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9.5.1 Finite Difference Discretization
We define weighted versions of the matrices
R
,
C
and
I
given in (4.14). For
w
:
N
k
×
N
k
R → R
define matrices in
R
with respect to
k
th coordinate direction
⎛
⎝
⎞
⎠
2
w(x
k,
1
)
−
w(x
k,
1
)
.
.
.
.
.
.
−
1
h
k
w(x
k,
2
)
R
w(x
k
)
:=
,
.
.
.
.
.
.
−
w(x
k,N
k
−
1
)
−
w(x
k,N
k
)
2
w(x
k,N
k
)
⎛
⎞
0
w(x
k,
1
)
⎝
⎠
.
.
.
.
.
.
1
2
h
k
−
w(x
k,
2
)
C
w(x
k
)
:=
,
.
.
.
.
.
.
w(x
k,N
k
−
1
)
−
w(x
k,N
k
)
0
as well as
diag
w(x
k,
1
), . . . , w(x
k,N
k
)
.
Here, we denote by
x
k,i
k
agridpointofagridinthe
k
th coordinate direction, i.e.
x
k,i
k
=
a
k
+
h
k
i
k
,
h
k
=
I
w(x
k
)
:=
b
k
−
a
k
N
k
+
1
. The following definition will help to simplify the
notation.
Definition
9.5.2
For
an
arbitrary
permutation
σ
:{
1
,...,d
}→{
1
,...,d
}
,
and matrices
X
w(x
k
)
,1
{
1
,...,d
}→{
σ(
1
),...,σ(d)
}
≤
k
≤
d
, we denote by
s
(
X
w(x
σ(
1
)
)
X
w(x
σ(d)
)
)
the sorted Kronecker product with factors sorted by
increasing indices, i.e.
s
X
w(x
σ(
1
)
)
⊗···⊗
X
w(x
σ(d)
)
:=
X
w(x
1
)
X
w(x
d
)
.
⊗···⊗
⊗···⊗
Using the finite difference quotients
δ
x
i
x
j
,
δ
x
i
on the grid
G
:=
(x
1
,i
1
,...,x
d,i
d
)
d
⊂
G
R
,
and proceeding exactly as in Sect. 8.4.1, we find that the finite difference matrix
G
SV
|
1
≤
i
k
≤
N
k
,
1
≤
k
≤
corresponding to (
9.27
) is given by
s
R
q
ii
(x
i
)
I
q
ii
(x
k
)
d
1
2
G
SV
:=
⊗
i
=
1
k
=
i
s
C
q
ij
(x
i
)
I
q
ij
(x
k
)
d
1
2
C
q
ij
(x
j
)
−
⊗
⊗
i,j
=
1
k/
∈{
i,j
}
j
=
i
s
C
μ
i
(x
i
)
I
μ
i
(x
k
)
d
I
c
k
(x
k
)
.
−
⊗
−
i
=
1
k
=
i
k
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