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}
u
L
2
(G)
=
C
2
u
1
2
min
{
C
−
1
,
1
2
L
2
(G)
+
u
2
2
≥
H
1
(G)
,
i.e. (
3.9
) holds with
C
3
=
0. Hence, according to Theorem
3.2.2
, the variational for-
mulation of the heat equation admits, for
u
0
∈
L
2
(G)
,
f
L
2
(J
H
−
1
(G))
a unique
∈
;
L
2
(J
H
0
(G))
H
1
(J
H
−
1
(G))
.
weak solution
u
∈
;
∩
;
e
−
λt
u
with
suitably chosen
λ
and multiply (
3.1
)by
e
−
λt
, we find that
v
satisfies the problem
We can always achieve
C
3
=
0in(
3.9
). If we substitute in (
3.1
)
v
=
e
−
λt
f,
∂
t
v
+
A
v
+
λv
=
in
J
×
G,
with
v(
0
,x)
=
u
0
(x)
in
G
. Choosing
λ>
0 large enough, the bilinear form
a(u,v)
+
λ(u, v)
satisfies (
3.9
) with
C
3
=
0.
3.3 Discretization
For the discretization we use the method of lines where first (
3.7
) is only discretized
in space to obtain a system of coupled ODEs which are solved in a second step.
Let
V
N
be a one-parameter family of subspaces
V
N
⊂
V
with finite dimension
N
=
dim
V
N
<
∞
. For each fixed
t
∈
J
we approximate the solution
u(t, x)
of (
3.7
)
by a function
u
N
(t)
V
N
be an approximation of
u
0
.
Then, the semidiscrete form of (
3.7
) is the initial value problem,
∈
V
N
. Furthermore, let
u
N,
0
∈
C
1
(J
Find
u
N
∈
;
V
N
)
such that for
t
∈
J
(∂
t
u
N
,v
N
)
H
+
a(u
N
,v
N
)
=
f, v
N
V
∗
,
V
,
∀
v
N
∈
V
N
,
(3.11)
u
N
(
0
)
=
u
N,
0
,
for the approximate solution function
u
N
(t)
:
J
→
V
N
.Let
V
N
be generated by a
finite element basis,
V
N
=
span
{
b
i
(x)
:
1
≤
i
≤
N
}
. We write
u
N
∈
V
N
in terms of
=
j
=
1
u
N,j
(t) b
j
(x)
, and obtain the matrix form of
the basis functions,
u
N
(t, x)
the semidiscretization (
3.11
)
C
1
(J
N
)
such that for
t
Find
u
N
∈
; R
∈
J,
M
u
N
(t)
+
A
u
N
(t)
=
f
(t) ,
(3.12)
u
N
(
0
)
=
u
0
,
where
u
0
denotes the coefficient vector of
u
N,
0
. The mass and stiffness matrices and
the load vector with respect to the basis of
V
N
are given by
M
ij
=
(b
j
,b
i
)
,
A
ij
=
a(b
j
,b
i
),
f
i
(t)
=
f, b
i
V
∗
,
V
,
(3.13)
H
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