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H
0
(G)
because
C
0
(G)
is dense in
H
0
(G)
.The
weak
∈
(
2.21
) also holds for all
v
or
variational formulation
of (
2.4
) reads:
C(J,H
0
(G))
C
1
(J, L
2
(G))
such that for
t
∈
∩
∈
Find
u
J
b
b
b
d
d
t
H
0
(G),
uv
d
x
+
∂
x
u∂
x
v
d
x
=
fv
d
x,
∀
v
∈
(2.22)
a
a
a
u(
0
,
·
)
=
u
0
,
L
2
(G)
. The finite element method
is based on the
Galerkin discretization
of (
2.22
). The idea is to project (
2.22
)toa
finite dimensional subspace
V
N
⊂
where we assume that the initial condition
u
0
∈
H
0
(G)
and to replace (
2.22
) by:
C
1
(J, V
N
),
such that for
t
Find
u
N
∈
∈
J
b
b
b
d
d
t
u
N
v
N
d
x
+
∂
x
u
N
∂
x
v
N
d
x
=
fv
N
d
x,
∀
v
N
∈
V
N
,
(2.23)
a
a
a
u
N
(
0
)
=
u
N,
0
,
where
u
N,
0
is an approximation of
u
0
in
V
N
. For example,
u
N,
0
=
P
N
u
0
,the
L
2
-
projection of
u
0
on
V
N
, satisfying
G
P
N
u
0
v
N
d
x
=
G
u
0
v
N
d
x
for all
v
N
∈
V
N
.
We show that (
2.23
) is equivalent to a linear system of ordinary differential equa-
tions (in time). Let
b
j
,
j
=
1
,...,N
be a basis of
V
N
. Since
u
N
(t,
·
)
∈
V
N
,we
have
N
u
N
(t, x)
=
u
N,j
(t)b
j
(x),
j
=
1
where
u
N,j
(t)
denote the time dependent coefficients of
u
N
with respect to the basis
of
V
N
. Inserting this series representations into (
2.23
) yields for
v
N
(x)
=
b
i
(x)
,
i
=
1
,...,N
,
b
b
b
d
d
t
u
N
(t, x)v
N
(x)
d
x
+
∂
x
u
N
(t, x)∂
x
v
N
(x)
d
x
=
f(t,x)v
N
(x)
d
x,
a
a
a
N
N
b
b
d
d
t
u
N,j
(t)b
j
(x)b
i
(x)
d
x
⇒
u
N,j
(t)b
j
(x)b
i
(x)
d
x
+
a
a
j
=
1
j
=
1
b
=
f(t,x)b
i
(x)
d
x,
a
1
u
N,j
u
N,j
N
N
b
b
b
j
(x)b
i
(x)
d
x
⇒
b
j
(x)b
i
(x)
d
x
+
a
a
j
=
j
=
1
b
=
f(t,x)b
i
(x)
d
x.
a
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