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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 1-forms by the anisotropy tensor, i.e. we deduce the the discrete action of the anisotropy tensor on primal 1-forms. 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.
(1) Centro de Investigación en Matemáticas, Calle Jalisco s/n, Guanajuato, GTO 36023, México. Email: esqueda,rherrera,botello@cimat.mx
(2) Departamento de Matemáticas, Universidad de Guanajuato, Guanajuato, GTO 36023, México. Email: valerocar@gmail.com
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 Navier-Stokes 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 1-form on the plane. In Section 3, we deduce the discretization of the pullback operator on primal 1-forms. 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 area-weight assignment for the nodes. In Section 6, we re-examine 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.
Acknowledgements. 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.
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 .
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Figure 1: Triangle |
Such a mesh has
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There is a well known boundary operator
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(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
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If we consider
then the map (1), which sends the oriented triangle to a sum of its oriented edges, is represented by the matrix
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while the map (2), which sends the oriented edges to sums of their oriented vertices, is represented by the matrix
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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 (see [8,6]). More precisely, suppose we have a function discretized by its values at the vertices
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Its discrete derivative, according to DEC, is
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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.
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It is precisely in this sense that, according to DEC,
Let
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The dual mesh of the primal mesh consisting of a single triangle is constructured as follows:
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Figure 2: Circumcenter |
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Figure 3: Dual segment |
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Figure 4: Dual quadrilateral |
For the Poisson equation in 2D, we need two matrices: one relating original edges to dual edges, and another relating vertices to dual cells.
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assigned to the segment ;
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assigned to the edge .
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assigned to the segment ;
In other words,
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assigned to the quadrilateral ;
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assigned to the quadrilateral ;
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assigned to the quadrilateral .
In other words,
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A differential 1-form on can be thought of as a function whose arguments are a point in and a vector in , i.e.
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and which is linear on the vector argument.
The dot product of allows us to see continuous 1-forms 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 1-form
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where . On the other hand, from a continuos 1-form we can build a vector field as follows
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where
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Given a tensor (a linear transformation) and the vector field associated to a 1-form as above, consider the vector field . For any vector , the value of the corresponding continuous 1-form is given by the product
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which is the value of the 1-form called the pullback of by the tensor .
In order to produce a primal 1-form from such a continuous 1-form 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
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This is equivalent to calculating the line integral (circulation) of the corresponding vector field along the path , , parametrizing the edge
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In Section 2, we dealt with the primal discretization of a continuous 1-form. Now we wish to understand the pullback of a primal 1-form. In order to do this, we will interpolate a primal 1-form using the Whitney interpolation forms to produce a continuous 1-form (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 Hassler Whitney in 1957 (see [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 1-forms.
First consider the Finite Element linear basis functions (or Whitney interpolation functions)
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and their differentials
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The Whitney interpolation 1-forms are
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These differential 1-forms are such that, along the edge
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i.e. or .
Now, suppose we have a primal 1-form given on our triangle by the collection of numbers
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We form a continuous 1-form as follows
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which is such that, for
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Now we will calculate
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for an arbitrary linear transformation
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Let
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and
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As it turns out,
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where
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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 1-forms is
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(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 pull-back operator.
In this section, we deduce the DEC formulae for the local flux, the local anisotropic flux and the local anisotropy operator for primal 1-forms.
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
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Thus, we have three Vector Calculus identities
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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
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Thus, if we want to find a discrete gradient vector of at the point , we need to solve the equations
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If
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then
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Now, if we were to find a discrete gradient vector of at the point , we need to solve the equations
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The vectors solving these equations is actually equal to . Indeed, consider
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so that
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(10) |
Thus, adding up (7) and (9) we get
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(11) |
Subtracting (8) from (10) we get
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(12) |
Since and are linearly independent and the two inner products in (11) and (12) vanish,
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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
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This means that the primal 1-form 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
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where and are the basis functions given in (3). Explicitly
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where
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is the area of the triangle, so that the FEML flux is given by
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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
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and recall the anisotropic Poisson equation
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First recall that, in Exterior Differential Calculus, for any ,
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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
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In order to produce a primal 1-form 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.
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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
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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 1-forms from Section 3. Thus, using (6), is the following matrix where we have set , i.e.
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where
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Consider the following figure, where denotes the anti-clockwise rotation
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Figure 5: Geometric interpretation of the entries of the anisotropy tensor discretization |
We have
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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
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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
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where and . Following the discretization of the discretized divergence operator [6], the corresponding local DEC discretization of the anisotropic Poisson equation is
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or equivalently
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(13) |
In order to simplify the notation, consider the lengths and areas defined in the Figure 6.
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Figure 6: Triangle |
Now, the discretized equation (13) looks as follows:
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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
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where is the matrix representing the anisotropic diffusion tensor, and the matrix is given explicitly by
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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
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Now, let us consider the first diagonal entry of the local FEML anisotropic Poisson diffusive matrix ,
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where is the anticlockwise rotation,
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In this notation, the diffusive term in local FEML is given as follows
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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.
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Figure 7: Circumscribed triangle. |
We have the following identities:
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For instance, we claim that
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Indeed,
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Thus,
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All we have to do now is show that
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Note that, since is orthogonal to , must be parallel to . Thus,
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(14) |
Now we are going to express in terms of . Let us consider
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where are coefficients to be determined. Taking inner products with and we get the two equations
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Solving for and
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Substituting all the relevant quantities in (14) we have, for instance, that the coefficient of is
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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
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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,
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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
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Figure 8: Negative (exterior) angles measured in an obtuse triangle. |
and some quantities can even be zero. For instance, if
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then
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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.
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Figure 9: Area weight assigned to |
The area is computed similarly, where the triangle is assigned negative area (see Figure 10).
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Figure 10: Area weight assigned to |
Note that for , the two triangles and both have positive areas (see Figure 11).
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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 [11,13,2].
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Figure 12: Square and inner circle with different conditions. |
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(13) | (13) | (c) |
Figure 13 |
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(a) | (b) | (c) |
Figure 14: Six of the meshes used in the first example. |
The numerical results for the maximum temperature value are exemplified in Table 7.1.
Mesh | nodes | elements | Max. Temp. Value | Max. Flux Magnitude | ||
DEC | FEML | DEC | FEML | |||
Figure 14(a) | 49 | 80 | 5.51836 | 5.53345 | 13.837 | 13.453 |
Figure 14(b) | 98 | 162 | 5.65826 | 5.66648 | 14.137 | 14.024 |
Figure 14(c) | 258 | 466 | 5.70585 | 5.71709 | 14.858 | 14.770 |
Figure 14(d) | 1,010 | 1,914 | 5.72103 | 5.72280 | 15.008 | 15.006 |
Figure 14(e) | 3,813 | 7,424 | 5.72725 | 5.72725 | 15.229 | 15.228 |
Figure 14(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 |
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(a) Contour Fill of Temperatures | (b) Contour Fill of Flux vectors on Elems |
Figure 15: Temperature and flux-magnitude distribution fields of the first example. |
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(a) | (b) |
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(c) | (d) |
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(e) | (f) |
Figure 16: Temperature and Flux magnitude graphs of the first example along a cross-section of the domain for different meshes. |
In Table 1, we show some global error metrics for different meshes. Figure 17 shows the error evolution in norm for this example.
Mesh | Nodes | norm | |
1 | 49 | 1.0537e-02 | 1.4438e-01 |
2 | 98 | 4.2447e-03 | 4.8558e-02 |
3 | 258 | 6.9781e-04 | 3.0390e-03 |
4 | 1,010 | 8.8386e-05 | 1.4877e-04 |
5 | 3,813 | 1.0736e-05 | 7.7369e-06 |
6 | 13,911 | 1.4422e-06 | 4.9791e-07 |
7 | 50,950 | 1.7582e-07 | 2.9608e-08 |
8 | 135,518 | 3.2621e-08 | 2.9233e-09 |
9 | 298,299 | 7.3566e-09 | 3.3610e-10 |
10 | 603,440 | 1.8577e-09 | 4.9496e-11 |
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Figure 17: DEC |
Let us solve the Poisson equation in a circle of radius one centered at the origin under the following conditions (see Figure 18):
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Figure 18: Disk of radius one. |
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(a) | (b) | (c) | (d) |
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(e) | (f) | ||
Figure 19: First six meshes used for unit disk. |
The numerical results for the maximum temperature value () are exemplified in Table 7.2 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 19(a) | 17 | 20 | 10.20014 | 10.19002 | 0.42133 | 0.43865 |
Figure 19(b) | 41 | 56 | 10.20007 | 10.19678 | 0.48544 | 0.49387 |
Figure 19(c) | 201 | 344 | 10.20012 | 10.20158 | 0.52470 | 0.52428 |
Figure 19(d) | 713 | 1304 | 10.20000 | 10.19969 | 0.54143 | 0.54224 |
Figure 19(e) | 2455 | 4660 | 10.20000 | 10.19990 | 0.54971 | 0.55138 |
Figure 19(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 |
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(a) Contour Fill of Temperatures | (b) Contour Fill of Flux vectors on Elems |
Figure 20: Temperature distribution and Flux magnitude fields for the finest mesh of the second example. |
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(a) | (b) |
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(c) | (d) |
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(e) | (f) |
Figure 21: Temperature and Flux magnitude graphs of the second example along a diameter of the circle for different meshes: mesh in Figure |
In Table 2, we show some global error metrics for different meshes. Figure 22 shows the error evolution in norm for this example.
Mesh | Nodes | norm | |
1 | 17 | 1.5818e-04 | 2.3555e-06 |
2 | 51 | 2.5395e-05 | 1.5639e-07 |
3 | 201 | 3.3643e-06 | 1.0517e-08 |
4 | 713 | 5.1563e-07 | 8.3543e-10 |
5 | 2,455 | 8.9235e-08 | 7.6073e-11 |
6 | 8,180 | 3.1731e-08 | 2.9858e-11 |
7 | 20,016 | 2.0217e-08 | 2.6580e-11 |
8 | 42,306 | 1.4421e-08 | 2.7062e-11 |
9 | 82,722 | 9.8533e-09 | 2.6164e-11 |
10 | 156,274 | 7.1352e-09 | 2.5954e-11 |
11 | 420,013 | 4.4277e-09 | 2.6151e-11 |
12 | 935,016 | 2.9635e-09 | 2.6003e-11 |
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Figure 22: DEC |
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Figure 23: Egg-like 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 |
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Figure 24: 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 |
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(a) | (b) | (c) |
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(d) | ||
Figure 25: Meshes for layered egg-like figure. |
The numerical results for the maximum temperature value () are exemplified in Table 7.3 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 25(a) | 342 | 616 | 2.79221 | 2.79854 | 18.41066 | 18.40573 |
Figure 25(b) | 1,259 | 2,384 | 2.83929 | 2.84727 | 18.93838 | 18.91532 |
Figure 25(c) | 4,467 | 8,668 | 2.85608 | 2.85717 | 19.13297 | 19.13193 |
Figure 25(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 |
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(a) Contour Fill of Temperatures | (b) Contour Fill of Flux vectors on Elems |
Figure 26: Temperature distribution and Flux magnitude fields for the finest mesh of the third example. |
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(a) | (b) |
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(c) | (d) |
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(e) | (f) |
Figure 27: Temperature and Flux magnitude graphs of the third example along a cross-section of the domain for different meshes: Mesh in Figure |
In Table 3, we show some global error metrics for different meshes. Figure 28 shows the error evolution in norm for this example.
Mesh | Nodes | norm | |
1 | 342 | 9.3638e-04 | 2.7786e-02 |
2 | 1,259 | 1.5173e-04 | 3.2731e-03 |
3 | 4,467 | 2.2110e-05 | 2.2937e-04 |
4 | 14,250 | 3.7233e-06 | 1.9151e-05 |
5 | 20,492 | 2.2311e-06 | 9.5930e-06 |
6 | 60,380 | 4.3769e-07 | 8.7330e-07 |
7 | 142,702 | 1.1664e-07 | 1.3926e-07 |
8 | 291,369 | 3.9764e-08 | 3.3314e-08 |
9 | 497,378 | 1.6680e-08 | 1.0275e-08 |
10 | 1,067,171 | 3.9594e-09 | 1.2190e-09 |
11 | 2,106,248 | 9.6949e-10 | 1.4415e-10 |
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Figure 28: DEC |
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 2-dimensional simplicial surfaces in 3D. Preliminary results on both problems are promising and competitive with FEML.
[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,
Published on 23/04/20
Submitted on 22/04/20
Licence: CC BY-NC-SA license
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