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. .
264 6. Finite Volume Method
.

.. This edge has to be
. removed from the
. . Delaunay triangulation.

Figure 6.7. Delaunay triangulation to the Voronoi diagram from Figure 6.5.

The implication

Voronoi diagram ⇒ Delaunay triangulation,
which we have just discussed, suggests that we ask about the converse
statement. We do not want to answer it completely at this point, but we
give the following sufficient condition.
Theorem 6.5 If a conforming triangulation of Ω (in the sense of finite
element methods) consists of nonobtuse triangles exclusively, then it is
a Delaunay triangulation, and the corresponding Voronoi diagram can be
constructed by means of the perpendicular bisectors of the triangles’ edges.
We mention that the centre of the circumcircle of a nonobtuse triangle is
located within the closure of that triangle.
In the analysis of the finite volume method, the following relation is
important.
, k ∈{1, 2, 3},
Lemma 6.6 Given a nonobtuse triangle K with vertices a i k
∩ K of the control volumes
then for the corresponding parts Ω i k ,K := Ω i k
, we have
Ω i k
1 1
|K|≤ |Ω i k ,K |≤ |K| , k ∈{1, 2, 3} .
4 2
The Donald diagram
In contrast to the Voronoi diagram, where the construction starts from a
given point set, the starting point here is a triangulation T h of Ω, which is
allowed to contain obtuse triangles.
,k ∈{1, 2, 3}. We define
Again, let K be a triangle with vertices a i k

Ω i k ,K := x ∈ K λ j (x) <λ k (x),j = k ,

(cf. (3.51)).
where λ k denote the barycentric coordinates with respect to a i k
1
Obviously, the barycentre satisfies a S = (a i 1 + a i 2 + a i 3 ), and (see, for
3
comparison, Lemma 6.6)
3 |Ω i k ,K | = |K| , k ∈{1, 2, 3} . (6.4)
This relation is a simple consequence of the geometric interpretation of
the barycentric coordinates as area coordinates given in Section 3.3. The