432
7 Variational Principles
(7.8.14) and boundary condition (7.8.15), the functional corresponding to the elliptic
Eq. (7.8.13) is
J [w] = (T w, w) − 2( f
∗
, w)
=
D
[−∇ · ( p∇w) + qw]wd D − 2
D
[ f + ∇ · ( p∇u 1 ) + qu 1 ]wd D
=
D
[ p(∇w)
2
+ qw
2
]d D +
Γ 2
σ w
2 dΓ − 2
D
[ f + ∇ · ( p∇u 1 ) + qu 1 ]wd D
=
D
[ p(∇w)
2
+ qw
2
]d D +
Γ 2
σ w
2 dΓ − 2
D
[ f + ∇ · ( p∇u 1 ) + qu 1 ]wd D
Substituting w = u − u 1 into the above expression, we obtain
J [w] =
D
{ p[∇(u − u 1 )]
2
+ q(u − u 1 )
2
}d D +
Γ 2
σ (u − u 1 )
2 dΓ
− 2
D
[ f + ∇ · ( p∇u 1 ) − qu 1 ](u − u 1 )d D
=
D
p[(∇u)
2
− 2∇u · ∇u 1 + (∇u 1 )
2
]d D
+
D
q(u
2
− 2uu 1 + u
2
1 )d D +
Γ 2
σ (u
2
− 2uu 1 + u
2
1 )dΓ
− 2
D
[ f + ∇ · ( p∇u 1 ) − qu 1 ](u − u 1 )d D
=
D
[ p(∇u)
2
+ qu
2
− 2 f u]d D +
Γ 2
σ u
2 dΓ
+
D
[ p(∇u 1 )
2
+ qu
2
1 + 2 f
∗ u 1 ]d D +
Γ 2
σ u
2
1 dΓ
− 2
D
( p∇u · ∇u 1 + quu 1 )d D − 2
Γ 2
σ uu 1 dΓ
− 2
D
[∇ · ( p∇u 1 ) − qu 1 ]ud D
(7.8.16)
Putting the sum of the later three terms in the above expression be −2J 1 [u, u 1 ],
then
J 1 [u, u 1 ] =
D
[ p∇u · ∇u 1 + quu 1 ]d D +
Γ 2
σ uu 1 dΓ +
D
[∇ · ( p∇u) − qu 1 ]ud D
=
D
[∇ · (up∇u 1 ) − u∇ · ( p∇u 1 )]d D +
Γ 2
σ uu 1 dΓ +
D
[u∇ · ( p∇u 1 )]d D
=
Γ
p
∂u 1
∂n
udΓ +
Γ 2
σ uu 1 dΓ
=
Γ 1
p
∂u 1
∂n
(w + u 1 )dΓ +
Γ 2
p
∂u 1
∂n
udΓ +
Γ 2
σ uu 1 dΓ