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(N
+
2
)
×
(N
+
2
)
∈ R
Hence, we can assemble the matrix
A
using the following summa-
tion
⎛
⎝
⎞
⎠
+
⎛
⎝
⎞
⎠
+···+
⎛
⎝
⎞
⎠
A
K
1
0
0
0
A
K
2
.
.
.
=
A
.
.
.
.
.
.
.
0
0
0
A
K
N
+
1
(3.23)
As for the stiffness matrix, we also decompose the load vector
f
into elemental
loads. For
f(t)
L
2
(G)
,wehave
∈
N
+
1
N
+
1
(f (t), b
i
)
=
f(t,x)b
i
(x)
d
x
=
f(t,x)b
i
(x)
d
x
=:
(f
l
(t), b
i
).
G
K
l
l
=
1
l
=
1
Using again the local shape function, we obtain for
i
=
1
,
2,
h
l
2
(f
l
(t), N
i
K
l
)
f(t,x)N
i
K
l
d
x
K
f
K
l
(t, ξ )N
i
=
=
d
ξ,
K
l
where
f
t,F
K
l
(ξ )
.
For general functions
f
, these integrals cannot be computed exactly and have to be
approximated by a numerical quadrature rule. To assemble the global load vector
f
,
we use
f
K
l
(t, ξ)
:=
⎛
⎝
⎞
⎠
+
⎛
⎝
⎞
⎠
+···+
⎛
⎝
⎞
⎠
f
K
1
0
.
0
0
f
K
2
.
0
0
.
0
f
K
N
+
1
f
=
.
H
0
(G)
we have
V
N
Remark 3.4.1
For
V
=
=
span
{
b
i
(x)
:
i
=
1
,...,N
}
, since
u(x
0
)
=
u(x
N
+
1
)
=
0. Hence, we get
N
degrees of freedom and the reduced ma-
trices
M
,
A
N
where the first and last rows and columns of
A
in (
3.23
)are
omitted. Similar considerations can be made for the vector
f
. If we have non-
homogeneous Dirichlet boundary condition,
u(t, a)
N
×
∈ R
=
φ
1
(t)
,
u(t, b)
=
φ
2
(t)
for
given functions
φ
1
,
φ
2
, we write
u(t, x)
=
w(t,x)
+
φ(t,x)
, where the function
φ(t,x)
satisfies the boundary conditions,
φ(t,a)
φ
2
(t)
.Now,
w
has again homogeneous Dirichlet boundary conditions and is the solution of
=
φ
1
(t)
,
φ(t,b)
=
d
d
t
w,v
V
∗
,
V
+
a(w,v)
=
f, v
V
∗
,
V
−
∂
t
φ,v
V
∗
,
V
−
a(φ,v).
Choosing
φ(t,x)
=
φ
1
(t)b
0
(x)
+
φ
2
(t)b
N
+
1
(x)
, we obtain the matrix form
M
w
N
(t)
+
A
w
N
(t)
=
l
(t),
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