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3.3. Element Types and Affine Equivalent Triangulations 121

P = P 2 (K) , (3.55)
Σ = {p (a i ) ,p (a ij ) , i =1,... ,d +1,i < j ≤ d +1} ,

where the a ij denote the midpoints of the edges (see Figure 3.5).
Since here we have
(d +1)(d +2)
|Σ| = =dim P,
2
it also suffices to specify the shape functions. They are given by

λ i (2λ i − 1) , i =1,... ,d +1 ,
4λ i λ j , i, j =1,... ,d +1 ,i < j .

. d = 2 . d = 3
.
. .
. . . . dim = 10

dim = 6 . . . .
. . .

Figure 3.5. Quadratic simplicial elements.

If we want to have polynomials of higher degree as local ansatz functions,
but still Lagrange elements, then degrees of freedom also arise in the interior
of K:
Finite Element: Cubic Ansatz on the Simplex
K = conv {a 1 ,... ,a d+1 } ,
P = P 3 (K) , (3.56)
Σ = {p(a i ),p(a i,i,j ),p(a i,j,k )} ,
where
2 1
a i,i,j := a i + a j for i, j =1,... ,d +1 ,i = j,
3 3
1
a i,j,k := (a i + a j + a k ) for i, j, k =1,...,d +1 ,i < j < k .
3
Since here |Σ| =dim P also holds, it is sufficient to specify the shape
functions, which is possible by
1
λ i (3λ i − 1)(3λ i − 2), i =1,... ,d +1 ,
2
9
λ i λ j (3λ i − 1), i, j =1,... ,d +1 ,i = j,
2