Small Trees for High Order Whitney Elements

In applied computations the need arises to define, for example, a discrete field with assigned curl or to represent a div-free field in a given discrete space. In the low degree case this need is often fulfilled by employing tree and co-tree techniques. The definition of tree and co-tree is thus revisited here in the frame of high order Whitney element reconstructions. We consider the case of fields that are reconstructed in a contractible polyhedral domain 
$$\varOmega \in \mathbb R^3$$

 Ω
 ∈
 
 
 ℝ
 
 
 3
 
 
 , with connected boundary ∂Ω, starting from their weights over suitable “small simplices” in a simplicial mesh 
$${\mathcal M}$$

 ℳ
 of the domain 
$$\bar \varOmega $$

 
 
 Ω
 
 ̄
 
 .

tree and co-tree approaches for r > 0 when fields in the high order Whitney spaces are represented on the basis of their weights on small simplices [7,9,10]. With this choice of degrees of freedom, the tree and co-tree concepts extend from r = 0 to r > 0 straightforwardly.

Basic Concepts
Let Ω ⊂ R 3 be a bounded polyhedral domain with Lipschitz boundary ∂Ω and M a simplicial mesh ofΩ. We denote by |A| the cardinality of the set A. For 0 ≤ k ≤ 3, let Δ k (T ) (resp. Δ k (M)) be the set of k-simplices of a mesh tetrahedron T (resp. of the mesh M). If we assign to each S ∈ Δ k (M) a real number c S we can define the k-chain c = S∈Δ 0 (M) c S S, i.e. a formal weighted sum of k-simplices S in M. One can add k-chains, namely (c +c) = S (c S +c S ) S, and multiply a k-chain by a scalar p, namely p c = S (p c S ) S. The set of all k-chains in M, here denoted C k (M), is a vector space, in one-to-one correspondence with the set of real vectors c = (c S ) S∈Δ k (M) . Each k-simplex S ∈ Δ k (M), can be associated with the elementary k-chain c with entries c S = 1 and cS = 0 forS = S. In the following we will use the same symbol S to denote the oriented k-simplex and the associated elementary k-chain.
The boundary operator ∂ takes a k-simplex S and returns the sum of all its (k − 1)-faces f with coefficient 1 or −1 depending of whether the orientation of the (k − 1)-face f matches or not with the orientation induced by that of the simplex S on f . Since the boundary operator is a linear mapping from C k (M) to C k−1 (M), it can be represented by a matrix ∂ of dimension |Δ k−1 (M)|×|Δ k (M)|, which is rather sparse, gathering the coefficients 0, −1, or +1. Note that in three dimensions, there are three nontrivial boundary operators acting, respectively, on edges, triangles and tetrahedra: ∂ 1 represented by the matrix G , ∂ 2 represented by R , and ∂ 3 represented by D . To fully specify ∂, we need to specify the boundary of each simplex S. By definition, we have , we have, respectively, The subscript is removed when there is no ambiguity, since the operator needed for a particular operation is indicated from the type of the operand (e.g., ∂ 3 when ∂ applies to tetrahedra). The notion of boundary can be extended to k-chains by linearity, ∂c = ∂( S∈Δ k (M) c S S) = S∈Δ k (M) c S ∂S.
We say that a k-chain c is closed if ∂ k c = 0. Non-trivial closed k-chains are called k-cycles and constitute the subspace Z k (M) = ker(∂ k ; C k (M)). A k-chain c is a boundary if it exists a (k + 1)-chain γ such that c = ∂ k+1 γ . The kboundaries constitute the subspace B k (M) = ∂ k+1 C k+1 (M). From the property ∂∂ = 0, we know that boundaries are cycles but not all cycles are boundaries, and we have is the homology spaces of order k of the mesh M, and the Betti' The presence of curl-free fields (resp. div-free fields) that are not the gradient of a scalar field (resp. the curl of a vector field) is indicated from the fact that b 1 = 0 (resp. b 2 = 0). We recall that Betti's numbers are topological invariants (i.e., they depend on the domain Ω up to a homeomorphism) and do not depend on the mesh M onΩ that is used to compute them (see [12] and an application in [11]).
For the high order case, we need to introduce some concepts of relative We thus talk about relative homology groups.
A k-cochain w (over the mesh M) is a linear mapping from C k (M) to R. They are discrete analogues to differential forms. For k > 0, the exterior derivative of the (k − 1)-form w is the k-form dw such that s dw = ∂s w for all s ∈ C k (M). With this simple equation relating the evaluation of dw on a simplex s to the evaluation of w on the boundary of this simplex, the exterior derivative is readily defined. We can naturally extend the notion of evaluation of a differential form w on an arbitrary chain by linearity: The operator d is the dual of the boundary operator ∂. As a corollary of the boundary operator property ∂∂ = 0, we have that dd = 0. Since we used arrays of dimension |Δ k (M)| to represent a k-cochain, the operator d can be represented by a matrix Again, we have one matrix for the exterior derivative operator for each simplex dimension. When a metric is introduced on the ambient affine space, the exterior derivative operator d stands for grad, curl, div, according to the value of k from 1 to 3, and it is represented by, respectively, G, R, D, the connectivity matrices of the mesh M.

Small Simplices, Weights and Potentials
We introduce the multi-index α = (α 0 , . . . , α s ) of s + 1 integers α i ≥ 0 and weight |α| = s i=1 α i . The set of multi-indices α with s + 1 components and weight r is denoted I(s + 1, r). We denote by v i the (Cartesian) coordinates of the node n i in R 3 . Given a multi-index α ∈ I(4, r), and a k-subsimplex S of T , the small simplex {α, S} is the k-simplex that belongs to the small tetrahedron with barycenter at the point of coordinates 3 , which is parallel and 1/(r + 1)homothetic to the (big) sub-simplex S of T . The notation {α, S} was first defined in [9]. The set of small tetrahedra of order r +1 > 1 can be visualized starting from the and connecting its points by edges parallel to those of T . (See, e.g., Fig. 1

.)
We denote by Λ k (Ω) the space of all smooth differential k-forms on Ω. The completion of Λ k (Ω) in the corresponding norm defines the Hilbert space be the space of so-called trimmed polynomial k-forms of degree r + 1 on T , with r ≥ 0, (as in [7]), and we define [4]).

Definition 1
The weights of a polynomial k-form u ∈ P − r+1 Λ k (T ), with 0 ≤ k ≤ 3 and r ≥ 0, are the scalar quantities on the small simplices {α, S} with α ∈ I(4, r) and S ∈ Δ k (T ). We now list some remarkable properties of the small simplices which are useful in the tree construction.

Property 1
The weights (1) of a Whitney k-form u ∈ P − r+1 Λ k (T ) on all the small simplex {α, S} of T are unisolvent, as stated in [7, Proposition 3.14]. The small simplices can thus support the degrees of freedom for fields u ∈ P − r+1 Λ k (T ), with 0 ≤ k ≤ 3 and r ≥ 0. Since the result on unisolvence holds true also by replacing T with F ∈ Δ n−1 (T ) then Tr F u ∈ P − r+1 Λ k (F ) is uniquely determined by the weights on small simplices in F . It thus follows that a locally defined u, with u |T ∈ P − r+1 Λ k (T ) and single-valued weights, is in H Λ k (Ω). We thus can use the weights on the small simplices {α, S} as degrees of freedom for the fields in the finite element space P − r+1 Λ k (M) being aware that their number is greater than the dimension of the space.

Property 3
The generated ( r+2 2 ) small faces on each face F of T , pave F together with the ( r+1 2 ) reversed triangles, denoted by ∇, contained in F . Similarly, the generated ( r+3 3 ) small tetrahedra contained in T pave T together with the ( r+2 3 ) octahedra, denoted by O, and the ( r+1 3 ) reversed tetrahedra, denoted by ⊥, contained in T , as shown in Fig. 1. Reversed octahedra and reversed tetrahedra are examples of "holes" in T (see [9,10]).
Property 4 Since homology is preserved by homotopy, in [10,Section 3.4], it is discussed the fact that the relative homology (i.e., the homology [modulo the holes' boundaries]), of the complex of small simplices is the same of the homology of M. This property is fundamental to build the tree for high order potentials when working with small simplices. The homology [modulo the holes' boundaries] can be translated in matrix notation, by showing that the boundary matrices associated with the small simplices, "modified" and "completed" (in a sense that we explain in the next section) by the relations [10, Proposition 3.5] are incidence matrices of a graph. To apply the theory presented in [10,Section 3.4] in a tetrahedron T ∈ Δ 3 (M), we need to introduce, for r > 0, two sets K 1 and K 2 of chains generated by the small simplices that belong to the boundary of some hole in T as follows: • K 1 are the chains generated by the boundary of the ( r+1 2 ) reversed triangle ∇ ⊂ F and that for each F ∈ Δ 2 (T ), and the boundary of the three faces out of four on the boundary ∂⊥ of each of the ( r 2 ) reversed tetrahedra ⊥ in T ; • K 2 are the chains generated by 4 out of 8 faces of the ( r+2 3 ) octahedra O in T . The involved faces are the small faces belonging to the boundary ∂O privated of ∂O ∩ (Δ 2 (T ) ∪ ∂⊥).
The two sets K 1 and K 2 satisfy the property ∂K 2 ⊂ K 1 , decisive to conclude that the relative homology [modulo the holes' boundaries] of the complex of the small simplices is the same as the homology of the original mesh M [10].

Trees and Graphs
As stated in [12], a directed graph G consists of two sets N and A of nodes and arcs, respectively, subjected to certain incidence relations, collected in the all-vertex incidence matrix M G ∈ Z |N |×|A| as follows: , if a ends in n, 0 , if a does not contain n.
An incidence matrix M of the graph G is any sub-matrix of M G with |N | − 1 rows and |A| columns. The node that corresponds to the row of M G that is not in M will be indicated as the reference node of G. A graph G is connected if there is a path between any two of its nodes. A tree T of a graph G is a connected acyclic subgraph of G. A spanning tree T s is a tree of G visiting all its nodes. Any connected graph G admits a spanning tree T s . We have now to particularize these notions for small simplices. In each tetrahedron T of the oriented mesh M, we consider the small mesh associated with L r+1 (T ) composed only of small tetrahedra, for a given r uniform all over the mesh M. The union of the small meshes for all T ∈ Δ 3 (M) is denoted M all .

A (Primal) Small Tree for the Gradient Problem
For r = 0, the graph G 1 has N = Δ 0 (M) and A = Δ 1 (M). The boundary matrix G is the all-vertex incidence matrix of the graph G 1 . Extracting a spanning 1-tree T 1 s from G 1 is equivalent to finding in G , minus one row, a submatrix of maximal rank (see [11] for a suitable and easy way of constructing T ). For r > 0, we have  Fig. 5 (fragmented visualization). Note that we can repeat this construction in the two-dimensional case.

A (Dual) Small Tree for the Divergence Problem
For r = 0, the graph G 2 is built on M * , the so-called dual mesh of M, as follows. Let us note that an internal face F ∈ Δ 2 (M) connects two adjacent tetrahedra T 1 , T 2 ∈ Δ 3 (M) whereas a boundary face F b ∈ Δ 2 (M) connects a tetrahedron T b ∈ Δ 3 (M) and the boundary ∂Ω. We can construct the following connected (dual) graph G 2 : the set of nodes, N , contains the barycenter of any tetrahedron T ∈ Δ 3 (M) together with one additional exterior node representing ∂Ω; the set of arcs, A, contains any face F ∈ Δ 2 (M). For r = 0, the matrix D associated with the boundary operator ∂ 3 , acting on C 3 (M), is an incidence matrix of the (dual) graph G 2 , with reference node the one corresponding to ∂Ω. Extracting a spanning tree T 2 s from G 2 is equivalent to finding in D a submatrix of maximal rank. For r > 0, let R 2 be the set of small faces chosen as follows: one small face for each octahedron O contained in K 2 (see the right side of Fig. 3 for the dashed small face in R 2 when r +1 = 2). To construct the graph G 2 for r > 0 we need to consider M * all , the dual mesh associated to M all , where nodes are the small tetrahedra and the arcs the small faces, apart from the ones in R 2 . To understand this, we can reason In green, the contribution of the small branches mapped from the green ones in the reference triangle. It is not necessary to report the red ones since they are either covered by the blue ones or omitted. The co-tree is in black blue tree). The small co-tree is in black. A similar construction can be repeated in 3D (both for k = 1 and k = 2) and it reflects the decomposition given, for instance, in [13] (Sect. 5). Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made. The images or other third party material in this chapter are included in the chapter's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.