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328 7. Discretization of Parabolic Problems

(4) # If A h is irreducible, then a grid point (x, t n ),x ∈ Ω h can be connected
via a chain of neighbours to every grid point (y, t k ),y ∈ Ω h and
0 ≤ k ≤ n.

ˆ
(5) (C h ) rs ≤ 0 for r =1,... , M 1 ,s = M 1 +1,... , M 2 :
ˆ
Analogously to (2), this follows from (5) for A h and (7.100).
∗ ˜
˜
(6) C r := M (C h ) rs =0 for r =1,... ,M:
s=1
Analogously to (7.102), we have
M

˜
˜
˜
C r = τA j := τ (A h ) jk ,
k=1
˜
so that the property is equivalent to the corresponding one of A h .
˜
(6) C r ≥ 0 for r =1,... , M
˜
is equivalent to (6) for A h by the above argument.
(7) For every s ∈ M 1 +1,... , M there exists an r ∈{1,... , M 1 } such
ˆ
that (C h ) rs =0:
Every listed boundary value should influence the solution: For the
0 N ˆ
values from ˆ u ,..., ˆ u h this is the case iff A h satisfies this property.
h
Furthermore, the “local” indices of the equation, where the boundary
values appear, are the same for each time level. For the values from
u 0 ∈ R M 1 the assertion follows from (7.100).
From the considerations we have the following theorem:
Theorem 7.28 Consider the one-step-theta method in the form (7.66).
ˆ
Let (7.100) hold. If the spatial discretization A h satisfies (1.32) (1), (2),
(3) (i),and (5),thena comparison principle holds:
n
(1) If for two sets of data f , u 0i and ˆ u ,n =0,... ,N and i =1, 2, we
i
hi
have
f ((n − 1+Θ)τ) ≤ f ((n − 1+Θ)τ) for n =1,... ,N ,
2
1
and
n n
u 01 ≤ u 02 ; , ˆ u h1 ≤ ˆ u h1 for n =0,... ,N,
then
n n
ˆ u h1 ≤ ˆ u h2 for n =1,... ,N
for the corresponding solutions.
n n
If ˆ u h1 = ˆ u h2 for n =1,... ,N, then condition (1.32) (5) can be
omitted.