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320 7. Discretization of Parabolic Problems

for i =1,... ,M. Then the following statements are equivalent:
(1) The one-step method is nonexpansive for the model problem (7.80)
and all eigenvalues λ i of A h .
(2) The one-step method is nonexpansive for the problem (7.41),with
respect to the norm · h induced by ·, · h .
(3) E h,τ h ≤ 1 in the matrix norm · h induced by the vector norm
· h.

Proof: We prove (1) ⇒ (3) ⇒ (2) ⇒ (1):
(1) ⇒ (3): According to (7.83) (1) is characterized by

|R(−λ i τ)|≤ 1 , (7.92)
for the eigenvalues λ i .
For the eigenvector, w i with eigenvalue λ i we have (7.91), and thus, for
M
c
an arbitrary u 0 = i=1 i w i ,
M
2 2
E h,τ u 0 = c i E h,τ w i
h h
i=1
M M
2 2 2 2
= c i R(−λ i τ)w i = c |R(−λ i τ)| w i ,
h i h
i=1 i=1
because of the orthogonality of the w i , and analogously,
M

2
2
2
u 0 = c w i ,
h
h
i
i=1
and finally, because of (7.92),
M

2 2 2 2
E h,τ u 0 ≤ c w i = u 0 ,
h i h h
i=1
which is assertion (3).
(3) ⇒ (2): is obvious.
(2) ⇒ (3):
|R(−λ i τ)| w i h = R(−λ i τ)w i h = E h,τ w i h ≤ w i h .

Thus, nonexpansiveness is often identical to what is (vaguely) called
stability:
Definition 7.25 A one-step method with a solution representation E h,τ
for q h = 0 is called stable with respect to the vector norm · h if
E h,τ h ≤ 1
in the induced matrix norm · h .