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=
Proof We proceed by induction on n . Inequality ( B.24 )for n
1 follows from
( B.20 ), ( B.21 ), if we note that the right hand side of ( B.20 ) is bounded by
C { u 0
2
2
L 1 ( 0 ,k
2
L 2 ( 0 ,k
for f S( 0 ,T) .
Assume now the assertion ( B.24 )istrueforsome n . Then, arguing as in the proof
of ( B.20 ), ( B.21 ), we obtain ( B.24 )for n
H + g
; H ) + h
; V ) }
+
1.
Next, we address the case (iii) in Theorem B.2.2 .
Lemma B.3.2 If ( B.15a ), ( B.15b ) holds , as k
0 we have
2
2
V Ck 2 .
u 1 P u 0
H + k u 1 P u 0
(B.26)
· H into ( B.23 ) to obtain
= P
u 0 K
Proof We use ( B.18 ) and insert v
k
0
C k
2
2
u 1 P
u 0
H +
k
u 1 P
u 0
V
u 1 P
u 0 H
g(τ)
H
d τ
k (h( 0 )
u 0 )
+
AP
u 0 ,u 1 P
k
0
d τ .
(B.27)
+|
u 1 P
u 0 | V
h(τ )
h( 0 )
Theassertion( B.26 ) follows from ( B.27 ) with the estimates
k
0
g(τ)
H
d τ
k
g
L ( 0 ,k ; H ) ,
k
0
Ck 3 / 2
h L 2 ( 0 ,k ; V ) .
h(τ )
h( 0 )
d τ
Remark B.3.3 Inserting v = u k, 1 into ( B.12b ), we obtain by similar arguments
2
2
V
u k, 2
u k, 1
H +
k
u k, 2
u k, 1
(f k, 1
k
A
u k, 1 ,u k, 2
u k, 1 )
Ck 2
d τ .
2 k
2
2
h (τ )
2
0 <τ < 2 k g(τ)
sup
H + h( 0 ) AP u 0
H +
(B.28)
0
For the proof of Theorem B.2.2 as well as for a priori error estimates, we require
the following perturbation results.
Lemma B.3.4 Assume 0
K
and we are given sequences
{
w m }
,
{
p m }
,
{
q m }
which
satisfy
,q m V for m
w 0 H
,w m + 1 K
,p m H
0 ,
(B.29)
and are such that , for m
0, it holds
w m + 1 w m + k A w m + 1 ,w m + 1 V , V k(p m + q m ,w m + 1 ).
(B.30)
Define , for M
1, with α as in ( B.4b ),
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