On he s abili y o app oxima ions o he S okes p oblem using
di e en ini e elemen spaces o each componen o he eloci y∗
F. Guill´en Gonz´alez†and J.R. Rod ´ıguez Gal ´an‡
Decembe 1, 2014
Abs ac
This pape s udies he s abili y o eloci y-p essu e mixed app oxima ions o he S okes
p oblem when di e en ini e elemen (FE) spaces o each componen o he eloci y ield a e
conside ed. We conside some new combina ions o con inuous FE educing he numbe o
deg ees o eedom in some eloci y componen s. Al hough he esul ing FE combina ions a e
no s able in gene al, by using he S enbe g’s mac o-elemen echnique, we show hei s abili y in
a wide amily o meshes (namely, in uni o mly uns uc u ed meshes). Mo eo e , a pos -p ocessing
is gi en in o de o con e any mesh amily in an uni o mly uns uc u ed mesh amily. Finally,
some 2Dand 3Dnume ical simula ions a e p o ided ag ee wi h he p e ious analysis.
Keywo ds. In -sup condi ion, incomp essible luids, mixed a ia ional o mula ions, ini e ele-
men s, mac o-elemen echnique.
AMS classi ica ion (2010). 35Q30, 65N12, 65N30, 76D07.
Con en s
1 In oduc ion 2
2 De ini ions and no a ions. P0app oxima ion o he p essu e 4
3 S abili y o he (P1,b,P1)–P1app oxima ion 5
3.1 The2Dcase......................................... 5
3.2 The 3Dcase......................................... 13
3.3 Nume icalsimula ions ................................... 17
4 Abou s abili y o (P2,P1)–P1FE 23
4.1 The2Dcase......................................... 23
4.2 Nume icalsimula ions ................................... 29
5 Ins abili y o (Q2,Q1)–Q1in s uc u ed meshes 31
∗This wo k has been pa ially suppo ed by p ojec MTM2012-32325 (MINECO, Spain). Also he second au ho
has been pa ially suppo ed by he esea ch g oup FQM-315 (Jun a de Andaluc´ıa).
†Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico and IMUS. Uni e sidad de Se illa. Ap do. 1160,
41080 Se illa (Spain). [email p o ec ed]
‡Depa amen o de Ma em´a icas. Uni e sidad de C´adiz. C.A.S.E.M. Pol´ıgono R´ıo San Ped o S/N, 11510 Pue o
Real, C´adiz (Spain). [email p o ec ed]
1
a Xi :1411.7930 1 [ma h.NA] 28 No 2014
6 Conclusions 33
1 In oduc ion
The main pu pose o his wo k is o analyze he s abili y o eloci y-p essu e mixed FE app ox-
ima ions o he S okes p oblem when he componen s o eloci y a e app oxima ed in di e en
spaces.
As desc ibed below, his is an issue o g ea in e es o he app oxima ion o some a ian s o
he Na ie -S okes equa ions (in pa icula o he app oxima ion o he hyd os a ic Na ie -S okes
equa ions go e ning he la ge scale ocean [GR14]) and, as a as we know, i has no been su icien ly
s udied in he li e a u e. In ac , al hough s abili y o eloci y-p essu e mixed FE app oxima ions
o he S okes p oblem is a ma e widely in es iga ed in he las decades, almos all o hese wo ks
ix he same FE space o all componen s o he eloci y ield, being he p essu e he only unknown
ha could be app oxima ed by a di e en space.
He e we conside he classical S okes p oblem in a domain Ω ⊂Rd(d= 2 o 3),
−∆w+∇p= in Ω,(1)
∇ · w= 0 in Ω,(2)
dis inguishing he componen s o he eloci y ield as ollows: w= (u, ), whe e u: Ω →Rd−1
and : Ω →R(ho izon al and e ical componen s, espec i ely). Also p: Ω →Ris he p essu e,
: Ω →Rdis he ec o o ex e nal o ces and, as usual, ∆ and ∇ · ep esen he Laplace and
di e gence ope a o s. Fo sake o simplici y, we ake cons an iscosi y (equal o one) and conside
homogeneous Di ichle bounda y condi ion,
w=0on ∂Ω.(3)
Le us in oduce usual no a ions: L2(Ω) ep esen s he space o he squa e in eg able unc ions
in Ω and ( , g) = RΩ g deno es i s scala p oduc , Hm(Ω) is he space o he unc ions whose
dis ibu ional de i a i es o o de up o m∈Nbelong o L2(Ω) and H1
0(Ω) is he subspace o
H1(Ω) whose unc ions anish on ∂Ω.
I one conside he bilinea o ms a(w,w)=(∇w,∇w) and b(p, w)=(p, ∇ · w), he a ia ional
o mula ion o he S okes p oblem (1)–(3) eads: ind (w, p)∈W×Psuch ha :
a(w,w)−b(p, w) = ZΩ
·w,∀w∈W,
b(p, w) = 0,∀p∈P,
whe e W=H1
0(Ω)dand P=L2
0(Ω) = p∈L2(Ω)/RΩp= 0. We assume in L2(Ω)d.
I is well known (see o ins ance [BBF13]) ha well-posedness o p e ious p oblem yields om
he coe ci i y o he bilinea o m a(·,·) and he in -sup condi ion o b(·,·), which holds o he
con inuous p oblem on W×P. Bu i we conside he disc e e p oblem, ind (wh, ph)∈Wh×Ph
such ha
a(wh,wh)−b(ph,wh) = ZΩ
·wh,∀wh∈Wh,(4)
b(ph,wh) = 0,∀ph∈Ph,(5)
whe e Wh⊂Wand Ph⊂Wa e amilies o FE spaces, al hough coe ci i y o a(·,·) in Wh×Phis
au oma ically inhe i ed, he con inuous in -sup condi ion is no enough o well posedness o (4)–(5).
2
In ac , each choice o Whand Ph equi es checking he ollowing disc e e coun e pa o he in -sup
condi ion (also called LBB condi ion): he e exis s β > 0 independen on hsuch ha
βkphkL2≤sup
06=wh∈WhRΩ(∇ · wh)ph
k∇whkL2
,∀ph∈Ph.(6)
In wha ollows, we will ake Wh=Uh×Vh, whe e Uhand Vha e, espec i ely, FE spaces
app oxima ing he ( wo componen s o ) ho izon al eloci y and he e ical eloci y. I Ω ⊂R2,Uh
is scala and is deno ed by Uh. I no ambigui y, we deno e by P1o P2 he co esponding con inuous
piecewise polynomial FE app oxima ion, while P0deno es discon inuous FE.
The case whe e Uhand Vha e app oxima ed by he same FE space has been widely s udied and
se e al combina ions (which we deno e by (Uh, Vh) – Ph, in he 2D case) a e known sa is ying he
disc e e in -sup condi ion (6). Tha is he case, o example, o (P1,b,P1,b) – P1(usually known as
P1+bubble o mini-elemen ), (P2,P2) – P1and (Q2,Q2) – Q1(Taylo -Hood elemen s), while some
o he s do no sa is y (6) in gene al (see o ins ance Sec ion 2and e e ences he ein). As usual,
P1,b deno es P1con inuous unc ions en iched by a “bubble” unc ion o each elemen and Qka e
con inuous enso Pk× Pkelemen s on quad angula meshes.
This wo k is cen e ed in he case whe e Uhand Vha e app oxima ed by di e en FE spaces,
o which (as a as we know) only he case (P2,P1) – P0has been add essed, see [S e90]. Some new
combina ions (Uh, Vh) – Pha e going o be s udied, showing (o e using) i s s abili y in wide mesh
amilies and ex ending he analysis o h ee-dimensional domains.
Speci ically, ou main con ibu ion is showing ha al hough he in -sup condi ion (6) does no
hold in gene al o he combina ions (P1,b,P1) – P1and (P2,P1) – P1(and also o he symme ic
cases (P1,P1,b) – P1and (P1,P2) – P1), i is sa is ied in nume ous mesh amilies, namely in uni-
o mly uns uc u ed meshes (see De ini ion 7). In ac , one can easily de ec i a gi en mesh is
no uns uc u ed and, wi h a sligh pos -p ocessing, ans o m i in o a uni o mly uns uc u ed one
(see Algo i hm 1below).
The e a e some o he well-known uns able elemen s ( o ins ance (P1,P1) – P0and (P1,b,P1) – P0,
see Sec ion 2) which ha e less deg ees o eedom han he “mini-elemen ” (P1,b,P1,b) – P1and ha
a e s able jus in some conc e e meshes. These meshes a e, in pa icula , uni o mly uns uc u ed
and indeed FE combina ions p o ided in his pape a e s able in a e y much la ge amily o meshes.
In his sense, (P1,b,P1) – P1cons i u es a “minimal-elemen ” because wi h less deg ees o eedom
han he “mini-elemen ” s abili y is p o ided o uns uc u ed meshes (o sligh ly pos -p ocessed
s uc u ed ones).
P e ious esul s a e ex ended o 3Ddomains by using (P1,b,P1,b,P1) – P1and e en (P1,b,P1,P1) – P1
(and he co esponding spaces eplacing P1,b by P2). Addi ionally, we p esen some nume ical es s
showing ha (P1,b,P1) – P1p ese es he accu acy O(h) while (P2,P1) – P1p esen s a loss o o -
de (only o de O(h)) wi h espec o he o de O(h2) o he classical Taylo -Hood app oxima ion
(P2,P2) – P1.
Ou in e es on he s abili y o aniso opic ho izon al/ e ical eloci y app oxima ion has been
mo i a ed by he s udy o some geophysical luid models go e ning he mo ion o la ge scale oceans
(see o ins ance [CB09] and e e ences he ein ci ed). In ac , hey lead o he Na ie -S okes
equa ions wi h aniso opic eddy iscosi ies in shallow domains o , a e escaling, o equi alen
models in adimensional domains [AG01,BL92]. In he linea s eady case, he ollowing Aniso opic
S okes p oblem a ises: 




−∆u+∇xp= in Ω,
−ε2∆ +∂zp=gin Ω,
∇ · (u, ) = 0 in Ω,
(AS)
3
whe e ε > 0 is he so-called aspec a io be ween e ical and ho izon al scales. When ε→0,
di usion o he e ical eloci y anishes [BL92] and a singula limi p oblem appea s, which is
called he Hyd os a ic S okes equa ions. This ype o p oblems also appea in he ansien nonlinea
case (called Hyd os a ic Na ie -S okes equa ions o P imi i e Equa ions o he ocean), see [AG01].
In a o hcoming wo k [GR14] (see also [Az´e94,Az´e96]) he well-posedness o he app oxima ions
in uni o mly uns uc u ed meshes o (AS) is de eloped ( alid when εis “small” o ze o). This heo y
is based on he S okes in -sup condi ion join ly o an addi ional in -sup condi ion needed o s abilize
he e ical eloci y. This new in -sup condi ion is sa is ied a disc e e le el when he p essu e
space is la ge enough wi h espec o he e ical eloci y one. Bu , o s anda d S okes FE spaces,
he when he numbe o deg ees o eedom o p essu e is oo la ge hen S okes s abili y is no
sa is ied. The ques ion is, a e he e in e media e possibili ies o app oxima ing eloci y-p essu e
mixed o mula ion sa is ying bo h s abili y cons ain s ? The answe gi en in his pape is yes, bu
o uni o mly uns uc u ed meshes.
This pape is o ganized as ollows. In Sec ion 2, we ecall some esul s abou app oxima ions wi h
discon inuous p essu e. In Sec ion 3we p o e ha (P1,b,P1) – P1is s able in uni o mly uns uc u ed
meshes and show i s ins abili y in some s uc u ed meshes. Resul s a e gene alized o he 3Dcase
and nume ical es s a e shown con i ming ou analysis. In Sec ion 4, we show ha (P2,P1) – P1is
also s able in mos uns uc u ed meshes, al hough he esul s a e sligh ly mo e es ic i e han o
(P1,b,P1) – P1app oxima ion. This heo y is suppo ed by some nume ical expe imen s, including
some 3D es s. Finally, in Sec ion 5, we show ins abili y o he combina ion (Q2,Q1) – Q1on
quad angula elemen s, bo h analy ical and compu a ionally.
2 De ini ions and no a ions. P0app oxima ion o he p essu e
Le us assume elsewhe e ha Ω ⊂Rd, wi h d= 2 o d= 3, is a domain wi h polygonal bounda y
and le Th⊂Rdbe a shape egula amily o iangula ions o Ω. Fu he mo e, le us suppose ha
This a simplicial mesh (i.e. composed o 2D iangles o 3D e ahed ons), unless o he wise s a ed.
In his sec ion we ecall some esul s abou discon inuous app oxima ion o p essu e. Speci ically,
he case d= 2 is conside ed and, o he app oxima ion o he disc e e S okes p oblem (4)–(5), he
ollowing P0discon inuous space o he app oxima ion o he p essu e is in oduced:
Ph={ph∈L2
0(Ω) / p|T∈ P0∀T∈ Th}.(7)
Abou eloci y, we de ine Wh=Uh×Vh, whe e Uhand Vha e con inuous FE spaces o ho izon al
and e ical eloci y, espec i ely. Indeed, we deal P1,P1,b o P2spaces o ho izon al eloci y and
P1 o e ical one. O cou se he symme ic choice is equi alen .
As esul , we conside he FE combina ions deno ed by (P1,P1) – P0, (P1b,P1) – P0and (P2,P1) – P0
whose s abili y has been su icien ly analyzed in li e a u e and we can summa ize as ollows:
(P2,P1) – P0is he minimal FE combina ion (wi h P0discon inuous p essu e) which is s able in
gene al meshes. Namely, le us conside
Vh={ h∈H1
0(Ω) ∩C0(Ω) / |T∈ P1,∀T∈ Th}(8)
•Ins abili y o (P1,P1) – P0app oxima ion, i.e. he case whe e Uh=Vh, is well known o occu
excep ing some speci ic meshes (called c isc oss meshes, see [QZ07]). They a e pa icula
cases o x–uns uc u ed and y–uns uc u ed meshes, see De ini ion 6.
•Ins abili y o (P1,b,P1) – P0, i.e. he case whe e
Uh={uh∈H1
0(Ω) ∩C0(Ω) / uh|T∈ P1,b(T),∀T∈ Th},(9)
4
can be easily educed o he abo e case, using he ac ha in eg als in iangles o de i a i es
o bubble unc ions anish. The case (P1,b,P1,b) – P0is simila .
•S abili y o (P2,P1) – P0, i.e. he case whe e
Uh={uh∈H1
0(Ω) ∩C0(Ω) / uh|T∈ P2,∀T∈ Th}(10)
while Vhand Pha e de ined espec i ely in (8) and (7), was shown by R. S enbe g in [S e90], us-
ing he mac o-elemen echnique (which was de eloped by himsel in ha e e ence and [S e84]).
3 S abili y o he (P1,b,P1)–P1app oxima ion
In wha ollows, con inuous piecewise linea p essu es a e selec ed:
Ph={ph∈L2
0(Ω) ∩C0(Ω) / p|T∈ P1,∀T∈ Th}.(11)
Using he mac o-elemen echnique, we s udy he 2D case, whe e Uhis a con inuous P1,b FE space,
as de ined in (9) and Vhis a con inuous P1space, as in (8). Resul s a e gene alized o he 3Dcase
(Sec ion 3.2) and some nume ical expe imen s a e shown, suppo ing hese esul s (Sec ion 3.3).
3.1 The 2D case
Fo he s udy o he (P1,b,P1) – P1app oxima ion, he mac o-elemen echnique is applied, ollowing
he seminal pape s o S enbe g [S e84,S e90] (specially he o me ). We show a local s abili y esul
(Theo em 3) and global s abili y esul (Theo em 4), whe e a gene aliza ion o S enbe g’s heo y
mus be in oduced.
A mac o-elemen pa i ioning, Mh, o a iangula ion Thin Ω is a amily o connec ed se s
M∈ Mhwhich a e union o a leas wo elemen s o Th. Fo each M∈ Mh, le U0,M and V0,M be
he spaces o unc ions in H1
0(M)∩C0(M) which a e, espec i ely, in P1,b and P1 o all T⊂M.
Le PMbe he space o unc ions in C0(M) which a e in P1 o e e y T⊂M.
De ini ion 1. We say ha (Uh, Vh)–Phis egula in a mac o-elemen Mi
The se NM=ph∈PM,ZM
(∂xuh+∂y h)ph= 0 ∀(uh, h)∈U0,M ×V0,M 
only consis s o he unc ions which a e cons an on M.
(12)
In o he case, (Uh, Vh)–Phis said singula in M.
S enbe g shown, oughly speaking, ha i his egula i y condi ion is sa is ied, independen ly o
he geome ical shape o he mac o-elemen s o some gi en pa i ioning (and unde sligh opological
es ic ions on mac o-elemen s, o mula ed in Theo em 3below) he global in -sup condi ion (6)
holds uni o mly (i.e. wi h a cons an βindependen o he mac o-elemen ).
Mo e speci ically, acco ding o he heo y o S enbe g, we in oduce he ollowing de ini ion:
De ini ion 2. A mac o-elemen Mis said o be equi alen (in he sense o S enbe g) o a e e -
ence mac o-elemen M∗i he e exis s a con inuous one- o-one applica ion FM:M∗→Mwhich
consis en ly maps elemen s con ained in M∗ o elemen s in M(see [S e84] o [S e90] o de ails).
The ollowing esul holds (see Theo em 3.1 in [S e90]):
5

Theo em 1. Suppose ha he e exis s a amily o mac o-elemen s, Mh, o Thwhich is composed
o a ixed se o equi alence (in he sense o S enbe g) classes Eno mac o-elemen s, n= 1, ..., N
and a posi i e in ege L(Nand Lindependen o h) such ha :
(M1) The condi ion (12) is sa is ied o e e y M∈ Mh(i.e. (Uh, Vh)–Phis egula in mac o-
elemen s).
(M2) Each M∈ Mhbelongs o one o he classes En,n= 1, ..., N.
(M3) I eis an edge (o ace) o an elemen o Thwhich is in e io o Ω, hen eis in e io o a
leas one and no mo e han Lmac o-elemen s o Mh.
Then he S okes disc e e in -sup condi ion (6) holds o (Uh, Vh)–Ph.
In his wo k we ocus on a conc e e se o amilies o mac o-elemen s, which a e de ined as union
o he elemen s o Thwhich sha e one only in e io e ex. Mo e p ecisely, le us deno e by ◦
Th he
se o all o he e ices o Thwhich a e in he in e io o Ω.
De ini ion 3.
•We say ha a mac o-elemen c
Mis cen e ed in a e ex q∈◦
Thi c
Mis he union o e e y
T∈ Thsuch ha qis a e ex o T.
•We de ine a e ex-cen e ed mac o-elemen pa i ioning o Th, deno ed by c
Mh, as any mac o-
elemen pa i ioning o Thsuch ha e e y c
M∈c
Mhis cen e ed in some e ex q∈◦
Th.
I c
Mis cen e ed in q0, we deno e by nq0
(o jus n ) he numbe o e ices qin c
Msuch ha
q6=q0. See ha n ma ches also he numbe o elemen s con ained in c
M. Fo example, Figu e 1
shows wo mac o-elemen s, each o which is cen e ed in a e ex, q0, wi h n = 5 in bo h cases.
Shape egula i y o Thimplies ha ∃N∈N(independen o h) such ha each e ex o This
in no mo e han Nelemen s. On he o he hand, i This a simplicial mesh amily in Rd, one has
d < nq
. The e o e:
d < nq
≤N, ∀q∈◦
Th.(13)
Fixed a e e ence mac o-elemen c
M∗(cen e ed in a e ex q∗∈◦
Th) and ixed n=nq∗
, we deno e
by b
En he amily o mac o-elemen s which a e equi alen (in he sense o S enbe g) o c
M∗. Thus
a mac o-elemen pa i ioning c
Mhis wha e e such ha e e y M∈c
Mhbelongs o some b
En, wi h
n∈N. He ea e we assume ha Thsa is ies he ollowing sligh ly es ic i e hypo hesis:
Assump ion 1. Each elemen T∈ Thhas a leas one e ex in he in e io o , Ω.
Acco ding o (13), o any simplex mesh sys em Th, one can build a leas one and a mos N−d
amilies o e ex-cen e ed mac o-elemen s, b
Ed+1,b
Ed+2 ..., b
EN. Assump ion 1ensu es ha Thcan
be co e ed by mac o elemen s o hese amilies.
Lemma 2. Any e ex-cen e ed mac o-elemen pa i ioning c
Mhsa is ies (M2) and (M3).
P oo . Only (M3) mus be sa is ied. I eis an edge (o ace) in he in e io o Ω, hen i is adjacen
o, a leas , one in e io e ex, q0and hen eis in e io o, a leas , he mac o-elemen cen e ed in
q0. On he o he hand eis in e io o (a mos ) dmac o-elemen s: ha one which a e cen e ed in
q0and hose ones cen e ed in qi,i= 1, ..., d −1, whe e qia e he o he e ices o e.
6
q0
q1
q2q3
q4
q5
T1
T2
T3
T4
T5M+
M−
(a) Mac oelemen ha can be spli by an
ho izon al line ( hen, i is y–s uc u ed and
(P1,b,P1) – P1is no egula in i ).
q0
q1
q2q3
q4
q5
1
2
3
4
5
T1
T2
T3
T4
T5
(b) Mac oelemen ha canno be spli by
any s aigh line (i is x–uns uc u ed and
y–uns uc u ed).
Figu e 1: Bubble mac oelemen s.
Thus (in meshes sa is ying Assump ion 1) Lemma 2and Theo em 1s a e ha accomplishing he
egula i y condi ion (M1) in some e ex-cen e ed mac o-elemen pa i ioning c
Mhis su icien o
in -sup condi ion (6). The d awback is ha , as i is shown below, no all o e ex-cen e ed mac o-
elemen s sa is y hypo hesis (M1). Nex we p o ide a esul cha ac e izing he mac o-elemen s o c
Mh
whe e ha hypo hesis (in ac (12)) holds o (P1,b,P1) – P1. The idea is checking he mac o-elemen
s uc u e in he ollowing sense.
De ini ion 4. Le Thbe a iangula ion o Ω⊂Rd(in p ac ice, d= 2 o 3) and le Mbe a
mac o-elemen in Th. We say ha Mcan be spli by an hype plane Πi he e exis s wo o he
mac o-elemen s M1and M2, composed o elemen s o Th, such ha M1∪M2=Mand M1∩M2⊂Π.
De ini ion 5. A mac o-elemen Mo a mesh This be said x–s uc u ed i Mcan be spli by an
hype plane x=C, o some C∈R. In o he case, i is said x–uns uc u ed.
No e ha a x–s uc u ed mac o-elemen can be spli by a s aigh line (a plane, in he 3Dcase)
which is o hogonal o he xaxis. Simila ly, a whole mesh This said x–s uc u ed i i can be
spli by an hype plane x=C,C∈R. Analogue de ini ions can be p o ided o y–s uc u ed and
z–s uc u ed mac o-elemen s o meshes.
Fo example:
•Figu e 1a ep esen s a 2Dmac o-elemen ha can be spli by an ho izon al line (and no by
a e ical line), hen i is y–s uc u ed (and x–uns uc u ed).
•The whole mesh in Figu e 3(i.e. all i s e ex-cen e ed mac o-elemen s) is x–s uc u ed and
y–s uc u ed.
•The mac o-elemen ep esen ed in Figu e 1b canno be spli by any ho izon al o e ical
s aigh line, he e o e i is x–uns uc u ed and y–uns uc u ed.
The ollowing esul ela es he egula i y in a mac o-elemen wi h i s s uc u e. In ac , i s a es
ha , in hose mac o-elemen s which a e uns uc u ed in some di ec ion, he bubble unc ions can
be sa ely emo ed om he co esponding componen o he eloci y ield.
Theo em 3. Le c
Mbe a mac o-elemen in a e ex-cen e ed pa i ioning c
Mh. Then:
7
1. The combina ion (P1,b,P1)–P1is egula in c
M(namely (12) holds o c
M) i and only i c
M
is y–uns uc u ed.
2. The combina ion (P1,P1,b)–P1is egula in c
Mi and only i c
Mis x–uns uc u ed.
This Theo em can also be in e p e ed as ollows: he s anda d mini-elemen (P1,b,P1,b) – P1is
“ oo ich” in hose mac o-elemen s which a e uns uc u ed in he xo ydi ec ion.
P oo . We show only he i s s a emen , because he second one is symme ic.
⇐Le c
M∈c
Mhbe a mac o-elemen wi h q0as unique in e io e ex. Le n =nq0
,qi(i= 1, ...n )
he numbe o elemen s a ound q0and i(i= 1, ..., n ) he bubble deg ees o eedom (ba ycen es
o he iangles in c
M). I c
Mis y–uns uc u ed hen he e a e no wo e ices qiand qjin c
M
ho izon ally aligned o q0. Fo each ph∈Nc
M, ha is Rc
M(∂xuh+∂y h)ph= 0, le us deno e he
shape unc ions
ph|Ti=ai+bix+ciy, ∀Ti∈c
M.
Fi s le us choose h= 0 and uh∈U0,
c
Msuch ha uh= 1 on a ba ycen e iand uh= 0 on all
o he deg ees o eedom in c
M( j,j= 1, ..., n ,j6=iand qk,k= 0, ..., n ), ha is uhis a bubble
unc ion on Ti. Then,
0 = Zc
M
(∂xuh+∂y h)ph=ZTi
∂xuhph=−ZTi
uh∂xph=−biZTi
uh,
hence
bi= 0 ∀i= 1, ..., n ,(14)
in pa icula ∂xph= 0 in c
M. Now, imposing he con inui y o phon q0and assuming, wi hou loss
o gene ali y, q0= (0,0), i is s aigh o wa d ha he e exis s a∈Rsuch ha
a1=... =an =a. (15)
On he o he hand, imposing he con inui y o phin he emaining e ices qi= (qx
i, qy
i)∈∂c
M
he common e ex o Tiand Ti+1 (deno ing Tn +1 =T1), one has:
qy
i(ci+1 −ci) = 0, i = 1, ..., n ,(16)
whe e (14) and (15) ha e been applied. The e o e i qy
i6= 0 o all i, ha is i he e is no e ex
ho izon ally aligned wi h q0, hen he e exis s c∈Rsuch ha
c1=... =cn =c. (17)
This equali y is also ue when he e is only one e ex ho izon ally aligned wi h q0, because in (17)
he e is one equa ion plus han unknowns c1, ..., cn .
Finally le us choose uh= 0 and h∈V0,
c
Mde ined as h= 1 in q0and h= 0 in all o he
deg ees o eedom (qi,i= 1, ..., n ). Applying he quad a u e o mula RT dx dy =|T|
3P3
i=1 (qT
i)
wi h qT
i he e ices o T, which is exac in P1,
0 = Zc
M
∂y hph=−
n
X
i=1 ZTi
h∂yph=−
n
X
i=1
ci
|Ti|
3
and conside ing (17) we conclude ci= 0 o all i. Consequen ly, phis cons an on c
Mand mac o-
elemen condi ion (12) holds.
8
⇒Assume he e a e wo o he e ices ho izon ally aligned wi h q0= (0,0) (wi hou loss o
gene ali y hey a e being indexed qjand qn ). F om (16) we only deduce:
c1=c2=... =cj,
cj+1 =cj+2 =... =cn ,
and in his case, o some c,ec∈R,
∂yph(x, y) = (ci y > 0,
eci y≤0.(18)
We a e going o use his idea o ob ain a coun e example. Indeed, now c
Mcan be spli in o wo
disjoin egions: c
M+={(x, y)∈c
M / y > 0}and c
M−={(x, y)∈c
M / y < 0}, each o which can
be w i en as union o iangles o c
M(see Figu e 1b). Fo any pa ame e s aand c∈R, le us de ine
ph∈Pc
Mas ollows:
ph(x, y) = 




a−c
|c
M+|yi (x, y)∈c
M+,
a+c
|c
M−|yi (x, y)∈c
M−,
whe e |R|deno es he a ea o a egion R⊂R2. E e y h∈V0,
c
Mcan be cha ac e ized as he P1
basis unc ion ha is equal o α= h(q0) in he e ex q0and 0 in all o he e ices in c
M. Hence
one has (applying he abo e mass-lumping quad a u e o mula):
Zc
M
∂y hph=−Zc
M
h∂yph=c
|c
M+|Zc
M+
h−c
|c
M−|Zc
M−
h
=c
|c
M+|X
T⊂
c
M+
|T|
3α−c
|c
M−|X
T⊂
c
M−
|T|
3α=c
3α−c
3α= 0
Then, aking in o accoun ha ∂xph= 0, he equali y Rc
M(∂xuh+∂y h)ph= 0 is sa is ied o all
(uh, h)∈U0,
c
M×V0,
c
M, he e o e (12) does no hold.
Now, we a e in e es ed in ex ending he local s abili y esul o Theo em 4 o a global esul o
he whole Th. Acco ding o Theo em 1and Lemma 2, in o de o apply he mac o-elemen heo y
o he pa icula amily c
Mh, i mus be shown ha mac o-elemen egula i y condi ion (12) holds
o all c
M∈c
Mh, namely o e e y mac o-elemen in he amilies b
Ed+1, ..., b
EN. The d awback is
ha y–s uc u ed mac o-elemen s a e pa o hese amilies, and egula i y condi ion is no sa is ied
o y–s uc u ed mac o-elemen s, acco ding o Theo em 4(in he case o (P1,b,P1) – P1).
No e ha e en i a conc e e mesh amily, Th, is buil whe e e e y c
Mis y–uns uc u ed (and
hen egula i y condi ion (12) is sa is ied o e e y c
M) one canno apply Theo em 1 o conclude
ha in -sup condi ion (5) holds. Indeed, in Theo em 1,Mhis composed o e e y o he equi alen
mac o-elemen s in he sense o S enbe g (De ini ion 2), independen ly o a conc e e mesh amily
(and hence including y–s uc u ed mac o-elemen s). In ac one could build an uns uc u ed mesh
amily con e ging o a s uc u ed one, hence s abili y is no sa is ied when h→0. Nume ical
es 8shows an example ela ed o his ac . The solu ion is conside meshes which a e uni o mly
uns uc u ed in he ollowing sense.
Le c
Mhbe a e ex-cen e ed mac o-elemen pa i ioning a mesh Th. Le c
M∈c
Mh, le qbe i s
in e io e ex and {T1, ..., Tn} he elemen s o Thcon ained in c
M. Fo i= 1, ..., n−1, we deno e by
σi he angle be ween he posi i e ho izon al semiaxis om qand he common edge o he iangles
Tiand Ti+1 o c
M. Simila ly, σnis de ined using he common edge o Tnand T1.
9
Finally, i he e ahed ons K1, ..., Kncan be connec ed h ough a chain o aces, T1,2, T2,3, ...,
Tn−1,n, ha a e no o hogonal o he plane z= 0, hen b1=b2=... =bn=b0and c1=c2=... =
cn=c0, o some b0,c0∈R.
Fo he ollowing esul , we use he concep o spli ing a mac o-elemen by a se o semi-planes.
This spli ing concep can be de ined in he sense o De ini ion 4.
Theo em 8. Le c
M∈c
Mhwi h one only in e io e ex, q0. Le 0be he e ical s aigh line
h ough q0and le Fbe he amily o he ( e ical) semi-planes bounded by 0.
1. I c
Mcanno be spli by semi-planes o F, hen (P1,P1,P1b)–P1is egula in c
M.
2. I c
Mcan be spli by only wo semi-planes o F, hen (P1,P1,P1b)–P1is egula in c
Mi and
only i hese wo semi-planes a e no aligned (i.e. hey do no o m a e ical plane spli ing
c
M).
3. I c
Mcan be spli by mo e han wo semi-planes o F, hen (P1,P1,P1b)–P1is no egula in
c
M.
P oo . Le phin Nc
Mwi h ph|Ti=ai+bix+ciy+diz,i= 1, ...nK. Wi hou loss o gene ali y, we
can assume q0= (0,0,0). Then, by con inui y in q0,a1=... =an. Le us choose u2
h= h= 0 in
c
Mwhile u1
his equal o 1 in q0and 0 in all o he deg ees o eedom. Using he quad a u e o mula
de ined in he e ices o e ahed ons (exac in P1), he ollowing condi ion is sa is ied:
0 = Zc
M
∂xu1
hph=−
nK
X
i=1
|Ki|bi.(24)
Applying he same a gumen o u1
h= h= 0 while u2
his equal o 1 in q0and 0 in all o he deg ees
o eedom:
0 = Zc
M
∂yu2
hph=−
nK
X
i=1
|Ki|ci.(25)
1. Since c
Mcanno be spli by semi-planes o F, hen any wo e ahed ons o c
Mcan be connec ed
by a chain o no e ical aces. Applying Lemma 7 o M0=c
M, one has ha phcan be w i en as
ph=a+bx +cy. Then, aking in o accoun (24) and (25), b= 0 and c= 0, hence pis cons an on
c
Mand he mac o-elemen condi ion (22) holds.
2. I c
Mcan be spli only by wo semi-planes, Π1and Π2o Fin o wo egions M0
1and M0
2 hen
(acco ding o Lemma 7), he e exis b1, c1, b2, c2∈Rsuch ha
ph|M0
1u=a+b1x+c1y,
ph|M0
2=a+b2x+c2y.
Le q0= (0,0,0), q1= (x1, y1, z1) and q2= (x2, y2, z1) be he e ices o an in e io e ical ace
T⊂Π1. The con inui y o phin Tmeans:
x1(b1−b2) + y1(c1−c2)=0,(26)
x2(b1−b2) + y2(c1−c2)=0.(27)
16

Bu , acco ding o Lemma 6,x1y1
x2y2= 0,
hen (26) and (27) a e educed o one only equa ion, e.g. (27).
Simila ly, i q0= (0,0,0), eq1= (ex1,ey1,ez1) and eq2= (ex2,ey2,ez1) a e he e ices o an in e io
e ical ace e
T⊂Π2, one only equa ion is ob ained om he con inui y o phin e
T:
ex1(b1−b2) + ey2(c1−c2)=0.(28)
⇐I he e ical semi-planes Π1and Π2a e no aligned hen he p ojec ions o (x1, y1, z1)∈Π1
and (ex1,ey1,ez1)∈Π2in z= 0 a e no aligned, ha is:
x1y1
ex1ey26= 0.
Hence, om (27)–(28), b1=b2and c1=c2. Using (24)–(25), we conclude b1=b2=c1=c2= 0,
and ph=ain c
M.
⇒I Π1and Π2a e aligned, (27) and (28) a e educed o one only equa ion which is no enough
o conclude b1=b2=c1=c2= 0. In ac , coun e examples o no cons an p essu es o Nc
M
sa is ying b16=b2o c16=c2, can be ob ained simila ly o he coun e example gi en in he p oo o
Theo em 5.
3. Le M0
1, ..., M0
nbe egions esul ing om spli ing c
Mby nsemi-planes (n > 2), Π1, ..., Πno F.
A guing as in he p e ious case one has ha , o each i= 1, ..., n,ph|M0
i=a+bix+ciyand he
con inui y o phin Πican be educed o one only equa ion:
xi(bi−bi+1) + yi(ci−ci+1)=0,
whe e qi= (xi, yi, zi)∈Πi. The e o e, we ha e a mos nindependen equa ions wi h 2nunknowns,
biand ci. Since n > 2, he equa ions (24) and (25) a e no enough o conclude bi=ci= 0 o all
i= 1, ..., n, because we ha e a mos n+2 independen equa ions bu 2nunknowns (and n+2 <2n
i n > 2).
Rema k 5. Using Theo em 1and Lemma 2, he mac o-elemen egula i y esul gi en in Theo em 8
can be used as a su icien condi ion assu ing he s abili y o (P1,P1,P1b) – P1in 3Dmeshes ha a e
uni o mly z–uns uc u ed, in he sense o meshes which uni o mly do no p esen a local s uc u e
like he one discussed in Theo em 8(spli ing by e ical semi-planes). In p ac ice, one can hope
ha he uns uc u ed 3Dmeshes ob ained wi h usual so wa e ools p esen his p ope y bu , in
any case, mesh may be es ed and e en ually sligh ly modi ied ( ia a gene aliza ion o Algo i hm 1)
o ensu ing s abili y.
3.3 Nume ical simula ions
Some compu a ional simula ions a e p o ided, which ag ee wi h he nume ical analysis o p e ious
sec ions.
Tes 1 ((P1,b,P1)–P1and he ca i y es in a x–s uc u ed and y–uns uc u ed mesh).
As i s nume ical es we ha e chosen, in he uni squa e Ω = (0,1)2, a x–s uc u ed and y–
uns uc u ed mesh like he one shown in Figu e 2, whe e he s abili y o (P1,b,P1) – P1is gua an eed
17
(a) (P1,b,P1,b) – P1. (b) (P1,b,P1) – P1. (c) (P1,P1,b) – P1.
Figu e 4: P essu e con ou plo s in a y–uns uc u ed mesh (lid-d i en ca i y es ).
(see Rema k 1). We ha e app oxima ed he solu ion o he S okes p oblem (1)–(3) wi h Di ichle
condi ions u= 1 on he op bounda y, u= 0 on he es o ∂Ω and = 0 on all ∂Ω.
Fo he p ac ical implemen a ion, ha speci ic mesh is gene a ed (wi h 15 ×15 edges, i.e. h≃
0.666) by a sc ip p og ammed using he Py hon language, which gene a es a mesh ile using he
o ma o he mesh gene a o Gmsh [GR09]. This ile is impo ed by he S okes sol e , which has
been p og ammed using he FEniCS FE sui e [LMW+12], aking ad an age o he acili ies o e ed
by his ool o se ing he bounda y condi ions on his speci ic mesh.
The p og am has been un o h ee choices: (P1,b,P1,b) – P1, (P1,b,P1) – P1and (P1,P1,b) – P1.
As expec ed, he las case p esen s he wo s beha iou o he p essu e, sugges ing he ins abili y
o (P1,P1,b) – P1in hese x–s uc u ed meshes (Figu e 4c). On he o he hand, he esul s o
(P1,P1,b) – P1a e simila o he mini-elemen (P1,b,P1,b) – P1(Figu es 4b and 4a, espec i ely), as
expec ed om he s abili y o (P1,b,P1) – P1in y–uns uc u ed meshes.
No ice ha in his es (and in he nex ones below) he S okes mixed scheme (4)-(5) has been
sligh ly al e ed by in oducing a s anda d p essu e penaliza ion in he di e gence equa ion:
ZΩ
(∂xu+∂y )p+εZΩ
p p = 0,∀p∈Ph,
whe e εis a small penaliza ion pa ame e (in his example, we ha e selec ed ε= 10−10). Wi h his
app oxima ion, he condi ion RΩp= 0 is sa is ied implici ly (we ob ain RΩp=−5.12 ·10−8) and
hence i is no longe necessa y o include i in he de ini ion o Ph. Then, we can ake Ph={ph∈
C0(Ω) / p|T∈ P1,∀T∈ Th}.
Tes 2 ((P1,b,P1)–P1and he ca i y es ). As he second nume ical es , he 2D S okes p ob-
lem (1)–(3) has been sol ed in he squa ed domain Ω = (0,1)2⊂R2, using (P1,b,P1) – P1FE
in wo di e en meshes: an uns uc u ed mesh wi h 3,792 iangles (buil wi h 20 edges on each
bounda y segmen , see Figu e 5a) and a s uc u ed mesh wi h he same numbe o bounda y edges
(Figu e 6a). Then, in bo h cases h≃1/20.
We ha e ixed he Neumann condi ion ν∂u
∂z = 1 on he op bounda y, while u= 0 on he es o ∂Ω
and = 0 on all ∂Ω. In his case, he implemen a ion has been de eloped in F eeFem++ [PHLHM].
In bo h simula ions (uns uc u ed and s uc u ed mesh) he eloci y iled p esen simila be-
ha io (Figu es 5b and 6b). Bu he p essu e unknown, ha is co ec in he uns uc u ed case
18
(a) Uns uc u ed mesh. (b) Veloci y ield. (c) P essu e.
Figu e 5: (P1,b,P1) – P1elemen s in a uns uc u ed mesh.
(a) S uc u ed mesh. (b) Veloci y ield. (c) P essu e.
Figu e 6: (P1,b,P1) – P1elemen s in a s uc u ed mesh.
19
Figu e 7: Veloci y and p essu e e o s o (P1,b,P1) – P1(le ) and (P1,b,P1,b) – P1( igh )
(Figu e 5c), p esen s clea non-physical oscilla ions in he uns uc u ed one (Figu e 6c) which has
a somewha “checke -boa d” s uc u e simila o classical uns able app oxima ions o S okes p ob-
lem [GR86]. Hence his expe imen con i m ha (P1,b,P1) – P1is s able in meshes which canno be
spli by any ho izon al segmen , as p edic ed by he 2Dnume ical analysis de eloped in Sec ion 3.1.
Tes 3 ((P1,b,P1)–P1and e o o de s). Thi dly, o gi e an es ima e o he con e gence o de s,
we ha e conside ed a 2D example in Ω = (0,1)2whose exac solu ion is:
u(x, y) = cos(2πx) sin(2πy)−sin(2πy), (x, y) = −u(y, x),(29)
p(x, y)=2π(cos(2πy)−cos(2πx)).(30)
No e ha ∂xu+∂y = 0, (u, )|∂Ω= 0 and RΩp= 0. The ex e nal o ce , is calcula ed so ha he
momen um equa ion (1) hold.
The p oblem has been sol ed in some uns uc u ed meshes ( ecall ha he heo y p edic s
ins abili y o (P1,b,P1) – P1in s uc u ed meshes), wi h h≃2−2, 2−3, ..., 2−8. The absolu e e o s
ob ained ha e been plo ed in Figu e 7a, while Table 1shows he alues o he con e gence o de s
associa ed o hose e o s, which a e calcula ed as log(eh2/eh1)/log(h2/h1), when h1< h2 a el
h ough he di e en mesh sizes.
Taking in o accoun e o es ima es (19), we can s a e he ollowing conclusions:
•Fo bo h componen s o he eloci y ield, op imal con e gence is ob ained, ha is o de O(h2)
in L2(Ω) and o de O(h) in H1(Ω). Fo p essu e, op imal o de O(h) in L2(Ω) is ob ained.
•The esul s a e e y simila (almos iden ical o he eloci y componen s) o he classical
mini-elemen , (P1,b,P1,b) – P1, whose e o s a e shown in Figu e 7b. I is signi ican o no e
ha he combina ion (P1,b,P1) – P1p ese es he e o o de s while equi es a mino numbe
o deg ees o eedom han (P1,b,P1,b) – P1and a smalle compu a ional e o . On he o he
hand, o app oach he Aniso opic S okes p oblem (see (AS) abo e), (P1,b,P1) – P1is s able
in mos meshes, while (P1,b,P1,b) – P1looses i s s abili y (c . [GR14]).
Tes 4 (Pos -p ecessing Algo i hm 1). We ha e used Algo i hm 1(wi h = 0.15) o modi ying
a s ongly s uc u ed mesh (de ined by 16 ×16 in e als in he uni squa e, see Figu e 8a), a i ing
20
h2−32−42−52−62−72−8
uku−uhkL21.099 1.709 1.952 1.876 1.988 2.019
ku−uhkH1
01.155 1.053 1.092 0.990 1.023 1.041
k − hkL21.914 2.076 2.071 1.981 2.059 2.058
k − hkH1
00.912 1.046 1.010 0.994 1.022 1.022
pkp−phkL20.558 1.817 0.808 1.414 0.809 0.988
Table 1: E o o de s o eloci ies and p essu e ((P1,b,P1) – P1FE)
a he uns uc u ed mesh gi en in Figu e 8b. P essu e o a s anda d ca i y-d i en es shows
ins abili ies o (P1,b,P1) – P1in he o iginal s uc u ed mesh (Figu e 8a) bu no in he pos -
p ocessed one (Figu e 8b).
Tes 5 (3D domain, (P1,b,P1,b,P1)–P1o (P1,b,P1,P1)–P1and ca i y es ). In he 3Dcase,
we ha e app oxima ed he solu ion o he S okes equa ions (1)–(3) in he cube domain Ω = (0,1)3⊂
R3, using bo h:
1. A s uc u ed mesh, cons uc ed by F eeFem++ [PHLHM] h ough he subdi ision in h ee
e ahed ons o each ones o he cubes esul ing o he di ision o Ω in o 323equal ec angula
pa allelepipeds.
2. a uns uc u ed mesh consis ing o 190,968 e ahed ons, cons uc ed by he mesh gene a o
Gmsh [GR09] and hen impo ed o F eeFem++.
The Di ichle condi ion u(x, y, z) = y(y−1), (x, y, z) = w(x, y, z) = 0 has been ixed on he op
bounda y, while w= 0 has been ixed on he es o ∂Ω (whe e w= (u, , w) deno es he solu ion).
The FE app oxima ions (P1,b,P1,b,P1,b) – P1, (P1,b,P1,b,P1) – P1and (P1,b,P1,P1) – P1ha e been
used in each one o he wo meshes and he esul ing su ace plo s o he app oxima ed p essu e,
ph, is shown in Figu e 9(s uc u ed mesh) and Figu e 10 (uns uc u ed one).
Owing o he choice o a colo map ha highligh s he small absolu e alues o ph, spu ious
oscilla ions o he p essu e a e e iden o (P1,b,P1,b,P1) – P1and (P1,b,P1,P1) – P1in he s uc u ed
mesh (Figu es 9b and 9c), whe e we ha e s a ed hei ins abili y. On he o he hand, Figu es 10b
and 10c sugges a co ec beha iou (excep ing some small oscilla ions on he op o he domain) i
he mesh is uns uc u ed, con i ming heo e ical esul s in Sec ion 3.2.
Tes 6 (3Ddomain, en iching wi h bubble in he co ec di ec ion). In his es i has
been conside ed a mesh Tho he cubic Ω = (0,1)3, which p esen s a clea s uc u e only in he
z-di ec ion. Namely, This z–s uc u ed while x–uns uc u ed and y–uns uc u ed.
In conc e e, we ha e conside ed a he op o he domain, S, an uns uc u ed iangula ion Th2o
S(de ined wi h 64 sub-in e als on ∂S, i.e. h≃0.015). Then a 3Dmesh, Th, has been cons uc ed,
de ining i om he ex ension o Th2along 16 laye s uni o mly dis ibu ed in he z-di ec ion.
We ha e p og amed he lid d i en ca i y es (as de ailed in p e ious es s) o (P1,b,P1,b,P1,b) – P1,
(P1,P1,P1,b) – P1and (P1,b,P1,P1) – P1FE combina ions. In he esul ing g aphics o p essu e,
shown in Figu e 11, i can be seen ha he wo i s cases (Figu es 11a and 11b) p esen a simila co -
ec quali a i e beha iou , while he hi d one (Figu e 11c) p esen s some non physical oscilla ions
(which, in his g aphics, a e ampli ied by he selec ion o he colo map).
21

(a) O iginal s uc u ed mesh (b) Pos -p ocessed uns uc-
u ed mesh
(c) P essu e in s uc u ed mesh (d) P essu e in pos -p ocessed
mesh
Figu e 8: Tes o Algo i hm 1in a s uc u ed mesh.
(a) (P1,b,P1,b,P1,b) – P1(b) (P1,b,P1,b,P1) – P1(c) (P1,b,P1,P1) – P1
Figu e 9: Compa ison o bubble 3DFE in a s uc u ed mesh.
22
(a) (P1,b,P1,b,P1,b) – P1(b) (P1,b,P1,b,P1) – P1(c) (P1,b,P1,P1) – P1
Figu e 10: Compa ison o bubble 3DFE in an uns uc u ed mesh.
(a) (P1,b,P1,b,P1,b) – P1(b) (P1,P1,P1,b) – P1(c) (P1,b,P1,P1) – P1
Figu e 11: Compa ison o bubble 3DFE in a z–s uc u ed mesh.
This beha iou ag ee wi h he heo y de eloped in Sec ion 3.2, which sugges he s abili y o
(P1,P1,P1,b) – P1. Indeed, o his mesh, we can hope ha he mac o-elemen s c
M∈c
Mh, a e
x–uns uc u ed and y–uns uc u ed, hence adding bubble unc ions o he i s componen s o he
eloci y ield is no manda o y. Bu hese mac o-elemen s a e z–s uc u ed and hen, o ob aining
egula i y, i is necessa y and su icien adding bubble unc ions o he hi d componen . Also due
o his ac , we canno hope s abili y o (P1,b,P1,P1) – P1, (see Figu e 11c).
4 Abou s abili y o (P2,P1)–P1FE
4.1 The 2D case
In his sec ion we in oduce a P2con inuous space o he app oxima ion o he ho izon al componen
o eloci y ield,
Uh={uh∈H1
0(Ω) ∩C0(Ω) / uh|T∈ P2,∀T∈ Th},
while Vhand Pha e app oxima ed by P1con inuous elemen s, as de ined in (8) and (11).
As in Sec ion 3.1, le us suppose Thsa is ying Assump ion 1and le c
Mhbe he a e ex-
cen e ed mac o-elemen pa i ioning o Th, as in Figu e 12. Le U0,M be he space o unc ions in
H1
0(M)∩C0(M) which a e P2in each elemen T⊂Mand le V0,M and PMas in Sec ion 3.1. In
23
q0
q1
q2
q3
q4
q5
s1
s2
s3s4
s5
T1
T2
T3
T4T5
Figu e 12: (P2,P1) – P1mac o-elemen wi h a unique in e io e ex, q0, and n = 5.
his amewo k, Lemma 2means ha a su icien condi ion o he disc e e in -sup condi ion (6)
is ha c
Mhsa is ies (12), namely (P2,P1) – P1is egula in e e y c
M∈c
Mh. The ollowing esul
cha ac e izes his condi ion in unc ion o he numbe o elemen s con ained in c
Mand he numbe
o e ices which a e ho izon ally aligned wi h q0. See ha his case is a li le mo e echnical han
he simila one (Theo em 3) p esen ed in Sec ion 3.1 (see Rema k 8 o u he de ails).
Theo em 9. Le c
M∈c
Mhbe a mac o-elemen which can be w i en as union o n elemen s which
sha e exac ly one in e io common e ex, q0.
1. I c
Mis y–s uc u ed (i.e. he e a e wo e ices ho izon ally aligned o q0), hen (P2,P1)–P1
is no egula in c
M, namely condi ion (12) does no hold.
2. I he e is jus one e ex in c
Mho izon ally aligned wi h q0, hen (P2,P1)–P1is egula in
M.
3. I he e is no e ex ho izon ally aligned wi h q0:
(a) I n is odd, hen (P2,P1)–P1is egula in c
M.
(b) I n is e en, hen (P2,P1)–P1is egula in c
Mi and only i he ollowing algeb aic
condi ion does no hold:
n
X
i=1
(−1)ico (σi)1
|Ti|+1
|Ti+1|= 0,(31)
whe e co (σi) = cos(σi)/sin(σi)is he co angen unc ion and σia e he angles in oduced
be o e De ini ion 6
Rema k 6. The case (P1,P2) – P1is analogue and, o b e i y, is no enounced he e.
P oo . We spli he p oo in o ou s eps.
24
S ep 1. Algeb aic cha ac e iza ion o Nc
M.
Le Ti,i= 1, ..., n , be he elemen s o c
M. I ph∈Nc
M, le us de ine he o m unc ions ph|Ti=
ai+bix+ciy. As in Figu e 12, le us deno e by q0 he only e ex in e io o c
M,qi,i= 1, ...n , he
e ices in ∂c
Mand si,i= 1, ..., n he midpoin s o he edges in he in e io o c
M.
Choosing h= 0 and uh∈U0,
c
Mde ined as uh= 1 on a midpoin siand uh= 0 on all o he
deg ees o eedom o c
M(sj,j= 1, ..., n ,j6=iand qk,k= 0, ..., n ), hen
0 = −Zc
M
∇ · (uh, h)ph=−Zc
M
∂xuhph=Zc
M
uh∂xph=
=biZTi
uh+bi+1 ZTi+1
uh=|Tj|
3bj+|Tj+1|
3bj+1,
whe e we ha e applied he quad a u e o mula RT =|T|
3P3
i=1 (sT
i),which is exac on P2i sT
i
a e he midpoin s o he edges o T. Again, we ha e iden i ied he index i=n + 1 wi h i= 1.
De ining αi=|Ti|, he ollowing linea sys em o biis ob ained:







α1α20. . . 0 0
0α2α3. . . 0 0
.
.
..
.
..
.
.....
.
..
.
.
000. . . αn −1αn
α10 0 . . . 0αn







.






b1
b2
.
.
.
bn −1
bn







=






0
0
.
.
.
0
0







(32)
A simple calcula ion shows ha , i b1=−b/α1 hen b2=b/α2,b3=−b/α3,. . . ,bn = (−1)n b/αn
and b1= (−1)n +1b/α1. The e o e, he solu ion o (32) can be cha ac e ized as ollows:



I n is odd, bi= 0,∀i= 1, ..., n .
I n is e en, bi=(−1)i
αi
b, ∀i= 1, ..., n ,∀b∈R.(33)
Taking uh= 1 in q0,uh= 0 on all o he deg ees o eedom in c
Mand h= 0 does no p o ide
any new in o ma ion, because he midpoin quad a u e o mula means ha RTiuh= 0 o all i.
Finally le us choose uh= 0 and h∈V0,
c
Mde ined as h= 1 on he in e io e ex, q0, and o
cou se h= 0 on all e ices on ∂c
M. Then 0 = ( h, ∂yph) = Pn
i=1 ciRTi h=Pn
i=1 ci|Ti|/3 hence
n
X
i=1
αici= 0.(34)
Now assuming, wi hou loss o gene ali y, q0= (0,0) and imposing he con inui y o phon q0, i
is s aigh o wa d ha
a1=... =an =: a.
I qi= (qx
i, qy
i)∈∂c
M,i= 1, ..., n , being q0and qi he common e ices o Tiand Ti+1 (Figu e 12),
con inui y o phin qimeans:
qy
i(ci+1 −ci) = qx
i(bi+1 −bi),∀i= 1, ..., n .(35)
25
(a) (P2,P2,P2) – P1(b) (P2,P2,P1) – P1(c) (P2,P1,P1) – P1
Figu e 18: Compa ing he p essu e in a 3Duns uc u ed mesh.
q11
q12
q13
q21
q22
q23
q31
q32
q33
Figu e 19: (Q2,Q1) – Q1mac oelemen
loss o gene ali y i can be assumed ha he cen al e ex is loca ed a he o igin o coo dina es;
i.e. q22 = (0,0) using he no a ion o Figu e 19.
Le us de ine he ollowing FE spaces in M:
U0,M ={uh∈H1
0(M)∩C0(M)/ uh|K∈ Q2,∀K∈M},
V0,M ={ h∈H1
0(M)∩C0(M)/ |K∈ Q1,∀K∈M}
PM={ph∈C0(M)/ p|K∈ Q1,∀K∈M}.
Le us conside he Q1 unc ion phde ined as
ph(x, y) = (a+cy i y > 0,
a−cy i y≤0,(39)
whe e a, c ∈R,c6= 0. Then each h∈V0,M can be w i en as α φ22 o any α∈R, whe e φ22 is he
Q1basis unc ion ha is equal o 1 in q22 and ze o in all o he e ices o M. Hence, i we apply
he 2D apezoidal ule (which is exac in Q1) and aking in o accoun ha ∂xph= 0,
ZM
(∂xuh+∂y h)ph=−ZM
h∂yph=−c
|M+|ZM+
αφ22 +c
|M−|ZM−
αφ22
=c
|M+|X
K⊂M+
|K|
4α−c
|M−|X
K⊂M−
|K|
4α=c
4α−c
4α= 0.
The e o e he combina ion (Q2,Q1) – Q1is no s able in quad angula mac o-elemen s.
As in Sec ion 4, i is no di icul o ex end he p e ious coun e example o demons a e ha
he e exis s a amily o no null p essu es ph∈Phsa is ying (∇ · (uh, h), ph) = 0 o all uh∈Uh
32

(a) (b)
Figu e 20: P essu e o (Q2,Q2) – Q1(le ) and (Q2,Q1) – Q1( igh ). The la e , p esen s spu ious
ocilla ions.
and h∈Vh. Hence, in ec angula s uc u ed meshes, he S okes in -sup condi ion (6) does no
hold o he (Q2,Q1) – Q1combina ion.
Tes 11 ((Q2,Q1)–Q1and ca i y es ). Nume ical expe imen s con i m he (Q2,Q1) – Q1in-
s abili y, as can be seen in Figu e 20. We conside he 2DS okes p oblem (1)–(2) in Ω = (0,1)2.
Di ichle bounda y condi ions o uha e been applied; u= 1 on he op {y= 1}and u= 0 on he
es o ∂Ω. Abou , a na u al bounda y condi ion ∂
∂n= 0 has been chosen on ∂Ω. Bo h s able
(Q2,Q2) – Q1and uns able (Q2,Q1) – Q1combina ions o FE ha e been es ed in a mesh composed
o 15 ×15 ec angula elemen s. As expec ed, Figu e 20 shows spu ious p essu e oscilla ions in he
la e case. This expe imen has been p og ammed in C++ wi h LibMesh [KPSC06], a pa allel FE
lib a y which suppo s na i ely he Q1and Q2elemen s.
6 Conclusions
In his pape we ha e de eloped wo new amilies o FE spaces app oxima ing he S okes p oblem,
whe e con inuous and piecewise linea unc ions P1a e en iched, only o one componen o he
eloci y, by ei he bubble P1,b o quad a ic polynomial P2. These combina ions ha e been deno ed,
in he 2Dcase, by (P1,b,P1) – P1and (P2,P1) – P1.
In o de o compa e he use o bubble unc ions, we ha e seen ha combina ion (P1,b,P1) – P1in
uni o mly uns uc u ed meshes is mo e e icien han he well-known mini-elemen (P1,b,P1,b) – P1,
because (P1,b,P1) – P1p ese es he i s o de accu a e o (P1,b,P1,b) – P1while equi es a mino
numbe o deg ees o eedom and hen a smalle compu a ional e o .
On he o he hand, we ha e also p o ed ha combina ion (P2,P1) – P1in mos uni o mly un-
s uc u ed meshes is s able bu i does no p ese e he second o de accu acy o he well-known
Taylo -Hood elemen (P2,P2) – P1, al hough (P2,P1) – P1uses a mino numbe o deg ees o ee-
dom.
Mo eo e , he s abili y cons ain s o (P2,P1) – P1a e mo e es ic i e han o (P1,b,P1) – P1,
because hey depend no only on he s uc u e o he mesh bu also depend on he algeb aic
condi ion (34). Hence, in p inciple, he numbe o meshes whe e (P2,P1) – P1is s able is sligh ly
less han in he (P1,b,P1) – P1case.
Finally, he same ype o esul s a e also deduced in 3Ddomains o he case o bubble unc ions.
33
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34