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We immediately have from (
3.26
), using Hölder's inequality,
Lemma 3.4.3
The L
2
-projection
L
2
(G)
S
1
T
P
N
:
→
defined in
(
3.26
)
is bounded
,
i
.
e
.
L
2
(G).
P
N
u
L
2
(G)
≤
u
L
2
(G)
,
∀
u
∈
(3.27)
3.5 Stability of the
θ
-Scheme
For the stability of (
3.14
), we prove that the finite element solutions satisfy an analog
of the estimate (
3.10
). For this section, we assume the uniform mesh width
h
in
space and constant time steps
k
=
T/M
. We define
a
:=
a(v,v)
1
2
.
v
(3.28)
V
N
In the analysis, we will use for
f
∈
the following notation:
(f, v
N
)
v
N
a
.
f
∗
:=
sup
v
N
∈
V
N
(3.29)
We will also need
λ
A
defined by
2
v
N
λ
A
:=
sup
v
N
∈
V
N
.
v
N
2
∗
1
In the case
2
≤
θ
≤
1, the
θ
-scheme is stable for any time step
k>
0, whereas in the
θ<
2
≤
case 0
the time step
k
must be sufficiently small.
θ<
2
,
assume
Proposition 3.5.1
In the case
0
≤
σ
:=
k(
1
−
2
θ)λ
A
<
2
.
(3.30)
Then
,
there are constants C
1
and C
2
independent of h and of k such that the se-
quence
u
N
}
M
m
{
of solutions of the θ -scheme
(
3.14
)
satisfies the stability estimate
=
0
M
−
1
M
−
1
u
m
+
θ
N
u
N
2
L
2
2
u
0
N
2
L
2
f
m
+
θ
2
∗
+
C
1
k
0
a
≤
+
C
2
k
0
,
(3.31)
m
=
m
=
1
where C
1
,C
2
satisfy in the case of
2
≤
θ
≤
1,
1
2
≥
0
<C
1
<
2
,
C
1
,
(3.32)
2
−
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