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1.4. Maximum Principles and Stability 41

can choose
u h2 := u h , f := A h u h2 , ˆ u h2 := 0 ,
2
u h1 := 0 , f := 0 , ˆ u h1 := 0
1
to conclude the assertion. Because of ˆ u hi := 0 for i =1, 2 the specific
ˆ
definition of A h plays no role.
A matrix with the property (1.39) is called inverse monotone.An
equivalent requirement is
−1
v h ≥ 0 ⇒ A v h ≥ 0 ,
h
and therefore by choosing the unit vectors as v h ,
A −1 ≥ 0 .
h
Inverse monotone matrices that also satisfy (1.32) (1), (2) are called M-
matrices.
Finally, we can weaken the assumptions for the validity of the comparison
principle.
Corollary 1.13 Suppose that A h ∈ R M 1 ,M 1 is inverse monotone and
(1.32) (5) holds. Let u h1 , u h2 ∈ R M 1 be solutions of

ˆ
A h u hi = −A h ˆ u hi + f i for i =1, 2
for given f , f ∈ R M 1 , ˆ u h1 , ˆ u h2 ∈ R M 2 that satisfy f ≤ f , ˆ u h1 ≤ ˆ u h2 .
1
2
2
1
Then
u h1 ≤ u h2 .
Proof: Multiplying the equation
ˆ
A h (u h1 − u h2 )= −A h (ˆ u h1 − ˆ u h2 )+ f − f
1 2
from the left by the matrix A −1 , we get
h
−1 ˆ −1
u h1 − u h2 = − A A h (ˆ u h1 − ˆ u h2 ) + A (f − f ) ≤ 0 .
h h 1 2
! ! ! ! !
≥0 ≤0 ≤0 ≥0 ≤0

The importance of Corollary 1.13 lies in the fact that there exist
˜
discretization methods, for which the matrix A h does not satisfy, e.g., con-
dition (1.32) (6), or (6) but A −1 ≥ 0. A typical example of such a method
∗
h
is the finite volume method described in Chapter 6.
In the following we denote by 1 a vector (of suitable dimension) whose
components are all equal to 1.
Theorem 1.14 We assume (1.32) (1)–(3), (4) ,(5). Furthermore, let
∗
(1) (2)
w , w ∈ R M 1 be given such that
h h
(1) (2) ˆ
A h w ≥ 1 , A h w ≥−A h1 . (1.40)
h h