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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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