Biomedical Engineering Reference
In-Depth Information
U
h
represents the standard
L
2
-inner
and
F ∈
as
((
F,
v
)) = (
f
,
v
)
,where
(
·,·
)
product.
With these definitions, we can give the following equivalent formulation of the
discrete bidomain system (5.27): given
f
L
2
U
h
such that
∈
(
Ω
)
,find
u
∈
U
h
a
bid
(
u
,
v
)=(
f
,
v
)
, ∀
v
∈
,
or in term of linear operators
A
u
=
F.
(5.31)
5.3.1.2 The two-level Additive Schwarz preconditioner
In this section, we construct and analyze a two-level overlapping Additive Schwarz
(AS) method for problem (5.31). Let
0
H
be a coarse shape-regular triangula-
T
=
T
tion of
Ω
consisting of
N
nonoverlapping hexahedral subdomains
Ω
m
,
m
=
1
, ...,
N
1
h
be a fine shape-regular triangu-
of diameter
H
m
and let
H
=
max
m
H
m
.Let
T
=
T
0
, consisting of hexahedral elements
lation nested in
T
τ
j
,
j
=
1
,...,
N
e
of diameter
h
j
and let
h
=
max
j
h
j
. An overlapping partition of
Ω
is then constructed using the
1
standard technique of adding to each subdomain
Ω
m
all the fine elements
τ
j
∈T
Ω
m
the overlapping sub-
within a distance
δ
from its boundary
∂Ω
m
. We denote by
Ω
m
, we associate
domain obtained by such extensions of
Ω
m
. With each subdomain
the following finite-element spaces
V
h
∈
Ω
\
Ω
m
}
V
m
.
Let
V
H
be the space of trilinear finite elements associated to the coarse triangulation
T
0
and define
V
m
:
=
{
u
i
∈
:
u
i
(
x
)=
0
x
and
U
m
:
=
V
m
×
V
H
:
V
H
:
U
H
:
V
H
×
V
H
=
{
u
e
∈
u
e
=
0
},
U
0
=
=
.
Ω
U
h
, whereas
U
m
is not a subset of
U
h
,
m
We remark that
U
0
⊂
=
1
,...,
N
. Define the
U
h
,
m
interpolation operators
I
m
:
U
m
→
=
1
,...,
N
,asgiven
u
i
−
u
e
u
=(
u
i
,
u
e
)
∈
U
m
,
I
m
u
=(
I
m
,
i
u
,
I
m
,
e
u
)
:
=
u
e
,
u
e
−
.
Ω
Ω
U
h
is simply the embedding operator. Defining the projection
The operator
I
0
:
U
0
→
operators
T
m
:
U
h
by
a
bid
(
T
m
u
→
U
m
,
v
)=
a
bid
(
u
,
I
m
v
)
∀
v
∈
U
m
,
I
m
T
m
, we can define the AS operator as
T
AS
=
and
T
m
=
B
AS
A
:
=
T
0
+
T
1
+
...
+
T
N
.
(5.32)
It is easy to see that the matrix form of this operator is
T
AS
=
B
AS
A ,
A
where
is the Bidomain stiffness matrix (5.28) and
B
AS
is the AS preconditioner
N
m
=
0
B
m
=
R
m
A
−
m
R
m
.
B
AS
=
(5.33)
Here, for
m
N
,
R
m
are boolean restriction matrices and
A
m
the local stiff-
ness matrices for the Bidomain problems restricted to the subdomain
=
1
,...,
Ω
m
, while for
