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3.3. Element Types and Affine Equivalent Triangulations 115

parameter h is a measure for the size of all elements and mostly chosen as

h =max diam (K) K ∈T h ;

that is, for instance, for triangles h is the length of the triangle’s largest
edge.
For a given vector space V h let

P K := {v| K | v ∈ V h } for K ∈T h , (3.40)
that is,

V h ⊂ v :Ω → R v| K ∈ P K for all K ∈T h .

In the example of “linear triangles” in (2.27) we have P K = P 1 ,the poly-
nomials of first order. In the following definitions the space P K will always
consist of polynomials or of smooth “polynomial-like” functions, such that
1
1
we can assume P K ⊂ H (K) ∩ C(K). Here, H (K) is an abbreviation for
1
H (int (K)). The same holds for similar notation.
As the following theorem shows, elements v ∈ V h of a conforming ansatz
space V h ⊂ V have therefore to be continuous :
1
Theorem 3.20 Suppose P K ⊂ H (K) ∩ C(K) for all K ∈T h .Then
¯
1
V h ⊂ C(Ω) ⇐⇒ V h ⊂ H (Ω)

and, respectively, for V 0h := v ∈ V h v =0 on ∂Ω ,

¯
1
V 0h ⊂ C(Ω) ⇐⇒ V 0h ⊂ H (Ω) .
0
Proof: See, for example, [9, Theorem 5.1 (p. 62)] or also Exercise 3.10.
¯
0
If V h ⊂ C(Ω), then we also speak of C -elements. Hence with this notion
we do not mean only the K ∈T h , but these provided with the local ansatz
space P K (and the degrees of freedom still to be introduced). For a bound-
2
ary value problem of fourth order, V h ⊂ H (Ω) and hence the requirement
1 ¯
V h ⊂ C (Ω) are necessary for a conforming finite element ansatz. There-
1
fore, this requires, analogously to Theorem 3.20, so-called C -elements.By
degrees of freedom we denote a finite number of values that are obtained
for some v ∈ P K from evaluating linear functionals on P K .The setof
these functionals is denoted by Σ K . In the following, these will basically
be the function values in fixed points of the element K, as in the example
of (2.27). We refer to these points as nodes. (Sometimes, this term is used
only for the vertices of the elements, which at least in our examples are
always nodes.) If the degrees of freedom are only function values, then we
speak of Lagrange elements and specify Σ by the corresponding nodes of
the element. Other possible degrees of freedom are values of derivatives in
fixed nodes or also integrals. Values of derivatives are necessary if we want
1
to obtain C -elements.