Biomedical Engineering Reference
In-Depth Information
N
m
{
Ω
km
}
(c) There exists a partition of unity
{
θ
km
}
, associated with
1
,which
=
satisfies
m
θ
km
=
1
,
H
0
(
Ω
km
)
∩
C
0
with
θ
km
∈
(
Ω
km
)
,
0
≤
θ
km
≤
1
,
¯
|
∇θ
km
|
L
∞
≤
C
/
δ
k
and
||
θ
km
−
θ
km
||
L
∞
(
K
)
≤
C
(
h
k
/
δ
k
)
,
(5.34)
k
and
¯
where K
∈T
θ
km
is the average of
θ
km
on K.
For
k
=
1
,...,
L
−
1and
m
=
1
, ...,
N
k
, we define the spaces
V
h
k
H
1
(
Ω
km
)
,
V
km
:
=
∩
U
km
=
V
km
×
V
km
.
Let
I
km
be the interpolation operators as in the two-level case. Defining the projec-
tions
T
km
:
U
h
→
U
km
by
a
bid
(
T
km
u
,
v
k
)=
a
bid
(
u
,
I
km
v
k
)
∀
v
k
∈
U
km
,
and
T
km
=
I
km
T
km
, we can introduce the Multilevel Additive Schwarz (MAS) oper-
ator
L
1
k
=
1
−
N
k
m
=
1
T
km
.
=
A
=
+
T
MAS
B
MAS
:
T
0
(5.35)
As mentioned before, approximate local solvers
T
km
could be used as well. The
preconditioner
B
MAS
is given by
N
k
m
=
1
R
km
A
−
1
L
1
k
=
1
−
R
0
A
−
1
B
MAS
=
R
0
+
km
R
km
,
0
Ω
km
at level
k
and
R
km
is the
where
A
km
is the local stiffness matrix of the subdomain
Ω
km
, see [129] for more details.
The following result for the multilevel case has been proved in [85].
restriction matrix from the finest level to subdomain
Theorem 5.
The condition number of the multilevel additive Schwarz operator
T
MAS
defined in (5.35) is bounded:
1
h
k
−
1
δ
k
κ
2
(
T
MAS
)
≤
C
max
k
+
,
=
1
,...,
L
−
1
where C is a constant independent of the mesh sizes h
k
and the number of levels L.
5.4 Numerical results: parallel scalability
The numerical experiments have been performed on the Linux Cluster IBM BCX of
the Cineca Consortium (www.cineca.it), with 5120 processors. Our FORTRAN code
is based on the parallel library PETSc, from the Argonne National Laboratory [4].
