Universität Regensburg Mathematik

Transcript

1 Unverstät Regensburg Matematk Fnte element approxmaton of a sarp nterface approac for gradent flow dynamcs of two-pase bomembranes Jon W. Barrett, Harald Garcke and Robert Nürnberg Preprnt Nr. 06/2017

2 Fnte Element Approxmaton of a Sarp Interface Approac for Gradent Flow Dynamcs of Two-Pase Bomembranes Jon W. Barrett Harald Garcke Robert Nürnberg Abstract A fnte element metod for te evoluton of a two-pase membrane n a sarp nterface formulaton s ntroduced. Te evoluton equatons are gven as an L 2 gradent flow of an energy nvolvng an elastc bendng energy and a lne energy. In te two pases Helfrc-type evoluton equatons are prescrbed, and on te nterface, an evolvng curve on an evolvng surface, gly nonlnear boundary condtons ave to old. Here we consder bot C 0 and C 1 matcng condtons for te surface at te nterface. A new weak formulaton s ntroduced, allowng for a stable semdscrete parametrc fnte element approxmaton of te governng equatons. In addton, we sow exstence and unqueness for a fully dscrete verson of te sceme. Numercal smulatons demonstrate tat te approac can deal wt a multtude of geometres. In partcular, te paper sows te frst computatons based on a sarp nterface descrpton, wc are not restrcted to te axsymmetrc case. Key words. parametrc fnte elements, Helfrc energy, spontaneous curvature, mult-pase membrane, lne energy, C 0 and C 1 matcng condtons AMS subject classfcatons. 35R01, 49Q10, 65M12, 65M60, 82B26, 92C10 1 Introducton Two-pase elastc membranes, consstng of coexstng flud domans, ave receved a lot of attenton n te last 20 years. Te nterest n two-pase membranes n partcular was trggered by te multtude of dfferent sapes observed n experments wt nomogeneous bomembranes and vescles. Bomembranes are typcally formed as a lpd Department of Matematcs, Imperal College London, London, SW7 2AZ, UK Fakultät für Matematk, Unverstät Regensburg, Regensburg, Germany 1

3 blayer, and often multple lpd components are nvolved, wc laterally can separate nto coexstng pases wt dfferent propertes. Among te complex morpologes tat appear are mcro-domans, wc resemble lpd rafts, and tese are of uge nterest n bology and medcne. As te tckness of te membrane s muc smaller tan ts lateral lengt scale, typcally te membrane s modelled as a two-dmensonal ypersurface n tree dmensonal Eucldean space. Te equlbrum sape of te membrane s obtaned by mnmzng an energy wc besdes oter contrbutons contans bendng energes nvolvng te mean curvature and te Gaussan curvature of te membrane. If dfferent pases occur, parameters n te curvature energy are nomogeneous, leadng to an nterestng free boundary problem as well as to a pletora of dfferent sapes. We refer to Baumgart et al. (2003), wo studed mult-component gant unlamellar vescles (GUVs) separatng nto dfferent pases. Tese autors were able to optcally resolve nteractons between te dfferent pases, ts curvature elastcty and te lne tenson of ts nterface. Tere ave been several studes on teoretcal and numercal aspects of two-pase membranes takng curvature elastcty and lne energy nto account, see e.g. Jülcer and Lpowsky (1993, 1996); Tu and Ou-Yang (2004); Baumgart et al. (2005); Wang and Du (2008); Das et al. (2009); Lowengrub et al. (2009); Ellott and Stnner (2010a,b, 2013); Helmers (2011, 2013); Coks et al. (2013); Mercker et al. (2013); Cox and Lowengrub (2015); Barrett et al. (2016a), wc we dscuss n te followng. Te by now classcal model for a one-pase membrane rests on te Canam Helfrc Evans elastc bendng energy 1 α (κ κ) 2 dh 2 +α G K dh 2, 2 Γ were Γ s a closed two-dmensonal ypersurface and H 2 denotes te two-dmensonal Hausdorff measure. Te mean curvature of Γ s denoted by κ, and K s ts Gaussan curvature. Te constants α and α G are bendng rgdtes, wle κ s te spontaneous curvature reflectng asymmetry n te membrane ntroduced, for nstance, by dfferent envronments on bot sdes of te membrane. In a fundamental work, Jülcer and Lpowsky (1993, 1996) generalzed te Canam Helfrc Evans model to two-pase membranes. Te geometry s now gven by two smoot surfaces Γ 1 and Γ 2, wt a common boundary γ. In general, te constants α, α G and κ take dfferent values n te two pases Γ 1 and Γ 2, wc we wll denote wt an ndex. On te curve γ lne tenson effects play an mportant role, and te total energy ntroduced by Jülcer and Lpowsky (1993, 1996) s gven as E((Γ ) 2 ) = [ 1 α 2 (κ κ ) 2 dh 2 +α G Γ Γ ] K dh 2 +ςh 1 (γ), (1.1) Γ were te constant ς R 0 denotes a possble lne tenson, and were an ndex {1,2} states tat quanttes suc as te curvatures and pyscal constants are evaluated wt respect to Γ. Of course, H 1 denotes te one-dmensonal Hausdorff measure. 2

4 In Jülcer and Lpowsky (1996) t s assumed tat te surface Γ = Γ 1 γ Γ 2 s a C 1 surface, meanng n partcular tat te normal to Γ s contnuous across te pase boundary γ. Te works of Helmers (2011, 2013, 2015), on te oter and, also allow for dscontnutes of te normal at γ. Te frst varaton of te energy E n (1.1) as been derved n Ellott and Stnner (2010a) for te C 1 case and n Wutz (2010) for te C 1 and te C 0 case. It s te goal of ts paper to develop a numercal metod for a gradent flow evoluton of te energy E. To be more precse, we wll consder an evoluton of te form V, χ Γ + V, χ = γ [ δ δγ E((Γ ) 2 ) ] ( χ). (1.2) Here V δ s te velocty of te surface, E s te frst varaton of te energy, χ s a test δγ vector feld on te surface related to drectons n wc one perturbs te gven surface Γ, and 0 s a gven constant. In addton,, Γ and, γ denote te L 2 nner products on te surface Γ and on te curve γ, respectvely. Te evoluton of te surface s ence gven as a steepest descent dynamcs wt respect to a wegted L 2 nner product tat combnes contrbutons from te surface and te curve. It wll turn out tat te governng equatons n te case were te surface s restrcted to be C 1 are V = [ α s κ α (κ κ ) 2 κ α (κ κ ) s ν 2 ] ν on Γ, (1.3) togeter wt te boundary condtons on γ = Γ : α 1 (κ κ 1 )+α G 1 κ γ. ν = α 2 (κ 2 κ 2 )+α G 2 κ γ. ν, (1.4a) [α ( s κ )] 2 1. µ [αg ]2 1 τ s +ς κ γ. ν = V. ν, (1.4b) 1 2 [α (κ κ ) 2 ] 2 1 +[α (κ κ )(κ κ γ. ν)] 2 1 +[α G ] 2 1τ 2 +ς κ γ. µ = V. µ. (1.4c) Equaton(1.3), wt s and s denotngtesurfacelaplacanandtesurfacegradent on Γ, respectvely, s Wllmore flow takng spontaneous curvature effects nto account. Te boundary condton (1.4a), wt κ γ denotng te curvature vector on, generalzes te equaton for te mean curvature n Naver boundary condtons, appearng for example n Deckelnck et al. (2017, (6)). Te equatons (1.4b,c), wt τ beng te geodesc torson of te curve on Γ(t) and wt [a ] 2 1 = a 2 a 1 denotng te jump of a across, appear n te case = 0 n Ellott and Stnner (2010b, (3.17), (3.18)), were addtonal terms to fx te surface areas and te enclosed volume appear. In te axsymmetrc case, te equatons (1.4a c) reduce to te equatons studed by Jülcer and Lpowsky (1996). Smlar condtons ave been derved by Tu and Ou-Yang (2004), and t as already been dscussed n Ellott and Stnner (2010b, Appendx B) tat tese autors mss one term. For postve te equatons (1.4b,c) gve rse to dynamc boundary condtons takng nto account an addtonal dsspaton mecansm at te boundary. A smlar condton for sem-free boundary condtons as been analyzed n Abels et al. (2016, (1.3)). For evolutons were te surface areas of Γ 1 and Γ 2, as well as te volume enclosed by Γ, are conserved, addtonal terms appear n (1.3) and (1.4c), see (2.15) and (2.19c), below. Moreover, n te case tat te surface Γ s just contnuous, te boundary condtons (1.4a c) ave to be replaced, and we refer to (2.18a,b), below, for te relevant equatons. 3

5 Numercally manly te C 1 case as been studed, wt te excepton of Helmers (2013), wo numercally studed C 0 surfaces wt knks n te axsymmetrc case wt te elp of a pase feld metod. In te C 1 case already Jülcer and Lpowsky (1996) computed several two-pase equlbrum sapes n te axsymmetrc case by solvng a governng boundary value problem for a system of ordnary dfferental equatons. Based on researc on model membranes, see Baumgart et al. (2003), t as now become possble to perform a systematc analyss of te nfluence of parameters also n te case of twopase coexstence. We refer to Baumgart et al.(2005), wo compared expermental vescle sapes wt sapes obtaned by solvng numercally te axsymmetrc sape equatons derved by Jülcer and Lpowsky (1996). In ts context, we also refer to Cox and Lowengrub(2015), wo, n contrast to te above works, also took te effect of spontaneous curvature nto account n te axsymmetrc case. Tey were able to sow tat spontaneous curvatures already n an axsymmetrc setup gve rse to a multtude of morpologes not seen n te case wtout spontaneous curvature. Almost all numercal results mentoned so far were for a sarp nterface setup. Anoter successful approac uses a pase feld to descrbe te two pases on te membrane. Lne energy n ts context s replaced by a Gnzburg Landau energy lke n te classcal Can Hllard teory. We refer to Wang and Du (2008); Lowengrub et al. (2009); Ellott and Stnner (2010a,b, 2013); Helmers (2011, 2013); Mercker et al. (2013); Mercker and Marcnak-Czocra (2015) for numercal results based on te pase feld approac. Te above papers use a gradent flow approac to obtan equlbrum sapes n te large tme lmt. An evoluton law usng a Can Hllard equaton on te membrane coupled to surface and bulk (Naver )Stokes equatons as been studed by te present autors n Barrett et al. (2016a). Rgorous analytcal results for two-pase elastc membranes are very lmted. So far only results for te axsymmetrc case are known. We refer to te work of Coks et al. (2013), wo sowed te exstence of global mnmzers for axsymmetrc mult-pase membranes, and te work of Helmers (2011, 2013, 2015), wo studed te sarp nterface lmt of te pase feld approac n an axsymmetrc stuaton. Exstence results for te evoluton problem are not avalable n te lterature so far and sould be addressed n te future. It s te goal of ts paper to ntroduce a fnte element approxmaton for a gradent flow dynamcs of te membrane energy E, wc s based on a sarp nterface approac. Instead of usng a pase feld on te membrane, we wll drectly dscretze te curve γ separatng te two pases Γ 1 and Γ 2. In tree dmensons te total surface Γ wll be dscretzed wt te elp of polyedral surfaces consstng of a unon of trangles. Te curve γ s dscretzed as a polygonal curve n R 3 ftted to te dscretzaton of Γ n te sense tat te polygonal curve s te boundary of te open polyedral sets Γ 1 and Γ 2. Te boundary condtons (1.4a c) are gly nonlnear and nvolve dervatves of an order up to tree wen formulated wt te elp of a parameterzaton. It s ence gly nontrval to dscretze tem n a pecewse lnear setup. In ts work, a splttng metod s used, wc bascally uses te poston vectors of te nodes and an approxmaton of te mean curvature vector as unknowns. Te approac n ts paper reles on a dscretzaton 4

6 of mean curvature leadng to good mes propertes. Ts dscretzaton was ntroduced by te present autors n Barrett et al. (2007, 2008) and as been prevously used for closed and open membranes, see Barrett et al. (2016b, 2017) and for elastc curvature flow of curves wt junctons, see Barrett et al. (2012). We wll use te varatonal structure of te problem to derve a dscretzaton wc wll turn out to be stable n a spatally dscrete and contnuous-n-tme semdscrete formulaton. In order to do so, we wll make use of an approprate Lagrangan and wll use deas of PDE constraned optmzaton. Te outlne of ts paper s as follows. In te subsequent secton we wll formulate te governng equatons wt all te detals. In Secton 3 a weak formulaton s ntroduced usng te calculus of PDE constraned optmzaton. A semdscrete dscretzaton s formulated n Secton 4. For ts sceme also energy decay propertes and conservaton propertes are sown. In Secton 5 a fully dscrete verson of te sceme s ntroduced, leadng to a lnear system at eac tme level, wc s sown to be unquely solvable. In Secton 6 we dscuss deas on ow to solve te resultng lnear algebra problems numercally. In Secton 7 we present several numercal results sowng tat te new approac allows t to approxmate solutons to te governng equatons also n gly nontrval geometres. In an appendx we sow tat te weak formulaton derved n ts work yelds n fact te strong formulaton for suffcently smoot evolutons. 2 Te governng equatons IntssectonweprecselyformulatetegovernngequatonsbotforteC 0 andtec 1 case. We always assume tat (Γ(t)) t [0,T] s an evolvng ypersurface wtout boundary n R d, d = 2,3, tat s parameterzed by x(,t) : Υ R d, were Υ R d s a gven reference manfold,.e. Γ(t) = x(υ,t). Ten V( q,t) := x t ( z,t) q = x( z,t) Γ(t) (2.1) defnes te velocty of Γ(t). In order to ntroduce te two-pase aspect, we consder te decomposton Γ(t) = Γ 1 (t) Γ 2 (t), were te nterors of Γ 1 (t) and Γ 2 (t) are dsjont and = Γ 1 (t) = Γ 2 (t). Weassume tateac Γ (t) s smoot, wt outer unt normal ν (t). See Fgure 1 for a sketc of te setup n te case d = 3. In partcular, we parameterze te two parts of te surface over fxed orented, compact, smoot reference manfolds Υ Υ,.e. we let Γ (t) = x(υ,t), = 1,2. Trougout ts paper we wll nvestgate two dfferent types of juncton condtons on : C 0 case : = Γ 1 (t) = Γ 2 (t), (2.2a) C 1 case : = Γ 1 (t) = Γ 2 (t) and ν 1 = ν 2 on. (2.2b) Of course, n te case (2.2b) t also olds tat µ 1 = µ 2, were µ denotes te outer conormal to Γ (t) on. 5

7 ν 2 ν 1 γ Γ 1 µ 1 Γ 2 µ 2 d s Fgure 1: Sketc of Γ = Γ 1 γ Γ 2 wt outer unt normals ν, conormals µ and tangent vector d s on γ for te case d = 3. In order to formulate te governng problems n more detal, we denote by s = ( s1,..., sd ) te surface gradent on Γ, and ten defne s χ = ( ) d sj χ k, as well as k,j=1 te Laplace Beltram operator s = s. s = d j=1 2 s j. We ten ntroduce te mean curvature vector as κ = κ ν = s d on Γ, (2.3) were d s te dentty functon on R d, and κ s te mean curvature of Γ,.e. te sum of te prncpal curvatures of Γ. In partcular, te prncpal curvatures κ,j, j = 1,...,d 1, togeter wt te egenvalue zero for te egenvector ν, are te d egenvalues of te symmetrc lnear map s ν : R d R d ; see e.g. Deckelnck et al. (2005, p. 152), were a dfferent sgn conventon s used. Te map s ν s also called te Wengarten map or sape operator. Te mean curvature κ and te Gaussan curvature K of Γ can now be stated as d 1 d 1 κ = κ,j = tr( s ν ) = s. ν and K = κ,j. (2.4) j=1 Trougout te paper te man case we are nterested n s d = 3, but t s often convenent to also dscuss te case d = 2 at te same tme. To ts end, we generalze te free energy (1.1) to [ E((Γ (t)) 2 ) = 1 α 2 (κ κ ) 2 dh d 1 +α G K dh ]+ςh d 1 d 2 (), Γ (t) Γ (t) (2.5) were κ and K are te mean and Gaussan curvatures of Γ (t), = 1,2, ς R 0 denotes a possble lne tenson, and α R >0 and α G R denote te bendng and Gaussan bendng rgdtes of Γ (t), = 1,2, respectvely. Here and trougout H k, k = 0,1,2, denotes te k-dmensonal Hausdorff measure n R d. 6 j=1

8 In te case d = 2, we always assume tat ς = α G 1 = α G 2 = 0. For te case d = 3, on te oter and, we menton tat te contrbutons [ 1 2 α Γ (t) κ 2 dh2 +α G Γ (t) K dh 2 ] to te energy (2.5) are postve semdefnte wt respect to te prncpal curvatures f α G [ 2α,0], = 1,2. (2.6) In te C 1 case, recall (2.2b), addng multples of 2 K dh 2 to te energy only canges te energy by a constant wc follows from te Gauss Bonnet teorem, see (2.11) below. Hence we obtantatte energy (2.5) canbebounded frombelow f α G max{α G 1,α G 2} 2α for = 1,2, wc wll old wenever mn{α 1,α 2 } 1 2 αg 1 α G 2. (2.7) We note tat te constrants(2.6) and(2.7) are lkely to ave mplcatons for te exstence and regularty teory of gradent flows for (2.5) n te C 0 and C 1 case, respectvely. In te case d = 3, smlarly to (2.3), fundamental to many approaces, wc numercally approxmate evolvng curves n a parametrc way, s te dentty d ss = κ γ on, (2.8) were κ γ s te curvature vector on. Here we coose te arclengt s of te curve suc tat µ = ( 1) ν d s on, (2.9) for = 1,2, denote te outer conormals to Γ (t) on. Note tat µ s a vector tat s perpendcular to te unt tangent d s on Γ (t) and les n te tangent space of Γ (t). Now (2.8) can be rewrtten as d ss = κ γ = ( κ γ. µ ) µ +( κ γ. ν ) ν on, (2.10) were κ γ. µ s te geodesc curvature and κ γ. ν s te normal curvature of on Γ (t), = 1,2. It ten follows from te Gauss Bonnet teorem, K dh 2 = 2πm(Γ (t))+ κ γ. µ dh 1, (2.11) Γ (t) were m(γ (t)) Z denotes te Euler caracterstc of Γ (t), tat te energy (2.5), s equvalent to E((Γ (t)) 2 ) = [ 1 α 2 Γ (t) (κ κ ) 2 dh 2 +α G [ ]] κ γ. µ dh 1 +2πm(Γ (t)) +ςh 1 (). (2.12) 7

9 We note tat we use a sgn for te conormal tat s dfferent from many autors n dfferental geometry, and ence we obtan a dfferent sgn n te Gauss Bonnet formula. Te wegted L 2 gradent flow, (1.2), of (2.5), for d = 2 or d = 3, ten leads to te evoluton law V. ν = α s κ α (κ κ ) 2 κ α (κ κ ) s ν 2 on Γ (t), = 1,2. (2.13) See te appendx for a dervaton of (2.13). In some cases, n partcular n applcatons for bomembranes, cf. Tu (2013), te surface areas of Γ 1 (t) and Γ 2 (t) need to stay constant durng te evoluton, as well as te volume enclosed by Γ(t). Here and trougout we use te termnology surface area and enclosed volume also for te case d = 2, wen te former s really curve lengt, and te latter means enclosed area. In ts case one can consder E λ ((Γ (t)) 2 ) = E((Γ (t)) 2 )+λv L d (Ω(t))+ λ A H d 1 (Γ (t)), (2.14) were Ω(t) denotes te nteror of Γ(t) and L d denotes te Lebesgue measure n R d. Here, λ A are Lagrange multplers for te area constrants, wc can be nterpreted as a surface tenson, and λ V s a Lagrange multpler for te volume constrant wc mgt be nterpreted as a pressure dfference. In ts case (2.13) s replaced by V. ν = α s κ α (κ κ ) 2 κ α (κ κ ) s ν 2 +λ A κ λ V on Γ (t), (2.15) for = 1,2. as In te case d = 3 we ntroduce te second fundamental form II of Γ (t), wc s gven II ( t 1, t 2 ) = [ t 1 ν ]. t 2 = [( s ν ) t 1 ]. t 2 on Γ (t), (2.16) for all tangental vectors t j, j = 1,2. We note tat II (, ) s a symmetrc blnear form, as s ν s symmetrc. In addton, we defne.e. τ = ( ν ) s. µ on. τ = II ( d s, µ ) on, (2.17) Stll consderng te case d = 3, n te C 0 juncton case, te boundary condtons on are gven by α (κ κ )+α G κ γ. ν = 0 on, = 1,2, (2.18a) [ ((α ( s κ ). µ α G (τ ) s ) ν ( 1 α ] 2 (κ κ ) 2 +α G K +λ A ) µ +ς κγ = V on. (2.18b) We note tat (2.18a) are two scalar condtons, wle (2.18b) gves rse to two condtons as µ, ν and κ γ are all perpendcular to te tangent space to. Expressng Γ 1 and 8

10 Γ 2 locally as two graps, we also obtan one condton for te egt functons stemmng from te C 0 condton. Altogeter we ave fve condtons, as s to be expected for a free boundary problem nvolvng fourt order operators on bot sdes of te free boundary. In ts context we also refer to Remark 2.1 n Barrett et al. (2012). In te C 1 juncton case, wen ν = ν 1 = ν 2 and µ = µ 2 = µ 1 on, te boundary condtons on for te dsspaton dynamcs (1.2), wt E replaced by E λ, are gven by [α (κ κ )] 2 1 +[αg ]2 1 κ γ. ν = 0 on, (2.19a) [α ( s κ )] 2 1. µ+ς κ γ. ν [α G ]2 1 τ s = V. ν on, (2.19b) [ 1 2 α (κ κ ) 2 +α (κ κ )(κ κ γ. ν) λ A ] 2 1 +[α G ] 2 1τ 2 +ς κ γ. µ = V. µ on, (2.19c) were τ = τ 2 = τ 1 s te geodesc torson of te curve on Γ(t). We note tat (2.19a c), n te case = 0, agree wt (3.16) (3.18) n Ellott and Stnner (2010b), see also Ellott and Stnner (2013, (2.7b,a,c)). In terms of countng te number of equatons, we see tat (2.19a c) are tree condtons, togeter wt one condton comng from ν 1 = ν 2 and one condton from te requrement tat te two pases matc up contnuously, leadng to fve condtons n total. We refer to te appendx for a dervaton of (2.15), (2.18a,b) and (2.19a c). Remark We note tat altoug te condtons (2.18a,b) and (2.19a c) were derved for te case d = 3, tey are also vald n te case d = 2 on recallng tat n ts case we set ς = α G 1 = αg 2 = 0. In partcular, (2.18a) ten smplfes to κ = κ on, = 1,2, wc s te same as te condton Barrett et al. (2012, (2.13c)) tat was derved by te autors for a C 0 juncton between two curves meetng n 2d. In addton, (2.18b) for d = 2 and = 0 collapses to Barrett et al. (2012, (2.13b)), modulo te dfferent sgn conventon employed tere. Smlarly, (2.19a) for d = 2 smplfes to α 1 (κ 1 κ 1 ) = α 2 (κ 2 κ 2 ) on, wc s te same as te condton Barrett et al. (2012, (2.18e)), modulo te dfferent sgn conventon employed tere, tat was derved by te autors for a C 1 juncton between two curves meetng n 2d. In addton, (2.19b,c) for d = 2 and = 0, collapse to Barrett et al. (2012, (2.18b,c)). 3 Weak formulaton On recallng (2.1), we defne te followng tme dervatve tat follows te parameterzaton x(,t) of Γ(t). Let ( t φ) Γ (t)= (φ t + V. φ) Γ (t) φ H 1 (Γ,T ), (3.1) were we ave defned te space-tme surfaces Γ,T := Γ (t) {t}, = 1,2, and Γ T := t [0,T] 9 t [0,T] Γ(t) {t}. (3.2)

11 Here we stress tat te defnton (3.1) s well-defned, even toug φ t and φ do not make sense separately for a functon φ H 1 (Γ,T ). For later use we note tat d dt ψ,φ Γ (t) = t ψ,φ Γ (t) +ψ, t φ Γ (t) + ψ φ, s. V ψ,φ H 1 (Γ,T ), Γ (t) (3.3) see Lemma 5.2 n Dzuk and Ellott (2013). Here, Γ (t) denotes te L 2 nner product on Γ (t), and, Γ(t) = 2, Γ (t). It mmedately follows from (3.3) tat d dt Hd 1 (Γ (t)) = s. V,1 Γ (t) = sd, s V Moreover, on recallng Lemma 2.1 from Deckelnck et al. (2005), t olds tat d dt Ld (Ω(t)) = V, ν For later use we note n addton tat s g dh d 1 = gκ ν dh d 1 + Γ (t) Γ (t) Γ (t) Γ (t). (3.4). (3.5) g µ dh d 2 g H 1 (Γ (t)), (3.6) see e.g. Teorem 2.10 n Dzuk and Ellott (2013) and Proposton 4.5 n Taylor (2011, p. 334). In ts secton we would lke to derve a weak formulaton for te L 2 gradent flow of E((Γ (t)) 2 ). To ts end, we need to consder varatons of te energy wt respect to Γ(t) = x(υ,t). Let H 1 γ(γ(t)) := {η L 2 (Γ(t)) : η Γ (t) H 1 (Γ (t)), = 1,2, (η Γ1 (t)) = (η Γ2 (t)) =: η H 1 ()}. (3.7) In addton, for any gven χ [H 1 γ (Γ(t))]d and for any ε (0,ε 0 ) for some ε 0 R >0, let Γ ε (t) := { Ψ( z,ε) : z Γ(t)}, were Ψ( z,0) = z and Ψ( z,0) = χ( z) z Γ(t). ε (3.8) Of course, we ave tat Γ ε (t) = Γ 1,ε (t) γ ε (t) Γ 2,ε (t), were Γ,ε (t) := { Ψ( z,ε) : z Γ (t)}, = 1,2, and γ ε (t) = Γ 1,ε (t) = Γ 2,ε (t). (3.9) Smlarly to(3.4), tefrst varatonofh d 1 (Γ (t)) wtrespect to Γ(t)ntedrecton χ [Hγ(Γ(t))] 1 d s gven by [ ] δ δγ Hd 1 (Γ (t)) ( χ) = d dε Hd 1 (Γ,ε (t)) ε=0 [ 1 = lm ε 0 ε H d 1 (Γ,ε (t)) H d 1 (Γ (t)) ] = sd, s χ, (3.10) 10 Γ (t)

12 see e.g. te proof of Lemma 1 n Dzuk (2008). For later use we note tat generalzed varants of (3.10) also old. Namely, we ave tat [ ] δ δγ w,1 Γ (t) ( χ) = d dε w,ε,1 Γ,ε (t) ε=0= w sd, s χ w L (Γ (t)), Γ (t) (3.11) were w,ε L (Γ,ε (t)), for any w L (Γ (t)), s defned by w,ε ( Ψ( z,ε)) = w ( z) z Γ (t), (3.12) and smlarly for w [L (Γ (t))] d. Ts defnton of w,ε yelds tat 0 ε w = 0, were 0 ε w ( z) = d dε w,ε( Ψ( z,ε)) ε=0 z Γ (t). (3.13) Ofcourse, (3.11)stefrst varatonanalogueof(3.3)wt w = ψ φ and ε 0 ψ = ε 0 φ = 0. Smlarly, t olds tat [ ] δ δγ w, ν Γ (t) ( χ) = d dε w,ε, ν,ε Γ,ε (t) ε=0 = ( w. ν ) sd, s χ + w, ε 0 ν Γ w (t) [L (Γ (t))] d, (3.14) Γ (t) were 0 ε w = 0 and ν,ε (t) denotes te unt normal on Γ,ε (t). In ts regard, we note te followng result concernng te varaton of ν, wt respect to Γ(t), n te drecton χ [H 1 γ(γ(t))] d : 0 ε ν = [ s χ] T ν on Γ (t) t ν = [ s V] T ν on Γ (t), (3.15) see Scmdt and Sculz (2010, Lemma 9). Fnally, we note tat for η [H 1 (Γ (t))] d t olds tat [ δ ] sd, s η ( χ) = d sd, s η,ε ε=0 = s. η, s. χ δγ Γ (t) dε Γ (t) Γ,ε (t) d [ ] + ( ν ) l ( ν ) m s ( η ) m, s ( χ) l Γ (t) ( s) m ( η ) l,( s ) l ( χ) m Γ (t) l,m=1 = s η, s χ Γ (t) + s. η, s. χ Γ (t) ( s η ) T,D( χ)( sd) T were 0 ε η = 0, see Lemma 2 and te proof of Lemma 3 n Dzuk (2008). Here Γ (t), (3.16) D( χ) := s χ+( s χ) T, (3.17) and we note tat our notaton s suc tat s χ = ( Γ χ) T, wt Γ χ = ( sl χ m ) d l,m=1 defned as n Dzuk (2008). It follows from (3.16) tat d sd, s η = s η, sv + s. η, s. dt V Γ (t) Γ (t) Γ (t) 11

13 ( s η) T,D( V)( sd) T Γ (t) η { ξ H 1 (Γ,T ) : t ξ = 0}. (3.18) In order to derve a sutable weak formulaton, we formally consder te frst varaton of (2.5) subject to te followng sde constrant, wc s nspred by te weak formulaton of (2.3), Q,θ κ, η Γ (t) + sd, s η = m, η η [H 1 (Γ (t))] d, = 1,2, (3.19) Γ (t) were θ [0,1] are fxed parameters, and were Q,θ are defned by Q,θ = θ Id+(1 θ ) ν ν on Γ (t). (3.20) Of course, (3.19) olds trvally on te contnuous level for κ = κ and for m beng te conormal µ. Moreover, n order to model a C 0 or C 1 contact we requre were C 1 = 0 for C 0 and C 1 = 1 for C 1. C 1 ( m 1 + m 2 ) = 0 on, (3.21) Smlarly to (3.19), we ntroduce te followng sde constrant, nspred by te weak formulaton of (2.8): κ γ, η + ds, η s = 0 η [H 1 ()] d. (3.22) Smlarly to (3.11) t olds tat [ ] δ δγ w,1 ( χ) = d dε w ε,1 γε(t) ε=0= wd s, χ s w L (), χ [H 1 γ (Γ(t))]d, (3.23) were ε 0 w = 0. Moreover, smlarly to (3.16), we note tat for η [Hγ 1(Γ(t))]d t olds tat [ δ ] ds, η s ( χ) = P γ η s, χ s δγ, (3.24) were 0 ε η = 0, and were P γ = Id d s d s on. (3.25) We defne te Lagrangan L((Γ (t), κ, m, y ) 2, κ γ, z, φ) = [ 1 2 α κ κ ν, κ κ ν Γ (t) +αg 12 κ γ, m ]

14 +ςh d 2 () κ γ, z ds, z s [ Q,θ κ, y + Γ (t) s d, s y +C 1 m 1 + m 2, φ ] m, y, Γ (t) were y [H 1 (Γ (t))] d and z [H 1 ()] d are Lagrange multplers for (3.19) and (3.22), respectvely. Smlarly, φ [L 2 ()] d s a Lagrange multpler for (3.21). We now want to compute te drecton of steepest descent f Γ of E((Γ (t)) 2 ), subject to te sde constrants (3.19), (3.22) and (3.21). Ts means tat f Γ needs to fulfll [ ] fγ δ, χ = Γ(t) δγ E(t) ( χ) χ [Hγ 1 (Γ(t))]d. (3.26) In partcular, on usng deas from te formal calculus of PDE constraned optmzaton, see e.g. Tröltzsc (2010), we can formally compute f Γ by requrng tat [ ] δ [ δγ L 1 ( χ) = lm (L(Γ ε 0 ε,ε (t), κ, m, y ) 2, κ γ, z, φ) [ δ δ κ 1 ] L ( ξ [ 1 1 ) = lm ε 0 ε [ ] δ L ( δ m 1 ζ 1 ) = lm 1 ε 0 ε [ ] δ 1 L ( η 1 ) = lm δ y 1 ε 0 ε L((Γ (t), κ, m, y ) 2, κ γ, z, φ) ] = fγ, χ, (3.27a) Γ(t) L(Γ 1 (t), κ 1 +ε ξ 1, m 1, y 1,Γ 2 (t), κ 2, κ 2, m 2, y 2, κ γ, z, φ) ] L((Γ (t), κ, m, y ) 2, κ γ, z, φ) = 0, (3.27b) [ L(Γ 1 (t), κ 1, m 1 +ε ζ 1, y 1,Γ 2 (t), κ 2, κ 2, m 2, y 2, κ γ, z, φ) ] L((Γ (t), κ, m, y ) 2, κ γ, z, φ) = 0, (3.27c) [ L(Γ 1 (t), κ 1, m 1, y 1 +ε η 1,Γ 2 (t), κ 2, κ 2, m 2, y 2, κ γ, z, φ) ] L((Γ (t), κ, m, y ) 2, κ γ, z, φ) = 0, (3.27d) for varatons χ [Hγ 1(Γ(t))]d, ξ 1 [L 2 (Γ 1 (t))] d, ζ 1 [L 2 ()] d and η 1 [L 2 (Γ 1 (t))] d ; and smlarly for te varatons for κ 2, m 2, y 2, κ γ, z and φ. On recallng (3.11) (3.16), ts yelds tat fγ, χ = Γ(t) [ s y, s χ + Γ(t) s. y, s. χ Γ(t) ( s y ) T,D( χ)( sd) T 1 [α 2 κ κ ν 2 2( κ.q,θ y )] sd, s χ + 0ε [Q,θ κ ], y ] ς ds, χ s Γ (t) 13 Γ (t) Γ (t) +α κ κ, 0 ε ν Γ (t)

15 + κ γ. z C 1( m 1 + m 2 ). φ χ [H 1 γ(γ(t))] d, (α G κ γ + y ). m, d s. χ s + P γ z s, χ s (3.28a) α ( κ κ ν ) Q,θ y = 0 on Γ (t), = 1,2, (3.28b) α G κ γ + y +C 1φ = 0 on, = 1,2, (3.28c) α G m z = 0 on, = 1,2, (3.28d) wt (3.19), (3.21) and (3.22). As 0 ε κ = 0, we ave tat 0 ε [Q,θ κ ] = (1 θ ) [ ( κ. 0 ε ν ) ν +( κ. ν ) 0 ε ν ]. (3.29) We observe tat (3.28b,c) mply tat Q,θ y = α κ α κ ν on Γ (t) and y +C 1 φ = α G κ γ on. (3.30) Let us now recover κ and κ γ n terms of te geometry agan. It mmedately follows from (3.19), (2.3) and (3.6) tat m = µ and Q,θ κ = κ = κ ν, wt te latter mplyng tat κ. ν = κ. (3.31) Hence we mmedately get κ = κ for θ (0,1]. For θ = 0, on te oter and, t follows from (3.30) and (3.31) tat α κ = [ y. ν +α κ ] ν, and so κ = κ ν = κ. Moreover, combnng (3.22) and (2.8) yelds tat κ γ = κ γ. Overall, we obtan from (3.30) tat Q,θ y = α (κ κ ) ν on Γ (t) and y +C 1 φ = α G κ γ on. (3.32) However, f θ (0,1], ten te two condtons n (3.32) are ncompatble n general f α G 0, snce tefrst condtonn(3.32)yelds tat y = α (κ κ ) ν. IfC 1 = 1tente two condtons are n general ncompatble even f α G = 0. Hence for general boundares and α G 0 we need to take θ = 0, at least locally at te boundary. Terefore t may be desrable to consder a varable θ L (Γ (t)). Te calculaton (3.28a d) remans vald provded tat 0 ε θ = 0. We wll make ts more rgorous on te dscrete level, see (4.20) below. Usng (3.15), (3.28c,d) n (3.28a) yelds te condensed verson fγ, χ = Γ(t) [ s y, s χ + Γ(t) s. y, s. χ Γ(t) ( s y ) T,D( χ)( sd) T 1 [α 2 κ κ ν 2 2( κ.q,θ y )] sd, s χ α κ κ,[ s χ] T ν Γ (t) Γ (t) (1 θ ) [ ] ] ( κ.[ s χ] T ν ) ν +( κ. ν )[ s χ] T ν, y Γ ς ds, χ (t) s 14 Γ (t)

16 + α G [ κ γ. m, d s. χ s + P γ ( m ) s, χ s ] χ [H 1 γ (Γ(t))]d, (3.33a) α ( κ κ ν ) Q,θ y = 0 on Γ (t), = 1,2, (3.33b) α G κ γ + y +C 1φ = 0 on, = 1,2, (3.33c) Q,θ κ, η + sd, s η = m, η η [H 1 (Γ (t))] d, = 1,2, Γ (t) Γ (t) (3.33d) C 1 ( m 1 + m 2 ) = 0 on, (3.33e) κ γ, η + ds, η s = 0 η [H 1 ()] d. (3.33f) Remark We recall from (3.32) and te dscusson below tat n general we requre θ = 0. If C 1 = 0 ten t follows from (3.33c) tat y = α G κ γ on, for = 1,2. Combnng ts wt (3.33b) for θ = 0 ten yelds tat (2.18a) olds. On te oter and, n te case of a C 1 juncton, wen C 1 = 1, ten (3.33e) mples tat µ 1 + µ 2 = 0 and ence tat ν 1 = ν 2 = ν on, and so t follows from (3.33b,c) wt θ = 0 tat α (κ κ )+α G κ γ. ν + φ. ν = 0 on, = 1,2, wc means tat (2.19a) olds. Te weak formulaton of a generalzed L 2 gradent flow of E((Γ (t)) 2 ) can ten be formulatedasfollows. Gven Γ (0), = 1,2, forallt (0,T]fndΓ (t) = x (Υ,t), = 1,2, wt V(t) [H 1 (Γ(t))] d, and κ (t) [L 2 (Γ (t))] d, y (t) [H 1 (Γ (t))] d, m (t) [H 1 ()] d, = 1,2, as well as κ γ [L 2 ()] d, z [L 2 ()] d, φ [L 2 ()] d suc tat V, χ + V, χ = fγ, χ χ [L 2 (Γ(t))] d. (3.34) Γ(t) Γ(t) and (3.33a f) old. Here we note tat = 0 recovers a weak formulaton for te standard L 2 gradent flow. As stated n (1.2), we allow for 0 n general, to allow for a dampng of te movement of te contact lne. In numercal smulatons suc a dampng often proves benefcal, as t suppresses possble oscllatons at te contact lne. On te oter and, suc a dsspaton mecansm at te boundary s probably also relevant n applcatons. 4 Semdscrete fnte element approxmaton Smlarly to Barrett et al. (2008), we ntroduce te followng dscrete spaces, based on te work of Dzuk (1991). Let Γ (t) R d be (d 1)-dmensonal polyedral surfaces,.e. unons of non-degenerate (d 1)-smplces wt no angng vertces (see Deckelnck et al. (2005, p. 164) for d = 3), approxmatng te surfaces Γ(t). In partcular, let 15

17 Γ (t) = J j=1 σ j (t), were {σ j (t)}j j=1 s a famly mutually dsjont open (d 1)-smplces wt vertces { q k (t)}k k=1. In analogy to te contnuous settng, we wrte Γ (t) = Γ 1 (t) γ (t) Γ 2(t), were γ (t) = Γ 1(t) = Γ 2(t). Here we let Γ (t) = J j=1 σ,j (t), wt vertces { q,k (t)}k k=1, = 1,2. We also assume tat γ (t) as te vertces { q γ,k (t)}kγ k=1. Clearly, t olds tat J = J 1 +J 2 and K = K 1 +K 2 K γ. Ten let V (Γ (t)) = { χ [C(Γ (t))] d : χ σ,j s lnear j = 1,...,J } = [W (Γ (t))]d, = 1,2, were W (Γ (t)) s te space of scalar contnuous pecewse lnear functons on Γ (t), wt {χ,k (,t)}k k=1 denotng te standard bass of W (Γ (t)),.e. In addton, let χ,k ( q,l (t),t) = δ kl k,l {1,...,K },t [0,T]. (4.1) V (Γ (t)) = { χ [C(Γ (t))] d : χ Γ (t) V (Γ (t)), = 1,2} = [W (Γ (t))] d. (4.2) We denote te bass functons of W (Γ (t)) by {χ k (,t)}k k=1. Moreover, let V (γ (t)) := { ψ [C(γ (t))] d : χ V (Γ (t)) χ γ (t)= ψ} =: [W (γ (t))] d, (4.3a) V 0 (Γ (t)) := { χ V (Γ (t)) : χ γ (t)= 0}, V 0(Γ (t)) := { χ V (Γ (t)) : χ γ (t)= 0}. (4.3b) (4.3c) We denote te bass functons of W (γ (t)) by {φ k (,t)}kγ k=1. We requre tat Γ (t) = X (Γ (0),t) wt X V (Γ (0)), and tat q k [H1 (0,T)] d, k = 1,...,K. For later purposes, we also ntroduce π (t) : C(Γ (t)) W (Γ (t)), te standard nterpolaton operator at te nodes { q,k (t)}k k=1, and smlarly π (t) : [C(Γ (t))]d V (Γ (t)), as well as e.g. π (t) : [C(Γ (t))] d V (Γ (t)). We denote te L 2 nner products on Γ (t), Γ (t) and and γ (t) by, Γ (t),, Γ (t) and, γ (t), respectvely. In addton, for pecewse contnuous functons, wt possble jumps across te edges of {σ,j }J j=1, we also ntroduce te mass lumped nner product J η,φ Γ (t) = j=1 J η,φ σ,j (t) := j=1 d 1 d Hd 1 (σ,j (t)) (ηφ)(( q,j k (t)) ), (4.4) were { q,j k (t)} d k=1 are te vertces of σ,j (t), and were we defne η(( q,j k (t)) ) := η( p). We naturally extend ts defnton to vector and tensor functons. lm σ j (t) p q,j k (t) We also defne te mass lumped nner products, Γ (t) and, γ (t) n te obvous way. Followng Dzuk and Ellott (2013, (5.23)), we defne te dscrete materal velocty for z Γ (t) by K [ ] d V ( z,t) := dt q k (t) χ k ( z,t). (4.5) k=1 16 k=1

18 For later use, we also ntroduce te fnte element spaces WT (Γ,T ) := {φ C(Γ,T ) : φ(,t) W (Γ (t)) t [0,T], φ( q,k ( ), ) H1 (0,T) k {1,...,K }}, were Γ,T := t [0,T] Γ (t) {t}, as well as te vector valued analogue V T(Γ,T ). In a smlar fason, we ntroduce WT (σ j,t ) and V T(σj,T ) va e.g. WT (σ j,t ) := {φ C(σ j,t ) : φ(,t) s lnear t [0,T], φ( q j k ( ), ) H 1 (0,T) k = 1,...,d}, were { q j k (t)} d k=1 are te vertces of σ j (t), and were σ j,t := t [0,T] σ j (t) {t}, for j {1,...,J}. Moreover, we defne te analogue varants WT (Γ T ) and V T (Γ T ) on Γ T = t [0,T] Γ (t) {t}, as well as WT (γ T ) and V T (γ T ) on γ T := t [0,T] γ (t) {t}, wt te scalar space for te latter e.g. beng gven by W T (γ T ) := {ψ C(γ T ) : χ W T (Γ T ) χ(,t) γ (t)= ψ(,t) t [0,T]}. Ten, smlarly to (3.1), we defne te dscrete materal dervatves on Γ (t) elementby-element va te equatons (, t φ) σ j (t)= (φ t + V. φ) σ j (t) φ WT (σ j,t ), j {1,...,J}. (4.6) On dfferentatng (4.1) wt respect to t, t mmedately follows tat, t χ k = 0 k {1,...,K}, (4.7) see also Dzuk and Ellott (2013, Lem. 5.5). It follows drectly from (4.7) tat, t φ(,t) = K χ k (,t) d dt φ k(t) on Γ (t) (4.8) k=1 for φ(,t) = K k=1 φ k(t)χ k (,t) W (Γ (t)). We recall from Dzuk and Ellott (2013, Lem. 5.6) tat d φ dh d 1 =, t φ+φ s. V dt dh d 1 φ W σj (t) σj (t) T (σ j,t ),j {1,...,J}. (4.9) Smlarly, we recall from Barrett et al. (2015, Lem. 3.1) tat d dt η,φ σj (t) =, t η,φ σj (t) +η,, t φ σj (t) +ηφ, s. V σj (t) η,φ WT (σ j,t ), (4.10) for all j {1,...,J}. Moreover, t olds tat d dt η,φ γ (t) =, t η,φ γ (t) +η,, t φ γ (t) +ηφ, d s. V s γ (t) η,φ W T (γ T ). (4.11) 17

19 We also note te dscrete verson of te tme dervatve varant of (3.16), d sd, s η dt Γ (t) = s η, sv + Γ (t) ( s η) T,D( V )( sd) T as well as te correspondng verson for γ (t), d ds, η s = P dt γ γ η s, Vs (t) γ (t) s. η, s. V Γ (t) Γ (t) η { ξ V T(Γ,T) :, t η { ξ V T (γ T ) :, t wc follows smlarly to (3.24). Here, smlarly to (3.25), we ave defned ξ = 0}, (4.12) ξ = 0}, (4.13) P γ = Id d s d s on γ (t). (4.14) Let ν denote te te outward unt normal to Γ (t), = 1,2, and smlarly let ν denote te te outward unt normal to Γ (t). For later use, we ntroduce te vertex normal functons ω (,t) V (Γ (t)) wt ω ( q,k(t),t) := 1 H d 1 (Λ,k (t)) j Θ,k H d 1 (σ,j(t)) ν σ,j (t), (4.15) were for k = 1,...,K we defne Θ,k := {j : q,k (t) σ,j (t)} and set Λ,k (t) := j Θ,k σ,j (t). Here we note tat z,w ν = z,w ω Γ (t) z V (Γ Γ (t) (t)), w W (Γ (t)). (4.16) In te analogous fason, we ntroduce te vertex normal functon ω (,t) V (Γ (t)),.e. we set ω ( q k (t),t) := 1 H d 1 (Λ k (t)) H d 1 (σj (t)) ν σ j (t), (4.17) were for k = 1,...,K we defne Θ k := {j : q k (t) σ j (t)} and set Λ k (t) := j Θ k σ j (t). Of course, t olds tat j Θ k z,w ν Γ (t) = z,w ω Γ (t) z V (Γ (t)), w W (Γ (t)). (4.18) It clearly follows from (4.16) and (4.18) tat z, ω = Γ (t) z, ω z V (Γ (t)). (4.19) Γ (t) 18

20 In addton, for a gven parameter θ [0,1] we ntroduce θ W (Γ (t)) and θ W (Γ (t)) suc tat { { θ ( q k(t),t) 0 q k = (t) γ (t), θ q k (t) and θ ( q 1 q k k(t),t) = (t) γ (t), γ (t), θ q k (t) (4.20) γ (t). Ten we ntroduce Q,θ [W (Γ (t))] d d and Q,θ k {1,...,K }, e.g. [W (Γ (t))] d d by settng, for Q,θ ( q,k (t),t) = θ ( q,k (t),t)id+(1 θ ( q,k (t),t)) ω ( q,k (t),t) ω ( q,k (t),t) ω ( q,k (t),t) 2, (4.21) were ere and trougout we assume tat ω ( q,k (t),t) 0 for k = 1,...,K and t [0,T]. Only n patologcal cases could ts assumpton be volated, and n practce ts never occurred. We note tat Q,θ z, v Γ (t) = z,q,θ v Γ (t) and Q,θ z, ω = z, ω Γ (t) (4.22) Γ (t) for all z, v V (Γ (t)), and analogously for θ n (4.22) replaced by θ. In addton, smlarly to (4.19), t olds tat z,q,θ ω = Γ (t) z,q,θ ω Γ (t) z V (Γ (t)). (4.23) Followng te approac n te contnuous settng, recall (2.12), (3.19), (3.22), we consder te frst varaton of te dscrete energy E ((Γ (t)) 2 ) := [ 1 α 2 κ κ ν 2,1 Γ (t) +αg [ κ ]] γ, m γ (t) +2πm(Γ (t)) +ςh d 2 (γ (t)), (4.24) were κ V (Γ (t)), m V (γ (t)), = 1,2, and κ γ V (γ (t)), subject to te sde constrants Q,θ κ, η + sd, s η = m, η η V (Γ Γ (t) Γ (t) γ (t) (t)), = 1,2, (4.25a) κ γ, χ + ds, χ γ (t) s = 0 χ V γ (γ (t)), (4.25b) (t) C 1 ( m 1 + m 2 ) = 0 on γ (t). (4.25c) Wen takng varatons of(4.25a), we need to compute varatons of te dscrete vertex normals ω. To ts end, for any gven χ V (Γ (t)) we ntroduce Γ ε (t) as n (3.8) and 19

21 0, ε defned by (3.13), bot wt Γ(t) replaced by Γ (t). We ten observe tat t follows from (4.16) wt w = 1 and te dscrete analogue of (3.14) tat z, 0, ε ω = z, 0, Γ (t) ε ν + ( z.( ν Γ (t) ω )) sd, s χ Γ (t) z V (Γ (t)), χ V (Γ (t)), (4.26) were ε 0, z = 0. In addton, we note tat for all ξ, η V (Γ (t)) wt ε 0, ξ = ε 0, η = 0 t olds tat 0, ε π [( ξ. )( ω ω η. )] ω [ ω = π G ( ξ, η). ] ε 0, ω on Γ (t), (4.27) were ( ) G ( ξ, η) = 1 ( ξ. ω ω ) η+( η. ω ) ξ 2 ( η. ω )( ξ. ω ) ω 2 ω 2. (4.28) It follows tat G ( ξ, η). ω = 0 ξ, η V (Γ (t)). (4.29) We defne te Lagrangan L (t) = [ 1 α 2 κ κ ν 2,1 Γ (t) +αg κ γ, Z γ (t) ds, Z s γ (t) [ (Q,θ κ, Y + Γ (t) +C 1 m 1 + m 2, Φ, γ (t) κ γ, m γ (t)] s d, s Y Γ (t) +ςh d 2 (γ (t)) m, Y ] γ (t) were κ V (Γ (t)) and κ γ V (γ (t)) satsfy (4.25a) and (4.25b), respectvely, wt Y V (Γ (t)) and Z V (γ (t)) beng te correspondng Lagrange multplers. Smlarly, Φ V (γ (t)) s a Lagrange multpler for (4.25c). On recallng te formal calculus of PDE constraned optmzaton, we obtan an L 2 gradent flow of E (t) subject to te sde constrants (4.25a c) by settng [ δ δγ L ]( χ) = 2 for χ V (Γ (t)), [ δ δ κ δ [ L ]( ϕ) = 0 for ϕ V (γ (t)), [ δ L ]( ξ) = 0 for ξ V (Γ (t)), [ δ δy Q,θ V, χ Γ (t) L ]( η) = 0 for η V (Γ (t)), L ]( φ) = 0 for φ V (γ (t)), leadng to Z = δ m δ κ γ 2 αg m, [ δ δz L ]( φ) = 0 for φ V (γ δ (t)) and [ δφ L ]( η) = 0 for η V (γ (t)). Here we recall te defnton of θ n (4.20). We employ ts doctored verson of θ n order to obtan exstence and unqueness for te fully dscrete approxmaton ntroduced n te next secton. See also Remark 3.1 n Barrett et al. (2012), were te analogue to 20

22 our stuaton ere corresponds to two curves meetng n te plane,.e. N = d = 2 n ter notaton. Overall ts gves rse to te followng semdscrete fnte element approxmaton of te gradent flow (3.34), were we ave noted te dscrete verson of (3.15), (4.26), (4.27), (4.29), varatonal versons of (4.10) (4.13) and tat ε 0, θ = 0. Gven Γ (0), fnd (Γ (t)) t (0,T] suc tat d Γ ( ) V T(Γ T ). In addton, for all t (0,T] fnd ( κ, Y ) [V (Γ (t))] 2, = 1,2, κ γ V (γ (t)), m V (γ (t)), = 1,2, and C 1 Φ V (γ (t)) suc tat Q,θ V, χ V, χ = 1 2 Γ (t) + [ sy, s χ Γ (t) + γ (t) s. Y, s. χ ( sy ) T,D( χ)( sd) T Γ (t) Γ (t) [α κ κ ν 2 2( Y.Q,θ κ )] s d, s χ Γ (t) α κ κ,[ s χ] T ν + (1 θ )( G Γ ( Y (t), κ ). ν ) sd, s χ (1 θ ) G ] ( Y, κ ),[ s χ] T ν + α G Γ (t) [ κ γ. m, d s. χ s + ] P γ γ ( m ) s, χ s (t) γ (t) ς ds, χ s χ V (Γ (t)), (4.30a) Q,θ κ, η + sd, s η = m, η η V (Γ Γ (t) Γ (t) γ (t) (t)), = 1,2, Γ (t) γ (t) (4.30b) κ γ, χ + ds, χ γ (t) s = 0 χ V γ (γ (t)), (4.30c) (t) C 1 ( m 1 + m 2 ) = 0 on γ (t), (4.30d) α G κ γ + Y +C 1Φ = 0 on γ (t), = 1,2, (4.30e) α ( κ κ ν ) Q,θ Y, ξ = 0 ξ V (Γ (t)), = 1,2. (4.30f) Γ (t) For later use we also observe tat coosng ξ = α 1 π [Q,θ η] n (4.30f) and combnng wt (4.30b), on recallng (4.22) and (4.16), yelds tat α 1 Q,θ Y,Q,θ η Γ (t) + s d, s η Γ (t) = m, η γ (t) κ ω, η Γ (t) η V (Γ (t)). (4.31) In order to be able to consder area and volume preservng varants of (4.30a f), we ntroduce te Lagrange multplers λ A, (t) R, = 1,2, and λ V, (t) R for te 21

23 constrants d dt Hd 1 (Γ (t)) = s. V,1 = sd, s V = 0, (4.32) Γ (t) Γ (t) were we recall (4.9), and d dt Ld (Ω (t)) = V, ν = V, ω = 0, (4.33) Γ (t) Γ (t) were we note a dscrete varant of (3.5) and (4.18). Here Ω (t) denotes te nteror of Γ (t). On recallng (4.19), (4.22) and (4.23), we can rewrte te constrant (4.33) as 0 = V, ω Γ (t) = V, ω = Γ (t) Q,θ V, ω = Γ (t) Q,θ V, ω. Γ (t) (4.34) Hence, on wrtng (4.30a) as Q,θ V, χ + V, χ = r, χ, Γ (t) γ (t) Γ (t) we consder Q,θ V, χ + V, χ = r, χ, Γ (t) γ (t) Γ (t) λ V, ω, χ λ A, Γ (t) sd, s χ (4.35) Γ (t) for all χ V (Γ (t)), were λ V, (t) R and λ A, (t) R, = 1,2, need to be determned. Of course, f we consder a volume preservng varant only, ten we let λ A, 1 (t) = λ A, 2 (t) = 0 and [ r λ V, (t) =, ω ] V, ω / ω, ω, (4.36) Γ (t) γ (t) Γ (t) wc we derved on coosng χ = ω n (4.35), and notng (4.34). For te general volume and area preservng flow, we ntroduce te projecton Π 0 : V (Γ (t)) V 0(Γ (t)) onto V 0(Γ (t)), recall (4.3b), and smlarly Π,0 : V (Γ (t)) V 0 (Γ (t)). We ntroduce te symmetrc blnear forms a,θ : V (Γ (t)) V (Γ (t)) R by settng a,θ( ζ, η) = Q,θ ζ, Π,0 η ζ, η V (Γ (t)), = 1,2, (4.37) Γ (t) 22

24 were we ave noted (4.22). It olds tat a,θ ( ζ, ζ) 0 for all ζ V (Γ (t)), wt te nequalty beng strct for Π,0 ζ 0 and θ > 0. Hence te Caucy Scwarz nequalty olds,.e. a,θ ( ζ, η) [a,θ ( ζ, ζ)] 1 2 [a,θ ( η, η)] 1 2 ζ, η V (Γ (t)), = 1,2, (4.38) wt strct nequalty n te case θ > 0 f Π,0ζ and Π,0 η are lnearly ndependent. Ten we note, on recallng (4.19), (4.30b) and (4.22), tat sd, s Π 0 ω = sd, s Π,0 ω Γ (t) = a Γ,θ ( κ, ω ) = ω (t), Π,0 κ Γ (t) (4.39a) and sd, s Π,0 κ = a,θ( κ, κ ). Γ (t) (4.39b) In addton, t follows from (4.19), (4.22) and (4.37) tat ω, Π 0 ω Γ (t) = ω, Π 0 ω Γ (t) = ω, Π,0 ω = Γ (t) a,θ ( ω, ω ). (4.40) Ten (4.35), (4.40) and (4.39a,b) yeld tat (λ V,,λ A, 1,λ A, 2 )(t) are suc tat 2 a,θ ( ω, ω ) a 1,θ ( κ 1, ω 1 ) a 2,θ ( κ 2, ω 2 ) λ V, (t) b 0 (t) a 1,θ ( κ 1, ω 1 ) a 1,θ ( κ 1, κ 1 ) 0 a 2,θ ( κ 2, ω 2 ) 0 a 2,θ ( κ 2, κ 2 ) λ A, λ A, 1 (t) 2 (t) = b 1 (t), b 2 (t) (4.41a) were b 0 (t) = b (t) = Π 0 V V, ω r, Π Γ (t) 0 ω, (4.41b) Γ (t) Π,0 V V,Q,θ κ Γ (t) + m, V r, Π γ,0 κ, = 1,2. (t) Γ (t) (4.41c) We note tat te matrx n (4.41a) s symmetrc and postve defnte as long as Π,0 ω and Π,0 [ Q,θ κ ] are lnearly ndependent, for = 1,2. Te rgt and sdes (4.41b,c) are obtaned by recallng (4.35), and on notng tat (4.22) and (4.34) mply tat Q,θ V, Π 0 ω = Γ (t) [ Q,θ V, ] Π 0 ω ω + Q Γ (t),θ V, ω Γ (t) 23

25 = V, Π 0 ω ω Γ (t) = wle (4.20), (4.22), (4.30b) and (4.32) yeld tat Q,θ V, Π,0 κ Π,0V,Q,θ κ = = Γ (t) = Π,0 V V,Q,θ κ Π,0 V V,Q,θ κ + Γ (t) + Γ (t) Γ (t) m, V γ (t) Π 0 V V, ω Γ (t), sd, s V Γ (t) (4.42) m, V. (4.43) γ (t) Weseetatonremovng telasttwo rowsandcolumnsn(4.41a),weobtananexpresson smlar to (4.36) for λ V, (t), but ere we test wt Π 0 ω as opposed to ω. Analogously, f we want to consder pase area preservatons only, ten removng te frst row and column n (4.41a) yelds a reduced system for te two Lagrange multplers λ A, (t), = 1,2. Te followng teorem establses tat (4.30a f) s ndeed a weak formulaton for a generalzed L 2 gradent flow of E (t) subject to te sde constrants (4.25a c). We wll also sow tat for θ = 0 te sceme produces conformal polyedral surfaces Γ 1 (t) and Γ 2 (t). Here we recall from Barrett et al. (2017), see also Barrett et al. (2008, 4.1), tat te open surfaces Γ (t), = 1,2, are conformal polyedral surfaces f { sd, s η = 0 η ξ V Γ (t) 0 (Γ (t)) : ξ( q,k (t)). ω ( q,k (t),t) = 0, k = 1,...,K }, = 1,2. (4.44) We recall from Barrett et al. (2008, 2017) tat conformal polyedral surfaces exbt good meses. Moreover, we recall tat n te case d = 2, conformal polyedral surfaces are equdstrbuted polygonal curves, see Barrett et al. (2007, 2011). Teorem Let θ [0,1], 0 and let {(Γ, κ 1, κ 2, Y1, Y2, κ γ, m 1, m 2, Φ )(t)} t [0,T] be a soluton to (4.30a f). In addton, we assume tat κ γ V T (γ T ), κ, π [Q,θ κ ] V T (Γ,T ), m V T (γ T ), = 1,2. Ten d dt E ((Γ (t))2 ) = Q,θ V, V V, V. (4.45) Γ (t) γ (t) Moreover, f θ = 0 ten Γ 1(t) and Γ 2(t) are open conformal polyedral surfaces for all t (0,T]. Proof. Takng te tme dervatve of (4.25a), were we coose dscrete test functons η suc tat, t η = 0, yelds for = 1,2 tat, t (Q,θ κ ), η Γ (t) 24

26 + [(Q,θ κ ). η] sd, sv ( s η) T,D( V )( sd) T Γ (t) + Γ (t) = sv, s η Γ (t) +, t m, η γ (t) + s. V, s. η m. η, d s. V s γ (t) Γ (t), (4.46) were we ave noted (4.10), (4.11), (4.12) and tat π [Q,θ κ ] V T (Γ,T ), m V T (γ T ), = 1,2. Smlarly, takng te tme dervatve of (4.25b) wt, t (4.11), (4.13) and κ γ V T (γ T ), tat χ = 0 yelds, on notng, t κ γ, χ + κ γ γ. χ, d s. Vs + P (t) γ γ χ s, Vs = 0. (4.47) (t) γ (t) Coosng χ = V n (4.30a), η = Y n (4.46), = 1,2, and combnng yelds, on notng te dscrete varant of (3.15), tat = + Q,θ V, V + V, V Γ (t) γ (t) [ 1 [α 2 κ κ ν 2 2Y.Q,θ κ ] sd, sv Γ (t) α κ κ,, t ν Γ (t) +, t (Q,θ κ ), Y (1 θ )( G ( Y, κ ). ν ) sd, sv ] (1 θ ) G ( Y, κ ),, t ν Γ (t) α G [, t [ κ γ. m, d s. V s + γ (t) +ς Γ (t) ds, V s P γ ( m ) s, V s Γ (t) + γ (t) γ (t) (Q,θ κ. Y ) s d, s V ] Γ (t) m, Y + m γ. Y, d s, ] Vs. (4.48) (t) γ (t) Coosng χ = 2 αg m n (4.47) and recallng (4.30d,e) and (4.11), t follows from (4.48) tat Q,θ V, V + V, V + Γ (t) γ (t) α κ κ,, t ν Γ (t) +, t (Q,θ κ ), Y (1 θ )( G ( Y, κ ). ν ) sd, sv (1 θ ) G ] ( Y, κ ),, t ν +ς Γ (t) [ 1 α 2 κ κ ν 2 sd, s V Γ (t) Γ (t) Γ (t) ds, V s Γ (t) 25

27 = = d dt α G [, t α G m, κ γ γ (t) + m,, t κ γ + κ γ γ. m, d s. ] Vs (t) γ (t) κ γ, m. (4.49) γ (t) We ave from (4.22), (4.30f) and (4.16) tat [ ], t (Q,θ κ ), Y α κ κ Γ (t),, t ν Γ (t) [ =, t κ,q,θ Y α κ κ Γ (t) κ ν,, +, t (Q,θ κ ) Q,θ, t κ, Y ] = Γ (t) t ν Γ (t) [ 1 α 2, t κ κ ν 2,1 +, Γ (t) t (Q,θ κ ) Q,θ, t κ, Y Γ (t) ]. (4.50) Combnng (4.49) and (4.50), on notng (4.10), (4.11), (4.24),, t θ = 0 (wc follows from (4.8) and (4.20)), κ V T(Γ,T ), = 1,2, ν σ,j ( ) V T(σ,j,T ), j = 1,...,J, = 1,2, (wc follows from te dscrete analogue of (3.15) and as d Γ ( ) V T (Γ T )) and te nvarance of m(γ (t)) under contnuous deformatons, yelds tat Q,θ V, V + V, V + d Γ (t) γ (t) dt E ((Γ (t)) 2 )+ were, on notng (4.21), P := (1 θ ) κ., t ω, Y 2. ω ω 2 Γ (t) + (1 θ )( κ. ω )( Y. ω ), ω., P = 0, (1 θ ) Y., t ω, κ. ω t ω ω 4 (1 θ )( G ( Y, κ ). ν ) s d, s V (1 θ ) G ( Y, κ ),, t ν Γ (t) Γ (t) ω 2 Γ (t) Γ (t), = 1,2. (4.51) It remans to sow tat P 1 and P 2 as defned n (4.51) vans. To see ts, we observe tat t follows from (4.29), (4.28) and te tme dervatve verson of (4.26) tat P = (1 θ ) G ( Y, κ ),, t ω Γ (t) 26

28 Ts proves te desred result (4.45). Γ 2 + (1 θ ) G ( Y, κ ).( ω ν ) sd, sv Γ (t) (1 θ ) G ( Y, κ ),, t ν = 0, = 1,2. (4.52) Γ (t) If θ = 0 ten t mmedately follows from (4.30b) tat (4.44) olds. Hence Γ 1 (t) and (t) are open conformal polyedral surfaces. Teorem Let θ [0,1], 0 and let {(Γ, κ 1, κ 2, Y1, Y2, κ γ, m 1, m 2, Φ,λ V,,λ A, 1, λ A, 2 )(t)} t [0,T] be a soluton to (4.35), (4.30b f) and (4.41a). In addton, we assume tat κ γ V T (γ T ), κ, π [Q κ,θ ] V T (Γ,T ), m V T (γ T ), = 1,2. Ten t olds tat as well as d dt E ((Γ (t))2 ) = Q,θ V, V V, V, (4.53) Γ (t) γ (t) d dt Hd 1 (Γ d (t)) = 0, = 1,2, dt Ld (Ω (t)) = 0, (4.54) were Ω (t) denotes te regon bounded by Γ (t). Moreover, f θ = 0 ten Γ 1 (t) and Γ 2 (t) are open conformal polyedral surfaces for all t (0,T]. Proof. We recall tat on coosng (λ V,,λ A, 1,λ A, 2 ) solvng te system (4.41a) yelds tat (4.32) and (4.33) old, and ence te desred results (4.54) old. Te stablty result (4.53) drectly follows from te proof of Teorem 4.1. In partcular, coosng χ = V n (4.35), on notng (4.32) and (4.33), yelds tat Q,θ V, V + V, V = r, V. Γ (t) γ (t) Γ (t) Combnng ts wt (4.46) yelds tat (4.48) olds, and te rest of te proof proceeds as tat of Teorem 4.1. Fnally, as n te proof of Teorem 4.1, for θ = 0 t follows from (4.30b) tat Γ 1(t) and Γ 2(t) are conformal polyedral surfaces. 5 Fully dscrete fnte element approxmaton In ts secton we consder a fully dscrete varant of te sceme (4.30a f) from Secton 4. To ts end, let 0 = t 0 < t 1 <... < t M 1 < t M = T beaparttonng of [0,T]nto possbly varable tme steps t m := t m+1 t m, m = 0,...,M 1. Let Γ m be a (d 1)-dmensonal polyedral surface, approxmatng Γ (t m ), m = 0,...,M, wt te two parts, = 1,2 and ter common boundary γ m. Followng Dzuk (1991), we now parameterze te new surface Γ m+1 over Γ m. Hence, we ntroduce te followng fnte element spaces. Let 27

29 Γ m = J j=1 σm j, were {σm j } J j=1 s a famly of mutually dsjont open trangles wt vertces. Ten for m = 0,...,M 1, let { q m k }K k=1 V (Γ m ) := { χ [C(Γ m )] d : χ σ m j s lnear j = 1,...,J} =: [W (Γ m )] d, (5.1) for m = 0,...,M 1. We denote te standard bass of W (Γ m ) by {χ m k }K k=1. In addton, smlarly to te semdscrete settng n Secton 4, we ntroduce te spaces W ( ) and V ( ), denotng te standard bass of W ( ) by {χ m,k }K k=1, as well as V (γ m ), and te nterpolaton operators π m : C(Γ m ) W (Γ m ) and smlarly π m : [C(Γ m )] d V (Γ m ). We also ntroduce te L 2 nner products, Γ m,, Γ m and, γ m, as well as ter mass lumped nner varants, Γ m,, Γ and, m γm. Smlarly to (4.15) and (4.17) we ntroduce te dscrete vertex normals ω m := K k=1 χm,k ωm,k V ( ) and ωm := K k=1 χm k ωm k V (Γ m ). We make te followng mld assumpton. (A) We assume for m = 0,...,M 1 tat H d 1 (σ m j ) > 0 for j = 1,...,J, and tat 0 { ω m,k : k = 1,...,K, = 1,2}. Moreover, n te case C 1 = 1 and θ = 0 we assume tat dmspan{ ω m,k : k = 1,...,K, = 1,2} = d, for m = 0,...,M 1. In addton, and smlarly to (4.20) and (4.21), we ntroduce θ m and θ m W (Γ m ), and ten Q m,θ m, Qm,θ [W (Γ m m )] d d, by settng e.g. Q m,θ m( qm,k ) = θm ( q,k m) Id + (1 θ m ( q,k m)) ωm,k 2 ω,k m ωm,k for k = 1,...,K, = 1,2. Smlarly to (4.28) and (4.14), we let ( ) G m ( ξ, η) = 1 ( ξ. ω ω m m 2 ) η+( η. ω m ) ξ 2 ( η. ωm )( ξ. ω m) ω m ω m 2 (5.2) and P m γ = Id d s d s on γ m. (5.3) On recallng (4.31), we consder te followng fully dscrete approxmaton of(4.30a f). For m = 0,...,M 1, fnd X m+1 V (Γ m ), ( Y m+1, m m+1 ) 2 V (Γ m 1 ) V (γ m ) V (Γ m 2 ) V (γ m ), κ m+1 γ V (γ m ) and C 1 Φ m+1 V (γ m ) suc tat X Q m m+1 d,θ, χ m sy m+1, s χ +α t m G ( m m+1 ) s, χ s γ m X +ς X m+1 s, χ s + γ m+1 d, χ m t m γ m [ = s. Y m, s. χ ( sy m ) T,D( χ)( sd) T 28

30 1 2 [α κ m κ ν m 2 2( Y m.q m,θ m κm )] sd, s χ Γ m α κ κ m,[ s χ] T ν m + (1 θ m )( G m ( Y m, κ m ). ν m ) sd, s χ (1 θ m ) G ] m ( Y m, κ m ),[ s χ] T ν m + α G [ κ m γ. m m, d s. χ s γ m + (Id+P m γ )( m m ) s, χ s γ m ] λ V,m ω m, χ Γ λ A,m m sd, s χ χ V (Γ m ), (5.4a) α 1 Q m,θ Y m+1 m,q m,θ m η + sx m+1, s η = m m+1, η κ γ m ω m, η η V ( ), = 1,2, (5.4b) κ m+1 γ, χ + X m+1 γ m s, χ s = 0 χ V γ (γ m ), (5.4c) m C 1 ( m m m m+1 2 ) = 0 on γ m, (5.4d) α G κ m+1 γ + Y m+1 +C 1 Φ m+1 = 0 on γ m, = 1,2, (5.4e) and set κ m+1 = α 1 π m [Q m,θ Y m+1 m ]+κ ω m and Γ m+1 = X m+1 ( ), = 1,2. For m 1 we notetat ere andtrougout, asno confuson canarse, we denote by κ m tefuncton z V ( ), defned by z( qm,k ) = κm ( qm 1,k ), k = 1 K, were κ m V(Γ m 1 ) s gven, and smlarly for e.g. Y m, m m and κ m γ. We note tat f C 1 = α1 G = αg 2 = 0 ten te weak conormals mm+1 play no role n te evoluton. However, for surface area conservaton tey do play a role also n tat case, see (5.5c) below. We also remark tat te parameter 0 as a stablzng effect on te evoluton of γ m. In practce, ts was partcularly useful for smulatons nvolvng surface area preservaton, and for C 0 experments wt Gaussan curvature energy contrbutons. Of course, (5.4a e) wt λ V,m = λ A,m 1 = λ A,m 2 = 0 corresponds to a fully dscrete approxmaton of(4.30a f), on recallng (4.31). For a fully dscrete approxmaton of te volume and/or surface area preservng flow, on recallng (4.41a c), we let (λ V,m,λ A,m 1,λ A,m 2 ) be te soluton of 2 am,θ ( ωm, ωm ) am 1,θ ( κm 1, ωm 1 ) am 2,θ ( κm 2, ωm 2 ) λ V,m b m a m 1,θ ( κm 1, ωm 1 ) am 1,θ ( κm 1, κm 1 ) 0 λ A,m 0 1 = b m 1, (5.5a) a m 2,θ ( κm 2, ω 2 m ) 0 a m 2,θ ( κm 2, κ m 2 ) λ A,m 2 b m 2 were, on notng te fully dscrete varant of (4.19), b m 0 = ( Π m,0 Id) d X m 1, ω m t m 1 29 sy m, s ( Π m 0 ωm )

31 = b m = = f m, Π m 0 ωm, Γ m ( Π m d,0 Id) X m 1, ω m f m, Π m,0 ωm t m 1 ( Π m,0 Id) d X m 1,Q m,θ t m κm m 1 s Y m, s ( Π m,0 κm ) sy m, s ( Π m,0 ω m ) ], (5.5b) ( Π m d,0 Id) X m 1,Q m,θ m κm t m 1 s Y m, s ( Π m,0 κm ) + f m, Π m,0 κm + f m, Π m,0 κm Here, for convenence, we ave re-wrtten (5.4a) as X Q m m+1 d,θ, χ m sy m+1, s χ t m Γ m X +ς X m+1 s, χ s + γ m+1 d, χ m t m = f m, χ λ V,m ω m, χ Γ m Γ m γ m m m, d X m 1 t m 1 Γ m, m m, d X m 1 +α G λ A,m sd, s χ t m 1 γ m γ m, = 1,2. (5.5c) ( m m+1 ) s, χ s γ m χ V (Γ m ), (5.6) and, analogously to (4.37), we ave defned a m,θ : V ( ) V ( ) R by settng a m,θ( ζ, η) = Q m,θ ζ, Π m,0 η ζ, η V (Γ m m ), = 1,2. As before, we note tat te matrx n (5.5a) s symmetrc postve defnte as long as Π m,0 ω m and Π m,0[ Q m,θ m κm ] are lnearly ndependent, for = 1,2. Teorem Let θ [0,1], 0 and α 1,α 2 > 0. Let te assumptons (A) old. Ten tere exsts a unque soluton X m+1 V (Γ m ), ( Y m+1, m m+1 ) 2 V (Γ m 1 ) V (γ m ) V (Γ m 2 ) V (γ m ), κ m+1 γ V (γ m ) and C 1 Φ m+1 V (γ m ) to (5.4a e). Proof. As (5.4a e) s lnear, exstence follows from unqueness. To nvestgate te latter, we consder te system: Fnd X V (Γ m ), ( Y, m ) 2 V (Γ m 1 ) V (γ m ) V (Γ m 2 ) V (γ m ), κ γ V (γ m ) and C 1 Φ V (γ m ) suc tat [ 1 t m Q m,θ m X, χ sy, s χ 30 +α G ( m ) s, χ s γ m ]

32 +ς Xs, χ s + X, χ = 0 χ V (Γ m ) (5.7a) γ m γ m α 1 Q m,θ Y m,q m,θ m η + sx, s η = m, η Γ m γ η V (Γ m m ), = 1,2, (5.7b) κ γ, χ γ + Xs, χ m s = 0 χ V (γ m ), (5.7c) γ m C 1 ( m 1 + m 2 ) = 0 on γ m, (5.7d) α G κ γ + Y +C 1 Φ = 0 on γ m, = 1,2. (5.7e) Coosng χ = X n (5.7a), η = Y n (5.7b), χ = m n (5.7c) and notng (5.7d,e), leads to 1 t m Q m,θ m = = X, X [ m, Y [ Y, m = +ς Xs, X s + X, X γ m α 1 γ m Q m,θ m Y,Q m,θ m Y +α G γ m m, κ γ γ m α 1 α 1 Q m,θ Y m,q m,θ Y m γ m α G ( m ) s, X s Q m,θ m Y,Q m,θ m Y ] γ m. (5.8) It follows from (5.8) and te defnton of Q m,θ tat X = 0 on γ m, and so (5.7c) mples m tat κ γ = 0. In addton, (5.8) yelds tat π m [Q m,θ Y m ] = 0, = 1,2. Hence, on addng te two equatons n (5.7b) wt η = X, we obtan tat sx, s X = 0, and so Γ X m = 0. Ten (5.7b) mples tat m 1 = m 2 = 0. Next, we ave from (5.7e), on recallng tat κ γ = 0, tat tere exsts a Y V (Γ m ) suc tat Y = Y Γ m, = 1,2. Coosng η = Y n (5.7a) yelds tat sy, s Y = 0, and ence Y s constant. If C 1 = 0 we Γ m mmedately obtan from (5.7e) tat Y = 0. If C 1 = 1, on te oter and, t follows tat Q m,θ m( qm,k ) Y = 0 for k = 1,...K, = 1,2, and ence Y. ω m ( qm,k ) = 0 k = 1,...K, = 1,2. (5.9) Te defnton of Q m,θm, recall te fully dscrete verson of (4.21), and (5.9) ten yeld tat θ Y = 0. Hence for θ (0,1] we mmedately obtan tat Y = 0, wle n te case θ = 0 t follows from assumpton (A) and (5.9) tat Y = 0. Fnally, we obtan tat Φ = 0 from (5.7e). ] 31

33 5.1 Implct treatment of volume and area conservaton In practce t can be advantageous to consder mplct Lagrange multplers (λ V,m+1,λ A,m+1 1,λ A,m+1 2 ) n order to obtan better dscrete volume and surface area conservatons. In partcular, we replace (5.6) wt X Q m m+1 d,θ, χ m sy m+1, s χ +α t m G ( m m+1 ) s, χ s γ m X +ς X m+1 s, χ s + γ m+1 d, χ m t m = f m, χ λ V,m+1 ω m, χ Γ m Γ m λ A,m+1 γ m s X m+1, s χ χ V (Γ m ), (5.10) andrequretecoupledsolutonsx m+1 V (Γ m ),( Y m+1, m m+1 ) 2 V (Γ m 1 ) V (γ m ) V (Γ m 2 ) V (γ m ), κ m+1 γ V (γ m ), C 1 Φ m+1 V (γ m ) and (λ V,m+1,λ A,m+1 1,λ A,m+1 2 ) R 3 to satsfy te nonlnear system (5.10), (5.4b e) as well as an adapted varant of (5.5a c), were te superscrpt m s replaced by m + 1 n all occurrences of m m, κm, Y m, λ V,m and λ A,m d X. In addton, m 1 X t m 1 n (5.5b,c) s replaced by m+1 d t m. In practce ts nonlnear system can be solved wt a fxed pont teraton as follows. Let (λ V,m+1,0,λ A,m+1,0 1,λ A,m+1,0 2 ) = (λ V,m,λ A,m 1,λ A,m 2 ) and X m+1,0 = d Γ m. Ten, for j 0, fndasoluton( X m+1,j+1, Y m+1,j+1, κ m+1,j+1 Γ, m m+1,j+1 )totelnear system (5.10), (5.4b e), wereanysuperscrpt m+1onleftandsdes sreplacedbym+1,j+1, andbym+1,j on te rgt and sde of (5.10). Ten let κ m+1,j+1 = α 1 π m [Q m,θ Y m+1,j+1 m ]+κ ω m be defned as usual, and compute (λ V,m+1,j+1,λ A,m+1,j+1 1,λ A,m+1,j+1 2 ) as te unque soluton to 2 am,θ ( ωm, ω m ) a m 1,θ ( κm+1,j+1 1, ω 1 m ) a m 2,θ ( κm+1,j+1 2, ω 2 m ) λ V,m+1,j+1 a m 1,θ ( κm+1,j+1 1, ω 1 m) am 1,θ ( κm+1,j+1 1, κ m+1,j+1 1 ) 0 λ A,m+1,j+1 a m 2,θ ( κm+1,j+1 2, ω 2 m ) 0 a m 2,θ ( κm+1,j+1 2, κ m+1,j ) λ A,m+1,j+1 2 were, = b m+1,j+1 0 = b m+1,j+1 0 b m+1,j+1 1 b m+1,j+1 2, ( Π m,0 Id) X m+1,j+1 d, ω m t m f m, Π m,0 ωm s Y m+1,j+1, s ( Π m,0 ωm ) (5.11a) ], (5.11b) 32

34 b m+1,j+1 =,Q m,θ m κm+1,j+1 ( Π m,0 Id) X m+1,j+1 d t m sy m+1,j+1, s ( Π m,0 κ m+1,j+1 ) and contnue te teraton untl + m m+1 X, m+1,j+1 d t m f m, Π m,0 κ m+1,j+1, = 1,2; λ V,m+1,j+1 λ V,m+1,j + λ A,m+1,j+1 1 λ A,m+1,j 1 + λ A,m+1,j+1 2 λ A,m+1,j 2 < γ m (5.11c) We remark tat te mplct sceme s cosen suc tat no new system matrces need to be assembled durng te fxed pont teraton. In partcular, all ntegrals are evaluated on te old nterfaces. But all te quanttes tat are calculated durng te lnear solves are treated mplctly,.e. X m+1, ( Y m+1, κ m+1, m m+1 ) 2, κm+1 γ, C 1 Φ m+1, as well as te Lagrange multplers. 6 Soluton metods Let us brefly outlne ow we solve te lnear system (5.4a e) n practce. Frst of all, smlarly to our approac n Barrett et al. (2010) for te numercal approxmaton of surface clusters wt trple juncton lnes, we reformulate (5.4a) as follows. On ntroducng te followng equvalent caracterzaton of V (Γ m ), recall (5.1), V (Γ m ) = {( η 1, η 2 ) 2 V ( ) : η 1 γ m= η 2 γ m}, (6.1) wecanrewrte(5.4a e)equvalentlyas: Fnd( X 1 m+1, X 2 m+1 ) V (Γ m ),( Y m+1, m m+1 ) 2 V (Γ m 1 ) V (γ m ) V (Γ m 2 ) V (γ m ), κ m+1 γ V (γ m ) and C 1 Φ m+1 V (γ m ) suc tat = Q m,θ m 1 2 X m+1 d, χ t m ς [ X m+1 ] s,[ χ ] s γ m [ s. Y m, s. χ s Y m+1, s χ X m+1 d, χ t m +α G γ m ( sy m ) T,D( χ )( sd) T [α κ m κ ν m 2 2( Y m.q m,θ m κm )] s d, s χ ( m m+1 ) s,[ χ ] s γ m α κ κ m,[ s χ ] T ν m Γ m 33

35 + (1 θ m )( G m ( Y m, κ m ). νm ) sd, s χ (1 θ m ) G ] m ( Y m, κ m ),[ s χ ] T ν m + α G [ κ m γ. mm, d s.[ χ ] s γ m + (Id+P m γ )( mm ) s,[ χ ] s γ m ] λ V,m ω m, χ λ A,m sd, s χ ( χ 1, χ 2 ) V (Γ m ) (6.2) and (5.4b e) old, were n (6.2) we ave used te fully dscrete verson of (4.19). Te above reformulaton s crucal for te constructon of fully practcal soluton metods, as t avods te use of te global fnte element space V (Γ m ). Wt te elp of (6.2), tsnowpossbletoworkwttebassoftesmpleproductfnteelement space V (Γ m ), on employng sutable projectons n te formulaton of te lnear problem. Ts constructon s smlar to e.g. te standard tecnque used for an ODE wt perodc boundary condtons. We recall from Barrett et al. (2017, (4.4a d)) te followng fnte element approxmaton for Wllmore flow of a sngle open surface Γ (t) wt free boundary condtons for Γ (t). For m = 0,...,M 1, fnd (δx m+1, Y m+1 ) V ( ) V ( ), wt X m+1 = d Γ m +δx m+1, and ( κ m+1 Γ, m m+1 ) [V ( )] 2 suc tat Q m,θ m +α G X m+1 d, χ t m sy m+1, s χ +ς [ m m+1 ] s, χ s = Γ m s. Y m, s. χ ( sy m ) T,D( χ)( sd) T α κ κ m,[ s χ] T ν m [ α κ m κ ν m 2 2Y m.q m,θ m κm (1 θ m )( G m ( Y m +α G λ A,m α 1 α G κ m+1 ] sd, s χ [ X m+1 ] s, χ s Γ m, κ m ). νm ) sd, s χ κ m Γ. m m, d s. χ s +α G (Id+P m Γ )[ m m ] s, χ s sd, s χ χ V ( ), Q m,θ m Y m+1 Γ + Y m+1,q m,θ m η, ϕ + s X m+1, s η = 0 ϕ V ( (1 θ m ) G m ( Y m, κ m ),[ s χ] T ν m = m m+1, η κ ω m, η η V ( ), (6.3a) (6.3b) ), (6.3c) 34

36 κ m+1 Γ, η + [ X m+1 ] s, η s = 0 η V ( ). (6.3d) Te correspondng lnear system from Barrett et al. (2017, (5.1)) s ten gven by A 1 MQ t m A ς 0 α G A Γ,Γ Y m+1 M Q 2 A 0 M Γ,Γ δx ( M Γ,Γ ) T 0 α G m+1 M Γ 0 κ m+1 0 ( A Γ Γ,Γ ) T M Γ 0 m m+1 [ B B + R] Y m +( A θ + A ς +λ A,m A) X m + b θ b α A = X m κm ω m 0. (6.4) ( A Γ,Γ ) T X m On replacng A ς wt ( 1 A 2 ς + 1 M ), were te defnton of 2 t M s clear from (6.2), and smlarly adaptng te frst entry n te rgt and sde of (6.4) to account for te term nvolvng λ V,m, we wrte (6.4) as B Z = g. Hence we can rewrte te lnear system for (6.2), (5.4b e) as P B B P Z Z 1 Z 2 Φ m+1 g 1 = P B g 2, 0 (6.5a) were 0 0 B 1 0 C 1M γ 0 B = 0, (6.5b) 0 0 B 2 C 1M γ 0 (0 0 0 C 1M γ ) (0 0 0 C 1M γ ) 0 and were M γ s a mass matrx on γ m. Moveover, P B and P Z are te ortogonal projectons tat encode te test and tral space V (Γ m ) n (6.2),.e. tey act on te frst and fft block row n (6.5b), and on te second entres of Z 1 and Z 2, respectvely. Te system(6.5a) can be effcently solved n practce wt a precondtoned BCGSTAB or GMRES teratve solver, were we employ te precondtoners B P Z 0 B2 1 0 P B and P Z B 1 P B 0 0 Id 35

37 for te cases C 1 = 0 and C 1 = 1, respectvely. Here we recall from Barrett et al. (2017) tat B 1 and B 2 are nvertble. Te nverses B1 1 and B2 1 can be computed wt te elp of te sparse factorzaton package UMFPACK, see Davs (2004). Smlarly, te nverse B 1, wc exsted n all our numercal tests, can also be computed wt te elp of UMFPACK. In practce we note tat te precondtoned Krylov subspace solvers usually take fewer tan ten teratons per tme step to converge. We stress tat te cosen precondtoners are crucal, as wtout approprate precondtonng te teratve solvers do not converge. Ts suggests tat te lnear systems (6.5a) are badly condtoned. 7 Numercal results We mplemented our fully dscrete fnte element approxmatons wtn te fnte element toolbox ALBERTA, see Scmdt and Sebert (2005). Te arsng systems of lnear equatons were solved wt te elp of te sparse factorzaton package UMFPACK, see Davs (2004). For te computatons nvolvng surface area preservng Wllmore flow, we always employ te mplct Lagrange multpler formulaton dscussed n 5.1. Te fully dscrete sceme (5.4a e) needs ntal data κ 0, Y 0, m 0, = 1,2, and κ 0 γ. Gven te ntal trangulaton Γ 0, we let m0 V (γ 0 ) be suc tat m 0, η γ 0 = µ 0, η γ 0 η V (γ 0 ), wt µ 0 denotng te conormal on Γ 0, = 1,2. In addton, we let κ 0 = 2 R ω0 (7.1) for smulatons were Γ (0) s part of a spere of radus R,.e. Γ (0) B R ( 0), and oterwse defne κ 0 V (Γ 0 ) to be te soluton of κ 0, η + sd, s η = m 0, η η V (Γ 0 γ 0 ). (7.2) Ten we defne Moreover, we let κ 0 γ V (γ 0 ) be suc tat Γ 0 Γ 0 Y 0 = α [ κ 0 κ ω 0 ]. (7.3) κ 0 γ, η + ds, η γ 0 s = 0 η V (γ 0 ). γ 0 Trougout ts secton we use unform tme steps t m = t, m = 0,...,M 1, and set t = 10 3 unless stated oterwse. In addton, unless stated oterwse, we fx α = 1 36

38 Fgure 2: (C 0 : κ 1 = κ 2 = 0, ς = 0.1) A plot of ( )2 a plot of te dscrete energy E m+1 ((Γ m ) 2 ). at tmes t = 0, 0.5, 1, 2. Below and κ = α G = 0, = 1,2, as well as ς = 0. At tmes we wll dscuss te dscrete energy of te numercal solutons, wc, smlarly to (4.24), s defned by E m+1 (( )2 ) := [ 1 α 2 κ m+1 κ ν m 2,1 +ςh d 2 (γ m ). +α G [ κ ]] m+1 γ, m m+1 +2πm(Γ m γ m ) Fnally, we fx θ = 0 trougout, unless oterwse stated. For te vsualzaton of our numercal results we wll use te colour red for Γ m 1, and te colour yellow for Γ m Te C 0 case In Fgure 2 we sow te evoluton of te outer sell of a torus joned wt two spercal caps. Here te two caps make up pase 1, wt te remander representng pase 2. Te ntal surface Γ 0 satsfes (J 1,J 2 ) = (2048,4096) and (K 1,K 2 ) = (1090,2112) and as maxmal dmensons 6 6 6,.e. up to translatons, te smallest cubod contanng Γ 0 s [0,6] 3. For te parameters κ 1 = κ 2 = 0 and ς = 0.1, te surface evolves towards a catenod. In Fgure 3 we sow te same evoluton for te values κ 1 = 2 and κ 2 = 0.5, wc s now markedly dfferent. Te same evoluton wt = 2, wc sows te slowng nfluence of > 0, s sown n Fgure 4. In botexperments te effect of te two dfferent spontaneous curvature values for te two pases can clearly be seen. Te same evoluton as n Fgure 4, but now for surface area preservng flow, s sown n Fgure 5. Here te observed relatve surface area loss s 0.12%. Te nterplay between te dfferent values of κ, te surface area constrants, and te C 0 attacment condton lead to an nterestng evoluton. A completely dfferent evoluton s obtaned wen we replace surface area 37

39 Fgure 3: (C 0 : κ 1 = 2, κ 2 = 0.5, ς = 0.1) A plot of ( ) 2 at tmes t = 0, 0.5, 1, 2. Below a plot of te dscrete energy E m+1 ((Γ m ) 2 ) Fgure 4: (C 0 : κ 1 = 2, κ 2 = 0.5, ς = 0.1, = 2) A plot of ( )2 t = 0, 0.5, 1, 2. Below a plot of te dscrete energy E m+1 ((Γ m ) 2 ). at tmes 38

40 Fgure 5: (C 0 : κ 1 = 2, κ 2 = 0.5, ς = 0.1, = 2) Surface area preservng flow. A plot of ( )2 at tmes t = 0, 0.5, 1, 2. Below a plot of te dscrete energy Em+1 ((Γ m ) 2 ) Fgure 6: (C 0 : κ 1 = 2, κ 2 = 0.5, ς = 0.1, = 2) Volume preservng flow. A plot of ( )2 at tmes t = 0, 0.5, 1, 2. Below a plot of te dscrete energy Em+1 ((Γ m ) 2 ). conservaton wt volume conservaton. Ts new smulaton s vsualzed n Fgure 6, were te observed relatve volume loss s 0.00%. A smulaton wt four dsconnected components for pase 1 s sown n Fgure 7. Te ntal surface Γ 0 satsfes (J 1,J 2 ) = (1816,4328) and (K 1,K 2 ) = (1000,2250) and as maxmal dmensons Te evoluton for te parameters κ 1 = κ 2 = 0 and ς = 1 goes towards a fournod. We now consder surface area preservng experments for setups were pase 1 s represented by sx or egt dsconnected components on te unt spere. For tese experments we use te tme step sze t = 10 4 and let κ 1 = 4, κ 2 = 2, ς = 1 and = 2. Te ntal surface Γ 0 n Fgure 8 satsfes (J 1,J 2 ) = (1032,7160) and (K 1,K 2 ) = (614,3668) and s an approxmaton of te unt spere. Pase 1 s made up of sx dsconnected com- 39

41 Fgure 7: (C 0 : κ 1 = κ 2 = 0, ς = 1) A plot of ( )2 a plot of te dscrete energy E m+1 ((Γ m ) 2 ). at tmes t = 0, 0.1, 0.5, 1. Below Fgure 8: (C 0 : κ 1 = 4, κ 2 = 2, ς = 1, = 2) Surface area preservng flow. A plot of ( )2 at tmes t = 0, 0.1, 0.3, Below a plot of te dscrete energy Em+1 ((Γ m ) 2 ). ponents. Here te observed relatve surface area loss s 0.36%. A smulaton wt egt dsconnected components for pase 1 s sown n Fgure 9. Te ntal surface Γ 0 satsfes (J 1,J 2 ) = (2048,6144) and (K 1,K 2 ) = (1184,3218). Here te observed relatve surface area loss s 0.28%. An example for volume and surface area preservng flow s sown n Fgure 10. Te ntal surface Γ 0 satsfes (J 1,J 2 ) = (2274,2274) and (K 1,K 2 ) = (1188,1188) and as maxmal dmensons In ts experment we coose κ 1 = κ 2 = 1, ς = 1 and = 2. Te relatve surface area loss for ts experment s 0.07%, wle te relatve volume loss s 0.00%. Te next set of experments llustrates te mpact of te Gaussan curvature energy. Te ntal surface Γ 0 s made up of two alves of an approxmaton of te unt spere 40

42 Fgure 9: (C 0 : κ 1 = 4, κ 2 = 2, ς = 1, = 2) Surface area preservng flow. A plot of ( ) 2 attmest = 0, 0.05, 0.1, BelowaplotoftedscreteenergyE m+1 ((Γ m ) 2 ) Fgure 10: (C 0 : κ 1 = κ 2 = 1, ς = 1, = 2) Volume and surface area preservng flow. A plot of ( )2 at tmes t = 0, 0.05, 0.1, 0.2. Below a plot of te dscrete energy E m+1 ((Γ m ) 2 ). 41

43 Fgure 11: (C 0 : κ 1 = κ 2 = 0, ς = 1, = 2) A plot of ( )2 at tmes t = 0, 0.01, 0.02, 0.1, 1. At tme t = 1 te evoluton as reaced a dsk. Below a plot of te dscrete energy E m+1 ((Γ m ) 2 ). and satsfes (J 1,J 2 ) = (2274,2274) and (K 1,K 2 ) = (1188,1188). An experment for κ 1 = κ 2 = 0, ς = 1 and = 2 s sown n Fgure 11. Te evoluton eventually reaces a slowly srnkng dsk. Coosng te parameters α1 G = αg 2 = 1, and usng te tme step sze t = 10 5, we obtan te smulaton n Fgure 12. We remark tat te condtons (2.6)trvallyold. Moreover, andncontrasttotec 1 case, anonzerogaussanbendng energy coeffcent as an nfluence on te evoluton even f α1 G = αg 2. In ts example we observe tat for a negatve α1 G = α2 G, te term 2 αg Γ K dh 2 for te ntal spere s negatve, and ence te evoluton remans convex trougout, n contrast to te evoluton n Fgure 11. Moreover, te evoluton n Fgure 12 s generally slower, snce large values of te Gaussan curvatures make 2 αg Γ K dh 2 more negatve. Repeatng te computaton for α1 G = 1 and αg 2 = 1.5 yelds te results n Fgure 13. We note once agan tat te condtons (2.6) old. For te evoluton n Fgure 13 we observe tat te curvature of pase 2 s decreasng slower due to te fact tat large values of Γ 2 K 2 dh 2 decrease te energy. 7.2 Te C 1 case We remark tat n te C 1 case, wt unform data α 1 = α 2 = α, κ 1 = κ 2 = κ and ς = = α G 1 = α G 2 = 0, our fnte element approxmaton collapses to te sceme from Barrett et al. (2016b) for te Wllmore flow of closed surfaces. Indeed, as a numercal ceck we confrmed tat Table 1 n Barrett et al. (2016b), for te approxmaton of te nonlnear ODE Barrett et al. (2016b, (5.1)), s reproduced exactly by our mplementaton 42