Civil Engineering Reference
In-Depth Information
7.11
Let
=
(
0
,
2
π)
2
be a square, and suppose
u
∈
H
0
()
is a weak solution
of
−
u
=
f
with
f
∈
L
2
()
. Using Problem 1.16, show that
u
∈
L
2
()
, and
then use the Cauchy-Schwarz inequality to show that all second derivatives lie in
L
2
, and thus
u
is an
H
2
function.
7.12
(A superconvergence property)
The boundary-value problem with the ordinary
differential equation
−
u
(x)
=
f(x),
x
∈
(
0
,
1
),
u(
0
)
=
u(
1
)
=
0
characterizes the solution of a variational problem with the bilinear form
1
u
v
dx.
a(u, v)
:
=
0
Let
u
h
be the solution in the set of piecewise linear functions on a partition of
(
0
,
1
)
,
and let
v
h
be the interpolant of
u
in the same set. Show that
u
h
=
v
h
by verifying
a(u
h
−
v
h
,w
h
)
=
0 for all piecewise linear
w
h
.
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