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3.3. Element Types and Affine Equivalent Triangulations 117

We start with simplicial finite elements, that is, elements whose basic
d
domain is a regular d-simplex of R . By this we mean the following:
d
Definition 3.21 Aset K ⊂ R is called a regular d-simplex if there exist
d
d + 1 distinct points a 1 ,... ,a d+1 ∈ R , the vertices of K, such that
are linearly independent (3.45)
a 2 − a 1 ,...,a d+1 − a 1
(that is, a 1 ,... ,a d+1 do not lie in a hyperplane) and

K = conv {a 1 ,...,a d+1 }
$ %
d+1 d+1

:= x = λ i a i
0 ≤ λ i (≤ 1) , λ i =1 (3.46)
i=1 i=1
$ %
d+1 d+1

= x = a 1 + λ i (a i − a 1 )
λ i ≥ 0 , λ i ≤ 1 .
i=2 i=2
A face of K is a (d − 1)-simplex defined by d points of {a 1 ,...,a d+1}.
The particular d-simplex
ˆ
K := conv {ˆa 1 ,..., ˆa d+1} with ˆa 1 =0 , ˆa i+1 = e i ,i =1,... ,d, (3.47)
is called the standard simplicial reference element.
In the case d = 2 we get a triangle with dim P 1 = 3 (cf. Lemma 2.10). The
faces are the 3 edges of the triangle. In the case d = 3 we get a tetrahedron
with dim P 1 = 4, the faces are the 4 triangle surfaces, and finally, in the
case d = 1 it is a line segment with dim P 1 = 2 and the two boundary
points as faces.
d
More precisely, a face is not interpreted as a subset of R , but of a
(d − 1)-dimensional space that, for instance, is spanned by the vectors
a 2 − a 1 ,...,a d − a 1 in the case of the defining points a 1 ,...,a d .
Sometimes, we also consider degenerate d-simplices, where the assump-
tion (3.45) of linear independence is dropped. We consider, for instance,
a line segment in the two-dimensional space as it arises as an edge of a
triangular element. In the one-dimensional parametrisation it is a regular
2
1-simplex, but in R a degenerate 2-simplex.
The unique coefficients λ i = λ i (x), i =1,... ,d + 1, in (3.46), are called
barycentric coordinates of x. This defines mappings λ i : K → R,i =
1,... ,d +1.
We consider a j as a column of a matrix; that is, for j =1,...,d, a j =
(a ij ) . The defining conditions for λ i = λ i (x) can be written as a
i=1,...,d
(d +1) × (d + 1) system of equations:

d+1


a ij λ j = x i 

j=1 x
d+1 ⇔ Bλ = 1 (3.48)


λ j = 1 

j=1