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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(11),8355-8367
PublishedOnlineNovemb er2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.1111884
Äu‡Âñ8+¡ECahn-Hilliard•§g
·Ak•{
XXX©©©ýýý
1
,•••••••••
2
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3
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Á‡
Cahn-Hilliard•§•oš‚5 ‡©•§§3Ôn§)Ô§zÆˆ‡+•Ñk2•A
^§ÏdïÄÙꊕ{äk¢SA^dŠ"©ÏL©ÛCahn-Hilliard•§˜«êŠ
‚ª§y²ÙØOÚÃ^‡Uþ-½5§¿…JÑ˜‡ÄuØO˜mÚžm
g·Aüѧ=‡Âñ8+¡E£superconvergentclusterrecovery§{¡•SCR¤•{§^
uꊦ)Cahn-Hilliard•§§TüÑ̇gŽ´ÄuØO(J5››‚Œ§l
Œ±kü$OŽ¤§•ÏLŽ~y²SCRŽ{p5Ú-½5"
'…c
ØO§Cahn-Hilliard•§§g·A§SCR§k•{
TheSCR-BasedAdaptiveFiniteElement
MethodfortheCahn-HilliardEquation
WenyanTian
1
,YaoyaoChen
2
,ZhaoxiaMeng
3
,Hong’enJia
1∗
1
CollegeofMathematics,TaiyuanUniversityofTechnology,TaiyuanShanxi
2
SchoolofMathematicsandStatistics,AnhuiNormalUniversity,WuhuAnhui
3
DepartmentofEnergyandPowerEngineering,ShanxiInstituteofEnergy,JinzhongShanxi
∗ÏÕŠö
©ÙÚ^:X©ý,•••,ŠŠ_,_÷.Äu‡Âñ8+¡ECahn-Hilliard•§g·Ak•{[J].A^êÆ
?Ð,2022,11(11):8355-8367.DOI:10.12677/aam.2022.1111884
X©ý
Received:Oct.28
th
,2022;accepted:Nov.23
rd
,2022;published:Nov.30
th
,2022
Abstract
TheCahn-Hilliardequationisafourth-ordernonlinearpartialdifferentialequation
withawiderangeofapplicationsinvariousfieldssuchasphysics,biology,andchem-
istry,soitisofpracticalapplicationtostudyitsnumericalmethods.Inthisstudy,
weanalyzedtheCahn-Hilliardequationinasecond-ordernumericalformat,demon-
strateditserrorestimateandunconditionalenergystability,andsuggestedaspatial
andtemporaladaptivestrategybasedontheposteriorerrorestimate,namelythe
superconvergentclusterrecovery(SCR)method,fornumericalsolutions.
Keywords
Error Estimate,TheCahn-Hilliard Equation,Adaptive,SCR,FiniteElement Method
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.2022.11118848357A^êÆ?Ð
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u(t
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Z
t
n
t
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ku
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t
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n
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u(t
n
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n−1
)
2
−u(t
n−
1
2
)).
DOI:10.12677/aam.2022.11118848358A^êÆ?Ð
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n
h
†u(t
n
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2
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2
(Ω))§k
ku
k
h
−u(t
k
)k+(
4t
ε
k
X
n=0
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n−
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2
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−µ(t
n−
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2
)k
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)
1
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1
(ε,u)4t
2
+K
2
(ε,u)h
2
),
Ù¥
C(ε,T) ∼exp(T\ε),
K
1
(ε,u) =
√
ε(ku
ttt
k
L
2
(0,T;L
2
)
+ku
tt
k
L
2
(0,T;H
2
)
)+
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√
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(ku
tt
k
L
2
(0,T;L
2
)
+ku
2
t
k
L
2
(0,T;L
2
)
),
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2
(ε,u) = ku
0
k
2
+
√
εku
t
k
L
2
(0,T;H
2
)
+
1
√
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kµk
C(0,T;H
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C(0,T;H
2
)
.
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n−
1
2
?f/ª(2.3)~(2.5)§
1
4t
(˜e
n
−˜e
n−1
,w
h
)+(∇¯e
n−
1
2
,∇w
h
) = (R
n
1
−
1
4t
(I−P
h
)(u(t
n
)−u(t
n−1
)),w
h
),
(¯e
n−
1
2
+ ˇe
n−
1
2
,v
h
) =
ε
2
(∇˜e
n
+∇˜e
n−1
,v
h
)+ε(4R
n
2
,v
h
)+
1
ε
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1
2
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n−1
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h
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k
2
−k˜e
n−1
k
2
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k¯e
n−
1
2
k
2
= 4t(R
n
1
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n
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n−1
)−(I−P
h
)(u(t
n
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n−1
),˜e
n
+ ˜e
n−1
)−24t(R
n
2
,¯e
n−
1
2
)
+
24t
ε
2
(f(u(t
n−
1
2
))−f(u
n
h
,u
n
n−1
),¯e
n−
1
2
)−
24t
ε
(ˇe
n−
1
2
,¯e
n−
1
2
)
:= I+II+III+IV+V,
¦^‘kεCauchyØªÚYoung’sØª§5OI,II,IIIÚVµ
I≤4tkR
n
kk˜e
n
+ ˜e
n−1
k≤
ε4t
4
2
Z
t
n
t
n−1
ku
ttt
(t)k
2
dt+
4t
2ε
k˜e
n
k
2
+
4t
2ε
k˜e
n−1
k
2
,
II≤k(I−P
h
)(u(t
n
)−u(t
n−1
))kk˜e
n
+ ˜e
n−1
k
≤
ε
2
Z
t
n
t
n−1
k(I−P
h
)u
t
(t)k
2
dt+
1
2ε
k˜e
n
k
2
+
1
2ε
k˜e
n−1
k
2
,
III≤34tεkR
n
2
k
2
+
4t
3ε
k¯e
n−
1
2
k
2
≤3ε4t
4
Z
t
n
t
n−1
ku
tt
(t)k
2
2
dt+
4t
3ε
k¯e
n−
1
2
k
2
,
V≤
24t
ε
(
√
3kˇe
n−
1
2
k)(
1
√
3
k¯e
n−
1
2
k) ≤
34t
ε
kˇe
n−
1
2
k
2
+
4t
3ε
k¯e
n−
1
2
k
2
.
DOI:10.12677/aam.2022.11118848359A^êÆ?Ð
X©ý
•{ü儧{•u(t
n
) = u
n
Úu(t
n−
1
2
) = u
n−
1
2
§©Û1o‘IVµ
IV=
24t
ε
2
(
u
n
h
−u
n−1
h
2
−u
n−
1
2
+(u
n−
1
2
)
3
−
(u
n
h
)
3
+(u
n
h
)
2
u
n−1
h
h
+u
n
h
(u
n−1
h
)
2
+(u
n−1
h
)
3
4
)
=
24t
ε
2
[(
u
n
h
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n−1
h
2
−u
n−
1
2
)+((u
n−
1
2
)
3
−(
u
n
−u
n−1
2
)
3
)+((
u
n
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n−1
2
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n
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n
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n
h
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n
h
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u
n−1
h
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n
h
(u
n−1
h
)
2
+(u
n−1
h
)
3
4
,¯e
n−
1
2
)]
:=
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ε
2
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1
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2
+IV
3
+IV
4
,¯e
n−
1
2
),
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g
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n
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3
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n
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u
n−1
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n
(u
n−1
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2
+(u
n−1
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3
4
.
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n
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t
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n
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n
t
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n
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n
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n
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1
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kξ
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4
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DOI:10.12677/aam.2022.11118848360A^êÆ?Ð
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k+k
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k+4t
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t
n
t
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ku
tt
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k))
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Z
t
n
t
n−1
ku
tt
(t)k
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Z
t
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t
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ku
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t
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k
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n
k
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k
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k
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k
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ë•©z
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