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where
2
2
1
2
v)
i
1
,i
2
:=
1
Q
kk
(δ
x
k
x
k
v)
i
1
,i
2
+
Q
12
(δ
x
1
x
2
v)
i
1
,i
2
−
μ
k
(δ
x
k
v)
i
1
,i
2
.
(
F
k
=
k
=
1
We write (
8.15
) in matrix form. Setting, for
N
:=
N
1
N
2
,
u
m
(u
1
,...,u
N
)
,
u
N
2
(i
1
−
1
)
+
i
2
:=
v
i
1
,i
2
,
=
=
j
we find that (
8.15
) is equivalent to
Find
u
m
+
1
N
∈ R
such that for
m
=
0
,...,M
−
1
,
I
θt
G
BS
u
m
+
1
=
I
θ)t
G
BS
u
m
,
(8.16)
+
−
(
1
−
u
0
=
u
0
,
where the matrices
I
and
G
BS
are given by
I
1
I
2
,
I
:=
⊗
1
2
Q
11
R
1
1
2
Q
11
I
1
G
BS
I
2
−
Q
12
C
1
C
2
R
2
:=
⊗
⊗
+
⊗
+
μ
1
C
1
I
2
+
μ
2
I
1
C
2
+
r
I
1
I
2
.
⊗
⊗
⊗
Herewith, we denote by
R
k
,
C
k
,
I
k
N
k
×
N
k
, the matrices given in (4.14) with
∈ R
respect to the coordinate direction
x
k
,
k
=
1
,
2.
8.4.2 Finite Element Discretization
In the two-dimensional case, the bilinear form
a
BS
(
·
,
·
)
in (
8.9
) simplifies to
1
Q
k
μ
k
r
2
2
1
2
a
BS
(ϕ, φ)
=
∂
x
ϕ∂
x
k
φ
d
x
+
∂
x
k
ϕφ
d
x
+
ϕφ
d
x.
G
G
G
,k
=
k
=
1
(8.17)
The discretization of the weak formulation (
8.13
) relies on finite element spaces
V
N
⊂
H
0
(G)
which are spanned by products of hat-functions. To be more precise,
for
N
1
,N
2
∈ N
we consider
V
N
:=
span
{
ϕ
j
(x
1
,x
2
)
|
1
≤
j
≤
N
1
N
2
}
=
span
{
ϕ
N
2
(i
1
−
1
)
+
i
2
(x
1
,x
2
)
|
1
≤
i
k
≤
N
k
,k
=
1
,
2
}
=
span
{
b
i
1
(x
1
)b
i
2
(x
2
)
|
1
≤
i
k
≤
N
k
,k
=
1
,
2
}
,
(8.18)
where
b
i
k
(x
k
)
are the univariate hat-functions as in (3.18). Note that dim
V
N
=
N
:=
N
1
N
2
. If the mesh sizes
h
k
=
2
R/(N
k
+
1
)
in both coordinate directions are con-
−
h
−
1
k
stant, we have
b
i
k
(x
k
)
=
max
{
0
,
1
|
x
k
−
x
k,i
k
|}
,
i
k
=
1
,...,N
k
, with mesh
points
x
k,i
k
:= −
R
+
h
k
i
k
,
i
k
=
0
,...,N
k
+
1.
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