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122 3. Finite Element Methods for Linear Elliptic Problems

27λ i λ j λ k , i,j,k =1,...,d +1 ,i < j < k .
Thus for d = 2 the value at the barycentre arises as a degree of freedom.
This, and in general the a i,j,k , i<j <k, can be dropped if the ansatz
space P is reduced (see [9, p. 70]).
All finite elements discussed so far have degrees of freedom that are
defined in convex combinations of the vertices. On the other hand, two
regular d-simplices can be mapped bijectively onto each other by a unique
affine-linear F,that is, F ∈P 1 such that as defining condition, the vertices
of the simplices should be mapped onto each other. If we choose, besides
ˆ
the general simplex K, the standard reference element K defined by (3.47),
ˆ
then F = F K : K → K is defined by
F(ˆx)= Bˆx + a 1 , (3.57)

where B =(a 2 − a 1 ,... ,a d+1 − a 1 ).
Since for F we have

d+1 d+1 d+1

F λ i ˆa i = λ i F(ˆa i )for λ i ≥ 0 , λ i =1 ,
i=1 i=1 i=1
F is indeed a bijection that maps the degrees of freedom onto each other as
ˆ
well as the faces of the simplices. Since the ansatz spaces P and P remain
invariant under the transformation F K , the finite elements introduced so
far are (in their respective classes) affine equivalent to each other and to
the reference element.
ˆ ˆ ˆ
Definition 3.22 Two Lagrange elements (K, P, Σ), (K, P, Σ) are called
ˆ
equivalent if there exists a bijective F : K → K such that
* +

ˆ
ˆ
F(ˆa) ˆa ∈ K generates a degree of freedom on K

= a a ∈ K generates a degree of freedom on K

(3.58)
and
* +

P = p : K → R p ◦ F ∈ P ˆ .

They are called affine equivalent if F is affine-linear.
Here we have formulated the definition in a more general way, since in
Section 3.8 elements with more general F will be introduced: For isopara-
metric elements the same functions F as in the ansatz space are admissible
for the transformation. From the elements discussed so far only the simplex
with linear ansatz is thus isoparametric. Hence, in the (affine) equivalent
case a transformation not only of the points is defined by
ˆ x = F −1 (x) ,
ˆ
ˆ
but also of the mappings, defined on K and K, (not only of P and P)is
given by
ˆ
ˆ v : K → R , ˆ v(ˆx):= v(F(ˆx))