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3.6.2 Convergence of the Finite Element Method
Assume uniform mesh width
h
in space and constant time steps
k
=
T/M
in time.
We show now that the computed sequence
{
u
N
}
converges, as
h
→
0 and
k
→
0, to
the exact solution of (
3.7
). We have
C
1
(J
H
2
(G))
C
3
(J
H
−
1
(G))
.
Let u
m
(x)
Theorem 3.6.5
Assume u
∈
;
∩
;
=
u(t
m
,x) and u
N
θ<
2
S
1
T
be as in
(
3.14
),
with V
N
=
.
Assume for
0
≤
also
(
3.30
).
Then
,
the following error bound holds
:
M
−
1
u
m
+
θ
N
u
M
u
N
2
u
m
+
θ
2
a
−
L
2
(G)
+
k
0
−
m
=
Ch
2
T
0
Ch
2
2
≤
max
0
T
u(t)
H
2
(G)
+
∂
t
u(s)
H
1
(G)
d
s
≤
t
≤
C
k
2
0
2
∗
∂
tt
u(s)
d
s
if
0
≤
θ
≤
1
,
+
k
4
0
(3.44)
1
2
∗
∂
ttt
u(s)
d
s
if θ
=
2
.
Remark 3.6.6
By the properties (
3.8
)-(
3.9
) (with
C
3
=
0), the norm
·
a
in (
3.28
)
is equivalent to the energy-norm
·
V
. Thus, we see from (
3.44
) that we have
u
M
u
N
V
=
O
−
+
k)
, i.e. first order convergence in the energy norm, provided
the solution
u(t, x)
is sufficiently smooth. However, one can also prove second order
convergence in the
L
2
-norm, i.e.
(h
u
M
u
N
L
2
(G)
=
O
(h
2
−
+
k)
,if
θ
∈[
0
,
1
]\{
1
/
2
}
,
u
M
u
N
L
2
(G)
=
O
(h
2
k
2
)
if
θ
and
1
/
2. Hence, for continuous, linear fi-
nite elements, we obtain the same convergence rates as for the finite difference dis-
cretization in Theorem 2.3.8.
−
+
=
The proof of Theorem
3.6.5
will be given in several steps. We define
e
N
:=
u
m
−
u
N
I
N
denotes nodal interpolant de-
fined in (
3.24
). Since we already estimated the consistency error
η
m
and consider the splitting (
3.34
), where now
u
m
−
I
N
u
m
,
=
we focus on
ξ
N
∈
V
N
.
ξ
N
}
m
are solutions of the θ -scheme
:
C
1
(J
Lemma 3.6.7
If u
∈
;
H)
,
the errors
{
Given ξ
N
:=
I
N
u
0
−
u
0
N
,
for m
1
find ξ
m
+
1
N
=
0
,...,M
−
∈
V
N
such that
∀
v
N
∈
V
N
:
−
ξ
N
,v
N
)
+
a
θξ
m
+
1
−
θ)ξ
N
,v
N
=
(r
m
,v
N
)
k
−
1
(ξ
m
+
1
N
+
(
1
(3.45)
N
where the residuals r
m
=
r
1
+
r
2
+
r
3
are given by
=
k
−
1
(u
m
+
1
u
m
+
θ
,v
N
,
(r
1
,v
N
)
u
m
)
−
−˙
=
k
−
1
(
u
m
), v
N
,
I
N
u
m
+
1
k
−
1
(u
m
+
1
(r
2
,v
N
)
−
I
N
u
m
)
−
−
(r
3
,v
N
)
I
N
u
m
+
θ
u
m
+
θ
,v
N
).
=
a(
−
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