Biomedical Engineering Reference
In-Depth Information
M
and
F
n
−
F
n
I
ion
(
v
n
M
,
w
n
+
1
c
n
+
1
I
app
]
[
−
,
)+
F
=
=
,
(5.29)
I
ion
(
v
n
M
,
w
n
+
1
c
n
+
1
I
app
]
M
[
,
)
−
where
v
n
M
=
u
e
. As in the continuous model,
v
n
M
is uniquely determined, while
u
i
and
u
e
are determined only up to the same additive time-dependent constant cho-
sen by imposing the condition
1
T
M
u
e
=
u
i
−
0. Hence, at each time step we have to
solve the large linear system (5.27), that, as shown in [29], is very ill-conditioned
and increases considerably the computational costs of the simulations.
5.3.1 Linear solvers for the Bidomain model
The advantage of using semi-implicit and/or operator-splitting methods is that they
only require the solution of linear systems at each time step. In order to devise effi-
cient iterative solvers for these linear systems, many different preconditioners have
been proposed: diagonal preconditioners [126], SSOR preconditioners [87], Block
Jacobi (BJ) preconditioners [29, 75, 148], Block triangular Monodomain-based pre-
conditioners [52], Multigrid preconditioners [3, 73, 75, 88, 89, 90, 131, 149, 150],
and Multilevel Schwarz preconditioners [85, 120]. These studies have shown that
Multigrid and Multilevel Schwarz methods yield efficient solvers for the discrete
Bidomain equations and improve considerably the performance of the simplest di-
agonal or block-diagonal preconditioner on several processors.
5.3.1.1 Functional reformulation of the semi-discrete Bidomain system
In this section, we introduce the notations and basic results that we will need in
the next sections in order to construct a two-level Schwarz preconditioner for the
Bidomain system and to prove its scalability. We will reformulate (5.27) as an elliptic
problem and we will adapt to it the standard abstract Schwarz theory described e. g.
in [129, 140]. In the rest of the section, we will denote by
v
u
generic functions in
the product space
U
h
defined below and we will denote their intra- and extracellular
components by
v
,
=(
v
i
,
v
e
)
,
u
=(
u
i
,
u
e
)
.
Let us introduce the spaces
U
h
V
h
×
V
h
V
h
V
h
:
1
T
M
u
e
=
=
,
with
=
{
u
e
∈
u
e
dx
=
0
},
(5.30)
Ω
=
Ω
∇
v
i
+
Ω
∇
v
e
+
Ω
(
equipped with the inner product
((
u
,
v
))
:
u
i
·
∇
u
e
·
∇
u
i
−
u
e
)(
.
We introduce the continuous and elliptic, with respect to the
v
i
−
v
e
)
and with the induced norm
|||·|||
|||·|||
-norm, bilinear
:
U
h
U
h
form
a
bid
(
·,·
)
×
→
R
defined by
a
bid
(
u
,
v
)
:
=
D
i
∇
u
i
·
∇
v
i
+
D
e
∇
u
e
·
∇
v
e
+
γ
Ω
(
u
i
−
u
e
)(
v
i
−
v
e
)
,
Ω
Ω
and the continuous linear form
f
:
U
h
U
h
×
→
R
defined as
(
f
,
v
)=(
f
,
v
i
)
−
(
f
,
v
e
)
U
h
:
U
h
U
h
as
∀
v
∈
.
We define the linear operator
A
→
U
h
((
A
u
,
v
)) =
a
bid
(
u
,
v
)
∀
v
∈
,
