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Some error estimates for the DEC method in the plane

07/15/2019
by   Ruben Carrillo, et al.
CIMAT
0

We show that the Discrete Exterior Calculus (DEC) method can be cast as the earlier box method for the Poisson problem in the plane. Consequently, error estimates are established, proving that the DEC method is comparable to the Finite Element Method with linear elements. We also discuss some virtues, others than convergence, of the DEC method.

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1. Introduction

Discrete Exterior Calculus is introduced in [6]

as a discrete version of Exterior Differential Calculus. Most mathematical objects in the continuous theory and the potential applications, have been explored actively with DEC. An application of interest in this work, is the numerical approximation of Partial Differential Equations (PDE). The attractive feature is that differential forms are the natural language for physical laws modeled by PDE.

At present, the method has proven successful. For instance, in [7] it is applied to solve Darcy flow and Poisson’s equation. As illustrated in [9], the method is natural for curved meshes. Therein, they show an application to the incompressible Navier-Stokes equations over surface simplicial meshes.

These numerical results are proof that the DEC method is a welcomed addition to PDE numerics. But, to our knowledge, theoretical results on convergence are scarce at best. There are works exploring numerical convergence such as the recent by Mohamed et al [8].

In [3], the authors show that in simple cases (e.g. flat geometry and regular meshes), the equations resulting from DEC are equivalent to classical numerical methods, finite difference or finite volume. It is well known that the latter involves a dual mesh in close analogy to the DEC method. This leads to the earlier paper of [1], where the box method, a finite volume type method, is analyzed. It is straightforward to join the dots and see that the box and DEC methods coincide for the Poisson equation and other simple self adjoint variants. Our purpose is to provide a proof of this fact while introducing the basics of the DEC method. Consequently a convergence result is established, namely the DEC method converges comparable to the Finite Element Method with linear elements. We do not aim for generality, thus our description of the DEC and BOX methods just provides enough details as to make the connection.

The outline is as follows.

In Section 2 we review the BOX method following closely [1]. Local aspects, box-wise, are stressed. In Section 3 we introduce cell complexes in DEC theory, and interpret the meshes in the FEM and BOX methods as the primal and dual cell complexes respectively. The discrete operators to approximate the Poisson equation are presented in Section 4. The proof of DEC equals BOX method is also carried out. Next, error estimates and first order convergence are established. A case for DEC is made, when classical methods are limited by geometric issues.

2. The box method for the Poisson equation in 2D

This section is a summary of [1].

2.1. Meshes

Let be a bounded polygonal domain in and its boundary. Let a Finte Element triangulation with vertices

To each vertex associate a region consisting of those triangles with as a vertex, as in Figure 1.

Figure 1. Region .

A box is a basic element of a dual mesh constructed as follows, see Figure 2. For each triangle select a point , later we shall restrict the mesh in order to select the circumcenter.

Figure 2. Triangle and box elements

The point is connected with straight line segments (, , ) to the edge midpoints of , (, , ). We obtain a partition of in three subregions (, , ). Sometimes for simplicity, denote both the set and its length (area).

We are led to a dual mesh for made out of boxes. For each vertex there is a corresponding box , consisting of the union of subregions in which have as a corner, see Figure 3.

Figure 3. Box

The straight line segments from the selected points to the edge midpoints are collected in a set of edges, .

We stress the following.


(2.1) Definition. For a box , let us define its boundary as the intersection of the topological boundary with the set of edges.


Notice that only for vertices in the open set the boundary is the topological one.

2.2. Function spaces

Let and denote the usual Sobolev spaces equipped with the norms

where

If the norm is equivalent to the energy norm

In we also define bilinear form

Following the linear Finite Element Method, let be the subspace of continuous piecewise linear polynomials associated with . Also consider

The subspace of , is the space of discontinuous piecewise constants with respect to the boxes.

Let the usual nodal basis for satisfying

Let the basis for

consisting of characteristic functions for

.

If , then for some unique scalars we have

Easily, the map

given by

is bijective. Moreover, and take the same value at the vertices of .


Let be the subspace of of functions that are zero in . For , the energy norm is defined by


A basic result is the following,


(2.2) Lemma. Let , and . Then

where is the outward pointing normal.

Proof. (Lemma 3 in[1])


An important observation is that the proof is local. That is, it suffices to show for , , namely

The equality is shown triangle by triangle for such that .

Let us assume that the triangle , it is proven that

Here

where is a cyclic permutation of .

2.3. The Poisson equation

Consider the Dirichlet Boundary Value problem,

where .

Let be the -weak solution of the Poisson equation. This solution satisfies

The FEM solution with linear elements is such that

In light of the lemma, let us introduce the bilinear form

Define by and the subspaces of and whose elements are zero on .


The box method consists on finding such that


The method leads to the linear system

where, by the proposition


(2.3) Remark. The stiffness matrix is identical to that arising from the Finite Element Method with continuous piecewise linear approximations. The corresponding linear systems differ only in the right hand side which are close in an average sense.

3. FEM and BOX meshes and cell complexes

In this section we describe the correspondence between FEM and BOX meshes and cell complexes.

3.1. Primal Complex

A dimensional simplex is the convex hull of geometrically independent points .

Let the -dimensional face with the vertex removed.


(3.1) Definition. A simplicial complex , is a collection of simplices in such that

  1. Every face of a simplex of is in .

  2. The intersection of any two simplices of is either a face of each of them or it is empty.

The union of all simplices of treated as a subset of is called the underlying space of and is denoted by .


(3.2) Definition. A simplicial complex of dimension is called a manifold-like simplicial complex if and only if is a manifold, with or without boundary. More precisely

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

  2. Each point on has a neighborhood homeomorphic to or dimensional half-space.


An oriented simplex is denoted by . Its orientation corresponds to the orientation of the corner basis at . Namely, . It is equivalent to state that two orderings of the vertices have the same orientation if they differ one from one another by an even permutation.

Let be a -dimensional face. We say that the induced orientation on this face is the same of if is even. If

is odd, it is the opposite.

Assume that two oriented -dimensional simplices, , share a face of dimension . They have the same orientation if the induced orientation of the shared face induced by is the opposite to that induced by .


(3.3) Definition. A manifold-like simplicial complex of dimension is called an oriented manifold-like simplicial complex if adjacent -simplices (i.e., those that share a common -face) have the same orientation (orient the shared -face oppositely) and simplices of dimensions and lower are oriented individually.


2D FEM interpretation:

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

  • For two adjacent triangles, orientation corresponds to set outward normals on the intersecting side, opposite to each other.

3.2. Dual Complex

Given the triangulation , a point is selected in the closure of each triangle in the box method. In the context of DEC, a natural choice is the circumcenter.


(3.4) Definition. The circumcenter of a -simplex is given by the center of the -circumsphere. We will denote the circumcenter of a simplex by . If the circumcenter of a simplex lies in its interior we call it a well-centered simplex. A simplicial complex all of whose simplices (of all dimensions) are well-centered will be called a well-centered simplicial complex.


Hereafter we shall consider triangulations corresponding to a well-centered simplicial complex.


(3.5) Definition. The circumcentric subdivision of a well-centered simplicial complex of dimension is denoted , and it is a simplicial complex with the same underlying space as and consisting of all simplices (each of which is called a subdivision simplex) of the form for (note that the index here is not dimension since it is a subscript). Here (i.e., is a proper face of for all ) and the are in .

Each subdivision simplex in a given simplex will be called a subdivision simplex of . Of these, a simplex () will be called a subdivision -simplex of .


(3.6) Circumcentric subdivision in 2D: Consider a simplicial complex with vertices , and , i.e., the complex consists of a triangle , its edges and its vertices. Then consists of the following elements:

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

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

  • (the -simplices of ): consists of triangles, for instance

    .


(3.7) Definition. Let be a well-centered manifold-like simplicial complex of dimension and let be one of its simplices. The circumcentric dual operator is given by

where the coefficient ensure the the orientation of is consistent with the orientation of the primal simplex, and the ambient volume-form.


The union of the interiors of the simplices in the definition of is the (circumcentric) duall cell also denoted by . We will call each -simplex an elementary dual simplex of . This is an -simplex in . The collection of dual cells is called the dual cell decomposition of . This is dual cell complex and will be denoted .


2D BOX interpretation:

If is a vertex in the triangulation of , then the dual cell of the simplex is the box .

4. DEC approximation of the Poisson equation

We shall use the FEM-BOX notation when appropriate.

4.1. Chains, Forms, Integrals and Derivatives

We start with an oriented manifold-like simplicial complex of dimension.


(4.1) Definition. The space of chains of simplices is given by

where is the number of simplices. is equipped with an Abelian group structure.


(4.2) Definition. The space of discrete forms (cochains) is the dual space of chains, namely


(4.3) Definition. The integral of a cochain over a chain is defined by


This bilinear pairing of cochain with chain is also denoted using bracket notation

The same construction can be applied to the dual cell complex. We denote the Abelian group of chains on the dual complex by

Also, the space of forms on the dual complex is denoted by


(4.4) Definition. The th boundary operator denoted by is the map

given by

where measn that is omitted.


(4.5) Definition. Let be a cochain. The th discrete exterior derivative of is the transpose of the st boundary operator

We have


The discrete exterior derivative operates on primal cochains. For dual cochains we have

The negative sign comes from the orientation on the dual mesh induced by the orientation on the primal mesh. See [2].

Two more facts. It is readily seen that

and the Stokes’ Theorem holds,

4.2. The Discrete Hodge star operator and the Poisson Equation

In the exterior calculus for smooth manifolds, the Hodge star, denoted , is an isomorphism between the space of forms and forms. It is apparent that a definition of a discrete Hodge star ought to involve this relation in correspondence with the fact that the dual of a simplex is a cell. This is accomplished with the bilinear pairing.


(4.6) Definition. The discrete Hodge star operator is a map

For a cochain it is defined as

for all simplices .


For all cochains , it follows that

Also the analogue for chains is valid. Namely,


The Poisson equation.


Let us denote by the measure of a set . Point sets are assigned unit measure. We restrict our discussion to the plane, and regard as a well-centered oriented manifold-like simplicial complex of dimension 2.

Assume that and are primal forms. Consequently, a DEC approximation of the Poisson equation will result on nodal values for .

Since is a dual form we may write the Poisson equation in DEC formalism as follows,

Let be a vertex in the triangulation . It is also a simplex . Applying the previous equation to its dual cell, we have

By the definition of Hodge star, the right hand side becomes

But , thus

Notice that the dual cell of is exactly the box ,

Hence, in box notation we have

For the left hand side we prove the following


(4.7) Theorem.

Proof.

Discrete derivatives are the transpose of boundary operators, thus

It is apparent that

By linearity of the boundary operator and the integral, it is enough to show the equality for a triangle , . Let be one of such triangles, or the primal simplex .

Let us discuss the orientation of the dual simplices and . For the former, consider the simplex . This is related to the area form , up to a sign determined by the relative orientation of and . Thus we have that

Thus we have that the correct orientation for the simplex is given by

Similarly,

and the orientation for the simplex is given by

We are led to compute in the simplex

By the definition of the dual cell operator

Applying the definition of the discrete Hodge star operator we obtain

Now

The result follows.


Consequently, in the dual cell we have

Assembling these block submatrices we obtain , the DEC solution of this system with zero boundary conditions.

4.3. Error estimates in energy norm

We show that the stiffness matrix of de BOX and DEC methods coincide.


Let us consider the triangle in Figure 4. Using the fact that

Figure 4.

and

we obtain by elementary geometry that

(1)

It is apparent that this is valid for every vertex in We are led to the following result.


(4.8) Theorem. Let be piecewise constant with respecto to . Then, there is a positive constant such that

Proof.

Multiplying the Poisson equation by and integrating by parts we obtain

Recall that the box solution satisfies

Thus , for piecewise constant with respecto to we have from equation (1) that , so satisfies

It follows that

Finally, continue with the proof of Theorem 1 and Corollary 1 in [1].

4.4. General triangulations

Let us assume that the triangle . In the box method, the line integral

depends only on and , the endpoints of the integration path. So the location of the distinguished point is arbitrary.

For the DEC method, we argue that the circumcenter need not be in the interior of .

Starting with the FEM local matrix computation, it is straightforward to see that

These expressions are valid regardless of the location of the circumcenter and can, indeed, take negative values. Such signs are essential for the calculations to work for general meshes. In [4] we tested the DEC method for bad quality meshes, in the FEM jargon. As expected, The numerical performance was of the first order in these not well centered meshes.

References

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  • [4] R. Herrera, S. Botello, H. Esqueda and M. A. Moreles, A geometric description of Discrete Exterior Calculus for general triangulations, Rev. int. metodos numer. calc. diseño ing. (Online first) URL https://www.scipedia.com/public/Herreraetal2018b
  • [5] Hirani, A. N. (2003). Discrete exterior calculus (Doctoral dissertation, California Institute of Technology).
  • [6] A. N. Hirani: “Discrete exterior calculus”. Diss. California Institute of Technology, 2003.
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  • [8] Mohamed, Mamdouh S., Anil N. Hirani, and Ravi Samtaney. ”Numerical convergence of discrete exterior calculus on arbitrary surface meshes.” International Journal for Computational Methods in Engineering Science and Mechanics 19.3 (2018): 194-206.
  • [9] Mohamed, Mamdouh S., Anil N. Hirani, and Ravi Samtaney. ”Discrete exterior calculus discretization of incompressible Navier?Stokes equations over surface simplicial meshes.” Journal of Computational Physics 312 (2016): 175-191.