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
)
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