Civil Engineering Reference
In-Depth Information
Fig. 24.
The values of the Clement interpolant in the (shaded) triangle
T
depend
on the values of the given function in its neighborhood
ω
T
Clement's Interpolation
The interpolation operator
I
h
in (6.5) can only be applied to
H
2
functions. On the
other hand, functions with less regularity can be approximated in some advanced
theories. Clement [1975] has constructed an interpolation process which applies to
H
1
functions. Typically this operator is used when features in
H
1
and
L
2
are to
be combined. The crucial point is that the interpolation error depends only on the
local mesh size. Thus, no power of
h
is lost, even if inverse estimates enter into the
analysis.
The operator is defined
nearly locally.
Let
T
h
be a shape-regular triangulation
of
. Given a node
x
j
, let
{
T
∈
T
h
;
x
j
∈
T
}
ω
j
:
=
ω
x
j
:
=
(
6
.
13
)
1
be the support of the shape function
v
j
∈
M
0
. Here
v
j
(x
k
)
=
δ
jk
. Furthermore, let
{
ω
j
;
x
j
∈
T
}
ω
T
:
=
(
6
.
14
)
be a neighborhood of
T
. Since
T
h
is assumed to be shape regular, the area can be
c(κ) h
T
. Moreover, the number of triangles that belong to
estimated by
µ(
ω
T
)
˜
≤
ω
T
is bounded.
6.9 Clement's Interpolation.
Let
T
h
be a shape-regular triangulation of . Then
there exists a linear mapping I
h
:
H
1
()
→
M
1
0
such that
v
−
I
h
v
m,T
≤
ch
1
−
m
for v
∈
H
1
(), m
=
v
1
, ω
T
0
,
1
,T
∈
T
h
T
v
−
I
h
v
0
,e
≤
ch
1
/
2
for v
∈
H
1
(), e
∈
∂T, T
∈
T
h
.
(
6
.
15
)
v
1
, ω
T
T
A simple construction is obtained by a combination of Clement's operator and
the procedures of Scott and Zhang [1990] or Yserentant [1990]. The construction is
Search WWH ::
Custom Search