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Q
11
2
r
M
1
A
BS
S
1
μ
1
B
1
M
2
−
Q
12
B
1
μ
2
M
1
)
B
2
=
+
+
⊗
+
(
+
⊗
+
Q
22
2
M
1
S
2
.
⊗
Proposition
8.4.2
can be extended to arbitrary dimensions
d>
2. The corresponding
finite element space
V
N
is spanned by the
d
-fold tensor products of univariate hat-
functions, i.e.
V
N
:=
span
{
ϕ
j
(x
1
,...,x
d
)
|
1
≤
j
≤
N
}
=
span
{
b
i
1
(x
1
)
···
b
i
d
(x
d
)
|
1
≤
i
k
≤
N
k
,k
=
1
,...,d
}
,
(8.19)
:=
k
=
1
N
k
. The index
j
in (
8.19
) is defined via the indices
with dim
V
N
=
N
i
1
,...,i
d
by
j
:=
d
−
1
1
d
−
k
1
N
+
k
(i
k
−
1
)
+
i
d
; compare with (
8.18
). The stiffness
k
=
=
matrix
A
BS
N
×
N
∈ R
becomes
i
=
1
Q
ii
d
i
−
1
d
1
2
A
BS
M
k
S
i
M
k
=
⊗
⊗
k
=
1
k
=
i
+
1
j
−
1
d
−
1
d
i
−
1
d
M
k
B
i
M
k
B
j
M
k
−
1
Q
ij
⊗
⊗
⊗
⊗
i
=
1
j
=
i
+
k
=
1
k
=
i
+
1
k
=
j
+
1
d
i
−
1
d
d
M
k
B
i
M
k
M
k
.
+
μ
k
⊗
⊗
+
r
(8.20)
k
=
1
k
=
1
k
=
i
+
1
k
=
1
If we furthermore discretize (
8.13
)bythe
θ
-scheme in time, we obtain the matrix
problem
Find
u
m
+
1
N
∈ R
such that for
m
=
0
,...,M
−
1
θt
A
BS
)
u
m
+
1
θ)t
A
BS
)
u
m
,
(
M
+
=
(
M
−
(
1
−
(8.21)
u
0
N
=
u
0
.
Note that if
N
1
= ··· =
N
d
, then the mesh width is the same in each coordinate
direction, and the matrices in the representation of
A
BS
M
1
satisfy
M
:=
= ··· =
M
d
,
B
S
d
. Hence, in this case, we have to
calculate only the matrices
M
,
B
and
S
, independent of the dimension
d
.
We give a convergence result that generalizes Theorem 3.6.5 to arbitrary dimen-
sions. For simplicity, we assume that the mesh width is the same in each coordinate
direction,
h
B
1
B
d
S
1
:=
=···=
and
S
:=
=···=
:=
h
1
=···=
h
d
.Let
u
:=
u
R
be the unique solution of (
8.13
), and let
u
N
(t
m
,
V
N
be its finite element approximation at time level
t
m
with correspond-
ing coefficient vector
u
m
·
)
∈
obtained from (
8.21
). As in the one-dimensional case, we
split the error
e
N
(x)
u
m
u
N
:=
u(t
m
,x)
−
u
N
(t
m
,x)
=:
−
as
e
N
=
u
m
−
P
N
u
m
+
P
N
u
m
−
u
N
=:
η
m
+
ξ
N
,
(8.22)
where
V
N
is a projector or a quasi-interpolant. The reason why we
cannot rely on a multivariate version of the nodal interpolant
P
N
:
V
→
I
N
as in (3.24)is
H
1
(G)
for
G
d
that functions
u
∈
V
=
⊂ R
are not necessarily continuous. In fact,
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