Civil Engineering Reference
In-Depth Information
1.8 Theorem.
Under the hypotheses of Theorem 1.7,
u
−
u
h
0
,
≤
ch
3
/
2
u
2
,
.
The error term
O
(h
3
/
2
)
arises from the pointwise estimate of the finite-
element functions
|
u
h
(x)
|≤
ch
2
for all
x
∈
; cf. (1.15). If we ap-
proximate the boundary with quadratic (instead of linear) functions, giving a one
higher power of
h
, the final result is improved by the same factor. This can be
achieved using isoparametric elements, for example.
|∇
u
h
(x)
|
Problems
1.9
Let
S
h
be an affine family of
C
0
elements. Show that in both the approximation
and inverse estimates,
·
2
,h
can be replaced by the mesh-dependent norm
T
j
v
2
,T
j
+
h
−
1
{
∂v
∂ν
2
ds.
2
h
2
|||
v
|||
:
=
e
m
e
m
}
Here
{
e
m
}
is the set of inter-element boundaries, and
·
denotes the jump of a
function.
Hint: In
H
2
(T
ref
)
,
and
2
ds
1
/
2
+
∂T
ref
|∇
v
|
2
2
,T
ref
v
v
are equivalent
2
,T
ref
norms.
1.10
The linear functional
L
u
appearing in the analysis of the Crouzeix-Raviart
element vanishes on the subset
H
0
()
by the definition of weak solutions. What
is wrong with the claim that
L
u
vanishes for all
w
∈
L
2
()
because of the density
of
H
0
()
in
L
2
()
?
1.11
If the stiffness matrices are computed by using numerical quadrature, then
only approximations
a
h
of the bilinear form are obtained. This holds also for
conforming elements. Estimate the influence on the error of the finite element
solution, given the estimate
2
1
|
a(v, v)
−
a
h
(v, v)
|≤
ε(h)
v
for all
v
∈
S
h
.
1.12
The Crouzeix-Raviart element has locally the same degrees of freedom as
the conforming
P
1
element
1
0
, i.e., the Courant triangle. Show that the (global)
dimension of the finite element spaces differ by a factor that is close to 3 if a
rectangular domain as in Fig. 9 is partitioned.
M
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