ESAIM: M2AN 43 (2009) 563–589 ESAIM: Ma hema ical Modelling and Nume ical Analysis
DOI: 10.1051/m2an/2009011 www.esaim-m2an.o g
STABILITY AND CONVERGENCE OF TWO DISCRETE SCHEMES
FOR A DEGENERATE SOLUTAL NON-ISOTHERMAL
PHASE-FIELD MODEL ∗
F ancisco Guill´
en-Gonz´
alez1and Juan Vicen e Gu i´
e ez-San ac eu1
Abs ac . We analyze wo nume ical schemes o Eule ype in ime and C0fini e-elemen ype wi h
P1-app oxima ion in space o sol ing a phase-field model o a bina y alloy wi h he mal p ope ies.
This model is w i en as a highly non-linea pa abolic sys em wi h h ee unknowns: phase-field, solu e
concen a ion and empe a u e, whe e he diffusion o he empe a u e and solu e concen a ion may
degene a e. The fi s scheme is nonlinea , uncondi ionally s able and con e gen . The o he scheme is
linea bu condi ionally s able and con e gen . A maximum p inciple is a oided in bo h schemes, using
a unca ion ope a o on he L2p ojec ion on o he P0fini e elemen o he disc e e concen a ion.
In addi ion, o he model when he hea conduc i i y and solu e diffusion coefficien s a e cons an s,
op imal e o es ima es o bo h schemes a e shown based on s abili y es ima es.
Ma hema ics Subjec Classi ica ion. 35Q72, 35K65, 65M12, 65M60.
Recei ed No embe 15, 2007. Re ised May 21, 2008 and Oc obe 1s , 2008.
Published online Ap il 30, 2009.
1. In oduc ion
1.1. The model
The phase-field me hod p o ides a ma hema ical desc ip ion o ee-bounda y p oblems associa ed o phys-
ical p ocesses wi h phase ansi ions. I pos ula es he exis ence o a unc ion, called he phase-field, whose
alue iden ifies he phase a a pa icula poin in space and ime. The me hod is pa icula ly sui able o cases
wi h complex g ow h s uc u es occu ing du ing phase ansi ions. The ma hema ical model s udied in his
wo k desc ibes he solidifica ion p ocess occu ing in a bina y alloy wi h empe a u e-dependen p ope ies. I
is based on a highly non-linea pa abolic sys em o pa ial diffe en ial equa ions wi h h ee dependen a iables:
phase-field, solu e concen a ion and empe a u e. Mo eo e , he empe a u e and concen a ion equa ion ha e
nonlinea degene a e diffusi i y.
Keywo ds and ph ases. Phase-field models, diffuse in e ace model, solidifica ion p ocess, degene a e pa abolic sys ems, backwa d
Eule schemes, fini e elemen s, s abili y, con e gence, e o es ima es.
∗This wo k has been pa ially suppo ed by DGI-MEC (Spain), G an MTM2006–07932 and CGCI MECD-DGU B azil/Spain,
G an 117/06.
1Dp o. E.D.A.N., Uni e si y o Se illa, Ap do. 1160, 41080 Se illa, Spain. [email p o ec ed]; [email p o ec ed]
A icle published by EDP Sciences c
EDP Sciences, SMAI 2009
564 F. GUILL´
EN-GONZ´
ALEZ AND J.V. GUTI´
ERREZ-SANTACREU
Le Ω ⊆Rd(d= 2 o 3) be a bounded domain wi h bounda y Γ. Deno e by [0,T] he ime in e al (T>0).
We use he no a ion Q=Ω×(0,T), Σ = Γ ×(0,T)andn(x) is he ou wa ds uni no mal ec o o Ω a he
poin x∈Γ.
A e some physical simplifica ions [5], we conside he ollowing diffe en ial p oblem, ela ed o a phase-field
model o a bina y alloy wi h he mal p ope ies [1]:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
αε2φ −ε2Δφ=1
2(φ−φ3)+β(θ−θAc−θB(1 −c)) in Q,
CVθ +l
2φ =∇·[K1(φ)∇θ]inQ,
c =∇·[K2(φ)(∇c+Mc(1 −c)∇φ)] in Q.
(1.1)
This model is comple ed wi h he Neumann bounda y condi ions
∂φ
∂nΣ=0,(K1(φ)∇θ)·nΣ=0,(K2(φ)∇c)·nΣ=0 (1.2)
and he ini ial condi ions
φ(x,0) = φ0(x),θ(x,0) = θ0(x),c(x,0) = c0(x)x∈Ω.(1.3)
The unknowns o his p oblem a e: φ:Q→R(phase-field) is he s a e a iable cha ac e izing he diffe en
phases so ha φ= 1 ep esen s he liquid phase and φ=−1 ep esen s he solid phase, θ:Q→Ris he
empe a u e o he ma e ial, c:Q→[0,1] (concen a ion) ep esen s he ac ion o one o he wo ma e ials
in he mix u e. The pa ame e α>0 is he elaxa ion scaling; he pa ame e βis gi en by β=ε[s]/3σ,whe e
ε>0 is he measu e o he in e ace wid h, σ he su ace ension and [s] he en opy densi y diffe ence be ween
phases; θA,θBa e he mel ing empe a u es o each o he wo ma e ials in he alloy; CV>0 is he specific
hea ; l>0 he la en hea ; K1≥0 he he mal conduc i i y; K2≥0 he solu e diffusi i y; M∈Ris a cons an
ela ed o he slopes o solid and liquid lines.
We will assume ha K1=K1(φ)andK2=K2(φ) a e wo globally Lipschi z con inuous unc ions sa is ying
0≤K1( )≤b1,0≤K2( )≤b2∀ ∈R,
wi h b1,b
2>0. In his sense, he p oblem is singula wi h espec o he empe a u e and concen a ion
when K1(φ)=0o K2(φ) = 0, espec i ely. As physically he diffusion o ma e ial in he solid phase can be
conside ed close o ze o [5]; his leads o a degene a e solu e diffusion. Such a phenomenon is included in his
model, assuming ha K2(φ)=0i φ=−1. On he o he hand, al hough he hea conduc i i y is nonze o
in bo h solid and liquid phases, we also conside a degene a e diffusion o he empe a u e wi h he aim o
conside ing a mo e gene al model. Mo eo e , his may help he de elopmen o nume ical me hods o sys ems
wi h simila cha ac e is ics.
The phase-field model o solidifica ion (1.1) is used o ea phenomena such as c ys al g ow h and he usion
o ma e ials.
Now we in oduce he defini ion o weak solu ions simila o ha gi en in [1,12] which ake in o accoun
he hea and solu e degene a e diffusi i y, espec i ely. Mo eo e , he maximum p inciple o he concen a ion
equa ion says us ha 0 ≤c≤1inQi 0 ≤c0≤1inΩ.
De ini ion 1.1. A iple (φ, θ, c) is called a weak solu ion o (1.1)–(1.3)in(0,T)i :
(1) φ∈L2(0,T;H2(Ω)) ∩L∞(0,T,H1(Ω)),φ
∈L2(Q),φ(0) = φ0,∂φ
∂n=0a.e. on Σ,
(2) θ∈L∞(0,T;L2(Ω)),θ
∈L2(0,T,H1(Ω)),θ(0) = θ0,
(3) c∈L∞(0,T;L2(Ω)),c
∈L2(0,T;H1(Ω)),c(0) = c0,0≤c≤1a.e. in Q,
(4) J1:= ∇(K1(φ)θ)−θ∇K1(φ)∈L2(Q),
(5) J2:= ∇(K2(φ)c)−c∇K2(φ)∈L2(Q),
STABILITY AND CONVERGENCE OF TWO DISCRETE SCHEMES 565
e i ying
αε2φ −ε2Δφ=1
2(φ−φ3)+β(θ−θAc−θB(1 −c)) a.e. in Q,
CVT
0
θ ,ηd +l
2T
0φ ,ηd +T
0J1,∇η=0,
T
0
c ,ηd +T
0J2,∇ηd +MT
0K2(φ)c(1 −c)∇φ, ∇η=0,
o each η∈L2(0,T;H1(Ω)). I , in addi ion, K1,K
2≥b0>0, hen θ,c ∈L2(0,T;H1(Ω)) and J1=K1(φ)∇θ,
J2=K1(φ)∇c.
He e and in wha ollows, ·,·deno es he inne p oduc in L2(Ω) and ·,· deno es he duali y be ween
H1(Ω)and H1(Ω).
1.2. Known esul s
In [1], he exis ence o weak solu ions o p oblem (1.1)–(1.3) bu wi h a cons an solu e diffusi i y (K2>0) is
ob ained ia he in oduc ion o a egula ized p oblem app oxima ing he degene a e he mal conduc i i y K1by
a s ic ly posi i e, egula unc ion ollowed by he de i a ion o sui able ap io ies ima es and he applica ion
o compac ness a gumen s. Mo e conc e ely, he ollowing exis ence esul was es ablished in [1].
Theo em 1.2. Le Ωbe an open bounded domain o Rd,d=2o 3, wi h smoo h bounda y Γ. Assume
φ0∈H1+γ(Ω) wi h 1/2<γ≤1such ha ∂φ0
∂n=0on Γ,θ0∈L2(Ω) and c0∈H1(Ω) such ha 0≤c0≤1a.e.
in Ω.Then, he eexis s(φ, θ, c)aweaksolu iono (1.1)–(1.3)(wi h K1>0a cons an ) in (0,T).
In addi ion, in [1] he au ho s say ha he hypo hesis φ0∈H1+γ(Ω) wi h 1/2<γ≤1 is no essen ial, and
he esul holds o φ0∈H1(Ω).
Scheid [12] p o ed he exis ence o weak solu ions, by using a simila me hodology o [1], o he ollowing
iso he mal phase-field model o a bina y alloy
αε2φ −ε2Δφ=F1(φ)+cF2(φ)inQ,
c =∇·[D1(φ)(∇c+D2(c, φ)∇φ)] in Q, (1.4)
which has a degene a e solu e diffusi i y D1(φ)≥0. The main difficul y o model (1.4) is he ea men o he
nonlinea e m in ol ing D1(φ)D2(c, φ)∇φin he concen a ion equa ion. Mo eo e , he maximum p inciple
o he phase-field a iable gi es −1≤φ≤1 unde he assump ions ha he abo e nonlinea i ies F1(φ)and
F2(φ) anishwhenφ=−1andφ=1.
E o es ima es o nonlinea nume ical schemes o iso he mal phase-field models ela ed o bina y alloys a e
gi en in [10] o a model as (1.4), and, in [4], conside ing aniso opic diffusion o he phase-field equa ion and
a mo e gene al igh -hand side in he phase-field equa ion, changing he e ms F1(φ)+cF2(φ)conside edin
[10]byS(c, φ) being a bounded, Lipschi z unc ion.
In [7], op imal e o es ima es a e gi en o a ully disc e e nonlinea nume ical scheme o a mo e simplified
phase-field model han (1.1) wi hou he concen a ion ( ha is one ma e ial is only conside ed) and wi h cons an
he mal conduc i i y K1>0, paying special a en ion on he dependency o he pa ame e ε. S abili y es ima es
independen o εa e p o ed o ksmall enough wi h espec o ε,andα, β a e cons an s depending on ε.I is
also shown some e o bounds depending only on a lowe polynomial o de o 1/ε. Mo eo e , e o es ima es
a e used o es ablish he con e gence o he ully disc e e scheme o solu ions o he sha p in e ace limi s unde
diffe en scaling hypo heses in i s coefficien s.
In [2], a ime-disc e e nonlinea scheme is p oposed o a phase-field p oblem again wi hou he concen a ion
a iable and eplacing in he equa ion o he empe a u e he e m l
2φ by he mo e gene al e m l
2 (θ,φ) ,
whe e is a gene ic unc ion sa is ying some adequa e p ope ies. Con e gence o his semi-disc e e in ime
scheme is p o ed, ob aining he exis ence and egula i y o solu ions o he limi p oblem.
566 F. GUILL´
EN-GONZ´
ALEZ AND J.V. GUTI´
ERREZ-SANTACREU
1.3. Main esul s o he pape
In his wo k we will conside wo nume ical schemes in o de o app oxima e p oblem (1.1) using con inuous
P1-fini e elemen s o he ee a iables (φ, θ, c). Since a maximum p inciple canno be e ified in gene al by he
disc e e concen a ion, we in oduce a unca ion ope a o on he L2p ojec ion on o P0, in o de o gua an ee
aL∞bound o some e ms in he disc e e concen a ion equa ion. A simila idea o unca ion, bu wi hou
he L2p ojec ion on o P0, has been used in [8] o a2DNa ie -S okes model wi h mass diffusion.
Fi s o all, we will p esen in Sec ion 2 he nonlinea nume ical scheme (2.1)–(2.3) which will be uncondi-
ionally s able and con e gen .
Theo em 1.3 (uncondi ionally s able, con e gen nonlinea scheme).Assume
φ0∈H1(Ω),θ
0∈L2(Ω) and c0∈L2(Ω) such ha 0≤c0≤1a.e. in Ω.
Le Ωbe such ha he H2- egula i y o he Neumann p oblem (3.19)holds. Le Thbe a egula , quasi-uni o m
amily o a polyhed al domain Ω. Then, he e exis s a con e gen subsequence o unc ions φh,k,θ
h,k and ch,k
associa ed o scheme (2.1)–(2.3)(see De . 3.5) owa ds a weak solu ion (φ, θ, c)o p oblem (1.1)–(1.3)in (0,T),
as (h, k)→0in he ollowing sense:
θh,k →θ, ch,k →c, in L∞(0,T;L2(Ω))-weak∗,
φh,k →φ, in L∞(0,T;H1(Ω))-weak∗,and in L2(0,T;H1(Ω))-s ong.
Second, we cons uc he linea nume ical scheme (6.1)–(6.3) which will be condi ionally s able and con e gen .
Theo em 1.4 (condi ionally s able, con e gen linea scheme).Assume he assump ions o Theo em 1.3 and
he cons ain
(S) lim
(h,k)→0
k
h=0.
Then, he e exis s a con e gen subsequence o unc ions φh,k,θh,k and ch,k associa ed o scheme (6.1)–(6.3)
(see De . 3.5) owa ds a weak solu ion (φ, θ, c)o p oblem (1.1)–(1.3)in (0,T),as(h, k)→0in he same sense
o Theo em 1.3.
A his poin , i is well o poin ou ha , in pa icula , he wo p e ious heo ems p o ide he exis ence
o weak solu ions o p oblem (1.1)–(1.3) unde hypo heses on he da a weake han hose imposed in [1](see
Thm. 1.2). To be mo e p ecise, he hypo hesis on c0is elaxed om c0∈H1(Ω) imposed in [1] oc0∈L2(Ω) as
was conside ed in [12]. Recall ha in [12] he iso he mal case is conside ed and in [1] he e is no degene a ion
in he solu e diffusi i y.
Finally, assuming bo h he hea conduc i i y and solu e diffusion coefficien s a e cons an s, e o es ima es
o o de O(k+h) a e shown owa ds a egula enough con inuous solu ion o (1.1)–(1.3).
Theo em 1.5. Unde hypo heses o Theo em 1.3 ( espec i ely Thm. 1.4), admi ing ha K1,K
2a e wo pos-
i i e cons an s and ha he e exis a con inuous solu ion (φ, θ, c) o (1.1)–(1.3)which e ifies he egula i y
assump ions (7.2), hen he disc e e solu ion o scheme (2.1)–(2.3)( espec i ely (6.1)–(6.3)) sa isfies o all
n<N
en+1
φ2
H1(Ω) +1
4ε2en+1
φ4
L4(Ω) +CV|en+1
θ|2+|en+1
c|2+αk
n

l=1 
el+1
φ−el
φ
k
2
+K1k
n

l=1
|∇el+1
θ|2+K2k
n

l=1
|∇el+1
c|2≤Ch2+k2,
STABILITY AND CONVERGENCE OF TWO DISCRETE SCHEMES 567
whe e Cis a cons an independen o hand k, and he e o s a e deno ed by en+1
φ=φn+1
h−φ( n+1),
en+1
θ=θn+1
h−θ( n+1)and en+1
c=cn+1
h−c( n+1).
The es o he pape is desc ibed as ollows. In Sec ion 2 he nonlinea scheme (2.1)–(2.3)isp esen ed,
ob aining i s uncondi ionally s abili y in Sec ion 3. In Sec ion 4some necessa y compac ness esul s a e p o ed,
passing o he limi in Sec ion 5and concluding he p oo o Theo em 1.3. In addi ion, a condi ionally s able
and con e gen linea scheme is s udied in Sec ion 6gi ing an ou line o he p oo o Theo em 1.4. Finally,
Sec ion 7is de o ed o s udying op imal e o es ima es o bo h schemes.
2. A nonlinea scheme
In wha ollows, le us conside a uni o m pa i ion n=nk o he ime in e al [0,T]wi hk=T/N he
ime s ep, le Ω ⊂Rd(d= 2 o 3) be a domain wi h polyhed al bounda y and Thbe a amily o iangula ions
o Ω wi h Ω= 
K∈Th
K.He eh:= max
K∈Th
hKwi h hK he diame e o K.Le Xhbe he fini e elemen subspace
o H1(Ω) u nished by globally con inuous, piecewise linea unc ions, ha is,
Xh={xh∈C0(Ω): xh|K∈P1(K),∀K∈T
h,}·
A fi s idea o app oxima e equa ion (1.1)1is
⎧
⎪
⎪
⎨
⎪
⎪
⎩
αφn+1
h−φn
h
k,x
h+∇φn+1
h,∇xh+1
2ε2(φn+1
h)3,x
h
=1
2ε2φn
h,x
h+β
ε2θn
h−θAcn
h−θB(1 −cn
h),x
h,∀xh∈Xh.
Bu , we add φn+1
h,x
h o he le -hand side and φn
h,x
h o he igh -hand side which will cancel each o he
in he limi as (h, k) go o ze o. The eason why we in oduce hese e ms is o ge s abili y cons an s only o
polynomial o de wi h espec o εa oiding exponen ial dependence. Conc e ely, since β=O(ε), we will ge
s abili y cons an s depending on 1/ε (see Rem. 3.2 below).
Then we p opose he ollowing scheme o app oxima e p oblem (1.1)–(1.3):
Ini ializa ion:Le (φ0
h,θ0
h,c
0
h)∈Xh×Xh×Xhbe sui able app oxima ions o (φ0,θ
0,c
0).
S ep n+1: Gi en(φn
h,θn
h,c
n
h)∈Xh×Xh×Xh.
Find φn+1
h∈Xhas a solu ion o he p oblem:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
αφn+1
h−φn
h
k,x
h+∇φn+1
h,∇xh+φn+1
h,x
h+1
2ε2(φn+1
h)3,x
h
=1
2ε2+1
φn
h,x
h+β
ε2θn
h−θAcn
h−θB(1 −cn
h),x
h,∀xh∈Xh.
(2.1)
Find θn+1
h∈Xhand cn+1
h∈Xhas solu ions o he decoupled a ia ional p oblems:
CVθn+1
h−θn
h
k,x
h+Kh
1(φn+1
h)∇θn+1
h,∇xh=−l
2φn+1
h−φn
h
k,x
h,∀xh∈Xh,(2.2)
⎧
⎪
⎨
⎪
⎩cn+1
h−cn
h
k,x
h+Kh
2(φn+1
h)∇cn+1
h,∇xh
=−MKh
2(φn+1
h)[P0cn
h]T(1 −[P0cn
h]T)∇φn
h,∇xh,∀xh∈Xh.
(2.3)

568 F. GUILL´
EN-GONZ´
ALEZ AND J.V. GUTI´
ERREZ-SANTACREU
He e Kh
1=K1+g1(h), Kh
2=K2+g2(h), whe e gia e posi i e unc ions o be chosen la e , and P0is he
L2o hogonal p ojec o on o X0
h,whe eX0
his he fini e elemen space o piecewise cons an unc ions, and [·]T
is a unca ion ope a o defined om X0
hin o X0
has ollows: Gi en xh∈X0
h, hen[xh]T∈X0
hsuch ha
∀K∈T
h,[xh]T|K=⎧
⎨
⎩
xh|Ki xh|K∈[0,1],
0i xh|K<0,
1i xh|K>1.
Since (2.2)and(2.3) a e quad a ic linea sys ems, i is easy o check he exis ence and uniqueness o solu ions.
On he o he hand, (2.1) is a disc e e nonlinea a ia ional p oblem and i s exis ence and uniqueness can be
p o ed as ollows: We define
J(φh)= α
2kΩ
|φh|2+1
2Ω|∇φh|2+|φh|2+1
8ε2Ω
|φh|4−Ω
gφ
h,(2.4)
whe e g=α
kφn
h+1
2ε2+1
φn
h+β
ε2θn
h−θAcn
h−θB(1 −cn
h). Clea ly, Jis a s ic ly con ex unc ional on Xh,
hen he minimum p oblem min
φh∈Xh
J(φh) has a unique solu ion cha ac e ized by i s Eule equa ion (2.1).
We will deno e by Cgene ic posi i e cons an s always independen o he disc e iza ion pa ame e s hand k.
3. APRIORI es ima es and weak con e gences
Le us add and sub ac he e m 1
2ε2φn+1
h,x
h o he le -hand side o (2.1) in o de o ew i e (2.1)
wi h espec o he so-called Ginzbu g-Landau unc ion (φ)= 1
2ε2φ2−1φwhich has he po en ial unc ion
F(φ)= 1
8ε2φ2−12, ha is, (φ)=∇φF(φ). Then, (2.1) is ew i en as:
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
αφn+1
h−φn
h
k,x
h+∇φn+1
h,∇xh+φn+1
h,x
h
+ (φn+1
h),x
h+1
2ε2φn+1
h−φn
h,x
h=φn
h,x
h
+β
ε2θn
h−θAcn
h−θB(1 −cn
h),x
h∀xh∈Xh.
(3.1)
I is easy o check ha i we selec φ0
h=Ihφ0,θ0
h=Ihθ0and c0
h=Ihc0as ini ial app oxima ions, whe e Ih
is an in e pola ion ope a o in o Xhsa is ying s abili y p ope ies in he L2,L4and H1no ms, i ollows ha
he e exis s a cons an C2>0 (independen o ε) such ha
2CVβ
lε2|θ0
h|2+|c0
h|2+1+φ0
h2
H1(Ω) +1
4ε2Ω
(|φ0
h|2−1)2≤C2
ε2·(3.2)
Fo ins ance, his is ue when Ihis he L2-p ojec o on o Xh,o Ihis he Cl´emen o Sco -Zhang egula iza ion
ope a o .
Le us deno e by |·| he L2(Ω)-no m and by ·
H1(Ω) he H1(Ω)-no m. Wi h such a no a ion we es ablish
he ollowing s abili y esul .
STABILITY AND CONVERGENCE OF TWO DISCRETE SCHEMES 569
Lemma 3.1. Assume φ0∈H1(Ω),θ0∈L2(Ω) and c0∈L2(Ω) such ha 0≤c0≤1a.e. in Ω. Then, o each k
such ha βk
ε2is sufficien ly small, he disc e e solu ion o scheme (2.1)–(2.3)sa isfies he ollowing es ima es:
(i)max
0≤n≤Nφn
h2
H1(Ω) ≤C, (ii)
N−1

n=0
φn+1
h−φn
h2
H1(Ω) ≤C, (iii)k
N−1

n=0 
φn+1
h−φn
h
k
2
≤C,
(i )max
0≤n≤N|θn
h|2≤C, ( )
N−1

n=0
|θn+1
h−θn
h|2≤C, ( i)k
N−1

n=0
|Kh
1(φn+1
h)∇θn+1
h|2≤C,
( ii)max
0≤n≤N|cn
h|2≤C, ( iii)
N−1

n=0
|cn+1
h−cn
h|2≤C, (ix)k
N−1

n=0
|Kh
2(φn+1
h)∇cn+1
h|2≤C,
whe e C>0depends on εand he da a (φ0,θ
0,c
0)bu is independen o (h, k).
P oo . Le xh=4β
lε
2kθn+1
hand xh=2kc
n+1
hbe es unc ions in (2.2)and(2.3), espec i ely. Now by using
he iden i y (a−b, 2a)=|a|2−|b|2+|a−b|2and bounding adequa ely he igh -hand side, we ha e
2CVβ
lε2|θn+1
h|2−|θn
h|2+|θn+1
h−θn
h|2+4β
lε2k|Kh
1(φn+1
h)∇θn+1
h|2
=−2β
ε2kφn+1
h−φn
h
k,θn
h+(θn+1
h−θn
h)
≤−
2β
ε2kφn+1
h−φn
h
k,θn
h+α
2k
φn+1
h−φn
h
k
2
+2β2
αε4k|θn+1
h−θn
h|2,
(3.3)
|cn+1
h|2−|cn
h|2+|cn+1
h−cn
h|2+k|Kh
2(φn+1
h)∇cn+1
h|2≤Ck|∇φn
h|2.(3.4)
By choosing βk
ε2sufficien ly small o con ol he las e m on he igh -hand side o (3.3), his inequali y
educes o
2CVβ
lε2|θn+1
h|2−|θn
h|2+1
2|θn+1
h−θn
h|2+4β
lε2k|Kh
1(φn+1
h)∇θn+1
h|2≤
−2β
ε2kφn+1
h−φn
h
k,θn
h+α
2k
φn+1
h−φn
h
k
2
·(3.5)
Nex , ake xh=2kφn+1
h−φn
h
kas a es unc ion in (2.1), i ollows ha
αk
φn+1
h−φn
h
k
2
+φn+1
h2
H1(Ω) −φn
h2
H1(Ω) +φn+1
h−φn
h2
H1(Ω)+2
 (φn+1
h),φ
n+1
h−φn
h
+1
ε2|φn+1
h−φn
h|2≤2β
ε2kθn
h,φn+1
h−φn
h
k+Ck|φn
h|2+Cβ2
αε4k(|cn
h|2+1),(3.6)
whe e Cis a cons an independen o ε.
570 F. GUILL´
EN-GONZ´
ALEZ AND J.V. GUTI´
ERREZ-SANTACREU
Now, using again he iden i y (a−b, a)=1
2(|a|2−|b|2+|a−b|2) wice, we can ew i e
2 (φn+1
h),φ
n+1
h−φn
h=1
2ε2Ω(φn+1
h)2−1(φn+1
h)2−(φn
h)2+(φn+1
h−φn
h)2
=2ΩF(φn+1
h)−F(φn
h)+ 1
8ε2((φn+1
h)2−(φn
h)2)2
+1
2ε2Ω(φn+1
h)2−1|φn+1
h−φn
h|2.(3.7)
No e ha he nega i e e m on he igh -hand side o (3.7) can be abso bed by he las e m on he le -hand
side o (3.6). This p ope y can be summa ized as
1
2ε2(φn+1
h)3−φn
hφn+1
h−φn
h≥F(φn+1
h)−F(φn
h) (3.8)
which is an app op ia e disc e e e sion o he equali y (φ)φ =F(φ) .
Nex , i we add up (3.4), (3.5)and(3.6), he fi s e m on he igh hand side o (3.5)and(3.6) disappea s,
and we ge
2CVβ
lε2|θn+1
h|2−|θn
h|2+1
2|θn+1
h−θn
h|2+4β
lε2k|Kh
1(φn+1
h)∇θn+1
h|2
+(|cn+1
h|2+1)−(|cn
h|2+1)+|cn+1
h−cn
h|2+k|Kh
2(φn+1
h)∇cn+1
h|2
+α
2k
φn+1
h−φn
h
k
2
+φn+1
h2
H1(Ω) −φn
h2
H1(Ω) +φn+1
h−φn
h2
H1(Ω)
+2ΩF(φn+1
h)−F(φn
h)≤Ckφn
h2
H1(Ω) +Cβ2
αε4k(|cn
h|2+1).
Finally, by summing o e n he disc e e G onwall lemma and he ini ial bound (3.2) p o ide he desi ed es i-
ma es, and his comple es he p oo . 
Rema k 3.2. Since he ini ial es ima es ha e o de O(ε−2), see (3.2), hen he s abili y es ima es ob ained in
Lemma 3.1 a e o o de O(ε−2eCβ2ε−4) o he a iables (φ, √β
εθ,c). As βis o o de O(ε), he o de educes o
O(ε−2eCε−2). In pa icula , hese es ima es would be independen o εi βwe e o o de O(ε2) and conside ing
an ini ial bound (3.2) independen o ε. Fu he mo e, i we unca e he disc e e concen a ion cn
hin (2.1)as
made in (2.3), ha is eplacing β
ε2θn
h−θAcn
h−θB(1 −cn
h),x
hby β
ε2θn
h−θA[P0cn
h]T−θB(1 −[P0cn
h]T),x
h,
his modified scheme has s abili y es ima es o o de O(ε−2+β2ε−4).
Rema k 3.3. Obse e ha in his nonlinea scheme, we ha e used a fi s -o de semi-implici app oxima ion o
he Ginzbu g-Landau unc ion (φ), which p o ides a s a iona y p oblem o sol e in each ime s ep, iden ified
wi h he c i ical poin o a con ex unc ional (see (2.4)). Mo eo e , his app oxima ion e ifies he p ope y (3.8).
Fo ins ance, i we use he fi s -o de implici app oxima ion (φn+1
h), hen he associa ed s a iona y p oblem
(o Allen-Cahn ype) is ela ed o he c i ical poin s o a non-con ex unc ional and he p ope y (3.8)inno
e ified, because a nega i e e m appea s on he igh -hand side. To be mo e conc e e, i ollows ha
(φn+1
h)φn+1
h−φn
h≥F(φn+1
h)−F(φn
h)−1
4ε2(φn+1
h−φn
h)2.
STABILITY AND CONVERGENCE OF TWO DISCRETE SCHEMES 571
Conside he linea ope a o Lh:Xh→Xhdefined as:
Lhφh,x
h=∇φh,∇xh+φh,x
h∀xh∈Xh.(3.9)
Then, he disc e e phase-field equa ion (2.1) can be ew i en as:
⎧
⎪
⎪
⎨
⎪
⎪
⎩φn+1
h−φn
h
k,x
h+1
αLhφn+1
h,x
h+1
2αε2(φn+1
h)3,x
h
=1
2αε2+1
αφn
h,x
h+β
αε2θn
h−θAcn
h−θB(1 −cn
h),x
h,∀xh∈Xh.
(3.10)
Taking xh=Lhφn+1
has a es unc ion in (3.10) and using he es ima es o Lemma 3.1, he ollowing esul
can be es ablished.
Co olla y 3.4. Unde he hypo heses o Lemma 3.1, i holds
k
N−1

n=0
|Lhφn+1
h|2≤C.
On he o he hand, since K1(·)≤b1and K2(·)≤b2, om( i)and(ix) o Lemma 3.1 we also ha e
k
N−1

n=0
|Kh
1(φn+1
h)∇θn+1
h|2≤C, k
N−1

n=0
|Kh
2(φn+1
h)∇cn+1
h|2≤C.
De ini ion 3.5. We define φh,k ( espec i ely 
φh,k) as he piecewise cons an unc ions in ime aking alues
φn+1
hon ( n,
n+1] ( espec i ely φn
h). Analogously, we define θh,k,
θh,k,andch,k,ch,k. Mo eo e , we define 
φh,k,

θh,k,ch,k ∈C0([0,T]; Xh) as he piecewise linea unc ions in ime such ha 
φh,k( n)=φn
h,
θh,k( n)=θn
h,
ch,k( n)=cn
h, espec i ely.
An easy consequence o he p e ious defini ion, Lemma 3.1 and Co olla y 3.4 is he ollowing esul .
Lemma 3.6. Unde he hypo heses o Lemma 3.1, he ollowing es ima es hold:
{θh,k}h,k,{
θh,k}h,k,{
θh,k}h,k is bounded in L∞(0,T;L2(Ω)),(3.11)
{ch,k}h,k,{ch,k}h,k,{ch,k}h,k is bounded in L∞(0,T;L2(Ω)),(3.12)
{φh,k}h,k,{
φh,k}h,k,{
φh,k}h,k is bounded in L∞(0,T;H1(Ω)),(3.13)
{Kh
1(φh,k)∇θh,k}h,k is bounded in L2(0,T;L2(Ω)),(3.14)
{Kh
2(φh,k)∇ch,k}h,k is bounded in L2(0,T;L2(Ω)),(3.15)
d
d 
φh,kh,k
is bounded in L2(0,T;L2(Ω)).(3.16)
{Lhφh,k}h,k is bounded in L2(0,T;L2(Ω)).(3.17)
578 F. GUILL´
EN-GONZ´
ALEZ AND J.V. GUTI´
ERREZ-SANTACREU
5. Passing o he limi
In o de o pass o he limi in he disc e e concen a ion equa ion, we will use he ollowing esul , which is
easy o p o e because equa ion (5.1) sa isfies he maximum p inciple:
Lemma 5.1. The ollowing wo sys ems a e equi alen :
c =∇·(K2(φ)[∇c+MT1
0c(1 −T1
0c)∇φ]) in Q, (5.1)
and
0≤c≤1,c
=∇·(K2(φ)[∇c+Mc(1 −c)∇φ]) in Q.
To pass o he limi in scheme (2.1)–(2.3), we ew i e he scheme as ollows: Taking xh=ηn+1
h∈Xha
sui able app oxima ion a ime n+1 o any unc ion η∈C0([0,T]; C∞
c(Ω)) such ha η(T) = 0 (clea ly ηN
h=0)
as a es unc ion in (2.1), (2.2)and(2.3), mul iplying by k, summing o e nand deno ing he unc ion ηh,k
simila ly o Defini ion 3.5, one a i es a
⎧
⎪
⎪
⎨
⎪
⎪
⎩
αε2T
0d
d 
φh,k,η
h,k+ε2T
0∇φh,k,∇ηh,k+ε2T
0φh,k −
φh,k,η
h,k
+1
2T
0(φh,k)3−
φh,k,η
h,k−βT
0
θh,k −θAch,k −θB(1 −ch,k),η
h,k=0,
CVT
0d
d 
θh,k,η
h,k+l
2T
0d
d 
φh,k,η
h,k+T
0Kh
1(φh,k)∇θh,k,∇ηh,k=0,
⎧
⎪
⎪
⎨
⎪
⎪
⎩
T
0d
d ch,k,η
h,k+T
0Kh
2(φh,k)∇ch,k,∇ηh,k
+MT
0Kh
2(φh,k)[P0ch,k]T(1 −[P0ch,k]T)∇
φh,k,∇ηh,k=0.
(5.2)
By applying all he con e gences al eady ob ained, he e a e no addi ional difficul ies in passing o he limi
ob aining ha (φ, θ, c) is a weak solu ion o (1.1). In pa icula , aking (h, k)→0 in he disc e e equa ion o
he concen a ion cand using (4.7), we a i e a he limi equa ion (5.1); hence 0 ≤c≤1andT1
0c=c. Finally,
he disc e e phase-field equa ion is e ified poin wise in Q hanks o he s ong egula i y o φ. The p oo o
Theo em 1.3 is finished.
Rema k 5.2. Asmen ionedinRema k3.2, aking[P0cn
h] ins ead o cn
hin he disc e e equa ion o he phase
field a iable (2.1) p o ides be e s abili y es ima es wi h espec o ε. Ne e heless we find ha he e is a limi
unc ion ϑ∈L∞(0,T;L2(Ω)) such ha [P0cn
h] ends o ϑweakly* in L∞(0,T;L2(Ω)), bu i is no clea how o
iden i y ϑwi h he limi unc ion c.Byusing(4.14)and ha 0≤c≤1a.e. Q (owing o he limi in (5.2)can
be aken as be o e), we can only deduce ha ϑ=ca.e. in 
Q,
Qbeing defined in he p oo o P oposi ion 4.3.
6. A condi ionally s able, con e gen linea scheme
In his sec ion we s udy a mo e explici scheme, whe e he nonlinea disc e e app oxima ion (2.1)o (1.1)1
is conside ed comple ely in he p e ious s ep ime, esul ing a linea (and decoupled) scheme. Con a y o he
p e ious nonlinea scheme, now o ob ain s abili y we will impose a cons ain on he disc e e pa ame e s.
Recall he defini ion o he Ginzbu g-Landau unc ion (φ)= 1
2ε2(φ2−1)φassocia ed o he po en ial
unc ion F(φ)= 1
8ε2(φ2−1)2. We p opose he ollowing linea scheme:
Ini ializa ion:Le (φ0
h,θ0
h,c
0
h)∈Xh×Xh×Xhbe sui able app oxima ions o (φ0,θ
0,c
0).
S ep n+1: Gi en(φn
h,θn
h,c
n
h)∈Xh×Xh×Xh.

STABILITY AND CONVERGENCE OF TWO DISCRETE SCHEMES 579
Find φn+1
h∈Xhas a solu ion o he p oblem:
αφn+1
h−φn
h
k,x
h+∇φn+1
h,∇xh+φn+1
h,x
h=− (φn
h),x
h+φn
h,x
h
+β
ε2θn
h−θAcn
h−θB(1 −cn
h),x
h,∀xh∈Xh.(6.1)
Find θn+1
h∈Xhand cn+1
h∈Xhas solu ions o he decoupled a ia ional p oblems:
CVθn+1
h−θn
h
k,x
h+Kh
1(φn+1
h)∇θn+1
h,∇xh=−l
2φn+1
h−φn
h
k,x
h,∀xh∈Xh,(6.2)
⎧
⎪
⎨
⎪
⎩cn+1
h−cn
h
k,x
h+Kh
2(φn+1
h)∇cn+1
h,∇xh
=−MKh
2(φn+1
h)[P0cn
h]T(1 −[P0cn
h]T)∇φn
h,∇xh,∀xh∈Xh.
(6.3)
The condi ional s abili y o scheme (6.1)–(6.3) will be ob ained by induc ion on he ime s ep n. Fi s o all,
we es ablish he ollowing esul which p o ides a basic ecu si e inequali y.
Lemma 6.1. Assume he cons ain :
(S) lim
(h,k)→0k/h =0.
I he e exis s a cons an Cd>0(independen o h,kand n,bu dependen onε) such ha
φn
h2
H1(Ω) +2CVβ
lε2|θn
h|2+|cn
h|2+1≤Cd,(6.4)
hen he ollowing inequali ies hold o (h, k)sufficien ly small (independen o n),
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
φn+1
h2
H1(Ω) −φn
h2
H1(Ω) +φn+1
h−φn
h2
H1(Ω)
+2CVβ
lε2|θn+1
h|2−|θn
h|2+1
2|θn+1
h−θn
h|2
+2ΩF(φn+1
h)−F(φn
h)+ 1
8ε2((φn+1
h)2−(φn
h)2)2
+α
2k
φn+1
h−φn
h
k
2
+2β
lε2k|Kh
1(φn+1
h)∇θn+1
h|2
≤R1k|φn
h|2+R2
β2
ε4k(|cn
h|2+1),
(6.5)
|cn+1
h|2−|cn
h|2+|cn+1
h−cn
h|2+k|Kh
2(φn+1
h)∇cn+1
h|2≤R1k|∇φn
h|2,(6.6)
whe e R1and R2a e posi i e cons an s independen o h,k,n,andε.
P oo . Fi s ly, we conside xh=2kφn+1
h−φn
h
kin (6.1) and bound he igh -hand side
3
2αk 
φn+1
h−φn
h
k
2
+φn+1
h2
H1(Ω) −φn
h2
H1(Ω) +φn+1
h−φn
h2
H1(Ω)
+2
 (φn
h),φ
n+1
h−φn
h≤2β
ε2kθn
h,φn+1
h−φn
h
k+C
αk|φn
h|2+Cβ2
αε4k(|cn
h|2+1).(6.7)
580 F. GUILL´
EN-GONZ´
ALEZ AND J.V. GUTI´
ERREZ-SANTACREU
Now, we handle he las e m on he le -hand side o (6.7) as ollows:
2φn+1
h−φn
h, (φn
h)=1
ε2φn+1
h−φn
h,((φn+1
h)2−1)φn
h+1
ε2φn+1
h−φn
h,((φn
h)2−(φn+1
h)2)φn
h:= I1−I2.
Nex ,wecon inue ew i ingI1as ollows:
I1=1
2ε2Ω
((φn+1
h)2−1)((φn+1
h)2−(φn
h)2−(φn+1
h−φn
h)2)
=1
4ε2Ω((φn+1
h)2−1)2−((φn
h)2−1)2+((φn+1
h)2−(φn
h)2)2
+k2
2ε2Ω
(1 −(φn+1
h)2)
φn+1
h−φn
h
k
2
·(6.8)
The e m I2is bounded as
I2≤C1
ε2k2φn
h2
L∞(Ω) 
φn+1
h−φn
h
k
2
+1
8ε2Ω
((φn+1
h)2−(φn
h)2)2.
The e o e, we ge om (6.7) and he p e ious compu a ions
3
2αk
φn+1
h−φn
h
k
2
+φn+1
h2
H1(Ω) −φn
h2
H1(Ω) +φn+1
h−φn
h2
H1(Ω)
+2ΩF(φn+1
h)−F(φn
h)+ 1
8ε2((φn+1
h)2−(φn
h)2)2+k2
2ε2
φn+1
h−φn
h
k
2
≤2β
ε2kθn
h,φn+1
h−φn
h
k+C
αk|φn
h|2+Cβ2
ε4αk(|cn
h|2+1)
+C1
ε2αkφn
h2
L∞(Ω) +φn+1
h2
L∞(Ω)k
φn+1
h−φn
h
k
2
≤2β
ε2kθn
h,φn+1
h−φn
h
k+C
αk|φn
h|2+Cβ2
ε4k(|cn
h|2+1)
+C1
ε2
k
hφn
h2
H1(Ω) +φn+1
h2
H1(Ω)k
φn+1
h−φn
h
k
2
,
(6.9)
whe e in he las line he in e se es ima e xhL∞(Ω) ≤Ch
−1/2xhH1(Ω) has been used.
Now we a e looking o he bound φn+1
hH1(Ω) ≤C1whe e C1>0 depends on he cons an Cdo hypo he-
sis (6.4) bu i will be independen o n. I will be ca ied ou by bounding φn+1
hH1(Ω) in e ms o φn
hH1(Ω),
|θn
h|and |cn
h|and using hypo hesis (6.4). Indeed, aking again xh=2kφn+1
h−φn
h
kas a es unc ion in (6.1),
STABILITY AND CONVERGENCE OF TWO DISCRETE SCHEMES 581
bu now bounding di ec ly he e m depending on (φn
h) on he igh -hand side, we ge
αk
φn+1
h−φn
h
k
2
+φn+1
h2
H1(Ω) −φn
h2
H1(Ω) +φn+1
h−φn
h2
H1(Ω)≤Cβ2
ε4αk|θn
h|2
+C
αk|φn
h|2+Cβ2
ε4αk(|cn
h|2+1)+Ck| (φn
h)|2
≤Cβ2
ε4αk|θn
h|2+C
αk|φn
h|2+Cβ2
ε4αk(|cn
h|2+1)
+C1
ε4kφn
h6
H1(Ω) +C1
ε4kφn
h2
H1(Ω).
In pa icula , by using hypo hesis (6.4), he p e ious inequali y says us
φn+1
h2
H1(Ω) ≤φn
h2
H1(Ω) +Ck
Cdβ
ε2+Cd+Cdβ2
ε4+C3
d
ε4+Cd
ε4≤φn
h2
H1(Ω) +C1(ε)k
wi h C1(ε) independen o h,kand n.
Thus, by using he p e ious es ima e in (6.9) and again hypo hesis (6.4), we ge
3
2αk
φn+1
h−φn
h
k
2
+φn+1
h2
H1(Ω) −φn
h2
H1(Ω) +φn+1
h−φn
h2
H1(Ω)
+2ΩF(φn+1
h)−F(φn
h)+ 1
8ε2((φn+1
h)2−(φn
h)2)2
≤2β
ε2kθn
h,φn+1
h−φn
h
k+C
αk|φn
h|2+Cβ2
αε4k(|cn
h|2+1)+Ck
h
1
ε2Cd+C1(ε)kk
φn+1
h−φn
h
k
2
·
By aking in o accoun he cons ain (S), in pa icula lim
(h,k)→0
k
h
1
ε2Cd+C1(ε)k= 0, so o any (h, k)small
enough such ha Ck
h
1
ε2Cd+C1(ε)k≤1
2α, he las e m on he igh -hand side can be abso bed, and emains
αk
φn+1
h−φn
h
k
2
+φn+1
h2
H1(Ω) −φn
h2
H1(Ω) +φn+1
h−φn
h2
H1(Ω)+2ΩF(φn+1
h)−F(φn
h)
+1
8ε2((φn+1
h)2−(φn
h)2)2≤2β
ε2kθn
h,φn+1
h−φn
h
k+C
αk|φn
h|2+Cβ2
αε4k(|cn
h|2+1).(6.10)
On he o he hand, ake xh=4β
lε2kθn+1
hin (6.2) o a i e a inequali y (3.5), ha is
2CVβ
lε2|θn+1
h|2−|θn
h|2+1
2|θn+1
h−θn
h|2+4β
lε2k|Kh
1(φn+1
h)∇θn+1
h|2≤
−2β
ε2kφn+1
h−φn
h
k,θn
h+α
2k
φn+1
h−φn
h
k
2
·(6.11)
Consequen ly, i suffices o add up (6.11)and(6.10) oge (6.5).
Finally, inequali y (6.6) is easily ob ained by es ing (6.3)bycn+1
hand bounding adequa ely as in he p oo
o Lemma 3.1.
582 F. GUILL´
EN-GONZ´
ALEZ AND J.V. GUTI´
ERREZ-SANTACREU
On he o he hand, we u n ou a en ion o he ini ial bound (3.2) which in pa icula e ifies hypo hesis (6.4)
imposed in Lemma 6.1. I is e y impo an in o de o gua an ee a co ec induc ion a gumen .
Now, we a e in posi ion o gi e he ollowing s abili y esul .
Lemma 6.2. Unde he hypo heses o Lemma 6.1, he disc e e solu ion o scheme (6.1)–(6.3)sa isfies he
ollowing es ima es:
(i)max
0≤n≤Nφn
h2
H1(Ω) ≤C, (ii)
N−1

n=0
φn+1
h−φn
h2
H1(Ω) ≤C, (iii)k
N−1

n=0 
φn+1
h−φn
h
k
2
≤C,
(i )max
0≤n≤N|θn
h|2≤C, ( )
N−1

n=0
|θn+1
h−θn
h|2≤C, ( i)k
N−1

n=0
|Kh
1(φn+1
h)∇θn+1
h|2≤C,
( ii)max
0≤n≤N|cn
h|2≤C, ( iii)
N−1

n=0
|cn+1
h−cn
h|2≤C, (ix)k
N−1

n=0
|Kh
2(φn+1
h)∇cn+1
h|2≤C,
whe e C>0is independen o (h, k)and depends on he da a (φ0,θ
0,c
0),α, β and ε.
P oo . Ob iously, i we le (6.5)and(6.6) hold o n=0, ..., N −1, we ge all he s a emen s o his lemma
by adding (6.5)and(6.6) and applying he disc e e G onwall lemma. The e o e, i suffices o p o e ha (6.5)
and (6.6) hold o n=0, ..., N −1.
Le us conside Cd=e
(R1+β2
ε4R2)TC2/ε2wi h C2>0 gi en in (3.2)andR1,R
2gi en in Lemma 6.1.As he
ini ial app oxima ions hold hypo hesis (6.4) o n= 0, inequali ies (6.5)and(6.6) a e sa isfied o n=0.
The final induc ion s ep can be easily seen by assuming ha inequali ies (6.5)and(6.6) hold o l=0, ..., n−1.
Then, adding up (6.5)and(6.6) om0 on−1, one has
2CVβ
lε2|θn
h|2+|cn
h|2+1+φn
h2
H1(Ω) +2Ω
F(φn
h)≤2CVβ
lε2|θ0
h|2+|c0
h|2+1+φ0
h2
H1(Ω)
+2Ω
F(φ0
h)+ k
n−1

l=0 R1φl
h2
H1+R2
β2
ε4(|cl
h|2+1)
.
Now, he disc e e G onwall lemma and (3.2) yield
2CVβ
lε2|θn
h|2+|cn
h|2+1+φn
h2
H1(Ω) +2Ω
F(φn
h)≤e(R1+β2
ε4R2)(n−1) k2CVβ
lε2|θ0
h|2+|c0
h|2
+1+φ0
h2
H1(Ω) +2Ω
F(φ0
h)≤e(R1+β2
ε4R2)TC2/ε2:= Cd.
Then, we find ha hypo hesis (6.4) is sa isfied. The e o e, in iew o Lemma 6.1, inequali ies (6.5)and(6.6)
hold. 
No e ha he s abili y es ima es ob ained in Lemma 6.2 a e o o de O((1/ε2)e(β2/ε4)) o he a iable
(φ, √β
εθ,c) as in he nonlinea scheme, bu now an adequa e cons ain o (h, k) small enough (depending
exponen ially on 1/ε) is necessa y ( ecall ha in he nonlinea scheme, only βk
ε2small enough was imposed).
To finish he p oo o Theo em 1.4 i is necessa y o p o e he con e gence o he linea scheme (6.1)–(6.3).
Bu , as he a gumen o his is simila o ha de eloped o he nonlinea scheme (2.1)–(2.3), i is le o he
eade . 
STABILITY AND CONVERGENCE OF TWO DISCRETE SCHEMES 583
7. E o es ima es o he non-degene a e case
In his sec ion we deal wi h he e o analysis o bo h linea and nonlinea scheme. The p esence o he
unca ion ope a o applied o he piecewise cons an ope a o P0makes nons anda d his e o analysis and
his pa icula unca ion is esponsible o o de O(h) in e o es ima es, al hough highe -o de fini e elemen s
we e conside ed.
In o de o be able o gua an ee a sufficien egula solu ion o p oblem (1.1)–(1.3) we assume he non-
degene a e case. Fo simplici y, we assume ha K1and K2a e posi i e cons an s, p o iding in pa icula
s anda d Neumann bounda y condi ions in (1.2).
Le {Th},0<h≤1, be a egula , quasi-uni o m amily o subdi isions o a polyhed al domain Ω ⊂Rm,
m= 2 o 3, whose bounda y Γ is such ha he p oblem
−Δu+u= in Ω,∂u
∂n= 0 on Γ (7.1)
holds he s abili y p ope y uH2(Ω) ≤| |, o each ∈L2(Ω). Recall ha , bo h p e ious hypo heses a e also
assumed in Theo em 1.3 o p o e he con e gence.
Define he global e o en
φ=φn
h−φ( n), en
θ=θn
h−θ( n), and en
c=cn
h−c( n). These e o s a e decomposed
in o a disc e e e o e·,d, and an in e pola ion e o e·,i as ollows
en
φ,d =φn
h−P1
hφ( n),e
n
φ,i =P1
hφ( n)−φ( n),
en
θ,d =θn
h−P0
hθ( n),e
n
θ,i =P0
hθ( n)−θ( n),
en
c,d =cn
h−P0
hc( n),e
n
c,i =P0
hc( n)−c( n),
whe e P1
h:H1→Xhis he H1-p ojec ion ope a o defined as
ψ−P1
hψ,xh+∇ψ−∇P1
hψ,∇xh=0 ∀xh∈Xh,
and P0
h:L2→Xhis he L2-p ojec ion ope a o defined as
ψ−P0
hψ,xh=0 ∀xh∈Xh.
Finally, le us ecall some app oxima ion p ope ies o P1
hand P0
h o be used la e on (see [6], P op. 1.134,
p. 73):
ψ−P0
hψH1(Ω) ≤ChψH2(Ω),ψ−P1
hψH1(Ω) ≤ChψH2(Ω),
|ψ−P0ψ|≤ChψH1(Ω),|ψ−P1
hψ|≤ChψH1(Ω).
In pa icula , om he las inequali y, one has
k|δ ψ( n+1)−δ P1
hψ( n+1)|2≤Ch2 n+1
n
ψ 2
H1(Ω),
whe eweuseδ o deno e he disc e e backwa d Eule ime de i a i e, ha is
δ ψ( n+1)=ψ( n+1)−ψ( n)
k·
No e ha , o he H1-in e pola ion e o o P0
h, a quasi-uni o m amily o fini e elemen s mus be assumed
and, o he L2-in e pola ion e o o P1
h, a duali y a gumen is equi ed whe e he ellip ic H2- egula i y o he
Ellip ic-Neumann p oblem (7.1)isimposed.

584 F. GUILL´
EN-GONZ´
ALEZ AND J.V. GUTI´
ERREZ-SANTACREU
Th oughou he sec ion we assume a egula solu ion o (1.1)–(1.3). Conc e ely, one assumes
φ∈L∞(0,T;W1,∞(Ω)) ∩L2(0,T;H2(Ω)),φ
∈L2(0,T;H1(Ω),φ
∈L2(0,T;L2(Ω)),
θ∈L∞(0,T;H1(Ω)) ∩L2(0,T,H2(Ω)),θ
∈L2(0,T;L2(Ω)),θ
∈L2(0,T;H1(Ω)),
c∈L∞(0,T;H1(Ω)) ∩L2(0,T,H2(Ω)),c
∈L2(0,T;L2(Ω)),c
∈L2(0,T;H1(Ω)).
(7.2)
7.1. E o es ima es o henonlinea scheme
We now s a e ou e o es ima es o he ully disc e e nonlinea scheme (2.1)–(2.3). I we compa e he exac
p oblem wi h he scheme and use he equali y a3−b3=(a−b)3+3ab(a−b), hen he e o equa ions a e
gi en by ⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
αδ en+1
φ,d ,x
h+∇en+1
φ,d ,∇xh+en+1
φ,d ,x
h+1
2ε2(en+1
φ,d )3,x
h
=−αδ en+1
φ,i ,x
h−1
2ε2(en+1
φ,i )3,x
h−3
2ε2P1
h(φ( n+1))φ( n+1)en+1
φ,i ,x
h
−3
2ε2φn+1
hP1
h(φ( n+1))en+1
φ,d ,x
h+1
2ε2en
φ,d +en
φ,i,x
h
+en
φ,d +en
φ,i,x
h+β
ε2en
θ,d,x
h−β
ε2(θA+θB)en
c,d,x
h+Rn+1
φ,x
h,
(7.3)
⎧
⎪
⎨
⎪
⎩
CVδ en+1
θ,d ,x
h+K1∇en+1
θ,d ,∇xh
=−K1∇en+1
θ,i ,∇xh−l
2δ en+1
φ,x
h+Rn+1
θ,x
h,
(7.4)
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
δ en+1
c,d ,x
h+K2∇en+1
c,d ,∇xh=−K2∇en+1
c,i ,∇xh
−MK
2[P0cn
h]T(1 −[P0cn
h]T)∇en
φ,∇xh
−MK
2(1 −[P0cn
h]T)en
cT∇φ( n),∇xh
+MK
2c( n)en
cT∇φ( n),∇xh+Rn+1
c,x
h,
(7.5)
whe e δ en+1 =(en+1 −en)/k,en
cT=[P0cn
h]T−c( n)and
Rn+1
φ=α
k n+1
n
( n− )φ ( )d +1
2ε2+1
 n+1
n
φ ( )d
+β
ε2 n+1
n
θ ( )d +β
ε2(θA+θB) n+1
n
c ( )d , (7.6)
Rn+1
θ=CV
k n
n
( n− )θ ( )d −l
2k n+1
n
( n− )φ ( )d ,
Rn+1
c=1
k n+1
n
( n− )c ( )d +∇·1−c( n+1)c( n+1) n+1
n
∇φ ( )
+1−(c( n)+c( n+1) n+1
n
c ( )∇φ( n).
Theo em 7.1. Unde he assump ions o Theo em 1.3, i he solu ion (φ, θ, c)o (1.1)–(1.3)sa isfies ha
(φ( ),θ( ),c( )) ∈H2(Ω)3 o all ∈[0,T]and he egula i y gi en in (7.2), hen he ollowing e o es ima es
hold o ksmall enough:
max
0≤n≤Nen+1
φ2
H1(Ω) +|en+1
θ|2+|en+1
c|2+k
N−1

n=0 |δ en+1
φ|2+|∇en+1
θ|2+|∇en+1
c|2≤Ch2+k2,(7.7)
STABILITY AND CONVERGENCE OF TWO DISCRETE SCHEMES 585
whe e he cons an C>0depends on he exac solu ion, bu is independen o (h, k).
P oo . Se xh=2kδ en+1
φ,d ∈Xhas a es unc ion in (7.3), and using he equali ies
a3(a−b)=1
2a2(a2−b2+(a−b)2)=1
4(a4−b4+(a2−b2)2)+1
2a2(a−b)2
o a=en+1
φand b=en
φ,wege
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
α2kδ en+1
φ,d 
2+en+1
φ,d 2
H1(Ω) −en
φ,d2
H1(Ω) +en+1
φ,d −en
φ,d2
H1(Ω)
+1
4ε2en+1
φ,d 4
L4(Ω) −en
φ,d4
L4(Ω) +|(en+1
φ,d )2−(en
φ,d)2|2+2|(en+1
φ,d )(en+1
φ,d −en
φ,d)|2
=−α2kδ en+1
φ,i ,δ
en+1
φ,d −1
ε2k(en+1
φ,i )3,δ
en+1
φ,d 
−3
ε2kP1
h(φ( n+1))φ( n+1)en+1
φ,i ,δ
en+1
φ,d −3
ε2kφn+1
hP1
h(φ( n+1))en+1
φ,d ,δ
en+1
φ,d 
+1
ε2ken
φ,d +en
φ,i,δ
en+1
φ,d +2ken
φ,d +en
φ,i,δ
en+1
φ,d 
+2β
ε2ken
θ,d,δ
en+1
φ,d −2β
ε2(θA+θB)ken
c,d,δ
en+1
φ,d +2kRn+1
φ,δ
en+1
φ,d :=
9

i=1
Ii.
(7.8)
Now, we mus bound each e m on he igh -hand side o (7.8). We jus ocus on he e ms I2,I3and I4:
I2≤Cλ1
ε4αken+1
φ,i 6
L6(Ω) +λkα|δ en+1
φ,d |2≤Cλ1
ε4αkh
6φ( n+1)6
H2(Ω) +λkα|δ en+1
φ,d |2,
I3≤3
ε2kP1
h(φ( n+1))L6(Ω)φ( n+1)L6(Ω)en+1
φ,i L6(Ω)|δ en+1
φ,d |
≤Cλ
1
ε4αkφ( n+1)4
H1(Ω)en+1
φ,i 2
H1(Ω) +λkα|δ en+1
φ,d |2
≤Cλ
1
ε4αkh
2φ( n+1)4
H1(Ω)φ( n+1)2
H2(Ω) +λkα|δ en+1
φ,d |2,
I4≤3
ε2kφn+1
hL6(Ω)P1
h(φ( n+1))L6(Ω)en+1
φ,d L6(Ω)|δ en+1
φ,d |
≤Cλ
1
ε4αkφ( n+1)2
H1(Ω)en+1
φ,d 2
H1(Ω) +λkα|δ en+1
φ,d |2.
In he las line, he s abili y es ima e φn+1
hL6(Ω) ≤Cgi eninLemma3.1 has been applied. No e ha he
p e ious inequali ies hold o any λ>0andCλ>0 a e diffe en cons an s o o de O(1/λ). Finally, he
cons an s Cλbounding I2and I3a e independen o ε, bu he cons an Cλ ela ed o I4depends on ε ia he
bound o Lemma 3.1.
The emainde o he e ms can be bounded mo e easily. Thus, by choosing λsmall enough, we ge
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αkδ en+1
φ,d 
2+en+1
φ,d 2
H1(Ω) −en
φ,d2
H1(Ω) +en+1
φ,d −en
φ,d2
H1(Ω)
+1
4ε2en+1
φ,d 4
L4(Ω) −en
φ,d4
L4(Ω) +|(en+1
φ,d )2−(en
φ,d)2|2+2|en+1
φ,d (en+1
φ,d −en
φ,d)|2
≤Ckh
2h4φ( n+1)6
H2(Ω) +φ( n+1)4
H1(Ω)φ( n+1)2
H2(Ω) +φ( n)2
H1(Ω)
+Ck 1
ε4αβ2|en
c,d|2+β2CV|en
θ,d|2+en
φ,d2
H1(Ω) +φ( n+1)2
H1(Ω)en+1
φ,d 2
H1(Ω)
+Ch
2 n+1
n
φ (s)2
H1(Ω) +Ck
2 n+1
n|φ (s)|2+|θ (s)|2+|c (s)|2+|φ (s)|2ds,
(7.9)
whe e C>0 a e diffe en cons an s independen o (h, k) and independen o he exac solu ion (φ, θ, c).
586 F. GUILL´
EN-GONZ´
ALEZ AND J.V. GUTI´
ERREZ-SANTACREU
We now es (7.4)wi h2ken+1
θ,d and bound he igh -hand side
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CV|en+1
θ,d |2−|en
θ,d|2+|en+1
θ,d −en
θ,d|2+2K1k|∇en+1
θ,d |2
=−2K1k∇en+1
θ,i ,∇en+1
θ,d −lkδ en+1
φ,d +δ en+1
φ,i ,e
n+1
θ,d +2kRn+1
θ,e
n+1
θ,d 
≤Ckh
2θ( n+1)2
H2(Ω) +K1k|∇en+1
θ,d |2+Ch
2 n+1
n
φ (s)2
H1(Ω)ds
+α
2k|δ en+1
φ,d |2+CC
Vk|en+1
θ,d |2+Ck
2 n+1
n
(θ (s)2
H1(Ω)+φ (s)2
H1(Ω))ds.
(7.10)
Le us now ake xh=2ken+1
c,d as a es unc ion in o (7.5),
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|en+1
c,d |2−|en
c,d|2+|en+1
c,d −en
c,d|2+2K2k|∇en+1
c,d |2
=−2K2k∇en+1
c,i ,∇en+1
c,d −2MK
2k[P0cn
h]T(1 −[P0cn
h]T)∇(en
φ,d +en
φ,i),∇en+1
c,d 
−2MK
2k(1 −[P0cn
h]T)en
cT∇φ( n),∇en+1
c,d 
+2MK
2kc( n)en
cT∇φ( n),∇en+1
c,d +2kRn+1
c,e
n+1
c,d .
We fi s ly bound he unca ed e o
|en
cT|2=|[P0cn
h]T−P0c( n)+P0c( n)−c( n)|2
≤C|[P0cn
h]T−P0c( n)|2+|P0c( n)−c( n)|2
≤C|P0cn
h−P0c( n)|2+h2|∇c( n)|2
≤C|en
c,d|2+|en
c,i|2+h2|∇c( n)|2
≤C|en
c,d|2+h2c( n)2
H1(Ω),
(7.11)
whe e in he las line we ha e used he s abili y p ope y |P0ψ|≤|ψ|. No e ha he in e pola ion e o
|P0c( n)−c( n)|appea ing in (7.11)isonlyo o de O(h), independen o he fini e elemen app oxima ion.
By i ue o (7.11), we bound
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|en+1
c,d |2−|en
c,d|2+|en+1
c,d −en
c,d|2+K2k|∇en+1
c,d |2
≤Ckh
2c( n+1)2
H2(Ω) +Ck|∇en
φ,d|2+Ckh
2∇φ( n)2
L∞(Ω)c( n)2
H1(Ω)
+Ckh
2φ( n)2
H2(Ω) +Ck∇φ( n)2
L∞(Ω)|en
c,d|2
+Ck
2 n+1
n
|∇φ (s)|2ds
+Ck
2∇φ( n+1)2
L∞ n+1
n
|c (s)|2ds+Ck
2 n+1
n
c (s)2
H1(Ω)ds.
(7.12)
Again, C>0 a e diffe en cons an s independen o (h, k) and independen o he exac solu ion (φ, θ, c).
STABILITY AND CONVERGENCE OF TWO DISCRETE SCHEMES 587
By adding (7.9), (7.10)and(7.12) and applying he gene alized disc e e G onwall lemma, we es ablish he
ollowing es ima e o all n<N and o ksmall enough:
en+1
φ,d 2
H1(Ω) +1
4ε2en+1
φ,d 4
L4(Ω) +CV|en+1
θ,d |2+|en+1
c,d |2+αk
n

l=0 δ el+1
φ,d 
2
+K1k
n

l=0
|∇el+1
θ,d |2+K2k
n

l=0
|∇el+1
c,d |2≤exp(C1T)C2h2+C3k2,(7.13)
whe e
C1=
C1+φ2
L2(0,T;W1,∞(Ω))
1−Ckmax{1,φ2
L∞(0,T;H1(Ω))}
C2=Cφ4
L∞(0,T;H1(Ω))φ2
L2(0,T;H2(Ω)) +φ2
L2(0,T;H2(Ω))
+φ 2
L2(0,T;H1(Ω)) +θ2
L2(0,T;H2(Ω))
+φ2
L2(0,T;W1,∞(Ω))c2
L∞(0,T;H1(Ω)) +c2
L2(0,T;H2(Ω)),
and
C3=Cφ 2
L2(0,T;L2(Ω)) +θ 2
L2(0,T;L2(Ω)) +c 2
L2(0,T;L2(Ω)) +φ 2
L2(0,T;L2(Ω))
+θ 2
L2(0;T;H1(Ω))+φ 2
L2(0,T;H1(Ω))+φ 2
L2(0,T;H1(Ω))
+φ2
L∞(0,T;W1,∞(Ω))c 2
L2(0,T;L2(Ω))ds+c 2
L2(0,T;H1(Ω)),
wi h C>0 independen o (h, k) and independen o he exac solu ion (φ, θ, c).
The same es ima es a e ob ained o he o al e o s by using he in e pola ion e o s; hence (7.7)canbe
deduced. 
Rema k 7.2. Obse e ha , o ob ain e o es ima es in he p e ious heo em, he mono ony p ope y (anal-
ogous o (3.8))
1
2ε2((en+1
φ,d )3−en
φ,d)(en+1
φ,d −en
φ,d)≥F(en+1
φ,d )−F(en
φ,d)
is no used, because he co esponding ini ial e m is
Ω
F(e0
φ,d)= 1
8ε2Ω
((e0
φ,d)2−1)2=O1
ε2·
The e o e, by using his p ope y, he e o es ima es o o de O(k+h) does no hold, because in his case he
final bound emains o o de O(k+h+1/ε2).
7.2. E o es ima es o helinea scheme
As he de i a ion o he e o equa ions o θn+1
hand cn+1
his exac ly he same as in he p e ious nonlinea
scheme, we only ea in his sec ion he e o equa ion o φn+1
h,whichisgi enby
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αδ en+1
φ,d ,x
h+∇en+1
φ,d ,∇xh+en+1
φ,d ,x
h+1
2ε2(en
φ,d)3,x
h
=−αδ en+1
φ,i ,x
h−1
2ε2(en
φ,i)3,x
h−3
2ε2P1
h(φ( n))φ( n)en
φ,i,x
h
−3
2ε2φn
hP1
h(φ( n))en
φ,d,x
h+1
2ε2en
φ,d +en
φ,i,x
h
+en
φ,d +en
φ,i,x
h+β
ε2en
θ,d −(θA+θB)en
c,d,x
h+Rn+1
φ+Sn+1
φ,x
h
(7.14)