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2.2. Linear Elements 57

allowed: not
allowed:

Figure 2.3. Triangulations.
The triangles of a triangulation will be numbered K 1 ,...,K N .The
subscript h indicates the fineness of the triangulation, e.g.,

h := max diam (K) K ∈T h ,

where diam (K):=sup |x − y| x, y ∈ K denotes the diameter of K.

Thus here h is the maximum length of the edges of all the triangles.
Sometimes, K ∈T h is also called a (geometric) element of the partition.
The vertices of the triangles are called the nodes, and they will be
numbered
a 1 ,a 2 ,... ,a M ,

i.e., a i =(x i ,y i ), i =1,... ,M,where M = M 1 + M 2 and
a 1 ,...,a M 1 ∈ Ω ,
(2.26)
a M 1 +1 ,...,a M ∈ ∂Ω .
This kind of arrangement of the nodes is chosen only for the sake
of simplicity of the notation and is not essential for the following
considerations.

a 10 a 8
• •

a 9

•
K 2
K 1
• K 5
a 11 a 1

K 3 K 4 K 6

• a
•

7

K 12 • a 2
•
K 10
a 3

K 11
K 8
•
a 4 K 7

K 9
•

a 5
•
a 6
Figure 2.4. A conforming triangulation with N = 12, M = 11, M 1 =3, M 2 =8.
An approximation of the boundary value problem (2.1), (2.2) with linear
finite elements on a given triangulation T h of Ω is obtained if the ansatz
space V h is defined as follows:

V h := u ∈ C(Ω) u| K ∈P 1 (K) for all K ∈T h ,u =0 on ∂Ω . (2.27)
¯