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Corollary 6.9 Under the assumptions of Lemma 6.8, we have for k ≡ 1,
∇uh , ∇vh for all uh , vh ∈ Vh .
= a0 (uh , vh )
h
0


Proof: It is su¬cient to verify the relation for vh = •i and arbitrary
i ∈ Λ. First, we see that

∇uh , ∇•i ∇uh · ∇•i dx .
=
0
K
K‚supp•i

Furthermore,

∇uh · ∇•i dx ∇•j · ∇•i dx
= uj
K K
j:‚K aj

∇•i · ∇•i dx + ∇•j · ∇•i dx .
= ui uj
K K
j=i:‚K aj

Since
1= •j
j:‚K aj

over K, it follows that

∇•i = ’ ∇•j ; (6.14)
j=i:‚K aj

that is, by means of Lemma 6.8,

∇uh · ∇•i dx (uj ’ ui ) ∇•j · ∇•i dx
=
K K
j=i:‚K aj
6.2. Finite Volume Method on Triangular Grids 271

mKij
(ui ’ uj )
= . (6.15)
dij
j=i:‚K aj

Summing over all K ‚ supp •i , we get
mij
∇uh , ∇•i (ui ’ uj ) 2
= a0 (uh , •i ) .
= h
0
dij
j∈Λi

Remark 6.10 By a more sophisticated argumentation it can be shown
that the above corollary remains valid if the di¬usion coe¬cient k is con-
stant on all triangles K ∈ Th and if the approximation µij is chosen
according to
±
1 k|K mK + k|K mK
 ij ij
k dσ = , mij > 0 ,
µij := (6.16)
mij “ij mij

 0, mij = 0 ,

where K, K are both triangles sharing the vertices ai , aj .

Treatment of Matrix-valued Di¬usion Coe¬cients
Corollary 6.9 and Remark 6.10 are valid only in the spatial dimension d =
2. However, for more general control volumes, higher spatial dimensions,
or not necessarily scalar di¬usion coe¬cients, weaker statements can be
proven.
As an example, we will state the following fact. As a by-product, we also
obtain an idea for how to derive discretizations in the case of matrix-valued
di¬usion coe¬cients. For a better distinction between the elements K of
the triangulation and the di¬usion coe¬cient, we keep the notation k for
the di¬usion coe¬cient, even if k is allowed to be a matrix-valued function
temporarily.
Lemma 6.11 Let Th be a conforming triangulation of „¦, where in the
case of the Voronoi diagram it is additionally required that all triangles be
nonobtuse. Furthermore, assume that the di¬usion matrix k : „¦ ’ R2,2 is
constant on the single elements of Th . Then for any i ∈ Λ and K ∈ Th we
have

(k∇uh ) · ∇•i dx = ’ (k∇uh ) · ν dσ for all uh ∈ Vh ,
‚„¦i ©K
K

where {„¦i }i∈Λ is either a Voronoi or a Donald diagram and ν denotes the
outer unit normal with respect to „¦i .
Without di¬culties, the proof can be carried over from the proof of a related
result in [20, Lemma 6.1].
Now we will show how to use this fact to formulate discretizations for the
case of matrix-valued di¬usion coe¬cients. Namely, using relation (6.14),
272 6. Finite Volume Method

we easily see that

(k∇uh ) · ν dσ uj (k∇•j ) · ν dσ
=
‚„¦i ©K ‚„¦i ©K
j:‚K aj

(uj ’ ui ) (k∇•j ) · ν dσ .
=
‚„¦i ©K
j=i:‚K aj

Summing over all triangles that lie in the support of •i , we obtain by
Lemma 6.11 the relation

(k∇uh ) · ∇•i dx = (ui ’ uj ) (k∇•j ) · ν dσ . (6.17)
„¦ ‚„¦i
j∈Λi

With the de¬nition
±
 dij (k∇•j ) · ν dσ , mij > 0 ,
µij := (6.18)
m
 ij ‚„¦i
0, mij = 0 ,
it follows that
mij
(k∇uh ) · ∇•i dx = µij (ui ’ uj ) .
dij
„¦ j∈Λi

Note that, in the case of Voronoi diagrams, (6.16) is a special case of the
choice (6.18).
Consequently, in order to obtain a discretization for the case of a matrix-
valued di¬usion coe¬cient, it is su¬cient to replace in the bilinear form bh
and, if the Voronoi diagram is used, also in a0 , the terms involving µij
h
according to formula (6.18).

Implementation of the Finite Volume Method
In principle, the ¬nite volume method can be implemented in di¬erent
ways. If the linear system of equations is implemented in a node-orientated
manner (as in ¬nite di¬erence methods), the entries of the system matrix
Ah and the components of the right-hand side q h can be taken directly
from (6.9).
On the other hand, an element-orientated assembling is possible, too.
This approach is preferable, especially in the case where an existing ¬nite
element program will be extended by a ¬nite volume module. The idea of
how to do this is suggested by equation (6.17). Namely, for any triangle
K ∈ Th , the restricted bilinear form ah,K with the appropriate de¬nition
of µij according to (6.18) is de¬ned as follows:
ah,K (uh , vh ) :=
± 
 
 
ui ’ uj
+ γij [rij ui + (1 ’ rij ) uj ] mij + ri ui mi ,
K K
vi µij
 j=i: 
dij
 
i∈Λ
‚K aj
6.2. Finite Volume Method on Triangular Grids 273

where mK := |„¦i © K|. Then the contribution of the triangle K to the
i
matrix entry (Ah )ij of the matrix Ah is equal to ah,K (•j , •i ). In the same
way, the right-hand side of (6.9) can be split elementwise.


6.2.4 Properties of the Discretization
Here we will give a short overview of basic properties of ¬nite volume
methods. For the sake of simplicity, we restrict ourselves to the case of a
constant scalar di¬usion coe¬cient k > 0. Then, in particular, it is useful
to set µij := k for all i ∈ Λ, j ∈ Λi .
Lemma 6.12 Suppose the approximations γij of νij · c|“ij satisfy γji =
’γij and the rij are de¬ned by (6.12) with a function R satisfying (P1).
Then we get for all uh , vh ∈ Vh ,
1
bh (uh , vh ) = ui vi γij mij
2
i∈Λ j∈Λi
1 1 1
rij ’ (ui ’ uj ) (vi ’ vj ) + (uj vi ’ ui vj ) γij mij .
+
2 2 2
i∈Λ j∈Λi


Proof: First, we observe that bh can be rewritten as follows:
1
vi (1 ’ rij ) uj ’ ’ rij ui γij mij
bh (uh , vh ) =
2
i∈Λ j∈Λi
(6.19)
1
+ ui vi γij mij .
2
i∈Λ j∈Λi

In the ¬rst term, we change the order of summation and rename the indices:
1
vj (1 ’ rji ) ui ’ ’ rji uj γji mji
bh (uh , vh ) =
2
i∈Λ j∈Λi

1
+ ui vi γij mij .
2
i∈Λ j∈Λi

Next we make use of the following relations, which easily result from dji =
dij and the assumptions on γij and rij :
1 1
(1 ’ rji ) γji = ’rij γij , ’ rji ’ rij
γji = γij .
2 2
So we get, due to mji = mij ,
1
vj ’rij ui ’ ’ rij uj γij mij
bh (uh , vh ) =
2
i∈Λ j∈Λi

1
+ ui vi γij mij .
2
i∈Λ j∈Λi
274 6. Finite Volume Method

Taking the arithmetic mean of both representations of bh , we arrive at
1
bh (uh , vh ) = ui vi γij mij
2
i∈Λ j∈Λi
1 1
(1 ’ rij ) uj vi ’ rij ui vj ’ ’ rji (ui vi + uj vj ) γij mij
+
2 2
i∈Λ j∈Λi
1 1
’ rij (uj vi + ui vj ’ ui vi ’ uj vj )
=
2 2
i∈Λ j∈Λi
1 1
(uj vi ’ ui vj ) γij mij +
+ ui vi γij mij .
2 2
i∈Λ j∈Λi

2

Corollary 6.13 Let c1 , c2 , ∇ · c ∈ C(„¦). Under the assumptions of
Lemma 6.12 and also assuming property (P2) for R, the bilinear form bh
satis¬es for all vh ∈ Vh the estimate
® 
1
vi ° (γij ’ νij · c) dσ » . (6.20)
bh (vh , vh ) ≥ ∇ · c dx +
2
2 „¦i “ij
i∈Λ j∈Λi



Proof: Due to rij ’ 1 γij ≥ 0, because of property (P2) in (6.13), it
2
immediately follows that
1 1
bh (vh , vh ) ≥ 2 2
vi γij mij = vi γij mij .
2 2
i∈Λ j∈Λi i∈Λ j∈Λi

For the inner sum, we can write

γij mij = γij dσ
“ij
j∈Λi j∈Λi

νij · c dσ + (γij ’ νij · c) dσ .
=
“ij “ij
j∈Λi j∈Λi

The ¬rst term can be rewritten as an integral over the boundary of „¦i , i.e.,

νij · c dσ = ν · c dσ .
“ij ‚„¦i
j∈Λi

By Gauss™s divergence theorem, it follows that

ν · c dσ = ∇ · c dx .
‚„¦i „¦i

2
6.2. Finite Volume Method on Triangular Grids 275

Remark 6.14 If the approximations γij are chosen according to (6.7),
then γji = ’γij , and (6.20) simpli¬es to
1
bh (vh , vh ) ≥ ∇ · c dx .
2
vi
2 „¦i
i∈Λ

Using a similar argument as in the treatment of the term j∈Λi γij mij
in the proof of Corollary 6.13, the value dh (vh , vh ) can be represented as
follows:
2 2
dh (vh , vh ) = ri vi mi = vi ri dx
„¦i
i∈Λ i∈Λ

(ri ’ r) dx .
2 2
= vi r dx + vi (6.21)
„¦i „¦i
i∈Λ i∈Λ

The second term vanishes if the approximations ri are de¬ned as in (6.8).
Theorem 6.15 Let the rij be de¬ned by (6.12) with R satisfying (P1) and
(P2). Suppose k > 0, c1 , c2 , ∇ · c, r ∈ C(„¦), r + 1 ∇ · c ≥ r0 = const ≥ 0 on
2
„¦ and that the approximations γij , respectively ri , are chosen according to
(6.7), respectively (6.8). Under the assumptions of Lemma 6.8, we have for
all vh ∈ Vh ,
2
ah (vh , vh ) ≥ k ∇vh , ∇vh vi mi = k |vh |1 + r0 vh
2 2
+ r0 0,h ;
0
i∈Λ

that is, the bilinear form ah is Vh -elliptic uniformly with respect to h.

Proof: We start with the consideration of a0 (vh , vh ). Due to Corollary 6.9,
h
the relation
2
a0 (vh , vh ) = k ∇vh , ∇vh = k |vh |1
h 0

holds. Furthermore, by Remark 6.14 and equation (6.21), we have
1
bh (vh , vh ) + dh (vh , vh ) ≥ ∇·c+r dx ≥ r0
2 2
vi vi mi .
2
„¦i
i∈Λ i∈Λ

Since by de¬nition,
ah (vh , vh ) = a0 (vh , vh ) + bh (vh , vh ) + dh (vh , vh ) ,
h

2
both relations yield the assertion.
Remark 6.16 Let the family of triangulations (Th )h be regular. Then the
norms de¬ned in (3.136) and in (6.10) and also the norms · 0,h and
· 0 are equivalent on Vh uniformly with respect to h; i.e., there exist two
constants C1 , C2 > 0 independent of h such that
¤v ¤ C2 v for all v ∈ Vh .
C1 v 0 0,h 0
276 6. Finite Volume Method

Proof: Due to Theorem 3.43 (i) only the uniform equivalence of the dis-
crete L2 -norms has to be shown. Denoting such an equivalence by ∼, we
=
have for v ∈ Vh with vi := v(ai ) for i ∈ Λ,
« 1/2
1/2
¬ ·
=¬ |„¦i,K |·
|vi |2 mi |vi |2
 
K∈Th :
i∈Λ i∈Λ
K©„¦i =…
« 1/2
¬ ·
∼¬ |vi |2 |K|1/2 ·
= 
K∈Th :
i∈Λ
K©„¦i =…
due to Lemma 6.6 or (6.4)
« 1/2

∼¬ ·
|K| |vi |2 
=
K∈Th i:
ai ∈K
« 1/2

∼¬ ·
|vi |2 

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