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326 7. Discretization of Parabolic Problems

i.e., a vector of dimension M 2 := M 1 +(N +1)M 2 , which may reduce to
ˆ u h = u 0 ∈ R M 1 .
With this notation we have
ˆ
p = −C h ˆ u h + e (7.99)
h
if we define
 
ˆ ˆ
−I + τΘA h τΘA h τΘA h O
. .
 . 
 O . . . . . . 
ˆ  
. . . .
C h =   ,
 . . . . . . . . 
 
ˆ
ˆ
O τΘA h τΘA h
 
τf(Θτ)
 
τf((1 + Θ)τ)
 
 
 . . 
e =  .  .
 
 . . 
 . 
τf((N − 1+Θ)τ)
In the following the validity of (1.32) or (1.32) for
∗
˜
ˆ
C h =(C h , C h )
will be investigated on the basis of corresponding properties of
ˆ
˜
A h =(A h , A h ) .
Note that even if A h is irreducible, the matrix C h is always reducible,
1
n
since u depends only on u ,..., u n−1 , but not on the future time levels.
h h h
(Therefore, (7.97) serves only for the theoretical analysis, but not for the
actual computation.)
In the following we assume that
τΘ(A h ) jj < 1 for j =1,... ,M 1 , (7.100)
which is always satisfied for the implicit Euler method (Θ = 1). Then:
(1) (C h ) rr > 0 for r =1,... , M 1
holds if (1) is valid for A h . Actually, also (A h ) jj > −1/(τΘ) would
be sufficient.
(2) (C h ) rs ≤ 0 for r, s =1,... , M 1 ,r = s:
If (2) is valid for A h , then only the nonpositivity of the diagonal
elements of the off-diagonal block of C h , −I + τΘA h , is in question.
This is ensured by (7.100) (weakened to “≤”).

M 1

(3) (i) C r := C h ≥ 0 for r =1,... , M 1 :
s=1 rs