From a cell model with active motion to a Hele–Shaw-like system: a numerical approach

FROM A CELL MODEL WITH ACTIVE MOTION TO A HELE-SHAW-LIKE SYSTEM. A NUMERICAL APPROACH FRANCISCO GUILL´ EN-GONZ´ ALEZ†AND JUAN VICENTE GUTI´ ERREZ-SANTACREU‡ Abs ac . In his pape we deal wi h he nume ical solu ion o a Hele–Shaw-like sys em ia a cell model wi h ac i e mo ion. Con e gence o app oxima ions is es ablished o well-posed ini ial da a. These da a a e chosen in such a way he ime de i a e is posi i e a he ini ial ime. The nume ical me hod is cons uc ed by means o a ini e elemen p ocedu e oge he wi h he use o a closed-nodal in eg a ion. This gi es ise o an algo i hm which p ese es posi i i y whene e a igh -angled iangula ion is conside ed. As a esul , uni o m-in- ime a p io i es ima es a e p o en which allows us o pass o limi owa ds a solu ion o he Hele–Shaw p oblem. 2010 Ma hema ics Subjec Classi ica ion. 92C50, 35B25, 35K55, 35Q92, 35R35, 76D27. Keywo ds. Fini e-elemen app oxima ion; nonlinea di usion; ee bounda y p oblems; Hele- Shaw lows. Con en s 1. In oduc ion 1 1.1. The models 1 1.2. No a ion 3 1.3. Ou line 3 2. Spa ial disc e iza ion 4 2.1. Fini e-elemen app oxima ion 4 2.2. The nume ical scheme 7 2.3. Main esul 8 3. P oo o Theo em 2.2 8 3.1. A p io i ene gy es ima es 8 3.2. Passing o he limi 12 4. An algo i hm on uns uc u ed meshes 17 5. Nume ical simula ion 19 5.1. Tempo al in eg a ion 19 5.2. Compu a ional expe imen s 19 Re e ences 22 1. In oduc ion 1.1. The models. Tumou cells a e ac i e mechanical sys ems ha a e able o p oduce o ces which cause andom mig a ion [3,8,14]. This mo emen is due o a he complica e mechanisms which occu inside cells and gi e ise o changes in cell shape. Ano he impo an mechanism unde which cells mo e is p essu e [5,8,13] as a consequence o space compe i ion gene a ed by Da e: No embe 9, 2018. This wo k was pa ially suppo ed by Minis e io de Econom´ıa y Compe i i idad unde Spanish g an MTM2015- 69875-P wi h he pa icipa ion o FEDER. 1 2 F. GUILL´ EN-GONZ ´ ALEZ AND J. V. GUTI´ ERREZ-SANTACREU cell p oli e a ion i sel . In he se ing up we ake in o conside a ion a e y simpli ied model which inco po a es he wo spa ial e ec s o desc ibing umou g ow h. Le Ω be a connec ed, open, bounded se o Rd, wi h d= 2 o 3, and [0, T] a ime in e al. Conside he cell model wi h ac i e mo ion [11] which consis s in inding a umou cell popula ion densi y n: Ω ×[0, T]→R+sa is ying (1) ∂ n−∇·(n∇p(n)) −ν∆n=n G(p(n)) in Ω ×(0, T), subjec o he (na u al) bounda y condi ion (2) ∇n·n= 0 on ∂Ω×(0, T), wi h nbeing he ou wa ds uni no mal ec o on he bounda y ∂Ω, and he ini ial condi ion (3) n| =0 =n0in Ω. He e p: [0,+∞)→[0,+∞) is de ined by (4) p=p(n) := k k−1nk−1∀n≥0,(k∈N, k ≥2), and G=G(p) is a unca ed dec easing unc ion such ha he e exis s Pmax >0 ( he homeos a ic p essu e) wi h (5) G(0) >0, G(p) = 0 ∀p≥Pmax >0,and G0(p)<0∀p∈(0, Pmax). In he abo e, Gs ands o he dec ease in he umuo cell g ow h a e when space is limi ed; he lack o space is go e ned by he local p essu e p, he pa ame e Pmax is he maximum p essu e h eshold ha umou cells can exceed be o e en e ing a quiescen s a e, and he pa ame e ν > 0 ep esen s he e ec o including he ac i e ( andom) mo ion o cells. I should be no ed ha he ela ionship o p(n) gi en in (4) is in e ible o n≥0: (6) n(p) := k−1 kp1/(k−1) ∀p≥0. In his wo k we assume ha {n0 k}k∈Nis a sequence o ini ial da a (3) o (1) such ha (7) 0 ≤p(n0 k)≤Pmax in Ω, and ha he e exis s a limi unc ion n0 ∞such ha (8) n0 k→n0 ∞in Lp(Ω)-s ongly o any p < ∞as k→ ∞. Consequen ly, de ining Nmax(k) := n(Pmax) wi h n(·) being gi en in (6), we ha e (9) 0 ≤n0 k≤Nmax(k) in Ω. om which we in e ha he e mus exis N0>0 such ha Nmax(k)≤N0. Unde he abo e assump ions, equa ion (1) gene a es a sequence o solu ions {nk}k∈Nwhich lead o a solu ion desc ibing he dynamics o umou g ow h as a ee-bounda y p oblem. To be mo e p ecise, he con e gence o he solu ions {nk}k∈No he ac i e mo ion cell model p oblem (1)-(3) owa ds a weak solu ion o a Hele–Shaw-like sys em, as he pa ame e kgoes o in ini y, was p o en in [11]. This limi sys em eads as ollows. Find n∞: Ω ×[0, T]→R+and p∞: Ω ×[0, T]→R+such ha (10) ∂ n∞−∆p∞−ν∆n∞=n∞G(p∞) in Ω ×(0, T), subjec o (11) n∞| =0 =n0 ∞in Ω, (12) ∇n∞·n= 0 and ∇p∞·n= 0 on ∂Ω×(0, T), APPROXIMATION OF A HELE-SHAW-LIKE SYSTEM 3 join ly o he complemen a y ela ion (13) p∞(∆p∞+G(p∞)) = 0 in Ω ×(0, T). The key poin in es ablishing con e gence is imposing ha ∂ nk(0) ≥0. Mo eo e , equa ion (10) is equi alen o sol ing (14) ∂ n∞−∇·(n∞∇p∞)−ν∆n∞=n∞G(p∞) in Ω ×(0, T). This equi alence will be accomplished due o he equali y ∇p∞=n∞∇p∞, which comes om he equali ies p∞∇n∞= 0 and p∞n∞=p∞. In his pape , we shall be conce ned wi h he con e gence o a ini e elemen scheme, he ime a iable being con inuous, o he ac i e mo ion cell model p oblem (1)-(3) owa ds he Hele-Shaw sys em (10)-(13) as he space disc e e pa ame e hgoes o ze o and kgoes o in ini y. 1.2. No a ion. We will assume he ollowing no a ion h oughou his pape . Le O ⊂ RM, wi h M≥1, be a Lebesgue-measu able se and le 1 ≤p≤ ∞. We deno e by Lp(O) he space o all Lesbegue-measu able eal- alued unc ions, :O → R, being p h-summable in O o p < ∞ o essen ially bounded o p=∞, and by k kLp(O)i s no m. When p= 2, he L2(O) space is a Hilbe space whose inne p oduc is deno ed by (·,·). To sho en he no a ion, he no m k·kL2(Ω) is abb e ia ed by k·k. Le α= (α1, α2, ..., αM)∈NMbe a mul i-index wi h |α|=α1+α2+... +αM, and le ∂αbe he di e en ial ope a o such ha ∂α=∂ ∂x1α1...∂ ∂xdαM. Fo m≥0 and 1 ≤p≤ ∞, we de ine Wm,p(O) o be he Sobole space o all unc ions whose mde i a i es a e in Lp(O), wi h he no m k kWm,p(O)= X |α|≤mk∂α kp Lp(O)  1/p o 1 ≤p < ∞, k kWm,p(O)= max |α|≤mk∂α kL∞(Ω), o p=∞, whe e ∂αis unde s ood in he dis ibu ional sense. Fo p = 2, Wm,2(O) will be deno ed by Hm(O). We also conside C∞(O) o be he space o unc ions con inuously di e en iable any numbe o imes, and C∞ c(O) o be he subspace o C∞(O) wi h compac suppo in O. Spaces o Bochne -measu able unc ions om a ime in e al [0, T] o a Banach space Xwill be deno ed as Lp(0, T;X) wi h k kL2(0,T ;X)=RT 0k (s)kp Xdsi 1 ≤p < ∞o k kL∞(0,T,X)= ess sups∈(0,T)k (s)kX<∞i p=∞. 1.3. Ou line. Nex we ske ch he emaining con en o his wo k. In sec ion 2 we p esen ou ini e-elemen spaces and some p elimina y esul mainly conce ning in e pola ion ope a o s. Fu - he mo e, we se ou ou ini e elemen nume ical me hod, whe e he ime a iable emains con in- uous, and he main esul o his pape . Nex is sec ion 3 which is de o ed o demons a ing he main esul . Fi s ly, a disc e e maximum p inciple o ini e-elemen app oxima ions is achie ed by assuming a pa i ion o he compu a ional domain being made up o igh -angled simplexes, and a p io i es ima es a e also es ablished independen o (h, k) wi h hbeing he space pa ame e associa ed o ou ini e-elemen space. As a esul , we a e able o p o e posi i i y o he ime de- i a i e o ini e-elemen app oxima ions. Then be e a p io i ene gy es ima es lead o ob aining compac ness o passing o he limi as (h, k)→(0,+∞). In sec ion 4, we p opose a a ian o ou nume ical algo i hm o nonob use iangula ions which keeps wi h a disc e e maximum p inciple 4 F. GUILL´ EN-GONZ ´ ALEZ AND J. V. GUTI´ ERREZ-SANTACREU and posi i e o he disc e e ime bu whose con e gence is no clea . Finally, in sec ion 4, some nume ical expe imen s a e p esen ed o s udying he beha io o se e al pa ame e s. 2. Spa ial disc e iza ion 2.1. Fini e-elemen app oxima ion. He ein we in oduce he hypo heses ha will be equi ed along his wo k. (H1) Le Ω be a bounded domain o Rd(d= 2 o 3) wi h a polygonal o polyhed al Lipschi z- con inuous bounda y. (H2) Le {Th}h>0be a amily o shape- egula , quasi-uni o m iangula ions o Ω made up o igh -angled simplexes being iangles in wo dimensions and e ahed a in h ee dimen- sions, so ha Ω = ∪K∈ThK, whe e h= maxK∈ThhK, wi h hKbeing he diame e o K. Fu he , le Nh={ai}i∈Ideno e he se o all he nodes o Th. (H3) Con o ming piecewise linea , ini e elemen spaces associa ed o Tha e assumed o app ox- ima ing H1(Ω). Le P1(K) be he se o linea polynomials on K; he space o con inuous, piecewise P1(K) polynomial unc ions on This hen deno ed as Nh=nh∈C0(Ω) : nh|K∈ P1(K)∀K∈ Th, whose Lag ange basis is deno ed by {ϕa}a∈Nh. We now gi e some auxilia y esul s o la e use. We begin by an in e se inequali y whose p oo can be ound in [4, Lem. 4.5.3] o [9, Lem. 1.138]. P oposi ion 2.1. Unde hypo heses (H1)–(H3), i ollows ha , (15) k∇nhkL2(K)≤Cin h−1 KknhkL2(K)∀K∈ Th,∀nh∈Nh, whe e Cin >0is a cons an independen o h. Le Ihbe he nodal in e pola ion ope a o om C0(Ω) o Nhand conside he disc e e inne p oduc (nh, nh)h=ZΩIh(nhnh) = X a∈Nh nh(a)nh(a)ZΩ ϕa o ∀nh, nh∈Nh, which induces he no m knhkh=p(nh, nh)hde ined on Nh. We ecall he ollowing local e o es ima e. See [4, Thm. 4.4.4] o [9, Thm. 1.103] o a p oo . P oposi ion 2.2. Unde hypo heses (H1)–(H3), i ollows ha , (16) kϕ−IhϕkL∞(K)≤Capph2 Kk∇2ϕkL∞(K)∀K∈ Th,∀ϕ∈W2,∞(K), whe e Capp >0is independen o h. We nex s a e he equi alence be ween he no ms k·khand k·kin Nhand a disc e e commu e app oxima ion p ope y o Ih. P oposi ion 2.3. Unde hypo heses (H1)–(H3), i ollows ha , o all nh, nh∈Nh, (17) knhk≤knhkh≤51/2knhk and (18) knhnh−Ih(nhnh)kL1(Ω) ≤Capphknhkk∇nhk, whe e Capp >0is independen o h. APPROXIMATION OF A HELE-SHAW-LIKE SYSTEM 5 P oo . We ha e knhk2=X a∈Nh n2 h(a)ZΩ ϕ2 a+X a6=e a∈Nh nh(a)nh(e a)ZΩ ϕaϕe a and knhk2 h=X a∈Nh n2 h(a)ZΩ ϕa. Since 1 = Pe a∈Nhϕe a, we w i e knhk2 h=X a,e a∈Nh n2 h(a)ZΩ ϕaϕe a=X a∈Nh n2 h(a)ZΩ ϕ2 a+X a6=e a∈Nh n2 h(a)ZΩ ϕaϕe a. Then knhk2 h−knhk2=X a>e a∈Nh (n2 h(a) + n2 h(e a)−2nh(a)nh(e a)) ZΩ ϕaϕe a =X a>e a∈Nh (nh(a)−nh(e a))2ZΩ ϕaϕe a≥0. F om he abo e equali y and Young’s inequali y, we ha e knhk2 h=knhk2+X a>e a∈Nh (nh(a)−nh(e a))2ZΩ ϕaϕe a ≤ knhk2+ 2 X a>e a∈Nh (n2 h(a) + n2 h(e a)) ZΩ ϕaϕe a =knhk2+ 2 X a∈Nh n2 h(a)ZΩ ϕaX e a<a ϕe a+ 2 X e a∈Nh n2 h(e a)ZΩ ϕe aX a>ea ϕa ≤ knhk2+ 4knhk2≤5knhk2. We now p o e (18). By using (16), we ob ain kIh(nhnh)−nhnhkL1(Ω) =X K∈ThkIh(nhnh)−nhnhkL∞(K)ZK 1 ≤Capp X K∈Th h2 Kk∇2(nhnh)kL∞(K)ZK 1. Since nh, nh∈P1(K) on K∈ Th, we w i e ∇2(nhnh)=2 d X i,j=1 ∂inh∂jnh. Then, om (15) and on no ing ha ∇nh,∇nha e piecewise cons an on each K∈ Th, we deduce ha kIh(nhnh)−nhnhkL1(Ω) ≤Capp X K∈Th h2 Kk∇nhkL∞(K)k∇nhkL∞(K)ZK 1 ≤Capp X K∈Th h2 KZK|∇nh||∇nh| ≤CappCin X K∈Th hKknhkL2(K)k∇nhkL2(K) ≤CappCin hknhkk∇nhk, om which we conclude ha (18) holds.  6 F. GUILL´ EN-GONZ ´ ALEZ AND J. V. GUTI´ ERREZ-SANTACREU We will need o use an (a e age) in e pola ion ope a o in o Nhwi h he ollowing p ope ies. In pa icula we use an ex ension o he Sco -Zhang in e pola ion ope a o o L1(Ω) unc ion. We e e o [15,10] and [2]. P oposi ion 2.4. Unde hypo heses (H1)–(H3), he e exis s an (a e age) in e pola ion ope a o Qh om L1(Ω) o Nhsuch ha (19) kQhψkWs,p(Ω) ≤Cs akψkWs,p(Ω) o s= 0,1and 1≤p≤ ∞, (20) kQh(ψ)−ψkWs,p(Ω) ≤Capph1+m−skψkWm+1,p(Ω) o 0≤s≤m≤1, and, o all ψ∈C∞(Ω) and nh∈Nh, (21) kQh(nhψ)−nhψkWs,p(Ω) ≤Capph1+m−sknhkWm,p(Ω)kψkWm+1,∞ o 0≤s≤m≤1. The key poin in p o ing a disc e e maximum p inciple is he ollowing p ope y which is ac- complished o igh -angled simplexes assumed in (H2). P oposi ion 2.5. Unde hypo heses (H1)–(H3), i ollows ha , o any diagonal nonnega i e ma ix D= diag(di)d i=1 (wi h di≥0), (22) D∇ϕa·∇ϕe a≤0a.e. in Ω i a6=e awi h a,e a∈ Nh. P oo . Fo e e y igh -angled d-simplex K∈ Tho e ices {ai}i=0,...,d wi h a0being he e ex suppo ing he igh angle, we deno e by Fai he opposi e ace o aiand by nai he ex e io ( o he d-simplex K) uni no mal ec o o he ace Fai. Le b Kbe he e e ence uni d-simplex wi h e ices b a0=0and b ai=ei,i= 1,··· , d, whe e {ei}i=1,··· ,d is he canonical basis o Rd. Le FKbe he in e ible a ine mapping ha maps b Kon o Kde ined by FKb x=a0+BKb x, whe e BK∈Rd×d is o hogonal. Le bϕb ai(b x) = ϕai(FKb x). Then we ha e b ∇bϕb ai=−1 d|b Fb ai| |b K|nb ai. In pa icula , nb ai=−eii i6= 0 and nb a0= [1,··· ,1]T. Thus, we ob ain b ∇bϕb ai·b ∇bϕb aj=1 d2|b Fb ai||b Fb aj| |b K|2nb ai·nb aj≤0 i i6=j. The e o e, by means o he change o a iable x=a0+BKb x, i ollows ha ∇ϕai=BKb ∇bϕb aiand hence D∇ϕai·∇ϕaj=DBKb ∇bϕb ai·BKb ∇bϕb aj=1 d2|b Fb ai||b Fb aj| |b K|2nT b aiBT KDBKnb aj≤0 i i6=j because, since BKis a o hogonal ma ix, he inne p oduc s de ined by Dand BT KDBKp ese es angles.  Rema k 2.1. When D=Idwi h Idbeing he d×diden i y ma ix, p ope y (22)can be p o ed o nonob use iangula ions [7]. Then p ope y (22)can be somewha seen a gene aliza ion es ic ed o igh -angled iangula ions. APPROXIMATION OF A HELE-SHAW-LIKE SYSTEM 7 Le us now in oduce he disc e e Laplacian associa ed o he mass-lumping scala p oduc (·,·)h. Fo any Σh∈Nh, le −e ∆hΣh∈Nhsol e (23) −(e ∆hΣh, nh)h= (∇Σh,∇nh)∀nh∈Nh. We end up wi h a compac ness esul [1, Lm. 2.4] needed in p o ing he equi alence be ween p oblems (10) and (14). Theo em 2.1. Assume ha (H1)-(H3) holds. Le 2d d+2 < ` < ∞. Suppose ha {ρh,k}h,k≥0⊂ L2(0, T;L2(Ω)) is such ha ρh,k( , ·)∈Nh o all ∈[0, T]and sa is ies kρh,kkH1(0,T ;L`(Ω)) +kρh,kkL∞(0,T ;L2(Ω))∩L2(0,T;H1(Ω)) +ke ∆hρh,kkL2(0,T ;L2(Ω)) ≤Cda . Then he e exis a subsequence {ρh,k}h,k>0(no elabeled) and a limi unc ion ρ, such ha ρh,k →ρin L2(0, T, H1(Ω))-s ongly as (h, k)→(0,+∞). He ea e Cwill deno e a gene ic cons an whose alue may change a each occu ence. This cons an may depend on he da a p oblem and he cons an s Cin ,Capp,Ccom and Cda . 2.2. The nume ical scheme. In o de o a oid dense echnical calcula ions, we assume o sim- plici y ha each elemen K∈ Thhas i s edges lined up wi h he axes. The nume ical scheme elies on a ini e-elemen me hod combined wi h a closed-nodal in eg a ion applied o he ime-de i a i e and p essu e-mig a ion e ms. Thus ou nume ical me hod which consis s in inding nh,k ∈C1([0, T]; Nh) such ha (24) (∂ nh,k, nh)h+ (∇Ih((nh,k)k),∇nh) + ν(∇nh,k,∇nh) = (G(p(nh,k))nh,k, nh)h∀n∈Nh nh,k(0) = n0 h,k, wi h p(nh,k) = k k−1(nh,k)k−1. Equi alen ly, we may w i e (24)1as (25) (∂ nh,k, nh)h+ (D(nh,k)∇nh,k,∇nh) + ν(∇nh,k,∇nh)=(G(p(nh,k))nh,k, nh)h, whe e D(nh,k) is a piecewise cons an , d×ddiagonal ma ix unc ion wi h espec o Thde ined as ollows. Le K∈ Thwi h e ices {ai}i=0,··· ,d whe e a0co esponds o he igh angle. Then (26) [D(nh,k)|K]ii =   (nh,k)k(ai)−(nh,k)k(a0) nh,k(ai)−nh,k(a0)i nh,k(ai)−nh,k(a0)6= 0, 0 i nh,k(ai)−nh,k(a0)=0. By he mean alue heo em, one can w i e (27) [D(nh,k)|K]ii =k(nh,k)k−1(ξi), whe e ξi=αai+ (1 −α)a0 o a ce ain α∈(0,1). The abo e choice o he sequence o {n0 h,k}h,k>0is as ollows. Le {n0 k}k∈N⊂H1(Ω) ∩L∞(Ω) sa is y (7) and (9). Then we selec n0 h,k =Qh(n0 k) so ha (28) 0 ≤n0 h,k(a)≤Nmax(k)∀a∈ Nh,k∇n0 h,kk ≤ Cs abk∇n0 kk, (29) n0 h,k →n0 kin H1(Ω)-s ongly as h→0. The e is an addi ional echnicali y ega ding he sequence o ini ial da a ha we mus conside : (H4) Assume {nh,k}h,k>0 o be such ha (30) −(∇Ih(n0 h,k)k,∇nh)−ν(∇n0 h,k,∇nh)+(G(p(n0 h,k))n0 h,k, nh)h≥0∀nh∈Nhwi h nh≥0. 8 F. GUILL´ EN-GONZ ´ ALEZ AND J. V. GUTI´ ERREZ-SANTACREU Rema k 2.2. This las condi ion is ela ed o imposing ∂ nh,k(0) ≥0which is c ucial o p o e he k→+∞limi . The exis ence and uniqueness o a solu ion o scheme (24) may be eadily jus i ied by Pica d’s heo em. To be mo e p ecise, one may p o e ha he e exis s a ime in e al [0, Th) o which p oblem (24) is uniquely sol able. As a consequence o a p io i ene gy es ima es, which we shall p o e in he nex sec ion, one deduces ha Th=T o all h > 0. 2.3. Main esul . We now a e eady o s a e ou main esul o his pape . We shall p o e ha scheme (24) p oduces a sequence o disc e e solu ions which sa i ies a p io i ene gy bounds uni o m wi h espec o (h, k) allowing us o pass o he limi as (h, k)→(0,+∞) owa ds weak solu ions o he Hele–Shaw-like sys em (10)-(13). Theo em 2.2. Assume ha (H1)-(H3) hold. Then he disc e e solu ion {(nh,k, ph,k)}h,k o (24) sa is ies he ollowing es ima es, o all a∈ Nhand ∈[0, T]: 0≤nh,k(a, )≤Nmax(k), 0≤p(nh,k(a, )) ≤Pmax, ∂ nh,k(a, )≥0, ∂ p(nh,k(a, )) ≥0. Fu he mo e, {nh,k,Ih((nh,k)k)}h,k con e ges owa ds weak solu ions (n∞, p∞)o p oblem (10)-(13) in he sense ha nh,k →n∞in L∞(0, T;H1(Ω))-weakly-?and in Lp((0, T)×Ω)-s ongly, and Ih((nh,k)k)→p∞in L∞(0, T;H1(Ω))-weakly-?and in Lp((0, T)×Ω)-s ongly, o any 1<p<∞p o ided ha (H5) k h →0as (h, k)→(0,+∞). 3. P oo o Theo em 2.2 3.1. A p io i ene gy es ima es. Ou goal is o p o e a p io i ene gy es ima es o he disc e e solu ion nh,k o (24) independen o (h, k). This i s lemma will be ocused on p o ing a disc e e maximum p inciple o nh,k based on he hypo hesis o igh -angled iangula ions. Mo eo e , some a p io i ene gy es ima es a e ob ained. Lemma 3.1. Assume ha (H1)-(H3) hold. Then he solu ion nh,k o scheme (24)sa is ies (31) 0 ≤nh,k(a, )≤Nmax(k)∀a∈ Nhand ∀ ≥0, and (32) knh,kkL∞(0,T ;L2(Ω)) +knh,kkL2(0,T ;H1(Ω)) ≤C, whe e C > 0is independen o (h, k). P oo . We i s p oceed o e i y (31). In doing so, we in oduce a modi ica ion o scheme (25) which unca es he nonlinea di usion e m as ollows: (33) (∂ nh,k, nh)h+ (D([nh,k]T)∇nh,k,∇nh) + ν(∇nh,k,∇nh) = (G(p([nh,k]T))nh,k, nh)h, whe e [nh,k]Tis he usual unca ion o nh,k om below by 0 and om abo e by Nmax(k). Again, by means o Pica d’s heo em, one has he exis ence and uniqueness o a solu ion nh,k o (33). Le nmin h,k =Ih(n− h,k)∈Nhbe de ined as nmin h,k =X a∈Nh n− h,k(a)ϕa, APPROXIMATION OF A HELE-SHAW-LIKE SYSTEM 9 whe e n− h,k(a) = min{0, nh,k(a)}. Analogously, one de ines nmax h,k =Ih(n+ h,k)∈Nhas nmax h,k =X a∈Nh n+ h,k(a)ϕa, whe e n+ h,k(a) = max{0, nh,k(a)}. No ice ha nh,k =nmin h,k +nmax h,k . On choosing nh=nmin h,k in (33), i ollows ha (34) 1 2 d d knmin h,k k2 h+ (D([nh,k]T)∇nh,k,∇nmin h,k ) + ν(∇nh,k,∇nmin h,k ) = kG(p([nh,k]T))1/2nmin h,k k2 h ≤G(0)knmin h,k k2 h. Nex obse e ha (D([nh,k]T)∇nh,k,∇nmin h,k ) = (D([nh,k]T)∇nmin h,k ,∇nmin h,k )+(D([nh,k]T)∇nmax h,k ,∇nmin h,k ) =kD([nh,k]T)1/2∇nmin h,k k2+X a6=˜ a∈Nh n− h,k(a)n+ h,k(e a)(D([nh,k]T)∇ϕa,∇ϕe a). Then, using he ac ha n− h,k(a)n+ h,k(e a)≤0 i a6=˜ aand ha D([nh,k]T) is a nonnega i e diagonal ma ix unc ion, one deduces, om (22), ha D([nh,k]T)∇ϕa·∇ϕe a≤0∀a6=e a∈ Nh and he eby (35) (D([nh,k]T)∇nh,k,∇nmin h,k )≥ kD([nh,k]T)1/2∇nmin h,k k2. Analogously, one ob ains (36) ν(∇nh,k,∇nmin h,k )≥νk∇nmin h,k k2, whe e we ha e used again (22) bu now o D=Id, wi h Idbeing he d×duni ma ix. Inse ing (35) and (36) in o (34) yields 1 2 d d knmin h,k k2 h+kD([nh,k]T)1/2∇nmin h,k k2+νk∇nmin h,k k2≤G(0)knmin h,k k2 h. By G ¨onwall’s lemma, we ha e nmin h,k ( )≡0 in Ω, o any ≥0, since nmin h,k (0) ≡0 in Ω; he eby his implies 0 ≤nh,k in (31). Fo he o he inequali y nh,k ≤Nmax(k) in (31), we p oceed in a simila ashion. In his case, one chooses nh= (nh,k −Nmax(k))max in (33) and akes in o accoun ha G(p([nh,k]T))nh,k(nh,k −Nmax(k))max ≡0 due o p([nh,k]T) = Pmax i nh,k ≥Nmax(k). I should be no ed ha any solu ion nh,k o he modi ied scheme (33) sa is ies he disc e e maximum p inciple (31), and consequen ly [nh,k]T≡nh,k; hence nh,k sa is ies he non- unca ed scheme (24) as well. Finally, by uniqueness o solu ions o scheme (24), he solu ion o (24) akes alues be ween 0 and Nmax(k); ha is (31). Now selec ing nh=nh,k in (25) and in oking G ¨onwall’s lemma, he ollowing ene gy es ima e holds, o all ∈[0, T]: (37) 1 2knh,k( )k2 h+ZT 0kD(nh,k)1/2∇nh,kk2+νZT 0k∇nh,kk2≤exp(2G(0)T)1 2kn0 h,kk2 h. Then he weak es ima es (32) a e deduced om (37) and (17).  A disc e e maximum p inciple o (nh,k)k−1and (nh,k)k ollows as a di ec consequence o (31). 16 F. GUILL´ EN-GONZ ´ ALEZ AND J. V. GUTI´ ERREZ-SANTACREU On es ing (57) agains Qh(Ih(nk h,k)ψ) wi h ψ∈C∞ c(Ω ×[0, T]) such ha ψ≥0, i ollows ha (59) ZT 0 (∂ nh,k ∗ρε,Qh(Ih(nk h,k)ψ))h=ZT 0 ((G(p(nh,k))nh,k)∗ρε,Qh(Ih(nk h,k)ψ))h −ZT 0 (∇(Σ(nh,k)∗ρε),∇Qh(Ih(nk h,k)ψ)). Since (∂ nh,k ∗ρε,Qh(Ih(nk h,k)ψ))h≥0, we ob ain (60) 0 ≤ZT 0 ((G(p(nh,k))nh,k)∗ρε,Qh(Ih(nk h,k)ψ))h−ZT 0 (∇(Σ(nh,k)∗ρε),∇Qh(Ih(nk h,k)ψ)). Taking he limi as (h, k)→(0,∞) yields (61) ZT 0 ((G(p(nh,k))nh,k)∗ρε,Qh(Ih(nk h,k)ψ))hd →ZT 0 ((G(p∞)n∞)∗ρε, p∞ψ)d and (62) ZT 0 (∇(Σ(nh,k)∗ρε),∇Qh(Ih(nk h,k)ψ))d →ZT 0 (∇((p∞+νn∞)∗ρε),∇(p∞ψ))d . In o de o p o e (61), we use he decomposi ion (uh, h)h= (uh, h) + (Ih(uh h)−uh h,1) o uh= (G(p(nh,k))nh,k)∗ρεand h=Qh(Ih(nk h,k)ψ) o w i e ZT 0 ((G(p(nh,k))nh,k)∗ρε,Qh(Ih(nk h,k)ψ))h=ZT 0 ((G(p(nh,k))nh,k)∗ρε,Qh(Ih(nk h,k)ψ)) +ZT 0 (Ih((G(p(nh,k)))nh,k)∗ρεQh(Ih(nk h,k)ψ)−(G(p(nh,k))nh,k)∗ρεQh(Ih(nk h,k)ψ),1). Then, i ollows om (21), (49) and (52) ha he i s e m con e ges o RT 0((G(p∞)n∞)∗ ρε, p∞ψ)d , and, on no ing ha k∇Qh(Ih(nk h,k)ψ)k ≤ Ck∇Ih(nk h,k)kkψkL∞+CkIh(nk h,k)kL∞k∇ψk om (19), and on ecalling (18) and (46), he second e m con e ges o ze o; he eby (61) holds. In o de o p o e (3.2.4), we w i e ZT 0 (∇(Σ(nh,k)∗ρε),∇Qh(Ih(nk h,k)ψ)) = ZT 0 (∇(Σ(nh,k)∗ρε),∇(Ih(nk h,k)ψ)) −ZT 0 (∇(Σ(nh,k)∗ρε),∇(Ih(nk h,k)ψ−Qh(Ih(nk h,k)ψ))). Then, i ollows om (58), (48) and (51) ha he i s e m con e ges o RT 0(∇((p∞+νn∞)∗ ρε),∇(p∞ψ))d , and on no ing ha k∇(Ih(nk h,k)ψ−Qh(Ih(nk h,k)ψ))k ≤ hk∇Ih(nk h,k)kkψkW2,∞(Ω) om (21) and on ecalling (46), he second e m con e ges o ze o; he eby (3.2.4) holds. Thus, by applying he p e ious con e gences (61) and o (60), we a i e a 0≤ZT 0 (G(p∞)n∞∗ρε, p∞ψ)−(∇((p∞+νn∞)∗ρε),∇(p∞ψ))d , and inally (55) holds by aking he limi as ε→0. APPROXIMATION OF A HELE-SHAW-LIKE SYSTEM 17 •We p oceed o p o e (56). W i e he i s e m on he igh -hand side o (59) as (63) ZT 0 (∂ nh,k ∗ρε,Qh(Ih(nk h,k)ψ))h=ZT 0 (∂ nh,k ∗ρε,Ih(nk h,k)ψ)h +ZT 0 (∂ nh,k ∗ρε,Qh(Ih(nk h,k)ψ)−Ih(nk h,k)ψ)h. These wo e ms a e handled as ollows. Fo he second e m o (63), we ha e, by (17), (21) and (41), ha ZT 0 (∂ nh,k ∗ρε,Qh(Ih(nk h,k)ψ)−Ih(nk h,k)ψ)hds→0 as (h, k)→(0,∞). Fo he i s e m o (63), we ha e ha , o each a∈ Nh, (∂ nh,k(a, )∗ρε)nk h,k(a, ) = nk h,k(a, )ZR ∂ nh,k(a, s)ρε( −s)ds =ZR nk h,k(a, s)∂ nh,k(a, s)ρε( −s)ds +ZR (nk h,k(a, )−nk h,k(a, s))∂ nh,k(a, s)ρε( −s)ds. On in eg a ing by pa s in ime and using (31) and (39), we ob ain ZR nk h,k(a, s)∂ nh,k(a, s)ρε( −s)ds=1 k+ 1 ZR nk+1 h,k (a, s)∂ ρε( −s)ds→0 as (h, k)→(0,∞). Fu he mo e, o s > , we ha e ha nk h,k(a, )−nk h,k(a, s)≤0 owing o (40). Then, i we choose supp(ρε)⊂(−ε, 0), hen ZR (nk h,k(a, )−nk h,k(a, s))∂ nh,k(a, s)ρε( −s)ds≤0. Le ing i s (h, k)→(0,∞) in (63) and hen ε→0, we ob ain (56) by epea ing he a gumen s ha led o (55). As a esul o (55) and (56), we no e ha (64) ZT 0 (G(p∞)n∞, p∞ψ)−(∇(p∞+νn∞),∇(p∞ψ))ds= 0 is sa is ied o all ψ∈C∞ c(Ω ×[0, T]) wi h ψ≥0, and he e o e i also holds o all ψ∈C∞ c(Ω × [0, T]). F om he ac ha p∞∇n∞= 0 and p∞≥0 a.e. in Ω×(0, T), we also deduce ha ∇p∞·∇n∞= 0 a.e. in Ω ×(0, T). As a consequence, he abo e a ia ional equa ion (64) is equi alen o ZT 0 (G(p∞)n∞, p∞ψ)−(∇p∞,∇(p∞ψ))ds= 0 which, aking in o accoun (53), implies (13) in he dis ibu ional sense. 4. An algo i hm on uns uc u ed meshes In o de o a oid using s uc u ed meshes, we p opose he ollowing scheme. Find nh,k ∈ C1([0, T]; Nh) such ha (65) (∂ nh,k, nh)h+k((nh,k)k−1∇nh,k,∇nh) + ν(∇nh,k,∇nh) = (G(p(nh,k))nh,k, nh)h∀nh∈Nh, nh,k(0) = n0 h,k. 18 F. GUILL´ EN-GONZ ´ ALEZ AND J. V. GUTI´ ERREZ-SANTACREU Equi alen ly, we may w i e (65)1as (∂ nh,k, nh)h+ (nh,k∇p(nh,k),∇nh) + ν(∇nh,k,∇nh) = (G(p(nh,k))nh,k, nh)h. He e he ini e-elemen space Nhis cons uc ed o e a amily o iangula ions {Th}h>0o Ω being shape- egula , quasi-uni o m and wi h acu e angles. This acu eness p ope y implies (22) o he pa icula case whe e Dis he d×diden i y ma ix [7]. We summa ize he p ope ies o scheme (65) in he ollowing heo em. Theo em 4.1. Suppose ha (H1)-(H4) a e sa is ied. Then scheme (65)sa is ies he ollowing p ope ies. Fo all a∈ Nhand ≥0, we ha e: 0≤nh,k(a, )≤Nmax(k) 0≤nk h,k(a, )≤PmaxNmax(k), ∂ nh,k(a, )≥0, ∂ nk h,k(a, )≥0, and he a p io i es ima es: knh,kkL∞(0,T ;L2(Ω))∩L2(0,T;H1(Ω)) ≤C, k∂ nh,kkL∞(0,T ;L1(Ω)) +k∂ nk h,kkL1(0,T ;L1(Ω)) ≤C, wi h C > 0being a cons an independen o (h, k). P oo . Full de ails o he p oo a e le o he in e es ed eade since i ollows mu a is mu andis he same a gumen s as o scheme (24).  Co olla y 4.1. Unde hypo heses (H1)-(H4), i ollows ha (66) X K∈ThZK>|∂xink h,k(x)|2+ZK<|∂xiIhnk h,k(x)|2dx≤C, whe e K>=(x∈K:nk−1 k,h (ξi) nk−1 h,k (x)>1) and K<=(x∈K:nk−1 k,h (ξi) nk−1 h,k ,(x)<1), wi h C > 0being a cons an independen o (h, k). P oo . Choose ¯nh=Ih(nk h,k) o ge (67) (∂ nh,k,Ih(nk h,k))h+((nh,k)k−1∇nh,k,∇Ih(nk h,k))+ν(∇nh,k,∇Ih(nk h,k)) = (G(p(nh,k))nh,k,Ih(nk h,k))h. I ollows immedia ely om (39) and (41) ha (68) (∂ nh,k,Ih(nk h,k))h≥0, and om (26) ha (69) ν(∇nh,k,∇Ih(nk h,k)) = ν(D(nh,k)∇nh,k,∇nh,k)≥0. Combining (67)-(69) yields on no ing (31) and (39) ha ((nh,k)k−1∇nh,k,∇Ih(nk h,k)) ≤G(0)|Ω|Nmax(k)2Pmax. APPROXIMATION OF A HELE-SHAW-LIKE SYSTEM 19 Finally, we in oke again (26) and ecall (27) o se k((nh,k)k−1∇nh,k,∇Ih(nk h,k)) = (∇nk h,k,∇Ih(nk h,k)) =X K∈ThZK nk−1 k,h (ξi) nk−1 h,k (x)|∂xink h,k(x)|2+nk−1 h,k (x) nk−1 k,h (ξi)|∂xiIhnk h,k(x)|2!dx. This comple es he p oo ia he de ini ions o K<and K>. Rema k 4.1. Un o una ely, con e gence o scheme (65)is no clea because es ima e (66)does no p o ide enough con ol o e he g adien o {nk h,k}h,k o {Ihnk h,k}h,k in o de o ob ain compac - ness and he e o e o pass o he limi as (k, h)→(0,+∞). 5. Nume ical simula ion 5.1. Tempo al in eg a ion. I is assumed he e o simplici y ha we ha e a uni o m pa i ion o [0, T] in o Mpieces, wi h ime s ep size τ=T/M and he ime alues ( m=mτ)M m=0. To simpli y he no a ion le us deno e δ nm+1 =nm+1 −nm τ. Fi s we p esen a i s -o de ime in eg a ion o scheme (65). Algo i hm 1: Linea semi-implici ime-s epping scheme S ep (m+ 1): Gi en nm h,k ∈Nh, ind nm+1 h,k ∈Nhsol ing he algeb aic linea sys em (70) (δ nm+1 h,k , nh)h+k((nm h,k)k−1∇nm+1 h,k ,∇nh) +ν(∇nm+1 h,k ,∇nh) = (G(p(nm h,k))nm h,k, n)h, o all nh∈Nh. 5.2. Compu a ional expe imen s. In his sec ion, we p esen se e al nume ical expe imen s o es he algo i hm p esen ed he ein. To do his, we conside he e olu ion o p oblem (1)-(3) wi h n0(x, y) = α e−(x2+y2) on he compu a ional domain Ω = (−10,10) ×(−10,10) wi h α > 0. In he nume ical se ing, we cons uc a s uc u ed iangula ion pa i ioning he edges o squa e in o 100 subin e als, co esponding wi h he mesh size h= 0.12582843 and he ime s ep size is τ= 10−5. The choice o he ime s ep τis such ha i helps o mi iga e he possibly nume ical de ia ion o he d-simplexes K∈ Th om he igh -angled s uc u e. The esul ing ma ix is s ic ly diagonally dominan . Ou in en ion is o illus a e he beha io exhibi ed by he solu ion o p oblem (1)-(3) when he di usion coe icien ν, he pa ame e kand he homeos a ic p essu e Pmax a y. We will se αand Pmax o be 1 and k o be 100 i no s a ed o he wise. Mo eo e , we conside G(p) = 200 πa c an(4(Pmax −p)+), 5.2.1. Analysis o he e ec o α(con ac ion/dila ion coe icien o he ini ial da um). In his es we choose α= 0.5 and 1. We a e in e es ed in compa ing he e olu ion o he densi y nand he p essu e p(n) when he maximum o he ini ial densi y akes di e en alues. In pa icula , we ha e o α= 0.5 ha he maximum alue o n0occu s only a he poin (0,0) and is 0.5, hence Nmax(k) emains below o 1. We hus obse e ha he maximum inc eases wi hou modi ying essen ially he exponen ial shape o he ini ial da um n0un il eaches Nmax(k) = 1. Once he densi y akes he alue 1 a = 0.01583 he measu e o poin s a which he densi y eaches he maximum g ows adially a ound (0,0) due o he ac ha he p essu e s a s inc easing and pushes o wa d he 20 F. GUILL´ EN-GONZ ´ ALEZ AND J. V. GUTI´ ERREZ-SANTACREU umo cells. Then he exponen ial s uc u e o he ini ial da um n0becomes a a eling wa e shape which mo es ou wa ds as inc eases. This beha io causes ha he e olu ion o he in e ace is delayed conce ning he case α= 1 as shown in Figu es 1and 2since he maximum alue 1 is eached om he beginning. Figu e 3 ep esen s he di e ence be ween he densi y and he p essu e a imes = 0.1, 0.2, 0.3 and 0.4, and indica es ha he p essu e is esponsible o he ad ance o he umo cells which is deduced om he annulus shape o he di e ence. Figu e 1. E olu ion o he densi y a imes = 0.1,0.2,0.3,0.4 o α= 0.5 ( op) and 1 (bo om). Figu e 2. E olu ion o he p essu e a imes = 0.1,0.2,0.3,0.4 o α= 0.5 ( op) and α= 1 (bo om). 5.2.2. Analysis o he e ec o ν(ac i e mo ion coe icien ). Now we se Pmax = 1 and ake di e en alues o ν= 0, 0.5 and 1. The e olu ion o he densi y nh,k is shown in Figu e 4whe e we see ha he eloci y o p opaga ion o he umo cells inc eases wi h espec o νas no ed o imes = 0.1, 0.2, 0.3 and 0.4. Mo eo e , no pa icula di e ences ha e been obse ed in he wid h o he in e ace be ween he umo and p e- umo cells o he di e en alues o ν. 5.2.3. Analysis o he e ec o k.In his simula ion we selec k= 10 and 1000. The i s hing we ha e no ed is ha he e is a dependence be ween kand τwhich has been aken 0.5·10−5. As can be seen in Figu e 5, he e a e no pa icula di e ences o k= 10 and 1000 a imes = 0.1, 0.2, 0.3 and 0.4. APPROXIMATION OF A HELE-SHAW-LIKE SYSTEM 21 Figu e 3. E olu ion o he di e ence be ween he densi y and p essu e a imes = 0.1,0.2,0.3,0.4 o α= 0.5 ( op) and α= 1 (bo om). Figu e 4. Compa ison o he densi y a imes = 0.1,0.2,0.3,0.4 o di e en ν= 0 ( op), 0.5 (middle), 1 (bo om). Figu e 5. Compa ison o he densi y a imes = 0.1,0.2,0.3,0.4 o di e en k= 10 ( op) and 1000 (bo om). 22 F. GUILL´ EN-GONZ ´ ALEZ AND J. V. GUTI´ ERREZ-SANTACREU 5.2.4. Analysis o he e ec o Pmax.Le us ake Pmax = 10 and 30. Figu e 6shows ha he dynamics is sensi i e o he di e en alues o he homeos a ic p essu e. We highligh ha , o Pmax = 30, he e olu ion o he in e phase is as e han he one o Pmax = 10. Mo eo e , he shape o he in e phase seems di e en as depic ed in Figu e 6 o imes = 0.1, 0.2, 0.3 and 0.4. Figu e 6. Compa ison o he densi y a imes = 0.1,0.2,0.3,0.4 o di e en Pmax = 10 ( op) and 30 (bo om). 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