Biomedical Engineering Reference
In-Depth Information
where
c
denotes the diffusion coefficient and
f
a source term. A more general form
of Eq. (
16.1
) is obtained by replacing the scalar
c
with a second-order tensor:
∇·
(
C
· ∇
u
)
+
f
=
0 .
(16.2)
However, currently attention is restricted to Eq. (
16.1
).
The essential boundary conditions along
u
read:
u
=
U
at
u
,
(16.3)
while the natural boundary conditions along
p
are given by
n
·
c
∇
u
=
P
at
p
,
(16.4)
with
n
the unit outward normal vector to the boundary
.
16.3
Divergence theorem and integration by parts
Let
n
be the unit outward normal to the boundary
of the domain
, and
φ
a
sufficiently smooth function on
, then
∇
φ
d
=
n
φ
d
.
(16.5)
If the function
φ
is replaced by a vector it can easily be derived:
n
·
φ
d
.
∇·
φ
d
=
(16.6)
Eq. (
16.6
) is known as the
divergence theorem
. For a proof of these equations,
see for example Adams [
1
].
Let both
φ
and
ψ
be sufficiently smooth functions on
, then
φ
∇
ψ
d
.
(
∇
φ
)
ψ
d
=
n
φψ
d
−
(16.7)
This is called integration by parts. To prove this we must integrate the product rule
of differentiation:
∇
(
φψ
)
=
(
∇
φ
)
ψ
+
φ
∇
ψ
,
(16.8)
to obtain
φ
∇
ψ
d
.
∇
(
φψ
)
d
=
(
∇
φ
)
ψ
d
+
(16.9)
