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Remark 10.6.1
If the Lévy measure satisfies
|
1
|
|
∞
z
ν(
d
z) <
, we have the gen-
z
|≤
erator
J
f )(x)
=
f
(x
+
z)k
(
−
1
)
(z)
d
z,
(
A
(f (x
+
z)
−
f (x))ν(
d
z)
=
R
R
and furthermore, if
,wehave(with
λ
=
R
1
ν(
d
z) <
∞
ν(
d
z)
)
|
z
|≤
J
f )(x)
(
A
=
f(x
+
z)k(z)
d
z
−
λf (x).
R
In both cases, similar discretization schemes can be derived.
10.6.2 Finite Element Discretization
We consider again the mesh (
10.26
) with uniform mesh width
h
and the finite el-
ement space
V
N
=
S
1
T
∩
H
0
(G)
with
S
1
T
=
span
{
b
i
(x)
:
i
=
1
,...,N
}
. We need
to compute the stiffness matrix
A
ij
=
a
J
(b
j
,b
i
)
, where the bilinear form
a
J
(
·
·
,
)
is
given by (
10.25
). We define auxiliary variables for
j
=
0
,
1
,
2
,...
,
h
(j
+
1
)h
k
j
k
(
−
2
)
(y
=
−
x)
d
y
d
x,
0
jh
(j
+
1
)h
h
k
j
k
(
−
2
)
(y
=
−
x)
d
y
d
x,
jh
0
and write for
j
≥
i
x
i
+
1
x
j
+
1
a
J
(b
j
,b
i
)
b
j
(y)b
i
(x)k
(
−
2
)
(y
=
−
x)
d
y
d
x
x
i
−
1
x
j
−
1
h
2
x
i
x
j
x
i
x
j
+
1
1
k
(
−
2
)
(y
k
(
−
2
)
(y
=
−
x)
d
y
d
x
−
−
x)
d
y
d
x
x
i
−
1
x
j
−
1
x
i
−
1
x
j
k
(
−
2
)
(y
−
x)
d
y
d
x
x
i
+
1
x
j
x
i
+
1
x
j
+
1
k
(
−
2
)
(y
−
x)
d
y
d
x
+
−
x
i
x
j
−
1
x
i
x
j
h
2
h
(j
−
i
+
1
)h
1
k
(
−
2
)
(y
−
x)
d
y
d
x
=
0
(j
−
i)h
h
(j
−
i
+
2
)h
k
(
−
2
)
(y
−
−
x)
d
y
d
x
0
(j
−
i
+
1
)h
h
(j
−
i)h
k
(
−
2
)
(y
−
−
x)
d
y
d
x
0
(j
−
i
−
1
)h
x)
d
y
d
x
.
h
(j
−
i
+
1
)h
k
(
−
2
)
(y
+
−
0
(j
−
i)h
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