hp-DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR PROBLEMS
V´
ı Dolejˇ
s´
ı
Cha les Uni e si y P ague
Facul y o Ma hema ics and Physics
Depa men o Nume ical Ma hema ics
Sokolo sk´
a 83, 186 75 P aha, Czech Republic
[email p o ec ed] .cuni.cz
Abs ac
We deal wi h a nume ical solu ion o nonlinea con ec ion-di usion p oblems wi h he aid o
he discon inuous Gale kin ini e elemen (DGFE) me hod. We p opose a new hp-adap a ion
echnique, which is based on a combina ion o a esiduum-noncon o mi y es ima o and a e-
gula i y indica o . The esiduum-noncon o mi y es ima o consis s o wo building blocks ( he
esiduum e o indica o and he alue o he noncon o mi y). The es ima o ma ks mesh ele-
men s o a e inemen . The egula i y indica o decides i he ma ked elemen s will be e ined
by h- o p- echnique. The esiduum-noncon o mi y es ima o as well as he egula i y indica o
a e easily compu able quan i ies. Mo eo e , he same echnique es ima es an algeb aic e o
a ising om an i e a i e solu ion o he co esponding nonlinea algeb aic sys em. The pe o -
mance o he p oposed hp-DGFE me hod is demons a ed by se e al nume ical examples.
Keywo ds: hp-discon inuous Gale kin ini e elemen me hod; esiduum-noncon o mi y indi-
ca o ; egula i y es ima o ; algeb aic e o .
In oduc ion
Ou aim is o de elop a su icien ly obus , e icien and accu a e nume ical scheme o he
simula ion o iscous comp essible lows. The discon inuous Gale kin ini e elemen (DGFE)
me hods ha e become e y popula nume ical echniques o he solu ion o he comp essible
Na ie -S okes equa ions. Recen p og ess o he use o he DG me hod o comp essible low
simula ions can be ound in [11].
In his pape , we deal wi h a model p oblem ep esen ed by a scala nonlinea con ec ion-
di usion equa ion. We ocus on a hp-adap a ion s a egy o DGFE me hods which signi ican ly
inc ease he accu acy and e iciency o he compu a ion, see, e.g. [2, 6, 8, 12, 13].
The p oposed s a egy is based on a combina ion o a esiduum-noncon o mi y es ima o
and a egula i y indica o . The esiduum-noncon o mi y es ima o gi es a lowe es ima e o
he e o measu e consis ing o he e o measu ed in a dual no m and he quan i y measu ing
a iola ion o he con o mi y o he solu ion. This es ima o is locally de ined o each mesh
elemen , i is easily compu able and i s implemen a ion is e y simple. The egula i y indica o
is based on he in eg a ion o in e elemen jumps o he app oxima e solu ion o e he elemen
bounda y. Taking in o accoun esul s om a p io i e o analysis, we de ine he egula i y
indica o . I his alue is smalle han one hen we apply a p- e inemen o he wise we use
ah- e inemen . Bo h e inemen s (hand p) a e only iso opic, an aniso opic adap a ion will be
a subjec o he u he esea ch.
56
1 P oblem Fo mula ion
1.1 Go e ning Equa ions
We conside a o mal nonlinea pa ial di e en ial equa ion
Lu =0 in Ω,(1a)
u=uDon
∂
ΩD,(1b)
Nu =gNon
∂
ΩN,(1c)
whe e u:Ω→Ris he unknown scala unc ion de ined on Ω∈Rd,d=2,3, Lis a o mal
second o de di e en ial ope a o and (1b) and (1c) o mally ep esen he Di ichle and Neu-
mann bounda y condi ions on he pa s o bounda y
∂
ΩDand
∂
ΩN, espec i ely. We assume
ha he e exi s a unique weak solu ion o (1) which we deno e again by u.
Le Th(h>0) be a pa i ion o he closu e Ωo he domain Ωin o a ini e numbe o closed
d-dimensional simplicies Kwi h mu ually disjoin in e io s. We call Th={K}K∈Tha iangu-
la ion o Ωand do no equi e he con o ming p ope ies om he ini e elemen me hod.
O e he iangula ion Thwe de ine he so-called
b oken Sobole space
Hs(Ω,Th):={ ; |K∈Hs(K)∀K∈Th},s≥0 (2)
wi h he semino m | |Hs(Ω,Th):=∑K∈Th| |2
Hs(K)1/2,whe e |·|Hs(K)deno es he semino m o
he Sobole space Hs(K),K∈Th.
Mo eo e , o each K∈Th, we assign a posi i e in ege pK(=local polynomial deg ee). Then
we de ine he se
ph:={pK,K∈Th}.(3)
We de ine he ini e dimensional subspace o H1(Ω,Th)which consis s o discon inuous piece-
wise polynomial unc ions associa ed wi h he ec o phby
Shp={ ; ∈L2(Ω), |K∈PpK(K)∀K∈Th},(4)
whe e PpK(K)deno es he space o all polynomials on Ko deg ee ≤pK,K∈Th.
We in oduce a o mal disc e iza ion o (1) wi h he aid o DGFE me hod. Hence, le
˜ch(u, ):H2(Ω,Th)×H2(Ω,Th)→R(5)
be he co esponding o m which is nonlinea wi h espec o i s i s a gumen and linea wi h
espec o he second one. We say ha he unc ion uh∈Shpis an app oxima e solu ion o (1),
i ˜ch(uh, h) = 0∀ h∈Shp.(6)
Mo eo e , we de ine he o m Nh:H1(Ω,Th)→Rby
Nh( ):=
2∑
Γ∈FI
hZΓh−1
Γ[[ ]]2dS+∑
Γ∈FD
hZΓh−1
Γ( −uD)2dS

1/2
,(7)
whe e uDis om (1b), FI
hdeno es he se o all in e io aces o Th,FD
hdeno es he se o all
aces o Thlying on
∂
ΩD,[[·]] is he jump o a unc ion om H1(Ω,Th)and hΓis he diame e
o a ace Γ.
Finally, we cha ac e ise he weak solu ion o (1).
57
Lemma 1.1 The ollowing implica ions a e alid:
i) Le u ∈H2(Ω)be he weak solu ion o (1) hen
˜ch(u, ) = 0∀ ∈H2(Ω,Th),(8a)
Nh(u) = 0.(8b)
ii) I u ∈H2(Ω,Th)sa is ies bo h condi ions o (8) hen u is he weak solu ion o (1).
The disc e e p oblem (6) ep esen s a sys em o Nh=dimShpnonlinea algeb aic equa ions.
We sol e i wi h he aid o a New on-like i e a i e me hod which gi es he solu ion ˜uh∈Shp
such ha ˜ch(˜uh, h)≈0∀ h∈Shp.
2 Residuum Es ima o
In his sec ion we in es iga e he disc e iza ion e o u−uhand he algeb aic e o ˜uh−uhin
a sui able (dual) no m and de ine es ima o s gi ing some in o ma ion abou hese e o s.
2.1 E o Measu e
Simila ly as in [3], ou p oposed e o measu e consis s o wo building blocks, which a e mo-
i a ed by Lemma 1.1, namely ela ions (8a) and (8b). Le X:=H2(Ω,Th)and k·kXis a no m
de ined on X, which will be speci ied la e . The i s building block is gi en by
Rh(uh):=sup
06= ∈X
˜ch(uh, )
k kX(9)
which de ines he esiduum e o in he dual no m o he app oxima e solu ion uh∈Shp⊂X
and i measu es a iola ion o (8a). Howe e , i is impossible o e alua e Rh(uh), since he
sup emum is aken o e an in ini e-dimensional space. The e o e, in ou app oach, we seek he
maximum o e some su icien ly la ge bu ini e dimension subspace o X, which is p esen ed
in Sec ion 2.2.
The second building block is based on (8b), which cha ac e ises a iola ion o he con o mi y
o he weak solu ion and a iola ion o he Di ichle bounda y condi ion. I is ep esen ed by he
alue Nh(uh)≥0 gi en by (7) which we call he noncon o mi y o he app oxima e solu ion.
In con as o Rh(uh) he quan i y Nh(uh)is di ec ly compu able om (7). Finally, ou e o
measu e is he sum o squa es o he esiduum e o and noncon o mi y, i.e.,
Eh(uh):= (Rh(uh)2+Nh(uh)2)1/2.(10)
Due o Lemma 1.1 we simply obse e ha Eh(uh) = 0 i and only i uh=u.
2.2 Global and Elemen Residuum Es ima o s
In he p e ious sec ion, we in oduced he e o measu e Eh(uh) = pRh(uh)2+Nh(uh)2o he
app oxima e solu ion uh∈Shp⊂X. Whe eas Nh(uh)is easy o e alua e, he quan i y Rh(uh)
has o be app oxima ed in a sui able way, which is p esen ed in his sec ion. Fo each K∈Th
and each in ege p≥0, we de ine he spaces
Sp
K:={
φ
h∈X,
φ
h|K∈Pp(K),
φ
h|Ω K=0}(11)
58
and S+
hp:={
φ
∈X;
φ
=∑
K∈Th
cK
φ
K,cK∈R,
φ
K∈SpK+1
K,K∈Th}.(12)
Ob iously, Shp⊂S+
hp⊂X.
Now, we de ine he elemen esiduum es ima o
ρ
h,K(uh):=sup
06=
ψ
h∈SpK+1
K
˜ch(uh,
ψ
h)
k
ψ
hkX=sup
ψ
h∈SpK+1
K,k
ψ
hkX=1
˜ch(uh,
ψ
h),uh∈X,(13)
o each K∈Thand he global esiduum es ima o
ρ
h(uh):=sup
06=
ψ
h∈S+
hp
˜ch(uh,
ψ
h)
k
ψ
hkX=sup
ψ
h∈S+
hp,k
ψ
hkX=1˜ch(uh,
ψ
h),uh∈X,(14)
which a e easily compu able quan i ies i k·kXis sui ably chosen, see Sec ion 2.4.
Ob iously, i u∈Xis he exac solu ion o (1) hen consis ency (8a) implies 0 =
ρ
h(u) =
ρ
h,K(u),K∈Th. Mo eo e , we ha e immedia ely a lowe bound
ρ
h(uh)≤Rh(uh),(15)
since
ρ
his he sup emum o e subspace S+
hp⊂X. Howe e , i is open i he e exis s an uppe
bound, i.e., Rh(uh)≤C
ρ
h(uh), whe e C>0. This will be he subjec o a u he esea ch.
2.3 Algeb aic Residuum Es ima o s
Simila ly as in he p e ious sec ion, we de ine he es ima o co esponding o he algeb aic
e o esiduum. Le ˜uh∈Shpbe he ou pu o he i e a i e p ocess used o he solu ion o (6).
We de ine he algeb aic esiduum es ima o
ρ
A
h(˜uh):=sup
06=
ψ
h∈Shp
˜ch(˜uh,
ψ
h)
k
ψ
hkX=sup
ψ
h∈Shp,k
ψ
hkX=1˜ch(uh,
ψ
h),(16)
which measu es he algeb aic e o ( he di e ence be ween ˜uhand uh), since
ρ
A
h(uh) = 0 due o
(6). The ela ions (14) and (16) gi e
ρ
A
h(˜uh)≤
ρ
h(˜uh), since in (14) he sup emum is aken o e
he la ge space.
We s op he i e a i e p ocess o he solu ion o he nonlinea algeb aic p oblem (6) i
ρ
A
h(˜uh)≤
βρ
h(˜uh),(17)
whe e
β
∈(0,1). In nume ical expe imen s we pu
β
≈0.01.
2.4 Choice o he No m k·kX
In o de o ensu e a as e alua ion o es ima o s
ρ
hand
ρ
A
h, we need o choose he no m k·kX
in a sui able way. We employ
k·kX:=
δ
k·k2
L2(Ω)+
ε
|·|2
H1(Ω,Th)1/2,(18)
which gua an ees ha
ρ
h(uh)2=∑
K∈Th
ρ
h,K(uh)2.(19)
He e
δ
and
ε
deno e he size o “con ec ion” and “di usion”, espec i ely.
The e o e, i is su icien o e alua e he elemen esiduum es ima o s
ρ
h,K,K∈Th. This is
a s anda d ask o seeking a cons ain ex ema o e SpK+1
Kwi h he cons ain k
ψ
hkX=1. This
can be done di ec ly e y as since he dimension o SpK+1
K,K∈This small.
59
2.5 Residuum-Noncon o mi y Es ima o s
We ha e al eady men ioned ha he second building block o he e o measu e is gi en by he
noncon o mi y Nh(uh)de ined by (7). Fo he pu pose o a mesh adap a ion, we de ine i s local
a ian
Nh,K( ):=
∑
Γ∈FI
h∩
∂
KZΓh−1
Γ[[ ]]2dS+∑
Γ∈FD
h∩
∂
KZΓh−1
Γ( −uD)2dS

1/2
, ∈H1(Ω,Th).
(20)
Ob iously, om (7) and (20), we ha e Nh( )2=∑K∈ThNh,K( )2.Finally, we de ine he local
and global esiduum-noncon o mi y es ima o s o he app oxima e solu ion uh∈Shpby
η
h,K(uh):=
ρ
h,K(uh)2+Nh,K(uh)21/2,K∈Th,(21a)
and
η
h(uh):=
ρ
h(uh)2+Nh(uh)21/2= ∑
K∈Th
η
h,K(uh)2!1/2
,(21b)
espec i ely. In i ue o (10), (15) and (21), we expec ha he global esiduum-noncon o mi y
es ima o
η
h(uh)app oxima es he e o measu e Eh(uh). In he ollowing we in oduce an adap-
a ion echnique which p oduces a hp-mesh and he co esponding app oxima e solu ion such
ha he es ima o
η
h(uh)is unde a gi en ole ance.
3hp-Adap a ion P ocess
In his sec ion, we p esen a new hp-adap i e DG echnique o he solu ion o (6). In Sec ion
2, we de ined he elemen and global esiduum-noncon o mi y es ima o s
η
h,Kand
η
h, espec-
i ely. We employ he no m k·kXgi en by (18) which gua an ees ha equali y (19) is alid. As
al eady men ioned, ou in e es is o ind he solu ion ˜uh∈Shpsuch ha
η
h(˜uh)≤
ω
,(22)
whe e
ω
>0 is a gi en ole ance.
Le Thbe a gi en mesh and ˜uh he co esponding app oxima ion o (6). We equi e ha
η
h,K(˜uh)≤
ω
√#Th∀K∈Th,(23)
whe e #Thdeno es he numbe o elemen s o Th. Ob iously, i (23) is sa is ied hen, due
o (21b), condi ion (22) is alid and he adap a ion p ocess s ops. O he wise, we ma k o
e inemen all K∈Th iola ing (23).
Fu he mo e, all ma ked elemen s will be e ined ei he by h- o by p-adap a ion, namely,
ei he we spli a gi en mo he elemen Kin o ou daugh e elemen s o we inc ease he deg ee
o polynomial app oxima ion o a gi en elemen . Thus he new mesh Tˆ
hand new se {ˆpK,K∈
Tˆ
h}a e c ea ed. We in e pola e he old solu ion on a new mesh and pe o m he nex adap a ion
s ep ill (22) is alid.
3.1 Regula i y Indica o
The es ima ion o he egula i y o he solu ion is an essen ial key o any hp-adap a ion s a egy.
Ou app oach is based on a measu e o in e -elemen jumps which is he base o he jump
indica o om [5] and he shock cap u ing echnique om [7].
60

We p opose he egula i y indica o
gK(uh):=R
∂
K∩Ω[[uh]]2dS
|K|h2pK−2
K
,K∈Th,(24)
whe e |K|is he a ea o K∈Th. I he exac solu ion is su icien ly egula , i.e., sK≥pK+1,
hen gK(uh)≈Oh2pK+1
K/h2
Kh2pK−2
K=O(h1
K).On he o he hand, i he exac solu ion is no
su icien ly egula , i.e., sK<pK+1 (⇔sK≤pK), hen gK(uh)≈Oh2sK−1
K/h2
Kh2pK−2
K=
O(h2
δ
−1
K),whe e
δ
=sK−pk≤0. Then we use he ollowing s a egy
gK(uh)≤1⇒solu ion is egula ⇒p- e inemen ,
gK(uh)>1⇒solu ion is i egula ⇒h- e inemen ,K∈Th.(25)
4 Nume ical Expe imen s
In he p e ious sec ions, we in oduced and de eloped he adap i e hp-DGFE me hod. We
demons a e i s pe o mance in his sec ion by se e al nume ical examples. Le ˜uhbe he ap-
p oxima e solu ion esul ing om an i e a i e me hod, i.e., he solu ion in luenced by he alge-
b aic e o . We deal wi h wo ollowing nume ical examples:
(E1) nonlinea con ec ion-di usion equa ion wi h a co ne singula i y om [9],
(E2) linea con ec ion-di usion equa ion wi h he s ong in e io laye and he exponen ial
bounda y laye om [10].
Fo he i s one, we know he exac solu ion and he e o e we a e able o e alua e he
compu a ional e o . We ca ied ou a hp-adap i e algo i hm s a ing on a mesh wi h he s ep
h0and P1polynomial app oxima ion. We e alua e ku−˜uhkX,Nh(˜uh)and
ρ
h(˜uh)wi h he
co esponding expe imen al o de s o con e gence (EOC) wi h espec o he numbe o deg ee
o eedom Nhde ined by
EOC =logehl+1−logehl
log(1/pNhl+1)−log(1/pNhl),l=1,2,..., (26)
Mo eo e , we e alua e he “e ec i i y index”
ie :=
η
h(˜uh)
ku−˜uhkX2+Nh(˜uh)21/2=
ρ
h(˜uh)2+Nh(˜uh)21/2
ku−˜uhkX2+Nh(˜uh)21/2.(27)
Le us no e ha he index ie is no a s anda d e ec i i y index since
ρ
h(˜uh)is he app oxima ion
o Rh(˜uh)and no o ku−˜uhkX. Ob iously, i Nh(˜uh)domina e
ρ
h(˜uh)and ku−˜uhkX hen
he index ie is close o one. In his case i would be also in e es ing o e alua e he a io
ρ
h(˜uh)/ku−˜uhkX.
4.1 (E1): Nonlinea Con ec ion-Di usion Equa ion wi h a Co ne Singula i y
We conside he scala nonlinea con ec ion-di usion equa ion
−∇·(K(u)∇u)−
∂
u2
∂
x1−
∂
u2
∂
x2=gin Ω:= (0,1)2,(28)
61
.00 .25 .50 .75 1.00
.00
.25
.50
.75
1.00
.00 .01 .03 .04 .05
.00
.01
.03
.04
.05
P
0
P
0
P
1
P
1
P
2
P
2
P
3
P
3
P
4
P
4
P
5
P
5
hp
Sou ce: Own
Fig. 1.Example (E1) gi en by (28) –(30) wi h
α
=−3/2: he inal g id wi h he co -
esponding deg ees o polynomial app oxima ion, he whole domain (le ) and i s de ail
(0,1/20)×(0,1/20)( igh )
whe e K(u)is he nonsymme ic ma ix gi en by
K(u) =
ε
2+a c an(u) (2−a c an(u))/4
0(4+a c an(u))/2.(29)
The pa ame e
ε
>0 plays a ole o an amoun o di usi i y and we pu
ε
=10−3. We p esc ibe
a Di ichle bounda y condi ion on
∂
Ωand se he sou ce e m gsuch ha he exac solu ion is
u(x1,x2) = (x2
1+x2
2)
α
/2x1x2(1−x1)(1−x2),
α
∈R.(30)
We p esen wo choices:
α
=4 and
α
=−3/2. I is possible o show (see [1]) ha u∈
H
κ
(Ω),
κ
∈(0,3+
α
), whe e H
κ
(Ω)deno es he Sobole -Slobode skii space o unc ions
wi h “non-in ege de i a i es”. Whe eas he choice
α
=4 gi es su icien ly egula solu ion,
he choice
α
=−3/2 leads o he solu ion wi h a singula i y a x1=x2=0. Nume ical exam-
ples p esen ed in [4], ca ied ou o a li le di e en p oblem, show ha his singula i y a oids
o achie e an o de o con e gence be e han O(h3/2)in he L2-no m and O(h1/2)in he
H1-semino m o any deg ee o polynomial app oxima ion. Ne e heless, he exac solu ion
is egula ou side o he singula i y.
Table 1 shows he esul s o p oblem (28) – (30) wi h
α
=4 and
α
=−3/2, namely he
alues o he e o ku−˜uhkX, noncon o mi y Nh(˜uh), esiduum e o es ima e
ρ
h(˜uh)wi h he
co esponding EOC, index ie and he compu a ional imes in seconds. We obse e ha he
compu a ional e o ku−˜uhkXcon e ge exponen ially o
α
=4 and signi ican ly as e han
O(h1/2) o
α
=−3/2. Fu he mo e, he index ie is e y close o one o inc easing Nhwhich
suppo s he accu acy o he me hod. A small inc ease o ie in Table 1 o
α
=4 o he las
adap a ion le el is caused by he ac ha we a e close o he machine accu acy.
Fu he mo e, Figu e 1 shows he inal hp-g id ob ained wi h he aid o he hp-DGFE algo-
i hm o
α
=−3/2. (The case
α
=4 is no in e es ing since only p- e inemen is ca ied ou
due o he egula i y o he exac solu ion.) We obse e ha he h-adap a ion was ca ied ou
in a small egion nea he singula i y. On he o he hand, he p-adap a ion appea s in egions
whe e he solu ion is egula .
62
Tab. 1.Example (E1) gi en by (28) –(30): e o ku−˜uhkX, noncon o mi y Nh(˜uh), esiduum
e o es ima e
ρ
h(˜uh)wi h he co esponding EOC, index ie and he compu a ional ime in
seconds
α
=4:
le #ThNhku−˜uhkXEOC Nh(˜uh)EOC
ρ
h(˜uh)EOC ie CPU(s)
0 128 384 6.45E-03 – 1.99E-02 – 5.58E-03 – 0.99 0.4
1 128 705 4.56E-04 8.72 3.07E-03 6.15 6.60E-04 7.03 1.01 0.7
2 128 384 6.45E-03 8.72 1.99E-02 6.15 5.58E-03 7.03 0.99 0.4
3 128 768 4.43E-04 7.73 3.01E-03 5.45 6.35E-04 6.27 1.01 0.7
4 128 1280 3.87E-05 9.54 2.75E-04 9.36 5.22E-05 9.78 1.01 1.0
5 128 1920 2.75E-06 13.05 1.73E-05 13.65 3.05E-06 14.02 1.00 1.4
6 128 2688 1.10E-07 19.12 7.04E-07 19.02 1.13E-07 19.59 1.00 2.2
7 128 3584 2.49E-09 26.35 1.62E-08 26.24 2.42E-09 26.72 1.00 3.4
8 128 4608 2.98E-11 35.22 2.23E-10 34.08 3.00E-11 34.92 1.00 5.3
9 128 5760 3.17E-15 82.03 2.02E-14 83.51 1.56E-14 67.83 1.25 9.0
α
=−3/2:
le #ThNhku−˜uhkXEOC Nh(˜uh)EOC
ρ
h(˜uh)EOC ie CPU(s)
0 128 384 1.32E-02 – 1.41E-01 – 4.52E-02 – 1.05 0.5
1 128 759 5.98E-03 2.32 6.70E-02 2.18 1.26E-02 3.75 1.01 0.8
2 128 919 5.50E-03 0.87 6.36E-02 0.55 6.26E-03 7.31 1.00 1.1
3 128 969 4.30E-03 9.31 5.52E-02 5.36 4.34E-03 13.81 1.00 1.4
4 134 1089 2.98E-03 6.29 3.96E-02 5.69 3.09E-03 5.86 1.00 1.6
5 140 1191 2.10E-03 7.81 2.75E-02 8.14 2.07E-03 8.91 1.00 1.9
6 152 1371 1.49E-03 4.82 1.93E-02 4.99 1.45E-03 5.03 1.00 2.1
7 158 1476 1.07E-03 9.07 1.37E-02 9.33 1.05E-03 8.72 1.00 2.5
8 164 1578 7.71E-04 9.78 9.78E-03 10.13 7.97E-04 8.34 1.00 2.9
9 176 1758 5.65E-04 5.75 7.04E-03 6.09 6.35E-04 4.21 1.00 3.3
10 188 1938 4.26E-04 5.81 5.15E-03 6.41 5.37E-04 3.43 1.00 3.8
11 200 2118 3.35E-04 5.41 3.87E-03 6.41 4.81E-04 2.48 1.00 4.3
Sou ce: Own
4.2 (E2): Linea Con ec ion-Di usion Equa ion wi h he S ong In e io Laye and he
Exponen ial Bounda y Laye
We conside he example om [10], gi en by
−
ε
△u+b1
∂
u
∂
x1+b2
∂
u
∂
x2=0 in Ω:= (0,1)2,(31)
whe e
ε
=10−8is a cons an di usion coe icien and (b1,b2) = (cos(−
π
/3),sin(−
π
/3)) is
he con ec ion. We p esc ibe he Di ichle bounda y condi ion on
∂
Ωby
uD(x1,x2) = 0 o x1=1 o x2≤0.7,
1 o he wise.(32)
The solu ion possesses an in e io laye in he di ec ion o he con ec ion s a ing in (x1,x2) =
(0,0.7)and con ains wo bounda y laye s along x1=0 and x2=0. On he bounda y x1=1 and
on he igh pa o he bounda y x2=0, exponen ial laye s a e de eloped. The wid h o laye s
a e p opo ional o
ε
.
Fo his case, he exac solu ion is piecewise cons an excep hin egions along he bounda y
and in e io laye , howe e , i s analy ical exp ession is unknown. The e o e he compu a ional
63
Tab. 2.Example (E2) gi en by (31) –(32) wi h
ε
=10−8: he app oxima ion o he e -
o k¯u−˜uhkX, noncon o mi y Nh(˜uh), esiduum e o es ima e
ρ
h(˜uh)wi h he co esponding
EOC, index ie and he compu a ional ime in seconds
le #ThNhk¯u−˜uhkXEOC Nh(˜uh)EOC
ρ
h(˜uh)EOC ie CPU(s)
0 128 384 1.25E-01 – 3.58E+00 – 1.03E+00 – 1.04 0.3
1 128 635 9.56E-02 1.08 3.57E+00 0.01 7.60E-01 1.20 1.02 0.6
2 167 1211 8.06E-02 0.53 5.03E+00 -1.06 7.60E-01 -0.00 1.01 1.0
3 245 1984 7.50E-02 0.30 7.09E+00 -1.39 5.85E-01 1.06 1.00 2.2
4 467 3921 6.35E-02 0.49 1.00E+01 -1.02 5.95E-01 -0.05 1.00 4.0
5 1025 9751 6.12E-02 0.08 1.42E+01 -0.76 6.89E-01 -0.32 1.00 9.4
6 2189 21663 4.51E-02 0.76 2.01E+01 -0.87 7.32E-01 -0.15 1.00 18.2
7 4910 54765 3.17E-02 0.76 2.84E+01 -0.75 6.75E-01 0.17 1.00 50.8
8 7733 106285 2.28E-02 0.99 2.86E+01 -0.02 5.38E-01 0.68 1.00 196.6
9 14240 221582 1.76E-02 0.71 2.86E+01 -0.00 5.20E-01 0.09 1.00 505.8
10 19103 310336 1.62E-02 0.50 2.86E+01 -0.00 5.16E-01 0.04 1.00 1046.7
11 17849 272175 1.61E-02 -0.05 2.86E+01 0.00 5.16E-01 -0.00 1.00 1163.8
12 17426 250641 1.61E-02 -0.01 2.86E+01 -0.00 5.16E-01 0.00 1.00 1226.7
13 17366 240586 1.61E-02 0.03 2.86E+01 -0.01 5.16E-01 0.00 1.00 1248.5
14 17288 236116 1.61E-02 -0.16 2.86E+01 0.02 5.16E-01 0.01 1.00 1290.4
15 17207 233366 1.61E-02 0.12 2.86E+01 -0.05 5.16E-01 0.01 1.00 1311.4
Sou ce: Own
e o ku−˜uhkXis app oxima ed by k¯u−˜uhkXwhe e ¯uis piecewise cons an unc ion co e-
sponding o he exac solu ion o (31) in he limi wi h
ε
→0. (The bounda y condi ions ha e
o be modi ied o cou se.)
Table 2 shows he esul s o compu a ions (E2) o p oblem (31) – (32), namely he alues
o he app oxima ion o he e o k¯u−˜uhkX, noncon o mi y Nh(˜uh), esiduum e o es ima e
ρ
h(˜uh)wi h he co esponding EOC, index ie and he compu a ional imes in seconds. The
alue k¯u−˜uhkXdoes no end o ze o, p obably due o he di e ence be ween he exac solu ion
uand i s app oxima ion ¯u. Mo eo e , he noncon o mi y Nh(˜uh)is high, i will dec ease wi h an
addi ional mesh adap a ion, howe e , he e we ace a p oblem wi h a oo high numbe o deg ee
o eedom and he limi s o ou compu e . I would be mo e e icien o apply an aniso opic
mesh adap a ion. Ne e heless, he esul s show ha he p esen edhp-me hodwo ks easonably,
bo h laye s a e well cap u ed.
Fu he mo e, Figu e 2 shows he inal hp-g id ob ained wi h he aid o he hp-DGFE al-
go i hm a e 5, 10 and 15 adap i e cycles. We obse e ha he h-adap a ion was ca ied ou
in egions nea bo h laye s. In egions, whe e he solu ion is cons an , he P0app oxima ion
is inally used. We also obse e ha he algo i hm is able o dec ease Nhwhen he hin laye s
a e well localized. Finally, Figu e 3 shows he de ail o he diagonal cu [0,0]→[1,1]o he
app oxima e solu ion a e l=5, l=10 and l=15 adap a ion cycles.
Conclusion
We p esen ed a hp-adap i e nume ical me hod o he solu ion o he second o de bounda y
alue p oblem. This app oach is based on a heu is ic app oxima ion o he e o measu ed
in a dual no m. Al hough a heo e ical jus i ica ion o his echnique is missing, nume ical
expe imen s show easonable compu a ional p ope ies.
64