Civil Engineering Reference
In-Depth Information
The Crouzeix-Raviart Element
The Crouzeix-Raviart element is the simplest nonconforming element for the
discretization of second order elliptic boundary-value problems. It is also called
the
nonconforming P
1
element
.
•
•
•
Fig. 28.
The Crouzeix-Raviart element or nonconforming
P
1
-element
1
∗
M
:
={
v
∈
L
2
()
;
v
|
T
is linear for every
T
∈
T
h
,
v
is continuous at the midpoints of the triangle edges
}
,
(
1
.
7
)
1
∗
,
0
1
M
:
={
v
∈
M
∗
;
v
=
0 at the midpoints of the edges on
∂
}
.
To solve the Poisson equation, let
for all
u, v
∈
H
1
()
+
M
1
a
h
(u, v)
:
=
T
∇
u
·∇
vdx
∗
,
0
,
T
∈
T
h
a
h
(v, v)
for all
v
∈
H
1
()
+
M
1
∗
v
h
:
=
,
0
.
=
T
∈
T
h
|
v
|
1
,T
, and it is called a
broken H
1
semi-norm
.
For simplicity, suppose
is a convex polyhedron. Then the problem is
H
2
-
regular, and
u
∈
H
2
()
.
Given
v
∈
H
2
()
, let
Iv
∈
M
2
h
2
By definition
v
:
∩
C
0
()
be the continuous piecewise linear
function which interpolates
v
at the vertices of the triangles. We denote edges of
the triangles by the letter
e
.
To apply Lemma 1.2, we compute
1
∗
,
0
L
u
(w
h
)
:
=
a
h
(u, w
h
)
−
, w
h
=
∇
u
∇
w
h
dx
−
fw
h
dx
T
∈
T
h
u w
h
dx
−
=
∂
ν
uw
h
ds
−
fw
h
dx
∂T
T
∈
T
h
T
=
∂
ν
uw
h
ds,
∂T
T
∈
T
h
1
∗
for
w
h
∈
M
−
u
=
f
holds in the weak
sense; cf. Example II.2.10. In addition, note that each interior edge appears twice
,
0
. Here we have used the fact that
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