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Inserting into (
3.46
), we get
M
−
1
e
m
+
θ
N
e
N
2
2
a
L
2
(G)
+
C
1
k
0
m
=
2
M
−
1
M
−
1
η
M
2
η
m
+
θ
2
ξ
N
2
ξ
m
+
θ
N
2
a
≤
L
2
(G)
+
C
1
k
0
a
+
L
2
(G)
+
C
1
k
0
m
=
m
=
≤
C
L
2
(G)
+
kC
θ
k
T
0
M
−
1
η
M
2
0
η
m
+
θ
2
a
+
ξ
N
2
2
∗
L
2
(G)
+
C
1
k
u(s)
d
s
m
=
T
M
−
1
2
∗
u
m
+
θ
−
I
N
u
m
+
θ
2
a
+
˙
u
−
I
N
˙
u
d
s
+
Ck
0
0
m
=
C
M
−
1
m
=
0
ξ
N
2
η
M
2
η
m
+
θ
2
a
≤
L
2
(G)
+
L
2
(G)
+
k
d
s
.
T
d
s
+
k
2
T
0
2
∗
2
∗
+
η
u(s)
0
−
I
N
u
are estimated using the interpolation esti-
mates of Proposition
3.6.1
. Furthermore, if
u
0
is approximated with the
L
2
-
projection, we have
=
(iv) The terms involving
η
u
ξ
N
L
2
(G)
=
I
N
u
0
−
u
N,
0
L
2
(G)
≤
I
N
u
0
−
u
0
L
2
(G)
+
u
0
−
u
N,
0
L
2
(G)
. Since
u
∈
C
1
(J
;
H
2
(G))
by assumption, we have
u
0
∈
H
2
(G)
.The
L
2
-projection
u
N,
0
is a quasi-optimal approximation of
u
0
,i.e.
Ch
2
u
0
H
2
(G)
. We obtain from Proposition
3.6.1
opti-
mal convergence rates with respect to
h
, and Theorem
3.6.5
is proved.
u
0
−
P
N
u
0
L
2
(G)
≤
3.7 Further Reading
The basic finite element method is, for example, described in Braess [24] and for
parabolic problems in detail by Thomée [154]. Error estimates in a very general
framework are also given in Ern and Guermond [64]. In this section, we only con-
sidered the
θ
-scheme for the time discretization. It is also possible to apply finite
elements for the time discretization as in Schötzau and Schwab [146, 147] where an
hp
-discontinuous Galerkin method is used. It yields exponential convergence rates
instead of only algebraic ones as in the
θ
-scheme.
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