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(x
−
x
N
)/h
N
+
1
if
x
∈
(x
N
,x
N
+
1
),
b
N
+
1
(x)
:=
(3.20)
0
else
.
If the mesh
T
is equidistant, i.e.
h
i
=
h
=
(b
−
a)/(N
+
1
)
, we can write
h
−
1
b
i
(x)
=
max
{
0
,
1
−
|
x
−
x
i
|}
,i
=
0
,...,N
+
1
.
Note that for a given subspace
S
1
T
, there are many different possible choices of basis
functions. The choice (
3.18
)-(
3.20
) are the basis functions with
smallest support
.
The FE subspace to approximate functions with homogeneous Dirichlet boundary
conditions is
S
1
S
1
H
0
(G)
T
,
0
:=
T
∩
=
span
{
b
i
(x)
:
i
=
1
,...,N
}
,
(3.21)
with dim
S
1
T
,
0
=
N
.
3.4.1 Elemental Forms and Assembly
·
·
·
·
=
+
We decompose
a(
,
)
(
3.16
)into
elemental bilinear forms a
l
(
,
)
,
l
1
,...,N
1,
α(x)b
j
(x)b
i
(x)
γ(x)b
j
(x)b
i
(x)
d
x
β(x)b
j
(x)b
i
(x)
a(b
j
,b
i
)
=
+
+
G
N
+
1
α(x)b
j
(x)b
i
(x)
γ(x)b
j
(x)b
i
(x)
d
x
β(x)b
j
(x)b
i
(x)
=
+
+
K
l
l
=
1
N
+
1
=:
a
l
(b
j
,b
i
).
l
=
1
The restrictions
b
i
|
K
l
,
i
=
l
−
1
,l
, are linear and given by the element shape func-
tions
N
K
l
:=
|
K
l
,
N
K
l
:=
|
K
l
,
l
=
+
1. The element stiffness
matrix
A
l
associated to
a
l
(
·
,
·
)
can be computed for each element independently. We
transform each element
K
l
∈
T
b
l
−
1
(x)
b
l
(x)
1
,...,N
to the so-called
reference element K
:=
(
−
1
,
1
)
via
an element mapping
1
2
(x
l
−
1
+
1
2
h
l
ξ,
∈
K,
K
l
x
=
F
K
l
(ξ )
:=
x
l
)
+
ξ
with derivative
F
K
l
(ξ )
=
h
l
/
2. Using the reference element shape functions
1
2
(
1
1
2
(
1
N
1
(ξ )
N
2
(ξ )
=
−
ξ),
=
+
ξ),
which are independent of the element
K
l
,wehaveforall
K
l
∈
T
,
N
K
l
(F
K
l
(ξ ))
=
N
1
(ξ ),
N
K
l
(F
K
l
(ξ ))
=
N
2
(ξ ).
The reference shape functions are shown in Fig.
3.1
.
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