Biomedical Engineering Reference
In-Depth Information
of
|∇
u
h
|
on
T
∈
T
h
by
|∇
u
T
|
and define regularized gradients by
|∇
u
T
|
ε
=
ε
2
+|∇
u
T
|
2
.
(11.12)
We will use the notation
u
p
=
u
h
(
x
p
), where
x
p
is the coordinate of the node
p
of triangulation
T
h
.
With these notations, we are ready to derive co-volume spatial discretization.
As is usual in finite volume methods [20, 34, 44], we integrate (11.11) over every
co-volume
p
,
i
=
1
,...,
M
.Weget
g
0
∇
u
n
|∇
u
n
−
1
dx
.
1
|∇
u
n
−
1
u
n
−
u
n
−
1
τ
dx
=
(11.13)
p
∇·
|
|
p
For the right-hand side of (11.13), using divergence theorem we get
g
0
∇
u
n
|∇
u
n
−
1
dx
=
g
0
|∇
u
n
−
1
∂
u
n
∂ν
p
∇·
ds
|
|
∂
p
g
0
|∇
u
n
−
1
∂
u
n
∂ν
ds
.
=
|
e
pq
q
∈
C
p
So we have an integral formulation of (11.11)
1
|∇
u
n
−
1
u
n
−
u
n
−
1
τ
g
0
|∇
u
n
−
1
∂
u
n
∂ν
dx
=
ds
(11.14)
|
|
p
e
pq
q
∈
C
p
expressing a “local mass balance” property of the scheme. Now the exact “fluxes”
on the right-hand side and “capacity function”
1
|∇
u
n
−
1
on the left-hand side (see
e.g. [34]) will be approximated numerically using piecewise linear reconstruc-
tion of
u
n
−
1
on triangulation
T
h
. If we denote
g
T
approximation of
g
0
on a triangle
T
∈
T
h
, then for the approximation of the right-hand side of (11.14), we get
|
⎛
⎝
T
∈
E
pq
⎞
u
q
−
u
p
h
pq
g
T
|∇
u
n
−
T
|
c
pq
⎠
,
(11.15)
q
∈
C
p
and the left-hand side of (11.14) is approximated by
M
p
m
(
p
)
u
p
−
u
n
−
1
p
(11.16)
,
τ
where
m
(
p
) is a measure in
IR
d
of co-volume
p
and either
1
|∇
u
n
−
1
m
(
T
∩
p
)
m
(
p
)
|∇
u
n
−
T
|
|∇
u
n
−
1
p
M
p
=
(11.17)
|
,
|=
p
T
∈
N
p
