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AdvancesinAppliedMathematics
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,2022,11(3),1389-1399
PublishedOnlineMarch2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.113151
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Quasi-UniformSuperconvergence
AnalysisofConformingGalerkin
FiniteElementMethodforthe
FourthOrderStationaryBi-Wave
SingularPerturbationProblem
YanmiWu
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,DongyangShi
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,2022,11(3):1389-1399.DOI:10.12677/aam.2022.113151
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1
InformationEngineeringUniversity,ZhengzhouHenan
2
SchoolofMathematicsandStatistics,ZhengzhouUniversity,ZhengzhouHenan
Received:Feb.28
th
,2022;accepted:Mar.22
nd
,2022;published:Mar.29
th
,2022
Abstract
In this paper, the conformingGalerkin finite element method is presented to solve the
fourthorderstationaryBi-wavesingularperturbationproblemsimulatinghightem-
peraturesuperconductordwavephenomenon.Firstly,theboundednessofthesolution
under thevariational scheme of itslinearproblemisanalyzed;Secondly, the existence
anduniquenessoftheapproximatesolutionforthenonlinearBi-waveproblemare
provedbyusingBrouwerfixedpointtheorem;Furthermore,basedonthehighac-
curacypropertyofBogner-Fox-Schmitelement,quasi-uniformsuperconvergenceand
supercloseerrorestimatesindependentofthenegativepoweroftheparameterinthe
energynormareobtained; Finally, thecorresponding numericalexamplesare provided
toverifythecorrectnessofthetheoreticalanalysis.
Keywords
Bi-WaveProblem,Bogner-Fox-SchmitElement,ExistenceandUniquenessofthe
Solution,Quasi-UniformSupercloseandSuperconvergence
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2022.1131511392
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h
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,θν
h
)+(
∇
(
ψ
−
˜
ψ
h
)
,
∇
ν
h
)+(
f
0
(
ψ
)(
ψ
−
˜
ψ
h
)
,ν
h
) = 0
.
(3.7)
-
ψ
−
˜
ψ
h
= (
ψ
−
I
h
ψ
)+(
I
h
ψ
−
˜
ψ
h
) :=
η
1
+
ξ
1
,
·
‚
ν
h
=
ξ
1
∈
V
h
0
§
Œ
k
ξ
1
k
2
W
+(
f
0
(
ψ
)(
I
h
ψ
−
˜
ψ
h
)
,ξ
1
) =
−
δ
(
θη
1
,θξ
1
)+(
∇
η
1
,
∇
ξ
1
)
−
(
f
0
(
ψ
)(
ψ
−
I
h
ψ
)
,ξ
1
)
.
5
¿
f
0
(
ψ
) =
λ
1
kψ
k
−
1
≥
0(
λ
1
>
0,
k>
0)
½
ö
f
0
(
ψ
) =
λ
2
e
ψ
>
0(
λ
2
>
0),
k
(
f
0
(
ψ
)(
I
h
ψ
−
˜
ψ
h
)
,I
h
ψ
−
˜
ψ
h
)
≥
0
.
(
Ü
(2.3)-(2.4)
ª
§
k
k
ξ
1
k
2
V
≤
Cδh
4
k
ψ
k
6
k
ξ
1
k
2
+
Ch
3
k
ψ
k
4
k∇
ξ
1
k
0
+
Ch
4
k
ψ
k
4
k
ξ
1
k
0
.
|
^
_
Ø
ª
Ú
Friedrichs
Ø
ª
§
Œ
k
ξ
1
k
2
V
≤
Cδh
3
k
ψ
k
6
k∇
ξ
1
k
0
+
Ch
3
|
ψ
|
4
k∇
ξ
1
k
0
+
Ch
4
k
ψ
k
4
k
ξ
1
k
0
,
?
§
k
k
ξ
1
k
V
≤
Ch
3
(
√
δ
k
ψ
k
6
+
k
ψ
k
4
)
.
DOI:10.12677/aam.2022.1131511393
A^
ê
Æ
?
Ð
Ç
ô
‹
§
œ
À
Ï
d
§
ψ
∈
H
6
(Ω)
§
·
‚
k
k
ψ
−
˜
ψ
h
k
V
≤
C
0
(
ψ
)(
√
δ
+1)
h
3
,
(3.8)
Ù
¥
§
C
0
(
ψ
)
´
˜
‡
†
h
Ú
δ
Ã
'
§
•
6
u
ψ
~
ê
"
é
u
α
h
∈
V
h
0
§
·
‚
½
Â
N
S
h
:
V
h
0
→
V
h
0
…
S
h
(
α
h
) =
ψ
l
§
÷
v
δ
(
θ
(
ψ
−
ψ
l
)
,θν
h
)+(
∇
(
ψ
−
ψ
l
)
,
∇
ν
h
)+(
f
0
(
ψ
)(
ψ
−
ψ
l
)
,ν
h
)
−
(
R
f
(
ψ
−
α
h
)
,ν
h
) = 0
.
(3.9)
-
ψ
−
ψ
l
= (
ψ
−
˜
ψ
h
)+(
˜
ψ
h
−
ψ
l
) :=
η
2
+
ξ
2
"
(
Ü
(3.7)
§
·
‚
ν
h
=
ξ
2
∈
V
h
0
§
Œ
k
ξ
2
k
2
V
+(
f
0
(
ψ
)(
˜
ψ
h
−
ψ
l
)
,ξ
2
) = (
R
f
(
ψ
−
α
h
)
,ξ
2
)
.
e
¡
§
·
‚
I
‡
y
²
N
S
h
k
˜
‡
½:
§
=
S
h
ò
˜
‡
O
˜
γ
(
˜
ψ
h
)
N
§
g
§
…
O
γ
1
(
˜
ψ
h
) =
{
α
h
∈
V
h
0
:
k
α
h
−
˜
ψ
h
k
V
≤
˜
γ
}
.
(3.10)
d
(3.8)
§
Œ
•
k
ψ
−
α
h
k
0
,
4
≤
C
(
k
ψ
−
˜
ψ
h
k
V
+
k
˜
ψ
h
−
α
h
k
V
)
≤
(
C
0
(
ψ
)(
√
δ
+1)
h
3
+ ˜
γ
)
.
(3.11)
|
^
(3.11)
§
·
‚
k
|
(
R
f
(
ψ
−
α
h
)
,ξ
2
)
|
≤k
1
2
f
00
(
ψ
+
(
α
h
−
ψ
))
k
0
,
4
k
ψ
−
α
h
k
2
0
,
4
k
ξ
2
k
0
,
4
≤
(
C
0
(
ψ
)(
√
δ
+1)
h
3
+ ˜
γ
)˜
γ
2
k
ξ
2
k
V
.
a
q
/
§
Š
â
(
f
0
(
ψ
)(
˜
ψ
h
−
ψ
l
)
,
˜
ψ
h
−
ψ
l
)
≥
0
,
k
k
ξ
2
k
V
≤
(
C
0
(
ψ
)(
√
δ
+1)
h
3
+ ˜
γ
)˜
γ
2
.
?
§
Œ
k
α
h
−
˜
ψ
h
k
V
≤k
ψ
l
−
˜
ψ
h
k
V
≤
C
1
(
ψ
)((
√
δ
+1)
h
3
+ ˜
γ
)˜
γ
2
.
Ù
¥
§
C
1
(
ψ
)
´
˜
‡
†
h
Ú
δ
Ã
'
§
•
6
u
C
0
(
ψ
)
~
ê
"
Ï
d
§
-
h
≤
(
C
1
(
ψ
))
−
2
§
À
˜
γ
=
C
1
(
ψ
)
h
§
k
k
α
h
−
˜
ψ
h
k
V
≤
˜
γ.
DOI:10.12677/aam.2022.1131511394
A^
ê
Æ
?
Ð
Ç
ô
‹
§
œ
À
=
§
é
u
¿
©
‚
º
€
h
,
S
h
ò
±
˜
ψ
h
•
¥
%
§
˜
γ
=
O
(
h
)
>
0
•
Œ
»
N
g
¥
"
e
5
§
·
‚
ò
Ï
L
y
²
S
h
´
O
˜
γ
(
˜
ψ
h
)
¥
˜
‡
Ø
N
5
y
²
¯
K
(3.3)
)
•
˜
5
"
¯
¢
þ
§
-
ψ
1
Ú
ψ
2
´
¯
K
(3.3)
ü
‡
Ø
Ó
)
§
k
k
ψ
i
−
˜
ψ
h
k
V
≤
C
2
(
ψ
)((
√
δ
+1)
h
3
+ ˜
γ
)˜
γ
2
,i
= 1
,
2
,
Ù
¥
C
2
(
ψ
)
´
˜
‡
†
h
Ú
δ
Ã
'
§
•
6
u
ψ
1
Ú
ψ
2
~
ê
"
é
u
?
¿
ν
h
∈
V
h
0
Ú
α
1
,α
2
∈
O
γ
1
(
˜
ψ
h
),
·
‚
k
δ
(
θ
(
ψ
1
−
ψ
2
)
,θν
h
)+(
∇
(
ψ
1
−
ψ
2
)
,
∇
ν
h
)+(
f
0
(
ψ
)(
ψ
1
−
ψ
2
)
,ν
h
)
−
(
R
f
(
ψ
−
α
1
)
−
R
f
(
ψ
−
α
2
)
,ν
h
) = 0
.
a
q
ƒ
c
O
§
Œ
k
ψ
1
−
ψ
2
k
V
≤
C
2
(
ψ
)((
√
δ
+1)
h
3
+ ˜
γ
)˜
γ
2
k
α
1
−
α
2
k
δ,h
.
À
h
≤
(
C
2
(
ψ
))
−
2
Ú
˜
γ
=
C
2
(
ψ
)
h
§
k
k
S
h
(
α
1
)
−
S
h
(
α
2
)
k
V
≤
((
√
δ
+1)
h
2
+1)
h
k
α
1
−
α
2
k
V
.
Ï
d
§
é
u
¿
©
h
,
S
h
´
O
˜
γ
(
˜
ψ
h
)
þ
Ø
N
"
,
˜
•
¡
§
·
‚
3
¯
K
(3.3)
¥
ν
h
=
ψ
h
∈
V
h
0
§
k
k
ψ
h
k
2
V
+(
f
(
ψ
h
)
,ψ
h
)
≤k
g
k
0
k
ψ
h
k
0
.
5
¿
a
.
(
I
) :
f
(
ψ
) =
λ
1
ψ
k
(
λ
1
>
0)
§
k
=
2
n
+1
2
m
+1
(
m,n
∈
N
)
ž
§
Œ
•
(
f
(
ψ
h
)
,ψ
h
)
≥
0
,
?
k
k
ψ
h
k
V
≤k
g
k
0
.
y
.
.
4.
[
˜
—
‡
%
C
Ú
‡
Â
ñ
Ø
O
y
3
·
‚
í
[
˜
—
‡
%
C
Ú
‡
Â
ñ
Ø
O
"
Ä
k
§
·
‚
ò
e
Ø
©
•
µ
ψ
−
ψ
h
= (
ψ
−
I
h
ψ
)+(
I
h
ψ
−
ψ
h
)
,
η
3
+
ξ
3
.
½
n
4.1.
-
ψ
Ú
ψ
h
©
O
´
¯
K
(3.2)
Ú
¯
K
(3.3)
)
"
b
ψ
∈
H
6
(Ω)
§
·
‚
k
k
I
h
ψ
−
ψ
h
k
V
≤
Ch
3
(
√
δ
k
ψ
k
6
+
k
ψ
k
4
)
.
(4.1)
DOI:10.12677/aam.2022.1131511395
A^
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Æ
?
Ð
Ç
ô
‹
§
œ
À
y
²
.
(
Ü
(3.2)
Ú
(3.3)
§
Œ
e
¡
Ø
•
§
µ
δ
(
θξ
3
,θν
h
)+(
∇
ξ
3
,
∇
ν
h
)+(
f
(
I
h
ψ
)
−
f
(
ψ
h
)
,ν
h
) =
−
δ
(
θη
3
,θν
h
)
−
(
∇
η
3
,
∇
ν
h
)
−
(
f
(
ψ
)
−
f
(
ψ
h
)
,ν
h
)
.
ν
h
=
ξ
3
∈
V
h
0
§
|
^
(3.1)
§
·
‚
k
k
ξ
3
k
2
V
+(
f
(
I
h
ψ
)
−
f
(
ψ
h
)
,ξ
3
)
≤
Cδh
4
k
ψ
k
6
k
ξ
3
k
2
+
Ch
3
k
ψ
k
4
k∇
ξ
3
k
0
+
Ch
4
k
ψ
k
4
k
ξ
3
k
0
.
5
¿
(
f
(
I
h
ψ
)
−
f
(
ψ
h
)
,ξ
3
)
≥
0
,
(
Ü
_
Ø
ª
Ú
Friedrichs
Ø
ª
,
Œ
•
k
ξ
3
k
2
V
≤
Cδh
3
k
ψ
k
6
k∇
ξ
3
k
0
+
Ch
3
k
ψ
k
4
k∇
ξ
3
k
0
+
Ch
4
k
ψ
k
4
k
ξ
3
k
0
.
Ï
d
§
k
k
ξ
3
k
V
≤
Ch
3
(
√
δ
k
ψ
k
6
+
k
ψ
k
4
)
.
y
.
"
·
‚
|
^
©
z
[16]
¥
Š
?
n
Ž
f
§
‰
Ñ
[
˜
—
N
‡
Â
ñ
Ø
O
"
½
n
4.2.
3
½
n
(4.1)
^
‡
e
,
·
‚
k
k
ψ
−
I
2
h
ψ
h
k
V
≤
Ch
3
(
√
δ
k
ψ
k
6
+
k
ψ
k
4
)(4.2)
5.
ê
Š
¢
3
!
¥
§
·
‚
3
«
•
Ω = (0
,
1)
×
(0
,
1)
þ
‰
Ñ
e
¡
ü
‡
ê
Š
Ž
~
"
Table1.
Numericalresultsof
ψ
for
δ
=10
∼
1
.
0
L
1.
δ
=10
∼
1
.
0
ž
§
ψ
ê
Š
(
J
δ
= 10
δ
= 1
.
0
n
×
n
k
ψ
−
ψ
h
k
V
Â
ñ
k
I
h
ψ
−
ψ
h
k
V
Â
ñ
k
ψ
−
ψ
h
k
V
Â
ñ
k
I
h
ψ
−
ψ
h
k
V
Â
ñ
4
×
44.0076e
−
02–8.5735e
−
01–4.0135e
−
02-8.5655e
−
01–
8
×
88.2861e
−
032.27401.7137e
−
012.26938.3250e
−
031.99131.7136e
−
012.3215
16
×
161.9219e
−
032.10822.1862e
−
022.97061.9247e
−
032.11282.1862e
−
022.9705
32
×
324.7210e
−
042.02532.6581e
−
033.03994.7227e
−
042.02702.6581e
−
033.0399
DOI:10.12677/aam.2022.1131511396
A^
ê
Æ
?
Ð
Ç
ô
‹
§
œ
À
Ž
~
1.
·
‚
ý
)
•
ψ
(
x,y
) =
sin
2
(
πx
)
sin
2
(
πy
)
§
Ù
¥
¼
ê
‘
g
Œ
d
T
ý
)
O
Ž
"
·
‚
3
L
1
∼
3
¥
©
O
Ñ
Ø
Ó
ë
ê
δ
= 10
−
4
∼
10
Ø
O
Š
Ú
Â
ñ
§
Œ
±
w
§
h
→
0
ž
§
k
ψ
−
ψ
h
k
V
´
±
O
(
h
2
)
•
`
„
Ç
Â
ñ
§
k
ψ
h
−
I
h
ψ
k
V
´
±
O
(
h
3
)
„
Ç
Â
ñ
§
ù
†
·
‚
n
Ø
©
Û
´
ƒ
Î
Ü
"
Ž
~
2.
·
‚
Ï
L
¼
ê
g
= 1
.
0
5
©
Û¯
K
(1.1)
"
•
Ä
°
(
)
ψ
´
™
•
§
À
J
δ
= 10
§
1
§
10
−
2
Ú
10
−
6
ž
§
·
‚
©
O
±
›
3
¿
©
1
16
×
1
16
e
ê
Š
)
ψ
h
ã
”
X
ã
1
¤
«
"
X
¤
Ï
"
§
‘
X
δ
Š
C
5
§
ù
)
ã
”
5
”
[7]
¥
ƒ
A
Ñ
t
¯
K
)
ã
”
"
Table2.
Numericalresultsof
ψ
for
δ
=10
−
1
∼
10
−
2
L
2.
δ
=10
−
1
∼
10
−
2
ž
§
ψ
ê
Š
(
J
δ
= 10
−
1
δ
= 10
−
2
n
×
n
k
ψ
−
ψ
h
k
V
Â
ñ
k
I
h
ψ
−
ψ
h
k
V
Â
ñ
k
ψ
−
ψ
h
k
V
Â
ñ
k
I
h
ψ
−
ψ
h
k
V
Â
ñ
4
×
44.1256e
−
02–8.5648e
−
01–4.5366e
−
02-8.5649e
−
01–
8
×
88.7351e
−
032.23971.7136e
−
012.32141.2035e
−
021.91441.7136e
−
012.3214
16
×
161.9582e
−
032.15732.1862e
−
022.97052.5140e
−
032.25922.1862e
−
022.9705
32
×
324.7430e
−
042.04562.6581e
−
033.03995.2190e
−
042.26812.6581e
−
033.0399
Table3.
Numericalresultsof
ψ
for
δ
=10
−
3
∼
δ
=10
−
4
L
3.
δ
=10
−
3
∼
δ
=10
−
4
ž
§
ψ
ê
Š
(
J
δ
= 10
−
3
δ
= 10
−
4
n
×
n
k
ψ
−
ψ
h
k
V
Â
ñ
k
I
h
ψ
−
ψ
h
k
V
Â
ñ
k
ψ
−
ψ
h
k
V
Â
ñ
k
I
h
ψ
−
ψ
h
k
V
Â
ñ
4
×
44.9008e
−
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