Civil Engineering Reference
In-Depth Information
Since the triangulation is assumed to be shape-regular, we could estimate the di-
ameters of all triangles in
ω
T
by
ch
T
, i.e. the diameter of
T
. The estimates are still
local. The minimal property of the
L
2
-projector
Q
h
implies
v
−
Q
h
v
≤
c
T
2
0
h
T
v
2
1
,T
.
(
7
.
15
)
Next from the Bramble-Hilbert lemma and a standard scaling argument we know
that there is a piecewise constant function
w
h
∈
M
0
(
T
h
)
such that
v
−
w
h
0
,T
≤
ch
T
|
v
|
1
,T
.
Now we apply an inverse estimate on each triangle:
2
1
2
1
,T
2
1
,T
|
Q
h
v
|
=
|
Q
h
v
|
=
|
Q
h
v
−
w
h
|
T
T
≤
c
T
h
−
2
2
0
,T
Q
h
v
−
w
h
T
2
h
−
T
0
,T
≤
c
T
2
0
,T
2
Q
h
v
−
v
+
v
−
w
h
≤
c
T
2
1
,T
2
v
=
c
v
1
.
Since
Q
h
v
0
≤
v
0
≤
v
1
, the proof is complete.
Note that Problem 6.15 illustrates that one has to be careful when dealing with
a projector for one norm and considering stability for another one.
Problems
7.10
Consider solving the boundary-value problem
−
u
=
1
)
2
2
,
0in
:
=
(
−
1
,
+
⊂ R
u(x, y)
=
xy
on
∂
using linear triangular elements on a regular triangular grid with 2
/h
∈ N
as in the
model problem 4.3. When the reduction to homogeneous boundary conditions as
in (2.21) is performed with
1
+
x
−
y
for
x
≥
y,
u
0
(x, y)
:
=
+
y
−
x
for
x
≤
y,
the finite element approximation at the grid points is
1
u
h
(x
i
,y
i
)
=
u(x
i
,y
i
)
=
x
i
y
i
.
(
7
.
16
)
Verify that the minimal value for the variational functional on
S
h
is
8
3
+
4
3
h
2
,
J(u
h
)
=
and hence,
J(u
h
)
is only an approximation.
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