Generic Geometries

Introduction

In the following we will give a definition of reference elements and subelement numbering. This is used to define geometries by prescibing a set of points in the space $\mathbf{R}^w$ .

The basic building block for these elements is given by a recursion formular which assignes to each set $E \subset \mathbf{R}^d$ either a pyramid element $E^\vert\subset \mathbf{R}^{d+1}$ and a prism element $E^\vert\subset \mathbf{R}^{d+1}$ with $E^\vert = \lbrace (x,\bar{x}) \mid x \in E, \bar{x} \in [0,1] \rbrace$ and $E^\circ = \lbrace ((1-\bar{x})x,\bar{x}) \mid x \in E, \bar{x} \in [0,1] \rbrace$ . The recursion starts with a single point $P_0=0\in\mathbf{R}^0$ .

For $ d=1,2,3 $ this leads to the following elements

: $L_1 = P_0^\vert = P_0^\circ = [0,1]$ is a line.
: $Q_2 = L_1^\vert$ is a cube and $S_2 = L_1^\circ$ is a simplex.
: $Q_3 = Q_2^\vert$ is a cube, $S_3 = S_2^\circ$ is a simplex, $\mathrm{pyramid}_3 = S_2^\vert$ is a pyramind, and $\mathrm{prism}_3 = Q_2^\circ$ is a prism.

In general if $ Q_d $

is a cube then $Q_d^\vert$ is also a cube and if $ S_d $

is a simplex then $S_d^\vert$ is also a simplex.

Based on the recursion formular we can also define a numbering of the subentities and also of the sub-subentities of $E^\vert$ or $E^\circ$ based on a numbering of $ E $ . For the subentities of codimension $ c $ we use the numbering

$E^\vert$ : we first numbers are assinged to the entities parallel to the axis in the same order as the subentites of the same codimension in ; then the subentities of codimension in the bottom followed by those in the top.
$E^\circ$ : in this case we first number the subentities of codimension in the bottom, followed by each subentity based on a subentity of codimension in .

For the subentity of codimension $ cc $

in a codimension $ c $

subentity

we use the numbering induced by the numbering the reference element corresponding to $ E' $

Here is a graphical representation of the reference elements:

One-dimensional reference element. For d=1 the simplex and cube are identical
Two-dimensional reference simplex (a.k.a. triangle)
Three-dimensional reference simplex (a.k.a. tetrahedron)

Face Numbering

Edge Numbering
Two-dimensional reference cube (a.k.a. quadrilateral)
Three-dimensional reference cube (a.k.a. hexahedron)

Face Numbering

Edge Numbering
Prism reference element

Face Numbering

Edge Numbering
Pyramid reference element

Face Numbering

Edge Numbering

In addition to the numbering and the corner coordinates of a reference element $\hat{E}$ we also define the barycenters $b(\hat{E})$ , the volume $|\hat{E}|$ and the normals $n_i(\hat{E})$ to all codimension one subelements.

The recursion formular is also used to define mappings from reference elements $ E $ to general polytop given by a set of coordinates for the corner points - together with the mapping $\Phi$ , the transpose of the jacobian $D\Phi^T(x)\in\mathbf{R}^{d,w}$ is also defined where $ d $ is the dimension of the reference element and $ w $ the dimension of the coordinates. This sufficies to define other necessary parts of a Dune geometry by LQ-decomposing $D\Phi^T$ : let $D\Phi^T(x) = LQ$ be given with a lower diagonal matrix $L\in \mathbf{R}^{d,d}$ and a matrix $Q\in\mathbf{R}^{d,w}$ which satisfies $ Q Q^T = I $ :

Jacobian inverse transpose
$D\Phi(x)^{-T}:=Q^T L^{-1}.$
Integration element
$\sqrt{\mathrm{det}(D\Phi(x)^T D\Phi(x))} = \sqrt{\mathrm{det}(LL^T)} = \Pi_{i=1}^d l_{ii}.$
Volume
$\int_{\hat{E}} \Pi_{i=1}^d l_{ii}(x) d\;x = |\hat{E}| \Pi_{i=1}^d l_{ii}(b(\hat{E})).$
(Here some assumptions on the degree of the integration element is used.)

The next sections describe the details of the construction.

Referemce Topology

We define the set $\mathcal{T}$ of reference topologies by the following rules:

$\mathcal{T}$ contains an element $\mathbf{p}$ that we call the point topology.
For $T \in \mathcal{T}$ , $\mathcal{T}$ contains an element $T^\vert$ that we call the prism over .
For $T \in \mathcal{T}$ , $\mathcal{T}$ contains an element $T^\circ$ that we call the pyramid over .

For each reference topology $T$

we define the following values:

Dimension: The point topology has dimension zero and the dimension of a prism or a pyramid topology over has dimension .
Size: For with we define the number through
- .
- If $T=t^\vert$ then $s_d(T)=2s_{d-1}(t)$ and for $c\in \{1,\dots,d-1\}$ we have $s_c(T)=s_{c}(t)+2s_{c-1}(t)$ .
- If $T=t^\circ$ then $s_d(T)=s_{d-1}(t)+1$ and for $c\in \{1,\dots,d-1\}$ we have $s_c(T)=s_{c-1}(t)+s_{c}(t)$ .
Subtopology: Given a reference topology of dimension and a codimension we now define the subtopology for :
- $S_{0,1}(T)=T$ and $S_{d,i}(T)=\mathbf{p}$
- For $T=t^\vert$ and $c\in \{1,\dots,d-1\}$ we define using the abbreviations $p=s_{c}(t),q=s_{c-1}(t)$
  $S_{c,i}(T) = \left\{\begin{array}{ll} S_{c,i}(t)^\vert & \mbox{for}\; i=1,\dots,p, \\ S_{c-1,i-p}(t) & \mbox{for}\; i=p+1,\dots,p+q, \\ S_{c-1,i-p-q}(t) & \mbox{for}\; i=p+q+1,\dots,p+2q. \end{array}\right.$
- For $T=t^\circ$ and $c\in \{1,\dots,d-1\}$ we define using the abbreviations $p=s_{c}(t),q=s_{c-1}(t)$
  $S_{c,i}(T) = \left\{\begin{array}{ll} S_{c-1,i}(t) & \mbox{for}\; i=1,\dots,q, \\ S_{c,i-q}(t)^\circ & \mbox{for}\; i=q+1,\dots,q+p. \end{array}\right.$

Referemce Domains

For each reference topology $T$

we assosiate the set of corners $\mathcal{C}(T):=(p_i(T))_{1=1}^{s_d(T)}\subset\mathbf{R}^d(T)$ defined through

$T=\mathbf{p}$ : $p_0(T)=0\in\mathbf{R}^0$
$T=t^\vert$ : $p_k(T)=(p_k(t),0),p_{d'+k}(T)=(p_k(t),1)$ for $k=1,\dots,d'$ , with .
$T=t^\circ$ : for $k=1,\dots,d-1$ and with

The convex hall of the set of points $\mathcal{C}(T)$ defines the reference domain $\mathrm{domain}(T)$ for the reference topology $T$

; it follows that

$\mathrm{domain}( \mathbf{p} ) := \lbrace 0 \rbrace \subset \mathbf{R}^0$ ,
$\mathrm{domain}( T^\vert ) := \mathrm{domain}( T ) \times [0,1]$ ,
$\mathrm{domain}( T^\circ ) := \lbrace ((1-\bar{x})x,\bar{x}) \mid x \in \mathrm{domain}( T ), \bar{x} \in [0,1] \rbrace$ .

Reference Elements and Mappings

A pair $E=(T,\Phi)$ of a topology $T$

and a map $\Phi:\mathrm{domain}( T ) \to \mathbf{R}^w$ with $w\geq d(T)$ is called an element.

The reference element is the pair $E(T)=(T,{\rm id})$ .

For a given set of points $\mathcal{C}:=(p_i)_{1=1}^{s_d(T)}\subset\mathbf{R}^w$ we define a mapping $\Phi_T(\mathcal{C},\cdot)\mathrm{domain}( T ) \to \mathbf{R}^w$ through $\Phi_T(\mathcal{C};p_i(T))=p_i$ for all $p_i(T)\in \mathcal{C}(T)$ . This mapping can be expressed using the recurive definition of the reference topologies through:

$\Phi_{\mathbf{p}}( (p_0) ; x ) = p_0$ ,
$\Phi_{T^\vert}( (p_1,\ldots,p_{s_d(T^\vert)} ; (x,\bar{x}) ) = (1 - \bar{x}) \Phi_T( p_1,\ldots,p_{s_d(T)} ; x ) + \bar{x} \Phi_T( p_{s_d(T)+1},\dots,p_{s_d(T^\vert)} ; x )$ with $x\in \mathrm{domain}( T )$ and $\bar{x}\in[0,1]$ .
$\Phi_{T^\circ}( p_1,\ldots,p_{s_d(T^\circ)}) ; (x,\bar{x}) ) = (1 - \bar{x}) \Phi_T( p_1,\ldots,p_{s_d(T)-1} ; x ) + \bar{x} p_{s_d(T^\circ)}$ with $x\in \mathrm{domain}( T )$ and $\bar{x}\in[0,1]$ .

Numbering of Subelements

Given a reference topology $T$

, a codimension $c\in\{0,\dots,d(T)\}$ and a subtopology $S_{c,i}(T)$ we define a subset of the corner set $\mathcal{C}(T)$ $\mathcal{C}_{c,i}(T)= (p_{k_j}(T))_{j=1}^{s_{d(S_{c,i}(T)}(S_{c,i}(T))}$ given by the subsequence $\mathcal{K}_{c,i}(T)=(k_j)_{j=1}^{s_{d(S_{c,i}(T)}(S_{c,i}(T))}$ of $\{1,dots,s_{d(T)}(T)\}$ :

$\mathcal{C}_{0,0}(T) = \mathcal{C}(T)$ , $\mathcal{C}_{d(T),i}(T) = (p_i(T))$ , and for $c\in\{1,\dots,d(T)-1\}$ we define $\mathcal{K}_{c,i}(T)=(k_j)_{j=1}^{s_{d(S_{c,i}(T)}(S_{c,i}(T))}$ through the recursion

$T=t^\vert$ :
For $i=1,\dots,s_c(T)$ we define $\mathcal{K}_{c,i}(T)=(k_1,\dots,k_n, k_1+s_{d(t)}(t)\dots,k_n+s_{d(t)}(t))$ with $\mathcal{K}_{c,i}(T)= \{k_1,\dots,k_n\}$ .
For $i=s_(T)+1,\dots,s_c(T)+s_{c+1}(T)$ we define $\mathcal{K}_{c,i}(T)=(k_1,\dots,k_n)$ with $\mathcal{K}_{c-1,i}(T)= \{k_1,\dots,k_n\}$ .
For $i=s_c(T)+s_{c+1}(T)+1,\dots,s_c(T)+2s_{c+1}(T)$ we define $\mathcal{K}_{c,i}(T)=(k_1+s_{d(t)}(t),\dots,k_n+s_{d(t)}(t))$ with $\mathcal{K}_{c-1,i}(T)= \{k_1,\dots,k_n\}$ .
$T=t^\circ$ :
For $i=1,\dots,s_{c-1}(T)$ we define $\mathcal{K}_{c,i}(T)=(k_1,\dots,k_n)$ with $\mathcal{K}_{c-1,i}(T)= \{k_1,\dots,k_n\}$ .
For $i=s_{c-1}(T)+1,\dots,s_{c-1}(T)+s_c(T)$ we define $\mathcal{K}_{c,i}(T)=(k_1,\dots,k_n,s_{d(T)}(T))$ with $\mathcal{K}_{c-1,i}(T)= \{k_1,\dots,k_n\}$ .

Given these subsets we define subreference elements $E_{c,i}(T)=(S_{c,i}(T),\Phi_{c,i}(T))$ of $E(T)$

given by the following mapping $\Phi_{c,i}(T,\cdot)=\Phi(S_{c,i}(T),\mathcal{C}_{c,i}(T),\cdot)$ .

Furthermore we define a numbering of the subreference elements of each subreference element in $T$ . This is the number $k=\mathrm{index}_{c,i,cc,ii}(T)$ for $c\in\{0,\dots,d(T)\}$ , $i\in\{1,\dots,s_c(T)\}$ , and $cc\in\{0,\dots,d(S_{c,i})\}$ , $ii\in\{1,\dots,s_{cc}(S_{c,i})\}$ for which

$\Phi_{c,i}(S_{c,i}(T))\circ \Phi_{cc,ii}(S_{cc,ii}(S_{c,i}(T))) = \Phi_{c+cc,k}(T).$