when the mesh is refined, the two methods converge to the same values.

==6 Conclusions==

DEC is a relatively recent discretization scheme for PDE 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:

<ol>

<li>We have presented 2D DEC in a simplified manner,  avoiding references to the theory of differential forms and motivating geometrically the new operators.  </li>
<li>We have carried out a numerical comparison between DEC and FEML by solving the 2D Poisson equation  on two cirved domains. The numerical experiments show the solutions obtained with DEC on coarse meshes are as good or better as those of FEML.  On the other hand, the experiements also show numerical convergence. </li>
<li>The computational cost of DEC is similar to that of FEML. </li>

</ol>

''Acknowledgements''. The second named author was partially supported by a grant of CONACYT, 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.

<span style="text-align: center; font-size: 75%;">

===BIBLIOGRAPHY===

<div id="cite-1"></div>
'''[[#citeF-1|[1]]]'''  D. Arnold, R. Falk, and R. Winther: “Finite element exterior calculus: From Hodge theory to numerical stability”, Bulletin of the American Mathematical Society 47 (2010), no. 2, 281?354.  

<div id="cite-2"></div>
'''[[#citeF-2|[2]]]'''  S. Botello, M.A. Moreles, E. Oñate: “Modulo de Aplicaciones del Método de los Elementos Finitos para resolver la ecuación de Poisson: MEFIPOISS.”   Aula CIMNE-CIMAT, Septiembre 2010, ISBN 978-84-96736-95-5.  

<div id="cite-3"></div>
'''[[#citeF-3|[3]]]'''  E. Cartan: ``Sur certaines expressions différentielles et le probleme de Pfaff". Annales Scientifiques de l'École Normale Supérieure. Série 3. Paris: Gauthier-Villars. 16: 239?332 (1899)   

<div id="cite-4"></div>
'''[[#citeF-4|[4]]]'''   K. Crane, F. de Goes, M. Desbrun, P. Schröder: “Digital geometry processing with discrete exterior calculus.” ACM SIGGRAPH 2013 Courses. ACM, 2013.  

<div id="cite-5"></div>
'''[[#citeF-5|[5]]]'''   I. Dassios, A. P. Jivkov, A. Abu-Muharib, P. James:  “A mathematical model for plasticity and damage: A discrete calculus formulation.” Journal of Computational and Applied Mathematics 312 (2017): 27-38.  

<div id="cite-6"></div>
'''[[#citeF-6|[6]]]'''  M. Desbrun, E. Kanso,  Y. Tong: “Discrete differential forms for computational modeling.” In SIGGRAPH 06: ACM SIGGRAPH 2006 Courses, pages 39&#8211;54, New York, NY, USA, 2006. ACM.  

<div id="cite-7"></div>
'''[[#citeF-7|[7]]]'''  M. Griebel, C. Rieger, A. Schier: “Upwind Schemes for Scalar Advection-Dominated Problems in the Discrete Exterior Calculus.” Transport Processes at Fluidic Interfaces. Birkhäuser, Cham, 2017. 145-175.  

<div id="cite-8"></div>
'''[[#citeF-8|[8]]]'''  A. N. Hirani: “Discrete exterior calculus”. Diss. California Institute of Technology, 2003.  

<div id="cite-9"></div>
'''[[#citeF-9|[9]]]'''  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): 151-169.  

<div id="cite-10"></div>
'''[[#citeF-10|[10]]]'''  A. N. Hirani, K. Kalyanaraman, E. B. VanderZee:  “Delaunay Hodge star”, Comput. Aided Des. 45 (2013) 540-544]  

<div id="cite-11"></div>
'''[[#citeF-11|[11]]]'''   M. S. Mohamed, A. N. Hirani, R. Samtaney:  “Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes.” Journal of Computational Physics 312 (2016): 175-191.  

<div id="cite-12"></div>
'''[[#citeF-12|[12]]]'''  E. Oñate:  “4 - 2D Solids. Linear Triangular and Rectangular Elements,” in Structural Analysis with the Finite Element Method.  Linear Statics, Volume 1: Basis and Solids, CIMNE-Springer, Barcelona, 2009. Pages 117-157, ISBN 978-1-4020-8733-2    

<div id="cite-13"></div>
'''[[#citeF-13|[13]]]'''   O. C. Zienkiewicz, R. L. Taylor and J. Z. Zhu: “3 - Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches,” In The Finite Element Method Set (Sixth Edition), Butterworth-Heinemann, Oxford, 2005, Pages 54-102, ISBN 9780750664318,   

</span>