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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(3),1389-1399
PublishedOnlineMarch2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.113151
o½~Bi-WaveÛÉįKN
Galerkink•[˜—‡Âñ©Û
ÇÇÇôôô‹‹‹
1
§§§œœœÀÀÀ
2
1
&Eó§ŒÆ,àHx²
2
x²ŒÆ§êƆÚOÆ,àHx²
ÂvFϵ2022c228F¶¹^Fϵ2022c322F¶uÙFϵ2022c329F
Á‡
3©¥§·‚|^NGalerkink••{¦)[p§‡NdÅy–o½~Bi-
waveÛÉįK"Äk§©ÛÙ‚5¯KC©‚ªe)k.5¶Ùg§|^Brouwer Ø
Ä:½ny²š‚5Bi-wave¯K%C)•3•˜5¶?§ÄuBogner-Fox-Schmit ü
p°Ý5Ÿ§3Uþ¿ÂeØ•6uëêKg˜[˜—‡%CÚ‡ÂñØO¶•
§·‚ÏLƒAꊎ~ynØ©Û(5"
'…c
Bi-Wave¯K§Bogner-Fox-Schmit§)•3•˜5§[˜—‡%CÚ‡Âñ5
Quasi-UniformSuperconvergence
AnalysisofConformingGalerkin
FiniteElementMethodforthe
FourthOrderStationaryBi-Wave
SingularPerturbationProblem
YanmiWu
1
,DongyangShi
2
©ÙÚ^:Çô‹,œÀ.o½~Bi-WaveÛÉįKNGalerkink•[˜—‡Âñ©Û[J].A^êÆ
?Ð,2022,11(3):1389-1399.DOI:10.12677/aam.2022.113151
Çô‹§œÀ
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.2022.1131511390A^êÆ?Ð
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DOI:10.12677/aam.2022.1131511391A^êÆ?Ð
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h
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,
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h
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h
ψ
0
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DOI:10.12677/aam.2022.1131511392A^êÆ?Ð
Çô‹§œÀ
e5·‚ò|^Brouwer ØÄ:½n5©Û¯K(3.3) )•3•˜5[15]"
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2m+1
(m,n∈N)§÷v
kψ
h
k
V
≤Ckgk
0
.(3.4)
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h
),ν
h
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h
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h
ψ)+(I
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ψ
h
) := η
1
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1
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h
ψ),ξ
1
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0
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h
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h
ψ−
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ψ
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kξ
1
k
2
V
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4
kψk
6
kξ
1
k
2
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4
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0
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DOI:10.12677/aam.2022.1131511393A^êÆ?Ð
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ψ
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k
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h
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l
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l
),θν
h
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l
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h
)+(f
0
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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
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k
V
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h
k
0,4
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˜
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h
k
V
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˜
ψ
h
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h
k
V
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0
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√
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)|
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1
2
f
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(ψ+(α
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0,4
kψ−α
h
k
2
0,4
kξ
2
k
0,4
≤(C
0
(ψ)(
√
δ+1)h
3
+ ˜γ)˜γ
2
kξ
2
k
V
.
aq/§Šâ
(f
0
(ψ)(
˜
ψ
h
−ψ
l
),
˜
ψ
h
−ψ
l
) ≥0,
k
kξ
2
k
V
≤(C
0
(ψ)(
√
δ+1)h
3
+ ˜γ)˜γ
2
.
?§Œ
kα
h
−
˜
ψ
h
k
V
≤kψ
l
−
˜
ψ
h
k
V
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1
(ψ)((
√
δ+1)h
3
+ ˜γ)˜γ
2
.
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1
(ψ) ´˜‡†h ÚδÃ'§•6uC
0
(ψ) ~ê"
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1
(ψ))
−2
§À˜γ= C
1
(ψ)h§k
kα
h
−
˜
ψ
h
k
V
≤˜γ.
DOI:10.12677/aam.2022.1131511394A^êÆ?Ð
Çô‹§œÀ
=§éu¿©‚º€h,S
h
ò±
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ψ
h
•¥%§˜γ= O(h) >0•Œ»Ng¥"
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h
´O
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(
˜
ψ
h
)¥˜‡Ø N5y²¯K(3.3) )•˜5"¯
¢þ§-ψ
1
Úψ
2
´¯K(3.3) ü‡ØÓ)§k
kψ
i
−
˜
ψ
h
k
V
≤C
2
(ψ)((
√
δ+1)h
3
+ ˜γ)˜γ
2
,i= 1,2,
Ù¥C
2
(ψ) ´˜‡†h ÚδÃ'§•6uψ
1
Úψ
2
~ê"
éu?¿ν
h
∈V
h0
Úα
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.
aqƒcO§Œ
kψ
1
−ψ
2
k
V
≤C
2
(ψ)((
√
δ+1)h
3
+ ˜γ)˜γ
2
kα
1
−α
2
k
δ,h
.
Àh≤(C
2
(ψ))
−2
Ú˜γ= C
2
(ψ)h§k
kS
h
(α
1
)−S
h
(α
2
)k
V
≤((
√
δ+1)h
2
+1)hkα
1
−α
2
k
V
.
Ïd§éu¿©h,S
h
´O
˜γ
(
˜
ψ
h
)þØ N"
,˜•¡§·‚3¯K(3.3) ¥ν
h
= ψ
h
∈V
h0
§k
kψ
h
k
2
V
+(f(ψ
h
),ψ
h
) ≤kgk
0
kψ
h
k
0
.
5¿a.(I) : f(ψ) = λ
1
ψ
k
(λ
1
>0)§k=
2n+1
2m+1
(m,n∈N) ž§Œ•
(f(ψ
h
),ψ
h
) ≥0,
?k
kψ
h
k
V
≤kgk
0
.
y..
4.[˜—‡%CÚ‡ÂñØO
y3·‚í[˜—‡%CÚ‡ÂñØO"Äk§·‚òeØ©•µ
ψ−ψ
h
= (ψ−I
h
ψ)+(I
h
ψ−ψ
h
) ,η
3
+ξ
3
.
½n4.1.-ψÚψ
h
©O´¯K(3.2) Ú¯K(3.3) )"bψ∈H
6
(Ω)§·‚k
kI
h
ψ−ψ
h
k
V
≤Ch
3
(
√
δkψk
6
+kψk
4
).(4.1)
DOI:10.12677/aam.2022.1131511395A^êÆ?Ð
Çô‹§œÀ
y².(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
h0
§|^(3.1)§·‚k
kξ
3
k
2
V
+(f(I
h
ψ)−f(ψ
h
),ξ
3
) ≤Cδh
4
kψk
6
kξ
3
k
2
+Ch
3
kψk
4
k∇ξ
3
k
0
+Ch
4
kψk
4
kξ
3
k
0
.
5¿
(f(I
h
ψ)−f(ψ
h
),ξ
3
) ≥0,
(Ü_ØªÚFriedrichs Øª,Œ•
kξ
3
k
2
V
≤Cδh
3
kψk
6
k∇ξ
3
k
0
+Ch
3
kψk
4
k∇ξ
3
k
0
+Ch
4
kψk
4
kξ
3
k
0
.
Ïd§k
kξ
3
k
V
≤Ch
3
(
√
δkψk
6
+kψk
4
).
y."
·‚|^©z[16] ¥Š?nŽf§‰Ñ[˜—N‡ÂñØO"
½n4.2.3½n(4.1)^‡e,·‚k
kψ−I
2h
ψ
h
k
V
≤Ch
3
(
√
δkψk
6
+kψk
4
)(4.2)
5.ꊢ
3!¥§·‚3«•Ω = (0,1)×(0,1) þ‰Ñe¡ü‡êŠŽ~"
Table1.Numericalresultsofψforδ=10∼1.0
L1.δ=10∼1.0ž§ψêŠ(J
δ= 10δ= 1.0
n×nkψ−ψ
h
k
V
ÂñkI
h
ψ−ψ
h
k
V
Âñkψ−ψ
h
k
V
ÂñkI
h
ψ−ψ
h
k
V
Âñ
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.2022.1131511396A^êÆ?Ð
Çô‹§œÀ
Ž~1.·‚ý)•ψ(x,y) = sin
2
(πx)sin
2
(πy)§Ù¥¼ê‘gŒdTý)OŽ"
·‚3L1 ∼3¥©OÑØÓëêδ= 10
−4
∼10 ØOŠÚÂñ§Œ±w§
h→0 ž§kψ−ψ
h
k
V
´±O(h
2
)•`„ÇÂñ§kψ
h
−I
h
ψk
V
´±O(h
3
)„ÇÂñ§ù†·
‚nØ©Û´ƒÎÜ"
Ž~2.·‚ÏL¼êg= 1.0 5©Û¯K(1.1)"
•Ä°()ψ´™•§ÀJδ= 10§1§10
−2
Ú10
−6
ž§·‚©O±›3¿©
1
16
×
1
16
e
êŠ)ψ
h
ã”Xã1¤«"X¤Ï"§‘XδŠC5§ù)ã”5”[7]
¥ƒAÑt¯K)ã”"
Table2.Numericalresultsofψforδ=10
−1
∼10
−2
L2.δ=10
−1
∼10
−2
ž§ψêŠ(J
δ= 10
−1
δ= 10
−2
n×nkψ−ψ
h
k
V
ÂñkI
h
ψ−ψ
h
k
V
Âñkψ−ψ
h
k
V
ÂñkI
h
ψ−ψ
h
k
V
Âñ
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.Numericalresultsofψforδ=10
−3
∼δ=10
−4
L3.δ=10
−3
∼δ=10
−4
ž§ψêŠ(J
δ= 10
−3
δ= 10
−4
n×nkψ−ψ
h
k
V
ÂñkI
h
ψ−ψ
h
k
V
Âñkψ−ψ
h
k
V
ÂñkI
h
ψ−ψ
h
k
V
Âñ
4×44.9008e−02–8.5650e−01–6.4473e−03-8.5651e−01–
8×81.8471e−021.40781.7136e−012.32151.3481e−032.25781.7136e−012.3215
16×166.1603e−031.58412.1862e−022.97052.9308e−042.20152.1862e−022.9705
32×321.4266e−032.11042.6581e−033.03995.9365e−052.30362.6581e−033.0399
Figure1.Thegraphicsofψ
h
when
1
16
×
1
16
att=0.1withδ=10,1.0,10
−2
and10
−6
,respectively
ã1.δ=10§1§10
−2
§10
−6
Ú¿©
1
16
×
1
16
žêŠ)ψ
h
ã”
DOI:10.12677/aam.2022.1131511397A^êÆ?Ð
Çô‹§œÀ
6.(Ø
©¥§·‚|^NGalerkink••{¦)[p§‡NdÅy–o½~Bi-wave
ÛÉįK"ÄuBrouwerØÄ:½n‰Ñš‚5Bi-wave¯KêŠ)•3•˜5y²§
¿(ÜBogner-Fox-Schmit üp°Ý5Ÿ§3Uþ¿Âe[˜—‡%CÚ‡Âñ(J"
•·‚‰Ñꊢy²ynØO(5"Ù¥§½n(4.2) ¥(4.2) ª´é©z[9] ¥
3Uþ‰ê¿ÂeÂñ•O(h
2
) ØOUõ§ùL²·‚Œ±^ •OŽ¤¼ƒÓ
%C°Ý"XÛò‘8ïÄ?˜Úò†ÛÉÄëêÃ'ØO=˜—
Âñ5ù˜´÷(J§ò´·‚8óŠ¥-:ïÄ••"
ë•©z
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