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

.
The basic building block for these elements is given by a recursion formular which assignes to each set
either a pyramid element
and a prism element
with
and
. The recursion starts with a single point
.
For
this leads to the following elements
:
is a line.
:
is a cube and
is a simplex.
:
is a cube,
is a simplex,
is a pyramind, and
is a prism.
In general if

is a cube then

is also a cube and if

is a simplex then

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
or
based on a numbering of
. For the subentities of codimension
we use the numbering
: 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.
: 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

in a codimension

subentity

we use the numbering induced by the numbering the reference element corresponding to

.
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
we also define the barycenters
, the volume
and the normals
to all codimension one subelements.
The recursion formular is also used to define mappings from reference elements
to general polytop given by a set of coordinates for the corner points - together with the mapping
, the transpose of the jacobian
is also defined where
is the dimension of the reference element and
the dimension of the coordinates. This sufficies to define other necessary parts of a Dune geometry by LQ-decomposing
: let
be given with a lower diagonal matrix
and a matrix
which satisfies
:
The next sections describe the details of the construction.
We define the set

of reference topologies by the following rules:
contains an element
that we call the point topology.- For
,
contains an element
that we call the prism over
. - For
,
contains an element
that we call the pyramid over
.
For each reference topology

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
then
and for
we have
. - If
then
and for
we have
.
- Subtopology: Given a reference topology
of dimension
and a codimension
we now define the subtopology
for
:
For each reference topology

we assosiate the set of corners

defined through
: 
:
for
, with
.
:
for
and
with 
The convex hall of the set of points

defines the reference domain

for the reference topology

; it follows that
A pair

of a topology

and a map

with

is called an element.
The reference element is the pair
.
For a given set of points
we define a mapping
through
for all
. This mapping can be expressed using the recurive definition of the reference topologies through:
Given a reference topology

, a codimension

and a subtopology

we define a subset of the corner set

given by the subsequence

of

:
,
, and for
we define
through the recursion
:
For
we define
with
.
For
we define
with
.
For
we define
with
.
:
For
we define
with
.
For
we define
with
.
Given these subsets we define subreference elements

of

given by the following mapping

.
Furthermore we define a numbering of the subreference elements of each subreference element in
. This is the number
for
,
, and
,
for which