A Proof of Some Lemmas in Section 3.3
A.1 Proof of Lemma 3.3
To prove the existence of
u
h
0
,
w
h
0
in (3.9), we use the following lemma (see [25, Lemma 1.4 of Chapter 2]).
Lemma A.1
Let 𝕏 be a finite-dimensional Hilbert space with inner product
(
⋅
,
⋅
)
and norm
∥
⋅
∥
.
Let ℙ be a continuous mapping from 𝕏 to itself such that, for a sufficiently large constant
q
′
>
0
,
(
P
(
ζ
)
,
ζ
)
>
0
for all
ζ
∈
X
such that
∥
ζ
∥
=
q
′
.
Then there exists
ζ
∈
X
,
∥
ζ
∥
≤
q
′
, such that
P
(
ζ
)
=
0
.
In our case,
X
=
S
h
k
:=
V
h
k
×
γ
1
2
Σ
h
k
.
Multiplying the last equation in (3.9) by 𝛾 and adding it to the second equation in (3.9), we have
(
G
h
(
X
)
,
χ
)
=
0
for all
χ
=
(
χ
1
,
γ
1
2
χ
2
)
∈
S
h
k
, where
X
:=
(
u
h
0
,
γ
1
2
w
h
0
)
and for any
ζ
=
(
ζ
1
,
γ
1
2
ζ
2
)
∈
S
h
k
,
G
h
(
ζ
)
∈
S
h
k
is defined using Riesz’s representation theorem by
(
G
h
(
ζ
)
,
χ
)
=
(
f
L
(
ζ
1
)
,
χ
1
)
+
γ
(
ζ
2
,
∇
χ
1
)
-
γ
∑
K
(
ζ
^
2
⋅
ν
,
χ
1
)
∂
K
-
(
q
h
0
,
χ
1
)
+
γ
(
ζ
2
,
χ
2
)
+
γ
(
ζ
1
,
∇
⋅
χ
2
)
-
γ
∑
K
(
ζ
^
1
,
ν
⋅
χ
2
)
∂
K
for all
χ
=
(
χ
1
,
γ
1
2
χ
2
)
∈
S
h
k
.
It is obvious that
G
h
is continuous.
Due to Lemma A.1,
G
h
(
ϖ
)
=
0
has a solution
ϖ
∈
B
q
′
=
{
χ
=
(
χ
1
,
γ
1
2
χ
2
)
∈
S
h
k
:
∥
χ
∥
2
=
∥
χ
1
∥
2
+
γ
∥
χ
2
∥
2
≤
q
′
2
}
if
(
G
h
(
χ
)
,
χ
)
>
0
for
∥
χ
∥
=
q
′
.
For more details, we refer readers to see [13] and [26, Chapter 13].
Recalling the choices of the fluxes in (2.3) and the boundary conditions in (2.4), there holds
(A.1)
∑
K
(
-
(
χ
1
,
∇
⋅
χ
2
)
K
+
(
χ
^
1
,
ν
⋅
χ
2
)
∂
K
)
=
∑
K
(
-
(
χ
1
,
ν
⋅
χ
2
)
∂
K
+
(
∇
χ
1
,
χ
2
)
K
+
(
χ
^
1
,
ν
⋅
χ
2
)
∂
K
)
=
-
∑
K
(
(
χ
^
2
⋅
ν
,
χ
1
)
∂
K
-
(
χ
2
,
∇
χ
1
)
K
)
.
Then we have
(
G
h
(
χ
)
,
χ
)
=
(
f
L
(
χ
1
)
,
χ
1
)
+
γ
∥
χ
2
∥
2
-
(
q
h
0
,
χ
1
)
=
(
χ
1
3
-
χ
1
,
χ
1
)
|
χ
1
|
≤
L
+
(
a
χ
1
+
a
′
,
χ
1
)
χ
1
<
-
L
+
(
a
χ
1
-
a
′
,
χ
1
)
χ
1
>
L
+
γ
∥
χ
2
∥
2
-
(
q
h
0
,
χ
1
)
≥
∥
χ
1
2
∥
|
χ
1
|
≤
L
2
-
1
2
∥
χ
1
2
∥
|
χ
1
|
≤
L
2
-
|
Ω
|
2
+
a
2
∥
χ
1
∥
|
χ
1
|
>
L
2
-
a
′
2
a
|
Ω
|
+
γ
∥
χ
2
∥
2
-
C
∥
q
h
0
∥
2
-
ε
0
∥
χ
1
∥
2
≥
1
2
|
Ω
|
∥
χ
1
∥
|
χ
1
|
≤
L
4
+
a
2
∥
χ
1
∥
|
χ
1
|
>
L
2
-
|
Ω
|
2
-
a
′
2
a
|
Ω
|
+
γ
∥
χ
2
∥
2
-
C
∥
q
h
0
∥
2
-
ε
0
∥
χ
1
∥
2
=
1
2
|
Ω
|
∥
χ
1
∥
|
χ
1
|
≤
L
4
-
ε
0
∥
χ
1
∥
|
χ
1
|
≤
L
2
+
(
a
2
-
ε
0
)
∥
χ
1
∥
|
χ
1
|
>
L
2
-
|
Ω
|
2
-
a
′
2
a
|
Ω
|
+
γ
∥
χ
2
∥
2
-
C
∥
q
h
0
∥
2
=
1
2
|
Ω
|
∥
χ
1
∥
|
χ
1
|
≤
L
4
-
ε
0
∥
χ
1
∥
|
χ
1
|
≤
L
2
+
(
a
2
-
ε
0
)
∥
χ
1
∥
|
χ
1
|
>
L
2
+
γ
∥
χ
2
∥
2
-
(
|
Ω
|
2
+
a
′
2
a
|
Ω
|
+
C
∥
q
h
0
∥
2
)
,
where
ε
0
is an arbitrarily small positive constant generated by Young’s inequality.
Since
|
Ω
|
,
a
,
a
′
,
∥
q
h
0
∥
are bounded and
∥
χ
∥
2
=
∥
χ
1
∥
|
χ
1
|
≤
L
2
+
∥
χ
1
∥
|
χ
1
|
>
L
2
+
γ
∥
χ
2
∥
2
=
q
′
2
,
it is easy to get that
(
G
h
(
χ
)
,
χ
)
is positive if
q
′
is large enough, provided by
ε
0
<
min
{
1
2
|
Ω
|
,
a
2
}
.
A.2 Proof of Lemma 3.4
We consider the following auxiliary equations:
(A.2)
{
∇
⋅
θ
*
=
δ
*
in
Ω
,
∇
ρ
*
=
θ
*
in
Ω
,
θ
*
⋅
ν
=
0
on
∂
Ω
,
with the regularity estimates
(A.3)
∥
ρ
*
∥
H
2
(
Ω
)
+
∥
θ
*
∥
H
1
(
Ω
)
≤
∥
δ
*
∥
L
2
(
Ω
)
.
If
u
h
1
0
,
w
h
1
0
and
u
h
2
0
,
w
h
2
0
all satisfy the equations in (3.9), then we have
(A.4)
(A.4a)
(
f
L
(
u
h
1
0
)
-
f
L
(
u
h
2
0
)
,
ξ
)
K
=
-
γ
(
w
h
1
0
-
w
h
2
0
,
∇
ξ
)
K
+
γ
(
(
w
^
h
1
0
-
w
^
h
2
0
)
⋅
ν
,
ξ
)
∂
K
,
(A.4b)
(
w
h
1
0
-
w
h
2
0
,
ϕ
)
K
=
-
(
u
h
1
0
-
u
h
2
0
,
∇
⋅
ϕ
)
K
+
(
u
^
h
1
0
-
u
^
h
2
0
,
ν
⋅
ϕ
)
∂
K
.
Letting
δ
*
=
u
h
1
0
-
u
h
2
0
, we get
(A.5)
(
u
h
1
0
-
u
h
2
0
,
u
h
1
0
-
u
h
2
0
)
K
=
(
u
h
1
0
-
u
h
2
0
,
∇
⋅
θ
*
)
K
=
(
u
h
1
0
-
u
h
2
0
,
∇
⋅
(
θ
*
-
Π
θ
*
)
)
K
+
(
u
h
1
0
-
u
h
2
0
,
∇
⋅
Π
θ
*
)
K
=
(
u
h
1
0
-
u
h
2
0
,
∇
⋅
(
θ
*
-
Π
θ
*
)
)
K
+
(
u
^
h
1
0
-
u
^
h
2
0
,
ν
⋅
Π
θ
*
)
∂
K
-
(
w
h
1
0
-
w
h
2
0
,
Π
θ
*
)
K
=
(
u
h
1
0
-
u
h
2
0
,
ν
⋅
(
θ
*
-
Π
θ
*
)
)
∂
K
-
(
u
^
h
1
0
-
u
^
h
2
0
,
ν
⋅
(
θ
*
-
Π
θ
*
)
)
∂
K
+
(
u
^
h
1
0
-
u
^
h
2
0
,
ν
⋅
θ
*
)
∂
K
-
(
w
h
1
0
-
w
h
2
0
,
Π
θ
*
-
θ
*
)
K
-
(
w
h
1
0
-
w
h
2
0
,
θ
*
)
K
,
where the third equality depends on equation (A.4b), and the last equality depends on the property of the interpolation operator Π in (3.3).
Due to
θ
*
=
∇
ρ
*
and equation (A.4a), we obtain
(A.6)
(
w
h
1
0
-
w
h
2
0
,
θ
*
)
K
=
(
w
h
1
0
-
w
h
2
0
,
∇
ρ
*
)
K
=
(
w
h
1
0
-
w
h
2
0
,
∇
(
ρ
*
-
P
ρ
*
)
)
K
+
(
w
h
1
0
-
w
h
2
0
,
∇
P
ρ
*
)
K
=
(
w
h
1
0
-
w
h
2
0
,
∇
(
ρ
*
-
P
ρ
*
)
)
K
+
(
w
^
h
1
0
-
w
^
h
2
0
⋅
ν
,
P
ρ
*
)
∂
K
-
1
γ
(
f
L
(
u
h
1
0
)
-
f
L
(
u
h
2
0
)
,
P
ρ
*
)
K
=
(
w
h
1
0
-
w
h
2
0
,
∇
(
ρ
*
-
P
ρ
*
)
)
K
+
(
w
^
h
1
0
-
w
^
h
2
0
⋅
ν
,
P
ρ
*
-
ρ
*
+
ρ
*
)
∂
K
-
1
γ
(
f
L
(
u
h
1
0
)
-
f
L
(
u
h
2
0
)
,
P
ρ
*
-
ρ
*
+
ρ
*
)
K
.
From (A.5) and (A.6), we have
∥
u
h
1
0
-
u
h
2
0
∥
K
2
=
(
u
h
1
0
-
u
h
2
0
,
ν
⋅
(
θ
*
-
Π
θ
*
)
)
∂
K
-
(
u
^
h
1
0
-
u
^
h
2
0
,
ν
⋅
(
θ
*
-
Π
θ
*
)
)
∂
K
+
(
u
^
h
1
0
-
u
^
h
2
0
,
ν
⋅
θ
*
)
∂
K
-
(
w
h
1
0
-
w
h
2
0
,
Π
θ
*
-
θ
*
)
K
-
(
w
h
1
0
-
w
h
2
0
,
∇
(
ρ
*
-
P
ρ
*
)
)
K
-
(
w
^
h
1
0
-
w
^
h
2
0
⋅
ν
,
P
ρ
*
-
ρ
*
+
ρ
*
)
∂
K
+
1
γ
(
f
L
(
u
h
1
0
)
-
f
L
(
u
h
2
0
)
,
P
ρ
*
-
ρ
*
+
ρ
*
)
K
=
(
u
h
1
0
-
u
h
2
0
,
ν
⋅
(
θ
*
-
Π
θ
*
)
)
∂
K
-
(
u
^
h
1
0
-
u
^
h
2
0
,
ν
⋅
(
θ
*
-
Π
θ
*
)
)
∂
K
+
(
u
^
h
1
0
-
u
^
h
2
0
,
ν
⋅
θ
*
)
∂
K
-
(
w
h
1
0
-
w
h
2
0
,
Π
θ
*
-
θ
*
)
K
-
(
w
h
1
0
-
w
h
2
0
,
∇
(
ρ
*
-
P
ρ
*
)
)
K
-
(
w
^
h
1
0
-
w
^
h
2
0
⋅
ν
,
P
ρ
*
-
ρ
*
+
ρ
*
)
∂
K
(A.7)
+
1
γ
(
f
L
(
u
h
1
0
)
-
f
L
(
u
h
2
0
)
,
P
ρ
*
-
ρ
*
)
K
+
1
γ
(
f
L
(
u
h
1
0
)
-
f
L
(
u
h
2
0
)
,
ρ
*
-
ρ
¯
*
+
ρ
¯
*
)
K
,
where
ρ
¯
*
=
1
|
Ω
|
∫
Ω
ρ
*
(
x
)
d
x
.
Due to the definition of
u
^
h
0
in (2.3), we have
(A.8)
∑
K
(
(
u
h
1
0
-
u
h
2
0
,
ν
⋅
(
θ
*
-
Π
θ
*
)
)
∂
K
-
(
u
^
h
1
0
-
u
^
h
2
0
,
ν
⋅
(
θ
*
-
Π
θ
*
)
)
∂
K
)
=
∑
K
(
u
h
1
0
-
u
h
2
0
,
ν
⋅
(
θ
^
*
-
Π
θ
^
*
)
)
∂
K
=
0
,
where
θ
^
*
=
θ
*
|
R
and the last equation depends on the property of the projection Π.
Because of the choice of
u
^
h
0
in (2.3), the boundary condition of
θ
*
in (A.2) and the continuity of
θ
*
, we have
(A.9)
∑
K
(
u
^
h
1
0
-
u
^
h
2
0
,
ν
⋅
θ
*
)
∂
K
=
0
.
Similarly,
(A.10)
∑
K
(
w
^
h
1
0
-
w
^
h
2
0
⋅
ν
,
ρ
*
)
∂
K
=
0
.
Taking
ξ
=
1
in (A.4a), it is easy to obtain
(A.11)
(
f
L
(
u
h
1
0
)
-
f
L
(
u
h
2
0
)
,
1
)
=
0
.
Inserting (A.8)–(A.11) into (A.7) and summing up equation (A.7) over all elements 𝐾, the following estimate holds:
(A.12)
∥
u
h
1
0
-
u
h
2
0
∥
2
=
-
(
w
h
1
0
-
w
h
2
0
,
Π
θ
*
-
θ
*
)
-
(
w
h
1
0
-
w
h
2
0
,
∇
(
ρ
*
-
P
ρ
*
)
)
-
∑
K
(
w
^
h
1
0
-
w
^
h
2
0
⋅
ν
,
P
ρ
*
-
ρ
*
)
∂
K
+
1
γ
(
f
L
(
u
h
1
0
)
-
f
L
(
u
h
2
0
)
,
P
ρ
*
-
ρ
*
)
+
1
γ
(
f
L
(
u
h
1
0
)
-
f
L
(
u
h
2
0
)
,
ρ
*
-
ρ
¯
*
)
≤
C
h
∥
θ
*
∥
H
1
(
Ω
)
∥
w
h
1
0
-
w
h
2
0
∥
+
C
h
∥
ρ
*
∥
H
2
(
Ω
)
∥
w
h
1
0
-
w
h
2
0
∥
+
C
h
∥
ρ
*
∥
H
1
(
Ω
)
∥
u
h
1
0
-
u
h
2
0
∥
,
where the above inequality depends on the trace inequalities for the polynomials and the Sobolev functions, the interpolation properties of 𝑃 and Π, the global Lipschitz continuity of
f
L
and the mean-value technique.
By choosing
ξ
=
u
h
1
0
-
u
h
2
0
,
ϕ
=
γ
(
w
h
1
0
-
w
h
2
0
)
in (A.4a), (A.4b), respectively, together with property (A.1), we have
∥
w
h
1
0
-
w
h
2
0
∥
2
+
(
f
L
(
u
h
1
0
)
-
f
L
(
u
h
2
0
)
,
u
h
1
0
-
u
h
2
0
)
=
0
,
which implies
(A.13)
∥
w
h
1
0
-
w
h
2
0
∥
2
≤
C
∥
u
h
1
0
-
u
h
2
0
∥
2
.
Inserting (A.13) into (A.12), and using the regularity estimates in (A.3), we deduce that
∥
u
h
1
0
-
u
h
2
0
∥
2
≤
C
h
∥
u
h
1
0
-
u
h
2
0
∥
2
,
i.e.
u
h
1
0
=
u
h
2
0
.
From (A.13), it is obvious to see that
w
h
1
0
=
w
h
2
0
.
Then we prove the uniqueness of
u
h
0
and
w
h
0
.
A.3 Proof of Lemma 3.5
From (3.9) and (3.13), we have the error equations
(A.14)
(A.14a)
(
e
q
0
,
ξ
)
K
=
(
f
L
(
u
0
)
-
f
L
(
u
h
0
)
,
ξ
)
K
+
γ
(
e
w
0
,
∇
ξ
)
K
-
γ
(
e
w
^
0
⋅
ν
,
ξ
)
∂
K
,
(A.14b)
(
e
w
0
,
ϕ
)
K
=
-
(
e
u
0
,
∇
⋅
ϕ
)
K
+
(
e
u
^
0
,
ν
⋅
ϕ
)
∂
K
.
We use a method similar to the proof for the uniqueness of
u
h
0
and
w
h
0
.
Taking
δ
*
=
u
0
-
u
h
0
in (A.2), with the property of Π and (A.14b), we have
(A.15)
(
e
u
0
,
e
u
0
)
K
=
(
e
u
0
,
∇
⋅
θ
*
)
K
=
(
e
u
0
,
∇
⋅
(
θ
*
-
Π
θ
*
)
)
K
+
(
e
u
0
,
∇
⋅
Π
θ
*
)
K
=
(
e
u
0
,
∇
⋅
(
θ
*
-
Π
θ
*
)
)
K
+
(
e
u
^
0
,
ν
⋅
Π
θ
*
)
∂
K
-
(
e
w
0
,
Π
θ
*
)
K
=
-
(
∇
(
u
0
-
P
u
0
)
,
θ
*
-
Π
θ
*
)
K
+
(
e
u
0
,
ν
⋅
(
θ
*
-
Π
θ
*
)
)
∂
K
+
(
e
u
^
0
,
ν
⋅
(
Π
θ
*
-
θ
*
)
)
∂
K
+
(
e
u
^
0
,
ν
⋅
θ
*
)
∂
K
-
(
e
w
0
,
Π
θ
*
-
θ
*
)
K
-
(
e
w
0
,
θ
*
)
K
=
-
(
∇
(
u
0
-
P
u
0
)
,
θ
*
-
Π
θ
*
)
K
-
(
u
h
0
-
u
^
h
0
,
ν
⋅
(
θ
*
-
Π
θ
*
)
)
∂
K
+
(
e
u
^
0
,
ν
⋅
θ
*
)
∂
K
-
(
e
w
0
,
Π
θ
*
-
θ
*
)
K
-
(
e
w
0
,
θ
*
)
K
.
Similarly to (A.6), we get
(A.16)
(
e
w
0
,
θ
*
)
K
=
(
e
w
0
,
∇
ρ
*
)
K
=
(
e
w
0
,
∇
(
ρ
*
-
P
ρ
*
)
)
K
+
(
e
w
0
,
∇
P
ρ
*
)
K
=
(
e
w
0
,
∇
(
ρ
*
-
P
ρ
*
)
)
K
+
(
e
w
^
0
⋅
ν
,
P
ρ
*
)
∂
K
-
1
γ
(
f
L
(
u
0
)
-
f
L
(
u
h
0
)
,
P
ρ
*
)
K
+
1
γ
(
e
q
0
,
P
ρ
*
)
K
=
(
e
w
0
,
∇
(
ρ
*
-
P
ρ
*
)
)
K
+
(
e
w
^
0
⋅
ν
,
P
ρ
*
-
ρ
*
+
ρ
*
)
∂
K
-
1
γ
(
f
L
(
u
0
)
-
f
L
(
u
h
0
)
,
P
ρ
*
-
ρ
*
)
K
-
1
γ
(
f
L
(
u
0
)
-
f
L
(
u
h
0
)
,
ρ
*
-
ρ
¯
*
+
ρ
¯
*
)
K
+
1
γ
(
e
q
0
,
P
ρ
*
-
ρ
*
+
ρ
*
-
ρ
¯
*
+
ρ
¯
*
)
K
.
By taking
ξ
=
-
P
e
u
0
,
ϕ
=
γ
Π
e
w
0
in (A.14a) and (A.14b), respectively, we have
γ
∥
Π
e
w
0
∥
2
=
-
γ
(
w
0
-
Π
w
0
,
Π
e
w
0
)
-
(
u
0
-
P
u
0
,
∇
⋅
Π
e
w
0
)
+
∑
K
(
u
^
0
-
P
u
^
0
,
ν
⋅
Π
e
w
0
)
∂
K
-
(
e
q
0
,
P
e
u
0
)
+
(
f
L
(
u
0
)
-
f
L
(
u
h
0
)
,
P
e
u
0
)
.
Due to the error estimate of
e
q
0
in (3.14), the superconvergent property of 𝑃 in (3.6) and the interpolation properties (3.2) and (3.4)–(3.5), we deduce that
(A.17)
∥
Π
e
w
0
∥
2
≤
C
h
2
k
+
2
+
C
∥
P
e
u
0
∥
2
.
By a method similar to (A.8)–(A.11), we obtain
∑
K
(
u
h
0
-
u
^
h
0
,
ν
⋅
(
θ
*
-
Π
θ
*
)
)
∂
K
=
∑
K
(
u
h
0
,
ν
⋅
(
θ
^
*
-
Π
θ
^
*
)
)
∂
K
=
0
,
∑
K
(
e
u
^
0
,
ν
⋅
θ
*
)
∂
K
=
0
,
∑
K
(
e
w
^
0
⋅
ν
,
ρ
*
)
∂
K
=
0
,
(
f
L
(
u
0
)
-
f
L
(
u
h
0
)
,
1
)
=
0
.
Inserting the above equations, the first equation in (3.10) and (A.16) into (A.15) and summing up over 𝐾, we have
(
e
u
0
,
e
u
0
)
=
-
(
∇
(
u
0
-
P
u
0
)
,
θ
*
-
Π
θ
*
)
-
(
e
w
0
,
Π
θ
*
-
θ
*
)
-
(
e
w
0
,
∇
(
ρ
*
-
P
ρ
*
)
)
-
∑
K
(
w
^
0
-
w
^
h
0
⋅
ν
,
P
ρ
*
-
ρ
*
)
∂
K
+
1
γ
(
f
L
(
u
0
)
-
f
L
(
u
h
0
)
,
P
ρ
*
-
ρ
*
+
ρ
*
-
ρ
¯
*
)
-
1
γ
(
e
q
0
,
P
ρ
*
-
ρ
*
+
ρ
*
-
ρ
¯
*
)
.
By the trace inequalities for the polynomials and the Sobolev functions, the interpolation properties of 𝑃 and Π, the global Lipschitz continuity of
f
L
and the mean-value technique, we get
∥
e
u
0
∥
2
≤
C
h
k
+
1
(
∥
θ
*
∥
H
1
+
∥
ρ
*
∥
H
2
(
Ω
)
)
+
C
h
(
∥
θ
*
∥
H
1
(
Ω
)
+
∥
ρ
*
∥
H
2
(
Ω
)
)
∥
Π
e
w
0
∥
+
C
h
∥
ρ
*
∥
H
2
(
Ω
)
∥
u
0
-
u
h
0
∥
≤
C
h
k
+
1
∥
u
0
-
u
h
0
∥
+
C
h
∥
u
0
-
u
h
0
∥
2
+
C
h
∥
u
0
-
u
h
0
∥
∥
P
e
u
0
∥
≤
C
h
k
+
1
∥
u
0
-
u
h
0
∥
+
C
h
∥
u
0
-
u
h
0
∥
2
,
where the second inequality depends on (A.17) and the regularity estimates in (A.3).
Then there hold the following estimates:
∥
P
e
u
0
∥
≤
C
h
k
+
1
,
∥
Π
e
w
0
∥
≤
C
h
k
+
1
.
If ℎ is sufficiently small and
k
+
1
>
d
2
, then it is easy to get that
∥
u
h
0
∥
∞
≤
∥
P
e
u
0
∥
∞
+
∥
u
0
-
P
u
0
∥
∞
+
∥
u
0
∥
∞
≤
C
h
-
d
2
h
k
+
1
+
∥
u
0
∥
∞
≤
L
.
The proof of Theorem 3.1 is complete.
B Proof of Some Lemmas in Section 4.3
B.1 Proof of Lemma 4.3
Taking
θ
=
Π
e
s
n
in equations (4.2b) and (4.3b), we have
(
e
s
n
,
Π
e
s
n
)
K
=
(
b
(
u
n
)
p
n
-
b
(
u
h
n
-
1
)
p
h
n
,
Π
e
s
n
)
K
.
Adding and subtracting
b
(
u
h
n
-
1
)
p
n
, we obtain
(
Π
e
s
n
,
Π
e
s
n
)
K
=
-
(
s
n
-
Π
s
n
,
Π
e
s
n
)
K
+
(
(
b
(
u
n
)
-
b
(
u
h
n
-
1
)
)
p
n
,
Π
e
s
n
)
K
+
(
(
p
n
-
p
h
n
)
b
(
u
h
n
-
1
)
,
Π
e
s
n
)
K
=
-
(
s
n
-
Π
s
n
,
Π
e
s
n
)
K
+
(
(
b
(
u
n
)
-
b
(
u
n
-
1
)
)
p
n
,
Π
e
s
n
)
K
+
(
(
b
(
u
n
-
1
)
-
b
(
u
h
n
-
1
)
)
p
n
,
Π
e
s
n
)
K
+
(
(
p
n
-
p
h
n
)
b
(
u
h
n
-
1
)
,
Π
e
s
n
)
K
.
Collecting the above equation over all elements 𝐾, with the interpolation properties of the projections 𝑃, Π and the property of 𝑏 in (1.2), there holds
∥
Π
e
s
n
∥
2
≤
C
h
2
k
+
2
+
C
τ
2
+
C
(
∥
P
e
u
n
-
1
∥
2
+
∥
Π
e
p
n
∥
2
)
,
where 𝐶 depends on
∥
s
n
∥
H
k
+
1
(
Ω
)
,
∥
u
n
-
1
∥
H
k
+
1
(
Ω
)
,
∥
p
n
∥
H
k
+
1
(
Ω
)
,
∥
p
n
∥
L
∞
(
Ω
)
and
∥
u
t
∥
L
∞
(
(
0
,
T
)
;
L
2
(
Ω
)
)
.
B.2 Proof of Lemma 4.4
Subtracting (2.2e) from (4.1e), we have
(B.1)
(
e
w
n
,
ϕ
)
K
=
-
(
e
u
n
,
∇
⋅
ϕ
)
K
+
(
e
u
^
n
,
ν
⋅
ϕ
)
∂
K
.
Choosing
ρ
=
P
e
u
n
,
θ
=
-
Π
e
w
n
,
ϕ
=
Π
e
s
n
in equations (4.2a), (4.2b), (4.3a), (4.3b), (B.1), we have the error equations
(
δ
t
e
u
n
,
P
e
u
n
)
K
=
-
(
e
s
n
,
∇
P
e
u
n
)
K
+
(
e
s
^
n
⋅
ν
,
P
e
u
n
)
∂
K
+
(
δ
t
u
n
-
u
t
n
,
P
e
u
n
)
K
,
-
(
e
s
n
,
Π
e
w
n
)
K
=
-
(
b
(
u
n
)
p
n
-
b
(
u
h
n
-
1
)
p
h
n
,
Π
e
w
n
)
K
,
(
e
w
n
,
Π
e
s
n
)
K
=
-
(
e
u
n
,
∇
⋅
Π
e
s
n
)
K
+
(
e
u
^
n
,
ν
⋅
Π
e
s
n
)
∂
K
.
According to the choices of the fluxes
s
^
h
n
,
u
^
h
n
in (2.3) and the boundary conditions in (2.4), we have
∑
K
[
-
(
Π
e
s
n
,
∇
P
e
u
n
)
K
+
(
Π
e
s
^
n
⋅
ν
,
P
e
u
n
)
∂
K
-
(
P
e
u
n
,
∇
⋅
Π
e
s
n
)
K
+
(
P
e
u
^
n
,
ν
⋅
Π
e
s
n
)
∂
K
]
=
0
.
Then, by the definition of the projection Π, we obtain
(
δ
t
P
e
u
n
,
P
e
u
n
)
=
-
(
δ
t
(
u
n
-
P
u
n
)
,
P
e
u
n
)
+
(
δ
t
u
n
-
u
t
n
,
P
e
u
n
)
+
(
s
n
-
Π
s
n
,
Π
e
w
n
)
-
(
(
b
(
u
n
)
p
n
-
b
(
u
h
n
-
1
)
p
h
n
)
,
Π
e
w
n
)
-
(
w
n
-
Π
w
n
,
Π
e
s
n
)
-
(
u
n
-
P
u
n
,
∇
⋅
Π
e
s
n
)
+
∑
K
(
u
^
n
-
P
u
^
n
,
Π
e
s
n
⋅
ν
)
∂
K
.
By the interpolation properties of the projections 𝑃 and Π in (3.2), (3.4)–(3.6), we get
∥
P
e
u
n
∥
2
-
∥
P
e
u
n
-
1
∥
2
2
τ
≤
C
h
2
k
+
2
+
C
τ
2
+
ε
(
∥
Π
e
s
n
∥
2
+
∥
Π
e
w
n
∥
2
+
∥
P
e
u
n
∥
2
)
+
C
(
∥
P
e
u
n
-
1
∥
2
+
∥
Π
e
p
n
∥
2
)
≤
C
h
2
k
+
2
+
C
τ
2
+
ε
(
∥
Π
e
w
n
∥
2
+
∥
P
e
u
n
∥
2
)
+
C
(
∥
P
e
u
n
-
1
∥
2
+
∥
Π
e
p
n
∥
2
)
,
where the last step depends on Lemma 4.3, and 𝐶 depends on 𝜀,
∥
u
t
∥
L
∞
(
(
0
,
T
)
;
H
k
+
1
(
Ω
)
)
,
∥
s
n
∥
H
k
+
1
(
Ω
)
,
∥
p
n
∥
H
k
+
1
(
Ω
)
,
∥
w
n
∥
H
k
+
1
(
Ω
)
,
∥
u
n
∥
H
k
+
2
(
Ω
)
,
∥
p
n
∥
L
∞
(
Ω
)
and
∥
u
t
t
∥
L
∞
(
(
0
,
T
)
;
L
2
(
Ω
)
)
.
Summing up the above inequality from 1 to 𝑛, we complete the proof of this lemma.
B.3 Proof of Lemma 4.5
Considering equations (4.2a), (4.2e), (4.3a), (4.3e) and taking
ρ
=
δ
t
P
e
u
n
,
ϕ
=
Π
e
s
n
, respectively, we have
(
δ
t
e
u
n
,
δ
t
P
e
u
n
)
K
=
-
(
e
s
n
,
δ
t
(
∇
P
e
u
n
)
)
K
+
(
e
s
^
n
⋅
ν
,
δ
t
P
e
u
n
)
∂
K
+
(
δ
t
u
n
-
u
t
n
,
δ
t
P
e
u
n
)
K
,
(
δ
t
e
w
n
,
Π
e
s
n
)
K
=
-
(
δ
t
e
u
n
,
∇
⋅
Π
e
s
n
)
K
+
(
δ
t
e
u
^
n
,
ν
⋅
Π
e
s
n
)
∂
K
.
It is easy to see that
(
δ
t
P
e
u
n
,
δ
t
P
e
u
n
)
=
-
(
δ
t
(
u
n
-
P
u
n
)
,
δ
t
P
e
u
n
)
+
(
δ
t
u
n
-
u
t
n
,
δ
t
P
e
u
n
)
-
(
δ
t
(
u
n
-
P
u
n
)
,
∇
⋅
Π
e
s
n
)
+
∑
K
(
δ
t
(
u
^
n
-
P
u
^
n
)
,
Π
e
s
n
⋅
ν
)
∂
K
-
(
δ
t
(
w
n
-
Π
w
n
+
Π
e
w
n
)
,
Π
e
s
n
)
.
From Lemma 4.3, we deduce that
∥
δ
t
P
e
u
n
∥
2
≤
C
h
2
k
+
2
+
C
τ
2
+
ε
∥
δ
t
Π
e
w
n
∥
2
+
C
(
∥
P
e
u
n
-
1
∥
2
+
∥
Π
e
p
n
∥
2
)
,
where 𝐶 depends on 𝜀,
∥
u
∥
L
∞
(
(
0
,
T
)
;
H
k
+
4
(
Ω
)
)
,
∥
u
∥
L
∞
(
(
0
,
T
)
;
W
3
,
∞
(
Ω
)
)
,
∥
u
t
∥
L
∞
(
(
0
,
T
)
;
H
k
+
2
(
Ω
)
)
,
∥
u
t
t
∥
L
∞
(
(
0
,
T
)
;
L
2
(
Ω
)
)
and
∥
w
t
∥
L
∞
(
(
0
,
T
)
;
H
k
+
1
(
Ω
)
)
.
B.4 Proof of Lemma 4.6
From equations (4.1c)–(4.1d) and (2.2c)–(2.2d), we have the error equations
(
e
p
n
,
η
)
K
=
-
(
e
q
n
,
∇
⋅
η
)
K
+
(
e
q
^
n
,
ν
⋅
η
)
∂
K
+
(
∇
(
u
n
r
n
-
u
n
-
1
r
n
)
,
η
)
K
,
(
e
q
n
,
φ
)
K
=
(
u
n
-
1
r
n
-
u
h
n
-
1
r
h
n
,
φ
)
K
+
γ
(
e
w
n
,
∇
φ
)
K
-
γ
(
e
w
^
n
⋅
ν
,
φ
)
∂
K
.
Choosing
η
=
γ
Π
e
w
n
,
φ
=
-
P
e
q
n
, respectively, with the definition of the projection Π, we have
(
P
e
q
n
,
P
e
q
n
)
=
γ
(
p
n
-
p
h
n
,
Π
e
w
n
)
+
γ
(
q
n
-
P
q
n
,
∇
⋅
Π
e
w
n
)
-
γ
∑
K
(
q
^
n
-
P
q
^
n
,
Π
e
w
n
⋅
ν
)
∂
K
-
γ
(
∇
(
u
n
r
n
-
u
n
-
1
r
n
)
,
Π
e
w
n
)
-
(
q
n
-
P
q
n
,
P
e
q
n
)
+
(
u
n
-
1
r
n
-
u
h
n
-
1
r
h
n
,
P
e
q
n
)
.
Depending on the equality of
u
n
-
1
r
n
-
u
h
n
-
1
r
h
n
=
u
n
-
1
r
n
-
u
h
n
-
1
r
n
+
u
h
n
-
1
r
n
-
u
h
n
-
1
r
h
n
, we get
∥
P
e
q
n
∥
2
≤
C
h
2
k
+
2
+
C
h
2
k
+
2
∥
u
h
n
-
1
∥
∞
2
+
C
τ
2
+
ε
∥
Π
e
w
n
∥
2
+
C
(
∥
P
e
u
n
-
1
∥
2
+
∥
u
h
n
-
1
∥
∞
2
∥
P
e
r
n
∥
2
+
∥
Π
e
p
n
∥
2
)
,
where 𝐶 depends on 𝜀,
∥
u
∥
L
∞
(
(
0
,
T
)
;
H
k
+
4
(
Ω
)
)
,
∥
u
t
∥
L
∞
(
(
0
,
T
)
;
H
1
(
Ω
)
)
and
∥
u
∥
∞
.
B.5 Proof of Lemma 4.7
Choosing the test functions
ρ
=
2
δ
t
P
e
q
1
,
θ
=
-
2
δ
t
Π
e
p
1
,
η
=
2
Π
e
s
1
,
φ
=
-
2
δ
t
P
e
u
1
,
ϕ
=
2
γ
δ
t
Π
e
w
1
,
ξ
=
δ
t
P
e
r
1
in (4.2), with the property of Π, we have the error equation
(B.2)
2
(
b
(
u
h
0
)
Π
e
p
1
,
δ
t
Π
e
p
1
)
+
2
γ
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
=
-
2
(
δ
t
(
u
1
-
P
u
1
)
,
δ
t
P
e
q
1
)
+
2
(
δ
t
u
1
-
u
t
1
,
δ
t
P
e
q
1
)
+
2
(
s
1
-
Π
s
1
,
δ
t
Π
e
p
1
)
-
2
(
(
b
(
u
1
)
-
b
(
u
h
0
)
)
p
1
,
δ
t
Π
e
p
1
)
-
2
(
b
(
u
h
0
)
(
p
1
-
Π
p
h
1
)
,
δ
t
Π
e
p
1
)
-
2
(
δ
t
(
p
1
-
Π
p
1
)
,
Π
e
s
1
)
-
2
(
δ
t
(
q
1
-
P
q
1
)
,
∇
⋅
Π
e
s
1
)
+
2
∑
K
(
δ
t
(
q
^
1
-
P
q
^
1
)
,
Π
e
s
1
⋅
ν
)
∂
K
+
2
(
∇
(
u
1
r
1
-
u
0
r
1
)
τ
,
Π
e
s
1
)
+
2
(
δ
t
(
q
1
-
P
q
1
)
,
δ
t
P
e
u
1
)
-
2
(
u
0
δ
t
r
1
-
u
h
0
δ
t
r
h
1
,
δ
t
P
e
u
1
)
-
2
γ
(
δ
t
(
w
1
-
Π
w
1
)
,
δ
t
Π
e
w
1
)
-
2
γ
(
δ
t
(
u
1
-
P
u
1
)
,
∇
⋅
δ
t
Π
e
w
1
)
+
2
γ
∑
K
(
δ
t
(
u
^
1
-
P
u
^
1
)
,
δ
t
Π
e
w
1
⋅
ν
)
∂
K
-
(
δ
t
(
r
1
-
P
r
1
)
,
δ
t
P
e
r
1
)
+
(
δ
t
r
1
-
r
t
1
,
δ
t
P
e
r
1
)
+
(
2
u
1
u
t
1
-
2
u
0
δ
t
u
1
+
2
u
0
δ
t
u
1
-
2
u
h
0
δ
t
u
h
1
,
δ
t
P
e
r
1
)
:=
ϱ
1
+
ϱ
2
+
ϱ
3
+
ϱ
4
,
where
ϱ
1
=
-
2
(
u
0
δ
t
r
1
-
u
h
0
δ
t
r
h
1
,
δ
t
P
e
u
1
)
+
(
2
u
0
δ
t
u
1
-
2
u
h
0
δ
t
u
h
1
,
δ
t
P
e
r
1
)
,
ϱ
2
=
-
2
(
(
b
(
u
1
)
-
b
(
u
h
0
)
)
p
1
,
δ
t
Π
e
p
1
)
-
2
(
b
(
u
h
0
)
(
p
1
-
Π
p
1
)
,
δ
t
Π
e
p
1
)
-
2
(
δ
t
(
u
1
-
P
u
1
)
,
δ
t
P
e
q
1
)
+
2
(
δ
t
u
1
-
u
t
1
,
δ
t
P
e
q
1
)
+
2
(
s
1
-
Π
s
1
,
δ
t
Π
e
p
1
)
,
ϱ
3
=
-
2
(
δ
t
(
p
1
-
Π
p
1
)
,
Π
e
s
1
)
+
2
(
δ
t
(
q
1
-
P
q
1
)
,
δ
t
P
e
u
1
)
-
2
γ
(
δ
t
(
w
1
-
Π
w
1
)
,
δ
t
Π
e
w
1
)
-
(
δ
t
(
r
1
-
P
r
1
)
,
δ
t
P
e
r
1
)
+
(
δ
t
r
1
-
r
t
1
,
δ
t
P
e
r
1
)
+
(
2
u
1
u
t
1
-
2
u
0
δ
t
u
1
,
δ
t
P
e
r
1
)
+
2
(
∇
(
u
1
r
1
-
u
0
r
1
)
τ
,
Π
e
s
1
)
,
ϱ
4
=
-
2
(
δ
t
(
q
1
-
P
q
1
)
,
∇
⋅
Π
e
s
1
)
+
2
∑
K
(
δ
t
(
q
^
1
-
P
q
^
1
)
,
Π
e
s
1
⋅
ν
)
∂
K
-
2
γ
(
δ
t
(
u
1
-
P
u
1
)
,
∇
⋅
δ
t
Π
e
w
1
)
+
2
γ
∑
K
(
δ
t
(
u
^
1
-
P
u
^
1
)
,
δ
t
Π
e
w
1
⋅
ν
)
∂
K
.
A simple mathematical inequality yields
(B.3)
∥
b
(
u
h
0
)
1
2
Π
e
p
1
∥
2
-
∥
b
(
u
h
0
)
1
2
Π
e
p
0
∥
2
τ
+
2
γ
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
=
2
(
b
(
u
h
0
)
Π
e
p
1
,
δ
t
Π
e
p
1
)
-
1
τ
∥
b
(
u
h
0
)
1
2
Π
e
p
1
-
b
(
u
h
0
)
1
2
Π
e
p
0
∥
2
+
2
γ
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
≤
2
(
b
(
u
h
0
)
Π
e
p
1
,
δ
t
Π
e
p
1
)
+
2
γ
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
=
ϱ
1
+
ϱ
2
+
ϱ
3
+
ϱ
4
,
where the last equality depends on (B.2).
Next we estimate
ϱ
1
,
ϱ
2
,
ϱ
3
and
ϱ
4
gradually.
Since
ϱ
1
=
-
2
(
(
δ
t
r
1
-
δ
t
r
h
1
)
u
h
0
+
(
u
0
-
u
h
0
)
δ
t
r
1
,
δ
t
P
e
u
1
)
+
2
(
(
δ
t
u
1
-
δ
t
u
h
1
)
u
h
0
+
(
u
0
-
u
h
0
)
δ
t
u
1
,
δ
t
P
e
r
1
)
=
-
2
(
(
δ
t
r
1
-
δ
t
P
r
1
)
u
h
0
+
(
u
0
-
u
h
0
)
δ
t
r
1
,
δ
t
P
e
u
1
)
+
2
(
(
δ
t
u
1
-
δ
t
P
u
1
)
u
h
0
+
(
u
0
-
u
h
0
)
δ
t
u
1
,
δ
t
P
e
r
1
)
,
together with (3.11) and Lemma 4.5, we get
|
ϱ
1
|
≤
C
h
2
k
+
2
+
C
∥
P
e
u
0
∥
2
+
ε
(
∥
δ
t
P
e
u
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
)
≤
C
h
2
k
+
2
+
C
(
∥
P
e
u
0
∥
2
+
∥
Π
e
p
1
∥
2
)
+
ε
(
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
)
.
By
b
(
u
1
)
-
b
(
u
h
0
)
=
b
(
u
1
)
-
b
(
u
0
)
+
b
(
u
0
)
-
b
(
u
h
0
)
, the interpolation properties of the projections 𝑃, Π and the error estimate of
P
e
u
0
in Theorem 3.1, we obtain
|
ϱ
2
|
≤
C
(
h
k
+
1
+
τ
)
∥
δ
t
P
e
q
1
∥
+
C
(
h
k
+
1
+
τ
)
∥
δ
t
Π
e
p
1
∥
.
For the estimate of
ϱ
3
, it is obvious to see that
|
ϱ
3
|
≤
C
h
2
k
+
2
+
C
τ
2
+
ε
(
∥
Π
e
s
1
∥
2
+
∥
δ
t
P
e
u
1
∥
2
+
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
)
+
C
∥
Π
e
s
1
∥
.
Using Lemma 4.3 and Lemma 4.5, we deduce that
|
ϱ
3
|
≤
C
h
2
k
+
2
+
C
τ
2
+
C
(
∥
P
e
u
0
∥
2
+
∥
Π
e
p
1
∥
2
)
+
ε
(
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
)
+
C
∥
Π
e
s
1
∥
.
In one dimension,
ϱ
4
=
0
.
In multi-dimension,
|
ϱ
4
|
≤
C
h
2
k
+
2
+
ε
(
∥
Π
e
s
1
∥
2
+
∥
δ
t
Π
e
w
1
∥
2
)
.
According to Lemma 4.3, we obtain
|
ϱ
4
|
≤
C
h
2
k
+
2
+
C
τ
2
+
C
(
∥
P
e
u
0
∥
2
+
∥
Π
e
p
1
∥
2
)
+
ε
∥
δ
t
Π
e
w
1
∥
2
.
Inserting the estimates of
ϱ
1
,
…
,
ϱ
4
into (B.3), with Theorem 3.1, we have
∥
b
(
u
h
0
)
1
2
Π
e
p
1
∥
2
-
∥
b
(
u
h
0
)
1
2
Π
e
p
0
∥
2
τ
+
2
γ
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
K
2
≤
C
h
2
k
+
2
+
C
τ
2
+
C
∥
Π
e
p
1
∥
2
+
ε
(
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
)
+
C
∥
Π
e
s
1
∥
+
C
(
h
k
+
1
+
τ
)
∥
δ
t
P
e
q
1
∥
+
C
(
h
k
+
1
+
τ
)
∥
δ
t
Π
e
p
1
∥
.
Recalling Theorem 3.1, Lemma 4.3, Lemma 4.6 and (3.11), there holds
∥
Π
e
p
1
∥
2
+
τ
∥
δ
t
Π
e
w
1
∥
2
+
τ
∥
δ
t
P
e
r
1
∥
2
≤
C
h
2
k
+
2
+
C
τ
2
+
C
τ
∥
Π
e
p
1
∥
2
+
ε
τ
(
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
)
+
C
τ
∥
Π
e
s
1
∥
+
C
(
h
k
+
1
+
τ
)
(
∥
P
e
q
1
∥
+
∥
Π
e
p
1
∥
)
+
∥
P
e
q
0
∥
2
+
∥
Π
e
p
0
∥
2
≤
C
h
2
k
+
2
+
C
τ
2
+
C
τ
∥
Π
e
p
1
∥
2
+
ε
τ
(
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
)
+
ε
(
∥
P
e
q
1
∥
2
+
∥
Π
e
p
1
∥
2
+
∥
Π
e
s
1
∥
2
)
≤
C
h
2
k
+
2
+
C
τ
2
+
C
τ
∥
Π
e
p
1
∥
2
+
ε
τ
(
∥
δ
t
Π
e
w
1
∥
2
+
∥
δ
t
P
e
r
1
∥
2
)
+
ε
(
∥
P
e
r
1
∥
2
+
∥
Π
e
w
1
∥
2
+
∥
Π
e
p
1
∥
2
)
.
From Lemma 4.2, it follows that
∥
Π
e
p
1
∥
2
+
τ
∥
δ
t
Π
e
w
1
∥
2
+
τ
∥
δ
t
P
e
r
1
∥
2
+
∥
Π
e
w
1
∥
2
+
∥
P
e
r
1
∥
2
≤
C
h
2
k
+
2
+
C
τ
2
+
ε
(
∥
P
e
r
1
∥
2
+
∥
Π
e
w
1
∥
2
)
+
ε
τ
∥
δ
t
P
e
r
1
∥
2
+
C
τ
∥
P
e
r
1
∥
2
+
∥
P
e
r
0
∥
2
+
ε
τ
∥
δ
t
Π
e
w
1
∥
2
+
C
τ
∥
Π
e
w
1
∥
2
+
∥
Π
e
w
0
∥
2
.
Noticing the initial data in Theorem 3.1, with sufficiently small 𝜏, we have
∥
Π
e
p
1
∥
2
+
τ
∥
δ
t
Π
e
w
1
∥
2
+
τ
∥
δ
t
P
e
r
1
∥
2
+
∥
Π
e
w
1
∥
2
+
∥
P
e
r
1
∥
2
≤
C
h
2
k
+
2
+
C
τ
2
,
and together with Lemma 4.3, Lemma 4.4, Lemma 4.5 and Lemma 4.6, we get the following estimates:
∥
Π
e
s
1
∥
2
+
∥
P
e
u
1
∥
2
+
∥
P
e
q
1
∥
2
+
τ
∥
δ
t
P
e
u
1
∥
2
≤
C
h
2
k
+
2
+
C
τ
2
,
where 𝐶 depends on 𝜀,
∥
u
∥
L
∞
(
(
0
,
T
)
;
H
k
+
4
(
Ω
)
)
,
∥
u
∥
L
∞
(
(
0
,
T
)
;
W
3
,
∞
(
Ω
)
)
,
∥
u
t
∥
L
∞
(
(
0
,
T
)
;
H
k
+
4
(
Ω
)
)
and
∥
u
t
t
∥
L
∞
(
(
0
,
T
)
;
L
2
(
Ω
)
)
.
B.6 Proof of Lemma 4.8
Similarly to the case of
n
=
1
, taking
ρ
=
2
δ
t
P
e
q
n
,
θ
=
-
2
δ
t
Π
e
p
n
,
η
=
2
Π
e
s
n
,
φ
=
-
2
δ
t
P
e
u
n
,
ϕ
=
2
γ
δ
t
Π
e
w
n
,
ξ
=
δ
t
P
e
r
n
in (4.3), there holds
(B.4)
2
(
b
(
u
h
n
-
1
)
Π
e
p
n
,
δ
t
Π
e
p
n
)
+
2
γ
∥
δ
t
Π
e
w
n
∥
2
+
∥
δ
t
P
e
r
n
∥
2
=
-
2
(
δ
t
(
u
n
-
P
u
n
)
,
δ
t
P
e
q
n
)
+
2
(
δ
t
u
n
-
u
t
n
,
δ
t
P
e
q
n
)
+
2
(
s
n
-
Π
s
n
,
δ
t
Π
e
p
n
)
-
2
(
(
b
(
u
n
)
-
b
(
u
h
n
-
1
)
)
p
n
,
δ
t
Π
e
p
n
)
-
2
(
b
(
u
h
n
-
1
)
(
p
n
-
Π
p
n
)
,
δ
t
Π
e
p
n
)
-
2
(
δ
t
(
p
n
-
Π
p
n
)
,
Π
e
s
n
)
-
2
(
δ
t
(
q
n
-
P
q
n
)
,
∇
⋅
Π
e
s
n
)
+
2
∑
K
(
δ
t
(
q
^
n
-
P
q
^
n
)
,
Π
e
s
n
⋅
ν
)
∂
K
+
2
(
∇
(
u
n
r
n
-
u
n
-
1
r
n
)
-
∇
(
u
n
-
1
r
n
-
1
-
u
n
-
2
r
n
-
1
)
τ
,
Π
e
s
n
)
+
2
(
δ
t
(
q
n
-
P
q
n
)
,
δ
t
P
e
u
n
)
-
2
(
u
n
-
1
r
n
-
u
n
-
2
r
n
-
1
τ
-
u
h
n
-
1
r
h
n
-
u
h
n
-
2
r
h
n
-
1
τ
,
δ
t
P
e
u
n
)
-
2
γ
(
δ
t
(
w
n
-
Π
w
n
)
,
δ
t
Π
e
w
n
)
-
2
γ
(
δ
t
(
u
n
-
P
u
n
)
,
∇
⋅
δ
t
Π
e
w
n
)
+
2
γ
∑
K
(
δ
t
(
u
^
n
-
P
u
^
n
)
,
δ
t
Π
e
w
n
⋅
ν
)
∂
K
-
(
δ
t
(
r
n
-
P
r
n
)
,
δ
t
P
e
r
n
)
+
(
δ
t
r
n
-
r
t
n
,
δ
t
P
e
r
n
)
+
(
2
u
n
u
t
n
-
2
u
n
-
1
δ
t
u
n
+
2
u
n
-
1
δ
t
u
n
-
2
u
h
n
-
1
δ
t
u
h
n
,
δ
t
P
e
r
n
)
:=
ϱ
5
+
ϱ
6
+
ϱ
7
+
ϱ
8
,
where
ϱ
5
=
-
2
(
u
n
-
1
r
n
-
u
n
-
2
r
n
-
1
τ
-
u
h
n
-
1
r
h
n
-
u
h
n
-
2
r
h
n
-
1
τ
,
δ
t
P
e
u
n
)
+
(
2
u
n
-
1
δ
t
u
n
-
2
u
h
n
-
1
δ
t
u
h
n
,
δ
t
P
e
r
n
)
,
ϱ
6
=
-
2
(
(
b
(
u
n
)
-
b
(
u
h
n
-
1
)
)
p
n
,
δ
t
Π
e
p
n
)
-
2
(
b
(
u
h
n
-
1
)
(
p
n
-
Π
p
n
)
,
δ
t
Π
e
p
n
)
-
2
(
δ
t
(
u
n
-
P
u
n
)
,
δ
t
P
e
q
n
)
+
2
(
δ
t
u
n
-
u
t
n
,
δ
t
P
e
q
n
)
+
2
(
s
n
-
Π
s
n
,
δ
t
Π
e
p
n
)
,
ϱ
7
=
-
2
(
δ
t
(
p
n
-
Π
p
n
)
,
Π
e
s
n
)
+
2
(
δ
t
(
q
n
-
P
q
n
)
,
δ
t
P
e
u
n
)
-
2
γ
(
δ
t
(
w
n
-
Π
w
n
)
,
δ
t
Π
e
w
n
)
-
(
δ
t
(
r
n
-
P
r
n
)
,
δ
t
P
e
r
n
)
+
(
δ
t
r
n
-
r
t
n
,
δ
t
P
e
r
n
)
+
(
2
u
n
u
t
n
-
2
u
n
-
1
δ
t
u
n
,
δ
t
P
e
r
n
)
+
2
(
∇
(
u
n
r
n
-
u
n
-
1
r
n
)
-
∇
(
u
n
-
1
r
n
-
1
-
u
n
-
2
r
n
-
1
)
τ
,
Π
e
s
n
)
,
ϱ
8
=
-
2
(
δ
t
(
q
n
-
P
q
n
)
,
∇
⋅
Π
e
s
n
)
+
2
∑
K
(
δ
t
(
q
^
n
-
P
q
^
n
)
,
Π
e
s
n
⋅
ν
)
∂
K
-
2
γ
(
δ
t
(
u
n
-
P
u
n
)
,
∇
⋅
δ
t
Π
e
w
n
)
+
2
γ
∑
K
(
δ
t
(
u
^
n
-
P
u
^
n
)
,
δ
t
Π
e
w
n
⋅
ν
)
∂
K
.
Using a method similar to (B.3), we have
(B.5)
(
b
(
u
h
n
-
1
)
Π
e
p
n
,
Π
e
p
n
)
-
(
b
(
u
h
n
-
2
)
Π
e
p
n
-
1
,
Π
e
p
n
-
1
)
τ
+
2
γ
∥
δ
t
Π
e
w
n
∥
2
+
∥
δ
t
P
e
r
n
∥
2
=
∥
b
(
u
h
n
-
1
)
1
2
Π
e
p
n
∥
2
-
∥
b
(
u
h
n
-
1
)
1
2
Π
e
p
n
-
1
∥
2
τ
+
(
b
(
u
h
n
-
1
)
-
b
(
u
h
n
-
2
)
τ
Π
e
p
n
-
1
,
Π
e
p
n
-
1
)
+
2
γ
∥
δ
t
Π
e
w
n
∥
2
+
∥
δ
t
P
e
r
n
∥
2
≤
2
(
b
(
u
h
n
-
1
)
Π
e
p
n
,
δ
t
Π
e
p
n
)
+
(
b
(
u
h
n
-
1
)
-
b
(
u
h
n
-
2
)
τ
Π
e
p
n
-
1
,
Π
e
p
n
-
1
)
+
2
γ
∥
δ
t
Π
e
w
n
∥
2
+
∥
δ
t
P
e
r
n
∥
2
≤
ϱ
5
+
ϱ
6
+
ϱ
7
+
ϱ
8
+
C
∥
δ
t
u
h
n
-
1
∥
L
∞
∥
Π
e
p
n
-
1
∥
2
,
where the last inequality depends on (B.4) and the smoothness assumption of 𝑏 in (1.2).
In view of
u
n
-
1
r
n
-
u
n
-
2
r
n
-
1
τ
-
u
h
n
-
1
r
h
n
-
u
h
n
-
2
r
h
n
-
1
τ
=
u
n
-
1
δ
t
r
n
+
r
n
-
1
δ
t
u
n
-
1
-
u
h
n
-
1
δ
t
r
h
n
-
r
h
n
-
1
δ
t
u
h
n
-
1
=
(
u
n
-
1
-
u
h
n
-
1
)
δ
t
r
n
+
u
h
n
-
1
(
δ
t
r
n
-
δ
t
r
h
n
)
+
(
δ
t
u
n
-
1
-
δ
t
u
h
n
-
1
)
r
n
-
1
+
δ
t
u
h
n
-
1
(
r
n
-
1
-
r
h
n
-
1
)
and
u
n
-
1
δ
t
u
n
-
u
h
n
-
1
δ
t
u
h
n
=
(
u
n
-
1
-
u
h
n
-
1
)
δ
t
u
n
+
u
h
n
-
1
(
δ
t
u
n
-
δ
t
u
h
n
)
,
together with Lemma 4.5 and the bound condition in (4.4), we get
ϱ
5
=
-
2
(
(
u
n
-
1
-
u
h
n
-
1
)
δ
t
r
n
+
u
h
n
-
1
(
δ
t
r
n
-
δ
t
r
h
n
)
,
δ
t
P
e
u
n
)
-
2
(
(
δ
t
u
n
-
1
-
δ
t
u
h
n
-
1
)
r
n
-
1
+
δ
t
u
h
n
-
1
(
r
n
-
1
-
r
h
n
-
1
)
,
δ
t
P
e
u
n
)
+
2
(
(
u
n
-
1
-
u
h
n
-
1
)
δ
t
u
n
,
δ
t
P
e
r
n
)
+
2
(
u
h
n
-
1
(
δ
t
u
n
-
δ
t
P
u
n
)
,
δ
t
P
e
r
n
)
≤
C
h
2
k
+
2
+
C
∥
P
e
u
n
-
1
∥
2
+
ε
(
∥
δ
t
u
h
n
-
1
∥
∞
2
∥
P
e
r
n
∥
2
+
∥
δ
t
P
e
r
n
∥
2
)
+
C
(
∥
δ
t
P
e
u
n
-
1
∥
2
+
∥
δ
t
P
e
u
n
∥
2
)
+
C
∥
δ
t
u
h
n
-
1
∥
∞
2
h
2
k
+
2
≤
C
h
2
k
+
2
+
ε
(
∥
δ
t
u
h
n
-
1
∥
∞
2
∥
P
e
r
n
∥
2
+
∥
δ
t
P
e
r
n
∥
2
)
+
C
(
∥
P
e
u
n
-
1
∥
2
+
∥
P
e
u
n
-
2
∥
2
)
+
C
(
∥
Π
e
p
n
-
1
∥
2
+
∥
Π
e
p
n
∥
2
)
+
ε
(
∥
δ
t
Π
e
w
n
-
1
∥
2
+
∥
δ
t
Π
e
w
n
∥
2
)
+
C
∥
δ
t
u
h
n
-
1
∥
∞
2
h
2
k
+
2
,
where 𝐶 depends on
𝜀,
∥
u
∥
L
∞
(
(
0
,
T
)
;
H
k
+
4
(
Ω
)
)
,
∥
u
∥
L
∞
(
(
0
,
T
)
;
W
3
,
∞
(
Ω
)
)
,
∥
u
t
∥
L
∞
(
(
0
,
T
)
;
H
k
+
2
(
Ω
)
)
,
∥
u
t
∥
∞
,
∥
u
t
t
∥
L
∞
(
(
0
,
T
)
;
L
2
(
Ω
)
)
.
Noting
(
b
(
u
h
n
-
1
)
(
p
n
-
Π
p
n
)
,
δ
t
Π
e
p
n
)
=
-
(
δ
t
(
b
(
u
h
n
-
1
)
(
p
n
-
Π
p
n
)
)
,
Π
e
p
n
-
1
)
+
δ
t
(
b
(
u
h
n
-
1
)
(
p
n
-
Π
p
n
)
,
Π
e
p
n
)
,
we obtain
ϱ
6
=
2
(
δ
t
(
b
(
u
h
n
-
1
)
(
p
n
-
Π
p
n
)
)
,
Π
e
p
n
-
1
)
-
2
δ
t
(
b
(
u
h
n
-
1
)
(
p
n
-
Π
p
n
)
,
Π
e
p
n
)
+
2
(
δ
t
(
(
b
(
u
n
)
-
b
(
u
h
n
-
1
)
)
p
n
)
,
Π
e
p
n
-
1
)
-
2
δ
t
(
(
b
(
u
n
)
-
b
(
u
h
n
-
1
)
)
p
n
,
Π
e
p
n
)
-
2
(
δ
t
(
s
n
-
Π
s
n
)
,
Π
e
p
n
-
1
)
+
2
δ
t
(
s
n
-
Π
s
n
,
Π
e
p
n
)
+
2
(
δ
t
(
δ
t
(
u
n
-
P
u
n
)
)
,
P
e
q
n
-
1
)
-
2
δ
t
(
δ
t
(
u
n
-
P
u
n
)
,
P
e
q
n
)
-
2
(
δ
t
(
δ
t
u
n
-
u
t
n
)
,
P
e
q
n
-
1
)
+
2
δ
t
(
δ
t
u
n
-
u
t
n
,
P
e
q
n
)
.
Through a simple mathematical calculation, we have
δ
t
(
b
(
u
h
n
-
1
)
(
p
n
-
Π
p
n
)
)
=
b
(
u
h
n
-
1
)
-
b
(
u
h
n
-
2
)
τ
(
p
n
-
Π
p
n
)
+
b
(
u
h
n
-
2
)
p
n
-
Π
p
n
-
(
p
n
-
1
-
Π
p
n
-
1
)
τ
,
and
(B.6)
δ
t
(
(
b
(
u
n
)
-
b
(
u
h
n
-
1
)
)
p
n
)
=
b
(
u
n
)
-
b
(
u
n
-
1
)
-
(
b
(
u
h
n
-
1
)
-
b
(
u
h
n
-
2
)
)
τ
p
n
+
(
b
(
u
n
-
1
)
-
b
(
u
h
n
-
2
)
)
p
n
-
p
n
-
1
τ
=
b
(
u
n
-
1
)
-
b
(
u
n
-
2
)
-
(
b
(
u
h
n
-
1
)
-
b
(
u
h
n
-
2
)
)
τ
p
n
+
(
b
(
u
n
-
1
)
-
b
(
u
h
n
-
2
)
)
p
n
-
p
n
-
1
τ
+
b
(
u
n
)
-
b
(
u
n
-
1
)
-
(
b
(
u
n
-
1
)
-
b
(
u
n
-
2
)
)
τ
p
n
.
Due to the smoothness assumption of 𝑏 and (B.6), we get
(B.7)
δ
t
(
(
b
(
u
n
)
-
b
(
u
h
n
-
1
)
)
p
n
)
=
b
′
(
μ
1
n
-
2
)
(
u
n
-
1
-
u
n
-
2
)
-
b
′
(
μ
2
n
-
2
)
(
u
h
n
-
1
-
u
h
n
-
2
)
τ
p
n
+
b
′
(
μ
3
n
-
1
)
(
u
n
-
u
n
-
1
)
-
b
′
(
μ
1
n
-
2
)
(
u
n
-
1
-
u
n
-
2
)
τ
p
n
+
(
b
(
u
n
-
1
)
-
b
(
u
h
n
-
2
)
)
p
n
-
p
n
-
1
τ
=
(
b
′
(
μ
1
n
-
2
)
-
b
′
(
μ
2
n
-
2
)
)
(
u
n
-
1
-
u
n
-
2
)
-
b
′
(
μ
2
n
-
2
)
(
u
n
-
1
-
u
n
-
2
-
(
u
h
n
-
1
-
u
h
n
-
2
)
)
τ
p
n
+
(
b
′
(
μ
3
n
-
1
)
-
b
′
(
μ
1
n
-
2
)
)
(
u
n
-
u
n
-
1
)
-
b
′
(
μ
1
n
-
2
)
(
u
n
-
u
n
-
1
-
(
u
n
-
1
-
u
n
-
2
)
)
τ
p
n
+
(
b
(
u
n
-
1
)
-
b
(
u
h
n
-
2
)
)
p
n
-
p
n
-
1
τ
,
where
μ
1
n
-
2
=
u
n
-
2
+
λ
1
n
-
2
(
u
n
-
1
-
u
n
-
2
)
,
μ
2
n
-
2
=
u
h
n
-
2
+
λ
2
n
-
2
(
u
h
n
-
1
-
u
h
n
-
2
)
,
μ
3
n
-
1
=
u
n
-
1
+
λ
3
n
-
1
(
u
n
-
u
n
-
1
)
,
0
<
λ
1
n
-
2
,
λ
2
n
-
2
,
λ
3
n
-
1
<
1
.
By
μ
1
n
-
2
-
μ
2
n
-
2
=
u
n
-
2
-
u
h
n
-
2
+
(
λ
1
n
-
2
-
λ
2
n
-
2
)
(
u
n
-
1
-
u
n
-
2
)
+
λ
2
n
-
2
(
u
n
-
1
-
u
n
-
2
-
(
u
h
n
-
1
-
u
h
n
-
2
)
)
,
μ
3
n
-
2
-
μ
1
n
-
2
=
u
n
-
1
-
u
n
-
2
+
(
λ
3
n
-
2
-
λ
1
n
-
2
)
(
u
n
-
u
n
-
1
)
+
λ
1
n
-
2
(
u
n
-
u
n
-
1
-
(
u
n
-
1
-
u
n
-
2
)
)
,
Taylor expansion for
b
′
(
μ
1
n
-
2
)
-
b
′
(
μ
2
n
-
2
)
and
b
′
(
μ
3
n
-
1
)
-
b
′
(
μ
1
n
-
2
)
in (B.7), the boundedness of
b
′
and
b
′′
in (1.2) and Lemma 4.5, we conclude that
|
ϱ
6
|
≤
C
h
2
k
+
2
+
C
τ
2
+
C
h
k
+
1
∥
δ
t
u
h
n
-
1
∥
L
∞
∥
Π
e
p
n
-
1
∥
+
C
h
k
+
1
∥
Π
e
p
n
-
1
∥
+
C
τ
∥
Π
e
p
n
-
1
∥
+
C
∥
P
e
u
n
-
2
∥
∥
Π
e
p
n
-
1
∥
+
C
∥
δ
t
P
e
u
n
-
1
∥
∥
Π
e
p
n
-
1
∥
+
ε
∥
P
e
q
n
-
1
∥
2
+
L
L
≤
C
h
2
k
+
2
+
C
τ
2
+
C
h
k
+
1
∥
δ
t
u
h
n
-
1
∥
L
∞
∥
Π
e
p
n
-
1
∥
+
C
(
∥
P
e
u
n
-
1
∥
2
+
∥
P
e
u
n
-
2
∥
2
+
∥
Π
e
p
n
-
1
∥
2
)
+
ε
∥
δ
t
Π
e
w
n
-
1
∥
2
+
ε
∥
P
e
q
n
-
1
∥
2
+
L
L
,
where
L
L
=
|
-
δ
t
(
b
(
u
h
n
-
1
)
(
p
n
-
Π
p
n
)
,
Π
e
p
n
)
-
δ
t
(
(
b
(
u
n
)
-
b
(
u
h
n
-
1
)
)
p
n
,
Π
e
p
n
)
+
δ
t
(
s
n
-
Π
s
n
,
Π
e
p
n
)
-
δ
t
(
δ
t
(
u
n
-
P
u
n
)
,
P
e
q
n
)
+
δ
t
(
δ
t
u
n
-
u
t
n
,
P
e
q
n
)
|
and 𝐶 depends on
𝜀,
∥
u
∥
L
∞
(
(
0
,
T
)
;
H
k
+
4
(
Ω
)
)
,
∥
u
∥
L
∞
(
(
0
,
T
)
;
W
3
,
∞
(
Ω
)
)
,
∥
u
t
∥
L
∞
(
(
0
,
T
)
;
H
k
+
4
(
Ω
)
)
,
∥
u
t
∥
L
∞
(
(
0
,
T
)
;
W
3
,
∞
(
Ω
)
)
and
∥
u
t
t
∥
L
∞
(
(
0
,
T
)
;
L
2
(
Ω
)
)
.
For
ϱ
7
, it is easy to see that
|
ϱ
7
|
≤
C
h
2
k
+
2
+
C
τ
2
+
ε
(
∥
Π
e
s
n
∥
2
+
∥
δ
t
P
e
u
n
∥
2
+
∥
δ
t
P
e
r
n
∥
2
+
∥
δ
t
Π
e
w
n
∥
2
)
.
Combining the above inequality with Lemma 4.3 and Lemma 4.5, we deduce that
|
ϱ
7
|
≤
C
h
2
k
+
2
+
C
(
∥
P
e
u
n
-
1
∥
2
+
∥
Π
e
p
n
∥
2
)
+
ε
(
∥
δ
t
Π
e
w
n
∥
2
+
∥
δ
t
P
e
r
n
∥
2
)
,
where 𝐶 depends on
𝜀,
∥
u
∥
L
∞
(
(
0
,
T
)
;
H
k
+
4
(
Ω
)
)
,
∥
u
∥
L
∞
(
(
0
,
T
)
;
W
3
,
∞
(
Ω
)
)
,
∥
u
t
∥
L
∞
(
(
0
,
T
)
;
H
k
+
4
(
Ω
)
)
and
∥
u
t
t
∥
L
∞
(
(
0
,
T
)
;
L
2
(
Ω
)
)
.
Similarly to the estimates of
ϱ
4
, we have
ϱ
8
=
0
in one dimension.
In multi-dimension,
|
ϱ
8
|
≤
C
h
2
k
+
2
+
ε
(
∥
Π
e
s
n
∥
2
+
∥
δ
t
Π
e
w
n
∥
2
)
.
According to Lemma 4.3, we obtain
|
ϱ
8
|
≤
C
h
2
k
+
2
+
C
(
∥
P
e
u
n
-
1
∥
2
+
∥
Π
e
p
n
∥
2
)
+
ε
∥
δ
t
Π
e
w
n
∥
2
,
where 𝐶 depends on
𝜀,
∥
u
∥
L
∞
(
(
0
,
T
)
;
H
k
+
4
(
Ω
)
)
,
∥
u
∥
L
∞
(
(
0
,
T
)
;
W
3
,
∞
(
Ω
)
)
and
∥
u
t
∥
L
∞
(
(
0
,
T
)
;
H
k
+
4
(
Ω
)
)
.
Inserting into (B.5) these estimates
ϱ
5
–
ϱ
8
, we have
(
b
(
u
h
n
-
1
)
Π
e
p
n
,
Π
e
p
n
)
-
(
b
(
u
h
n
-
2
)
Π
e
p
n
-
1
,
Π
e
p
n
-
1
)
τ
+
2
γ
∥
δ
t
Π
e
w
n
∥
2
+
∥
δ
t
P
e
r
n
∥
2
≤
C
h
2
k
+
2
+
C
τ
2
+
C
(
∥
P
e
u
n
-
1
∥
2
+
∥
P
e
u
n
-
2
∥
2
+
∥
Π
e
p
n
∥
2
+
∥
Π
e
p
n
-
1
∥
2
)
+
C
∥
δ
t
u
h
n
-
1
∥
L
∞
∥
Π
e
p
n
-
1
∥
2
+
ε
(
∥
δ
t
Π
e
w
n
∥
2
+
∥
δ
t
Π
e
w
n
-
1
∥
2
+
∥
δ
t
P
e
r
n
∥
2
)
+
ε
∥
δ
t
u
h
n
-
1
∥
∞
2
∥
P
e
r
n
∥
2
+
C
∥
δ
t
u
h
n
-
1
∥
∞
2
h
2
k
+
2
+
ε
∥
P
e
q
n
-
1
∥
2
+
L
L
,
which complete the proof of this lemma.