February 21, 2000
Notes for Lecture #8
Finite element method
The nite element approach is based on an expansion of the unknown electrostatic potential
in terms of known functions of xed shape
f
ij
(
x; y
)
g
.In two dimensions this expansion
takes the form:
4
"
0
(
x; y
)=
X
ij
ij
ij
(
x; y
)
;
(1)
where
ij
represents the amplitude associated with the shape function
ij
(
x; y
). The ampli-
tude values can be determined for a given solution of the Poisson equation:
r
2
(4
"
0
(
x; y
)) = 4
(
x; y
)
;
(2)
by solving a linear algebra problem of the form
X
ij
M
kl;ij
ij
=
G
kl
;
(3)
where
M
kl;ij
Z
dx
Z
dy
r
kl
(
x; y
)
r
ij
(
x; y
)and
G
kl
Z
dx
Z
dy
kl
(
x; y
)4
(
x; y
)
:
(4)
In obtaining this result, wehave assumed that the boundary values vanish. In order for this
result to be useful, we need to be able evaluate the integrals for
M
kl;ij
and for
G
kl
. In the
latter case, we need to know the form of the charge density. The form of
M
kl;ij
only depends
upon the form of the shap e functions. If we take these functions to be:
ij
(
x; y
)
X
i
(
x
)
Y
j
(
y
)
;
(5)
where
X
i
(
x
)
(
1
j
x
x
i
j
h
for
x
i
h
x
x
i
+
h
0 otherwise
;
(6)
and
Y
j
(
y
) has a similar expression in the variable
y
. Then
M
kl;ij
Z
dx
Z
dy
"
d
X
k
(
x
)
dx
d
X
i
(
x
)
dx
Y
l
(
y
)
Y
j
(
y
)+
X
k
(
x
)
X
i
(
x
)
d
Y
l
(
y
)
dy
d
Y
j
(
y
)
dy
#
:
(7)
There are four types of non-trivial contributions to these values:
Z
x
i
+
h
x
i
h
(
X
i
(
x
))
2
dx
=
h
Z
1
1
(1
j
u
j
)
2
du
=
2
h
3
;
(8)
Z
x
i
+
h
x
i
h
(
X
i
(
x
)
X
i
+1
(
x
))
dx
=
h
Z
1
0
(1
u
)
udu
=
h
6
;
(9)
