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114 3. Finite Element Methods for Linear Elliptic Problems

3.3 Element Types and Affine Equivalent
Triangulations

In order to be able to exploit the theory developed in Sections 3.1 and 3.2
we make the assumption that Ω is a Lipschitz domain.
The finite element discretization of the boundary value problem (3.12)
with the boundary conditions (3.18)–(3.20) corresponds to performing a
Galerkin approximation (cf. (2.23)) of the variational equation (3.35) with
the bilinear form a and the linear form b, supposed to be defined as in
1
(3.31), and some w ∈ H (Ω) with the property w = g 3 on Γ 3 .The solution
of the weak formulation of the boundary value problem is then given by
˜ u := u + w,if u denotes the solution of the variational equation (3.35).
Since the bilinear form a is in general not symmetric, (2.21) and (2.23),
respectively (the variational equation), are no longer equivalent to (2.22)
and (2.24), respectively (the minimization problem), so that in the following
we pursue only the first, more general, ansatz.
The Galerkin approximation of the variational equation (3.35) reads as
follows: Find some u ∈ V h such that

˜
a(u h ,v)= b(v) − a(w, v)= b(v) for all v ∈ V h . (3.39)
The space V h that is to be defined has to satisfy V h ⊂ V . Therefore, we
speak of a conforming finite element discretization, whereas for a non-
conforming discretization this property, for instance, can be violated. The
ansatz space is defined piecewise with respect to a triangulation T h of Ω
with the goal of getting small supports for the basis functions. A trian-
gulation in two space dimensions consisting of triangles has already been
defined in definition (2.25). The generalization in d space dimensions reads
as follows:

d
Definition 3.19 A triangulation T h of a set Ω ⊂ R consists of a finite
number of subsets K of Ω with the following properties:
(T1) Every K ∈T h is closed.
(T2) For every K ∈T h its nonempty interior int (K) is a Lipschitz domain.
K.
(T3) Ω= ∪ K∈T h
(T4) For different K 1 and K 2 of T h the intersection of int (K 1 )and int (K 2 )
is empty.

The sets K ∈T h , which are called somewhat inaccurately elements in the
following, form a nonoverlapping decomposition of Ω. Here the formulation
is chosen in such a general way, since in Section 3.8 elements with curved
boundaries will also be considered. In Definition 3.19 some condition, which
corresponds to the property (3) of definition (2.25), is still missing. In the
following this will be formulated specifically for each element type. The