Biomedical Engineering Reference
In-Depth Information
m
0
R
0
is the fine-to-coarse restriction matrix and
A
0
is the coarse Bidomain stiff-
ness matrix associated with the coarse space
U
0
=
=
U
H
. More general projection-like
operators
T
m
associated with approximate bilinear forms and inexact local solvers
could be used as well, see [129, 140]. The use of the
B
AS
preconditioner (5.33) for
the iterative solution of the Bidomain discrete system (5.31) can also be regarded as
a way for replacing (5.31) with the preconditioned system
T
AS
u
=
g
,
=
where
g
B
AS
f
, which can be accelerated by a Krylov subspace method.
The following result extend the classical overlapping Schwarz analysis to the
condition number of
T
AS
, see [85].
Theorem 4.
The condition number of the 2-level additive Schwarz operator
T
AS
de-
fined in (5.32) is bounded by
C
1
H
δ
κ
2
(
T
AS
)
≤
+
,
with C independent of h
,
H
,
δ
.
5.3.1.3 The multilevel Additive Schwarz preconditioner
The method described in the previous section uses two levels with a coarse and a
fine mesh. The smaller is
H
, the smaller are the local problems, while the algebraic
linear system corresponding to the problem in
U
0
becomes larger. We can then apply
recursively this two-level technique to the coarse problem and obtain an additive
multilevel method for the Bidomain system. We refer to the original work of Dryja
and Widlund [41], Zhang [159] and Dryja, Sarkis and Widlund [42] for an overview
of these methods for scalar elliptic problems, including multilevel diagonal scaling
variants and different choices of coarse spaces.
Let us consider
L
≥
2 rather than just two mesh levels: on each level
k
=
k
with elements of char-
0
,
1
,...,
L
−
1,
Ω
is discretized with a shape-regular mesh
T
k
k
−
1
, with level
L
acteristic size
h
k
.
T
is a refinement of
T
−
1 being the finest,
hence
h
L
−
1
=
h
and
h
0
=
H
. On each level
k
=
0
,...,
L
−
1, we define a finite-
V
H
, and the spaces
V
h
k
,
U
h
k
as in
element space
V
h
k
, with
V
h
L
−
1
V
h
and
V
h
0
=
=
N
k
m
=
1
{
Ω
km
}
the previous section. We introduce
L
−
1 sets of overlapping subdomains
for
k
1, such that on each level there is an overlapping decomposition
Ω
=
N
m
=
1
Ω
km
and we denote with
=
1
,...,
L
−
δ
k
the overlap at level
k
. As in [159], we make
the following assumption about the sets
{
Ω
km
}
.
Ω
=
N
m
=
1
Ω
km
satisfies the
Assumption 5.3.1.
On each level the decomposition
following:
(a)
∂Ω
km
aligns with the boundaries of level k elements, i.e.
Ω
km
is the union of
Ω
km
)=O
level k elements. In addition, diameter(
(
h
k
−
1
)
.
N
m
{
Ω
km
}
(b) The subdomains
form a finite covering of
Ω
, with a covering con-
=
1
N
m
{
Ω
km
}
stant N
c
, i.e. we can color
1
using at most N
c
colors in such a way that
subdomains of the same color are disjoint.
=
