Civil Engineering Reference
In-Depth Information
6.16
Prove the Bramble-Hilbert lemma for
t
=
1 by choosing
Iv
to be the constant
function
vdx
dx
Iv
:
=
.
6.17
Consider the situation as in the construction of Clements' operator. We modify
the definition of the operator
Q
j
:
L
2
(ω
j
)
→
P
0
by the rule
Q
j
v
:
1
=
v(x
j
)
if
v
|
ω
j
∈
M
0
(
T
),
(
6
.
23
)
i.e., if the restriction of
v
to
ω
j
is a finite element function with the present grid.
Show that also in this case
v
−
Q
j
v
0
,ω
j
≤
h
j
|
v
|
1
,ω
j
with
c
depending only on the shape parameter of the triangulation of
ω
j
.
Hint:
The modification has an advantage. If the given function coincides with a
piecewise linear function on a subdomain
˜
, then the projector reproduces
v
at the
nodes in the interior of
˜
.
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