Database Reference
In-Depth Information
N
d
, we consider the family of
Now, with the multi-index l
¼
(
l
1
,
...
,
l
d
)
∈
standard regular grids
N
d
Ω
l
, l
∈
ð
7
:
9
Þ
:¼
2
l
1
. That is,
2
l
d
on
Ω
l
is equidis-
tant with respect to each coordinate direction, but, in general, has different mesh
sizes in the different coordinate directions. The grid points contained in a grid
Ω
with mesh size
h
l
:¼ j
l
1
; ...;
j
l
d
; ...;
Ω
l
are
the points
x
l
,
j
:¼ x
l
1
,
j
; ...;
x
l
d
ð
7
:
10
Þ
. On each grid
Þ :¼
2
l
1
2
l
d
with
h
l
:¼ h
l
1
; ...;
ð
h
l
d
; ...;
Ω
l
, we define the space
V
1
of piecewise
d
-linear functions
n
o
,2
l
t
,
t ¼
1,
V
l
:¼
span
ϕ
1
,
j
,
j
t
¼
0,
...
...
,
d
ð
7
:
11
Þ
which is spanned by the usual
d
-dimensional piecewise
d
-linear hat functions
ϕ
1
,
j
ðÞ:¼
Y
d
t¼
1
ϕ
l
t
,
j
t
xðÞ:
ð
7
:
12
Þ
Here, the one-dimensional functions
ϕ
l
t
,
j
t
xðÞ
with support
x
l
t
,
j
t
h
l
t
,
x
l
t
,
j
t
þ h
l
t
¼
ð Þh
l
½
(e.g., restricted to [0,1]) can be created from a unique
one-dimensional mother function
ð
j
t
1
Þh
l
t
,
j
t
þ
1
ϕ
(
x
)
1
jj x
∈
1, 1 ,
½
ϕ
ðÞ:¼
ð
7
:
13
Þ
0
otherwise
by dilatation and translation, i.e.,
x
t
j
t
h
l
t
ϕ
l
t
,
j
t
xðÞ:¼ ϕ
:
ð
7
:
14
Þ
h
l
t
N
d
indicates
the level of a grid or a space or a function, respectively, whereas the multi-index
j
∈
In the previous definition and in the following, the multi-index l
N
d
denotes the location of a given grid point
x
l,j
or of the respective basis
function
ϕ
1,j
(x), respectively.
Now, we can define the difference spaces
∈
[
d
W
1
¼ V
1
=
V
1
e
t
,
ð
7
:
15
Þ
t¼
1
where e
1
denotes the
t
th unit vector. To complete this definition, we formally set
V
1
¼
0if
l
t
¼
1 for at least one
t
∈
{1,
...
,
d
}. In other words,
W
1
consists of
