Biomedical Engineering Reference
In-Depth Information
E
H
u
e
,
V
m
def
=
w
n
,
Ω
H
+
χ
m
V
m
,
Ω
H
+
τ
σ
u
e
,
Ω
H
+
τ
σ
V
m
2
2
2
0
2
0
2
0
2
0
i
∇
i
∇
,
Ω
H
m
=
0
τ
σ
u
m
+
e
n
−
1
e
2
0
+
2
∇
,
Ω
H
,
and similarly for the rest.
w
n
V
m
,
u
e
,
u
T
)
Theorem 3.
Assume that the hypotheses of Theorem 2 hold and let
(
,
be given by Algorithm 2. Then for
γ
≥
2
C
ti
and
1
≤
n
≤
N the following estimate
holds
h
u
T
E
H
u
e
,
V
m
+
τ
γσ
T
2
0
,
Σ
m
=
0
τ
σ
n
−
1
u
m
+
T
t
T
m
=
0
τ
u
m
+
1
n
−
1
u
m
+
e
,
Ω
T
+
γσ
2
2
0
2
0
+
T
∇
−
T
,
Σ
h
E
H
u
e
,
V
m
+
τ
γσ
h
u
T
t
T
,
Σ
+
τ
σ
u
T
1
m
=
0
τ
I
m
+
1
n
−
2
2
0
2
0
2
0
T
∇
,
Ω
T
+
,
Ω
H
.
app
Therefore, the explicit heart-torso coupling of Algorithm 2 is energy-stable provides
τ
=
O
(
h
)
.
Remark 10.
The proof of the above result does not make use of any numerical dis-
sipation apart from the one directly provided by the explicit Robin-Robin splitting
(4.21). Note that this is particularly well adapted to the heart-torso coupling (4.8),
since the quasi-static elliptic equations (4.8)
3
,
4
do not generate numerical dissipation
in time.
We conclude this subsection with a few numerical illustrations of the accuracy of
Algorithm 2. Fig. 4.8 shows that the superior stability properties and computational
cost reduction featured by the proposed Robin heart-torso decoupling schemes come
with a price: a condition
h
2
τ
=
O
(
)
is required to guarantee an overall convergence
rate
h
involved in the explicit Robin treatment intro-
duces a non-standard coupling between the space and time discretizations, which
has a consistency of
O
(
h
)
. Indeed, the penalty 1
/
.
In spite of that, Fig. 4.9 shows that the proposed Robin splittings are able to pro-
vide accurate 12-lead ECG signals, both for a healthy and a pathological condition.
Note that this is a major advantage with respect to the conventional heart-torso un-
coupling approximation, which (for a similar computational cost) provides inaccu-
rate ECG signals (see Fig. 4.6). Somehow the discretization error introduced by the
Robin heart-torso decoupling is negligible with respect to the modelling error in-
volved in the heart-torso uncoupling approximation.
O
(
τ
/
h
)