On the stability of approximations for the Stokes problem using different finite element spaces for each component of the velocity

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 Re e ences [AG01] P. Az´e ad and F. Guill´en. Ma hema ical jus i ica ion o he hyd os a ic app oxima ion in he p imi i e equa ions o geophysical luid dynamics. Siam J. Ma h. Ana., 33(4):847–859, 2001. [AR09] Douglas N. A nold and Ma ie E. Rognes. S abili y o Lag ange elemen s o he mixed Laplacian. Calcolo, 2009. [Az´e94] P. Az´e ad. Analyse e app oxima ion du p obl`eme de S okes dans un bassin peu p o ond. C. R. Acad. Sci. Pa is S´e . I Ma h., 318(1):53–58, 1994. [Az´e96] P. Az´e ad. Analyse des qua ions de Na ie -S okes en bassin peu p o ond e de l’qua ion de anspo . PhD hesis, Neuch el, 1996. [BBF13] Daniele Bo i, F anco B ezzi, and Michel Fo in. Mixed Fini e Elemen Me hods and Applica ions, ol- ume 44 o Sp inge Se ies in Compu a ional Ma hema ics. Sp inge Be lin Heidelbe g, Be lin, Heidelbe g, 2013. [BF91] F. B ezzi and M. Fo in. Mixed and Hyb id Fini e Elemen Me hods. Sp inge -Ve lag, New-Yo k, 1991. [BL92] O. Besson and M.R. Laydi. Some es ima es o he aniso opic Na ie -S okes equa ions and o he hyd os a ic app oxima ion. Ma h. Mod. and Num. Anal, Vol. 26(7):855–865, 1992. [CB09] B. Cushman-Roisin and J. M. Becke s. In oduc ion o Geophysical Fluid Dynamics - Physical and Nume ical Aspec s. Academic P ess, 2009. [GR86] V. Gi aul and P.-A. Ra ia . Fini e elemen me hods o Na ie -S okes equa ions. Sp inge -Ve lag, 1986. [GR09] C. Geuzaine and J.-F. Remacle. Gmsh: A 3-D ini e elemen mesh gene a o wi h buil -in p e- and pos -p ocessing acili ies. In e na ional Jou nal o Nume ical Me hods in Enginee ing, 79:1309 – 1331, 2009. [GR14] F. Guill´en-Gonz´alez and J.R. Rod ´ıguez-Gal ´an. Analysis o he hyd os a ic s okes p oblem and ini e- elemen app oxima ion in uns uc u ed meshes. Nume . Ma h., 2014. Accep ed, doi 10.1007/s00211-014- 0663-8. [KPSC06] B.S. Ki k, J.W. Pe e son, R.H. S ogne , and G.F. Ca ey. libmesh: a c++ lib a y o pa allel adap i e mesh e inemen /coa sening simula ions. Eng. wi h Compu ., 22(3):237–254, Decembe 2006. [LMW+12] Ande s Logg, Ken -And e Ma dal, Ga h N. Wells, e al. Au oma ed Solu ion o Di e en ial Equa ions by he Fini e Elemen Me hod. Sp inge , 2012. [PHLHM] O. Pi onneau, F. Hech , A. Le Hya ic, and J. Mo ice. F eeFEM++, h p://www. ee em.o g/. [QZ07] J. Qin and S. Zhang. S abili y and app oximabili y o he P1-P0 elemen o S okes equa ions. In e na- ional Jou nal o Nume ical Me hods in Fluids, 54:497–515, 2007. [S e84] R. S enbe g. Analysis o mixed ini e elemen me hods o he S okes p oblem: A uni ied app oach. Ma h Compu , 42(165):9–23, Janua y 1984. [S e90] R. S enbe g. A echnique o analysing ini e elemen s me hods o iscuous incomp essible low. In e - na ional Jou nal o Nume ical Me hods in Fluids, 11:835–948, 1990. 34