Information Technology Reference
In-Depth Information
=
=
where
i, j
1
,...,N
.Let
k
m
,
m
1
,...,M
, be a sequence of (not necessarily
0,
t
m
:=
i
=
1
k
i
such that
t
M
=
T
. Applying
the
θ
-scheme, we obtain the fully discrete form
equal sized) time steps and set
t
0
:=
Find
u
N
∈
V
N
such that for
m
=
1
,...,M,
k
−
m
(u
N
−
u
m
−
1
,v
N
)
H
+
a(u
m
−
1
+
θ
,v
N
)
=
f
m
−
1
+
θ
,v
N
V
∗
,
V
,
∀
v
N
∈
V
N
,
N
N
u
0
N
=
u
N,
0
,
(3.14)
where
u
m
+
θ
N
θ)u
N
(t
m
)
and
f
m
+
θ
=
θu
N
(t
m
+
1
)
+
(
1
−
=
θf (t
m
+
1
)
+
(
1
−
θ)f(t
m
)
.
We can again write (
3.14
) in matrix notation,
Find
u
N
∈ R
N
such that for
m
=
1
,...,M,
k
m
θ
A
)
u
N
=
θ)
A
)
u
m
−
1
N
k
m
(θ
f
m
θ)
f
m
−
1
),
(
M
+
(
M
−
k
m
(
1
−
+
+
(
1
−
(3.15)
u
0
N
=
u
0
.
In the next section, we discuss the implementation of the matrix form (
3.15
).
3.4 Implementation of the Matrix Form
Let
G
(a, b)
. We describe a scheme to calculate the stiffness matrix
A
in case the
corresponding bilinear form
a(
=
·
,
·
)
has the form
α(x)ϕ
(x)φ
(x)
γ(x)ϕ(x)φ(x)
d
x,
β(x)ϕ
(x)φ(x)
a(ϕ,φ)
=
+
+
(3.16)
G
and the finite element subspace
V
N
consists of continuous, piecewise linear func-
tions. Let
T
={
a
=
x
0
<x
1
<x
2
<
···
<x
N
+
1
=
b
}
be an arbitrary mesh on
G
.
Setting
K
l
:=
(x
l
−
1
,x
l
)
,
h
l
:= |
K
l
|=
x
l
−
x
l
−
1
,
l
=
1
,...,N
+
1, we can also write
N
+
1
K
l
}
T
={
. Define
l
=
1
T
:=
u(x)
|
K
l
is linear on
K
l
∈
T
.
S
1
C
0
(G)
∈
:
u
(3.17)
A basis for
S
1
T
=
span
{
b
i
(x)
:
i
=
0
,...,N
+
1
}
is given by the so-called
hat-
functions
where, for 1
≤
i
≤
N
,
⎧
⎨
(x
−
x
i
−
1
)/h
i
if
x
∈
(x
i
−
1
,x
i
]
,
b
i
(x)
:=
(x
i
+
1
−
x)/h
i
+
1
if
x
∈
(x
i
,x
i
+
1
),
(3.18)
⎩
0
else
,
and
(x
1
−
x)/h
0
if
x
∈
(x
0
,x
1
),
b
0
(x)
:=
(3.19)
0
else
,
Search WWH ::
Custom Search